In this contribution we present a least-squares finite element formulation to model steady-state flow of incompressible, non-Newtonian fluids in three dimensions including data assimilation. The approach is based on the incompressible Navier–Stokes equations and the nonlinear viscosity is considered by means of the Carreau–Yasuda viscosity model, which can account for shear-thickening and shear-thinning behavior of generalized Newtonian fluids. We outline the procedure how to integrate given data into the numerical solution of flow problems without additional computational cost using the least-squares FEM. Assimilation of experimental data provides the opportunity to reduce model errors resulting in a solution which more closely approximates reality. Furthermore, the preprocessing of the available data using Kriging interpolation is also described briefly. The presented formulation is first validated by investigating the flow in a cube with an exact solution without data assimilation. Convergence is evaluated based on the error in velocities and pressure compared to the exact solution. Then the effect of data assimilation is shown by modeling blood flow through a carotid bifurcation model and integrating data either along lines or over entire cross-sectional areas. The improvement of the numerical solution by means of data assimilation is revealed by comparing the calculated velocity profiles with experimental and numerical reference values. © 2022

We construct two-dimensional, two-phase random heterogeneous microstructures by stochastic simulation using the planar Boolean model, which is a random collection of overlapping grains. The structures obtained are discretized using finite elements. A heterogeneous Neo-Hooke law is assumed for the phases of the microstructure, and tension tests are simulated for ensembles of microstructure samples. We determine effective material parameters, i.e., the effective Lamé moduli λ∗ and μ∗, on the macroscale by fitting a macroscopic material model to the microscopic stress data, using stress averaging over many microstructure samples. The effective parameters λ∗ and μ∗ are considered as functions of the microscale material parameters and the geometric parameters of the Boolean model including the grain shape. We also consider the size of the Representative Volume Element (RVE) given a precision and an ensemble size. We use structured and unstructured meshes and also provide a comparison with the FE2 method. © 2022, The Author(s).

This work presents a mixed least-squares finite element formulation for rate-independent elasto-plasticity at finite strains. In this context, the stress-displacement formulation is defined by the L2(B) -norm minimization of a first-order system of differential equations written in residual form. The utilization of the least-squares method (LSM) provides some well-known advantages. For the proposed rate-independent elasto-plastic material law a straight forward application of the LSM leads to discontinuities within the first variation of the formulation, based on the non-smoothness of the constitutive relation. Therefore, a modification by means of a modified first variation is necessary to guarantee a continuous weak form, which is done in terms of the considered test spaces. In addition to that an antisymmetric displacement gradient in the test space is added to the formulation due to a not a priori fulfillment of the stress symmetry condition, which results from the stress approximation with Raviart-Thomas functions. The resulting formulation is validated by a numerical test and compared to a standard displacement finite element formulation. © 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.

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).

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 Mechanik

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.

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.

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.

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.

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.

We investigate theoretically and numerically the use of the least-squares finite-element method (LSFEM) to approach data-assimilation problems for the steady-state, incompressible Navier-Stokes equations. Our LSFEM discretization is based on a stress-velocity-pressure (S-V-P) first-order formulation, using discrete counterparts of the Sobolev spaces H(div) ×H1 ×L2 for the variables respectively. In general, S-V-P formulations are promising when the stresses are of special interest, e.g., for non-Newtonian, multiphase or turbulent flows. Resolution of the system is via minimization of a least-squares functional representing the magnitude of the residual of the equations. A simple and immediate approach to extend this solver to data assimilation is to add a data-discrepancy term to the functional. Whereas most data assimilation techniques require a large number of evaluations of the forward simulation and are therefore very expensive, the approach proposed in this work uniquely has the same cost as a single forward run. However, the question arises: what is the statistical model implied by this choice? We answer this within the Bayesian framework, establishing the latent background covariance model and the likelihood. Further we demonstrate that—in the linear case—the method is equivalent to application of the Kalman filter, and derive the posterior covariance. We practically demonstrate the capabilities of our method on a backward-facing step case. Our LSFEM formulation (without data) is shown to have good approximation quality, even on relatively coarse meshes—in particular with respect to mass conservation and reattachment location. Adding limited velocity measurements from experiment, we show that the method is able to correct for discretization error on very coarse meshes, as well as correct for the influence of unknown and uncertain boundary conditions. © 2018 by Begell House, Inc.

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.

