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Adapting meshfree Galerkin schemes for representing highly anisotropic fields
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Maloney, Samuel A. (2023) Adapting meshfree Galerkin schemes for representing highly anisotropic fields. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3985079
Abstract
Two methods for representing highly anisotropic fields are presented, based on partially and fully meshfree Galerkin formulations. In both, a mapping function is used to provide information about the local direction of the anisotropy, with one of the global coordinates chosen to parameterise the ‘parallel’ position along the mapping in a one-to-one manner. In the first method, dubbed FCIFEM, standard unstructured finite element meshes are used on planes of constant parallel coordinate to represent the necessary small-scale variations perpendicular to the mapping direction, with large spacings then possible between these planes because of the small variation along the mapping. This greatly reduces the number of degrees of freedom required to represent fields in this space and the associated computational cost of simulations involving such fields. No mesh connectivity is defined between planes, and field-aligned basis functions are constructed using the mapping function to extend the standard finite element bases into the full domain. In the second method, dubbed FCIMLS, the meshfree moving least squares (MLS) formalism is used to compute bases capable of representing arbitrarily high-order polynomials. A similar arrangement of nodes into distantly spaced planes is used to help ensure full domain coverage, but now with no defined mesh connectivity required between any nodes. Integration of the bases is addressed with reference to methods developed for other fully meshfree methods, and the schemes (as well as other similar element-free Galerkin schemes) are shown to be locally conservative under certain conditions. Prototype code is developed in Python and used to validate the methods with robust convergence of several test problems being demonstrated. A significant decrease in the number of degrees of freedom required for a given level of accuracy is achieved for model problems with a moderate degree of anisotropy.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QC Physics | ||||
Library of Congress Subject Headings (LCSH): | Anisotropy, Galerkin methods, Meshfree methods (Numerical analysis), Finite element method | ||||
Official Date: | October 2023 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Department of Physics | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | McMillan, Ben F. | ||||
Format of File: | |||||
Extent: | xi, 96 pages : illustrations | ||||
Language: | eng |
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