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Sharif Rahman - One of the best experts on this subject based on the ideXlab platform.

  • Probabilistic Analysis of Dynamic Systems with Complex- valued Eigensolutions
    49th AIAA ASME ASCE AHS ASC Structures Structural Dynamics and Materials Conference <br> 16th AIAA ASME AHS Adaptive Structures Conference<br, 2008
    Co-Authors: Sharif Rahman
    Abstract:

    This article presents a dimensional decomposition method for calculating the probabilistic characteristics of complex-valued eigenvalues and eigenvectors of linear, stochastic, dynamic systems. The method involves a novel function decomposition allowing lower-dimensional approximations of Eigensolutions, Lagrange interpolation of lowerdimensional component functions, and Monte Carlo simulation. Compared with the commonly used perturbation method, neither the assumption of small input variability nor the calculation of the derivatives of Eigensolutions is required by the method developed. Results of numerical examples from linear stochastic dynamics indicate that the decomposition method provides excellent estimates of the moments and/or probability densities of eigenvalues and eigenvectors for various cases, including large statistical variations of input.

  • Stochastic dynamic systems with complex‐valued Eigensolutions
    International Journal for Numerical Methods in Engineering, 2007
    Co-Authors: Sharif Rahman
    Abstract:

    A dimensional decomposition method is presented for calculating the probabilistic characteristics of complex-valued eigenvalues and eigenvectors of linear, stochastic, dynamic systems. The method involves a function decomposition allowing lower-dimensional approximations of Eigensolutions, Lagrange interpolation of lower-dimensional component functions, and Monte Carlo simulation. Compared with the commonly used perturbation method, neither the assumption of small input variability nor the calculation of the derivatives of Eigensolutions is required by the method developed. Results of numerical examples from linear stochastic dynamics indicate that the decomposition method provides excellent estimates of the moments and/or probability densities of eigenvalues and eigenvectors for various cases including large statistical variations of input. Copyright © 2007 John Wiley & Sons, Ltd.

  • stochastic dynamic systems with complex valued Eigensolutions
    International Journal for Numerical Methods in Engineering, 2007
    Co-Authors: Sharif Rahman
    Abstract:

    A dimensional decomposition method is presented for calculating the probabilistic characteristics of complex-valued eigenvalues and eigenvectors of linear, stochastic, dynamic systems. The method involves a function decomposition allowing lower-dimensional approximations of Eigensolutions, Lagrange interpolation of lower-dimensional component functions, and Monte Carlo simulation. Compared with the commonly used perturbation method, neither the assumption of small input variability nor the calculation of the derivatives of Eigensolutions is required by the method developed. Results of numerical examples from linear stochastic dynamics indicate that the decomposition method provides excellent estimates of the moments and/or probability densities of eigenvalues and eigenvectors for various cases including large statistical variations of input. Copyright © 2007 John Wiley & Sons, Ltd.

Jeng-tzong Chen - One of the best experts on this subject based on the ideXlab platform.

  • True and Spurious Eigensolutions of an Elliptical Membrane by Using the Nondimensional Dynamic Influence Function Method
    Journal of Vibration and Acoustics, 2014
    Co-Authors: Jeng-tzong Chen, Ying-te Lee, Jia-wei Lee, Wen-che Lee
    Abstract:

    In this paper, we employ the nondimensional dynamic influence function (NDIF) method to solve the free vibration problem of an elliptical membrane. It is found that the spurious Eigensolutions appear in the Dirichlet problem by using the double-layer potential approach. Besides, the spurious Eigensolutions also occur in the Neumann problem if the single-layer potential approach is utilized. Owing to the appearance of spurious Eigensolutions accompanied with true Eigensolutions, singular value decomposition (SVD) updating techniques are employed to extract out true and spurious eigenvalues. Since the circulant property in the discrete system is broken, the analytical prediction for the spurious solution is achieved by using the indirect boundary integral formulation. To analytically study the eigenproblems containing the elliptical boundaries, the fundamental solution is expanded into a degenerate kernel by using the elliptical coordinates and the unknown coefficients are expanded by using the eigenfunction expansion. True and spurious eigenvalues are simultaneously found to be the zeros of the modified Mathieu functions of the first kind for the Dirichlet problem when using the single-layer potential formulation, while both true and spurious eigenvalues appear to be the zeros of the derivative of modified Mathieu function for the Neumann problem by using the double-layer potential formulation. By choosing only the imaginary-part kernel in the indirect boundary integral equation method (BIEM) to solve the eigenproblem of an elliptical membrane, spurious Eigensolutions also appear at the same position with those of NDIF since boundary distribution can be lumped. The NDIF method can be seen as a special case of the indirect BIEM by lumping the boundary distribution. Both the analytical study and the numerical experiments match well with the same true and spurious solutions. [DOI: 10.1115/1.4026354]

  • Analytical and numerical investigation for true and spurious Eigensolutions of an elliptical membrane using the real-part dual BIEM/BEM
    Meccanica, 2011
    Co-Authors: Jeng-tzong Chen, Jia-wei Lee, Shyue-yuh Leu
    Abstract:

