#### Prof. Dr.-Ing. Jörg Schröder

Institute of Mechanics

University of Duisburg-Essen

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- j.schroeder@uni-due.de
- +49 201 183 2708
- personal website

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**3D-multilayer simulation of microstructure and mechanical properties of porous materials by selective sintering**

Zhou, X. and Yang, Y. and Bharech, S. and Lin, B. and Schröder, J. and Xu, B.-X.*GAMM Mitteilungen*44 (2021)This work presents multilayer phase-field simulation of selective sintering process and the calculation of effective mechanical properties and residual stress of the microstructure using the finite element method. The dependence of the effective properties and residual stress on the process parameters, such as beam power and scan speed, are analyzed. Significant partial melting of powders is observed for large beam power and low scan speed, which results in low porosity of the microstructure. Nonlinear relationship between the effective mechanical properties and process parameters is observed. The increasing rate of effective mechanical properties decreases with increasing beam power, while increases with decreasing scan speed. The dependence of effective Young's modulus and Poisson's ratio on porosity are well established using power law models. Stress concentrations are found at the necking region of powders and the intensity increases with the level of partial melting, which results in increasing residual stress in the microstructure. The numerical results reveal quantitatively the process-microstructure-property relation, which implies the feasibility of the subsequent data-driven approach. © 2021 The Authors. GAMM - Mitteilungen published by Wiley-VCH GmbH.view abstract 10.1002/gamm.202100017 **A comparative study of micro-mechanical models for fiber pullout behavior of reinforced high performance concrete**

Storm, J. and Pise, M. and Brands, D. and Schröder, J. and Kaliske, M.*Engineering Fracture Mechanics*243 (2021)The pseudo-ductile material behavior of fiber reinforced high performance concrete is mainly characterized by the fiber pullout process. Thereby, complex fiber–concrete interactions, i.e. interface debonding, concrete micro cracks, slippage, adhesion and further unknown processes, are commonly investigated in single-fiber pullout tests. The study in this contribution is based on the experimental results of Gebuhr etal., (2019) and compares three different numerical models applied to the fiber pullout test. An accurate and efficient model for fiber pullout behavior forms the basis for the prediction of the overall behavior by means of composite models or multi-scale approaches in subsequent studies. © 2020 Elsevier Ltdview abstract 10.1016/j.engfracmech.2020.107506 **A Selection of Benchmark Problems in Solid Mechanics and Applied Mathematics**

Schröder, J. and Wick, T. and Reese, S. and Wriggers, P. and Müller, R. and Kollmannsberger, S. and Kästner, M. and Schwarz, A. and Igelbüscher, M. and Viebahn, N. and Bayat, H.R. and Wulfinghoff, S. and Mang, K. and Rank, E. and Bog, T. and D’Angella, D. and Elhaddad, M. and Hennig, P. and Düster, A. and Garhuom, W. and Hubrich, S. and Walloth, M. and Wollner, W. and Kuhn, C. and Heister, T.*Archives of Computational Methods in Engineering*28 (2021)In this contribution we provide benchmark problems in the field of computational solid mechanics. In detail, we address classical fields as elasticity, incompressibility, material interfaces, thin structures and plasticity at finite deformations. For this we describe explicit setups of the benchmarks and introduce the numerical schemes. For the computations the various participating groups use different (mixed) Galerkin finite element and isogeometric analysis formulations. Some programming codes are available open-source. The output is measured in terms of carefully designed quantities of interest that allow for a comparison of other models, discretizations, and implementations. Furthermore, computational robustness is shown in terms of mesh refinement studies. This paper presents benchmarks, which were developed within the Priority Programme of the German Research Foundation ‘SPP 1748 Reliable Simulation Techniques in Solid Mechanics—Development of Non-Standard Discretisation Methods, Mechanical and Mathematical Analysis’. © 2020, The Author(s).view abstract 10.1007/s11831-020-09477-3 **Experimental and Numerical Investigations on the Development and Stability of Residual Stresses Arising from Hot Forming Processes**

Behrens, B.-A. and Schröder, J. and Wester, H. and Brands, D. and Uebing, S. and Kock, C.*Minerals, Metals and Materials Series*(2021)Residual stresses are an important issue as they affect both the manufacturing process as well as the performance of the final parts. Taking the whole process chain of hot forming into account, the integrated heat treatment provided by a defined temperature profile during cooling of the parts offers a great potential for the targeted adjustment of the desired residual stress state. The aim of this work is the investigation of technological reproducibility and stability of residual stresses arising from the thermomechanical forming process. For this purpose, a long-term study of residual stresses on hot-formed components is conducted. In order to develop finite element models for hot forming, a comprehensive thermomechanical material characterisation with special focus on phase transformation effects is performed. The numerical model is validated by means of a comparison between residual stress states determined with X-ray diffraction on experimentally processed components and predicted residual stresses from the simulations. © 2021, The Minerals, Metals & Materials Society.view abstract 10.1007/978-3-030-75381-8_192 **Frazil ice in the antarctic marginal ice zone**

Paul, F. and Mielke, T. and Schwarz, C. and Schröder, J. and Rampai, T. and Skatulla, S. and Audh, R.R. and Hepworth, E. and Vichi, M. and Lupascu, D.C.*Journal of Marine Science and Engineering*9 (2021)Frazil ice, consisting of loose disc-shaped ice crystals, is the first ice that forms in the annual cycle in the marginal ice zone (MIZ) of the Antarctic. A sufficient number of frazil ice crystals form the surface “grease ice” layer, playing a fundamental role in the freezing processes in the MIZ. As soon as the ocean waves are sufficiently damped by a frazil ice cover, a closed ice cover can form. In this article, we investigate the rheological properties of frazil ice, which has a crucial influence on the growth of sea ice in the MIZ. An in situ test setup for measuring temperature and rheological properties was developed. Frazil ice shows shear thinning flow behavior. The presented measurements enable real-data-founded modelling of the annual ice cycle in the MIZ. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.view abstract 10.3390/jmse9060647 **Numerical characterization of residual stresses in a four-point-bending experiment of textured duplex stainless steel**

Maassen, S.F. and Erdle, H. and Pulvermacher, S. and Brands, D. and Böhlke, T. and Gibmeier, J. and Schröder, J.*Archive of Applied Mechanics*(2021)The resulting shapes in production processes of metal components are strongly influenced by deformation induced residual stresses. Dual-phase steels are commonly used for industrial application of, e.g., forged or deep-drawn structural parts. This is due to their ability to handle high plastic deformations, while retaining desired stiffness for the products. In order to influence the resulting shape as well as component characteristics positively it is important to predict the distribution of phase-specific residual stresses which occur on the microscale of the material. In this contribution a comparative study is presented, where two approaches for the numerical simulation of residual stresses are applied. On the one hand a numerically efficient mean field theory is used to estimate on the grain level the total strain, the plastic strains and the eigenstrains based on macroscopic stress, strain and stiffness data. An alternative ansatz relies on a Taylor approximation for the grain level strains. Both approaches are applied to the corrosion-resistant duplex steel X2CrNiMoN22-5-3 (1.4462), which consists of a ferritic and an austenitic phase with the same volume fraction. Mean field and Taylor approximation strategies are implemented for usage in three dimensional solid finite element analysis and a geometrically exact Euler–Bernoulli beam for the simulation of a four-point-bending test. The predicted residual stresses are compared to experimental data from bending experiments for the phase-specific residual stresses/strains which have been determined by neutron diffraction over the bending height of the specimen. © 2021, The Author(s).view abstract 10.1007/s00419-021-01931-3 **Numerical process design for targeted residual stress adjustment in hot bulk formed components taking into account macro- and microscale [Numerische Prozessauslegung zur gezielten Eigenspannungseinstellung in warmmassivumgeformten Bauteilen unter Berücksichtigung von Makro- und Mikroskala]**

Behrens, B.-A. and Schröder, J. and Brands, D. and Brunotte, K. and Wester, H. and Scheunemann, L. and Uebing, S. and Kock, C.*Forschung im Ingenieurwesen/Engineering Research*(2021)The aim of this work is the adjustment of an advantageous compressive residual stress profile in hot-formed components by intelligent process control with tailored cooling from the forging heat. The feasibility and potential are demonstrated in a hot forming process in which cylindrical specimen with an eccentric hole are formed at 1000 °C and subsequently cooled in water from the forging heat. Previous work shows that tensile residual stresses occur in the specimen formed in this way from the material 1.3505. Using the presented multi-scale FE models, an alternative process variant is analysed in this work, where advantageous compressive residual stresses can be generated instead of tensile residual stresses through tailored cooling from the forming heat in the specimen. The tailored cooling is achieved by partially exposing the specimen to a water-air spray. In this way, the local plastification can be influenced by inhomogeneous strains due to thermal and transformation-induced effects in order to customise the resulting residual stress distribution. The scientific challenge of this work is to generate different residual stresses in the surface of the specimen without changing the geometrical and microstructural properties. It is demonstrated that influencing the residual stresses and even reversing the stress sign is possible using smart process control during cooling. © 2021, The Author(s).view abstract 10.1007/s10010-021-00482-x **Numerical treatment of small strain single crystal plasticity based on the infeasible primal-dual interior point method**

Scheunemann, L. and Nigro, P.S.B. and Schröder, J.*International Journal of Solids and Structures*232 (2021)In this contribution, a small strain single crystal plasticity framework in the context of an infeasible primal–dual interior point method (IPDIPM) is discussed with a focus on the numerical treatment. Related to rate-independent algorithms in the field of single-crystal plasticity, the use of the IPDIPM to solve the constrained optimization problem offers the advantage that it handles the naturally arising redundancy in the slip system intrisically through a barrier term. This formulation penalizes the approach of the unfeasible domain, whereas the penalization term gradually approaches zero in the algorithm. This paper focusses on the numerical treatment and presents different tangent operator formulations and compares their convergency behavior in a numerical example. © 2021 Elsevier Ltdview abstract 10.1016/j.ijsolstr.2021.111149 **On the simultaneous use of simple geometrically exact shear-rigid rod and shell finite elements**

Costa e Silva, C. and Maassen, S.F. and Pimenta, P.M. and Schröder, J.*Computational Mechanics*(2021)This work addresses simultaneous use of geometrically exact shear-rigid rod and shell finite elements and describes both models within the same framework. Parameterization of the rotation field is performed by Rodrigues rotation vector, which makes the incremental updating of the rotational variables remarkably simple. For the rod element, cubic Hermitian interpolation for the displacements together with quadratic Lagrange interpolation for the incremental torsion angle were employed, while, for the triangular shell element, a complete quadratic Lagrange interpolation was used. The internal incremental torsion angle resulting from the displacement field within the shell element is then made compatible with the boundary incremental torsion angle of the shell element by an internal Lagrange multiplier. The compatibility between contiguous shell elements as well rod elements is mastered in the standard way by simply connecting nodes. This technique is an important contribution of the work, whose performance is illustrated by several numerical examples. © 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH, DE part of Springer Nature.view abstract 10.1007/s00466-020-01967-2 **Residual stresses in hot bulk formed parts: microscopic stress analysis for austenite-to-martensite phase transformation**

Uebing, S. and Brands, D. and Scheunemann, L. and Schröder, J.*Archive of Applied Mechanics*(2021)In this contribution, the two-scale analysis of residual stress states in a hot bulk formed part with subsequent cooling in the framework of the FE 2-method is presented. The induction of specific residual stress states in order to improve a component’s properties is an area of current research. In general, residual stresses can be induced inside a component in different ways, e.g., quenching, phase transformation in hot forming processes or dislocation movements. It is widely known that different types of residual stresses can be characterized based on the scale the type acts on. In addition to the macroscopic residual stress analysis, in which residual stresses of first type are considered, this contribution specifically analyzes the microscopic residual stress evolution as a consequence of the cooling of the component. © 2021, The Author(s).view abstract 10.1007/s00419-021-01921-5 **Residual stresses in hot bulk formed parts: two-scale approach for austenite-to-martensite phase transformation**

Uebing, S. and Brands, D. and Scheunemann, L. and Schröder, J.*Archive of Applied Mechanics*(2021)In production engineering, current research focuses on the induction of targeted residual stress states in components in order to improve their properties. Therein, the combination of experiment and simulation plays an important role. In this contribution, a focus is laid on the investigation of hot forming processes with subsequent cooling. A numerical approach is presented to analyze the distribution of residual stresses resulting from cooling of a cylinder with an eccentric hole made of chromium-alloyed steel. The occurring phase transformation, which is evoked by cooling, is considered in order to compute residual stress distributions inside the material. © 2021, The Author(s).view abstract 10.1007/s00419-020-01836-7 **Residual Stresses in Hot Bulk Formed Parts—A Phenomenological Approach for the Austenite-to-Martensite Phase Transformation**

Uebing, S. and Brands, D. and Scheunemann, L. and Kock, C. and Wester, H. and Behrens, B.-A. and Schröder, J.*Minerals, Metals and Materials Series*(2021)In production engineering, current research focuses on the induction of targeted residual stress states in components in order to improve their properties rather than follow the usual path of minimizing residual stresses to prevent failure. In this contribution, a focus is laid on the investigation of the subsequent cooling process of hot bulk formed parts. Such cooling of a component leads to a microscopic phase transformation, which has to be considered in order to compute residual stresses inside the material. A numerical approach based on a phenomenological macroscopic material model is presented to depict the related stress evolution. © 2021, The Minerals, Metals & Materials Society.view abstract 10.1007/978-3-030-75381-8_196 **A Concept for the Extension of the Assumed Stress Finite Element Method to Hyperelasticity**

Viebahn, N. and Schröder, J. and Wriggers, P.*CISM International Centre for Mechanical Sciences, Courses and Lectures*597 (2020)The proposed work extends the well-known assumed stress elements to the framework of hyperelasticity. In order to obtain the constitutive relationship, a nonlinear set of equations is solved implicitly on element level. A numerical verification, where two assumed stress elements are compared to classical enhanced assumed strain elements, depicts the reliability and efficiency of the proposed concept. This work is closely related to the publication of Viebahn et al. (2019) © 2020, CISM International Centre for Mechanical Sciences.view abstract 10.1007/978-3-030-33520-5_4 **A Fully Nonlinear Beam Model of Bernoulli–Euler Type**

de Mattos Pimenta, P. and Maassen, S. and da Costa e Silva, C. and Schröder, J.*CISM International Centre for Mechanical Sciences, Courses and Lectures*597 (2020)This work presents a geometrically exact Bernoulli–Euler rod model. In contrast to Pimenta (1993b), Pimenta and Yojo (1993), Pimenta (1996), Pimenta and Campello (2001), where the hypothesis considered was Timoshenko’s, this approach is based on the Bernoulli–Euler theory for rods, so that transversal shear deformation is not accounted for. Energetically conjugated cross-sectional stresses and strains are defined. The fact that both the first Piola–Kirchhoff stress tensor and the deformation gradient appear again as primary variables is also appealing. A straight reference configuration is assumed for the rod, but, in the same way, as in Pimenta (1996), Pimenta and Campello (2009), initially curved rods can be accomplished, if one regards the initial configuration as a stress-free deformed state from the straight position. Consequently, the use of convective non-Cartesian coordinate systems is not necessary, and only components on orthogonal frames are employed. A cross section is considered to undergo a rigid body motion and parameterization of the rotation field is done by the rotation tensor with the Rodrigues formula that makes the updating of the rotational variables very simple. This parametrization can be seen in Pimenta et al. (2008), Campello et al. (2011). A simple formula for the incremental Rodrigues parameters in function of the displacements derivative and the torsion angle is also settled down. A 2-node finite element with Cubic Hermitian interpolation for the displacements, together with a linear approximation for the torsion angle, is displayed within the usual Finite Element Method, leading to adequate C1 continuity. © 2020, CISM International Centre for Mechanical Sciences.view abstract 10.1007/978-3-030-33520-5_5 **A generalized micromorphic approach accounting for variation and dispersion of preferred material directions**

