Computer-based simulation of PDEs involves approximating the unknowns and relies on suitable description of geometrical entities such as the computational domain and its properties. The Finite Element Method (FEM) is by large the most popular technique for the computer-based simulation of PDEs and hinges on the assumption that discretized domain and unknown fields are both represented by piecewise polynomials, on tetrahedral or hexahedral partitions.
In reality, the simulation of PDEs is a brick within a workflow where, at the beginning, the geometrical entities are created, described and manipulated with a geometry processor, often through Computer-Aided Design systems (CAD), and then used as input in Computer-Aided Engineering systems (CAE) where they are handled and processed for the simulation (often also called analysis). The output of the simulation is then visualized and possibly interpreted with an engineering perspective. This workflow is often repeated many times as part of an optimization loop. We quickly realize that new paradigms have to be identified in order to optimize this workflow as a whole. The representation of geometric entities has been studied in the field of geometric modeling, and often the requirements of shape design are different from those of simulation, which is based on numerical methods for PDEs, developed as a branch of mathematics (and mechanics). The use of FEM on CAD geometries (which are mainly represented through their boundaries) calls then for (re-)meshing and re-interpolation techniques that are computationally expensive and result in non-exact geometries as well as inaccurate solutions.
By attacking this bottleneck, this project develops innovative mathematical tools for the numerical solutions of PDEs, with the goal of considerably improving the efficiency and the robustness of the overall workflow. This goal is interdisciplinary in nature, and calls upon the tight interaction of researchers from the communities of geometry processing and of numerical analysis of PDEs, in order to find a common ground for the representation and use of geometric quantities. The research has to be developed from both sides, with a common objective:
(i) PDE solvers have to be developed / extended in such a way that the class of geometries and tessellation that they can handle, without losing accuracy or robustness, is as wide as possible. We refer to such a class as the class of analysis suitable geometric models or ASGMs.
(ii) geometric modeling techniques are to be developed in order to construct and manipulate ASGMs.
Thanks to the common work on the definition and rules of ASGMs that lies at the very core of CHANGE, the geometry processing becomes PDE-aware, and vice versa.
The result of CHANGE will reshape the design to analysis pipeline, which is at the very basis of the conception of practically any industrial project. Thus, if on the short term CHANGE’s results consist of mathematical theorems, proofs and academic testing, on the medium-to-long term the results of CHANGE may drastically reduce the time-to-production on any industrial manufactured objected and will contribute to the development of computational twins. Thus, the optimization cycle of industrialized product will be improved allowing for savings of materials, lighting of structures, while safety will be improved and certified by mathematically coherent model descriptions.
