Implementation of neural networks on Lagrangian mesh based simulations

SupervisorĀ : Prof. F. Avellan
Co-Supervisor: Dr. Pascal Clausen
Assistant: Alexis Papagiannopoulos
ThemeĀ : Neural networks, Lagrangian mesh, mesh optimization

Finite Element Method (FEM) is used as numerical approach for solving Partial Differential Equations (PDE) that describe physical phenomena (e.g deformation, fluid flow). By applying this method, the geometry of the object under study is discretized into a mesh consisting of smaller subsets called elements (triangles, tetrahedra etc.). In a Lagrangian framework, during a simulation the points of the mesh move according to the assigned governing equation. The shape and size of the elements play an essential role to the accuracy of the numerical solution acquired by FEM ([1]). As a result, the motion of the mesh can produce badly shaped or inverted elements which impact strongly the accuracy of the solution. Therefore, before solving the PDE, the geometry of those elements needs to be fixed (remeshing) ([2]).
To improve the speed of the remeshing process, a first approach involves the use of machine learning techniques (neural networks). With a remesher prototype at hand (written in C++), the student’s work will be focused on integrating trained neural networks to the existing code, adapt them to the remeshing routine and optimize them for better performance. Moreover, the student will run simulations and perform a quantitative study to compare the speed and robustness with the existing remesher.

Mesh optimization

[1] Jonathan Shewchuk. What is a good linear finite element? interpolation, conditioning, anisotropy, and quality measures (preprint). University of California at Berkeley, 73, 2002.

[2] Pascal Clausen, Martin Wicke, Jonathan R. Shewchuk, and James F.O’Brien. Simulating liquids and solid-liquid interactions with la-grangian meshes. ACM Trans. Graph., 32(2):17:1{17:15, April 2013. ISSN 0730-0301. doi: 10.1145/2451236.2451243. URL : http://doi.acm.org/10.1145/2451236.2451243.