Horn-Von Hoegen, M. and Skatulla, S. and Schröder, J.*Computers and Structures*232 (2020)Materials exhibiting a heterogeneous and non-uniform composition in terms of elastic and anisotropic properties such as biological tissues require special efforts to accurately describe their constitutive behavior. In contrast to classical models, micromorphic formulations can predict the macroscopically observable material response as originated from distinct scale-dependent micro-structural deformation mechanisms. This is facilitated by additional independent degrees of freedom and associated additional strain and stress quantities. Here, a generalized continuum is mathematically constructed from a macro-continuum and a micro-continuum which are both adequately coupled on kinematics and constitutive levels as well as by micro-boundary conditions. In view of biomechanical modeling, the potential of the formulation is studied for a number of academic examples characterized by an anisotropic material composition to elucidate the micromorphic material response as compared with the one obtained using a classical continuum mechanics approach. The results demonstrate the ability of the generalized continuum approach to address non-affine elastic reorientation of the preferred material direction in the macro-space and its dispersion in the micro-space as affecting deformation, strain and stress on the macroscopic level. In particular, if the anisotropy in the micromorphic formulation is solely linked to the extra degrees of freedom and associated strain and stress measures, the deformation for small and large strains is shown to be distinctly different to the classical response. Together with the ability to implicitly account for scale-dependent higher-order deformation effects in the constitutive law the proposed generalized micromorphic formulation provides an advanced description, especially for fibrous biological materials. © 2017 Elsevier Ltd.view abstract 10.1016/j.compstruc.2017.11.013 **A mixed least-squares finite element formulation with explicit consideration of the balance of moment of momentum, a numerical study**

Igelbüscher, M. and Schröder, J. and Schwarz, A.*GAMM Mitteilungen*43 (2020)Important conditions in structural analysis are the fulfillment of the balance of linear momentum (vanishing resultant forces) and the balance of angular momentum (vanishing resultant moment), which is not a priori satisfied for arbitrary element formulations. In this contribution, we analyze a mixed least-squares (LS) finite element formulation for linear elasticity with explicit consideration of the balance of angular momentum. The considered stress-displacement (σ − u) formulation is based on the squared L2(ℬ)-norm minimization of the residuals of a first-order system of differential equations. The formulation is constructed by means of two residuals, that is, the balance of linear momentum and the constitutive equation. Motivated by the crucial point of weighting factors within LS formulations, a scale independent formulation is constructed. The displacement approximation is performed by standard Lagrange polynomials and the stress approximation with Raviart-Thomas functions. The latter ansatz functions do not a priori fulfill the symmetry of the Cauchy stress tensor. Therefore, a redundant residual, the balance of angular momentum ((x − x0) × (divσ + f) + axl[σ − σT]), is introduced and the results are discussed from the engineering point of view, especially for coarse mesh discretizations. However, this formulation shows an improvement compared to standard LS σ − u formulations, which is considered here in a numerical study. © 2019 The Authors. GAMM - Mitteilungen published by Wiley-VCH Verlag GmbH & Co. KGaA on behalf of Gesellschaft für Angewandte Mathematik und Mechanikview abstract 10.1002/gamm.202000009 **A novel algorithm for rate independent small strain crystal plasticity based on the infeasible primal-dual interior point method**

Scheunemann, L. and Nigro, P.S.B. and Schröder, J. and Pimenta, P.M.*International Journal of Plasticity*124 (2020)Single crystal plasticity plays a major role in the analysis of material anisotropy and texture evolution, treats each crystalline grain individually. The polycrystalline material response is obtained upon considering a structure consisting of various individual grains, often also considering interface effects at the grain boundaries. On the individual grain level, single crystal plasticity can be treated in the mathematical framework of multi-surface plasticity, leading to a constrained optimization problem wherein multiple constraints are defined as yield criteria on the different slip systems. In this work, we present a new algorithm for the solution of the constrained optimization problem based on the Infeasible Primal Dual Interior Point method (IPDIPM). The main motivation herein is the handling of the ill-posed problem without the use of simple perturbation technique, see e.g. Miehe and Schröder [2001]. The proposed algorithm, involving slack variables, is developed for the framework of small strain single crystal plasticity. The use of slack variables therein stabilizes the conventional method and allows for a temporary violation of the constraint condition during the optimization. Moreover, all slip systems are considered simultaneously, omitting an iterative active set search. Several numerical examples are simulated to show the performance of the developed algorithm. © 2019 Elsevier Ltd.view abstract 10.1016/j.ijplas.2019.07.020 **A simple finite element for the geometrically exact analysis of Bernoulli–Euler rods**

da Costa e Silva, C. and Maassen, S.F. and Pimenta, P.M. and Schröder, J.*Computational Mechanics*65 (2020)This work develops a simple finite element for the geometrically exact analysis of Bernoulli–Euler rods. Transversal shear deformation is not accounted for. Energetically conjugated cross-sectional stresses and strains are defined. A straight reference configuration is assumed for the rod. The cross-section undergoes a rigid body motion. A rotation tensor with the Rodrigues formula is used to describe the rotation, which makes the updating of the rotational variables very simple. A formula for the Rodrigues parameters in function of the displacements derivative and the torsion angle is for the first time settled down. The consistent connection between elements is thoroughly discussed, and an appropriate approach is developed. Cubic Hermitian interpolation for the displacements together with linear Lagrange interpolation for the torsion incremental angle were employed within the usual Finite Element Method, leading to adequate C1 continuity. A set of numerical benchmark examples illustrates the usefulness of the formulation and numerical implementation. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.view abstract 10.1007/s00466-019-01800-5 **Corrigendum to ‘Strong converse magnetoelectric effect in (Ba,Ca)(Zr,Ti)O3-NiFe2O4 multiferroics: A relationship between phase-connectivity and interface coupling’ (Acta Materialia (2018) 144 (305–313), (S1359645417309072), (10.1016/j.actamat.2017.10.048))**

Naveed-Ul-Haq, M. and Shvartsman, V.V. and Trivedi, H. and Salamon, S. and Webers, S. and Wende, H. and Hagemann, U. and Schröder, J. and Lupascu, D.C.*Acta Materialia*187 (2020)The authors regret to inform the readers of Acta Materialia that Figure 10 of the article contains mistakes, none of which change the overall interpretation of the data nor the main point of the paper, but which should be rectified, nonetheless. 1. There was a calculation error during the conversion from applied voltage to the electric field and finally to the converse ME coupling coefficient. This was amended, with the correct data being shown in Table 1, to more clearly show the composition-dependent change to the reader. The overall interpretation remains unaffected, with the sample BCZTNFO30 displaying the highest ME coefficient.2. The value of the ME coefficient for the BCZTNFO50 sample was recorded at a different frequency (1.7 Hz) than the other compositions (8 Hz), which lead to a strongly deviating result for this sample. In order to obtain a dataset with consistent parameters, a single data point from the magnetic field dependent measurement regime was used, the results of which are shown in Figure 11 of the original manuscript. The slope and thus the converse ME coupling coefficient were calculated from a linear function between this point and the point of origin, which is justified by the fact that our measurement values show high stability and precision, making it highly unlikely for significant deviations to result from this approximation. The combination of parameters (magnetic field, temperature, electric field) represents the optimum for this sample, as also used for the original measurement, in addition to the now matching frequency of 8 Hz. This updated data point has also been included in Table 1 and Figure 1.3. Furthermore, the dataset of the BCZTNFO20 sample currently shown in Figure 10 of the manuscript is not the correct one, evident by the lower point density, higher amount of noise and the deviating frequency of 11 Hz. This dataset was replaced with the correct one for this sample, as shown in Figure 2, recorded at the same point density and frequency (8 Hz) as all the other samples of this set.From this updated set of data, new values of ME coefficients were calculated: 16.9(6) ps/m for BCZTNFO20, 90.1(6) ps/m for BCZTNFO30, 34.9(2) ps/m for BCZTNFO40 and 25.4 ps/m for BCZTNFO50. This does not change the conclusions that were drawn in the manuscript, as sample BCTNFO30 still has the highest value of the converse ME coupling coefficient. However, all of the values are increased at least twofold due to the corrected voltage conversion, clearly showing that this NFO-based sample set exhibits greatly improved ME response compared to previously used CFO-based materials. Furthermore, thanks to the now correct coefficient for BCZTNFO50, the composition dependent peak is much clearer, with the coupling coefficient consistently dropping when we move away from the optimum BCZTNFO30 composition to either side (see Figure 1), giving more merit to our original interpretation. The significant change of the measurement signal for BCZTNFO50 we observed when going from 1.7 Hz to 8 Hz is most probably caused by the fact that this sample has a lower resistivity of only 1 GΩ compared to the 30 GΩ of BCZTNFO30 (see Figure 3). This presumably induces a strong increase of frequency-dependence of the ME coefficient through Maxwell-Wagner relaxation due to the different dielectric permittivities of the two constituents that make up the sample, further exacerbated by the higher temperature at which this measurement was performed. These findings were discussed in greater detail in the dissertation of one of the co-authors [1]. Another minor correction needs to be made regarding Figure 1 of the original manuscript: There are typo mistakes on all four panels of Figure 1, where the phase of BCZT was incorrectly labeled as P4/mmm, which should be labeled as P4/mm. The same should be understood for Table 1 and wherever P4/mmm is mentioned in the text. The authors would like to apologise for any inconvenience caused. © 2020 Acta Materialia Inc.view abstract 10.1016/j.actamat.2020.01.054 **Engineering Notes on Concepts of the Finite Element Method for Elliptic Problems**

Schröder, J.*CISM International Centre for Mechanical Sciences, Courses and Lectures*597 (2020)In this contribution, we discuss some basic mechanical and mathematical features of the finite element technology for elliptic boundary value problems. Originating from an engineering perspective, we will introduce step by step of some basic mathematical concepts in order to set a basis for a deeper discussion of the rigorous mathematical approaches. In this context, we consider the boundedness of functions, the classification of the smoothness of functions, classical and mixed variational formulations as well as the $$H^{-1}$$-FEM in linear elasticity. Another focus is on the analysis of saddle point problems occurring in several mixed finite element formulations, especially on the solvability and stability of the associated discretized versions. © 2020, CISM International Centre for Mechanical Sciences.view abstract 10.1007/978-3-030-33520-5_1 **Exasteel: Towards a virtual laboratory for the multiscale simulation of dual-phase steel using high-performance computing**

Klawonn, A. and Lanser, M. and Uran, M. and Rheinbach, O. and Köhler, S. and Schröder, J. and Scheunemann, L. and Brands, D. and Balzani, D. and Gandhi, A. and Wellein, G. and Wittmann, M. and Schenk, O. and Janalík, R.*Lecture Notes in Computational Science and Engineering*136 (2020)We present a numerical two-scale simulation approach of the Nakajima test for dual-phase steel using the software package FE2TI, a highly scalable implementation of the well known homogenization method FE2. We consider the incorporation of contact constraints using the penalty method as well as the sample sheet geometries and adequate boundary conditions. Additional software features such as a simple load step strategy and prediction of an initial value by linear extrapolation are introduced. The macroscopic material behavior of dual-phase steel strongly depends on its microstructure and has to be incorporated for an accurate solution. For a reasonable computational effort, the concept of statistically similar representative volume elements (SSRVEs) is presented. Furthermore, the highly scalable nonlinear domain decomposition methods NL-FETI-DP and nonlinear BDDC are introduced and weak scaling results are shown. These methods can be used, e.g., for the solution of the microscopic problems. Additionally, some remarks on sparse direct solvers are given, especially to PARDISO. Finally, we come up with a computationally derived Forming Limit Curve (FLC). © The Author(s) 2020.view abstract 10.1007/978-3-030-47956-5_13 **Implicit time discretization schemes for mixed least-squares finite element formulations**

Averweg, S. and Schwarz, A. and Nisters, C. and Schröder, J.*Computer Methods in Applied Mechanics and Engineering*368 (2020)This work is an extension of the ideas in Averweg et al. (2019) with the focus on a detailed investigation of implicit time discretization schemes to model instationary fluid flow, based on the incompressible Navier–Stokes equations, and linear elastodynamic structural behavior. The variational approaches for fluid and solid mechanics are based on a mixed least-squares finite element method. The L2-norm minimization of the residuals of the constructed first-order systems of the governing differential equations is based on two-field stress–velocity (SV) functionals. For the time discretization of the SV-fluid formulation, four different types of implicit integration schemes are investigated, namely the Houbolt method, the Crank–Nicolson method and two explicit, singly diagonally implicit Runge–Kutta methods (ESDIRK). The SV-formulation for the solid is discretized applying the Houbolt method. The presented time integration schemes are validated investigating an unsteady fluid flow and an elastodynamic structural benchmark. Since both (fluid and solid) SV formulations are discretized using conforming finite element spaces in H(div) and H1, respectively, the inherent fulfillment of coupling conditions, when modeling fluid–structure interaction problems, is given a priori. Therefore, the applicability is also examined by two simplified FSI problems for small deformations, in order to represent the main characteristics of the presented approach. © 2020 Elsevier B.V.view abstract 10.1016/j.cma.2020.113111 **Micromechanical modeling of DP600 steel: From microstructure to the sheet metal forming process**

Vajragupta, N. and Maassen, S. and Clausmeyer, T. and Brands, D. and Schröder, J. and Hartmaier, A.*Procedia Manufacturing*47 (2020)This study proposes a micromechanical modeling scheme to predict relevant mechanical behavior of DP600 steel for the sheet metal forming process. This study can be divided into two parts which are the prediction of the advanced anisotropic initial yield function by means of microstructure-based simulations and the investigation of microstructure changes during the sheet metal forming process. Firstly, based on the quantitative microstructure characterization of DP600 steel by EBSD analysis, the obtained statistical information of important microstructural features is used to generate a microstructure model with the help of an advanced dynamic microstructure generator (ADMG), which combines a particle simulation method with radical Voronoi tessellation. In the next step, finite element simulations with a non-local crystal plasticity model for the individual grains are conducted. With the help of these simulations, the crystal plasticity parameters are adapted to match the experiments. The resulting parameterized microstructure model of DP600 steel is then applied to various loading conditions to investigate the corresponding mechanical responses. For the second part, macroscopic simulations of the bending process are performed and local deformation fields of the location of interest are captured and imposed as boundary conditions on the microstructure model to study the changes in the microstructural features. © 2020 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the 23rd International Conference on Material Forming.view abstract 10.1016/j.promfg.2020.04.347 **Modeling of freezing processes of ice floes within the framework of the TPM**

Schwarz, A. and Bluhm, J. and Schröder, J.*Acta Mechanica*231 (2020)Sea ice is floating ice which is formed by the freezing of ocean water in the polar regions of the Earth, i.e., the Arctic and the Antarctic. Thus, a closed smooth ice surface can be formed consisting of these ice configurations. In recent years, the simulation of sea ice evolution, especially for the use in climate models, became more important, see, for instance, Danilov et al. (Geosci Model Dev 8:1747–1761, 2015) and the references therein. In the present paper, a coupled macroscopic model based on the Theory of Porous Media is introduced in view of the finite element simulation of the coalescence of ice floes due to freezing in calm sea and weather conditions. Attention is paid to the description of the temperature development, the determination of energy, enthalpy, specific heat and mass exchange between water and ice as well as volume deformations due to ice formation during freezing. The main idea is based on a theoretically motivated evolution equation for the phase transition of ice and water, which guarantees the thermodynamical consistency. Numerical examples show that the simplified model is indeed capable of simulating the temperature development and energetic effects during phase change. © 2020, Springer-Verlag GmbH Austria, part of Springer Nature.view abstract 10.1007/s00707-020-02686-8 **Preface GAMM Mitteilungen**

Schröder, J. and Wick, T.*GAMM Mitteilungen*43 (2020)view abstract 10.1002/gamm.202000010 **3D magnetostrictive Preisach model for the analysis of magneto-electric composites**

Labusch, M. and Schröder, J. and Lupascu, D.C.*Archive of Applied Mechanics*89 (2019)In the present contribution, we compare numerical simulations of magneto-electric composites with experimental measurements. The coupling between electric polarization and magnetization of such materials can improve the operation of sensors and actuators and can enable new technical devices. These composites consist of a ferroelectric and a magnetostrictive phase and generate the ME coupling as a strain-induced product property. However, the responses of the individual phases are highly nonlinear. It is important to predict the behaviors of both phases in an appropriate manner to ensure a realistic prediction of the magneto-electric coefficient. Therefore, we simulate the nonlinear ferroelectric hysteresis curves based on the ferroelectric and ferroelastic switching behavior of the spontaneous polarization directions on the submicroscopic level. Therefore, a switching criterion considering the change in the free energy is evaluated. For the simulation of the magnetostrictive behavior, we derive in this contribution a three-dimensional Preisach model. For this, the classical scalar Preisach model acts on a rotational time-dependent magnetization vector. After a homogenization approach within the finite-element (FE 2)-method, the effective macroscopic hysteresis curves are obtained. Furthermore, the magneto-electric coefficient is obtained from the homogenization and compared with experimental measurements in terms of magnitude and nonlinear behavior. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.view abstract 10.1007/s00419-019-01529-w **A finite element formulation for freezing and thawing processes of ice within the framework of the tpm**

