MSPA-MCS: Automatic Geometric Simplification

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ABSTRACTPI: Stephen Vavasis proposal: 0434338The team members investigate new algorithms for automatic simplification of complex geometries arising in solution of boundary and initial value problems from engineering and biomedical applications. Geometric simplification allows the use of finite-element meshes with many fewer elements, which greatly speeds up the solution time of partial differential equations. Powerful new tools from hyperbolic geometry are applied to the simplification problem. These tools enable linear-time estimation of the effect on the solution of a change in the boundary using purely geometric analysis and without prior information about the solution. The simplification algorithm is based on a partitioning of the original geometry into simpler shapes.Brain injury is an enormous medical problem in the U.S., with 1.5 million new cases per year, $56 billion spent on medical expenses, and countless lives affected adversely. Life prediction for composite materials like concrete, asphalt and laminates, similarly have significant public-safety and economic consequences. These two disparate problems share in common the property that they are amenable to computer modeling and analysis. Furthermore, for accurate results, both applications require computer modeling of complex geometric shapes. Unfortunately, complex geometric shapes require time-consuming computations on large high-performance computers and thus limit the practical applicability of computer modeling. The project invents computer methods that can quickly and accurately simplify complex geometries. "Accurately" in this context means that the simplification is guaranteed not to change the outcome of the simulation by more than a small percentage. Therefore, the research project leads to much faster analysis of problems with complex geometry. In the long run, this leads to better understanding of the consequences and perhaps therapy for traumatic brain injury and better life-prediction capability for complex structural materials.