MSPA-MCS: Automatic Geometric Simplification

0434338团队成员研究在解决工程和生物医学应用中的边界和初值问题时出现的复杂几何形状自动简化的新算法。几何简化允许使用单元少得多的有限元网格，这大大加快了偏微分方程组的求解速度。将双曲几何中强大的新工具应用于简化问题。这些工具能够在没有关于解的先验信息的情况下，使用纯粹的几何分析对边界变化对解的影响进行线性时间估计。简化算法是基于将原始几何形状划分为更简单的形状。在美国，脑损伤是一个巨大的医疗问题，每年有150万新病例，560亿美元的医疗费用，无数人的生命受到不利影响。混凝土、沥青和层合板等复合材料的寿命预测，同样会对公共安全和经济产生重大影响。这两个完全不同的问题有一个共同的特点，那就是它们适合于计算机建模和分析。此外，为了获得准确的结果，这两种应用都需要对复杂的几何形状进行计算机建模。遗憾的是，复杂的几何形状需要在大型高性能计算机上进行耗时的计算，从而限制了计算机建模的实际适用性。该项目发明了可以快速准确地简化复杂几何图形的计算机方法。在这种情况下，“准确”意味着简化保证不会改变模拟的结果超过一个小的百分比。因此，该研究项目可以更快地分析具有复杂几何图形的问题。从长远来看，这将导致更好地理解创伤性脑损伤的后果，或许还会导致治疗，以及对复杂结构材料更好的寿命预测能力。

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.