Collaborative Proposal: SGER: Conforming Polygonal Finite Elements

在这个小补助金探索性研究提案中，追求有限元方法的一个新的和新颖的方向。 多边形有限元在网格生成方面提供了更大的灵活性，并且更适合于涉及材料域拓扑结构的显著变化的固体力学应用。在这个建议中，最近的进展，无网格近似，计算几何，计算机图形学将被用来构建符合试验和测试近似多边形有限元。 将开发多边形单元网格生成的计算算法（优化方案），并将对椭圆边值问题有限元技术的精度和收敛性进行数值研究。本文的研究结果将为有限元研究开辟新的途径，使多边形有限元方法在固体力学和计算材料科学中得到更广泛的应用，并为多边形上的体积规则的建立提供动力.

In this Small Grant Exploratory Research proposal, a new and novel direction in finite element methodology is pursued. Polygonal finite elements provide greater flexibility in mesh generation and are better-suited for applications in solid mechanics which involve a significant change in the topology of the material domain. In this proposal, recent advances in meshfree approximations, computational geometry, and computer graphics will be used to construct conforming trial and test approximations onpolygonal finite elements. Computational algorithms (optimization scheme) for mesh generation with polygonal elements will be developed, and numerical studies on the accuracy and convergenceof the finite element technique will be conducted forelliptic boundary-value problems. The results of this study can potentially open-up new pathways in finite element research, lead to the wider usage of polygonal finite element methods in solid mechanicsand computational materials science, and also provide impetus for the construction of cubature rules on polytopes.