Schwarz, A. and Bluhm, J. and Schröder, J.*Advances in Engineering Materials, Structures and Systems: Innovations, Mechanics and Applications - Proceedings of the 7th International Conference on Structural Engineering, Mechanics and Computation, 2019*(2019)In recent years the simulation of sea ice evolution, e.g. for the use in climate models, became more important, see for instance Danilov et al. (2015) and the references therein. In many contributions, the ice motion, which goes back to the findings in Hibler III (1979), is investigated. There, a numerical model for the simulation of sea ice circulation and thickness evolution on the basis of an evolution equation is explicitly described. In the present contribution, a coupled finite element method based on the Theory of Porous Media (TPM), see e.g. Bowen (1980) and Bowen (1982), for the direct modeling of phase transition of ice and water is presented. In detail, we investigate the ice deformation, the temperature development and the evolution of energy, enthalpy and mass exchange between the constituents. The main idea is based on a theoretically motivated evolution equation for the phase transition of ice and water, which guarantees the thermodynamical consistency. The resulting finite element is a four-field formulation in terms of ice displacements, liquid pressure, volume fraction of ice and temperature. Here, we make use of a quadratic interpolation of the ice displacements and a linear interpolation for the other degrees of freedom. We present first numerical examples, which examines freezing processes. © 2019 Taylor & Francis Group, London, UK.view abstract 10.1201/9780429426506-48 **A review on modelling of brine transport mechanisms in sea ice**

Schröder, J. and Cook, A.D. and Skatulla, S. and Machutchon, K. and Ricken, T.*Advances in Engineering Materials, Structures and Systems: Innovations, Mechanics and Applications - Proceedings of the 7th International Conference on Structural Engineering, Mechanics and Computation, 2019*(2019)It is evident that the sea ice cycle, from its formation to its melt, is governed by a complex interaction of the ocean, atmosphere and surrounding continents. Once sea water begins to freeze, physical, biological and chemical processes have implications on the evolution of the sea ice morphology (Thomas, 2017). The distinguishing factor between fresh and sea water ice is brine inclusions that get trapped within the ice pores during freezing. Salt inclusions within frozen ice influence the salinity of the seawater as well as the physical properties of the sea ice (Hunke et al., 2010). These brine inclusions form part of a dynamic process within the ice characterized by the movement of brine and phase transition which are the foundation of many of its physical properties (Hunke et al., 2010). Brine removal subsequently begins to occur due to vertical gravity drainage into the underlying ocean water. This review article provides an overview of models that have attempted to describe this complex system. It also introduces a biphasic model being developed based on the Theory of Porous Media (TPM) which considers a solid phase for the pore structure of the ice matrix as well as a liquid phase for the brine inclusions, respectively. The TPM framework is able to describe and study the various desalination mechanisms that are significant in aiding the salt flux into the Southern Ocean. This will foster understanding of brine rejection and how it is linked to the porous microstructure of Antarctic sea ice. © 2019 Taylor & Francis Group, London, UK.view abstract 10.1201/9780429426506-62 **A two-scale homogenization analysis of porous magneto-electric two-phase composites**

Labusch, M. and Schröder, J. and Lupascu, D.C.*Archive of Applied Mechanics*89 (2019)A computational homogenization analysis for the simulation of porous magneto-electric composite materials is presented. These materials combine two or more ferroic states with each other enabling a coupling between magnetization and electric polarization. This magneto-electric coupling finds application in sensor technology or data storage devices. Since most single-phase multiferroics show coupling at very low temperatures beyond technically relevant applications, two-phase composites, consisting of a ferroelectric and a ferromagnetic phases, are manufactured. They generate a strain-induced magneto-electric coupling at room temperature. The performance and reliability of these materials is influenced by defects or pores, which can arise during the manufacturing process. We analyze the impact of pores on the magnitude of the magneto-electric coupling coefficient. In order to determine the effective properties of the composite, a two-scale finite element (FE 2) homogenization approach is performed. It combines the macroscopic and microscopic scale by direct incorporation of the microscopic morphology. We derive the basic equations for the localization and the homogenization of the individual field variables and give an algorithmic expression for the effective tangent moduli. We discuss the influence of pores on the magneto-electric coupling in two-phase composites by analyzing numerical examples. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.view abstract 10.1007/s00419-018-01500-1 **Advanced modeling of ferroic materials**

Schröder, J. and Lupascu, D.C.*Archive of Applied Mechanics*89 (2019)view abstract 10.1007/s00419-019-01550-z **An extension of assumed stress finite elements to a general hyperelastic framework**

Viebahn, N. and Schröder, J. and Wriggers, P.*Advanced Modeling and Simulation in Engineering Sciences*6 (2019)Assumed stress finite elements are known for their extraordinary good performance in the framework of linear elasticity. In this contribution we propose a mixed variational formulation of the Hellinger–Reissner type for hyperelasticity. A family of hexahedral shaped elements is considered with a classical trilinear interpolation of the displacements and different piecewise discontinuous interpolation schemes for the stresses. The performance and stability of the new elements are investigated and demonstrated by the analysis of several benchmark problems. In addition the results are compared to well known enhanced assumed strain elements. © 2019, The Author(s).view abstract 10.1186/s40323-019-0133-z **Analysis and evaluation of the pull-out behavior of hooked steel fibers embedded in high and ultra-high performance concrete for calibration of numerical models**

Gebuhr, G. and Pise, M. and Sarhil, M. and Anders, S. and Brands, D. and Schröder, J.*Structural Concrete*20 (2019)This paper aims at extending the existing knowledge regarding the pull-out behavior of single steel fibers embedded in high- and ultra-high-strength concretes with compressive strengths exceeding 100 MPa. Apart from the compressive strength, straight fibers, and fibers with hooked-ends as well as different embedded lengths are considered. The experiments have shown that the bond strength for straight fibers increases with an increasing compressive strength, mechanical anchorage in terms of hooks multiply the load-bearing capacity. For hooked end fibers in ultra-high performance concrete (UHPC) fiber rupture occurred. Special attention was paid to characterize scatter adequately. Especially for hooked-end fibers in UHPC conventional slip-wise averaging does not represent the maximum load well. Therefore, a more precise approach basing on characteristic points is introduced. The experimental results are presented to be used for calibration and validation of numerical models. As an example, an elasto-plastic phase-field model using a Drucker-Prager yield condition is developed, which represents the pull-out behavior of straight fibers satisfyingly. © 2019 fib. International Federation for Structural Concreteview abstract 10.1002/suco.201900034 **Comparative study on time-integrator schemes in a least-squares sea ice finite element formulation**

Nisters, C. and Schröder, J. and Niekamp, R. and Ricken, T.*Advances in Engineering Materials, Structures and Systems: Innovations, Mechanics and Applications - Proceedings of the 7th International Conference on Structural Engineering, Mechanics and Computation, 2019*(2019)The motion of sea ice in large scales of several thousand kilometers is modeled by the viscous-plastic (VP) sea ice rheology. The sea ice motion model is based on the findings of Hibler III (1979), who introduced a numerical model for the simulation of sea ice circulation and thickness evolution over a seasonal cycle. The velocity and stress fields, as well as the sea ice thickness and sea ice concentration, are included in the model. Recent research on a finite element implementation of the model is devoted to formulations based on the (mixed) Galerkin variational approach. Here, special treatments are necessary regarding the stabilization of the numerical complex scheme. It is therefore suggested to utilize a mixed least-squares formulation to overcome possible numerical drawbacks. The least-squares finite element method is well established, especially in the branch of fluid mechanics, see e.g. Jiang (1998), Cai et al. (2004) and Bochev & Gunzburger (2009). A significant advantage of the method is its applicability to first-order systems, such that it results in stable and robust formulations also for not self-adjoint operators like in the Navier-Stokes equations. The presented least-squares finite element formulations are based on the instationary sea ice equations including two tracer equations of transient convection type. Therefore, the mixed least-squares approach includes four primary fields, which are the stress tensor σ, the velocity field v and two scalar tracers Aice and Hice. Different time-integrators are investigated regarding accuracy and stability with a particular focus on the treatment of the tracer equations. The investigation of the formulation with the help of a boundary value problem is provided. © 2019 Taylor & Francis Group, London, UK.view abstract 10.1201/9780429426506-49 **Deterioration development of steel fibre reinforced high performance concrete in low-cycle fatigue**

Pise, M. and Sahril, M. and Brands, D. and Schröder, J. and Gebuhr, G. and Anders, S.In this contribution, the deterioration behaviour of steel fibre reinforced high performance concrete (HPC) beams in flexural tests is described and analysed. The focus is drawn on quasi-static low-cycle tests in an attempt to derive the progression of stiffness, plastic deformation, and the hysteresis energy of individual load cycles depending on the amount of fibres used per specimen. To achieve this goal, different fibre quantities from plain concrete up to 1.5 vol.-% are looked into. Additionally, a prospect is given on the usability of the described deterioration indicators to predict the concrete’s performance in high-cycle loading processes. Lastly, a short outline for the applicability of the acquired data on a numeric model using a phase-field approach as described in Pise et al. (2018) will be given. © 2019 Taylor & Francis Group, London, UK.view abstract 10.1201/9780429426506-249 **Direct and inverse identification of constitutive parameters from the structure of soft tissues. Part 2: dispersed arrangement of collagen fibers**

Horn-Von Hoegen, M. and Marino, M. and Schröder, J. and Wriggers, P.*Biomechanics and Modeling in Mechanobiology*18 (2019)This paper investigates on the relationship between the arrangement of collagen fibers within soft tissues and parameters of constitutive models. Starting from numerical experiments based on biaxial loading conditions, the study addresses both the direct (from structure to mechanics) and the inverse (from mechanics to structure) problems, solved introducing optimization problems for model calibration and regression analysis. A campaign of parametric analyses is conducted in order to consider fibers distributions with different main orientation and angular dispersion. Different anisotropic constitutive models are employed, accounting for fibers dispersion either with a generalized structural approach or with an increasing number of strain energy terms. Benchmark data sets, toward which constitutive models are fitted, are built by employing a multiscale description of fiber nonlinearities and accounting for fibers dispersion with an angular integration method. Results show how the optimal values of constitutive parameters obtained from model calibration vary as a function of fibers arrangement and testing protocol. Moreover, the fitting capabilities of constitutive models are discussed. A novel strategy for model calibration is also proposed, in order to obtain a robust accuracy with respect to different loading conditions starting from a low number of mechanical tests. Furthermore, novel results useful for the inverse determination of the mean angle and the variance of fibers distribution are obtained. Therefore, the study contributes: to better design procedures for model calibration; to account for mechanical alterations due to remodeling mechanisms; and to gain structural information in a nondestructive way. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.view abstract 10.1007/s10237-019-01119-3 **Elasto-plastic phase-field model of hydraulic fracture in saturated binary porous media**

Pise, M. and Bluhm, J. and Schröder, J.*International Journal for Multiscale Computational Engineering*17 (2019)In many fields of engineering, especially in geo sciences and rock mechanics, the theoretical and numerical modeling of hydraulic fracturing of porous materials plays an important role. Hydraulic fracturing is a well-known technology in which porous materials are fractured by a pressurized liquid. The process involves the pressure injection of a fracking fluid (primarily water, often enriched with filling materials and thickening agents) and accompanied by crack nucle-ation and propagation, as well as mass transport. This article presents a macroscopic model based on the Theory of Porous Media (TPM). For simplification, an incompressible binary model consisting of the solid and liquid phases is used. The development of the damage of the elastic-plastic solid phase is controlled by an evolution equation, which corresponds to known diffusive phase-field models within a continuum mechanical framework. A numerical example shows that the simplified model is indeed capable of simulating hydraulic fracturing of porous media. © 2019 by Begell House, Inc.view abstract 10.1615/IntJMultCompEng.2019027066 **Experimental and numerical investigations of the development of residual stresses in thermo-mechanically processed Cr-alloyed steel 1.3505**

Behrens, B.-A. and Schröder, J. and Brands, D. and Scheunemann, L. and Niekamp, R. and Chugreev, A. and Sarhil, M. and Uebing, S. and Kock, C.*Metals*9 (2019)Residual stresses in components are a central issue in almost every manufacturing process, as they influence the performance of the final part. Regarding hot forming processes, there is a great potential for defining a targeted residual stress state, as many adjustment parameters, such as deformation state or temperature profile, are available that influence residual stresses. To ensure appropriate numerical modeling of residual stresses in hot forming processes, comprehensive material characterization and suitable multiscale Finite Element (FE) simulations are required. In this paper, experimental and numerical investigations of thermo-mechanically processed steel alloy 1.3505 (DIN 100Cr6) are presented that serve as a basis for further optimization of numerically modeled residual stresses. For this purpose, cylindrical upsetting tests at high temperature with subsequently cooling of the parts in the media air or water are carried out. Additionally, the process is simulated on the macroscale and compared to the results based on the experimental investigations. Therefore, the experimentally processed specimens are examined regarding the resulting microstructure, distortions, and residual stresses. For the investigation on a smaller scale, a numerical model is set up based on the state-data of the macroscopic simulation and experiments, simulating the transformation of the microstructure using phase-field theory and FE analysis on micro- and meso-scopic level. © 2019 by the author. Licensee MDPI, Basel, Switzerland.view abstract 10.3390/met9040480 **FEM analysis of a multiferroic nanocomposite: comparison of experimental data and numerical simulation**

Labusch, M. and Lemke, V. and Schmitz-Antoniak, C. and Schröder, J. and Webers, S. and Wende, H.*Archive of Applied Mechanics*89 (2019)In this contribution, we analyze the properties of two-phase magneto-electric (ME) composites. Such ME composite materials have raised scientific attention in the last decades due to many possible applications in a wide range of technical devices. Since the effective magneto-electric properties significantly depend on the microscopic morphology, we investigate in more detail the changes in the in-plane polarization due to an applied magnetic field. It was shown in previous works that it is possible to grow vertically aligned nanopillars of magnetostrictive cobalt ferrite in a piezoelectric barium titanate matrix by pulsed laser deposition. Based on x-ray linear dichroism, the displacements of titanate ions in the matrix material can be measured due to an applied magnetic field near the boundary of the interface between the matrix and the nanopillars. Here, we focus on (1–3) fiber-induced composites, based on previous experiments, where cobalt ferrite nanopillars are embedded in a barium titanate matrix. In the numerical simulations, we adjusted the boundary value problem to match the experimental setup and compare the results with previously made assumptions of the in-plane polarizations. A further focus is taken on the deformation behavior of the nanopillar over its whole height. Such considerations validate the assumption of the distortion of the nanopillars under an external magnetic field. Furthermore, we analyze the resulting magneto-electric coupling coefficient. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.view abstract 10.1007/s00419-019-01534-z **Introduction from the New Editor-in-Chief**

Schröder, J.*Archive of Applied Mechanics*89 (2019)view abstract 10.1007/s00419-019-01580-7 **Low-order locking-free mixed finite element formulation with approximation of the minors of the deformation gradient**

Kraus, A. and Wriggers, P. and Viebahn, N. and Schröder, J.*International Journal for Numerical Methods in Engineering*120 (2019)In this work, a low-order mixed finite element formulation for three-dimensional nonlinear elastic problems is presented. The main goal of this paper is to develop a robust and efficient element formulation to overcome locking arising in the cases of hyperelastic bending, quasi-incompressibility, and anisotropy. For this, a low-order discretisation of a five-field Hu-Washizu functional written in terms of the minors of the Cauchy-Green tensor is used. For the tested boundary value problems, the proposed element formulation is more accurate and computational efficient than comparable element formulations. © 2019 John Wiley & Sons, Ltd.view abstract 10.1002/nme.6168 **Model order reduction with Galerkin projection applied to nonlinear optimization with infeasible primal-dual interior point method**

Nigro, P.S.B. and Simões, E.T. and Pimenta, P.M. and Schröder, J.*International Journal for Numerical Methods in Engineering*120 (2019)It is not new that model order reduction (MOR) methods are employed in almost all fields of engineering to reduce the processing time of complex computational simulations. At the same time, interior point methods (IPMs), a method to deal with inequality constraint problems (which is little explored in engineering), can be applied in many fields such as plasticity theory, contact mechanics, micromechanics, and topology optimization. In this work, a MOR based in Galerkin projection is coupled with the infeasible primal-dual IPM. Such research concentrates on how to develop a Galerkin projection in one field with the interior point method; the combination of both methods, coupled with Schur complement, permits to solve this MOR similar to problems without constraints, leading to new approaches to adaptive strategies. Moreover, this research develops an analysis of error from the Galerkin projection related to the primal and dual variables. Finally, this work also suggests an adaptive strategy to alternate the Galerkin projection operator, between primal and dual variable, according to the error during the processing of a problem. © 2019 The Authors. International Journal for Numerical Methods in Engineering Published by John Wiley & Sons Ltd.view abstract 10.1002/nme.6181 **Modified mixed least-squares finite element formulations for small and finite strain plasticity**

