Direct simplicial dynamics: Simulations in biomechanics

Abstract

We develop a method for calculating mechanics of surfaces, of bodies, and of bodies with surfaces, directly with the geometric discretization of such bodies into simplices, rather than discretizing the partial differential equations of continuum mechanics in either finite element or finite difference approximations. We have applied this approach to selected simulations in biomechanics. Simplices have exactly the degrees of freedom to describe symmetric tensors, and our approach constructs the mechanics of media without having to apply a Galerkin basis of piecewise affine maps to differential equations. We also formulate a discrete shape operator as a symmetric affine matrix which may be used to describe bending energies, including anisotropy, for surfaces in a systematic way, analogous to the use of the metric tensor for strain elasticity. Given the uncertainties and complexities of biological tissue mechanics that would make formulating differential equations, constitutive properties, and boundary conditions difficult, we offer direct simplicial mechanics as an alternative for selected applications worthy of consideration. The present version is focused on statics, but we indicate an approach to dynamical phenomena. The methods detailed in this paper have been incorporated into software intended for commercial applications of simulations in biomechanics of tissue and full details on the mathematical method and the formulas needed for implementation are provided here. Several illustrative examples of simulations are given.