Locking-free finite element methods for thin plates and shells /
In this dissertation two locking-free numerical methods for
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| Format: | Thesis Book |
| Language: | English |
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[Place of publication not identified] :
[publisher not identified] ;
1997.
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| Subjects: | |
| Online Access: | http://proxy.library.tamu.edu/login?url=http://proquest.umi.com/pqdweb?did=736584691&sid=1&Fmt=2&clientId=2945&RQT=309&VName=PQD |
| Summary: | In this dissertation two locking-free numerical methods for thin plates and shells are proposed. We start with an overview of the basic theory of finite element methods, including mixed methods and least squares methods. Locking phenomena are discussed, followed by a description of the framework of a popularly used technique to construct and analyze locking-free methods. The mathematical models of plates and shells are reviewed, with emphases on the Reissner-Mindlin plate model and the Naghdi shell model. The first method we propose is a negative norm least squares finite element method solving the Reissner- Mindlin model of thin plate bending problems. The reformulation of Brezzi and Fortin is employed to avoid locking. The two major advantages of this method are the simple, balanced elements and the natural, easy preconditioning. The construction of the discrete negative norm is discussed. Part of the Brezzi-Fortin reformulation is a Stokes-like problem with a singular perturbation. Although it is a saddle point problem, the finite element subspaces do not need to satisfy any inf-sup condition, stability is achieved by an inexpensive extra term in the scheme. To implement the second equation of the Stokes-like problem with the least squares method, a discrete L' norm is used. A quasi-optimal error estimate is proven for the Stokeslike problem and an error estimate result is presented for the complete system. The preconditioning method is a natural outcome of the stability theorem. Numerical experiments are reported. The second method is a mixed finite element method for the Naghdi shell model. It is believed to be the first mixed method mathematically proven to be free of locking, without any restrictive or unrealistic conditions. We give a locking free reformulation which avoids the presence of geometric coefficients multiplying any derivative terms in the off- diagonal bilinear forms. For this reformulation, we can prove a stability condition by choosing appropriate finite element spaces and using a new method to deal with the remaining geometric coefficients. This stability condition is proposed here in this thesis. It is weaker then the standard inf-sup (or LBB) condition, but sufficient to prove a quasi-optimal error estimate for many saddle point problems which have a lower-right diagonal term, including our reformulation of the Naghdi shell model. The error estimate is independent of the thickness, and quasi-optimal for the approximation spaces. |
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| Item Description: | Vita. "Major Subject: Mathematics". |
| Physical Description: | viii, 82 leaves : illustrations ; 28 cm. Issued also on microfiche from University Microfilms Inc. |
| Bibliography: | Includes bibliographical references: pages 77-80. |