Igelbüscher, M. and Schwarz, A. and Steeger, K. and Schröder, J.*International Journal for Numerical Methods in Engineering*117 (2019)In this contribution, we propose mixed least-squares finite element formulations for elastoplastic material behavior. The resulting two-field formulations depending on displacements and stresses are given through the (Formula presented.) -norm minimization of the residuals of the first-order system of differential equations. The residuals are the balance of momentum and the constitutive equation. The advantage of using mixed methods for an elastoplastic material description lies in the direct approximation of the stresses as an unknown variable. In addition to the standard least-squares formulation, an extension of the least-squares functional as well as a modified formulation is done. The modification by means of a varied first variation of the functional is necessary to guarantee a continuous weak form, which is not automatically given within the elastoplastic least-squares approach. For the stress approximation, vector-valued Raviart-Thomas functions are chosen. On the other hand, standard Lagrange polynomials are taken into account for the approximation of the displacements. We consider classical J2 plasticity for a small and a large deformation model for the proposed formulations. For the description of the elastic material response, we choose for the small strain model Hooke's law and for finite deformations a hyperelastic model of Neo-Hookean type. The underlying plastic material response is defined by an isotropic von Mises yield criterion with linear hardening. © 2018 John Wiley & Sons, Ltd.view abstract 10.1002/nme.5951 **Numerical calibration of elasto-plastic phase-field modeling of fracture for experimental pullout tests of single steel fibers embedded in high-performance concrete**

Gebuhr, G. and Anders, S. and Pise, M. and Brands, D. and Sarhil, M. and Schröder, J.In this contribution, a constitutive framework of elasto-plastic phase-field model of fracture is applied to simulation of pullout test of single steel fiber embedded in high-performance concrete and is compared to experimental results. For the description of the mechanical behavior the Drucker-Prager plasticity model is used. The aim is to examine the pullout behavior of a single steel fiber and its influence on the overall material behavior. Therefore, the mechanical behavior of high-performance concrete is studied in experiments. The predictive capability of the above mentioned model is analyzed in detail by simulating pullout tests and calibrated using experimental data. © 2019 Taylor & Francis Group, London, UK.view abstract 10.1201/9780429426506-240 **A 3D magnetostrictive preisach model for the simulation of magneto-electric composites on multiple scales**

Schröder, J. and Labusch, M.*Lecture Notes in Applied and Computational Mechanics*86 (2018)In this contribution we derive a three dimensional ferroelectric Preisach model based on an orientation distribution function. Therefore, the classical scalar Preisach model is modified and applied on the individual orientations, which are uniformly distributed in the three dimensional space. This model is used to simulate the behavior of magneto-electric (ME) composites. Such effective multiferroic materials combine two or more ferroic characteristics and can exhibit a coupling between electric polarization and magnetization. Since most of the single-phase ME materials exhibit a weak magneto-electric coupling at low temperatures, two-phase ME composites produce an ME coupling at room temperature. The basic idea for the manufacturing of ME composites is to use the interaction of the ferroelectric and magnetostrictive phases in order to generate strain-induced ME properties. However, in contrast to single-phase multiferroics, the ME coefficient of composites significantly depends on the microscopic morphology and the electro- as well as magneto-mechanical properties of the individual constituents. Therefore, we implemented the 3D Preisach model into the FE 2 -method in order to depict the realistic ferroelectric behavior and directly incorporate the microstructure by the consideration of underlying representative volume elements. © Springer International Publishing AG 2018.view abstract 10.1007/978-3-319-65463-8_15 **A comparative study of mixed least-squares FEMs for the incompressible navier-stokes equations**

Schwarz, A. and Nickaeen, M. and Serdas, S. and Nisters, C. and Ouazzi, A. and Schröder, J. and Turek, S.*International Journal of Computational Science and Engineering*17 (2018)In the present contribution, we compare (quantitatively) different mixed least-squares finite element methods (LSFEMs) with respect to computational costs and accuracy. Various first-order systems are derived based on the residual forms of the equilibrium equation and the continuity condition. The first formulation under consideration is a div-grad first-order system resulting in a three-field formulation with total stresses, velocities, and pressure (S-V-P) as unknowns. Here, the variables are approximated in H(div) × H1 × L2 on triangles and in H1 × H1 × L2 on quadrilaterals. In addition to that a reduced stress-velocity (S-V) formulation is derived and investigated. S-V-P and S-V formulations are promising approaches when the stresses are of special interest, e.g., for non-Newtonian, multiphase or turbulent flows. The main focus of the work is drawn to performance and accuracy aspects on the one side for finite elements with different interpolation orders and on the other side on the usage of efficient solvers, for instance of Krylov-space or multigrid type. © 2018 Inderscience Enterprises Ltd.view abstract 10.1504/ijcse.2018.094421 **A simple and efficient Hellinger–Reissner type mixed finite element for nearly incompressible elasticity**

Viebahn, N. and Steeger, K. and Schröder, J.*Computer Methods in Applied Mechanics and Engineering*340 (2018)The mixed finite element method is a promising approach in order to overcome locking phenomena of classical displacement based finite elements in the (nearly) incompressible regime for the elasticity problem. In this work we present a novel element based on Hellinger–Reissner's principle for linear elasticity. Essential for the construction of the element is a restriction of the solution space for the stresses, resulting in a formulation with displacements in H1(B) and stresses in H(div,B). The symmetry of the stresses is achieved in a weak sense, without the necessity of additional degrees of freedom. This formulation leads for the case of lowest order interpolation to a very efficient and robust finite element, even satisfying the numerical inf–sup test in two and three dimensions. © 2018 Elsevier B.V.view abstract 10.1016/j.cma.2018.06.001 **An FE2-scheme for magneto-electro-mechanically coupled boundary value problems**

Labusch, M. and Schröder, J. and Keip, M.-A.*CISM International Centre for Mechanical Sciences, Courses and Lectures*581 (2018)The magneto-electric coupling in materials can find several applications in future technologies and could furthermore improve devices, for instance in data storage media. Since all natural single-phase magneto-electric (ME) multiferroics and most of the synthetic single-phase ME materials show a magneto-electric coupling far below room temperature, composite materials are manufactured which consist of a ferroelectric and a magnetostrictive phase. Both phases interact with each other in consequence of transferred deformations, such that these composites produce strain-induced ME properties at room temperature. In order to predict a realistic material behavior, it is necessary to implement appropriate numerical models to reflect the natural behavior of the single phases. This chapter will focus on the numerical implementation of different numerical models for the description of ferroic materials into the two-scale finite element homogenization approach (FE2-method) for the simulation of magneto-electro-mechanically coupled boundary value problems. In detail, we investigate the magneto-electric response in consideration of piezoelectric, electrostrictive and ferroelectric models in combination with a piezomagnetic phase. Reliable results for the ME coefficient are obtained by using a ferroelectric model which takes into account the microscopic properties over single tetragonal unit cells, which are allocated in the three dimensional space based on an orientation distribution function. As a result of switching-events of microscopic polarization vectors, we obtain the macroscopic dielectric and butterfly hysteresis curves of the ferroelectric phase. Thereby, the model for the computations of the ME coefficient is physically motivated. © 2018, CISM International Centre for Mechanical Sciences.view abstract 10.1007/978-3-319-68883-1_5 **Different approaches for mixed LSFEMs in hyperelasticity: Application of logarithmic deformation measures**

Schwarz, A. and Steeger, K. and Igelbüscher, M. and Schröder, J.*International Journal for Numerical Methods in Engineering*115 (2018)We present geometrically nonlinear formulations based on a mixed least-squares finite element method. The L2-norm minimization of the residuals of the given first-order system of differential equations leads to a functional, which is a two-field formulation dependent on displacements and stresses. Based thereon, we discuss and investigate two mixed formulations. Both approaches make use of the fact that the stress symmetry condition is not fulfilled a priori due to the row-wise stress approximation with vector-valued functions belonging to a Raviart-Thomas space, which guarantees a conforming discretization of H(div). In general, the advantages of using the least-squares finite element method lie, for example, in an a posteriori error estimator without additional costs or in the fact that the choice of the polynomial interpolation order is not restricted by the Ladyzhenskaya-Babuška-Brezzi condition (inf-sup condition). We apply a hyperelastic material model with logarithmic deformation measures and investigate various benchmark problems, adaptive mesh refinement, computational costs, and accuracy. Copyright © 2018 John Wiley & Sons, Ltd.view abstract 10.1002/nme.5838 **Direct and inverse identification of constitutive parameters from the structure of soft tissues. Part 1: micro- and nanostructure of collagen fibers**

Marino, M. and Horn-Von Hoegen, M. and Schröder, J. and Wriggers, P.*Biomechanics and Modeling in Mechanobiology*17 (2018)Soft tissues are characterized by a nonlinear mechanical response, highly affected by the multiscale structure of collagen fibers. The effectiveness and the calibration of constitutive models play a major role on the reliability and the applicability of computational models in biomechanics. This paper presents a procedure for the identification of the relationship between collagen-related structural features in soft tissues with model parameters of classical polynomial- and exponential-based constitutive models. Histological features at microscale, as well as biochemical and biophysical properties at nanoscale, are addressed by employing a multiscale structural description of soft tissue mechanics as benchmark data set. Both the direct (from structure to parameters) and the inverse (from parameters to structure) problem are addressed. Suitable optimization problems are introduced for accurate numerical and approximated analytical direct relationships. The inverse identification has been addressed by providing also a measure of the reliability of the computed estimates. Results show the effectiveness of the proposed strategies and allow to discuss the fitting capabilities of classical constitutive approaches in terms of parameters identification. © 2018, Springer-Verlag GmbH Germany, part of Springer Nature.view abstract 10.1007/s10237-018-1009-8 **Finite and virtual element formulations for large strain anisotropic material with inextensive fibers**

Wriggers, P. and Hudobivnik, B. and Schröder, J.*Lecture Notes in Applied and Computational Mechanics*86 (2018)Anisotropic material with inextensive or nearly inextensible fibers introduce constraints in the mathematical formulations of the underlying differential equations from mechanics. This is always the case when fibers with high stiffness in a certain direction are present and a relatively weak matrix material is supporting these fibers. In numerical solution schemes like the finite element method or the virtual element method the presence of constraints—in this case associated to a possible fiber inextensibility compared to a matrix—lead to so called locking-phenomena. This can be overcome by special interpolation schemes as has been discussed extensively for volume constraints like incompressibility as well as contact constraints. For anisotropic material behaviour the most severe case is related to inextensible fibers. In this paper a mixed method is developed for finite elements and virtual elements that can handle anisotropic materials with inextensive and nearly inextensive fibers. For this purpose a classical ansatz, known from the modeling of volume constraint is adopted leading stable elements that can be used in the finite strain regime. © Springer International Publishing AG 2018.view abstract 10.1007/978-3-319-65463-8_11 **Fundamentals of magneto-electro-mechanical couplings: Continuum formulations and invariant requirements**

Schröder, J.*CISM International Centre for Mechanical Sciences, Courses and Lectures*581 (2018)Couplings of magnetic and electric fields in materials could allow for promising applications in medical and information technology. In this contribution, we recapitulate well-known aspects of magneto-electro-mechanical properties and their couplings. At first, we echo basic aspects of electricity and magnetism and Maxwell’s equations. Secondly, we summarize the governing equations for electrostatics and magnetostatics, point out the properties of physical fields across internal surfaces, and discuss the work-energy theorem of electrodynamics, the so-called Poynting‘s theorem. Thirdly, we will discuss some fundamental concepts of magneto-electro-mechanical couplings in matter. Here, we will formulate thermodynamic potentials depending on different basic variables in order to be flexible with a view to different modeling aspects. Afterwards, we discuss aspects of form-invariance of physical laws under coordinate transformations: Lorentz invariance, Galilean transformation and time reversal. Here, we focus on piezoelectric as well as on magnetic symmetry groups and give remarks on classical invariant theory suitable for coordinate-invariant modeling of thermodynamical potentials. © 2018, CISM International Centre for Mechanical Sciences.view abstract 10.1007/978-3-319-68883-1_1 **Multiscale modeling of electroactive polymer composites**

Keip, M.-A. and Schröder, J.*CISM International Centre for Mechanical Sciences, Courses and Lectures*581 (2018)Electroactive polymer composites are materials that consist of an elastomeric matrix and dispersed high-dielectric-modulus or metallic inclusions. The addition of the inclusions generally leads to a significant enhancement of the electrostatic actuation or, more generally, of the overall electro-mechanical coupling. This enhancement is mainly due to the contrast of dielectric moduli of the individual phases, which induces fluctuations of the electric field in the matrix material. The present contribution aims at the derivation and implementation of a multiscale homogenization framework for the macroscopic simulation of electroactive polymer composites with explicit consideration of their microscopic structure. This is achieved through the development of a two-scale computational homogenization approach for electro-mechanically coupled solids at finite deformations. The microscopic part of the problem is defined on a representative volume element that is attached at each integration point of the macroscopic domain. In order to derive energetically consistent transition conditions between the scales a generalized form of the Hill-Mandel condition extended to electro-elastic phenomena at large deformations is exploited. An efficient solution of the macroscopic boundary value problem is guaranteed by means of an algorithmically consistent tangent. The method is applied to the simulation of different dielectric polymer-ceramic composites, which are analzyed with regard to their effective actuation properties. In addition to that, an example of a multiscale electro-mechanical actuator at large deformations is presented. © 2018, CISM International Centre for Mechanical Sciences.view abstract 10.1007/978-3-319-68883-1_6 **Remarks on a fluid-structure interaction scheme based on the least-squares finite element method at small strains**

Nisters, C. and Schwarz, A. and Averweg, S. and Schröder, J.*Advanced Structured Materials*80 (2018)The present contribution introduces a least-squares finite element method (LSFEM) based fluid-structure interaction (FSI) approach. The proposed method is based on the formulation of mixed finite elements in terms of stresses and velocities for both the fluid and the solid regime. The LSFEM offers the advantage of a flexibility to construct functionals with sophisticated physical quantities as e.g. stresses, velocities and displacements. The approximation of the stresses and velocities in suitable spaces, namely in the spaces H (div) and H1, respectively, leads to the inherent fulfillment of the coupling conditions of a FSI method. A numerical example considering an incompressible, linear elastic material behavior at small deformations and the incompressible Navier–Stokes equations demonstrates the applicability of the LSFEM-FSI method. © 2018, Springer International Publishing AG.view abstract 10.1007/978-3-319-70563-7_12 **Semiconductor effects in ferroelectrics**

Lupascu, D.C. and Anusca, I. and Etier, M. and Gao, Y. and Lackner, G. and Nazrabi, A. and Sanlialp, M. and Trivedi, H. and Ul-Haq, N. and Schröder, J.*CISM International Centre for Mechanical Sciences, Courses and Lectures*581 (2018)In this textbook ferroelectrics have so far been dealt with as insulators. External electric fields can and will induce polarization in any insulating material. This is dielectricity. On top of this, pyroelectrics exhibit a temperature dependent spontaneous electric polarization, namely a crystallographic phase transition which is polar. It disappears above the Curie-point. Below the Curie point, external electric fields can rotate or alter the direction of this spontaneous polarization. If this becomes a remanent state, the material is ferroelectric and exhibits electric hysteresis. Another aspect in these materials is the fact that electrical insulation is a stretchable term. While metals are well defined and offer conductivity down to very low temperatures, already semi-metals will turn partly insulating at low temperature. Semiconductors are typically insulating in a certain low temperature range (energetically ≲ 1/10 kT) above which thermal excitation of charge carriers into the conduction band will induce a finite conductivity. The energetic band gap determines this barrier and the exponential tail of the Fermi-Dirac distribution determines the number of charge carriers in the conduction band as well as the missing electrons (termed holes) in the valence band. Typical ferroelectrics exhibit band gaps that turn the material insulating at room temperature. This is the case for most oxides. Ferroelectric sulfides typically display much lower band gaps and turn conducting at or even below room temperature already. Another aspect enters when one considers that external electroding is always necessary to drive a ferroelectric capacitor. In the context of a semiconductor picture we deal with a classical Schottky barrier. Grain boundaries play another particular role in polycrystalline materials. This may even lead to positive temperature coefficient resistor (PTCR) characteristics. In this lecture we will draw the connection between a ferroelectric, its semiconductor character, point defects, and their overall interactions. Particularly the inner and outer boundaries of crystallites become subject to band bending, 2D-conducting planes, space charge regions, and diverse other effects. Also optical effects as well as fatigue depend on the semiconductor and defect induced properties. We intend to give the newcomer access to this complex field which has seen a peak in understanding in the late 70ies of the 20th century experiencing a certain revival recently due to a number of exciting findings associated with domain walls. Furthermore, magnetoelectric composites have recently been found to display peculiar electrical effects related to their semiconductor character rather than the magnetic part of their properties. © 2018, CISM International Centre for Mechanical Sciences.view abstract 10.1007/978-3-319-68883-1_3 **Stress-velocity mixed least-squares FEMs for the time-dependent incompressible navier-stokes equations**

