PP 1748/1: Foundation and Application of Generalized Mixed FEM Towards Nonlinear Problems in Solid Mechanics

The research of this project aims at the mathematical foundation and the engineering application of generalised mixed FEM as well as the development and the analysis of new non-standard methods that yield guaranteed results for nonlinear problems in solid mechanics. The practical applications in computational engineering is the focus of the Workgroup LUH at the Leibniz University Hannover in cooperation with the Workgroup HU at the Humboldt Universität zu Berlin with focus on mathematical foundation of the novel discretization schemes. The joint target is the effective and reliable simulation in nonlinear continuum mechanics with development of adaptive numerical discretizations based on ultraweak formulations between nonconforming, mixed and discontinuous Galerkin or Petrov-Galerkin Finite Element Methods. In the first funding period, the workgroup LUH developed different discontinuous discretization methods. An efficient extension/enhancement of the original discontinuous Galerkin Finite Element Method (dG FEM) avoids shear-locking effects and volumetric-locking for (nearly) incompressible and elasto-plastic material behaviour. The workgroup HU developed and analysed a discontinuous Petrov-Galerkin (dPG) FEM for a nonlinear model problem in collaboration with the workgroup LUH and proved optimal convergence rates of adaptive dPG and least-squares methods for linear elastic problems. Further topics of research were guaranteed error bounds for pointwise symmetric discretizations in linear elasticity and the analysis of nonconforming FEM for polyconvex materials.

The research of this project aims at the mathematical foundation and the engineering application of generalised mixed FEM as well as the development and the analysis of new non-standard methods that yield guaranteed results for nonlinear problems in solid mechanics. The practical applications in computational engineering is the focus of the Workgroup LUH at the Leibniz University Hannover in cooperation with the Workgroup HU at the Humboldt Universität zu Berlin with focus on mathematical foundation of the novel discretization schemes. The joint target is the effective and reliable simulation in nonlinear continuum mechanics with development of adaptive numerical discretizations based on ultraweak formulations between nonconforming, mixed and discontinuous Galerkin or Petrov-Galerkin Finite Element Methods. In the first funding period, the workgroup LUH developed different discontinuous discretization methods. An efficient extension/enhancement of the original discontinuous Galerkin Finite Element Method (dG FEM) avoids shear-locking effects and volumetric-locking for (nearly) incompressible and elasto-plastic material behaviour. The workgroup HU developed and analysed a discontinuous Petrov-Galerkin (dPG) FEM for a nonlinear model problem in collaboration with the workgroup LUH and proved optimal convergence rates of adaptive dPG and least-squares methods for linear elastic problems. Further topics of research were guaranteed error bounds for pointwise symmetric discretizations in linear elasticity and the analysis of nonconforming FEM for polyconvex materials.

Duration of Project

Start date: 09/2014

End date: 11/2019

Research Areas

Publications

P. Bringmann and C. Carstensen. An adaptive least-squares FEM for the Stokes equations with optimal convergence rates. Numer. Math., 135(2):459–492, 2017.

P. Bringmann and C. Carstensen. h-adaptive least-squares finite element methods for the 2D Stokes equations of any order with optimal convergence rates. Comput. Math. Appl., 74(8):1923–1939, 2017.

P. Bringmann, C. Carstensen, D. Gallistl, F. Hellwig, D. Peterseim, S. Puttkammer, H. Rabus, and J. Storn. Towards adaptive discontinuous Petrov-Galerkin methods. PAMM. Proc. Appl. Math. Mech., 16(1):741–742, 2016.

P. Bringmann, C. Carstensen, and G. Starke. An adaptive least-squares FEM for linear elasticity with optimal convergence rates. SIAM J. Numer. Anal., 56(1):428–447, 2018.

C. Carstensen, P. Bringmann, F. Hellwig, and P. Wriggers. Nonlinear discontinuous petrov-galerkin methods. Numer. Math., 139(3):529–561, 2018.

C. Carstensen, L. Demkowicz, and J. Gopalakrishnan. Breaking spaces and forms for the DPG method and applications including Maxwell equations. Comput. Math. Appl., 72(3):494–522, 2016.

C. Carstensen and F. Hellwig. Low-order discontinuous Petrov-Galerkin finite element methods for linear elasticity. SIAM J. Numer. Anal., 54(6):3388–3410, 2016.

C. Carstensen and F. Hellwig. Optimal convergence rates for adaptive lowest-orderdiscontinuous Petrov-Galerkin schemes. SIAM J. Numer. Anal., 56(2):1091–1111, 2018.

C. Carstensen, E. J. Park, and P. Bringmann. Convergence of natural adaptive least squares FEMs. Numer. Math., 136(4):1097–1115, 2017.

C. Carstensen, D. Peterseim, and A. Schröder. The norm of a discretized gradient in H(div)* for a posteriori finite element error analysis. Numer. Math., 132:519–539, 2016.

C. Carstensen and H. Rabus. Axioms of adaptivity with separate marking for data resolution. SIAM J. Numer. Anal., 55(6):2644–2665, 2017.

C. Carstensen and S. Puttkammer. A low-order discontinuous Petrov-Galerkin method for the stokes equations. Numer. Math., 140(1):1–34, 2018.

C. Carstensen and J. Storn. Asymptotic exactness of the least-squares finite element residual. SIAM J. Numer. Anal., 56(4):2008–2028, 2018.

T. Steiner and P. Wriggers. A primal discontinuous Petrov-Galerkin finite element method for linear elasticity. PAMM. Proc. Appl. Math. Mech., 17(1):83–86, 2017.

T. Steiner, P. Wriggers, and S. Löhnert. A discontinuous galerkin finite element method for linear elasticity using a mixed integration scheme to circumvent shear-locking. PAMM. Proc. Appl. Math. Mech., 16:769–770, 2016.