In this contribution three different mixed least-squares finite element methods (LSFEMs) are investigated with respect to accuracy and efficiency with simultaneous consideration of regular and adaptive meshing strategies. The deliberations are made for the incompressible Navier–Stokes equations. The generally known first-order div–grad system in terms of the (total) stress, velocity and pressure (SVP) formulation is the basis for two further div–grad least-squares formulations in terms of (total) stress and velocity (SV), whereby both formulations are derived from different fountainheads. The extended SV formulation is an enrichment of the known stress–velocity formulation proposed by Cai et al. (2004). The second SV formulation is based on the substitution of the pressure in the SVP formulation, thus it is based on a discontinuous pressure interpolation. Advantage of the SV formulations is a smaller system matrix size due to the reduction of the relevant degrees of freedom. Nevertheless, the approximation quality has to last the demand of the established least-squares formulations. The drawback of a poor mass conservation is well-investigated for the LSFEM and known to overcome by using high-order interpolations for all solution variables. Therefore, the main attention of this contribution is focused on accuracy, while aiming for high efficiency, which is intrinsic for the fundamental idea in here. In the light of efficient LSFEM solutions the advantage of the inherent a posteriori error estimator of the method is used for deliberations on different marking strategies in an h-type adaptive mesh refinement. © 2018 Elsevier B.V.

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.

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.

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.

The performance of least-squares finite element formulations for geometrically linear and nonlinear problems is investigated in this work. We consider different elastic material behaviors as, e.g., quasi-incompressibility and transverse isotropy. Basis for the provided element formulations is a first-order system of differential equations consisting of the residual forms of the balance of momentum, a constitutive relation, and a (redundant) residual enforcing a stronger control of the balance of moment of momentum. The sum of the squared L-2(B)-norms of the residuals leads to a functional, which is the basis for the related minimization problem. As unknown fields the displacements (approximated in W-1,W-p(B)) and the stresses (approximated in W-q(div, B)) are chosen. Here, the choice of the polynomial orders of the interpolation functions for the displacements and stresses is not restricted by the so-called LBB condition; they can be chosen independently. Numerical examples for the proposed formulations are presented and compared to standard and mixed Galerkin formulations.

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.

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.

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.

Atypical haemolytic-uremic syndrome (aHUS) is, in most cases, due to hereditary or acquired defects in complement regulation and a life-threatening disease. Despite the rapidly grown knowledge about the primary defects in aHUS, the pathogenesis that links complement dysregulation with microthrombus formation in aHUS is still unknown. Thus, we examined the glomerular microvascular expression of pro- and antithrombotic genes. Glomeruli were microdissected from 12 archival paraffin-embedded biopsies with aHUS and from seven control biopsies. Glomerular mRNA expression was quantified by single real-time PCR reactions after preamplification. In addition immunostains were performed for plasminogen activator inhibitor 1 (PAI-1) and for tissue plasminogen activator (tPA). Results were compared between cases and controls and with clinical data. Glomeruli in aHUS had increased mRNA expression of antifibrinolytic, prothrombotic PAI-1, antithrombotic thrombomodulin (THBD) and CD73 and decreased expression of profibrinolytic, antithrombotic tPA compared to controls. Impaired fibrinolysis due to increased microvascular expression of the antifibrinolytic PAI-1 in combination with the decreased expression of the profibrinolytic tPA seems to be a final common pathway in renal thrombotic microangiopathy that is also effective in aHUS. The concomitant induction of antithrombotic transcripts likely indicates counterregulatory efforts, demonstrating that the capillary bed is not a passive victim of complement attack. Future research should investigate if and how complement activation could induce the reported shift in the expression of PAI-1 and tPA.

Background. Thrombotic microangiopathy (TMA) in renal transplants (rTx-TMA) is a serious complication and is usually either recurrent TMA (RecTMA) due to humoral rejection (HR-TMA) or due to calcineurin inhibitor toxicity (CNI-TMA). Although the triggers are known, our knowledge about the thrombogenic transcriptome changes in the microvessels is rudimentary. Methods. We examined the expression of several prothrombotic and antithrombotic genes in 25 biopsies with rTx-TMA (6 RecTMA, 9 HR-TMA, and 10 CNI-TMA) and 8 controls. RNA from microdissected glomeruli of paraffin-embedded tissue was isolated and mRNA transcripts were quantified with real-time polymerase chain reaction after preamplification. Results were correlated with clinicopathologic parameters. Results. Glomerular mRNA expression of KLF2, KLF4, and tPA was lower and that of PAI-1 was higher in rTx-TMA than in the controls. Glomerular mRNA expression of KLF2 and KLF4 correlated with that of tPA and inversely with that of PAI-1 in rTx-TMA. The mRNA expression of complement regulators CD46 and CD59 were higher in rTx-TMA than in the controls. Only in HR-TMA were glomerular ADAMTS13 and CD55 down-regulated. Conclusions. The glomerular capillary bed seems to contribute to all subtypes of rTx-TMA by down-regulation of the endothelial transcription factors KLF2 and KLF4, indicating dedifferentiation with subsequent up-regulation of PAI-1 and down-regulation of tPA, resulting in inhibition of local fibrinolysis. Decreased glomerular expression of ADAMTS13 and CD55 could be an additional pathway toward microthrombosis exclusively in HR-TMA.

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.

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.