Large-Scale Structures in Acoustics and Electromagnetics Proceedings of a Symposium (1996) / Chapter Skim
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1 HIGH-ORDER, MULTILEVEL, ADAPTIVE TIME-DOMAIN METHODS FOR STRUCTURAL ACOUSTICS SIMULATIONS
1 HIGH-ORDER, MULTILEVEL, ADAPTIVE TIME-DOMAIN METHODS FOR STRUCTURAL ACOUSTICS SIMULATIONS
Pages 5-49
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From page 5... ...
In recent times, attempts at large-scale simulations involving high lea-values have revealed limitations of many classical methods and have led to the consideration of new and alternative approaches, including time-domain techniques. This paper describes a new class of high-order methods which employ adaptive hpversion finite-element methods implemented on a special data structure designed to capture multiple-scale response.
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Mathematically, the problem can be adequately modeled by a system of linear conservation laws, which, with further manipulations, can be reduced to the wave equation for the acoustical pressure coupled to the equations of elastodynamics for the structure. Traditionally, approaches for the computer simulations of structural acoustics phenomena have been primarily based on frequency-domain formulations of such problems, in which a Fourier transformation of the dynamic equations is made, and this results in a system of partial-differential equations in complex-valued field variables with frequencydependent coefficients.
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with prescribed velocities and pressure, respectively, I:o denotes the truncated exterior boundary, and rfs denotes the part of the boundary which is in contact with the elastic body. We further denote rfS = rSf = rc, and we use superscripts s andfto distinguish between variables having Q(s)
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at time t s(u~s') = (so ~ = CDu~sj = the stress tensor evaluated at the displacement ups' ps = mass density of the elastic body 6 x 6 symmetric positive definite matrix of elastic constants the unit outward normal (at the interface, it is the unit outward normal to the elastic body)
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In addition, since the acoustic fluid is inviscid, we must have zero tangential traction on the elastic body on Fc. We notice that the pressure p in (1.1)
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,w(s) ~#rc=w unrig / \ Figure 1.2 A submerged elastic structure.
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. The space X, when endowed with the following inner product, (U.
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3. Within the above formalism, the initial boundary value problem of structural acoustics can be reinterpreted as an abstract second order Cauchy problem: -+Au=0 O<~ |
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| dt U+AU=0 0<~$ ~ U=UO Here UO specif~esinitialconditions, UO=(vO,uOT)
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is to be solved numerically by using high-order Taylor-Galerkin methods and hpadaptive finite element methods. Velocity-Stress Formulation The second approach consists of converting the system of equations of elastodynamics into an equivalent first-order system, and coupling it with (1.1)
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Again, (1.41) is solved numerically by using high-order Taylor-Galerkin methods and hpadaptive finite element methods.
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In the classical method of lines, an approximation in space variables converts the original initial-boundary-value problem into a system of ordinary differential equations (ODEs) , which next is discretized in time using one of many time integration schemes for ODE.
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A particular choice of zero diagonal elements of M is made with an eye on wellposedness of a typical one-stage problem and a possible splitting of the operator defining the left-hand side of each stage. To make the i-th stage solution Zj m-th order accurate, it is necessary to satisfy the order conditions for Hi and to make the previous stage solutions Zi-l'Zi-2, .~,Zl, at least of the order m-l, (otherwise, some coefficients have to be set to O; e.g., if Zi-~ is of the order m-2, then, necessarily pii-~ = 0 )
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are referred to herein as the order conditions. It is important to realize that the particular forms of the coefficient matrices M and N limit the attainable order of (direct)
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However, any computational domain is necessarily bounded, so it is essential to introduce appropriate boundary conditions at the boundaries. In particular, we are interested in radiating boundary conditions that have the following features: (i)
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the boundary conditions fit into the framework of our Taylor-Galerkin schemes. We first develop radiating boundary conditions for the scalar wave equation and then deduce boundary conditions suitable for the equivalent Ist-order hyperbolic system.
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, we get the first radiating boundary condition Ad: ~lPIx=a (P,x +P,`~|x=a 0 Using the next approximation to (1 _ :2 / (i92 )
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To get stable boundary operators of arbitrary order, we consider the Pade approximation to (1 _ £,2 / cv2 ) 1/2 of the form (1 42 / m2)
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~/2 rather than Pade approximations may result in strongly unstable boundary conditions (Enquist and Maj da, 1979)
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is nothing but the system of equations of linear acoustics in appropriately defined nondimensional variables. Then the boundary operator ~2 ~ ~2P| x=a bitt +Pxt 2 P,YY]
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(1.92) On Owe define the following three categories of shape functions: · Node functions.
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: VI K ~ PK } (1.99) The space Xhp possesses standard interpolation properties of tap-finite element methods (see, e.g., Babuska and Suri, 1987)
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nodes. In the figure, open circles denote active vertex nodes which support bilinear shape fixations, open rectangles denote edge and interior nodes which support various high-order shape functions, and darkened circles and rectangles denote constrained nodes.
