Lamination-based efficient treatment of weak discontinuities for non-conforming finite element meshes

TL;DR

This work tackles the challenge of representing interfaces with weak discontinuities on non-conforming finite-element meshes by introducing the laminated element technique (LET). LET treats any element cut by an interface as a laminate of the two phases, with per-element volume fraction and lamination orientation derived from a level-set representation, and uses a simple macro–micro transition to obtain the homogenized response without adding global degrees of freedom. Across elastic, elastic-plastic, and hyperelastic examples in 2D and 3D, LET consistently improves accuracy over two common non-conforming approaches and yields smooth dependence on moving interfaces, though it does not outperform X-FEM in convergence rate. The approach is simple to implement at the element level and is well-suited to complex geometries and potential moving-interface problems, offering a practical alternative for fast, robust simulations in computational solid mechanics.

Abstract

When modelling discontinuities (interfaces) using the finite element method, the standard approach is to use a conforming finite-element mesh in which the mesh matches the interfaces. However, this approach can prove cumbersome if the geometry is complex, in particular in 3D. In this work, we develop an efficient technique for a non-conforming finite-element treatment of weak discontinuities by using laminated microstructures. The approach is inspired by the so-called composite voxel technique that has been developed for FFT-based spectral solvers in computational homogenization. The idea behind the method is rather simple. Each finite element that is cut by an interface is treated as a simple laminate with the volume fraction of the phases and the lamination orientation determined in terms of the actual geometrical arrangement of the interface within the element. The approach is illustrated by several computational examples relevant to the micromechanics of heterogeneous materials. Elastic and elastic-plastic materials at small and finite strain are considered in the examples. The performance of the proposed method is compared to two alternative, simple methods showing that the new approach is in most cases superior to them while maintaining the simplicity.

Figures (18)

• Figure 1: Laminated element technique (LET): the element that is cut by an interface is treated as a simple laminate with the volume fraction and lamination orientation specified by the actual geometry of the interface within the element. $\bm{N}$ is the unit normal to the interface.
• Figure 2: Discretization approaches employed in this work: (a) conforming mesh, (b) element-level assignement (ELA), (c) Gauss-point-level assignement (GPLA), and (d) laminated element technique (LET). In ELA (resp. GPLA), the whole element (resp. Gauss point) belongs to a single phase that is determined by the value of the level-set function in the element centre (resp. at the Gauss point).
• Figure 3: Elastic inclusion problem: (a) scheme of the problem; (b) computational domain with a regular (non-conforming) mesh of quadrilateral elements ($16\times16$ elements, $h=0.125$). The interface $\Sigma$ is approximated by the zero level set, $\phi^h=0$. The arrows in panel (b) represent the nodal forces applied to the boundary nodes, which are calculated from the traction resulting from the analytical solution.
• Figure 4: Elastic inclusion problem: rate of convergence in energy norm for (a) soft inclusion ($E_2/E_1=10$) and (b) hard inclusion ($E_2/E_1=0.005$). The results obtained for a matching mesh of four-node quadrilateral elements are labelled 'MMsh'. The results for X-FEM are taken from Moes2003, and those for A-FEM from Essongue2020.
• Figure 5: Elastic inclusion problem: rate of convergence in $L_2$ norm for (a) soft inclusion ($E_2/E_1=10$) and (b) hard inclusion ($E_2/E_1=0.005$).