#### Prof. Dr. Gerhard Starke

Numerical Mathematics

University of Duisburg-Essen

##### Contact

- gerhard[dot]starke[at]uni-due[dot]de
- +49 201 183 2339
- personal website

##### Author IDs

- Scopus: 7006220147

##### Hub

**Multigrid for the dual formulation of the frictionless Signorini problem**

Rovi, G. and Kober, B. and Starke, G. and Krause, R.*International Journal for Numerical Methods in Engineering*124 (2023)view abstract 10.1002/nme.7214 **Least-Squares Finite Element Formulation for Finite Strain Elasto-Plasticity**

Igelbüscher, M. and Schröder, J. and Schwarz, A. and Starke, G.*Lecture Notes in Applied and Computational Mechanics*98 (2022)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.view abstract 10.1007/978-3-030-92672-4_6 **Stress Equilibration for Hyperelastic Models**

Bertrand, F. and Moldenhauer, M. and Starke, G.*Lecture Notes in Applied and Computational Mechanics*98 (2022)Stress equilibration is investigated for hyperelastic deformation models in this contribution. From the displacement-pressure approximation computed with a stable finite element pair, an H(div ) -conforming approximation to the first Piola-Kirchhoff stress tensor is computed. This is done in the usual way in a vertex-patch-wise manner involving local problems of small dimension. The corresponding reconstructed Cauchy stress is not symmetric but its skew-symmetric part is controlled by the computed correction. This difference between the reconstructed stress and the stress approximation obtained directly from the Galerkin approximation also serves as an upper bound for the discretization error. These properties are illustrated by computational experiments for an incompressible rigid block loaded on one half of its top boundary. © 2022, The Author(s), under exclusive license to Springer Nature Switzerland AG.view abstract 10.1007/978-3-030-92672-4_4 **Stress-Based Methods for Quasi-Variational Inequalities Associated with Frictional Contact**

Kober, B. and Starke, G. and Krause, R. and Rovi, G.*International Series of Numerical Mathematics*172 (2022)The stress-based formulation of elastic contact with Coulomb friction in the form of a quasi-variational inequality is investigated. Weakly symmetric stress approximations are constructed using a finite element combination on the basis of Raviart–Thomas spaces of next-to-lowest order. An error estimator is derived based on a displacement reconstruction and proved to be reliable under certain assumptions on the solution formulated in terms of a norm equivalence in the trace space H1∕2(Γ). Numerical results illustrate the effectiveness of the adaptive refinement strategy for a Hertzian frictional contact problem in the compressible as well as in the incompressible case. © 2022, Springer Nature Switzerland AG.view abstract 10.1007/978-3-030-79393-7_18 **Weakly symmetric stress equilibration and a posteriori error estimation for linear elasticity**

Bertrand, F. and Kober, B. and Moldenhauer, M. and Starke, G.*Numerical Methods for Partial Differential Equations*(2021)This paper proposes and analyzes a posteriori error estimator based on stress equilibration for linear elasticity with emphasis on the behavior for (nearly) incompressible materials. It is based on an H(div)-conforming, weakly symmetric stress reconstruction from the displacement-pressure approximation computed with a stable finite element pair. Our focus is on the Taylor-Hood combination of continuous finite element spaces of polynomial degrees k + 1 and k for the displacement and the pressure, respectively. This weak symmetry allows us to prove that the resulting error estimator constitutes a guaranteed upper bound for the error with a constant that depends only on local constants associated with the patches and thus on the shape regularity of the triangulation. It does not involve global constants like those from Korn's in equality which may become very large depending on the location and type of the boundary conditions. Local efficiency, also uniformly in the incompressible limit, is deduced from the upper bound by the residual error estimator. Numerical results for the popular Cook's membrane test problem confirm the theoretical predictions. © 2021 The Authors. Numerical Methods for Partial Differential Equations published by Wiley Periodicals LLC.view abstract 10.1002/num.22741 **A posteriori error estimates by weakly symmetric stress reconstruction for the Biot problem**

