Second-Order Compatible-Strain Mixed Finite Elements for 2D Compressible Nonlinear Elasticity

TL;DR

We develop explicit, second-order compatible-strain finite elements for 2D compressible nonlinear elasticity by formulating a three-field Hu-Washizu functional and constructing explicit shape functions for the displacement gradient and first Piola–Kirchhoff stress. The method uses covariant/contravariant Piola transforms along with mid-nodes and pseudo-nodes to enforce edge-continuity constraints, producing stable solutions without hourglass or locking in near-incompressible regimes. Numerical experiments across homogeneous and inhomogeneous tests, Cook's membrane, rubber sealing, and perforated-block tension demonstrate strong convergence, good agreement with reference results, and robustness across mesh resolutions without the need for pressure DOFs. The framework provides explicit local-to-global shape-function mappings and can be extended to incompressible materials and other element families, offering a practical path for high-order CSFEs in nonlinear elasticity.

Abstract

In recent years, a new class of mixed finite elements -- compatible-strain mixed finite elements (CSMFEs) -- has emerged that uses the differential complex of nonlinear elasticity. Their excellent performance in benchmark problems, such as numerical stability for modeling large deformations in near-incompressible solids, makes them a promising choice for solving engineering problems. Explicit forms exist for various shape functions of first-order CSMFEs. In contrast, existing second-order CSMFEs evaluate shape functions using numerical integration. In this paper, we formulate second-order CSMFEs with explicit shape functions for the displacement gradient and stress tensor. Concepts of vector calculus that stem from exterior calculus are presented and used to provide efficient forms for shape functions in the natural coordinate system. Covariant and contravariant Piola transformations are then applied to transform the shape functions to the physical space. Mid-nodes and pseudo-nodes are used to enforce the continuity constraints for the displacement gradient and stress tensor over the boundaries of elements. The formulation of the proposed second-order CSMFEs and technical aspects regarding their implementation are discussed in detail. Several benchmark problems are solved to compare the performance of CSMFEs with first-order CSMFEs and other second-order elements that rely on numerical integration. It is shown that the proposed CSMFEs are numerically stable for modeling near-incompressible solids in the finite strain regime.

Figures (29)

• Figure 1: Two-simplex $\mathscr{T}$ with vertices $v_{i}$ and edges $e_{i},i=1,2,3$. Tangent and normal vectors for each edge are denoted by $\mathbf{t}_{i}$ and $\mathbf{n}_{i}$, respectively. The orientation of edges is induced by the way the vertices are labeled.
• Figure 2: (a) Global shape functions $\mathbf{V}^{e}_{1},\mathbf{V}^{e}_{2},\mathbf{V}^{e}_{3}\in\mathcal{P}^{c}_{2}\left(T\mathcal{T}\right)$ on the edge $e$ shared by two simplices $\mathscr{T}_{1}$ and $\mathscr{T}_{2}$. (b) Local shape functions $\mathbf{v}^{\mathscr{T}}_{1},\mathbf{v}^{\mathscr{T}}_{2},\mathbf{v}^{\mathscr{T}}_{3}\in\mathcal{P}^{c}_{2}\left(T\mathscr{T}\right)$ on the simplex $\mathscr{T}$. These shape functions are used for interpolating the displacement gradient.
• Figure 3: (a) Global shape functions $\mathbf{V}^{e}_{1},\mathbf{V}^{e}_{2},\mathbf{V}^{e}_{3}\in\mathcal{P}^{d}_{2}\left(T\mathcal{T}\right)$ on the edge $e$ shared by two simplices $\mathscr{T}_{1}$ and $\mathscr{T}_{2}$. (b) Local shape functions $\mathbf{v}^{\mathscr{T}}_{1},\mathbf{v}^{\mathscr{T}}_{2},\mathbf{v}^{\mathscr{T}}_{3}\in\mathcal{P}^{d}_{2}\left(T\mathscr{T}\right)$ on the simplex $\mathscr{T}$. These shape functions are used for interpolating the stress tensor.
• Figure 4: (a) Global shape functions $\mathbf{V}^{e}_{1},\mathbf{V}^{e}_{2}\in\mathcal{P}^{c-}_{2}\left(T\mathcal{T}\right)$ on the edge $e$ shared by two simplices $\mathscr{T}_{1}$ and $\mathscr{T}_{2}$. (b) Local shape functions $\mathbf{v}^{\mathscr{T}}_{1},\mathbf{v}^{\mathscr{T}}_{2}\in\mathcal{P}^{c-}_{2}\left(T\mathscr{T}\right)$ on the simplex $\mathscr{T}$. These shape functions are used for interpolating the displacement gradient.
• Figure 5: (a) Global shape functions $\mathbf{V}^{e}_{1},\mathbf{V}^{e}_{2}\in\mathcal{P}^{d-}_{2}\left(T\mathcal{T}\right)$ on the edge $e$ shared by two simplices $\mathscr{T}_{1}$ and $\mathscr{T}_{2}$. (b) Local shape functions $\mathbf{v}^{\mathscr{T}}_{1},\mathbf{v}^{\mathscr{T}}_{2}\in\mathcal{P}^{d-}_{2}\left(T\mathscr{T}\right)$ on the simplex $\mathscr{T}$. These shape functions are used for interpolating the stress tensor.

Theorems & Definitions (16)

• Lemma 1
• proof
• Example 1
• Lemma 2
• Corollary 2.1
• proof
• Example 2
• Lemma 3
• Corollary 3.1
• proof