Dynamic analysis of structural systems using component modes

Hurty, Walter C.

doi:10.2514/3.2947

Cited by 1,028 publications

(298 citation statements)

References 7 publications

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“…The CMS is as a substructuring method which was already developed in 1960 by Hurty [44] for the solution of large scale eigenvalue problems arising in structural engineering analysis. The idea behind CMS is to substructure the spatial domain of the PDE eigenvalue problem into subdomains, and to approximate the sought eigensolutions of the global problem by using eigensolutions of problems that are defined on the smaller subdomains.…”

Section: Automated Multi-level Substructuringmentioning

confidence: 99%

Solving an elliptic PDE eigenvalue problem via automated multi-level substructuring and hierarchical matrices

Gerds

Grasedyck

2013

Comput. Visual Sci.

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To solve an elliptic PDE eigenvalue problem in practice, typically the finite element discretisation is used. From approximation theory it is known that only the smaller eigenvalues and their corresponding eigenfunctions can be well approximated by the finite element discretisation because the approximation error increases with increasing size of the eigenvalue. The number of well approximable eigenvalues or eigenfunctions, however, is unknown. In this work asymptotic estimates of these quantities are derived. For example, it is shown that for three-dimensional problems under certain smoothness assumptions on the data only the smallest Θ(N 2/5 ) eigenvalues and only the eigenfunctions associated to the smallest Θ(N 1/4 ) eigenvalues can be well approximated by the finite element discretisation, when the N -dimensional finite element spaces of piecewise affine functions with uniform mesh refinement are used.To solve the discretised elliptic PDE eigenvalue problem and to compute all well approximable eigenvalues and eigenfunctions, a new method is introduced which combines a recursive version of the automated multi-level substructuring (short AMLS) method with the concept of hierarchical matrices (short H-matrices). AMLS is a domain decomposition technique for the solution of elliptic PDE eigenvalue problems where, after some transformation, a reduced eigenvalue problem is derived whose eigensolutions deliver approximations of the sought eigensolutions of the original problem.Whereas the classical AMLS method is very efficient for elliptic PDE eigenvalue problems posed in two dimensions, it is getting very expensive for three-dimensional problems, due to the fact that it computes the reduced eigenvalue problem via dense matrix operations. This problem is resolved by the use of hierarchical matrices. H-matrices are a data-sparse approximation of dense matrices which, e.g., result from the inversion of the stiffness matrix that is associated to the finite element discretisation of an elliptic PDE operator. The big advantage of H-matrices is that they provide matrix arithmetic with almost linear complexity. This fast H-matrix arithmetic is used for the computation of the reduced eigenvalue problem. Beside this, the size of the reduced eigenvalue problem is bounded by a new recursive version of AMLS which further reduces the costs for the computation and the solution of this problem. Altogether this leads to a new method which is well-suited for three-dimensional problems and which allows us to compute a large amount of eigenpair approximations in optimal complexity.

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Section: Automated Multi-level Substructuringmentioning

confidence: 99%

Solving an elliptic PDE eigenvalue problem via automated multi-level substructuring and hierarchical matrices

Gerds

Grasedyck

2013

Comput. Visual Sci.

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“…Such an approximations can be very rough and one may be tempted to correct them in some way. This techniques is taken to the limit and exploited in a quite effective way in the Automated Multi-Level Substructuring (AMLS) algorithm [2,7]. In AMLS, the subspaces are corrected by adding correction terms from the interface variables.…”

Section: Philippe and Saadmentioning

confidence: 99%

On correction equations and domain decomposition for computing invariant subspaces

Philippe

Saad

2007

Computer Methods in Applied Mechanics and Engineering

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“…In this approach, a standard static condensation procedure is performed to eliminate an eigenmodal expansion of the component-interior "bubble" degrees of freedom in terms of many fewer interface, or "port," degrees of freedom associated with a Schur complement system. Although the original CMS methods [9,13] do not consider reduction of the degrees of freedom associated with the ports, more recent work considers several port economizations (or interface reduction strategies): an eigenmode expansion (with subsequent truncation) for the port degrees of freedom is proposed in [6,12]; an adaptive port reduction procedure based on a posteriori error estimators for the port reduction is proposed in [18]; and an alternative port reduction approach, with a different bubble function approximation space, is proposed for time-dependent problems in [3].…”

Section: Introductionmentioning

confidence: 99%

“…A popular approach to component-based analysis is the classical component mode synthesis (CMS) method [9,13]. In this approach, a standard static condensation procedure is performed to eliminate an eigenmodal expansion of the component-interior "bubble" degrees of freedom in terms of many fewer interface, or "port," degrees of freedom associated with a Schur complement system.…”

Section: Introductionmentioning

confidence: 99%

Port reduction in parametrized component static condensation: approximation and a posteriori error estimation

Eftang

Patera

2013

Numerical Meth Engineering

104

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We introduce a port (interface) approximation and a posteriori error bound framework for a general component-based static condensation method in the context of parameter-dependent linear elliptic partial differential equations. The key ingredients are i) efficient empirical port approximation spaces -the dimensions of these spaces may be chosen small in order to reduce the computational cost associated with formation and solution of the static condensation system, and ii) a computationally tractable a posteriori error bound realized through a non-conforming approximation and associated conditionerthe error in the global system approximation, or in a scalar output quantity, may be bounded relatively sharply with respect to the underlying finite element discretization.Our approximation and a posteriori error bound framework is of particular computational relevance for the static condensation reduced basis element (SCRBE) method. We provide several numerical examples within the SCRBE context which serve to demonstrate the convergence rate of our port approximation procedure as well as the efficacy of our port reduction error bounds.

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Dynamic analysis of structural systems using component modes

Cited by 1,028 publications

References 7 publications

Solving an elliptic PDE eigenvalue problem via automated multi-level substructuring and hierarchical matrices

Solving an elliptic PDE eigenvalue problem via automated multi-level substructuring and hierarchical matrices

On correction equations and domain decomposition for computing invariant subspaces

Port reduction in parametrized component static condensation: approximation and a posteriori error estimation

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