A boundary condition in Padé series for frequency‐domain solution of wave propagation in unbounded domains

Song, Chongmin; Bazyar, Mohammad Hossein

doi:10.1002/nme.1852

Cited by 32 publications

(19 citation statements)

References 52 publications

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“…Figure 4 shows the pressure distribution along the dam height acting on the rigid dam, in which the pressure p = − with fluid density , = H /c and the magnitude of pressure | p| is normalized by a 0 H . Analytical solutions [25] and solutions from Equation (1) in the full matrix form are marked with circles and rectangles, respectively, while solutions from the diagonal Equation (20) are plotted by solid line. The three results are exactly the same.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…In numerical methods, dams are often discretized into solid finite elements by using finite element method (FEM), while the reservoir is either directly modeled by boundary 598 S. M. LI methods were used to evaluate the dynamic stiffness matrix, which avoid evaluating integral convolution equations, but evaluation errors increased with frequency increasing so that results at high frequencies were not acceptable [19]. To evaluate accurately high frequencies behaviors of the dynamic stiffness matrix, a Pade series was presented to analyze out-of-plane motion of circular cavity embedded in full-plane through using the SBFEM alone [20]. Good results were obtained at high frequencies, but results at low frequencies were inferior even if a high-order Pade series was used.…”

Section: Introductionmentioning

confidence: 99%

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Diagonalization procedure for scaled boundary finite element method in modeling semi‐infinite reservoir with uniform cross‐section

2009

Numerical Meth Engineering

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SUMMARYTo improve the ability of the scaled boundary finite element method (SBFEM) in the dynamic analysis of dam-reservoir interaction problems in the time domain, a diagonalization procedure was proposed, in which the SBFEM was used to model the reservoir with uniform cross-section. First, SBFEM formulations in the full matrix form in the frequency and time domains were outlined to describe the semi-infinite reservoir. No sediments and the reservoir bottom absorption were considered. Second, a generalized eigenproblem consisting of coefficient matrices of the SBFEM was constructed and analyzed to obtain corresponding eigenvalues and eigenvectors. Finally, using these eigenvalues and eigenvectors to normalize the SBFEM formulations yielded diagonal SBFEM formulations. A diagonal dynamic stiffness matrix and a diagonal dynamic mass matrix were derived. An efficient method was presented to evaluate them. In this method, no Riccati equation and Lyapunov equations needed solving and no Schur decomposition was required, which resulted in great computational costs saving. The correctness and efficiency of the diagonalization procedure were verified by numerical examples in the frequency and time domains, but the diagonalization procedure is only applicable for the SBFEM formulation whose scaling center is located at infinity.

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Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Diagonalization procedure for scaled boundary finite element method in modeling semi‐infinite reservoir with uniform cross‐section

2009

Numerical Meth Engineering

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“…Bazyar and Song extended the SBFEM to model time‐harmonic and transient response of non‐homogeneous elastic unbounded domains. They later developed new techniques to enhance the solution procedures for modeling unbounded domains in frequency and time domains . The method was later employed to model steady‐state seepage problems .…”

Section: Introductionmentioning

confidence: 99%

Transient seepage analysis in zoned anisotropic soils based on the scaled boundary finite‐element method

Bazyar

Talebi

2014

Num Anal Meth Geomechanics

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SUMMARY The scaled boundary finite‐element method, a semi‐analytical computational scheme primarily developed for dynamic stiffness of unbounded domains, is applied to the analysis of unsteady seepage flow problems. This method is based on the finite‐element technology and gains the advantages of the boundary element method as well. Only boundary of the domain is discretized, no fundamental solution is required and singularity problems can be modeled rigorously. Anisotropic and non‐homogeneous materials satisfying similarity are modeled with no additional efforts. In this study, firstly, formulation of the method for the transient seepage flow problems is derived followed by its solution procedures. The accuracy, simplicity and applicability of the method are demonstrated via four numerical examples of transient seepage flow – three of them are available in the literature. Homogenous, non‐homogenous, isotropic and anisotropic material properties are considered to show the versatility of the technique. Excellent agreement with the finite‐element method is observed. The method out‐performs the finite‐element method in modeling singularity points. Copyright © 2014 John Wiley & Sons, Ltd.

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“…Alternative procedures, which aim at avoiding the convolution integral altogether by developing the SBFEM directly in the time domain, have been proposed recently. Song and Bazyar presented a Padé approximation of the dynamic stiffness matrix of an unbounded medium in the frequency‐domain, which has a large range and a high rate of convergence. Bazyar and Song then developed a high‐order local transmitting boundary based on a continued‐fraction solution of the dynamic stiffness matrix and applied it to two‐dimensional problems.…”

Section: Introductionmentioning

confidence: 99%

A high‐order approach for modelling transient wave propagation problems using the scaled boundary finite element method

Chen

Birk

Song

et al. 2013

Numerical Meth Engineering

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A high-order time-domain approach for wave propagation in bounded and unbounded domains is proposed. It is based on the scaled boundary FEM, which excels in modelling unbounded domains and singularities. The dynamic stiffness matrices of bounded and unbounded domains are expressed as continued-fraction expansions, which leads to accurate results with only about three terms per wavelength. An improved continued-fraction approach for bounded domains is proposed, which yields numerically more robust time-domain formulations. The coefficient matrices of the corresponding continued-fraction expansion are determined recursively. The resulting solution is suitable for systems with many DOFs as it converges over the whole frequency range, even for high orders of expansion. A scheme for coupling the proposed improved high-order time-domain formulation for bounded domains with a high-order transmitting boundary suggested previously is also proposed. In the time-domain, the coupled model corresponds to equations of motion with symmetric, banded and frequency-independent coefficient matrices, which can be solved efficiently using standard time-integration schemes. Numerical examples for modal and time-domain analysis are presented to demonstrate the increased robustness, efficiency and accuracy of the proposed method.

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A boundary condition in Padé series for frequency‐domain solution of wave propagation in unbounded domains

Cited by 32 publications

References 52 publications

Diagonalization procedure for scaled boundary finite element method in modeling semi‐infinite reservoir with uniform cross‐section

Diagonalization procedure for scaled boundary finite element method in modeling semi‐infinite reservoir with uniform cross‐section

Transient seepage analysis in zoned anisotropic soils based on the scaled boundary finite‐element method

A high‐order approach for modelling transient wave propagation problems using the scaled boundary finite element method

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