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Dictyostelium discoideum (Dd) is a model organism, studied for reasons from cell movement to chemotaxis to human disease control. Creating a computer model of the life cycle of Dd has garnered great interest, one part of which is the Aggregation Stage, where thousands of amoeba gather together to form a slug. Chemotaxis is the mechanism through which this is accomplished. This thesis develops two- and three-dimensional alternating direction implicit code which solves the diffusion equation on an adaptive grid. The calculated values for both two and three dimensions are checked against the actual solution and error results are provided. Comparisons are made between the coarse grid with refinement case and a fine grid without refinement case. Also, a non-negativity condition for two dimensions is derived to give a bound on the three major parameters: the diffusion coefficient and the spatial and time discretizations.

Ray tracing is a well-know technique for producing realistic graphics. However, the time necessary to generate images is unacceptably long. When producing the many frames that are necessary for animations, the time is magnified. Many methods have been proposed to reduce the calculations necessary in ray tracing. Much of the effort has attempted to reduce the number of rays cast or to reduce the number of intersection calculations. Both of these techniques exploit spatial coherence. These acceleration techniques are expanded not only to exploit spatial coherence but also to exploit temporal coherence in order to reduce calculations by treating animation information as a whole as opposed to isolating calculations to each individual frame. Techniques for exploiting temporal coherence are explored along with associated temporal bounding methods. By first ray tracing a temporally expanded scene, we are able to avoid traversal calculations in associated frames where object intersection is limited. This reduces the rendering times of the associated frames.

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A set of algorithms building a program package for the generation of twoandthree-dimensional unstructured/hybrid grids around complex geometrieshas been developed. The unstructured part of the grid generator is based on the advancing frontalgorithm. Tetrahedra of variable size, as well as directionally stretched tetrahedracan be generated by specification of a proper background grid, initiallygenerated by a Delaunay algorithm. A marching layer prismatic grid generation algorithm has been developedfor the generation of grids for viscous flows. The algorithm is able to handleregions of narrow gaps, as well as concave regions. The body surface is describedby a triangular unstructured surface grid. The subsequent grid layers in theprismatic grid are marched away from the body by an algebraic procedurecombined with an optimization procedure, resulting in a semi-structured gridof prismatic cells. Adaptive computations using remeshing have been done with use of a gradientsensor. Several key-variables can be monitored simultaneously. The sensorindicates that only the key-variables with the largest gradients give a substantialcontribution to the sensor. The sensor gives directionally stretched grids. An algorithm for the surface definition of curved surfaces using a biharmonicequation has been developed. This representation of the surface canbe used both for projection of the new surface nodes in h-refinement, and theinitial generation of the surface grid. For unsteady flows an algorithm has been developed for the deformationof hybrid grids, based on the solution of the biharmonic equation for the deformationfield. The main advantage of the grid deformation algorithm is that itcan handle large deformations. It also produces a smooth deformation distributionfor cells which are very skewed or stretched. This is necessary in orderto handle the very thin cells in the prismatic layers. The algorithms have been applied to complex three-dimensional geometries,and the influence of the grid quality on the accuracy for a finite volumeflow solver has been studied for some simpler generic geometries. / QC 20100812