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Ph.D. Theses

Efficient Methods for Solving Biomechanical Equations

By Toshiro K. Ohsumi
Advisor: Joseph E. Flaherty
April 24, 2003

Several steps are taken to produce a problem solving environment (PSE) 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 used 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 PSE 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 simulating the formation of artificial arteries were solved using h-adaptivity and mesh motion in two dimensions. The effectivity of the adaptive procedures which were developed are shown. A modern object-oriented code closer to the ideal of a PSE, Trellis, was developed and used to solve the ABT equations in three-dimensions using h-adaptivity and mesh motion for the hexahedral intersitital cell traction assay and wound healing problems, hitherto unsolved. Warm-restarts in DASPK were implemented in Trellis and shown to greatly speed up adaptive computations.

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