$hp$-FEM for Elastoplasticity & $hp$-Adaptivity Based on Local Error Reductions
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913222189121536 |
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| author | Bammer, Patrick |
| author_facet | Bammer, Patrick |
| contents | The first part of the cumulative thesis contains the numerical analysis of different $hp$-finite element discretizations related to two different weak formulations of a model problem in elastoplasticity with linearly kinematic hardening. Thereby, the weak formulation either takes the form of a variational inequality of the second kind, including a non-differentiable plasticity functional, or represents a mixed formulation, in which the non-smooth plasticity functional is resolved by a Lagrange multiplier. As the non-differentiability of the plasticity functional causes many difficulties in the numerical analysis and the computation of a discrete solution it seems advantageous to consider discretizations of the mixed formulation. In a first work, an a priori error analysis of an higher-order finite element discretization of the mixed formulation (explicitly including the discretization of the Lagrange multiplier) is presented. The relations between the three different $hp$-discretizations are studied in a second work where also a reliable a posteriori error estimator that also satisfies some (local) efficiency estimates is derived. In a third work, an efficient semi-smooth Newton solver is proposed, which is obtained by reformulating a discretization of the mixed formulation as a system of decoupled nonlinear equations. The second part of the thesis introduces a new $hp$-adaptive algorithm for solving variational equations, in which the automatic mesh refinement does not rely on the use of an a posteriori error estimator or smoothness indicators but is based on comparing locally predicted error reductions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_01875 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $hp$-FEM for Elastoplasticity & $hp$-Adaptivity Based on Local Error Reductions Bammer, Patrick Numerical Analysis 65N30, 65N50 The first part of the cumulative thesis contains the numerical analysis of different $hp$-finite element discretizations related to two different weak formulations of a model problem in elastoplasticity with linearly kinematic hardening. Thereby, the weak formulation either takes the form of a variational inequality of the second kind, including a non-differentiable plasticity functional, or represents a mixed formulation, in which the non-smooth plasticity functional is resolved by a Lagrange multiplier. As the non-differentiability of the plasticity functional causes many difficulties in the numerical analysis and the computation of a discrete solution it seems advantageous to consider discretizations of the mixed formulation. In a first work, an a priori error analysis of an higher-order finite element discretization of the mixed formulation (explicitly including the discretization of the Lagrange multiplier) is presented. The relations between the three different $hp$-discretizations are studied in a second work where also a reliable a posteriori error estimator that also satisfies some (local) efficiency estimates is derived. In a third work, an efficient semi-smooth Newton solver is proposed, which is obtained by reformulating a discretization of the mixed formulation as a system of decoupled nonlinear equations. The second part of the thesis introduces a new $hp$-adaptive algorithm for solving variational equations, in which the automatic mesh refinement does not rely on the use of an a posteriori error estimator or smoothness indicators but is based on comparing locally predicted error reductions. |
| title | $hp$-FEM for Elastoplasticity & $hp$-Adaptivity Based on Local Error Reductions |
| topic | Numerical Analysis 65N30, 65N50 |
| url | https://arxiv.org/abs/2402.01875 |