Efficient methods for solving biomechanical equations.

摘要

Several steps are taken to produce a partial differential equation solving environment (PDESE) for solving time-dependent nonlinear biomechanical problems using adaptive finite element methods (FEMs) in one-, two-, and three-dimensions. We develop several methods to increase the efficiency of FEMs used to solve systems of partial differential equations arising in biomechanical models. The table look-up method is used to increase the speed of quadrature procedures specific to FEMs by interpolating the non-polynomial part of the integrand then using a pre-computed table of values. An enhanced prototype of a PDESE using h-adaptivity was used to solve a reaction-diffusion system of equations simulating the spread of Lyme disease taking into account vector dynamics. Procedures to correct sliver elements generated by mesh motion required for domain reshaping were developed and used in a h-adaptive two-dimensional micro-scale fluid pump simulation. The anisotropic biphasic theory (ABT) equations of Barocas and Tranquillo describing the formation of artificial arteries were solved using h-adaptivity and mesh motion in two dimensions. The efficacy of the adaptive procedures that were developed is shown. A modern object-oriented code closer to the ideal of a PDESE, Trellis, was enhanced and used to solve the ABT equations in three dimensions using h-adaptivity and mesh motion for the hexahedral interstitial cell traction assay (ICTA) and wound healing problems, hitherto unsolved. Warm restarts in DASPK were implemented within Trellis and shown to speed up adaptive computations by a factor of four for the ICTA problem.