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Four-field mixed finite elements for incompressible nonlinear elasticity

Santiago Badia, Wei Li, Ricardo Ruiz-Baier

Abstract

We present a stable finite element method for incompressible nonlinear elasticity based on a four-field mixed formulation involving the displacement, displacement gradient, first Piola--Kirchhoff stress and pressure. Unlike existing four-field mixed formulations, such as the compatible strain mixed finite element method (CSFEM), the proposed approach employs a discontinuous displacement field and requires no stabilisation in either 2D or 3D. A Newton--Raphson linearisation is derived and finite element pairs satisfying the relevant inf-sup conditions are identified. To recover accurate continuous displacement fields, an efficient postprocessing technique is further introduced. We establish the well-posedness of the linearised continuous problem together with a priori error estimates for the discrete formulation. Extensive numerical experiments in both 2D and 3D demonstrate optimal or even super convergence rates and enhanced robustness, particularly in 3D where CSFEM typically requires stabilisation.

Four-field mixed finite elements for incompressible nonlinear elasticity

Abstract

We present a stable finite element method for incompressible nonlinear elasticity based on a four-field mixed formulation involving the displacement, displacement gradient, first Piola--Kirchhoff stress and pressure. Unlike existing four-field mixed formulations, such as the compatible strain mixed finite element method (CSFEM), the proposed approach employs a discontinuous displacement field and requires no stabilisation in either 2D or 3D. A Newton--Raphson linearisation is derived and finite element pairs satisfying the relevant inf-sup conditions are identified. To recover accurate continuous displacement fields, an efficient postprocessing technique is further introduced. We establish the well-posedness of the linearised continuous problem together with a priori error estimates for the discrete formulation. Extensive numerical experiments in both 2D and 3D demonstrate optimal or even super convergence rates and enhanced robustness, particularly in 3D where CSFEM typically requires stabilisation.
Paper Structure (22 sections, 8 theorems, 86 equations, 14 figures, 2 tables)

This paper contains 22 sections, 8 theorems, 86 equations, 14 figures, 2 tables.

Table of Contents

  1. Introduction
  2. A mixed formulation for incompressible nonlinear elasticity
  3. Notation
  4. Incompressible nonlinear elasticity
  5. A four-field mixed formulation
  6. Linearisation
  7. Finite element discretisation
  8. A mixed finite element method for incompressible nonlinear elasticity
  9. Finite elements
  10. Discontinuous displacement correction
  11. Continuous and discrete analysis of the linearised problem
  12. Well-posedness analysis of the linearised problem
  13. Discrete solvability analysis for the linearised problem
  14. Error estimates for the discrete linearised problem
  15. Error estimates for the post-processed displacement field
  16. ...and 7 more sections

Key Result

Theorem 4.1

Denote $\mathbf{V}_2 \doteq \ker(B_2)$. Assume that the following conditions hold: Then, for any given linear functionals $F_1$, $F_2$ and $F_3$, there exists a unique solution $(\boldsymbol{k},(\boldsymbol{\sigma},p),\boldsymbol{u})\in \mathbb{L}^2(\Omega)\times [\mathbb{H}^{\mathbf{0}}(\operatorname*{\mathbf{div}},\Omega) \times \mathrm{L}^2(\Omega)]\times\mathbf{L}^2(\Omega)$ t

Figures (14)

  • Figure 1: Illustration of element-wise dof for the proposed stable pairs in 2D.
  • Figure 2: Geometries and displacement boundary conditions for the inflation problems. Left: a 2D cylindrical shell; right: a 3D hollow spherical ball.
  • Figure 3: Error convergence of the fe solution errors versus mesh size for various fe pairs in typical inflation problems. The displacement boundary data parameter is taken as $\lambda=3$. Reference slopes: 1 (dotted lines), 2 (dashed lines), 3 (dot-dashed lines), 4 (solid lines). Panels: (a) 2D results, (b) 3D results.
  • Figure 4: Error convergence of the csfem solution errors versus mesh size in the 2D inflation problem. The displacement boundary intensity parameter is $\lambda=3$. Reference slopes: 1 (dotted lines), 2 (dashed lines).
  • Figure 5: Geometries and traction boundary conditions for the Cook's membrane problems. Left: a 2D membrane; right: a 3D membrane.
  • ...and 9 more figures

Theorems & Definitions (14)

  • Theorem 4.1
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • Lemma 4.3
  • proof
  • ...and 4 more