Schwarz, A. and Nisters, C. and Averweg, S. and Schröder, J.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*10665 LNCS (2018)In this article a mixed least-squares finite element method (LSFEM) for the time-dependent incompressible Navier-Stokes equations is proposed and investigated. The formulation is based on the incompressible Navier-Stokes equations consisting of the balance of momentum and the continuity equations. In order to obtain a first-order system the Cauchy stress tensor is introduced as an additional variable to the system of equations. From this stress-velocity-pressure approach a stress-velocity formulation is derived by adding a redundant residual to the functional without additional variables in order to strengthen specific physical relations, e.g. mass conservation. We account for implementation aspects of triangular mixed finite elements especially regarding the approximation used for H(div) × H1 and the discretization in time using the Newmark method. Finally, we present the flow past a cylinder benchmark problem in order to demonstrate the derived stress-velocity least-squares formulation. © Springer International Publishing AG 2018.view abstract 10.1007/978-3-319-73441-5_14 **Strong converse magnetoelectric effect in (Ba,Ca)(Zr,Ti)O3 - NiFe2O4 multiferroics: A relationship between phase-connectivity and interface coupling**

Naveed-Ul-Haq, M. and Shvartsman, V.V. and Trivedi, H. and Salamon, S. and Webers, S. and Wende, H. and Hagemann, U. and Schröder, J. and Lupascu, D.C.*Acta Materialia*144 (2018)Studying multiferroic magnetoelectrics has been a focus field for the last decade and a half, and the exploration of new materials is one of the several aspects of this quest. Here we report on the synthesis and characterization of NiFe2O4-based multiferroic composites which employ (Ba,Ca)(Zr,Ti)O3 as the ferroelectric/piezoelectric component and NiFe2O4 as the magnetostrictive phase. We find that these composites show excellent magnetoelectric properties. Especially the composite with 30 vol% of NiFe2O4 has a converse ME coefficient approximately two times larger than the previously reported one for BaTiO3-CoFe2O4 composites. A relationship between the phase connectivity within these composites and the ME properties was explored by the time of flight secondary ion mass microscopy. We believe that our investigation will be helpful for the design of magnetoelectric materials as components of sensors and memory devices. © 2017 Acta Materialia Inc.view abstract 10.1016/j.actamat.2017.10.048 **A Prange–Hellinger–Reissner type finite element formulation for small strain elasto-plasticity**

Schröder, J. and Igelbüscher, M. and Schwarz, A. and Starke, G.*Computer Methods in Applied Mechanics and Engineering*317 (2017)In this contribution we propose a mixed variational formulation of the Prange–Hellinger–Reissner type for elasto-plasticity at small strains. Here, the displacements and the stresses are interpolated independently, which are balanced within the variational functional by the relation of the elastic strains and the partial derivative of the complementary stored energy with respect to the stresses. For the elasto-plastic material behavior a von Mises yield criterion is considered, where we restrict ourselves w.l.o.g. to linear isotropic hardening. In the proposed formulation we enforce the constraints arising from plasticity point-wise in contrast to the element-wise realization of the plastic return mapping algorithm suggested in Simo et al. (1989). The performance of the new formulation is demonstrated by the analysis of several benchmark problems. Here, we compare the point-wise treatment of elasto-plasticity with the original element-wise formulation of Simo et al. (1989). Furthermore, we derive an algorithmic consistent treatment for plane stress as well as for plane strain condition. © 2016 Elsevier B.V.view abstract 10.1016/j.cma.2016.12.005 **A simple triangular finite element for nonlinear thin shells: statics, dynamics and anisotropy**

Viebahn, N. and Pimenta, P.M. and Schröder, J.*Computational Mechanics*59 (2017)This work presents a simple finite element implementation of a geometrically exact and fully nonlinear Kirchhoff–Love shell model. Thus, the kinematics are based on a deformation gradient written in terms of the first- and second-order derivatives of the displacements. The resulting finite element formulation provides C1-continuity using a penalty approach, which penalizes the kinking at the edges of neighboring elements. This approach enables the application of well-known C0-continuous interpolations for the displacements, which leads to a simple finite element formulation, where the only unknowns are the nodal displacements. On the basis of polyconvex strain energy functions, the numerical framework for the simulation of isotropic and anisotropic thin shells is presented. A consistent plane stress condition is incorporated at the constitutive level of the model. A triangular finite element, with a quadratic interpolation for the displacements and a one-point integration for the enforcement of the C1-continuity at the element interfaces leads to a robust shell element. Due to the simple nature of the element, even complex geometries can be meshed easily, which include folded and branched shells. The reliability and flexibility of the element formulation is shown in a couple of numerical examples, including also time dependent boundary value problems. A plane reference configuration is assumed for the shell mid-surface, but initially curved shells can be accomplished if one regards the initial configuration as a stress-free deformed state from the plane position, as done in previous works. © 2016, Springer-Verlag Berlin Heidelberg.view abstract 10.1007/s00466-016-1343-6 **Electro-chemical aspects of IPMCs within the framework of the theory of porous media**

Leichsenring, P. and Serdas, S. and Wallmersperger, T. and Bluhm, J. and Schröder, J.*Smart Materials and Structures*26 (2017)Ionomeric polymer-metal composites (IPMCs) consist of an ionomer with bound anionic groups and mobile counterions. They are plated with noble impermeable metal cover layers. By application of an electric voltage, a transport of the mobile ions towards the respective electrode occurs. Due to local electrostatic and ionic forces, a local deformation of the IPMC can be observed. Therefore IPMCs are promising candidates for electrochemical transducers. In the present research, the chemo-electro-mechanical behavior of IPMCs is described within the framework of the theory of porous media. First, the field equations are derived with respect to the second law of thermodynamics. Second, a reduced set of equations for the chemo-electric behavior is formulated and discretized by applying the finite element method. In the numerical investigations a parametric study of the time and space dependent behavior is carried out in order to quantify the influence of different material compositions. Based on this study, the characteristic response of IPMC to the application of an electric voltage can be predicted. Concluding, the obtained computational framework is an excellent tool for the design of electrochemical transducers. © 2017 IOP Publishing Ltd.view abstract 10.1088/1361-665X/aa590e **On the stability analysis of hyperelastic boundary value problems using three- and two-field mixed finite element formulations**

Schröder, J. and Viebahn, N. and Wriggers, P. and Auricchio, F. and Steeger, K.*Computational Mechanics*(2017)In this work we investigate different mixed finite element formulations for the detection of critical loads for the possible occurrence of bifurcation and limit points. In detail, three- and two-field formulations for incompressible and quasi-incompressible materials are analyzed. In order to apply various penalty functions for the volume dilatation in displacement/pressure mixed elements we propose a new consistent scheme capturing the non linearities of the penalty constraints. It is shown that for all mixed formulations, which can be reduced to a generalized displacement scheme, a straight forward stability analysis is possible. However, problems based on the classical saddle-point structure require a different analyses based on the change of the signature of the underlying matrix system. The basis of these investigations is the work from Auricchio et al. (Comput Methods Appl Mech Eng 194:1075–1092, 2005, Comput Mech 52:1153–1167, 2013). © 2017 Springer-Verlag Berlin Heidelbergview abstract 10.1007/s00466-017-1415-2 **The exponentiated Hencky energy: anisotropic extension and case studies**

Schröder, J. and Horn-Von Hoegen, M.. and Neff, P.*Computational Mechanics*(2017)In this paper we propose an anisotropic extension of the isotropic exponentiated Hencky energy, based on logarithmic strain invariants. Unlike other elastic formulations, the isotropic exponentiated Hencky elastic energy has been derived solely on differential geometric grounds, involving the geodesic distance of the deformation gradient (Formula presented.) to the group of rotations. We formally extend this approach towards anisotropy by defining additional anisotropic logarithmic strain invariants with the help of suitable structural tensors and consider our findings for selected case studies. © 2017 Springer-Verlag GmbH Germanyview abstract 10.1007/s00466-017-1466-4 **A novel mixed finite element for finite anisotropic elasticity; the SKA-element Simplified Kinematics for Anisotropy**

Schröder, J. and Viebahn, N. and Balzani, D. and Wriggers, P.*Computer Methods in Applied Mechanics and Engineering*310 (2016)A variety of numerical approximation schemes for boundary value problems suffer from so-called locking-phenomena. It is well known that in such cases several finite element formulations exhibit poor convergence rates in the basic variables. A serious locking phenomenon can be observed in the case of anisotropic elasticity, due to high stiffness in preferred directions. The main goal of this paper is to overcome this locking problem in anisotropic hyperelasticity by introducing a novel mixed variational framework. Therefore we split the strain energy into two main parts, an isotropic and an anisotropic part. For the isotropic part we can apply different well-established approximation schemes and for the anisotropic part we apply a constant approximation of the deformation gradient or the right Cauchy–Green tensor. This additional constraint is attached to the strain energy function by a second-order tensorial Lagrange-multiplier, governed by a Simplified Kinematic for the Anisotropic part. As a matter of fact, for the tested boundary value problems the SKA-element based on quadratic ansatz functions for the displacements, performs excellent and behaves more robust than competitive formulations. © 2016 Elsevier B.V.view abstract 10.1016/j.cma.2016.06.029 **Algorithmic two-scale transition for magneto-electro-mechanically coupled problems. FE2-scheme: Localization and homogenization**

Schröder, J. and Labusch, M. and Keip, M.-A.*Computer Methods in Applied Mechanics and Engineering*302 (2016)In this paper we present a computational homogenization procedure for the simulation of magneto-electro-mechanically coupled boundary value problems (bvp)s on two scales. We derive the basic equations for the localization and the homogenization of the individual field variables and give an algorithmic expression for the effective tangent moduli. The resulting algorithmic two-scale transition procedure is implemented into an FE2-method, which allows us to compute macroscopic boundary value problems in consideration of attached microscopic representative volume elements. The challenge in the simulation of magneto-electro-mechanically coupled materials is the modeling of the complicated interactions between magnetical, electrical and mechanical quantities on both scales. A primer example for such interactions is given by magneto-electric composites. Thus, we apply the presented method to the two-scale simulation of a variety of versions of magneto-electric composites. In detail, we consider two-phase composites composed of (i) piezomagnetic and piezoelectric, (ii) piezomagnetic and non-linear electrostrictive, as well as (iii) piezomagnetic and non-linear dissipative ferroelectric phase. Based on numerical examples, we discuss aspects of the applicability, the numerical stability as well as the predictive capability of the proposed method. © 2015 Elsevier B.V.view abstract 10.1016/j.cma.2015.10.005 **An algorithmic scheme for the automated calculation of fiber orientations in arterial walls**

Fausten, S. and Balzani, D. and Schröder, J.*Computational Mechanics*58 (2016)We propose an algorithmic scheme for the numerical calculation of fiber orientations in arterial walls. The basic assumption behind the procedure is that the fiber orientations are mainly governed by the principal tensile stress directions resulting in an improved load transfer within the artery as a consequence of the redistribution of stresses. This reflects the biological motivation that soft tissues continuously adapt to their mechanical environment in order to optimize their load-bearing capacities. The algorithmic scheme proposed here enhances efficiency of the general procedure given in Hariton et al. (Biomech Model Mechanobiol 6(3):163-175, 2007), which consists of repeatedly identifying a favored fiber orientation based on the principal tensile stresses under a certain loading scenario, and then re-calculating the stresses for that loading scenario with the modified favored fiber orientation. Since the method still depends on a highly accurate stress approximation of the finite element formulation, which is not straightforward to obtain in particular for incompressible and highly anisotropic materials, furthermore, a modified model is introduced. This model defines the favored fiber orientation not only in terms of the local principal stresses, but in terms of the volume averages of the principal stresses computed over individual finite elements. Thereby, the influence of imperfect stress approximations can be weakened leading to a stabilized convergence of the reorientation procedure and a more reasonable fiber orientation with less numerical noise. The performance of the proposed fiber reorientation scheme is investigated with respect to different finite element formulations and different favored fiber orientation models, Hariton et al. (Biomech Model Mechanobiol 6(3):163-175, 2007) and Cyron and Humphrey (Math Mech Solids 1-17, 2014). In addition, it is applied to calculate the fiber orientation in a patient-specific arterial geometry.view abstract 10.1007/s00466-016-1321-z **An engineering tool to estimate eigenstresses in three-dimensional patient-specific arteries**

Schröder, J. and Horn-von Hoegen, M.*Computer Methods in Applied Mechanics and Engineering*306 (2016)In this contribution we propose an engineering based approach to incorporate eigenstress distributions in arteries. Eigenstresses are known for a variety of biological tissues. In the case of arterial walls the so called opening angle experiment is frequently used to characterize the residual strains/stresses. Assuming that the transmural stress distribution of suitable stress measures should be smooth, a residual stress tensor is directly estimated from the Cauchy stresses. These stress measures are constructed in such a way that the basic biophysiological characteristics, like the fiber orientations which are taken into account by suitable structural tensors, are adequately respected. In order to achieve reasonable eigenstress distributions in the arteries the local stress measures should approach spatial-sector averaged values. These ideas are imbedded in an algorithm to update the residual stress tensor incrementally. An ideal tube and degenerated patient-specific arteries from medical imaging are used to demonstrate the performance of the proposed procedure and to estimate the opening angle. © 2016 Elsevier B.V.view abstract 10.1016/j.cma.2016.03.020 **Computational modeling of dual-phase steels based on representative three-dimensional microstructures obtained from EBSD data**

Brands, D. and Balzani, D. and Scheunemann, L. and Schröder, J. and Richter, H. and Raabe, D.*Archive of Applied Mechanics*86 (2016)The microstructure of dual-phase steels consisting of a ferrite matrix with embedded martensite inclusions is the main contributor to the mechanical properties such as high ultimate tensile strength, high work hardening rate, and good ductility. Due to the composite structure and the wide field of applications of this steel type, a wide interest exists in corresponding virtual computational experiments. For a reliable modeling, the microstructure should be included. For that reason, in this paper we follow a computational strategy based on the definition of a representative volume element (RVE). These RVEs will be constructed by a set of tomographic measurements and mechanical tests. In order to arrive at more efficient numerical schemes, we also construct statistically similar RVEs, which are characterized by a lower complexity compared with the real microstructure but which represent the overall material behavior accurately. In addition to the morphology of the microstructure, the austenite–martensite transformation during the steel production has a relevant influence on the mechanical properties and is considered in this contribution. This transformation induces a volume expansion of the martensite phase. A further effect is determined in nanoindentation test, where it turns out that the hardness in the ferrite phase increases exponentially when approaching the martensitic inclusion. To capture these gradient properties in the computational model, the volumetric expansion is applied to the martensite phase, and the arising equivalent plastic strain distribution in the ferrite phase serves as basis for a locally graded modification of the ferritic yield curve. Good accordance of the model considering the gradient yield behavior in the ferrite phase is observed in the numerical simulations with experimental data. © 2015, Springer-Verlag Berlin Heidelberg.view abstract 10.1007/s00419-015-1044-1 **Continuum mechanical description of an extrinsic and autonomous self-healing material based on the theory of porous media**

Specht, S. and Bluhm, J. and Schröder, J.*Advances in Polymer Science*273 (2016)Polymers and polymeric composites are used in many engineering applications, but these materials can spontaneously lose structural integrity as a result of microdamage caused by stress peaks during service. This internal microdamage is hard to detect and nearly impossible to repair. To extend the lifetime of such materials and save maintenance costs, self-healing mechanisms can be applied that are able to repair internal microdamage during the usual service load. This can be realized, for example, by incorporating microcapsules filled with monomer and dispersed catalysts into the polymeric matrix material. If a crack occurs, the monomer flows into the crack, reacts with the catalysts, and closes the crack. This contribution focuses on the development of a thermodynamically consistent constitutive model that is able to describe the damage and healing behavior of a microcapsule-based self-healing material. The material under investigation is an epoxy matrix with microencapsulated dicyclopentadiene healing agents and dispersed Grubbs’ catalysts. The simulation of such a multiphase material is numerically very expensive if the microstructure is to be completely resolved. To overcome this, a homogenization technique can be applied to decrease the computational costs of the simulation. Here, the theoretical framework is based on the theory of porous media, which is a macroscopic continuum mechanical homogenization approach. The developed five-phase model consists of solid matrix material with dispersed catalysts, liquid healing agents, solidified healed material, and gas phase. A discontinuous damage model is used for the description of the damage behavior, and healing is simulated by a phase transition between the liquid-like healing agents and the solidified healed material. Applicability of the developed model is shown by means of numerical simulations of the global damage and healing behavior of a tapered double cantilever beam, as well as simulations of the flow behavior of the healing agents at the microscale. © Springer International Publishing Switzerland 2016.view abstract 10.1007/12_2015_338 **Implementation of incremental variational formulations based on the numerical calculation of derivatives using hyper dual numbers**