    The paper performs analytical and numerical investigation of the true and spurious Eigensolutions of an elliptical membrane using the real-part boundary integral equation method (BIEM) following the successful work on a circular case by using the dual boundary element method (BEM) (Kuo et al. in Int. J. Numer. Methods Eng. 48:1401–1422, 2000). We extend to the elliptical case in this paper. To analytically study the eigenproblems of an elliptical membrane, the elliptical coordinates and Mathieu functions are adopted. The fundamental solution is expanded into the degenerate kernel by using the elliptical coordinates and the boundary densities are expanded by using the eigenfunction expansion. The Jacobian terms may exist in the degenerate kernel, boundary density and boundary contour integration but they can cancel each other out. Therefore, the orthogonal relations are reserved in the boundary contour integral. It is interesting to find that the BIEM using the real or the imaginary-part kernel to deal with an elliptical membrane yields spurious Eigensolutions. This finding agrees with those corresponding to the circular case. The spurious eigenvalues in the real-part BIEM are found to be the zeros of the mth-order (even or odd) modified Mathieu functions of the second kind or their derivatives. To verify this finding, the BEM is implemented. Furthermore, the commercial finite-element code ABAQUS is also utilized to provide Eigensolutions for comparisons. It is found that good agreement is obtained.

  • Eigensolutions of multiply connected membranes using the method of fundamental solutions
    Engineering Analysis With Boundary Elements, 2005
    Co-Authors: Jeng-tzong Chen, I. L. Chen, Ying-te Lee
    Abstract:

    In this paper, the method of fundamental solutions (MFS) of single and double-layer potential approaches for solving the eigenfrequencies of multiply connected membranes is proposed. By employing the fundamental solution, the coefficients of influence matrices are easily determined. The spurious eigensolution accompanied by the true eigensolution appears. It is found that the spurious eigensolution using the MFS depends on the location of the inner boundary where the sources are distributed. To verify this finding, the true and spurious eigenvalues in an annular domain are analytically studied using the degenerate kernels and circulants for an annular membrane. In order to obtain the true eigensolution, the singular value decomposition (SVD) updating techniques and the Burton and Miller method are utilized to filter out the spurious Eigensolutions. Two examples are demonstrated analytically and numerically to see the validity of the present method.

  • study on the true and spurious Eigensolutions of two dimensional cavities using the dual multiple reciprocity method
    Engineering Analysis With Boundary Elements, 2003
    Co-Authors: Jeng-tzong Chen, Shyhrong Kuo, I L Chung, C X Huang
    Abstract:

    In this paper, true and spurious Eigensolutions for a circular cavity using the dual multiple reciprocity method (MRM) are analytically derived and numerically verified by the developed program. The roots of spurious eigenequation are found analytically by using symbolic manipulation software. A more efficient method is proposed by choosing a fewer number of equations from the dual MRM instead of all of the equations in the dual MRM. Numerical experiments are performed by using dual MRM program for comparison purposes. A circular cavity of radius 1 m with Neumann boundary conditions is considered, and the results match very well between the theoretical prediction and the numerical experiments for the first four true eigenvalues and the first two spurious eigenvalues. Also, a noncircular case of square cavity is numerically implemented. The true Eigensolutions can be easily solved by the dual MRM program in conjunction with the singular value decomposition technique. At the same time, the boundary modes and the multiplicities of the true eigenvalues can also be determined.

  • Eigensolution of annular membrane using the method of fundamental solutions
    2003
    Co-Authors: Ying-te Lee, I. L. Chen, Jeng-tzong Chen
    Abstract:

    In this paper, the method of fundamental solutions (MFS) for solving the eigenfrequencies of annular membrane is proposed. By employing the fundamental solution, the coefficients of influence matrices are easily determined. The spurious eigensolution in conjunction with the true eigensolution appears. It is found that the spurious eigensolution using the MFS depends on the location of the inner boundary where the sources are distributed. To verify this finding, the true and spurious eigenvalues in an annular domain are analytically studied using the degenerate kernel and circulant. In order to obtain the true eigensolution, the singular value decomposition (SVD) updating technique and the Burton & Miller method are utilized to filter out the spurious Eigensolutions. One example is demonstrated analytically and numerically to see the validity of the present method.

Ga-ping Wang - One of the best experts on this subject based on the ideXlab platform.

  • Analytical and numerical methods of symplectic system for Stokes flow in two-dimensional rectangular domain
    Applied Mathematics and Mechanics, 2008
    Co-Authors: Ga-ping Wang, Faming Sun
    Abstract:

    In this paper, a new analytical method of symplectic system, Hamiltonian system, is introduced for solving the problem of the Stokes flow in a two-dimensional rectangular domain. In the system, the fundamental problem is reduced to an eigenvalue and eigensolution problem. The solution and boundary conditions can be expanded by Eigensolutions using adjoint relationships of the symplectic ortho-normalization between the Eigensolutions. A closed method of the symplectic eigensolution is presented based on completeness of the symplectic eigensolution space. The results show that fundamental flows can be described by zero eigenvalue Eigensolutions, and local effects by nonzero eigenvalue Eigensolutions. Numerical examples give various flows in a rectangular domain and show effectiveness of the method for solving a variety of problems. Meanwhile, the method can be used in solving other problems.

