The numerical solution of partial differential equations (PDEs) is ubiquitous in science and engineering applications, including simulation of elastoplastic deformations, fluids, and light scattering. The finite element method (FEM) is the most commonly used discretization of PDEs, especially in the context of structural and thermal analysis, due to its generality and rich selection of off-the-shelf commercial implementations. Ideally, a PDE solver should be a ``black box'': the user provides as input the domain boundary, boundary conditions, and the governing equations, and the code computes the value of the solution at a set of user-specified points of the input domain. This is surprisingly far from being the case for all existing open-source or commercial software, despite the research efforts in this direction and the large academic and industrial interest. To a large extent, this is due to treating meshing and FEM basis construction as two disjoint problems, often exposing the user to the technical issues of interfacing the meshing software with FEM basis construction, both of which, strictly speaking, are technical issues internal to the solver. This state of matters presents a fundamental problem for applications that require fully automatic, robust processing of large collections of meshes of varying sizes, an increasingly common situation as large collections of geometric data become available. This proposal introduces an integrated pipeline, considering meshing and element design as a single challenge, and developing a software platform to enable black box analysis on complex geometric models represented as point clouds, triangle meshes, or CAD (Computer Aided Design) models, opening the door to new shape design technique to a wide range of new applications in sciences and engineering.
This project proposes to develop a set of software components based on a set of novel approaches the investigators have developed combined with "filtered" use of rational or multi-precision numerical representations to handle robustness problems while maintaining practical performance. The proposed set of geometry processing techniques, while slower than existing ones, are fully robust in a sense of always produce a valid result with minimal assumptions on the input. The geometric toolkit will allow to automatically convert geometrical data in the form of range scans, CAD models, or voxel grids into a surface or volumetric representation, directly usable in widely used open-source finite element method (FEM) packages. It will include mesh generation, in addition to tetrahedral meshes, for other common types of discretizations: hexahedral meshes, and hex-dominant hybrid meshes. The key innovation is to achieve numerical robustness with minimal added algorithmic complexity by carefully mixing higher precision representations for the critical part, while relying on standard fixed-precision floating point representation for the rest and designing algorithms amenable to this approach. As in overwhelming majority of cases higher accuracy is needed for a vanishingly small fraction of computation, this approach allows the users to achieve sensible running time while ensuring output validity and algorithmic correctness on imperfect, real world data. Secondly, the invetigators will integrate FEM basis construction with meshing decoupling accuracy from mesh quality. The software toolkit developed in this proposal has potential for a major impact in all domains that require computational simulation of physical phenomena in complex geometries, enabling the automation of data acquisition, reconstruction, and simulation pipelines. The expectation of this project is that the outcome will not only be a reduction in human time, but the opportunity to fully automate this pipeline will open new research venues. The release of all the software with a MPL2 license will facilitate integration of the results of the work into commercial software, in addition to academic/non-profit research use.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
