Implementation and adaptivity of a space-time finite element method for structural dynamics

Li, Xiangdong; Wiberg, Nils‐Erik

doi:10.1016/s0045-7825(97)00207-7

Cited by 111 publications

(80 citation statements)

References 17 publications

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“…Then, the time-discrete variational problem seeks a pair {u for all test pairs {ϕ 1 , ϕ 0 } ∈ W k × W k , where in this particular case we can take W k := V k . Again by adding up the two equations (3.26), we obtain a compact expression for the semidiscrete equations: 27) with the bilinear form and force term defined, respectively, as follows:…”

Section: )mentioning

confidence: 99%

“…For the approximation of problem (3.17) by the cG(1)/dG(0) method (3.31), we have the following a posteriori error representation: 27) with arbitrary {ϕ The proof follows the same line of argument as that of Corollary 4.3 and is therefore omitted. From this error representation we can then derive an a posteriori error estimate analogous to the one derived before in Corollary 4.4 for the cG(1)/cG(1) method.…”

Section: 22mentioning

confidence: 99%

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Adaptive Galerkin Finite Element Methods for the Wave Equation

Bangerth

Geiger

Rannacher

2010

Comput. Methods Appl. Math.

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-This paper gives an overview of adaptive discretization methods for linear second-order hyperbolic problems such as the acoustic or the elastic wave equation. The emphasis is on Galerkin-type methods for spatial as well as temporal discretization, which also include variants of the Crank-Nicolson and the Newmark finite difference schemes. The adaptive choice of space and time meshes follows the principle of "goaloriented" adaptivity which is based on a posteriori error estimation employing the solutions of auxiliary dual problems.2000 Mathematics Subject Classification: 35L05, 65M50, 65M60, 74S05.

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Section: )mentioning

confidence: 99%

Section: 22mentioning

confidence: 99%

Adaptive Galerkin Finite Element Methods for the Wave Equation

Bangerth

Geiger

Rannacher

2010

Comput. Methods Appl. Math.

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“…Commonly used adaptive algorithms for time dependent problems, see, e.g., [3,50], perform an adaptive refinement process using a prescribed tolerance in every time step. This refinement process is independent of previous and subsequent time steps.…”

Section: Adaptive Algorithmmentioning

confidence: 99%

“…In general, a posteriori error estimates for second order hyperbolic problems are possible for two different discretisation approaches. One of them uses space time Galerkin methods for discretisation and applies similar techniques for error control as in the static case ( [2][3][4][5]). The other one is based on finite differences in time and finite elements in space.…”

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confidence: 99%

Space adaptive finite element methods for dynamic Signorini problems

2009

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Space adaptive techniques for dynamic Signorini problems are discussed. For discretisation, the Newmark method in time and low order finite elements in space are used. For the global discretisation error in space, an a posteriori error estimate is derived on the basis of the semidiscrete problem in mixed form. This approach relies on an auxiliary problem, which takes the form of a variational equation. An adaptive method based on the estimate is applied to improve the finite element approximation. Numerical results illustrate the performance of the presented method.

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“…The first approach is spacetime finite element method (1) - (4) , in which the spatial and temporal domains are simultaneously discretized and the basis functions are generally adopted to be continuous in space and discontinuous at the discrete time levels. These methods are generally used to solve problems exhibiting discontinuities or sharp gradients in their solutions.…”

Section: Introductionmentioning

confidence: 99%

A Step-by-Step Time Integration Method Based on the Principle of Minimum Transformed Energy

Liu

Zhang

et al. 2005

JSME Int. J., Ser. C

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A step-by-step time integration method is presented for dynamic response analysis based on Benthien-Gurtin's principle of minimum transformed energy in linear elastodynamics. First of all, a single convolution-type (Gurtin-type) functional in the real space is obtained from the functional in Laplace space. Then the concrete functional, that can be used to construct time integration methods by adopting some interpolation functions, is established after successively spatial and temporal discretization. The cubic Hermite interpolation functions in temporal domain are adopted to show the procedure of constructing time integration method and the new numerical method is developed finally. The specific stability characteristics about the unconditional and conditional stabilities of the numerical method are investigated in detail. The numerical examples show that the algorithm is of satisfying accuracy and is an effective method for the numerical calculations of dynamic response problems in engineering.

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Implementation and adaptivity of a space-time finite element method for structural dynamics

Cited by 111 publications

References 17 publications

Adaptive Galerkin Finite Element Methods for the Wave Equation

Adaptive Galerkin Finite Element Methods for the Wave Equation

Space adaptive finite element methods for dynamic Signorini problems

A Step-by-Step Time Integration Method Based on the Principle of Minimum Transformed Energy

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