Cut Finite Element Methods on Overlapping Meshes: Analysis and Applications
Doctoral thesis, 2021
This thesis deals with both analysis and applications of cut finite element methods (CutFEMs) on overlapping meshes. By overlapping meshes we mean a mesh hierarchy with a background mesh at the bottom and one or more overlapping meshes that are stacked on top of it. Overlapping meshes can be used as an alternative to costly remeshing for problems with changing geometry. The main content of the thesis is the five appended papers. The thesis consists of an analysis part and an applications part.
In the analysis part (Paper I and Paper II), we consider cut finite element methods on overlapping meshes for a time-dependent parabolic model problem: the heat equation on two overlapping meshes, where one mesh is allowed to move around on top of the other. In Paper I, the overlapping mesh is prescribed a cG(1) movement, meaning that its location as a function of time is continuous and piecewise linear. The cG(1) mesh movement results in a space-time discretization for which existing analysis methodologies either fail or are unsuitable. We therefore propose, to the best of our knowledge, a new energy analysis framework that is general enough to be applicable to the current setting. In Paper II, the overlapping mesh is prescribed a dG(0) movement, meaning that its location as a function of time is discontinuous and piecewise constant. The dG(0) mesh movement results in a space-time discretization for which existing analysis methodologies work with some modifications to handle the shift in the overlapping mesh's location at discrete times.
The applications part (Paper III, IV, and V) concerns cut finite element methods on overlapping meshes for stationary PDE-problems. We consider two potential applications for CutFEM on overlapping meshes. The first application, presented in Paper III, presents methodology for evaluating configurations of buildings based on wind and view. The wind model is based on a CutFEM on overlapping meshes for Stokes equations. The second application, presented in Paper IV and Paper V, concerns a software application (app). The app lets a user define and solve physical problems governed by PDEs in an immersive and interactive augmented reality environment.
In the analysis part (Paper I and Paper II), we consider cut finite element methods on overlapping meshes for a time-dependent parabolic model problem: the heat equation on two overlapping meshes, where one mesh is allowed to move around on top of the other. In Paper I, the overlapping mesh is prescribed a cG(1) movement, meaning that its location as a function of time is continuous and piecewise linear. The cG(1) mesh movement results in a space-time discretization for which existing analysis methodologies either fail or are unsuitable. We therefore propose, to the best of our knowledge, a new energy analysis framework that is general enough to be applicable to the current setting. In Paper II, the overlapping mesh is prescribed a dG(0) movement, meaning that its location as a function of time is discontinuous and piecewise constant. The dG(0) mesh movement results in a space-time discretization for which existing analysis methodologies work with some modifications to handle the shift in the overlapping mesh's location at discrete times.
The applications part (Paper III, IV, and V) concerns cut finite element methods on overlapping meshes for stationary PDE-problems. We consider two potential applications for CutFEM on overlapping meshes. The first application, presented in Paper III, presents methodology for evaluating configurations of buildings based on wind and view. The wind model is based on a CutFEM on overlapping meshes for Stokes equations. The second application, presented in Paper IV and Paper V, concerns a software application (app). The app lets a user define and solve physical problems governed by PDEs in an immersive and interactive augmented reality environment.
parabolic problem
multi-objective optimization
multi-mesh
energy analysis
CutFEM
augmented reality
overlapping meshes
moving meshes
Sal MC, M-huset, Hörsalsvägen 5
Opponent: Senior researcher Miguel A. Fernández, Inria, Frankrike
Author
Carl Lundholm
Chalmers, Mathematical Sciences, Applied Mathematics and Statistics
Mats G. Larson, Anders Logg, Carl Lundholm A cut finite element method for the heat equation on overlapping meshes: Energy analysis for cG(1) mesh movement
Mats G. Larson, Carl Lundholm A cut finite element method for the heat equation on overlapping meshes: L2-analysis for dG(0) mesh movement
Anders Logg, Christian Valdemar Lorenzen, Carl Lundholm Multi-mesh multi-objective optimization with application to a model problem in urban design
Solving Poisson’s Equation on the Microsoft HoloLens
Proceedings of the ACM Symposium on Virtual Reality Software and Technology, VRST,;(2017)
Paper in proceeding
Finite element simulation of physical systems in augmented reality
Advances in Engineering Software,;Vol. 149(2020)
Journal article
A more efficient way to approximate physical phenomena
To mathematically describe physical phenomena such as heat transfer and fluid flow, one generally uses a certain type of equations called differential equations. Solving differential equations is therefore valuable for making adequate predictions of nature, which is important in many areas of science and engineering. However, differential equations can be quite hard or even impossible to solve exactly. A compromise is therefore to solve them approximatively instead. One way of doing this is with the finite element method (FEM).
A key component of FEM is the computational mesh. The mesh is a partition of the domain in which the differential equation is to be solved. If the domain changes, a new mesh might have to be generated which increases the computational cost. An alternative to costly remeshing is to use cut finite element methods (CutFEMs). With CutFEM, the mesh and the domain may be completely unrelated. This means that the same mesh may be used for changing domains. A way to increase the sophistication of CutFEM further is to use overlapping meshes, meaning a background mesh with one or more overlapping meshes on top of it.
This thesis presents both analysis and applications of CutFEMs on overlapping meshes. The analysis concerns time-dependent problems and the main contribution is a new analysis framework that is sufficiently robust to handle such problems. The applications concern stationary problems.
To mathematically describe physical phenomena such as heat transfer and fluid flow, one generally uses a certain type of equations called differential equations. Solving differential equations is therefore valuable for making adequate predictions of nature, which is important in many areas of science and engineering. However, differential equations can be quite hard or even impossible to solve exactly. A compromise is therefore to solve them approximatively instead. One way of doing this is with the finite element method (FEM).
A key component of FEM is the computational mesh. The mesh is a partition of the domain in which the differential equation is to be solved. If the domain changes, a new mesh might have to be generated which increases the computational cost. An alternative to costly remeshing is to use cut finite element methods (CutFEMs). With CutFEM, the mesh and the domain may be completely unrelated. This means that the same mesh may be used for changing domains. A way to increase the sophistication of CutFEM further is to use overlapping meshes, meaning a background mesh with one or more overlapping meshes on top of it.
This thesis presents both analysis and applications of CutFEMs on overlapping meshes. The analysis concerns time-dependent problems and the main contribution is a new analysis framework that is sufficiently robust to handle such problems. The applications concern stationary problems.
Areas of Advance
Building Futures (2010-2018)
Subject Categories
Computational Mathematics
ISBN
978-91-7905-519-6
Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 4986
Publisher
Chalmers
Sal MC, M-huset, Hörsalsvägen 5
Opponent: Senior researcher Miguel A. Fernández, Inria, Frankrike