Implementation of a Modified Moving Least Squares Approximation for Predicting Soft Tissue Deformation Using a Meshless Method

In applications where the organic soft tissue undergoes large deformations, traditional finite element methods can fail due to element distortion. In this context, meshless methods, which require no mesh for defining the interpolation field, can offer stable solutions. In meshless method, the moving least square (MLS) shape functions have been widely used for approximating the unknown field functions using the scattered field nodes. However, the classical MLS places strict requirements on the nodal distributions inside the support domain in order to maintain the non-singularity of the moment matrix. These limitations are preventing the practical use of higher order polynomial basis in classical MLS for randomly distributed nodes despite their capability for more accurate approximation of complex deformation fields. A modified moving least squares (MMLS) approximation has been recently developed by ISML. This paper assesses the interpolation capabilities of the MMLS. The proposed meshless method based on MMLS is used for computing the extension of a soft tissue sample and for a brain deformation simulation in 2D. The results are compared with the commercial finite element software ABAQUS. The simulation results demonstrate the superior performance of the MMLS over classical MLS with linear basis functions in terms of accuracy of the solution.