A novel multipoint stress control volume method for linear elasticity on quadrilateral grids

TL;DR

The paper introduces a novel multipoint stress control volume method for linear elasticity on quadrilateral grids that is locally conservative and locking-free. By weakly enforcing stress symmetry with a Lagrange multiplier and using piecewise-constant approximations for stress, displacement, and rotation, the method localizes the stress bilinear form around vertices, allowing stress elimination and yielding a cell-centered system (with two variants depending on the rotation space). It provides rigorous error analysis showing optimal $L^2$-convergence for stress and rotation and $L^2$-superconvergence for displacement, with extensive 2D and 3D numerical tests confirming robustness to heterogeneous coefficients and nearly incompressible materials. The approach offers computational efficiency through block-diagonal stress systems and potential for multigrid solvers, while maintaining accuracy on structured and unstructured grids, making it well-suited for practical elasticity problems.

Abstract

In this paper, we develop a novel control volume method that is locally conservative and locking-free for linear elasticity problem on quadrilateral grids. The symmetry of stress is weakly imposed through the introduction of a Lagrange multiplier. As such, the method involves three unknowns: stress, displacement and rotation. To ensure the well-posedness of the scheme, a pair of carefully defined finite element spaces is used for the stress, displacement and rotation such that the inf-sup condition holds. An appealing feature of the method is that piecewise constant functions are used for the approximations of stress, displacement and rotation, which greatly simplifies the implementation. In particular, the stress space is defined delicately such that the stress bilinear form is localized around each vertex, which allows for the local elimination of the stress, resulting in a cell-centered system. By choosing different definitions of the space for rotation, we develop two variants of the method. In particular, the first method uses a constant function for rotation over the interaction region, which allows for further elimination and results in a cell-centered system involving displacement only. A rigorous error analysis is performed for the proposed scheme. We show the optimal convergence for $L^2$-error of the stress and rotation. Moreover, we can also prove the superconvergence for $L^2$-error of displacement. Extensive numerical simulations indicate that our method is efficient and accurate, and can handle problems with discontinuous coefficients.

Figures (6)

• Figure 1: The macro-element cell is divided into four subcells (left); the black solid lines represent the primal edges and the red dashed lines represent the dual edges (inner subcell edges). The interaction region is formed by the union of four subcells $E_i, i=1,\cdots,4$ sharing the common vertex (right). The filled dots denote the cell displacement $\{\bm{u}_i\}$ and the blue dots denote the stress $\{\underline{\sigma}_i\}$.
• Figure 2: Two subcells $E_1$ and $E_2$ associated with a common face $F$ in 3D, $u_1$ and $u_2$ are the centers of two macro elements, each black dot denotes a cell displacement, cyan dot denotes the stress $\underline{\sigma}_1$, the dashed edges are edges in dual grid, the solid edges are edges in macro-element.
• Figure 3: One interaction region with four subcells numbered $E_i, i=1,\cdots,4$, in the reference space. The filled dots denote the cell pressure $\{p_i\}$ and the blue dots denote the velocity $\{\bm{u}_i\}$.
• Figure 4: residual distribution, $h=1/32$, example 1.
• Figure 5: Examples of unstructured meshes.

Theorems & Definitions (26)

• Lemma 3.1
• proof
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Lemma 4.1
• Lemma 4.2
• proof
• Lemma 4.3
• proof