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Finite element approximation of general second order hyperbolic equations
Dissertation (MSc)--University of Pretoria, 2011.
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| Format: | Thesis |
| Language: | English |
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University of Pretoria
2013
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| _version_ | 1867613467080392704 |
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| access_status_str | Open Access |
| author2 | Janse van Rensburg, N.F. (Nicolaas) |
| author_browse | Janse van Rensburg, N.F. (Nicolaas) |
| author_facet | Janse van Rensburg, N.F. (Nicolaas) |
| collection | Thesis |
| dc_rights_str_mv | © 2011, University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria. |
| description | Dissertation (MSc)--University of Pretoria, 2011. |
| format | Thesis |
| id | oai:repository.up.ac.za:2263/30641 |
| institution | University of Pretoria (South Africa) |
| language | English |
| last_indexed | 2026-06-10T12:36:36.461Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UPSpace — University of Pretoria Institutional Repository |
| publishDate | 2013 |
| publishDateRange | 2013 |
| publishDateSort | 2013 |
| publisher | University of Pretoria |
| publisherStr | University of Pretoria |
| record_format | dspace |
| source_str | UPSpace — University of Pretoria Institutional Repository |
| spelling | oai:repository.up.ac.za:2263/30641 Finite element approximation of general second order hyperbolic equations Janse van Rensburg, N.F. (Nicolaas) madelein.basson@up.ac.za Basson, Madelein UCTD Finite element method Wave equation Damping Galerkin approximation Timoshenko beam Convergence Second order hyperbolic equations Dissertation (MSc)--University of Pretoria, 2011. The vibration of elastic bodies and structures consisting of elastic bodies is an active research field in engineering. A mathematical model for such a vibrating system is a complex system of partial differential equations. In this dissertation only linear cases are considered and the variational form is of the same type as the variational form for the well known wave equation in each case. The generalized problem is to solve a second order differential equation for a function with values in a Hilbert space. This problem is referred to as the general second order hyperbolic equation in this dissertation. Obviously numerical approximation of solutions is inevitable. The finite element method proved to be ideal. It is used for steady-state problems, eigenvalue problems and dynamic problems. This dissertation is a study of the state of the theory for dynamic problems: convergence of the finite element approximation and error estimates. The approach is to split the approximation problem. First the convergence for the semi-discrete problem is considered followed by the convergence of the fully discrete approximation to the solution of the semi-discrete problem. The focus is on the (continuous) Galerkin method. However one chapter is devoted to the application of the Trotter-Kato theorem and brief discussions on the mixed finite element method and discontinuous Galerkin method are included. An in depth study was made of two articles and the relevant part of a PhD thesis. From a further five articles the useful parts (for this dissertation) were studied in depth. Five more articles were read but only to determine their contribution to the theory. In this dissertation the theory for the Galerkin method is presented as a unit; the material from the sources are merged, duplication eliminated and similarities and differences noted. By standardizing the notation and expanding proofs the presentation is (hopefully) more readable than the original publications. Other notable contributions in this dissertation are the generalization of certain assumptions for wider application, weakening of some assumptions, the generalization of the main result in the 1976 article of Baker, application of Dupont's method (1973) to the Timoshenko beam with boundary damping and application of the Trotter-Kato theorem to the general second order hyperbolic equation. The main conclusion is that the general results are not satisfactory. Apart from the application of the Trotter-Kato theorem, where error estimates are in general not available, the theory is not sufficiently general to cover all linear vibration models. It is also clear from examples that the assumptions in the theory are too restrictive. The dissertation does not only provide an overview of the theory of the Galerkin method for dynamic problems but the presentation provides a basis for future research. Mathematics and Applied Mathematics Restricted Natural and Agricultural Sciences 2013-09-09T07:20:58Z 2012-05-15 2013-09-09T07:20:58Z 2012-04-13 2011-12 2012-01-11 Dissertation Basson, M 2011, Finite element approximation of general second order hyperbolic equations, MSc dissertation, University of Pretoria, Pretoria, viewed yymmdd < http://upetd.up.ac.za/thesis/available/etd-01112012-152355 / > C12/4/52/gm http://hdl.handle.net/2263/30641 http://upetd.up.ac.za/thesis/available/etd-01112012-152355/ en © 2011, University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria. application/pdf University of Pretoria |
| spellingShingle | UCTD Finite element method Wave equation Damping Galerkin approximation Timoshenko beam Convergence Second order hyperbolic equations Finite element approximation of general second order hyperbolic equations |
| title | Finite element approximation of general second order hyperbolic equations |
| title_full | Finite element approximation of general second order hyperbolic equations |
| title_fullStr | Finite element approximation of general second order hyperbolic equations |
| title_full_unstemmed | Finite element approximation of general second order hyperbolic equations |
| title_short | Finite element approximation of general second order hyperbolic equations |
| title_sort | finite element approximation of general second order hyperbolic equations |
| topic | UCTD Finite element method Wave equation Damping Galerkin approximation Timoshenko beam Convergence Second order hyperbolic equations |
| url | http://hdl.handle.net/2263/30641 http://upetd.up.ac.za/thesis/available/etd-01112012-152355/ |