Dynamic Analysis of Structures and Structural Systems

Beskos, D.E.

doi:10.1007/978-3-7091-2790-2_1

Cited by 16 publications

(13 citation statements)

References 87 publications

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“…Detailed information about this procedure can be found in the book by Dominguez (1993). For a more extensive literature review on dynamic BE formulations, the reader is referred to Beskos (1987Beskos ( , 1997Beskos ( , 2003. Frequent applications are soil-structure-interaction problems (e.g.…”

Section: Viscoelastic Direct Boundary Element Formulation In Time Domainmentioning

confidence: 99%

Boundary Element Methods for the Dynamic Analysis of Elastic, Viscoelastic, and Piezoelectric Solids

Gaul

Kögl

Moser

et al. 2004

Encyclopedia of Computational Mechanics

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As an alternative to domain discretization methods, the boundary element method (BEM) provides a powerful tool for the solution of single‐field and multifield initial value/boundary value problems. These are encountered, for example, in the analysis of elastic solids and compressible fluids and their interaction at contacting boundaries. As a basis for a boundary element analysis of these problems, one can employ singular boundary integral equations as well as multifield variational formulations. To eliminate the domain integrals from these formulations, different types of fundamental solutions are employed as weighting and approximation functions. When using fundamental solutions that fulfill the dynamic field equations, one obtains singular boundary integral equations and all domain integrals are eliminated. Such a formulation in time domain is presented for the calculation of solids with generalized viscoelastic constitutive behavior. The time convolution is numerically approximated by an efficient convolution quadrature that uses the Laplace transformed fundamental solution. Infinite domains can be easily treated because the fundamental solution fulfills the Sommerfeld radiation condition. If the fundamental solutions that are employed in the boundary element formulation fulfill only the static part of the field equation, the representation formula contains both boundary and domain integrals. Several techniques are available for transforming these domain integrals to the boundary, two of which will be presented here. The first technique is the Dual Reciprocity Method, which uses particular solutions to achieve an approximate transformation of the domain integral to the boundary. This technique is very versatile and will be applied to anisotropic elastodynamics and the coupled field problem of dynamic piezoelectricity. The second technique provides an analytical transformation to the boundary by constructing a so‐called fundamental solution of higher order and will be applied – in the framework of the hybrid boundary element method – to elastodynamics. Both techniques yield mass matrices for the domain integral, similar to those obtained with the finite element method (FEM). Since static fundamental solutions are employed, the system matrices do not depend on frequency or time, which makes the formulations especially suited for the calculation of the eigenvalue problem, transient and forced vibration analysis. Direct boundary element formulations based upon the collocation method lead to nonsymmetric matrices, which is a disadvantage when the matrices are coupled with symmetric finite element matrices. A symmetric system of equations can be obtained by starting from a variational principle. This will be demonstrated by means of elastodynamics, for which a boundary element formulation is derived from a multifield version of Hamilton's principle. A superposition of static fundamental solutions is used to approximate the field in the domain, and a fundamental solution of higher order is employed for transforming the domain integral over the inertia term to the boundary. The resulting mass matrix is symmetric and positive‐definite. The formulation involves the calculation of singular boundary integrals if the load points of the fundamental solutions are placed on the nodes of the boundary discretization. To avoid the occurrence of singular integrals, the load points are placed outside the domain. This way, only regular integrals have to be evaluated numerically, and the resulting boundary element formulation becomes nonsingular. The boundary element formulations described here, along with their implementation, are the result of recent progress in boundary elements research. The numerical examples indicate that for linear dynamic field problems, the presented methods can be seen as alternatives to well‐established domain discretization methods, in particular the finite element method. In the interest of brevity, aspects of numerically efficient implementations, error estimation, and adaptivity have not been addressed.