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A FREQUENCY-DOMAIN APPROACH For completeness, we briefly summarize parallel work we have done on the analysis of structural acoustic problems using new coupled boundary-element and finite-element approaches in which the problem is formulated in the frequency domain. Frequency-domain formulations are obtained by Fourier transformations of the time-domain equations and variables.
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and the exterior domain Q(f ) ' and Or andpinC represent amplitudes of normal stress, tangential stress, and incident wave pressure at the frequency lo, respectively.
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and -DTEDu - pS6~)
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, the implementation for such nrocerl~re~ is complicated and the accuracy of its result its ~.~nilv very boor - rim ~ ~ J ~ d ~ _ ~ ~ ~ ~ The method used in this study to avoid hypersingular integral kernels associated with (1.115) was proposed in Karafiat et al.
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We consider the problem of diffraction of a plane wave consisting of a short train of pulses, by an elastic shell, see Figure 1.5. The problem was solved for the following data: Acoustic fluid Small signal sound speed, c0 = 1500 m l s Ambient mass density, p f = 7700 kg I m3 32
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Figures 1.6 through 1.16 show the computed spatial waveforms of various acoustical and elastic variables in nondimensional form. Figure 1.6 shows the initial finite element mesh and Figure 1.7 shows the initial condition function in the form of contour maps.
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The problem was solved using a 2-stage 4th-order Taylor-Galerkin method with stability parameter p=0.471, constant time step '\t=0.01, and a fixed finite element mesh. Figure 1.18 shows the finite element mesh, and the consecutive Figures 1.19, 1.20, 1.21, and 1.22 present the computed pressure distributions at time stages t = 1, 1.5, 2.0, and 2.5, respectively.
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1 1 1 1 1 1 1 1 ~ I I ! 1 1 1 1 ~ 1 I j 1 1 ~ ~ ~ 1 -.0052 -.0028 Figure 1.7 Elastic scattering problem.
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lOOE~2 .0035 .0075 Figure 1.9 Elastic scattering problem. The pressure at time t = 1.0.
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10 Elastic scattering problem. The pressure at time t = 1.5.
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~=-.1059902 ~=.1408~6 Figure 1.13 Elastic sc~edug problem. The 8 - 8 component of stress tensor in dashc shed ~ time ~ = 1.E 38
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Different shadings correspond to different element spectral orders p. D.O.F=10003 Figure 1.15 Elastic scattering problem.
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/ ., . I , ~ Fern ~ ~0.5m y BOUNDARY CONDITIONS rv zero normal velocity rOO : approximate non-reflexive ret, : Vn = f(t)
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\ / / / / / / 6 7 8 ~D.O.F=3413 Figure 1 . 19 The vibrating cylinder problem.
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\ .~ Figure 1.21 The vibrating cylinder problem. Operator ~2 .
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Pressure distribution at time t = 2.5. / / \\ Figure 1.23 The vibrating cylinder problem.
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Figure 1.25 compares the numerical solution and corresponding analytic solution for the pressure distribution around the surface of the cylinder with ka = 400 and Jp/~= p . With a uniform mesh of 1,024 quadratic elements and 2,048 degrees of freedom, the result illustrated in Figure 1.25 shows a good correlation between the numerical solutions and analytic solution and the L2-norm of relative error, computed by ^h 2 Irc (P - He )
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Because of symmetry, only the solution from 0° to 180° around the cylindrical boundary is plotted. To verify this, the coupled elastic cylinder and fluid model given in Example 1 is solved for the same incident wave given in (1.125)
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. 0 1 `,`` I,\\,,, 2 3 4 l elastic thin shell 5 6 7 8 9 10 11 12 13 14 wave number k Figure 1.26 The resonant frequencies of the coupled elastic cylinder and fluid model.
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1 o -1 (/ it_ 1 1 1 1 1 1 , , , , 1 , , , , l o 2 pressure 0.005 0.00464286 0.00428571 0.00392857 0.00357143 0.00321429 0.00285714 0.0025 0.00214286 0.00178571 0.00142857 0.00107143 0.000714286 0.000357143 -0.000357143 -0.000714286 -0.00107143 -0.00142857 -0.00178571 -0.00214286 -0.0025 -0.00285714 -0.00321429 -0.00357143 _0.00392857 -0.00428571 -0.00464286 -0.005 Figure 1.27 The acoustic pressure contours for the rigid scattering (upper half) and elastic scattering (lower half)
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Miller, 1971, "The application of integral equation methods to the numerical solution of some exterior boundary-value problems," Proc.
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Geng, l 991 a, "Solution of elastic scattering problems in linear acoustics using hp boundary element methods," J Comput.
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