Tanaka, M. and Balzani, D. and Schröder, J.*Computer Methods in Applied Mechanics and Engineering*301 (2016)In this paper, novel implementation schemes for the automatic calculation of internal variables, stresses and consistent tangent moduli for incremental variational formulations (IVFs) describing inelastic material behavior are proposed. IVFs recast inelasticity theory as an equivalent optimization problem where the incremental stress potential within a discrete time interval is minimized in order to obtain the values of internal variables. In the so-called Multilevel Newton-Raphson method for the inelasticity theory, this minimization problem is typically solved by using second derivatives with respect to the internal variables. In addition to that, to calculate the stresses and moduli further second derivatives with respect to deformation tensors are required. Compared with classical formulations such as the return mapping method, the IVFs are relatively new and their implementation is much less documented. Furthermore, higher order derivatives are required in the algorithms demanding increased implementation efforts. Therefore, even though IVFs are mathematically and physically elegant, their application is not standard. Here, novel approaches for the implementation of IVFs using HDNs of second and higher order are presented to arrive at a fully automatic and robust scheme with computer accuracy. The proposed formulations are quite general and can be applied to a broad range of different constitutive models, which means that once the proposed schemes are implemented as a framework, any other dissipative material model can be implemented in a straightforward way by solely modifying the constitutive functions. These include the Helmholtz free energy function, the dissipation potential function and additional side constraints such as e.g. the yield function in the case of plasticity. Its uncomplicated implementation for associative finite strain elasto-plasticity and performance is illustrated by some representative numerical examples. © 2015 Elsevier B.V.view abstract 10.1016/j.cma.2015.12.010 **Modeling of Microstructure Evolution with Dynamic Recrystallization in Finite Element Simulations of Martensitic Steel**

Baron, T.J. and Khlopkov, K. and Pretorius, T. and Balzani, D. and Brands, D. and Schröder, J.*Steel Research International*87 (2016)A metallurgical material description of the flow behavior for finite element (FE) simulations was developed. During hot compression tests, the dynamic microstructure evolution is modeled on the example of high-strength martensitic steel MS-W 1200. Compression tests at 900-1000 °C with a strain rate of 0.1 s-1 on fine-grain and coarse-grain samples were performed. An analysis of the flow behavior identified a strong correlation between the dynamic recrystallization kinetics and the initial microstructure. The regression analysis has been used to determine correction factors of the new model to describe the dynamic recrystallization. A good agreement between FE simulation and measurement shows the validity of the new model. A metallurgical material description of the flow behavior for finite element (FE) simulations is developed. During hot compression tests, the dynamic microstructure evolution is modeled on the example of high-strength martensitic steel MS-W 1200. An analysis of the flow behavior identifies a strong correlation between the dynamic recrystallization kinetics and the initial microstructure. © 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim.view abstract 10.1002/srin.201400576 **Numerical modeling of fluid–structure interaction in arteries with anisotropic polyconvex hyperelastic and anisotropic viscoelastic material models at finite strains**

Balzani, D. and Deparis, S. and Fausten, S. and Forti, D. and Heinlein, A. and Klawonn, A. and Quarteroni, A. and Rheinbach, O. and Schröder, J.*International Journal for Numerical Methods in Biomedical Engineering*32 (2016)The accurate prediction of transmural stresses in arterial walls requires on the one hand robust and efficient numerical schemes for the solution of boundary value problems including fluid–structure interactions and on the other hand the use of a material model for the vessel wall that is able to capture the relevant features of the material behavior. One of the main contributions of this paper is the application of a highly nonlinear, polyconvex anisotropic structural model for the solid in the context of fluid–structure interaction, together with a suitable discretization. Additionally, the influence of viscoelasticity is investigated. The fluid–structure interaction problem is solved using a monolithic approach; that is, the nonlinear system is solved (after time and space discretizations) as a whole without splitting among its components. The linearized block systems are solved iteratively using parallel domain decomposition preconditioners. A simple – but nonsymmetric – curved geometry is proposed that is demonstrated to be suitable as a benchmark testbed for fluid–structure interaction simulations in biomechanics where nonlinear structural models are used. Based on the curved benchmark geometry, the influence of different material models, spatial discretizations, and meshes of varying refinement is investigated. It turns out that often-used standard displacement elements with linear shape functions are not sufficient to provide good approximations of the arterial wall stresses, whereas for standard displacement elements or F-bar formulations with quadratic shape functions, suitable results are obtained. For the time discretization, a second-order backward differentiation formula scheme is used. It is shown that the curved geometry enables the analysis of non-rotationally symmetric distributions of the mechanical fields. For instance, the maximal shear stresses in the fluid–structure interface are found to be higher in the inner curve that corresponds to clinical observations indicating a high plaque nucleation probability at such locations. Copyright © 2015 John Wiley & Sons, Ltd. Copyright © 2015 John Wiley & Sons, Ltd.view abstract 10.1002/cnm.2756 **One-way and fully-coupled FE2 methods for heterogeneous elasticity and plasticity problems: Parallel scalability and an application to thermo-elastoplasticity of dual-phase steels**

Balzani, D. and Gandhi, A. and Klawonn, A. and Lanser, M. and Rheinbach, O. and Schröder, J.*Lecture Notes in Computational Science and Engineering*113 (2016)In this paper, aspects of the two-scale simulation of dual-phase steels are considered. First, we present two-scale simulations applying a top-down oneway coupling to a full thermo-elastoplastic model in order to study the emerging temperature field. We find that, for our purposes, the consideration of thermomechanics at the microscale is not necessary. Second, we present highly parallel fully-coupled two-scale FE2 simulations, now neglecting temperature, using up to 458;752 cores of the JUQUEEN supercomputer at Forschungszentrum Jülich. The strong and weak parallel scalability results obtained for heterogeneous nonlinear hyperelasticity exemplify the massively parallel potential of the FE2 multiscale method. © Springer International Publishing Switzerland 2016.view abstract 10.1007/978-3-319-40528-5_5 **Robust numerical schemes for an efficient implementation of tangent matrices: Application to hyperelasticity, inelastic standard dissipative materials and thermo-mechanics at finite strains**

Tanaka, M. and Balzani, D. and Schröder, J.*Lecture Notes in Applied and Computational Mechanics*81 (2016)In this contribution robust numerical schemes for an efficient implementation of tangent matrices in finite strain problems are presented and their performance is investigated through the analysis of hyperelastic materials, inelastic standard dissipative materials in the context of incremental variational formulations, and thermo-mechanics. The schemes are based on highly accurate and robust numerical differentiation approaches which use non-real numbers, i.e., complex variables and hyper-dual numbers. The main advantage of these approaches are that, contrary to the classical finite difference scheme, no round-off errors in the perturbations due to floating-point arithmetics exist within the calculation of the tangent matrices. This results in a method which is independent of perturbation values (in case of complex step derivative approximations if sufficiently small perturbations are chosen). An efficient algorithmic treatment is presented which enables a straightforward implementation of the method in any standard finite-element program. By means of hyperelastic, finite strain elastoplastic, and thermo-elastoplastic boundary value problems, the performance of the proposed approaches is analyzed. © Springer International Publishing Switzerland 2016.view abstract 10.1007/978-3-319-39022-2_1 **Special Issue on Phase Field Approaches to Fracture: In Memory of Professor Christian Miehe (1956–2016)**

Keip, M.-A. and Kiefer, B. and Schröder, J. and Linder, C.*Computer Methods in Applied Mechanics and Engineering*312 (2016)view abstract 10.1016/j.cma.2016.09.042 **Theoretical framework of modeling of ionic EAPs within the Theory of Porous Media**

Bluhm, J. and Serdas, S. and Schröder, J.*Archive of Applied Mechanics*86 (2016)A thermo-electromechanical formulation for the description of ionic electroactive polymers is derived within the framework of the Theory of Porous Media. The model consists of an electrically charged porous solid saturated with an ionic solution. The saturated porous medium is assumed to be incompressible. Different constituents following different kinematic paths are considered such as solid, fluid, anions, cations and free charges. The electromechanical and the electrodynamic field equations are discussed. Based on the second law of thermodynamics, a consistent model is developed. With respect to the closure problem of the model, the needed constitutive relations and evolution equations are presented. © 2016, Springer-Verlag Berlin Heidelberg.view abstract 10.1007/s00419-015-1110-8 **A highly accurate 1st- and 2nd-order differentiation scheme for hyperelastic material models based on hyper-dual numbers**

Tanaka, M. and Sasagawa, T. and Omote, R. and Fujikawa, M. and Balzani, D. and Schröder, J.*Computer Methods in Applied Mechanics and Engineering*283 (2015)In this paper we propose a numerical scheme for the calculation of stresses and corresponding consistent tangent moduli for hyperelastic material models, which are derived in terms of the first and second derivatives of a strain energy function. This numerical scheme provides a compact model-independent framework, which means that once the framework is implemented, any other hyperelastic material model can be incorporated by solely modifying the energy function. The method is based on the numerical calculation of strain energy derivatives using hyper-dual numbers and thus referred to as hyper-dual step derivative (HDSD). The HDSD does neither suffer from roundoff errors nor from truncation errors and is thereby a highly accurate method with high stability being insensitive to perturbation values. Furthermore, it enables the calculation of derivatives of arbitrary order. This is a great advantage compared to other numerical approaches as, e.g., the finite difference approximation which is highly sensitive with respect to the perturbation value and which thus only yields accurate approximations for a small regime of perturbation values. Another alternative, the complex-step derivative approximation enables highly accurate derivatives for a wide range of small perturbation values, but it only provides first derivatives and is thus not able to calculate stresses and moduli at once. In this paper, representative numerical examples using an anisotropic model are provided showing the performance of the proposed method. In detail, an introductory example shows the insensitivity with respect to the perturbation values and the higher accuracy compared to the finite difference scheme. Furthermore, examples demonstrate the robustness and simple implementation of the HDSD scheme in finite element software. It turns out that the higher accuracy compared with other approaches can still be achieved in reasonable computing time. © 2014 Elsevier B.V.view abstract 10.1016/j.cma.2014.08.020 **Computation of non-linear magneto-electric product properties of 0-3 composites**

Schröder, J. and Labusch, M. and Keip, M.-A. and Kiefer, B. and Brands, D. and Lupascu, D.C.*GAMM Mitteilungen*38 (2015)The magneto-electric (ME) coupling of multiferroic materials is of high interest for a variety of advanced applications like in data storage or sensor technology. Since the ME coupling of single-phase multiferroics is too low for technical applications, the manufacturing of composite structures becomes relevant. These composites generate the effective ME coupling as a strain-induced product property. Several experiments on composite multiferroics showed remarkable ME coefficients that are orders of magnitudes higher than those of single-phase materials. The present paper investigates the arising effective product properties of two-phase ME composites by simulating the coupling behavior using a two-scale finite element (FE2) homogenization approach. By means of this method, microstructures with different volume fractions of the individual phases and associated macroscopic ME coupling coefficients are considered. We investigate the influence of different magnetization states by means of the non-linear dissipative magnetostriction material model originally established in [1]. © 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.view abstract 10.1002/gamm.201510002 **Construction of statistically similar RVEs**

Scheunemann, L. and Balzani, D. and Brands, D. and Schröder, J.*Lecture Notes in Applied and Computational Mechanics*78 (2015)In modern engineering, micro-heterogeneous materials are designed to satisfy the needs and challenges in a wide field of technical applications. The effective mechanical behavior of these materials is influenced by the inherent microstructure and therein the interaction and individual behavior of the underlying phases. Computational homogenization approaches, such as the FE2 method have been found to be a suitable tool for the consideration of the influences of the microstructure. However, when real microstructures are considered, high computational costs arise from the complex morphology of the microstructure. Statistically similar RVEs (SSRVEs) can be used as an alternative, which are constructed to possess similar statistical properties as the realmicrostructure but are defined by a lower level of complexity. These SSRVEs are obtained from a minimization of differences of statistical measures and mechanical behavior compared with a real microstructure in a staggered optimization scheme, where the inner optimization ensures statistical similarity and the outer optimization problem controls themechanical comparativity of the SSRVE and the real microstructure. The performance of SSRVEs may vary with the utilized statistical measures and the parameterization of the microstructure of the SSRVE.With regard to an efficient construction of SSRVEs, it is necessary to consider statistical measures which can be computed in reasonable time and which provide sufficient information of the real microstructure.Minkowski functionals are analyzed as possible basis for statistical descriptors of microstructures and compared with other well-known statistical measures to investigate the performance. In order to emphasize the general importance of considering microstructural features by more sophisticated measures than basic ones, i.e. volume fraction, an analysis of upper bounds on the error of statistical measures and mechanical response is presented. © Springer International Publishing Switzerland 2015.view abstract 10.1007/978-3-319-18242-1_9 **Coordinate-invariant phase field modeling of ferro-electrics, part I: Model formulation and single-crystal simulations**

Schrade, D. and Keip, M.-A. and Thai, H. and Schröder, J. and Svendsen, B. and Müller, R. and Gross, D.*GAMM Mitteilungen*38 (2015)An electro-mechanically coupled phase field model for ferroelectric domain evolution is introduced. Based on Gurtin's concept of a microforce balance, a generalized Ginzburg-Landau evolution equation is derived from the second law of thermodynamics. The thermodynamic potential is formulated for transversely isotropic material behavior by adopting a coordinateinvariant formulation. The model is reduced to 2D and implemented into a finite element framework. The numerical simulations concern the microstructure evolution in mechanically clamped BaTiO3 single-crystals. In the second part of this contribution Keip et al. [1], the poling behavior of ferroelectric composites and polycrystals is investigated with regard to size effects and the influence of a discontinuous order parameter field across grain boundaries. © 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.view abstract 10.1002/gamm.201510005 **Coordinate-invariant phase field modeling of ferro-electrics, part II: Application to composites and poly-crystals**

Keip, M.-A. and Schrade, D. and Thai, H. and Schröder, J. and Svendsen, B. and Müller, R. and Gross, D.*GAMM Mitteilungen*38 (2015)This paper deals with the application of the model presented in the first part Schrade et al. [1] to ferroelectric composites filled with electrically conducting inclusions as well as to ferroelectric polycrystals. Composites are analyzed through the use of a computational homogenization framework for phase field methods proposed in Zäh & Miehe [2]. This will give insights into the coupled phenomena taking place on the microscale and on their relation to the overall behavior. Both will be of special interest for the development of advanced composite materials with tailored properties like, for example, particulate magneto-electric composites, which are composed of a ferroelectric matrix and magnetic rare-earth elements or metals. Furthermore, we analyze the behavior of ferroelectric polycrystals with a focus on size effects. This will enable us to reveal preferred microstructure configurations depending on the system and grain size. In addition to that, it will serve as basis for the extraction of the directional properties of polycrystals with respect to their switching behavior in the different grains of the polycrystal. Associated simulations could then be used to supply coarser models with the needed directional informations. (© 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) © 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.view abstract 10.1002/gamm.201510006 **Design of 3D statistically similar Representative Volume Elements based on Minkowski functionals**

Scheunemann, L. and Balzani, D. and Brands, D. and Schröder, J.*Mechanics of Materials*90 (2015)In this paper an extended optimization procedure is proposed for the construction of statistically similar RVEs (SSRVEs) which are defined as artificial microstructures showing a lower complexity than the associated real microstructures. This enables a computationally efficient discretization required for numerical calculations of microscopic boundary value problems and leads therefore to more efficient computational two-scale schemes. The optimization procedure is staggered and consists of an outer and an inner optimization problem. The outer problem treats different types of morphology parameterizations, different sets of statistical measures and different sets of weighting factors needed in the inner problem to minimize differences of mechanical errors that compare the response of the SSRVE with a target (real) microstructure. The inner problem minimizes differences of statistical measures describing the microstructure morphology for fixed parameterization type, statistical measures and weighting factors. The main contribution here is the analysis of new microstructure descriptors based on tensor-valued Minkowski functionals, whose numerical calculation requires less time compared to e.g. lineal-path functions. Thereby, a more efficient inner optimization problem can be realized and thus, an automated solution of the outer optimization problem becomes more practicable. Representative examples demonstrate the performance of the proposed method. It turns out that the evaluation of objective functions formulated in terms of the Minkowski functionals is almost 2000 times faster than functions taking into account lineal-path functions. © 2015 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.mechmat.2015.03.005 **Electrochemical strain microscopy time spectroscopy: Model and experiment on LiMn**2 O4