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Section: Viscoelastic Direct Boundary Element Formulation In Time Domainmentioning

confidence: 99%

Boundary Element Methods for the Dynamic Analysis of Elastic, Viscoelastic, and Piezoelectric Solids

Gaul

Kögl

Moser

et al. 2004

Encyclopedia of Computational Mechanics

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“…Many details are given, e.g., by Zienkiewicz et al [25,26], who were among the first suggesting a ''mariage à la mode -the best of two worlds'', by von Estorff and Prabucki [13], von Estorff and Antes [14], Belytschko and Lu [4], Yu et al [23], or Rizos and Wang [21]. A rather complete overview is provided by Beskos [5][6][7]. It should be mentioned, that in most cases the BEM has been used to model those parts of the investigated bodies which are of semi-infinite extension, while finite parts were represented with the FEM.…”

Section: Introductionmentioning

confidence: 97%

Iterative coupling of BEM and FEM for nonlinear dynamic analyses

Soares

Estorff

Mansur

2004

Computational Mechanics

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The present work deals with the iterative coupling of boundary element and finite element methods. First, the domain of the original problem is subdivided into two subdomains, which are separately modeled by the FEM and BEM. Thus the special features and advantages of the two methodologies can be taken into account. Then, prescribing arbitrary transient boundary conditions, a successive renewal of the variables on the interface between the two subdomains is performed through an iterative procedure until the final convergence is achieved. In the case of local nonlinearities within the finite element subdomain, it is straightforward to perform the iterative coupling together with the iterations needed to solve the nonlinear system. The procedure turns out to be very efficient. Moreover, a special formulation allows taking into account different durations of the time steps in each subdomain.The numerical simulation of arbitrarily shaped continuous bodies subjected to harmonic or transient loads remains, despite much effort and progress over the last decades, a challenging area of research. In most cases, discrete techniques, such as the finite element method (FEM) and the boundary element method (BEM) have been employed and continuously further developed with respect to accuracy and efficiency. Both methodologies can be formulated in the time domain or in the frequency domain, and each approach has relative benefits and limitations. The finite element method, for instance, is well suited for inhomogeneous and anisotropic materials as well as for dealing with the nonlinear behavior of a body. For systems with infinite extension and regions of high stress concentration, however, the use of the boundary element method is by far more advantageous. More details are given, e.g., by Hughes [17] or Bathe [2] for the FEM and by Becker [3] or Dominguez [10] in the case of the BEM.In fact, it did not take long until some researchers started to combine the FEM and the BEM in order to profit from their respective advantages by trying to evade their disadvantages. Up to now, quite a few publications concentrate on such coupling approaches. Many details are given, e.g., by Zienkiewicz et al. [25,26], who were among the first suggesting a ''mariage à la mode -the best of two worlds'', by von Estorff and Prabucki [13], von Estorff and Antes [14], Belytschko and Lu [4], Yu et al. [23], or Rizos and Wang [21]. A rather complete overview is provided by Beskos [ [5][6][7]. It should be mentioned, that in most cases the BEM has been used to model those parts of the investigated bodies which are of semi-infinite extension, while finite parts were represented with the FEM. Moreover, it has been assumed that the considered systems behave linearly, which means that only elastic material and small displacements are considered.The FEM/BEM coupling in the time domain has also been successfully used to take into account nonlinear effects. Thus Pavlatos and Beskos [20] as well as Yazdchi et al. [24], for instance, modeled an inelastic structure ...

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“…The Boundary Element Method (BEM) has developed into a powerful numerical method for solving inelastic transient dynamic problems [1][2][3][4]. Because of acceleration and initial stress terms in the governing equation, the direct application of the BEM to the inelastic transient dynamic problems generates domain integrals in addition to the usual surface integrals [1].…”

Section: Introductionmentioning

confidence: 99%

A BEM formulation for inelastic transient dynamic analysis using domain decomposition and particular integrals

2010

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This study deals with the domain decomposition method and particular integrals for multi-region inelastic transient dynamic analysis. The particular integral formulation for single-region inelastic transient dynamic analysis is obtained by eliminating the acceleration volume integral and treating the initial stress term by volume cell. The Houbolt time integration scheme is used for the timemarching process. The Newton-Raphson algorithm for plastic multiplier is used to solve the system equation. In order to extend to multi-region problems, the domain decomposition method is examined. The domain of the original problem is subdivided into subregions. The interface boundary conditions are updated by using the iterative coupling employing Schwarz algorithm. Numerical results of two example problems are given to demonstrate the validity and accuracy of the present formulation.

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Dynamic Analysis of Structures and Structural Systems

Cited by 16 publications

References 87 publications

Boundary Element Methods for the Dynamic Analysis of Elastic, Viscoelastic, and Piezoelectric Solids

Boundary Element Methods for the Dynamic Analysis of Elastic, Viscoelastic, and Piezoelectric Solids

Iterative coupling of BEM and FEM for nonlinear dynamic analyses

A BEM formulation for inelastic transient dynamic analysis using domain decomposition and particular integrals

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