Second order Method for Solving 3D Elasticity Equations with Complex and Sharp Interfaces

TL;DR

This paper presents a second-order accurate matched interface and boundary (MIB) method for solving complex 3D elasticity interface problems involving multiple materials with discontinuities and irregular geometries.

Contribution

The work introduces a novel MIB scheme that effectively handles cross derivatives and interface jump conditions in 3D elasticity problems with complex interfaces.

Findings

Achieves second-order convergence in error norms.

Validates effectiveness across various discontinuity and geometry scenarios.

Handles large contrasts in material properties successfully.

Abstract

Elastic materials are ubiquitous in nature and indispensable components in man-made devices and equipments. When a device or equipment involves composite or multiple elastic materials, elasticity interface problems come into play. The solution of three dimensional (3D) elasticity interface problems is significantly more difficult than that of elliptic counterparts due to the coupled vector components and cross derivatives in the governing elasticity equation. This work introduces the matched interface and boundary (MIB) method for solving 3D elasticity interface problems. The proposed MIB method utilizes fictitious values on irregular grid points near the material interface to replace function values in the discretization so that the elasticity equation can be discretized using the standard finite difference schemes as if there were no material interface. The interface jump conditions are rigorously enforced on the intersecting points between the interface and the mesh lines. Such an enforcement determines the fictitious values. A number of new technique are developed to construct efficient MIB schemes for dealing with cross derivative in coupled governing equations. The proposed method is extensively validated over both weak and strong discontinuity of the solution, both piecewise constant and position-dependent material parameters, both smooth and nonsmooth interface geometries, and both small and large contrasts in the Poisson's ratio and shear modulus across the interface. Numerical experiments indicate that the present MIB method is of second order convergence in both $L_\infty$ and $L_2$ error norms.