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Galerkin Difference Methods and Appl...
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Jacangelo, John.
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Galerkin Difference Methods and Applications to Wave Equations.
紀錄類型:
書目-電子資源 : Monograph/item
正題名/作者:
Galerkin Difference Methods and Applications to Wave Equations./
作者:
Jacangelo, John.
出版者:
Ann Arbor : ProQuest Dissertations & Theses, : 2019,
面頁冊數:
125 p.
附註:
Source: Dissertations Abstracts International, Volume: 81-02, Section: B.
Contained By:
Dissertations Abstracts International81-02B.
標題:
Applied mathematics. -
電子資源:
http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=13806695
ISBN:
9781085564236
Galerkin Difference Methods and Applications to Wave Equations.
Jacangelo, John.
Galerkin Difference Methods and Applications to Wave Equations.
- Ann Arbor : ProQuest Dissertations & Theses, 2019 - 125 p.
Source: Dissertations Abstracts International, Volume: 81-02, Section: B.
Thesis (Ph.D.)--Rensselaer Polytechnic Institute, 2019.
This item must not be sold to any third party vendors.
The Galerkin Difference (GD) method, a finite element method built using standard Galerkin projection but employing nonstandard basis functions, was originally developed for one space dimension in [J. W. Banks and T. Hagstrom, On Galerkin difference methods, J. Comput. Phys., 313 (2016), pp. 310-327]. The C0 basis was derived by considering standard piecewise continuous polynomial interpolation. The resulting GD approximations were found to have excellent properties both in terms of their accuracy and computational efficiency. Here the method is extended to two space dimensions and to higher derivative operators.To reach two space dimensions, a tensor product construction is leveraged. Theoretical and computational evidence shows the method behaves as expected for the acoustic wave equation. For the elastic wave equation, the approximations are found to be at least as accurate as predicted. In the special case of a free surface the scheme is more accurate than expected, exhibiting unexpected superconvergence. Extension to curvilinear mapped grids is also considered for acoustics. In all cases, the use of a tensor product construction allows for efficient solution of the linear system involving the mass matrix, which implies optimal linear time solutions with respect to the number of degrees of freedom.Additionally, we further extend GD by considering higher derivative operators, such as those commonly found in beam or plate models of solid mechanics. These higher-order PDEs necessitate higher continuity basis functions. To derive this smoother basis, we introduce the Difference Spline, and subsequently the d-Difference Spline, which is a locally constructed C1 (Cd) polynomial interpolant using only discrete data at p + 1 consecutive grid points. We show that the D-1-Spline interpolant is a p-th order accurate approximation, and basisfunctions associated with each grid point are derived. The basis is then used in a standard weak-form finite element approximation of the PDEs, and classical finite element theory shows that the method is p-th order accurate in the L2 norm. Numerical convergence studiesin the Euler-Bernoulli Beam and the Kirchhoff-Love Plate are preformed and verify the theory.
ISBN: 9781085564236Subjects--Topical Terms:
2122814
Applied mathematics.
Subjects--Index Terms:
Biharmonic equations
Galerkin Difference Methods and Applications to Wave Equations.
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The Galerkin Difference (GD) method, a finite element method built using standard Galerkin projection but employing nonstandard basis functions, was originally developed for one space dimension in [J. W. Banks and T. Hagstrom, On Galerkin difference methods, J. Comput. Phys., 313 (2016), pp. 310-327]. The C0 basis was derived by considering standard piecewise continuous polynomial interpolation. The resulting GD approximations were found to have excellent properties both in terms of their accuracy and computational efficiency. Here the method is extended to two space dimensions and to higher derivative operators.To reach two space dimensions, a tensor product construction is leveraged. Theoretical and computational evidence shows the method behaves as expected for the acoustic wave equation. For the elastic wave equation, the approximations are found to be at least as accurate as predicted. In the special case of a free surface the scheme is more accurate than expected, exhibiting unexpected superconvergence. Extension to curvilinear mapped grids is also considered for acoustics. In all cases, the use of a tensor product construction allows for efficient solution of the linear system involving the mass matrix, which implies optimal linear time solutions with respect to the number of degrees of freedom.Additionally, we further extend GD by considering higher derivative operators, such as those commonly found in beam or plate models of solid mechanics. These higher-order PDEs necessitate higher continuity basis functions. To derive this smoother basis, we introduce the Difference Spline, and subsequently the d-Difference Spline, which is a locally constructed C1 (Cd) polynomial interpolant using only discrete data at p + 1 consecutive grid points. We show that the D-1-Spline interpolant is a p-th order accurate approximation, and basisfunctions associated with each grid point are derived. The basis is then used in a standard weak-form finite element approximation of the PDEs, and classical finite element theory shows that the method is p-th order accurate in the L2 norm. Numerical convergence studiesin the Euler-Bernoulli Beam and the Kirchhoff-Love Plate are preformed and verify the theory.
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