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An error-estimate-free and remapping-free variational mesh refinement and coarsening method for dissipative solids at finite strains. (English) Zbl 1155.74412

Summary: A variational \(h\)-adaptive finite element formulation is proposed. The distinguishing feature of this method is that mesh refinement and coarsening are governed by the same minimization principle characterizing the underlying physical problem. Hence, no error estimates are invoked at any stage of the adaption procedure. As a consequence, linearity of the problem and a corresponding Hilbert-space functional framework are not required and the proposed formulation can be applied to highly non-linear phenomena. The basic strategy is to refine (respectively, unrefine) the spatial discretization locally if such refinement (respectively, unrefinement) results in a sufficiently large reduction (respectively, sufficiently small increase) in the energy. This strategy leads to an adaption algorithm having \(O(N)\) complexity. Local refinement is effected by edge-bisection and local unrefinement by the deletion of terminal vertices. Dissipation is accounted for within a time-discretized variational framework resulting in an incremental potential energy. In addition, the entire hierarchy of successive refinements is stored and the internal state of parent elements is updated so that no mesh-transfer operator is required upon unrefinement. The versatility and robustness of the resulting variational adaptive finite element formulation is illustrated by means of selected numerical examples.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74C15 Large-strain, rate-independent theories of plasticity (including nonlinear plasticity)

Software:

ALBERT
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Full Text: DOI

References:

