Large-Scale Structures in Acoustics and Electromagnetics Proceedings of a Symposium (1996) / Chapter Skim
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9 DESIGN AND ANALYSIS OF FINITE ELEMENT METHODS FOR TRANSIENT AND TIME-HARMONIC STRUCTURAL ACOUSTICS
9 DESIGN AND ANALYSIS OF FINITE ELEMENT METHODS FOR TRANSIENT AND TIME-HARMONIC STRUCTURAL ACOUSTICS
Pages 183-203
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From page 183... ...
The resulting space-time approach, therefore, permits the development of a finite element method for transient structural acoustics with the desired combination of increased stability and high accuracy. Additionally, the proposed time-discontinuous Galerkin space-time method provides a natural variational setting for the incorporation of high-order accurate nonreflecting boundary conditions that are local in time.
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From page 184... ...
In the second part, we investigate a multilevel preconditioning approach that is based on the in-version of the hierarchical finite element method. THE TRANSIENT STRUCTURAL ACOUSTICS PROBLEM Consider the coupled system illustrated in Figure 9.1, consisting of the computational domain Q = Qf A Qs ~ composed of a fluid domain Qf, and structural domain Qs .
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is the radiation boundary condition imposed on the artificial boundary TOO, which approximates the asymptotic behavior of the solution at infinity, as described by the Sommerfeld radiation condition. Also, appropriate initial conditions are assumed corresponding to the above coupled second-order system of hyperbolic equations.
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~ 1 ~ 1 1 1 1 1 1 1 1 ~ 1 1 1 1 where UP denotes the space of pth-order polynomials and C° denotes the space of continuous functions. For clarity, only natural boundary conditions are employed.
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Q . A natural measure of stability for the coupled structural acoustics problem is the total energy for the system: E(u, ¢)
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These jump operators weakly enforce initial conditions across time slabs and are crucial for obtaining an unconditionally stable algorithm for unstructured space-time finite element discretizations with highorder interpolations. The specific form of these jump operators is designed such that a natural norm emanates from the variational equation and satisfies a strong coercivity condition.
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Time-dependent boundary conditions are then obtained through the application of an inverse Fourier transform. This new sequence of local time-dependent boundary conditions provides increasing accuracy with order N
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Convergence Analysis To study the convergence rates of the space-time finite element formulation for the exterior structural acoustics problem the following space-time mesh size parameters are introduced. For the structural domain Qs~h5= max~cL/\t,~c}where cL is the dilatational wave speed and Arc and At are maximum element diameters in space and time, respectively.
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This norm emanates naturally from the coupled fluid-structure variational equations (9.13)
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. 0 100 1000 h-i Figure 9.3 Convergence of the numerical error employing the Q2 element; h = hf is the element mesh size parameter.
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ITERATIVE SOLUTION OF LARGE-SCALE TIME-HARMONIC PROBLEMS In the second part of this paper, we consider efficient methods for the solution of large-scale matrix problems that arise from finite element methods for structural acoustics. Direct solution techniques become expensive for solving large systems of linear equations due to excessive growth in computational and storage requirements.
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~ -I \ \ ';~N ) Figure 9.4 Scattering from a geometrically complex rigid structure due to point source.
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basis functions at the coarsest level, level 1, consist of the nodal basis functions, and (2) the hierarchical basis at any level j > 1 consists of nodal basis functions corresponding to nodes in that level which are not present in any of the coarser levels, together with the hierarchical basis for level j- 1.
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,...,yrn(x) }T denote column vectors comprising nodal and hierarchical basis functions respectively.
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Each Pk essentially has the same structure as Pi (shown in (9.321~. Now consider the solution of our nodal basis finite element equations Ed = f In order to achieve improved conditioning of the matrix equations, we start by employing hierarchical basis functions in the bilinear form a(.,.)
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The hierarchical basis preconditioners, therefore, can also be employed in conjunction with highly storage-efficient matrix-free implementations of a gradient-type iterative algorithm. Matrix-free iterative approaches involve calculating characteristic computational kernels required in the iterative method without the explicit assembly of any global matrix, which enables substantial storage reductions.
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(9.41) in the above equations, p is the unknown acoustic pressure in the fluid domain A, k is the wavenumber, and n denotes unit outward normals from respective boundaries.
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Table 9.1 summarizes the number of iterations required for convergence using various preconditioners. Observe that the unpreconditioned algorithm, denoted byM/, suffers substantial deterioration in iteration count as the mesh size decreases.
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tar' A.' i, 10000 1E5 Number of unknowns (b) Next, we examine the effect of increasing frequency on the performance of various preconditioners.
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error estimates for self-adaptive solution strategies for unstructured space-time discretizations, and the implementation of high-order accurate time-dependent nonreflecting boundary conditions. Furthermore, through the use of acoustic velocity potential and structural displacement as the solution variables, the space-time method is unconditionally stable and converges at an optimal rate in a noun that is stronger than the total energy norm.
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From page 203... ...
Thompson, L.L., and P.M. Pinsky, 1 995b, "A space-time finite element method for structural acoustics in infinite domains, part ii: Exact time-dependent non-reflecting boundary conditions," Comput.
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