Amanieu, H.-Y. and Thai, H.N.M. and Luchkin, S.Yu. and Rosato, D. and Lupascu, D.C. and Keip, M.-A. and Schröder, J. and Kholkin, A.L.*Journal of Applied Physics*118 (2015)Electrochemical Strain Microscopy (ESM) can provide useful information on ionic diffusion in solids at the local scale. In this work, a finite element model of ESM measurements was developed and applied to commercial lithium manganese (III,IV) oxide (LiMn<inf>2</inf>O<inf>4</inf>) particles. ESM time spectroscopy was used, where a direct current (DC) voltage pulse locally disturbs the spatial distribution of mobile ions. After the pulse is off, the ions return to equilibrium at a rate which depends on the Li diffusivity in the material. At each stage, Li diffusivity is monitored by measuring the ESM response to a small alternative current (AC) voltage simultaneously applied to the tip. The model separates two different mechanisms, one linked to the response to DC bias and another one related to the AC excitation. It is argued that the second one is not diffusion-driven but is rather a contribution of the sum of several mechanisms with at least one depending on the lithium ion concentration explaining the relaxation process. With proper fitting of this decay, diffusion coefficients of lithium hosts could be extracted. Additionally, the effect of phase transition in LiMn<inf>2</inf>O<inf>4</inf> is taken into account, explaining some experimental observations. © 2015 AIP Publishing LLC.view abstract 10.1063/1.4927747 **Modeling of self-healing effects in polymeric composites**

Bluhm, J. and Specht, S. and Schröder, J.*Archive of Applied Mechanics*85 (2015)Polymers and polymer composites are used in many engineering applications, but they can loose a high rate of stiffness and strength due to internal micro cracks/damages during their lifetime cycle. These damages are very hard to detect and nearly impossible to repair. To avoid failure due to such damages, a self-healing system is considered where microencapsulated healing agents and catalysts are embedded in the polymer matrix. For the numerical simulation of such a self-healing material, a thermodynamically consistent multiphase model, based on the Theory of Porous Media, is developed in this contribution. The different phases of the model are the solid matrix material with embedded catalysts, the liquid healing agents, the solid healed material and the gas phase, which represents the volume fraction of the micro cracks in the model. For the description of the healing mechanism, a mass exchange between the liquid healing agents and the solid healed material, in consideration of the change of the aggregate state, is introduced, which depends on the local concentration of catalysts in the polymer matrix. The applicability of the developed model is shown by means of numerical test simulations of a tapered double cantilever beam. © 2014, Springer-Verlag Berlin Heidelberg.view abstract 10.1007/s00419-014-0946-7 **Numerical calculation of thermo-mechanical problems at large strains based on complex step derivative approximation of tangent stiffness matrices**

Balzani, D. and Gandhi, A. and Tanaka, M. and Schröder, J.*Computational Mechanics*55 (2015)In this paper a robust approximation scheme for the numerical calculation of tangent stiffness matrices is presented in the context of nonlinear thermo-mechanical finite element problems and its performance is analyzed. The scheme extends the approach proposed in Kim et al. (Comput Methods Appl Mech Eng 200:403–413, 2011) and Tanaka et al. (Comput Methods Appl Mech Eng 269:454–470, 2014 and bases on applying the complex-step-derivative approximation to the linearizations of the weak forms of the balance of linear momentum and the balance of energy. By incorporating consistent perturbations along the imaginary axis to the displacement as well as thermal degrees of freedom, we demonstrate that numerical tangent stiffness matrices can be obtained with accuracy up to computer precision leading to quadratically converging schemes. The main advantage of this approach is that contrary to the classical forward difference scheme no round-off errors due to floating-point arithmetics exist within the calculation of the tangent stiffness. This enables arbitrarily small perturbation values and therefore leads to robust schemes even when choosing small values. An efficient algorithmic treatment is presented which enables a straightforward implementation of the method in any standard finite-element program. By means of thermo-elastic and thermo-elastoplastic boundary value problems at finite strains the performance of the proposed approach is analyzed. © 2015, Springer-Verlag Berlin Heidelberg.view abstract 10.1007/s00466-015-1139-0 **A first-order system least squares method for hyperelasticity**

Müller, B. and Starke, G. and Schwarz, A. and Schröder, J.*SIAM Journal on Scientific Computing*36 (2014)A least squares mixed finite element method for deformations of hyperelastic materials using stress and displacement as process variables is presented and studied. The method is investigated in detail for the specific case of a neo-Hookean material law and is based on the representation of the strain-stress relation. A formulation is derived for compressible materials and shown to remain valid in the incompressible limit, automatically enforcing the incompressibility constraint. The mapping properties of the first-order system operator are studied in appropriate Sobolev spaces. Under the assumption of a locally unique solution with sufficient regularity, it is proved that the firstorder least squares residual constitutes an upper bound for the error measured in a suitable norm, provided that the finite element approximation is sufficiently close. The method is tested numerically in a plane strain situation using next-to-lowest-order Raviart-Thomas elements for the stress tensor and conforming quadratic elements for the displacement components. The improvement of the stress representation is demonstrated by the evaluation of the boundary traction approximation. © 2014 Society for Industrial and Applied Mathematics.view abstract 10.1137/130937573 **A novel scheme for the approximation of residual stresses in arterial walls**

Schröder, J. and Brinkhues, S.*Archive of Applied Mechanics*84 (2014)In this contribution, a novel approach for the modeling of residual stresses in human arteries is proposed. The starting point in a variety of contributions is the opening angle of the section of an artery as a consequence of a longitudinal cut. In contrast to this, we focus directly on the current stress state within the arterial wall. To be more precise, we analyze the gradients of suitable invariant stress measures in thickness direction of the arterial wall. As an underlying optimization criterion, we assume that these gradients have to be smoothed between their inner and outer margins of the individual layers in an appropriate way. In order to do this, we define suitable radial sections for the media and adventitia, where this condition has to be enforced independently. The efficiency of the proposed model is demonstrated by means of a patient-specific cross section of a diseased artery in a two-dimensional simulation. © 2014 Springer-Verlag Berlin Heidelberg.view abstract 10.1007/s00419-014-0838-x **An invariant formulation for phase field models in ferroelectrics**

Schrade, D. and Müller, R. and Gross, D. and Keip, M.-A. and Thai, H. and Schröder, J.*International Journal of Solids and Structures*51 (2014)This paper introduces an electro-mechanically coupled phase field model for ferroelectric domain evolution based on an invariant formulation for transversely isotropic piezoelectric material behavior. The thermodynamic framework rests upon Gurtin's notion of a micro-force system in conjunction with an associated micro-force balance. This leads to a formulation of the second law, from which a generalized Ginzburg-Landau evolution equation is derived. The invariant formulation of the thermodynamic potential provides a consistent way to obtain the order parameter dependent elastic stiffness, piezoelectric, and dielectric tensor. The model is reduced to 2d and implemented into a finite element framework. The material constants used in the simulations are adapted to meet the thermodynamic condition of a vanishing micro-force. It is found that the thermodynamic potential taken from the literature has to be extended in order to avoid a loss of positive definiteness of the stiffness and the dielectric tensor. The small-signal response is investigated in the presence and in the absence of the additional regularizing terms in the potential. The simulations show the pathological behavior of the model in case these terms are not taken into account. The paper closes with microstructure simulations concerning a ferroelectric nanodot subjected to an electric field, a cracked single crystal, and a ferroelectric bi-crystal. © 2014 Elsevier Ltd. All rights reserved.view abstract 10.1016/j.ijsolstr.2014.02.021 **Construction of statistically similar representative volume elements - Comparative study regarding different statistical descriptors**

Scheunemann, L. and Schröder, J. and Balzani, D. and Brands, D.*Procedia Engineering*81 (2014)Advanced high strength steels, such as dual-phase steel (DP steel), provide advantages for engineering applications compared to conventional high strength steel. The main constituents of DP steel on the microscopic level are martensitic inclusions embedded in a ferritic matrix. A way to include these heterogeneities on the microscale into the modeling of the material is the FE2- method. Herein, in every integration point of a macroscopic finite element problem a microscopic boundary value problem is attached, which consists of a representative volume element (RVE) often defined as a segment of a real microstructure. From this representation, high computational costs arise due to the complexity of the discretization which can be circumvented by the use of a Statistically Similar RVE (SSRVE), which is governed by similar statistical features as the real target microstructure but shows a lower complexity. For the construction of such SSRVEs, an optimization problem is constructed which consists of a least-square functional taking into account the differences of statistical measures evaluated for the real microstructure and the SSRVE. This functional is minimized to identify the SSRVE for which the similarity in a statistical sense is optimal. The choice of the statistical measures considered in the least-square functional however play an important role. We focus on the construction of SSRVEs based on the volume fraction, lineal-path function and spectral density and check the performance in virtual tests. Here the response of the individual SSRVEs is compared with the real target microstructure. Further higher order measures, some specific Minkowski functionals, are investigated regarding their applicability and efficiency in the optimization process. © 2014 The Authors. Published by Elsevier Ltd.view abstract 10.1016/j.proeng.2014.10.157 **Construction of two- and three-dimensional statistically similar RVEs for coupled micro-macro simulations**

Balzani, D. and Scheunemann, L. and Brands, D. and Schröder, J.*Computational Mechanics*54 (2014)In this paper a method is presented for the construction of two- and three-dimensional statistically similar representative volume elements (SSRVEs) that may be used in computational two-scale calculations. These SSRVEs are obtained by minimizing a least-square functional defined in terms of deviations of statistical measures describing the microstructure morphology and mechanical macroscopic quantities computed for a random target microstructure and for the SSRVE. It is shown that such SSRVEs serve as lower bounds in a statistical sense with respect to the difference of microstructure morphology. Moreover, an upper bound is defined by the maximum of the least-square functional. A staggered optimization procedure is proposed enabling a more efficient construction of SSRVEs. In an inner optimization problem we ensure that the statistical similarity of the microstructure morphology in the SSRVE compared with a target microstructure is as high as possible. Then, in an outer optimization problem we analyze mechanical stress–strain curves. As an example for the proposed method two- and three-dimensional SSRVEs are constructed for real microstructure data of a dual-phase steel. By comparing their mechanical response with the one of the real microstructure the performance of the method is documented. It turns out that the quality of the SSRVEs improves and converges to some limit value as the microstructure complexity of the SSRVE increases. This converging behavior gives reason to expect an optimal SSRVE at the limit for a chosen type of microstructure parameterization and set of statistical measures. © 2014, Springer-Verlag Berlin Heidelberg.view abstract 10.1007/s00466-014-1057-6 **Product properties of a two-phase magneto-electric composite: Synthesis and numerical modeling**

Labusch, M. and Etier, M. and Lupascu, D.C. and Schröder, J. and Keip, M.-A.*Computational Mechanics*54 (2014)Magneto-electric (ME) materials are of high interest for a variety of advanced applications like in data storage and sensor technology. Due to the low ME coupling in natural materials, composite structures become relevant which generate the effective ME coupling as a strain-mediated product property. In this framework, it seems to be possible to achieve effective ME coefficients that can be exploited technologically. The present contribution investigates the realization of particulate ME composites with a focus on their experimental and computational characterization. We will show that different states of pre-polarizations of the ferroelectric material have a decisive influence on the overall obtainable ME coefficient. Details on the synthesis of two-phase composite microstructures consisting of a barium titanate matrix and cobalt ferrite inclusions will be discussed. Subsequently we will employ computational homogenization in order to determine the effective properties of the experimental composite numerically. We investigate the influence of different states of pre-polarization on the resulting ME-coefficients. For the numerical incorporation of the pre-polarization we use a heuristic method. © 2014 Springer-Verlag Berlin Heidelberg.view abstract 10.1007/s00466-014-1031-3 **Robust numerical calculation of tangent moduli at finite strains based on complex-step derivative approximation and its application to localization analysis**

Tanaka, M. and Fujikawa, M. and Balzani, D. and Schröder, J.*Computer Methods in Applied Mechanics and Engineering*269 (2014)An extremely robust and efficient numerical approximation of material and spatial tangent moduli at finite strains is presented that can be easily implemented within standard FEM software. This method is based on the complex-step derivative approximation (CSDA) approach. The CSDA is proved to be of second order accurate and it does not suffer from roundoff errors in floating point arithmetics that limit the accuracy of other classical numerical approaches as e.g. finite difference approximation. Therefore, the CSDA can provide approximations extremely similar to analytical solutions when perturbation values are chosen close to machine precision. Implementation details of the robust numerical approximation of tangent moduli from stress calculations using the CSDA are given and their performance is illustrated through representative examples involving finite deformations. In addition to that, we focus on the determination of material instabilities. Therefore, an accompanying localization analysis is performed, where the acoustic tensor is directly computed from the approximation of the moduli. It is shown that classical numerical approximations are sensitive with respect to the perturbation value such that material instabilities may be artificially detected just as a result of slightly changing the perturbation. On the other hand, the CSDA approach provides high-accurate and robust approximations within a wide range of perturbation values such that the material instabilities can be detected precisely. © 2013 Elsevier B.V.view abstract 10.1016/j.cma.2013.11.005 **Two-scale computational homogenization of electro-elasticity at finite strains**

Keip, M. A. and Steinmann, P. and Schröder, J.*Computer Methods in Applied Mechanics and Engineering*278 (2014)This contribution addresses a two-scale computational homogenization framework for the simulation of electro-active solids at finite strains. A generalized form of the Hill-Mandel condition is employed for the derivation of energetically consistent transition conditions between the scales. The continuum mechanical formulation is implemented into a two-scale finite element environment, in which we attach a microscopic representative volume element at each integration point of the macroscopic domain. In order to allow for an efficient solution of the macroscopic boundary value problem an algorithmically consistent tangent of the macroscopic problem is derived. The method will be applied to the analysis of dielectric polymer-ceramic composites, where we determine the effective actuation of composites with different microstructures. Furthermore, we show the applicability of the proposed method to the computation of two-scale electro-mechanically coupled boundary value problems in consideration of large deformations. (C) 2014 Elsevier B.V. All rights reserved.view abstract 10.1016/j.cma.2014.04.020 **Two-scale modeling of DP steel incorporating distributed properties inside micro-constituents**

Schröder, J. and Gandhi, A. and Balzani, D.*Procedia Engineering*81 (2014)Advanced High Strength Steels (AHSS) are increasingly used in the industry due to their excellent strength and formability properties enabling weight savings. In this wide class of steel we restrict ourselves to the modeling of Dual Phase (DP) steels which are, at the microscale, characterized by a hard martensitic inclusion phase embedded in a soft ferritic matrix phase. During the production process the martensite transforms from austenite by rapidly cooling down the material and thereby causing a volume jump leading to initial plastic strains associated with eigenstresses of higher order. A technique to incorporate theses distributed properties in the ferrite matrix is proposed and implemented using the direct micro-macro transition approach. © 2014 The Authors. Published by Elsevier Ltd.view abstract 10.1016/j.proeng.2014.10.162 **Weighted overconstrained least-squares mixed finite elements for static and dynamic problems in quasi-incompressible elasticity**

Schwarz, A. and Steeger, K. and Schröder, J.*Computational Mechanics*54 (2014)The main goal of this contribution is the improvement of the approximation quality of least-squares mixed finite elements for static and dynamic problems in quasi-incompressible elasticity. Compared with other variational approaches as for example the Galerkin method, the main drawback of least-squares formulations is the unsatisfying approximation quality in terms of accuracy and robustness. Here, lower-order elements are especially affected, see e.g. [33]. In order to circumvent these problems, we introduce overconstrained first-order systems with suited weights. We consider different mixed least-squares formulations depending on stresses and displacements with a maximal cubical polynomial interpolation. For the continuous approximation of the stresses Raviart-Thomas elements are used, while for the displacements standard conforming elements are employed. Some numerical benchmarks are presented in order to validate the performance and efficiency of the proposed formulations. © 2014 Springer-Verlag Berlin Heidelberg.view abstract 10.1007/s00466-014-1009-1 **A finite element method for contact using a third medium**

