$hp$-FEM for Elastoplasticity & $hp$-Adaptivity Based on Local Error Reductions

TL;DR

The work targets elastoplasticity with linearly kinematic hardening by developing $hp$-finite element methods for both a variational-inequality and a mixed formulation. It delivers (i) a priori error analysis for higher-order mixed discretizations including the Lagrange multiplier, (ii) a reliable residual-based a posteriori error estimator with local efficiency, and (iii) a semi-smooth Newton solver leveraging decoupled nonlinear systems. The second part introduces an $hp$-adaptive algorithm based on locally predicted energy reductions, which steers refinement without traditional a posteriori indicators. Collectively, the results enable robust, high-order hp-adaptive strategies that achieve exponential convergence on elastoplastic problems with low regularity, benefiting applications in metal forming and concrete deformation.

Abstract

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.

Figures (5)

• Figure 1: Two-dimensional shape functions constructed via images of integrated Legendre polynomials.
• Figure 2: The stress vector $\mathfrak{s}_{\mathfrak{n}}(\mathfrak{x},t)$.
• Figure 3: Stress versus strain for an elastoplastic material.
• Figure 4: Refinement of $\widehat{Q}$ with respect to $\hat{\mathfrak{z}}\in(-1,1)^2$ and corresponding refinement of $Q = \mathfrak{F}_Q(\widehat{Q})$.
• Figure 5: A $p$-enrichment (left) vs. an $hp$-refinement (right).

Theorems & Definitions (19)

• theorem 1: [P2, Thm. 1]
• theorem 2: [P3, Thm. 4]
• theorem 3: [P3, Thm. 2]
• theorem 4: [P2, Thm. 7]
• theorem 5: [P1, Thm. 2]
• theorem 6: [P1, Thm. 3]
• theorem 7: [P2, Thm. 6]
• theorem 8: [P2, Thm. 8]
• theorem 9: [P2, Thm. 9]
• theorem 10: [P2, Thm. 11]