[1] Mosler, Variational h-adaption in finite deformation elasticity and plasticity, International Journal for Numerical Methods in Engineering 72 (5) pp 505– (2007) · Zbl 1194.74451
[2] Ortiz, Nonconvex energy minimisation and dislocation in ductile single crystals, Journal of the Mechanics and Physics of Solids 47 pp 397– (1999) · Zbl 0964.74012
[3] Ortiz, The variational formulation of viscoplastic constitutive updates, Computer Methods in Applied Mechanics and Engineering 171 pp 419– (1999) · Zbl 0938.74016
[4] Radovitzky, Error estimation and adaptive meshing in strongly non-linear dynamic problems, Computer Methods in Applied Mechanics and Engineering 172 pp 203– (1999) · Zbl 0957.74058
[5] Aubry, The mechanics of deformation-induced subgrain-dislocation structures in metallic crystals at large strains, Proceedings of the Royal Society of London A 451 pp 3131– (2003) · Zbl 1041.74506
[6] Carstensen, Non-convex potentials and microstructures in finite-strain plasticity, Proceedings of the Royal Society of London A 458 pp 299– (2002) · Zbl 1008.74016
[7] Miehe, Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation, International Journal for Numerical Methods in Engineering 55 pp 1285– (2002) · Zbl 1027.74056
[8] Yang, A variational formulation of the coupled thermo-mechanical boundary-value problem for general dissipative solids, Journal of the Mechanics and Physics of Solids 54 pp 401– (2006) · Zbl 1120.74367
[9] McNeice, Optimization of finite-element grids based on minumum potential-energy, Journal of Engineering for Industry-Transactions of the ASME 95 (1) pp 186– (1973) · doi:10.1115/1.3438097
[10] Felippa, Numerical experiments in finite element grid optimization by direct energy search, Applied Mathematical Modelling 1 pp 93– (1976) · Zbl 0348.65094
[11] Eshelby, The force on an elastic singularity, Philosophical Transactions of the Royal Society of London A 244 pp 87– (1951) · Zbl 0043.44102
[12] Eshelby, The elastic energy-momentum tensor, Journal of Elasticity 5 pp 321– (1975) · Zbl 0323.73011
[13] Braun, Configurational forces induced by finite element discretization, Proceedings of the Estonian Academy of Sciences, Physics, Mathematics 46 pp 24– (1997) · Zbl 0920.73359
[14] Mueller, On material forces and finite element discretizations, Computational Mechanics 29 pp 52– (2002) · Zbl 1053.74048
[15] Thoutireddy P. Variational arbitrary Lagrangian-Eulerian method. Ph.D. Thesis, California Institute of Technology, Pasadena, U.S.A., 2003.
[16] Thoutireddy, A variational r-adaption and shape-optimization method for finite-deformation elasticity, International Journal for Numerical Methods in Engineering 61 pp 1– (2004) · Zbl 1079.74597
[17] Kuhl, An ALE formulation based on spatial and material settings of continuum mechanics. Part 1: generic hyperelastic formulation, Computer Methods in Applied Mechanics and Engineering 193 (39-41) pp 4207– (2004) · Zbl 1068.74078
[18] Askes, An ALE formulation based on spatial and material settings of continuum mechanics. Part 2: classification and applications, Computer Methods in Applied Mechanics and Engineering 193 (39-41) pp 4223– (2004)
[19] Mosler, On the numerical implementation of variational arbitrary Lagrangian-Eulerian (VALE) formulations, International Journal for Numerical Methods in Engineering 67 (9) pp 1272– (2006) · Zbl 1113.74073
[20] Mosler, On the Numerical Modeling of Localized Material Failure at Finite Strains by Means of Variational Mesh Adaption and Cohesive Elements (2007)
[21] Rivara, Local modification of meshes for adaptive and/or multigrid finite-element methods, Journal of Computational and Applied Mathematics 36 pp 79– (1991) · Zbl 0733.65075
[22] Rivara, A 3-D refinement algorithm suitable for adaptive and multi-grid techniques, Communications in Applied Numerical Methods 8 pp 281– (1992) · Zbl 0755.65115
[23] Bänsch, Local mesh refinement in 2 and 3 dimensions, Impact of Computing in Science and Engineering 3 pp 181– (1991) · Zbl 0744.65074
[24] Ortiz, Adaptive mesh refinement in strain localization problems, Computer Methods in Applied Mechanics and Engineering 90 pp 781– (1991)
[25] Molinari, Three-dimensional adaptive meshing by subdivision and edge-collapse in finite-deformation dynamic plasticity problems with application to adiabatic shear banding, International Journal for Numerical Methods in Engineering 53 pp 1101– (2002)
[26] Rivara, New longest-edge algorithms for the refinement and/or improvement of unstructured triangulations, International Journal for Numerical Methods in Engineering 40 pp 3313– (1997) · Zbl 0980.65144
[27] Mitchell, Adaptive refinement for arbitrary finite-element spaces with hirachical bases, International Journal for Numerical Methods in Engineering 40 pp 3313– (1997)
[28] Kossaczký, A recursive approach to local mesh refinement in two and three dimensions, Journal of Computational and Applied Mathematics 55 pp 275– (1994) · Zbl 0823.65119
[29] Maubach, Local bisection refinement for N-simplicial grids generated by reflection, SIAM Journal on Scientific Computing 16 pp 210– (1995) · Zbl 0816.65090
[30] Bänsch, An adaptive finite-element strategy for the three-dimensional time-dependent Navier-Stokes equations, Journal of Computational and Applied Mathematics 36 pp 3– (1991) · Zbl 0727.76078
[31] Schmidt A, Siebert KG. ALBERT: an adaptive hierarchical finite element toolbox. Technical Report, Institut für Angewandte Mathematik, Albert-Ludwigs-Universität Freiburg, 2000.
[32] Ern, Theory and Practice of Finite Elements (2004) · Zbl 1059.65103 · doi:10.1007/978-1-4757-4355-5
[33] Verfürth, A Review of Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques (1996) · Zbl 0853.65108
[34] Ainsworth, A Posterior Error Estimation in Finite Element Analysis (2000) · Zbl 1008.65076 · doi:10.1002/9781118032824
[35] Zienkiewicz, The Mathematics of Finite Elements and Applications (1982)
[36] Cuitiño, A material-independent method for extending stress update algorithms from small-strain plasticity to finite plasticity with multiplicative kinematics, Engineering Computations 9 pp 437– (1992)
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