Geometric modeling for simulation of complex physical phenomena raises many challenges including algorithmic efficiency, practicality, scalability, robustness, theoretical guarantees, and compatibility with the emerging numerical methods. We study solutions for geometric discretization problems for spatial domains (encountered in conventional scientific computing) and for space-time domains (motivated by the next-generation numerical methods being developed for solving PDEs directly in the space-time continuum). Our approach combines the strengths of theoretical algorithms (time complexity, output size optimality, and quality guarantees) and practical heuristics (ease of implementation, performance in practice, scalability). Two broad classes of problems are studied: (i) We develop fast, sequential and parallel algorithms and software to compute premium-quality, size-optimal, simplicial and cubical meshes of spatial domains which can evolve as the simulation progress for isotropic and anisotropic problems. (ii) We develop scalable, provably-good meshing algorithms and software to compute space-time triangulations which enables us to perform simulations directly in the space-time domain. The algorithms and the software tools developed within this project are being integrated with applications and contributing to the fundamental research in engineering, scientific computing, solid modeling, computer-aided design, graphics, geographic information systems, computational biology, visualization and molecular modeling. As a result, this project has broader impact across a number of scientific, medical and industrial fields. Moreover, the project has academic impact through the inclusion of underrepresented groups, the development of interdisciplinary courses which focus on linking fundamental concepts in theoretical areas such as graph theory, geometry and topology to application problems in biology and engineering.

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University of Florida
United States
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