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

摘要

Elastic materials are ubiquitous in nature and indispensable components inman-made devices and equipments. When a device or equipment involves compositeor multiple elastic materials, elasticity interface problems come into play.The solution of three dimensional (3D) elasticity interface problems issignificantly more difficult than that of elliptic counterparts due to thecoupled vector components and cross derivatives in the governing elasticityequation. This work introduces the matched interface and boundary (MIB) methodfor solving 3D elasticity interface problems. The proposed MIB method utilizesfictitious values on irregular grid points near the material interface toreplace function values in the discretization so that the elasticity equationcan be discretized using the standard finite difference schemes as if therewere no material interface. The interface jump conditions are rigorouslyenforced on the intersecting points between the interface and the mesh lines.Such an enforcement determines the fictitious values. A number of new techniqueare developed to construct efficient MIB schemes for dealing with crossderivative in coupled governing equations. The proposed method is extensivelyvalidated over both weak and strong discontinuity of the solution, bothpiecewise constant and position-dependent material parameters, both smooth andnonsmooth interface geometries, and both small and large contrasts in thePoisson's ratio and shear modulus across the interface. Numerical experimentsindicate that the present MIB method is of second order convergence in both$L_\infty$ and $L_2$ error norms.