Multiscale Asymptotic Analysis and Numerical Simulation for the Second Order Helmholtz Equations with Rapidly Oscillating Coefficients Over General Convex Domains

Cao, Liqun; Cui, Junzhi; Dechao, Zhu

doi:10.1137/s0036142900376110

Cited by 54 publications

(12 citation statements)

References 11 publications

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“…Furthermore [10][11][12][13][14] gave specific computational methods, in which the rate control of honeycomb strength was based on the non-recoverable crush densification. In this paper we outline the development of soft solid concept(P2) for recoverable controlled fluid-structure interaction from 2-dimensional [3] to 3dimensional cases [1,10,15] in positive definite finite element analysis (FEA) [16] schemes.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear property of the visco-elastic-plastic material in the impact problem

Hou

2009

J. Shanghai Univ.(Engl. Ed.)

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Section: Introductionmentioning

confidence: 99%

Nonlinear property of the visco-elastic-plastic material in the impact problem

Hou

2009

J. Shanghai Univ.(Engl. Ed.)

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“…Though material heterogeneity has been examined in a number of models, such as fiber bundle model (DANIELS, 1945;COLEMAN, 1958;CURTIN, 1997), coupled pattern mapping model WEI et al, 2000), network model etc., the mechanisms underlying the catastrophic transition through various spatial scales have not been clearly revealed in computational simulations (CAO et al, 2002). Therefore, it is of the deepest interest to reveal the characteristic features in the catastrophe transition and to seek its underlying mechanism.…”

Section: Introductionmentioning

confidence: 99%

Catastrophic Rupture Induced Damage Coalescence in Heterogeneous Brittle Media

荣峰

汪海英

夏蒙棼

et al. 2006

Pure appl. geophys.

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In heterogeneous brittle media, the evolution of damage is strongly influenced by the multiscale coupling effect. To better understand this effect, we perform a detailed investigation of the damage evolution, with particular attention focused on the catastrophe transition. We use an adaptive multiscale finite-element model (MFEM) to simulate the damage evolution and the catastrophic failure of heterogeneous brittle media. Both plane stress and plane strain cases are investigated for a heterogeneous medium whose initial shear strength follows the Weibull distribution. Damage is induced through the application of the Coulomb failure criterion to each element, and the element mesh is refined where the failure criterion is met. We found that as damage accumulates, there is a stronger and stronger nonlinear increase in stress and the stress redistribution distance. The coupling of the dynamic stress redistribution and the heterogeneity at different scales result in an inverse cascade of damage cluster size, which represents rapid coalescence of damage at the catastrophe transition.

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“…For elliptic problems with multiple scale solutions that result from rapidly oscillating coefficients, this method has been shown to be effective in obtaining globally accurate solutions [Hou et al (1999); Hou and Wu (1997)]. Other related multiscale methods include those presented in Babuška et al [1994]; Brezzi and Russo [1994]; Hughes [1995]; Dorobantu and Engquist [1998]; Matache et al [2000]; Cao et al [2002]; and E and Engquist [2003]. Here, we apply the idea of building special basis functions to capture the correct local behavior to other more difficult situations.…”

Section: Introductionmentioning

confidence: 99%

Multiscale Numerical Methods for Singularly Perturbed Convection-Diffusion Equations

Park

Hou

2004

Int. J. Comput. Methods

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We present an efficient and robust approach in the finite element framework for numerical solutions that exhibit multiscale behavior, with applications to singularly perturbed convection-diffusion problems. The first type of equation we study is the convectiondominated convection-diffusion equation, with periodic or random coefficients; the second type of equation is an elliptic equation with singularities due to discontinuous coefficients and non-smooth boundaries. In both cases, standard methods for purely hyperbolic or elliptic problems perform poorly due to sharp boundary and internal layers in the solution.We propose a framework in which the finite element basis functions are designed to capture the local small-scale behavior correctly. When the structure of the layers can be determined locally, we apply the multiscale finite element method, in which we solve the corresponding homogeneous equation on each element to capture the small scale features of the differential operator. We demonstrate the effectiveness of this method by computing the enhanced diffusivity scaling for a passive scalar in the cellular flow. We also carry out the asymptotic error analysis for its convergence rate and perform numerical experiments for verification. For a random flow with nonlocal layer structure, we use a variational principle to gain additional information in our attempt to design asymptotic basis functions. We also apply the same framework for elliptic equations with discontinuous coefficients or non-smooth boundaries. In that case, we construct local basis function near singularities using infinite element method in order to resolve extreme singularity. Numerical results on problems with various singularities confirm the efficiency and accuracy of this approach.

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Multiscale Asymptotic Analysis and Numerical Simulation for the Second Order Helmholtz Equations with Rapidly Oscillating Coefficients Over General Convex Domains

Cited by 54 publications

References 11 publications

Nonlinear property of the visco-elastic-plastic material in the impact problem

Nonlinear property of the visco-elastic-plastic material in the impact problem

Catastrophic Rupture Induced Damage Coalescence in Heterogeneous Brittle Media

Multiscale Numerical Methods for Singularly Perturbed Convection-Diffusion Equations

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