Wriggers, P. and Schröder, J. and Schwarz, A.*Computational Mechanics*52 (2013)The numerical simulation of contact problems is nowadays a standard procedure in many engineering applications. The contact constraints are usually formulated using either the Lagrange multiplier, the penalty approach or variants of both methodologies. The aim of this paper is to introduce a new scheme that is based on a space filling mesh in which the contacting bodies can move and interact. To be able to account for the contact constraints, the property of the medium, that imbeds the bodies coming into contact, has to change with respect to the movements of the bodies. Within this approach the medium will be formulated as an isotropic/anisotropic material with changing characteristics and directions. In this paper we will derive a new finite element formulation that is based on the above mentioned ideas. The formulation is presented for large deformation analysis and frictionless contact. © 2013 Springer-Verlag Berlin Heidelberg.view abstract 10.1007/s00466-013-0848-5 **A Simultaneous Augmented Lagrange Approach for the Simulation of Soft Biological Tissue**

Böse, D. and Brinkhues, S. and Erbel, R. and Klawonn, A. and Rheinbach, O. and Schröder, J.*Lecture Notes in Computational Science and Engineering*91 (2013)In this paper, we consider the elastic deformation of arterial walls as occurring, e.g., in the process of a balloon angioplasty, a common treatment in the case of atherosclerosis. Soft biological tissue is an almost incompressible material. To account for this property in finite element simulations commonly used free energy functions contain terms penalizing volumetric changes. The incorporation of such penalty terms can, unfortunately, spoil the convergence of the nonlinear iteration scheme, i.e., of Newton's method, as well as of iterative solvers applied for the solution of the linearized systems of equations. We show that the augmented Lagrange method can improve the convergence of the linear and nonlinear iteration schemes while, at the same time, implementing a guaranteed bound for the volumetric change. Our finite element model of an atherosclerotic arterial segment, see Fig. 1, is constructed from intravascular ultrasound images; for details see [4]. © Springer-Verlag Berlin Heidelberg 2013.view abstract 10.1007/978-3-642-35275-1_43 **Augmented Lagrange methods for quasi-incompressible materials-Applications to soft biological tissue**

Brinkhues, S. and Klawonn, A. and Rheinbach, O. and Schröder, J.*International Journal for Numerical Methods in Biomedical Engineering*29 (2013)Arterial walls in the healthy physiological regime are characterized by quasi-incompressible, anisotropic, hyperelastic material behavior. Polyconvex material functions representing such materials typically incorporate a penalty function to account for the incompressibility. Unfortunately, the penalty will affect the conditioning of the stiffness matrices. For high penalty parameters, the performance of iterative solvers will degrade, and when direct solvers are used, the quality of the solutions will deteriorate. In this paper, an augmented Lagrange approach is used to cope with the quasi-incompressibility condition. Here, the penalty parameter can be chosen much smaller, and as a consequence, the arising linear systems of equations have better properties. An improved convergence is then observed for the finite element tearing and interconnecting-dual primal domain decomposition method, which is used as an iterative solver. Numerical results for an arterial geometry obtained from ultrasound imaging are presented. © 2012 John Wiley & Sons, Ltd.view abstract 10.1002/cnm.2504 **Finite strain viscoelasticity: How to consistently couple discretizations in time and space on quadrature-point level for full order p ≥ 2 and a considerable speed-up**

Eidel, B. and Stumpf, F. and Schröder, J.*Computational Mechanics*52 (2013)In computational viscoelasticity, the spatial finite element discretization for the global solution of the weak form of the balance of momentum is coupled to the temporal discretization for solving local initial value problems (IVP) of viscoelastic flow. In this contribution we show that this global-local or space-time coupling is consistent, if the total strain tensor as the coupling quantity exhibits the same approximation order p in time as the Runge-Kutta (RK) integration algorithm. To this end we construct interpolation polynomials, based on data at tn+1, tn, ⋯, tn+2-p, p ≥ 2, which provide consistent strain data at RK stages. This is a generalization of the idea proposed in (Eidel and Kuhn, Int J Numer Methods Eng 87(11):1046-1073, 2011). For lower-order strain interpolation, time integration exhibits order reduction and therefore low efficiency. For consistent strain interpolation, the adapted RK methods up to p=4 obtain full convergence order and thus approve the novel concept of consistency. High speed-up factors substantiate the improved efficiency compared with Backward-Euler. © 2013 Springer-Verlag Berlin Heidelberg.view abstract 10.1007/s00466-012-0823-6 **Parallel simulation of patient-specific atherosclerotic arteries for the enhancement of intravascular ultrasound diagnostics**

Balzani, D. and Böse, D. and Brands, D. and Erbel, R. and Klawonn, A. and Rheinbach, O. and Schröder, J.*Engineering Computations (Swansea, Wales)*29 (2012)Purpose - The purpose of this paper is to present a computational framework for the simulation of patient-specific atherosclerotic arterial walls. Such simulations provide information regarding the mechanical stress distribution inside the arterial wall and may therefore enable improved medical indications for or against medical treatment. In detail, the paper aims to provide a framework which takes into account patient-specific geometric models obtained by in vivo measurements, as well as a fast solution strategy, giving realistic numerical results obtained in reasonable time. Design/methodology/approach - A method is proposed for the construction of three-dimensional geometrical models of atherosclerotic arteries based on intravascular ultrasound virtual histology data combined with angiographic X-ray images, which are obtained on a routine basis in the diagnostics and medical treatment of cardiovascular diseases. These models serve as a basis for finite element simulations where a large number of unknowns need to be calculated in reasonable time. Therefore, the finite element tearing and interconnecting-dual primal (FETI-DP) domain decomposition method is applied, to achieve an efficient parallel solution strategy. Findings - It is shown that three-dimensional models of patient-specific atherosclerotic arteries can be constructed from intravascular ultrasound virtual histology data. Furthermore, the application of the FETI-DP domain decomposition method leads to a fast numerical framework. In a numerical example, the importance of three-dimensional models and thereby fast solution algorithms is illustrated by showing that two-dimensional approximations differ significantly from the 3D solution. Originality/value - The decision for or against intravascular medical treatment of atherosclerotic arteries strongly depends on the mechanical situation of the arterial wall. The framework presented in this paper provides computer simulations of stress distributions, which therefore enable improved indications for medical methods of treatment. © Emerald Group Publishing Limited.view abstract 10.1108/02644401211271645 **Two-scale homogenization of electromechanically coupled boundary value problems: Consistent linearization and applications**

Schröder, J. and Keip, M.-A.*Computational Mechanics*50 (2012)The contribution addresses a direct micro-macro transition procedure for electromechanically coupled boundary value problems. The two-scale homogenization approach is implemented into a so-called FE 2-method which allows for the computation of macroscopic boundary value problems in consideration of microscopic representative volume elements. The resulting formulation is applicable to the computation of linear as well as nonlinear problems. In the present paper, linear piezoelectric as well as nonlinear electrostrictive material behavior are investigated, where the constitutive equations on the microscale are derived from suitable thermodynamic potentials. The proposed direct homogenization procedure can also be applied for the computation of effective elastic, piezoelectric, dielectric, and electrostrictive material properties. © 2012 Springer-Verlag.view abstract 10.1007/s00466-012-0715-9 **A new mixed finite element based on different approximations of the minors of deformation tensors**

Schröder, J. and Wriggers, P. and Balzani, D.*Computer Methods in Applied Mechanics and Engineering*200 (2011)Finite element formulations for arbitrary hyperelastic strain energy functions that are characterized by a locking-free behavior for incompressible materials, a good bending performance and accurate solutions for coarse meshes need still attention. Therefore, the main goal of this contribution is to provide an improved mixed finite element for quasi-incompressible finite elasticity. Based on the knowledge that the minors of the deformation gradient play a major role for the transformation of infinitesimal line-, area- and volume elements, as well as in the formulation of polyconvex strain energy functions a mixed finite element with different interpolation orders of the terms related to the minors is developed. Due to the formulation it is possible to condensate the mixed element formulation at element level to a pure displacement form. Examples show the performance and robustness of the element. © 2011 Elsevier B.V.view abstract 10.1016/j.cma.2011.08.009 **Analysis of a modified first-order system least squares method for linear elasticity with improved momentum balance**

Starke, G. and Schwarz, A. and Schröder, J.*SIAM Journal on Numerical Analysis*49 (2011)A modified first-order system least squares formulation for linear elasticity, obtained by adding the antisymmetric displacement gradient in the test space, is analyzed. This approach leads to surprisingly small momentum balance error compared to standard least squares approaches. It is shown that the modified least squares formulation is well posed and its performance is illustrated by adaptive finite element computation based on using a closely related least squares functional as a posteriori error estimator. The results of our numerical computations show that, for the modified least squares approach, the momentum balance error converges at a much faster rate than the overall error. We prove that this is due to a strong connection of the stress approximation to that obtained from a mixed formulation based on the Hellinger-Reissner principle (with exact local momentum balance). The practical significance is that our proposed approach is almost momentum-conservative at a smaller number of degrees of freedom than mixed approximations with exact local momentum balance. © 2011 Society for Industrial and Applied Mathematics.view abstract 10.1137/100799149 **Approximation of random microstructures by periodic statistically similar representative volume elements based on lineal-path functions**

Schröder, J. and Balzani, D. and Brands, D.*Archive of Applied Mechanics*81 (2011)For the direct incorporation of micromechanical information into macroscopic boundary value problems, the FE2-method provides a suitable numerical framework. Here, an additional microscopic boundary value problem, based on evaluations of representative volume elements (RVEs), is attached to each Gauss point of the discretized macrostructure. However, for real random heterogeneous microstructures the choice of a "large" RVE with a huge number of inclusions is much too time-consuming for the simulation of complex macroscopic boundary value problems, especially when history-dependent constitutive laws are adapted for the description of individual phases of the mircostructure. Therefore, we propose a method for the construction of statistically similar RVEs (SSRVEs), which have much less complexity but reflect the essential morphological attributes of the microscale. If this procedure is prosperous, we arrive at the conclusion that the SSRVEs can be discretized with significantly less degrees of freedom than the original microstructure. The basic idea for the design of such SSRVEs is to minimize a least-square functional taking into account suitable statistical measures, which characterize the inclusion morphology. It turns out that the combination of the volume fraction and the spectral density seems not to be sufficient. Therefore, a hybrid reconstruction method, which takes into account the lineal-path function additionally, is proposed that yields promising realizations of the SSRVEs. In order to demonstrate the performance of the proposed procedure, we analyze several representative numerical examples. © 2010 Springer-Verlag.view abstract 10.1007/s00419-010-0462-3 **Effective electromechanical properties of heterogeneous piezoelectrics**

Keip, M.-A. and Schröder, J.*Lecture Notes in Applied and Computational Mechanics*59 LNACM (2011)The present contribution discusses a two-scale homogenization procedure for the continuum mechanical modeling of heterogeneous electro-mechanically coupled materials. The direct meso-macro formulation is implemented into an FE 2-homogenization environment, which allows for the computation of a macroscopic boundary value problem in consideration of attached heterogeneous representative volume elements at each macroscopic point. The resulting homogenization approach is capable of computing the effective elastic, piezoelectric, and dielectric properties of electro-mechanically coupled materials in consideration of arbitrary mesostructures. © 2011 Springer-Verlag Berlin Heidelberg.view abstract 10.1007/978-3-642-22738-7_6 **Multiscale modeling of electro-mechanically coupled materials: Homogenization procedure and computation of overall moduli**

Schröder, J. and Keip, M.-A.*IUTAM Bookseries*24 (2011)In this contribution, a meso-macro transition procedure for electromechanically coupled materials is presented. The utilized mesoscopic material model will be introduced and implemented into an FE2- homogenization approach. The resulting two-scale formulation is capable to compute macroscopic boundary value problems under consideration of attached heterogeneous representative volume elements at each macroscopic point. The presented direct homogenization procedure also allows for the efficient computation of effective electro-mechanical material parameters. © Springer Science+Business Media B.V. 2011.view abstract 10.1007/978-90-481-9887-0-25 **A modified least-squares mixed finite element with improved momentum balance**

Schwarz, A. and Schröder, J. and Starke, G.*International Journal for Numerical Methods in Engineering*81 (2010)The main goal of this contribution is to provide an improved mixed finite element for quasi-incompressible linear elasticity. Based on a classical least-squares formulation, a modified weak form with displacements and stresses as process variables is derived. This weak form is the basis for a finite element with an advanced fulfillment of the momentum balance and therefore with a better performance. For the continuous approximation of stresses and displacements on the triangular and tetrahedral elements, lowest-order Raviart-Thomas and linear standard Lagrange interpolations can be used. It is shown that coercivity and continuity of the resulting asymmetric bilinear form could be established with respect to appropriate norms. Further on, details about the implementation of the least-squares mixed finite elements are given and some numerical examples are presented in order to demonstrate the performance of the proposed formulation. © 2009 John Wiley & Sons, Ltd.view abstract 10.1002/nme.2692 **FE 2-simulation of microheterogeneous steels based on statistically similar RVEs**

Balzani, D. and Schröder, J. and Brands, D.*IUTAM Bookseries*21 (2010)A main problem of direct homogenization methods is the high computational cost, when we have to deal with large random microstructures. This leads to a large number of history variables which needs a large amount of memory, and moreover a high computation time. We focus on random microstructures consisting of a continuous matrix phase with a high number of embedded inclusions. In this contribution a method is presented for the construction of statistically similar representative volume elements (SSRVEs) which are characterized by a much less complexity than usual random RVEs in order to obtain an efficient simulation tool. The basic idea of the underlying procedure is to find a simplified SSRVE, whose selected statistical measures under consideration are as close as possible to the ones of the original microstructure. © 2010 Springer Science+Business Media B.V.view abstract 10.1007/978-90-481-9195-6-2 **On the mechanical modeling of anisotropic biological soft tissue and iterative parallel solution strategies**

Balzani, D. and Brands, D. and Klawonn, A. and Rheinbach, O. and Schröder, J.*Archive of Applied Mechanics*80 (2010)Biological soft tissues appearing in arterial walls are characterized by a nearly incompressible, anisotropic, hyperelastic material behavior in the physiological range of deformations. For the representation of such materials we apply a polyconvex strain energy function in order to ensure the existence of minimizers and in order to satisfy the Legendre-Hadamard condition automatically. The 3D discretization results in a large system of equations; therefore, a parallel algorithm is applied to solve the equilibrium problem. Domain decomposition methods like the Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) method are designed to solve large linear systems of equations, that arise from the discretization of partial differential equations, on parallel computers. Their numerical and parallel scalability, as well as their robustness, also in the incompressible limit, has been shown theoretically and in numerical simulations. We are using a dual-primal FETI method to solve nonlinear, anisotropic elasticity problems for 3D models of arterial walls and present some preliminary numerical results. © 2009 Springer-Verlag.view abstract 10.1007/s00419-009-0379-x **Polyconvex Energies for Trigonal, Tetragonal and Cubic Symmetry Groups**

Schröder, J. and Neff, P. and Ebbing, V.*Iutam Symposium on Variational Concepts with Applications to the Mechanics of Materials*21 (2010)In large strain elasticity the existence of minimizers is guaranteed if the variational functional to be minimized is sequentially weakly lower semicontinuous (s.w.l.s.) and coercive. Therefore, for the description of hyperelastic materials poly-convex functions which are always s.w.l.s. should be preferably used. A variety of isotropic and anisotropic polyconvex energies, in particular for the triclinic, monoclinic, rhombic and transversely isotropic symmetry groups, already exist. In this contribution we propose a new approach for the description of trigonal, tetragonal and cubic hyperelastic materials in the framework of polyconvexity. The anisotropy of the material is described by invariants in terms of the right Cauchy Green tensor and a specific fourth-order structural tensor. In order to show the adaptability of the introduced polyconvex energies for the approximation of real anisotropic material behavior we focus on the fitting of a trigonal fourth-order tangent moduli near the reference state to experimental data.view abstract 10.1007/978-90-481-9195-6_17

#### SFB 270: HoMMage - Hysteresis design of magnetic materials for efficient energy conversion

#### SPP 1748: Reliable Simulation Techniques in Solid Mechanics. Development of Non-standard Discretization Methods, Mechanical and Mathematical Analysis (concluded)

#### SPP 1713: Chemomechanics: Strong coupling of thermo-chemical and thermo-mechanical states in applied materials (concluded)

#### SPP 1648: SPPEXA - Software for Exascale Computing - Project EXASTEEL: Bridging Scales for Multiphase Steels (concluded)

#### FOR 1509: Ferroic Functional Materials (concluded)

#### Phase-field and variational update formulations for modeling of material instabilities in metal elasto-plasticity (concluded)

#### FOR 797: Analysis and computation of microstructure in finite plasticity (concluded)

#### Effiziente Simulationstechniken für robuste Least-Squares FEM in der Fluiddynamik (concluded)

#### Biomechanics

#### boundary value problems

#### continuum mechanics

#### Coupled Problems

#### elasticity

#### electro- and magneto-mechanics

#### finite element method

#### finite element method

#### microstructure

#### modelling and simulation

#### multiscale modeling

#### numerical methods