  • An application of the symplectic system in two-dimensional viscoelasticity
    International Journal of Engineering Science, 2006
    Co-Authors: Weixiang Zhang, Ga-ping Wang
    Abstract:

    This paper redescribes fundamental problem of the two-dimensional viscoelasticity in symplectic system. With the aid of the symplectic character and integral transformation, solutions of duality equations are obtained, or Saint-Venant solutions of extension and bend and local solutions of boundary effects. Thus the original problem is reduced to finding zero eigenvalue Eigensolutions and non-zero eigenvalue Eigensolutions. Meanwhile, adjoint relationships of the symplectic orthogonality in the Laplace domain are generalized to in the time domain. After obtaining fundamental Eigensolutions, the problem can be discussed in the eigensolution space of the time domain without the need of the Laplace transformation and inverse one. As its application, a direct method is shown and some examples are discussed, which reveal relations between the creep or relaxation and Eigensolutions. The symplectic method and numerical method provide an idea for other researching as well.

Ying-te Lee - One of the best experts on this subject based on the ideXlab platform.

  • True and Spurious Eigensolutions of an Elliptical Membrane by Using the Nondimensional Dynamic Influence Function Method
    Journal of Vibration and Acoustics, 2014
    Co-Authors: Jeng-tzong Chen, Ying-te Lee, Jia-wei Lee, Wen-che Lee
    Abstract:

    In this paper, we employ the nondimensional dynamic influence function (NDIF) method to solve the free vibration problem of an elliptical membrane. It is found that the spurious Eigensolutions appear in the Dirichlet problem by using the double-layer potential approach. Besides, the spurious Eigensolutions also occur in the Neumann problem if the single-layer potential approach is utilized. Owing to the appearance of spurious Eigensolutions accompanied with true Eigensolutions, singular value decomposition (SVD) updating techniques are employed to extract out true and spurious eigenvalues. Since the circulant property in the discrete system is broken, the analytical prediction for the spurious solution is achieved by using the indirect boundary integral formulation. To analytically study the eigenproblems containing the elliptical boundaries, the fundamental solution is expanded into a degenerate kernel by using the elliptical coordinates and the unknown coefficients are expanded by using the eigenfunction expansion. True and spurious eigenvalues are simultaneously found to be the zeros of the modified Mathieu functions of the first kind for the Dirichlet problem when using the single-layer potential formulation, while both true and spurious eigenvalues appear to be the zeros of the derivative of modified Mathieu function for the Neumann problem by using the double-layer potential formulation. By choosing only the imaginary-part kernel in the indirect boundary integral equation method (BIEM) to solve the eigenproblem of an elliptical membrane, spurious Eigensolutions also appear at the same position with those of NDIF since boundary distribution can be lumped. The NDIF method can be seen as a special case of the indirect BIEM by lumping the boundary distribution. Both the analytical study and the numerical experiments match well with the same true and spurious solutions. [DOI: 10.1115/1.4026354]

  • Eigensolutions of multiply connected membranes using the method of fundamental solutions
    Engineering Analysis With Boundary Elements, 2005
    Co-Authors: Jeng-tzong Chen, I. L. Chen, Ying-te Lee
    Abstract:

    In this paper, the method of fundamental solutions (MFS) of single and double-layer potential approaches for solving the eigenfrequencies of multiply connected membranes is proposed. By employing the fundamental solution, the coefficients of influence matrices are easily determined. The spurious eigensolution accompanied by the true eigensolution appears. It is found that the spurious eigensolution using the MFS depends on the location of the inner boundary where the sources are distributed. To verify this finding, the true and spurious eigenvalues in an annular domain are analytically studied using the degenerate kernels and circulants for an annular membrane. In order to obtain the true eigensolution, the singular value decomposition (SVD) updating techniques and the Burton and Miller method are utilized to filter out the spurious Eigensolutions. Two examples are demonstrated analytically and numerically to see the validity of the present method.

  • Eigensolution of annular membrane using the method of fundamental solutions
    2003
    Co-Authors: Ying-te Lee, I. L. Chen, Jeng-tzong Chen
    Abstract:

    In this paper, the method of fundamental solutions (MFS) for solving the eigenfrequencies of annular membrane is proposed. By employing the fundamental solution, the coefficients of influence matrices are easily determined. The spurious eigensolution in conjunction with the true eigensolution appears. It is found that the spurious eigensolution using the MFS depends on the location of the inner boundary where the sources are distributed. To verify this finding, the true and spurious eigenvalues in an annular domain are analytically studied using the degenerate kernel and circulant. In order to obtain the true eigensolution, the singular value decomposition (SVD) updating technique and the Burton & Miller method are utilized to filter out the spurious Eigensolutions. One example is demonstrated analytically and numerically to see the validity of the present method.

J. Kirkhope - One of the best experts on this subject based on the ideXlab platform.