Bertrand, F. and Starke, G.*Computers and Mathematics with Applications*(2020)A posteriori error estimates are constructed for the three-field variational formulation of the Biot problem involving the displacements, the total pressure and the fluid pressure. The discretization under focus is the H1(Ω)-conforming Taylor–Hood finite element combination, consisting of polynomial degrees k+1 for the displacements and the fluid pressure and k for the total pressure. An a posteriori error estimator is derived on the basis of H(div)-conforming reconstructions of the stress and flux approximations. The symmetry of the reconstructed stress is allowed to be satisfied only weakly. The reconstructions can be performed locally on a set of vertex patches and lead to a guaranteed upper bound for the error with a constant that depends only on local constants associated with the patches and thus on the shape regularity of the triangulation. Particular emphasis is given to nearly incompressible materials and the error estimates hold uniformly in the incompressible limit. Numerical results on the L-shaped domain confirm the theory and the suitable use of the error estimator in adaptive strategies. © 2020view abstract 10.1016/j.camwa.2020.10.011 **Equilibrated Stress Reconstruction and a Posteriori Error Estimation for Linear Elasticity**

Bertrand, F. and Kober, B. and Moldenhauer, M. and Starke, G.*CISM International Centre for Mechanical Sciences, Courses and Lectures*597 (2020)Based on the displacement–pressure approximation computed with a stable finite element pair, a stress equilibration procedure for linear elasticity is proposed. Our focus is on the Taylor–Hood finite element space, with emphasis on the behavior for (nearly) incompressible materials. From a combination of displacement in the standard continuous finite element spaces of polynomial degrees k+1 and pressure in the standard continuous finite element spaces of polynomial degrees k, we construct an H(div)-conforming, weakly symmetric stress reconstruction. Explicit formulas are first given for a flux reconstruction and then for the stress reconstruction. © 2020, CISM International Centre for Mechanical Sciences.view abstract 10.1007/978-3-030-33520-5_3 **Weakly symmetric stress equilibration for hyperelastic material models**

Bertrand, F. and Moldenhauer, M. and Starke, G.*GAMM Mitteilungen*43 (2020)A stress equilibration procedure for hyperelastic material models is proposed and analyzed in this paper. Based on the displacement-pressure approximation computed with a stable finite element pair, it constructs, in a vertex-patch-wise manner, an H(div)-conforming approximation to the first Piola-Kirchhoff stress. This is done in such a way that its associated Cauchy stress is weakly symmetric in the sense that its antisymmetric part is zero tested against continuous piecewise linear functions. Our main result is the identification of the subspace of test functions perpendicular to the range of the local equilibration system on each patch which turn out to be rigid body modes associated with the current configuration. Momentum balance properties are investigated analytically and numerically and the resulting stress reconstruction is shown to provide improved results for surface traction forces by computational experiments. © 2019 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheimview abstract 10.1002/gamm.202000007 **A Posteriori Error Estimation for Planar Linear Elasticity by Stress Reconstruction**

Bertrand, F. and Moldenhauer, M. and Starke, G.*Computational Methods in Applied Mathematics*19 (2019)The nonconforming triangular piecewise quadratic finite element space by Fortin and Soulie can be used for the displacement approximation and its combination with discontinuous piecewise linear pressure elements is known to constitute a stable combination for incompressible linear elasticity computations. In this contribution, we extend the stress reconstruction procedure and resulting guaranteed a posteriori error estimator developed by Ainsworth, Allendes, Barrenechea and Rankin [2] and by Kim [18] to linear elasticity. In order to get a guaranteed reliability bound with respect to the energy norm involving only known constants, two modifications are carried out: (i) the stress reconstruction in next-to-lowest order Raviart-Thomas spaces is modified in such a way that its anti-symmetric part vanishes in average on each element; (ii) the auxiliary conforming approximation is constructed under the constraint that its divergence coincides with the one for the nonconforming approximation. An important aspect of our construction is that all results hold uniformly in the incompressible limit. Global efficiency is also shown and the effectiveness is illustrated by adaptive computations involving different Lamé parameters including the incompressible limit case. © 2019 Walter de Gruyter GmbH, Berlin/Boston 2019.view abstract 10.1515/cmam-2018-0004 **Strong vs. weak symmetry in stress-based mixed finite element methods for linear elasticity**

