A space-time finite element method for structural acoustics in infinite domains part 1: Formulation, stability and convergence

Thompson, Lonny L.; Pinsky, Peter M.

doi:10.1016/0045-7825(95)00955-8

Cited by 37 publications

(32 citation statements)

References 26 publications

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“…Essentially, the continuity of u andu is weakly enforced by means of energy inner products (cf. [13] and [22] for details). Hence, the variational form of the problem given in Equations (1)-(3) may be written as follows.…”

Section: Variational Formulationmentioning

confidence: 98%

“…Therein, unconditional stability has been proved for a Galerkin least-squares version of this method for elastodynamic problems. In [22], it has been extended to exterior Helmholtz problems. The governing differential equation is multiplied by time derivatives of the weighting functions w and integrated over each space-time slab.…”

Section: Variational Formulationmentioning

confidence: 99%

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A space–time discontinuous Galerkin method for the solution of the wave equation in the time domain

Petersen

Farhat

Tezaur

2008

Numerical Meth Engineering

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SUMMARYIn recent years, the focus of research in the field of computational acoustics has shifted to the medium frequency regime and multiscale wave propagation. This has led to the development of new concepts including the discontinuous enrichment method. Its basic principle is the incorporation of features of the governing partial differential equation in the approximation. In this contribution, this concept is adapted for the simulation of transient problems governed by the wave equation. We present a space-time discontinuous Galerkin method with Lagrange multipliers, where the shape approximation in space and time is based on solutions of the homogeneous wave equation. The use of hierarchical wave-like basis functions is enabled by means of a variational formulation that allows for discontinuities in both the spatial and the temporal discretizations. Numerical examples in one space dimension demonstrate the outstanding performance of the proposed method compared with conventional space-time finite element methods.

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Section: Variational Formulationmentioning

confidence: 98%

Section: Variational Formulationmentioning

confidence: 99%

A space–time discontinuous Galerkin method for the solution of the wave equation in the time domain

Petersen

Farhat

Tezaur

2008

Numerical Meth Engineering

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“…One of the reasons for that is that low-order ÿnite element methods exhibit poor dispersion properties [10], while higher-order classical ÿnite elements raise some troublesome problems like the occurrence of spurious waves. Recently, space-time and Galerkin= least-squares ÿnite element methods, related to Hamilton's principle, have been introduced both for acoustic and full elastic wave propagation [11][12][13] with some success, even though application of these methods to realistic seismological problems have still to be shown.…”

Section: Introductionmentioning

confidence: 99%

The spectral element method for elastic wave equations—application to 2-D and 3-D seismic problems

Komatitsch¹,

Vilotte²,

Vai³

et al. 1999

Int. J. Numer. Meth. Engng.

168

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SUMMARYA spectral element method for the approximate solution of linear elastodynamic equations, set in a weak form, is shown to provide an e cient tool for simulating elastic wave propagation in realistic geological structures in two-and three-dimensional geometries. The computational domain is discretized into quadrangles, or hexahedra, deÿned with respect to a reference unit domain by an invertible local mapping. Inside each reference element, the numerical integration is based on the tensor-product of a Gauss -Lobatto -Legendre 1-D quadrature and the solution is expanded onto a discrete polynomial basis using Lagrange interpolants. As a result, the mass matrix is always diagonal, which drastically reduces the computational cost and allows an e cient parallel implementation. Absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains. The time discretization is based on an energy-momentum conserving scheme that can be put into a classical explicit-implicit predictor=multicorrector format. Long term energy conservation and stability properties are illustrated as well as the e ciency of the absorbing conditions. The accuracy of the method is shown by comparing the spectral element results to numerical solutions of some classical two-dimensional problems obtained by other methods. The potentiality of the method is then illustrated by studying a simple three-dimensional model. Very accurate modelling of Rayleigh wave propagation and surface di raction is obtained at a low computational cost. The method is shown to provide an e cient tool to study the di raction of elastic waves and the large ampliÿcation of ground motion caused by three-dimensional surface topographies.

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“…One could, for example, use boundary elements of infinite size (but with an artificial, finite, centroid), or use mapping/squashing functions to map an infinite space into a new, finite one. These and other strategies put forward in the FEM community (e.g., [1,7,6,17]), however, are beyond the scope of this article.…”

Section: Exact Markov Model Of Approximate Optimisermentioning

confidence: 99%

Continuous Optimisation Theory Made Easy? Finite-Element Models of Evolutionary Strategies, Genetic Algorithms and Particle Swarm Optimizers

Poli¹,

Langdon²,

Clerc³

et al. 2007

Foundations of Genetic Algorithms

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Abstract. We propose a method to build discrete Markov chain models of continuous stochastic optimisers that can approximate them on arbitrary continuous problems to any precision. We discretise the objective function using a finite element method grid which produces corresponding distinct states in the search algorithm. Iterating the transition matrix gives precise information about the behaviour of the optimiser at each generation, including the probability of it finding the global optima or being deceived. The approach is tested on a (1+1)-ES, a bare bones PSO and a real-valued GA. The predictions are remarkably accurate.

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A space-time finite element method for structural acoustics in infinite domains part 1: Formulation, stability and convergence

Cited by 37 publications

References 26 publications

A space–time discontinuous Galerkin method for the solution of the wave equation in the time domain

A space–time discontinuous Galerkin method for the solution of the wave equation in the time domain

The spectral element method for elastic wave equations—application to 2-D and 3-D seismic problems

Continuous Optimisation Theory Made Easy? Finite-Element Models of Evolutionary Strategies, Genetic Algorithms and Particle Swarm Optimizers

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