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A Hybrid High-Order method for finite elastoplastic deformations within a logarithmic strain framework

Mickaël Abbas, Alexandre Ern, Nicolas Pignet

TL;DR

This work extends Hybrid High-Order (HHO) methods to finite elastoplastic deformations within a logarithmic strain framework, yielding a primal, locking-free formulation that operates on polyhedral meshes with non-matching interfaces. It combines face-based polynomials and locally eliminated cell unknowns via static condensation, paired with a local gradient reconstruction and stabilization to ensure equilibrated tractions and a symmetric elastoplastic tangent modulus $\mathbb{A}_{ep}$. The methodology is demonstrated through 2D and 3D benchmarks (necking, Cook's membrane, torsion, quasi-incompressible sphere) with comparisons to analytical solutions and Code_Aster, showing robust performance and avoidance of volumetric locking even under incompressible plastic flow. The results indicate that HHO achieves high accuracy with moderate DOFs and general mesh compatibility, offering a promising, scalable tool for industrial elastoplastic simulations and avenues for future extensions to non-local plasticity and contact.

Abstract

We devise and evaluate numerically a Hybrid High-Order (HHO) method for finite plasticity within a logarithmic strain framework. The HHO method uses as discrete unknowns piecewise polynomials of order $k\ge1$ on the mesh skeleton, together with cell-based polynomials that can be eliminated locally by static condensation. The HHO method leads to a primal formulation, supports polyhedral meshes with non-matching interfaces, is free of volumetric locking, the integration of the behavior law is performed only at cell-based quadrature nodes, and the tangent matrix in Newton's method is symmetric. Moreover, the principle of virtual work is satisfied locally with equilibrated tractions. Various two- and three-dimensional benchmarks are presented, as well as comparison against known solutions with an industrial software using conforming and mixed finite elements.

A Hybrid High-Order method for finite elastoplastic deformations within a logarithmic strain framework

TL;DR

This work extends Hybrid High-Order (HHO) methods to finite elastoplastic deformations within a logarithmic strain framework, yielding a primal, locking-free formulation that operates on polyhedral meshes with non-matching interfaces. It combines face-based polynomials and locally eliminated cell unknowns via static condensation, paired with a local gradient reconstruction and stabilization to ensure equilibrated tractions and a symmetric elastoplastic tangent modulus . The methodology is demonstrated through 2D and 3D benchmarks (necking, Cook's membrane, torsion, quasi-incompressible sphere) with comparisons to analytical solutions and Code_Aster, showing robust performance and avoidance of volumetric locking even under incompressible plastic flow. The results indicate that HHO achieves high accuracy with moderate DOFs and general mesh compatibility, offering a promising, scalable tool for industrial elastoplastic simulations and avenues for future extensions to non-local plasticity and contact.

Abstract

We devise and evaluate numerically a Hybrid High-Order (HHO) method for finite plasticity within a logarithmic strain framework. The HHO method uses as discrete unknowns piecewise polynomials of order on the mesh skeleton, together with cell-based polynomials that can be eliminated locally by static condensation. The HHO method leads to a primal formulation, supports polyhedral meshes with non-matching interfaces, is free of volumetric locking, the integration of the behavior law is performed only at cell-based quadrature nodes, and the tangent matrix in Newton's method is symmetric. Moreover, the principle of virtual work is satisfied locally with equilibrated tractions. Various two- and three-dimensional benchmarks are presented, as well as comparison against known solutions with an industrial software using conforming and mixed finite elements.

Paper Structure

This paper contains 24 sections, 3 theorems, 44 equations, 17 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Plasticity model
  3. Kinematics
  4. Constitutive logarithmic strain model
  5. Finite elastoplasticity model problem in incremental form
  6. Algorithmic aspects
  7. Example: nonlinear isotropic hardening with a von Mises yield criterion
  8. The Hybrid High-Order method
  9. Discrete setting
  10. Local gradient reconstruction and stabilization
  11. Global discrete problem
  12. Discrete principle of virtual work
  13. Nonlinear solver and implementation
  14. Numerical examples
  15. Necking of a 2D rectangular bar
  16. ...and 9 more sections

Key Result

Lemma 3

Let the gradient reconstruction operator be defined by eq_reconstruction_grad and the stabilization operator be defined by eq:stabHDG or eq:stabHHO. Let $\gamma_{{\partial T}}$ be defined below eq:snorme. Then, we have the following properties: (i) Boundedness: there exists $\alpha_\sharp< +\infty$, (ii) Stability: there exists $\alpha_\flat > 0$, uniform w.r.t. $h$, such that, for all $T \in \mat

Figures (17)

  • Figure 1: Face (black) and cell (gray) degrees of freedom in $U^{k,l}_T$ for different values of the pair $(k,l)$ in the two-dimensional case (each dot represents a degree of freedom which is not necessarily a point evaluation).
  • Figure 2: Necking of a 2D rectangular bar: (a) Geometry and boundary conditions (dimensions in $\textrm{mm}$). For symmetric reasons only the upper right-quarter of the bar is considered (b) Mesh composed of 400 quadrangles used for the computations. (c) Vertical reaction versus imposed displacement for the different methods (all the curves overlap except that for Q1) .
  • Figure 3: Necking of a 2D rectangular bar: Equivalent plastic strain $p$ at the quadrature points on the final configuration for the different methods.
  • Figure 4: Necking of a 2D rectangular bar: trace of the Cauchy stress tensor $\sigma$ (in $\textrm{MPa}$) at the quadrature points on the final configuration for the different methods.
  • Figure 5: Cook's membrane: (a) Geometry and boundary conditions (dimensions in $\textrm{mm}$). (b) Convergence of the vertical displacement of the point $A$ (in $\textrm{mm}$) vs. the number of degrees of freedom for Q1, Q2, Q2_RI, UPG, and HHO methods.
  • ...and 12 more figures

Theorems & Definitions (5)

  • Lemma 3: Boundedness and stability
  • Remark 4: HDG-type stabilization
  • Lemma 5: Equilibrated tractions
  • Theorem 6: Coercivity
  • Remark 7