Kober, B. and Starke, G.*Lecture Notes in Computational Science and Engineering*126 (2019)Based on the Hellinger-Reissner principle, accurate stress approximations can be computed directly in suitable H(div)-like finite element spaces treating conservation of momentum and the symmetry of the stress tensor as constraints. Two stress finite element spaces of polynomial degree 2 which were proposed in this context will be compared and relations between the two will be established. The first approach uses Raviart-Thomas spaces of next-to-lowest degree and is therefore H(div)-conforming but produces only weakly symmetric stresses. The stresses obtained from the second approach satisfy symmetry exactly but are nonconforming with respect to H(div). It is shown how the latter finite element space can be derived by augmenting the componentwise next-to-lowest Raviart-Thomas space with suitable bubbles. However, the convergence order of the resulting stress approximation is reduced from two to one as will be confirmed by numerical results. Finally, the weak stress symmetry property of the first approach is discussed in more detail and a post-processing procedure for the construction of stresses which are element-wise symmetric on average is proposed. © Springer Nature Switzerland AG 2019.view abstract 10.1007/978-3-319-96415-7_28 **An adaptive least-squares fem for linear elasticity with optimal convergence rates**

Bringmann, P. and Carstensen, C. and Starke, G.*SIAM Journal on Numerical Analysis*56 (2018)Adaptive mesh-refining is of particular importance in computational mechanics and established here for the lowest-order locking-free least-squares finite element scheme which solely employs conforming P1 approximations for the displacement and lowest-order Raviart–Thomas approximations for the stress variables. This forms a competitive discretization in particular in three-dimensional linear elasticity with traction boundary conditions although the stress approximation does not satisfy the symmetry condition exactly. The paper introduces an adaptive mesh-refining algorithm based on separate marking and exact solve with some novel explicit a posteriori error estimator and proves optimal convergence rates. The point is robustness in the sense that the crucial input parameters Θ for the Dörfler marking and κ for the separate marking as well as the equivalence constants in the asymptotic convergence rates do not degenerate as the Lamé parameter λ tends to 8. © 2018 Society for Industrial and Applied Mathematics.view abstract 10.1137/16M1083797 **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 **An adaptive least-squares mixed finite element method for the Signorini problem**

Krause, R. and Müller, B. and Starke, G.*Numerical Methods for Partial Differential Equations*33 (2017)We present and analyze a least squares formulation for contact problems in linear elasticity which employs both, displacements and stresses, as independent variables. As a consequence, we obtain stability and high accuracy of our discretization also in the incompressible limit. Moreover, our formulation gives rise to a reliable and efficient a posteriori error estimator. To incorporate the contact constraints, the first-order system least squares functional is augmented by a contact boundary functional which implements the associated complementarity condition. The bilinear form related to the augmented functional is shown to be coercive and therefore constitutes an upper bound, up to a constant, for the error in displacements and stresses in H1(Ω)d × H(div, Ω)d. This implies the reliability of the functional to be used as an a posteriori error estimator in an adaptive framework. The efficiency of the use of the functional as an a posteriori error estimator is monitored by the local proportion of the boundary functional term with respect to the overall functional. Computational results using standard conforming linear finite elements for the displacement approximation combined with lowest-order Raviart-Thomas elements for the stress tensor show the effectiveness of our approach in an adaptive framework for two-dimensional and three-dimensional Hertzian contact problems. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 276–289, 2017. © 2016 Wiley Periodicals, Inc.view abstract 10.1002/num.22086 **Dev-Div- and DevSym-DevCurl-inequalities for incompatible square tensor fields with mixed boundary conditions**

