Stable Meshless Methods for Soft-Tissue Simulation

Core Team Members: Jack S. Hale, Team Legato, University of Luxembourg.
Collaborators: Alejandro A. Ortiz, Universidad de Chile. Christian J. Cyron, Yale University.

« Supported by the Fonds National de la Recherche, Luxembourg (#6693582) »
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In this project we are working to develop robust inf-sup stable meshfree numerical methods for simulating soft tissue. Our experiments show that meshfree methods are significantly more robust with respect to extremely large deformations than finite elements, although at increased computational cost. Meshfree methods also reduce the burden of mesh construction. Our method only requires a set of nodal positions and a Delaunay tessellation of the domain. Because the basis functions are not posed on the tessellation the method works even with poor quality and distorted nodal layouts.

3D nearly-incompressible elastic punch problem solved using second order maximum entropy basis functions with bubbles for the displacements and first order maximum entropy basis functions for pressure.

3D nearly-incompressible elastic punch problem solved using second order maximum entropy basis functions with bubbles for the displacements and first order maximum entropy basis functions for pressure.

A critical difficulty is satisfying the well-known inf-sup stability condition. Inspired by the classic MINI element of Arnold, we are developing a local patch projection method enriched with bubble functions that satisfies the inf-sup condition. Additionally we have developed novel integration techniques that allow the method to pass the patch test to machine precision and achieve optimal order convergence.

You can find more technical details in the following paper:

Meshfree volume-averaged nodal projection method for nearly-incompressible elasticity using meshfree and bubble basis functions