Error Estimates for Finite Element Simulations Using Neural Networks

In this work residual error estimates are constructed using Neural Networks for Finite Element Method. These can be used to do adaptive mesh refinement. Two neural networks are developed the Multilayer Perceptron and the Transformer model. The error estimates are made for 1d poisson equations but the idea will generalise to higher dimensions as...

Finite Element Method for two dimensional Westervelt Equation

This essay shows a two dimensional implementation of the finite element method for the Westervelt equation. To do this the finite element method is first applied to the linear wave equation, then to non-linear diffusion and finally to the Westervelt equation. Both an element by element and a faster vectorized implementation are given for the...

Modelling Magnetic Phase Transitions: with Density Functional Theory and the Finite Element Method

The present thesis discusses two modelling endeavours that serve to provide insights into both the magnetic and structural dynamics associated with a first order magnetic phase transition. <br/>Firstly, density functional theory has been used to model the lithiation of a supercell of 2x2x2 the conventional unit cell of the ferromagnet Fe2P....

An elastic model for meshfitting for the Level-set Method

The numerical errors involving the application of the level-set method to a problem can be minimized by fitting a mesh (usually a triangular mesh in two dimensions) to the zero level-set curve defined by the level-set function. These numerical errors depend on two things: the size and skewness of the triangles in the mesh. To fit the mesh, Timo...

Meshfitting for the Level-set Method

In applying the level-set method in the context of a finite-element method, errors can be minimized by adjusting the mesh to the shape of the level-set curve. The size of the different types of errors that occur depend on the goodness of fit to the zero levelset curve, the skewness of the triangles and the size of the triangles of the mesh. To...

Finite Element Method Applied to the One-dimensional Westervelt Equation

In this thesis we researched the applicability, properties and efficiency of the finite element method to solve the one-dimensional Westervelt equation, which describes nonlinear plane wave propagation. The goal was to investigate whether this lesser-known solution method has advantages or disadvantages compared to more commonly used solution...