Method of Fundamental Solutions for Plate Vibrations in Multiply Connected Domains

Tsai, Chia‐Cheng; Young, D. L.; Fan, Chia-Ming

doi:10.1017/s1727719100000885

Cited by 10 publications

(6 citation statements)

References 22 publications

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“…It is quite important for us to find a stable and efficient numerical algorithm to solve Eq. (15). Remember that the concerned problem here is an inverse problem which is highly ill-posed such that we need to use some numerical techniques to avoid possible numerical instability.…”

Section: The Exponentially Convergent Scalar Homotopy Algorithm (Ecsha)mentioning

confidence: 99%

“…In order to avoid the mesh generation and numerical integration, many meshless methods are proposed recently, such as the MFS [3,6,[13][14][15][16], the radial basis function collocation methods [17][18][19], the MCTM [20][21][22][23][24][25], etc. For the current problem, since positions of some boundary points are unknown, it is then nature for us to adopt the boundary-type meshless scheme to solve the problem.…”

Section: Introductionmentioning

confidence: 99%

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The Modified Collocation Trefftz Method and Exponentially Convergent Scalar Homotopy Algorithm for the Inverse Boundary Determination Problem for the Biharmonic Equation

Chan

Fan

2013

J. mech.

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In this paper, the modified collocation Trefftz method (MCTM) and the exponentially convergent scalar homotopy algorithm (ECSHA) are adopted to analyze the inverse boundary determination problem governed by the biharmonic equation. The position for part of the boundary with given boundary condition is unknown and the position for the rest of the boundary with overspecified Cauchy boundary conditions is given a priori. Since the spatial position for portion of boundary is not given a priori, it is extremely difficult to solve such a boundary determination problem by any numerical scheme. In order to stably solve the boundary determination problem, the MCTM will be adopted in this study owing to that it can avoid the generation of mesh grid and numerical integration. When this problem is modeled by MCTM, a system of nonlinear algebraic equations will be formed and then be solved by ECSHA. Some numerical examples will be provided to demonstrate the ability and accuracy of the proposed scheme. In addition, the stability of the proposed meshless method will be tested by adding some noise into the prescribed boundary conditions and then to see how does that affect the numerical results.

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Section: The Exponentially Convergent Scalar Homotopy Algorithm (Ecsha)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Modified Collocation Trefftz Method and Exponentially Convergent Scalar Homotopy Algorithm for the Inverse Boundary Determination Problem for the Biharmonic Equation

Chan

Fan

2013

J. mech.

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“…In their work, the moving least-squares (MLS) interpolation was used to construct shape functions through a set of nodes arbitrarily distributed in the domain. Tsai et al [9] applied the method of fundamental solutions (MFS) to solve eigenfrequencies of plate vibration of multiply connected domains. The complex-valued MFS combined with the mix potential methods were utilized to avoid the spurious eigenvalues in their paper.…”

Section: Introductionmentioning

confidence: 99%

Mixed boundary node method for free vibration analysis of rectangular plates with variable thickness and general boundary conditions

Huang¹,

Jian²,

Tang³

et al. 2013

Boundary Elements and Other Mesh Reduction Methods XXXVI

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A mixed boundary node method is proposed for analyzing the free vibration of rectangular plates with variable thickness and general boundary conditions. The fundamental differential equations expressed by mixed variables are established and the unknown variables exist only on the partial boundary nodes. The relationship between the variables on the internal nodes and the pointed boundary nodes can be determined through using local regional integration and scanning technology. By utilizing boundary conditions, the discrete solutions for deflection of the plate with a concentrated load and arbitrary variable thickness are obtained and used to establish the eigen-value problem of matrix of the free vibration problem of plate. The convergence and accuracy of the numerical solutions for the natural frequency parameter calculated by the proposed method are investigated. The frequency parameters and their modes of free vibration are shown for some rectangular plates.

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“…Detailed descriptions and comparisons of the MQ method and the MFS can be found in [9]. Specially, the MFS performed better for eigenproblems [10,11].…”

Section: Introductionmentioning

confidence: 99%

The Method of Fundamental Solutions with Dual Reciprocity for thin Plates on Winkler Foundations with Arbitrary Loadings

Tsai

2008

J. mech.

Self Cite

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This paper describes the combination of the method of fundamental solutions (MFS) and the dual reciprocity method (DRM) as a meshless numerical method to solve problems of thin plates resting on Winkler foundations under arbitrary loadings, where the DRM is based on the augmented polyharmonic splines constructed by splines and monomials. In the solution procedure, the arbitrary distributed loading is first approximated by the augmented polyharmonic splines (APS) and thus the desired particular solution can be represented by the corresponding analytical particular solutions of the APS. Thereafter, the complementary solution is solved formally by the MFS. In the mathematical derivations, the real coefficient operator in the governing equation is decomposed into two complex coefficient operators. In other words, the solutions obtained by the MFS-DRM are first treated in terms of these complex coefficient operators and then converted to real numbers in suitable ways. Furthermore, the boundary conditions of lateral displacement, slope, normal moment, and effective shear force are all given explicitly for the particular solutions of APS as well as the kernels of MFS. Finally, numerical experiments are carried out to validate these analytical formulas.

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Method of Fundamental Solutions for Plate Vibrations in Multiply Connected Domains

Cited by 10 publications

References 22 publications

The Modified Collocation Trefftz Method and Exponentially Convergent Scalar Homotopy Algorithm for the Inverse Boundary Determination Problem for the Biharmonic Equation

The Modified Collocation Trefftz Method and Exponentially Convergent Scalar Homotopy Algorithm for the Inverse Boundary Determination Problem for the Biharmonic Equation

Mixed boundary node method for free vibration analysis of rectangular plates with variable thickness and general boundary conditions

The Method of Fundamental Solutions with Dual Reciprocity for thin Plates on Winkler Foundations with Arbitrary Loadings

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