Bauer, S. and Neff, P. and Pauly, D. and Starke, G.*ESAIM - Control, Optimisation and Calculus of Variations*22 (2016)Let Ω ⊂ ℝn, n ≥ 2, be a bounded Lipschitz domain and 1 < q < ∞. We prove the inequality ∥T∥Lq(Ω) ≤ CDD (∥ dev T∥Lq(Ω) + ∥ Div T∥Lq(Ω)) being valid for tensor fields T : Ω → ℝnxn with a normal boundary condition on some open and non-empty part Γν of the boundary ∂Ω. Here dev T = T - 1/n tr (T) · Id denotes the deviatoric part of the tensor T and Div is the divergence row-wise. Furthermore, we prove ∥T∥L2(Ω) ≤ CDSC (∥ dev sym T∥L2(Ω) + ∥ Curl T∥L2(Ω)) if n ≥ 3, ∥T∥L2(Ω) ≤ CDSDC (∥ dev sym T∥L2(Ω) + ∥ dev Curl T∥L2(Ω)) if n = 3, being valid for tensor fields T with a tangential boundary condition on some open and non-empty part Γτ of ∂Ω. Here, sym T = 1/2 (T + TT) denotes the symmetric part of T and Curl is the rotation row-wise. © EDP Sciences, SMAI 2015.view abstract 10.1051/cocv/2014068 **Parametric Raviart-Thomas elements for mixed methods on domains with curved surfaces**

Bertrand, F. and Starke, G.*SIAM Journal on Numerical Analysis*54 (2016)The finite element approximation on curved boundaries using parametric Raviart-Thomas spaces is studied in the context of the mixed formulation of Poisson's equation as a saddlepoint system. It is shown that optimal-order convergence is retained on domains with piecewise Ck+2 boundary for the parametric Raviart-Thomas space of degree k ≥ 0 under the usual regularity assumptions. This extends the analysis in [F. Bertrand, S. Münzenmaier, and G. Starke, SIAM J. Numer. Anal., 52 (2014), pp. 3165-3180] from the first-order system least squares formulation to mixed approaches of saddle-point type. In addition, a detailed proof of the crucial estimate in three dimensions is given which handles some complications not present in the two-dimensional case. Moreover, the appropriate treatment of inhomogeneous ux boundary conditions is discussed. The results are confirmed by computational results which also demonstrate that optimal-order convergence is not achieved, in general, if standard Raviart-Thomas elements are used instead of the parametric spaces. © 2016 Society for Industrial and Applied Mathematics.view abstract 10.1137/15M1045442 **Stress-Based Finite Element Methods in Linear and Nonlinear Solid Mechanics**

Muller, B. and Starke, G.*Advanced Finite Element Technologies*566 (2016)A comparison of stress-based finite element methods is given for the prototype problem of linear elasticity and then extended to finite-strain hyperelasticity. Of particular interest is the accuracy of traction forces in reasonable Sobolev norms with an emphasis on uniform approximation behavior in the incompressible limit. The mixed formulation of Hellinger-Reissner type leading to a saddle-point problem as well as a first-order system least-squares approach are investigated and the strong connections between these two methods are studied. In addition, we also discuss stress reconstruction techniques based on displacement approximations by nonconforming finite elements.view abstract 10.1007/978-3-319-31925-4_4 **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 **First-order system least squares on curved boundaries: Higher-order Raviart-Thomas elements**

Bertrand, F. and Münzenmaier, S. and Starke, G.*SIAM Journal on Numerical Analysis*52 (2014)With this paper, our investigation of the finite element approximation on curved boundaries using Raviart-Thomas spaces in the context of first-order system least squares methods is continued and extended to the higher-order case. It is shown that the optimal order of convergence is retained from the lowest-order case if a parametric version of Raviart-Thomas elements is used. This is illustrated numerically for an elliptic boundary value problem involving a circular boundary curve. © 2014 Societ y for Industrial and Applied Mathematics.view abstract 10.1137/130948902 **First-order system least squares on curved boundaries: Lowest-order Raviart-Thomas elements**

