Stabilised Fine Element Methods For Fictitious Domain Problems
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Stabilised Fine Element Methods for Fictitious Domain Problems
This thesis deals with the solution of the Laplace and heat equations on complicated domains. The approach follows the idea of the fictitious domain method, in which a larger (simpler) domain is introduced with the idea of avoiding the use of meshes that resolve the geometry. The first part of the thesis is dedicated to propose and analyse a new stabilised finite element method for the heat equation. The analysis, not available to date, is based on the introduction of a new projected initial condition that satisfies the boundary conditions of the original problem weakly. This allows us to prove inconditional stability and optimal convergence of the solution, thus avoiding the restriction linking the time discretisation and mesh width parameters present in previous references. In the second part of this thesis the methodology has been adapted and extended to cover the case in which the problem at hand is posed in a domain containing several inclusions of small size. For this case, the usual fictitious domain approach is no longer applicable, and then a new method that compensates for the lack of stability of the original one is proposed, analysed and tested numerically. The numerical analysis has been carried out for the steady state case, but its applicability to time dependent problems is sketched and shown by means of numerical experiments.
Geometrically Unfitted Finite Element Methods and Applications
This book provides a snapshot of the state of the art of the rapidly evolving field of integration of geometric data in finite element computations. The contributions to this volume, based on research presented at the UCL workshop on the topic in January 2016, include three review papers on core topics such as fictitious domain methods for elasticity, trace finite element methods for partial differential equations defined on surfaces, and Nitsche’s method for contact problems. Five chapters present original research articles on related theoretical topics, including Lagrange multiplier methods, interface problems, bulk-surface coupling, and approximation of partial differential equations on moving domains. Finally, two chapters discuss advanced applications such as crack propagation or flow in fractured poroelastic media. This is the first volume that provides a comprehensive overview of the field of unfitted finite element methods, including recent techniques such as cutFEM, traceFEM, ghost penalty, and augmented Lagrangian techniques. It is aimed at researchers in applied mathematics, scientific computing or computational engineering.
Fundamentals of Enriched Finite Element Methods
Fundamentals of Enriched Finite Element Methods provides an overview of the different enriched finite element methods, detailed instruction on their use, and also looks at their real-world applications, recommending in what situations they're best implemented. It starts with a concise background on the theory required to understand the underlying functioning principles behind enriched finite element methods before outlining detailed instruction on implementation of the techniques in standard displacement-based finite element codes. The strengths and weaknesses of each are discussed, as are computer implementation details, including a standalone generalized finite element package, written in Python. The applications of the methods to a range of scenarios, including multi-phase, fracture, multiscale, and immersed boundary (fictitious domain) problems are covered, and readers can find ready-to-use code, simulation videos, and other useful resources on the companion website to the book. - Reviews various enriched finite element methods, providing pros, cons, and scenarios forbest use - Provides step-by-step instruction on implementing these methods - Covers the theory of general and enriched finite element methods