Automated Geometric Modeling and Analysis

点击翻译按钮获取中文摘要

Partial differential equations (PDEs) are ubiquitously used in the mathematical description of our world, from the movement of bouncing balls to the complex behavior of an electron in an atom. Due to their versatility, PDEs are also used in countless disciplines to compute analyses; i.e., to use of a computer to solve the PDE in order to obtain a virtual simulated behavior. For instance, the solution of PDEs is used in architecture to evaluate structural soundness, in medicine to anticipate and study the effects of treatments, in biology to compute non-observable quantities, and in quantum physics to describe the wave state of a system. The finite element method is the most commonly used method to solve PDEs, particularly in disciplines related to structural and thermal analysis or fluid dynamics. A PDE solver should not require any knowledge related to mathematics or computer science. A user should be asked to provide only the input domain boundaries, the governing model, and the boundary conditions. With this setup, a PDE solver system should compute the solution for every point in the domain and present it to the user. Surprisingly, this is not the case for existing commercial and open-source software, where tedious, unintuitive, and time-consuming manual interactions are needed to obtain a result. For instance, to get a solution from a given input model, it takes weeks of manually removing triangles and moving vertices to fill the input model with tetrahedra. Such manual procedures pose fundamental problems when processing a large number of simulations, a practice that became popular in recent years with machine learning. When processing large collections, a fully automated PDE solver is necessary; it is inconceivable to manually tweak parameters and hope to obtain a solution. Our long-term goal is to develop an intuitive and easy-to-use analysis pipeline that requires only the input boundary, the governing equations, and the boundary conditions. With this essential input, the PDE solver will produce a valid, efficient, and robust solution. Developing such a pipeline is an ambitious effort that requires considering different physical behaviors (e.g., mechanical deformation, fluids, contacts, waves) and fields (e.g., mathematics, physics, computer science). Thus, in the next five years, we plan to focus on the following objectives: 1) generate robust meshes from smooth curved domains; 2) develop new criteria to compensate for low-quality curved meshes; 3) extend collision response to curved geometries; 4) address complex physical phenomena in fluids or fluid-structure interaction; and 5) integrate the new robust curved pipeline in existing packages. My HQPs will work on real-world problems and develop practical solutions; this is fundamental for both industry and for research. I believe that my research's intersection with other fields will benefit other researchers worldwide and increase the prestige and visibility of Canadian research.