Bertrand, F. and Münzenmaier, S. and Starke, G.*SIAM Journal on Numerical Analysis*52 (2014)The effect of interpolated edges of curved boundaries on Raviart-Thomas finite element approximations is studied in this paper in the context of first-order system least squares methods. In particular, it is shown that an optimal order of convergence is achieved for lowestorder elements on a polygonal domain. This is illustrated numerically for an elliptic boundary value problem involving circular curves. The computational results also show that a polygonal approximation is not sufficient to achieve convergence of optimal order in the higher-order case. © 2014 Society for Industrial and Applied Mathematics.view abstract 10.1137/13091720X **New Poincaré-type inequalities**

Bauer, S. and Neff, P. and Pauly, D. and Starke, G.*Comptes Rendus Mathematique*352 (2014)We present some Poincaré-type inequalities for quadratic matrix fields with applications e.g. in gradient plasticity or fluid dynamics. In particular, applications to the pseudostress-velocity formulation of the stationary Stokes problem and to infinitesimal gradient plasticity are discussed. © 2013.view abstract 10.1016/j.crma.2013.11.017 **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 **Efficient solvers for mixed finite element discretizations of nonlinear problems in solid mechanics**

Starke, G.*Lecture Notes in Applied and Computational Mechanics*57 (2011)A common goal of our projects in the three phases of GRK 615 was, among other issues, the development of efficient solvers for different mixed finite element approaches to nonlinear problems in solid mechanics. In the first phase, the PEERS ('plane elasticity element with reduced symmetry') was studied for elastoplastic deformationmodels. The nonlinear algebraic systems were solved with a fixed point iteration leading to a linear elasticity problem in each step which was treated by suitable constraint preconditioners.The treatment of elastoplastic deformations by least squaresmixed finite elementmethodswas the subject of the project in the second phase. In particular, appropriate regularizations for the non-smoothness of the nonlinear problems were investigated. In the third phase, the least squares finite element formulation of contact problems was studied. For the Signorini problem, the quadratic minimization problems under affine constraints were treated by an active set strategy. Preconditioned conjugate gradient iterations for a null space formulation were used for the systems arising in each step. © 2011 Springer-Verlag Berlin Heidelberg.view abstract 10.1007/978-3-642-20490-6-8 **First-order system least squares for coupled stokes-darcy flow**

Münzenmaier, S. and Starke, G.*SIAM Journal on Numerical Analysis*49 (2011)The coupled problem with Stokes flow in one subdomain and a Darcy flow model in a second subdomain is studied in this paper. Both flow problems are treated as first-order systems, involving pseudostress and velocity in the Stokes case and using a flux-pressure formulation in the Darcy subdomain as process variables, respectively. The Beavers-Joseph-Saffman interface conditions are treated by an appropriate interface functional which is added to the least squares functional associated with the subdomain problems. A combination of H(div)-conforming Raviart-Thomas and standard H1-conforming elements is used for the Stokes as well as for the Darcy subsystem. The homogeneous least squares functional is shown to be equivalent to an appropriate norm allowing the use of standard finite element approximation estimates. It also establishes the fact that the local evaluation of the least squares functional itself constitutes an a posteriori error estimator to be used for adaptive refinement strategies. © 2011 Society for Industrial and Applied Mathematics.view abstract 10.1137/100805108 **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 **Adaptive least squares finite element methods in elasto-plasticity**

Starke, G.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*5910 LNCS (2010)In computational mechanics applications, one is often interested not only in accurate approximations for the displacements but also for the stress tensor. Least squares finite element methods are perfectly suited for such problems since they approximate both process variables simultaneously in suitable finite element spaces. We consider an least squares formulation for the incremental formulation of elasto-plasticity using a plastic flow rule of von Mises type. The nonlinear least squares functional constitutes an a posteriori error estimator on which an adaptive refinement strategy may be based. The variational formulation under plane strain and plane stress conditions is investigated in detail. Standard conforming elements are used for the displacement approximation while the stress is represented by Raviart-Thomas elements. The algebraic least squares problems arising from the finite element discretization are nonlinear and nonsmooth and may be solved by generalized Newton methods. © 2010 Springer-Verlag Berlin Heidelberg.view abstract 10.1007/978-3-642-12535-5_80

#### elasticity

#### elastoplasticity

#### finite element method

#### mechanical properties

#### numerical methods

#### solid mechanics