Use of trigonometric terms in the finite element method with application to vibrating membranes

Milsted, M.G.; Hutchinson, J. R.

doi:10.1016/s0022-460x(74)80089-1

Cited by 32 publications

(8 citation statements)

References 4 publications

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“…Exact solutions are available in the literature for this membrane [9]. In order to see the manner of convergence of the solutions, the membrane is discretized into one triangular Fourier p-element and the number of trigonometric terms p is varied.…”

Section: Resultsmentioning

confidence: 99%

“…Table 3 clearly shows that a very fast convergence from above to the exact values occurs as the number of trigonometric terms in increased from two to "ve and the values for p"5 are in excellent agreement with the exact ones. In Table 3, there is an obvious typographical error in reference [9]. The exact value for the 10th frequency parameter was given as 19)120 when it should have been 19)110.…”

Section: Resultsmentioning

confidence: 99%

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A TRIANGULAR FOURIER p -ELEMENT FOR THE ANALYSIS OF MEMBRANE VIBRATIONS

Houmat

2000

Journal of Sound and Vibration

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

A TRIANGULAR FOURIER p -ELEMENT FOR THE ANALYSIS OF MEMBRANE VIBRATIONS

Houmat

2000

Journal of Sound and Vibration

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“…Notice that the potential part of the Hamiltonian is obtained by simply "collocating" the potential V (x, y) on the grid, an operation with a limited computational price. The result shown in (10) corresponds to the matrix element of the Hamiltonian operator Ĥ between two grid points, (k, k ) and (j, j ), which can be selected using two integer values K and J, as shown in (6). Following this procedure the solution of the Schrödinger (Helmholtz) equation on the uniform grid generated by the LSF corresponds to the diagonalization of a (N − 1) 2 × (N − 1) 2 square matrix, whose elements are given by eq.…”

Section: The Methodsmentioning

confidence: 99%

Solving the Helmholtz equation for membranes of arbitrary shape

Amore

2008

Preprint

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I calculate the modes of vibration of membranes of arbitrary shape using a collocation approach based on Little Sinc Functions. The matrix representation of the PDE obtained using this method is explicit and it does not require the calculation of integrals. To illustrate the virtues of this approach, I have considered a large number of examples, part of them taken from the literature, and part of them new. When possible, I have tested the accuracy of these results by comparing them with the exact results (when available) or with results from the literature. In particular, in the case of the L-shaped membrane, the first example discussed in the paper, I show that it is possible to extrapolate the results obtained with different grid sizes to obtain higly precise results. Finally, I also show that the present collocation technique can be easily combined with conformal mapping to provide numerical approximations to the energies which quite rapidly converge to the exact results.

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“…The finite-element method has been used in [98] for Land H-shapes, in [20] for circular and square waveguides with quadruple ridges, in 126] for various crosses, in [65] for "bent" waveguides (L-shapes at angles other then 90), in [82] for rectangles with rounded corners, in [32 for shapes given by polar coordinates such as portions of spirals, and in [89] for limaons and cardioids. It is also used in the interesting paper [84] where mode shapes are followed as parameters are varied to change rectangles into ellipses via hyperellipses and into parabolas via superellipses.…”

Section: I=1mentioning

confidence: 99%

Eigenvalues of the Laplacian in Two Dimensions

Kuttler¹,

Sigillito²

1984

SIAM Rev.

232

159

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The eigenvalue problem for the Laplace operator in two dimensions is classical in mathematics and physics. Nevertheless, computational methods for estimating the eigenvalues are still of much current interest, particularly in applications to acoustic and electromagnetic waveguides. Although our primary interest is with the computational methods, there are a number of theoretical results on the behavior of the eigenvalues and eigenfunctions that are useful for understanding the methods and, in addition, are of interest in themselves. These results are discussed first and then the various computational methods that have been used to estimate the eigenvalues are reviewed with particular emphasis on methods that give error bounds. Some of the more powerful techniques available are illustrated by applying them to a model problem. 3. Elementary solutions. We quote Rayleigh 128 who says, "The theory of the free vibrations of a membrane was first successfully considered by Poisson 116]. His theory in the case of the rectangle left little to be desired." For the rectangle 0 _< x _< a, 0 <_ y <_ b, the eigenfunctions are

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Use of trigonometric terms in the finite element method with application to vibrating membranes

Cited by 32 publications

References 4 publications

A TRIANGULAR FOURIER p -ELEMENT FOR THE ANALYSIS OF MEMBRANE VIBRATIONS

A TRIANGULAR FOURIER p -ELEMENT FOR THE ANALYSIS OF MEMBRANE VIBRATIONS

Solving the Helmholtz equation for membranes of arbitrary shape

Eigenvalues of the Laplacian in Two Dimensions

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