**J. Tinsley Oden**

**Andrzej Safjan**

**Po Geng**

**Leszek Demkowicz**

University of Texas at Austin

Traditional approaches toward the computer simulation of structural acoustics phenomena have been based primarily on frequency-domain formulations of such problems, and these have dominated both the scientific literature and available analysis software in structural acoustics for many years. These classical approaches have been very successful for a limited class of problems, particularly those involving scales (wavelength) that are large relative to the characteristic dimensions of the structures. In recent times, attempts at large-scale simulations involving high |

Much of structural acoustics is concerned with the classical problem of interaction of the motion of an elastic structure with that of a perfect inviscid fluid into which the structure is immersed. Small perturbations in the velocity or pressure fields of the fluid create waves that impinge upon the structure and scatter into the fluid-structure domain while small motions of the structure may radiate energy into

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Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium
1
High-Order, Multilevel, Adaptive Time-Domain Methods For StructuralAcoustics Simulations
J. Tinsley Oden
Andrzej Safjan
Po Geng
Leszek Demkowicz
University of Texas at Austin
Traditional approaches toward the computer simulation of structural acoustics phenomena have been based primarily on frequency-domain formulations of such problems, and these have dominated both the scientific literature and available analysis software in structural acoustics for many years. These classical approaches have been very successful for a limited class of problems, particularly those involving scales (wavelength) that are large relative to the characteristic dimensions of the structures. In recent times, attempts at large-scale simulations involving high ka-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 hp-version finite-element methods implemented on a special data structure designed to capture multiple-scale response. This is accomplished through the use of multilevel mesh and spectral representations, a posteriori error estimation, adaptive error control, and algorithms for treating non-reflective boundary conditions. Applications to several model problems are presented. In addition, new coupled finite-element and boundary-element methods employing frequency-domain strategies are also discussed, and comparisons with the time-domain strategies are given.
INTRODUCTION
Much of structural acoustics is concerned with the classical problem of interaction of the motion of an elastic structure with that of a perfect inviscid fluid into which the structure is immersed. Small perturbations in the velocity or pressure fields of the fluid create waves that impinge upon the structure and scatter into the fluid-structure domain while small motions of the structure may radiate energy into

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the surrounding fluid media. The radiation and scattering of waves in such fluid-structure interaction has been the subject of research for many decades.
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 frequency-dependent coefficients. The fluid may be modeled with the exterior Helmholtz equation or, as is commonly done, by an equivalent boundary-integral equation. These classical approaches have been very successful for a large and important class of problems, particularly those involving scales which are not too small in relation to characteristic dimensions of the structure. In recent times, attempts at large-scale simulations involving high ka values (k being the wave number and a the characteristic dimension) have encountered formidable computational difficulties and have exposed serious and fundamental limitations of some classical frequency-domain approaches.
In the present work, some new high-order time-domain methods are presented that have proven to be effective for simulating certain classes of problems in structural acoustics and that may have potential for overcoming many of the shortcomings of the frequency-domain methods for high ka. These techniques are built around several special ideas and algorithms:
the structural acoustics problem is formulated as a hyperbolic system of conservation laws and this leads to an abstract Cauchy problem in common Hilbert space settings in which the spectral theory of linear operators is applicable;
new, high-order, multistage Taylor-Galerkin (TG) approximations of these equations in the time-domain are constructed which deliver high-order temporal accuracy with unconditional stability;
spatial approximations are constructed using hp-finite element methods in which local mesh-size h and local spectral order p are varied in a way to adaptively control error, which provides multi-level approximations of the unknowns;
a posteriori error estimates are constructed over each element and time step to provide a running account of solution quality and to provide a basis for the adaptive control strategy;
applications to representative model problems reveal that the methods are robust and provide remarkably accurate results for a class of two-dimensional cases.
In addition to the time-domain methods, a brief account of a coupled boundary element/finite element method is also given. This approach also employs hp-methods and is based on a new version of the Burton-Miller integral equations for the acoustical fluid. Results obtained by using large-scale parallel computers for such problems are presented.
We also make comparisons of time-domain and frequency-domain approaches for a model class of problems. In these computations, some advantages of the time-domain approach over traditional frequency-domain approaches are observed.

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A FLUID-SOLID INTERACTION PROBLEM
Statement of the Problem
Our investigation focuses on the simulation of structural acoustics phenomena using classical models of acoustics and elastic bodies characterized by systems of hyperbolic equations. The elastic structure is modeled by the classical Navier-Lamé equations of linear elasticity, and the fluid is a perfect acoustical medium modeled as a quiescent, inviscid fluid. The mathematical setting is described as follows, and is depicted in Figure 1.1.
Figure 1.1 The solid-structure interaction problem.
Let be a region occupied by an elastic body and and let its boundary consist of three distinct parts, where Гu and Гt are portions of with prescribed displacements and tractions, respectively, and Гsf is the the portion of subjected to the action of the fluid (the contact surface). Similarly, let be a region occupied by acoustic fluid and let the boundary consist of four distinct parts, , where Гv and Гp are portions of with prescribed velocities and pressure, respectively, denotes the truncated exterior boundary, and Гfs denotes the part of the boundary which is in contact with the elastic body. We further denote Гfs = Гsf = Гc' and we use superscripts s and f to distinguish between variables having Ω(s) and Ω(f) as domains.
The fluid-solid interaction problem consists of solving:
• Equations of linear acoustics:

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• Equations of linear elastodynamics:
• Contact conditions:
Here:
= u(s) (x,t) is the displacement vector at particle at time t
s(u(s)) = (sij) = CDu(s) = the stress tensor evaluated at the displacement u(s)
ρs = mass density of the elastic body
C = 6 × 6 symmetric positive definite matrix of elastic constants
n = (ni) = the unit outward normal (at the interface, it is the unit outward normal to the elastic body)
p = p(x,t) is the acoustic pressure at at time t
= v(f) (x,t) is the velocity in the acoustic fluid
ρf = mass density of the acoustic fluid
c0 = small signal sound speed in the acoustic fluid
= boundary operator at the truncation boundary
D = generalized gradient operator defined below

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Problem (1.1)-(1.2) is completed by specifying initial conditions:
Here are given boundary and initial data.
Equations (1.1) and (1.2) are well known, so we comment only on the interface conditions (1.3). On Гc, we have continuity of the normal component of velocity in both media, and continuity of the normal tractions. In addition, since the acoustic fluid is inviscid, we must have zero tangential traction on the elastic body on Гc.
We notice that the pressure p in (1.1) is a primitive variable which is coupled with stresses s of (1.2) through contact conditions, and the stress is not a primitive variable in (1.2). These incompatibilities prompt us to seek alternative formulations of the problem.
Velocity-Displacement Formulation
The formulation is similar to that reported in Hubert and Palencia (1989). The principal steps in the proposed methodology are as follows.
1. Introducing an auxiliary variable u(f),
and assuming the compatibility condition
equations (1.1) are transformed to the following form:
In addition, we make the usual assumption that u(f) is a potential vector field:
for some scalar field Φ.
Finally, the transition conditions (1.3) take the form

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and the fluid-solid interaction problems amount to solving equations (1.2), (1.8), (1.9), (1.10).
2. Weak formulation. We now restrict attention to the case in which and , as indicated in Figure 1.2. Then Ω(f) and Ω(s) are assumed to be disjoint open connected sets with smooth boundaries which intersect in a C2 manifold Гc of dimension N - 1. Let u denote the displacement vector in the whole region , and u(f) and u(s) denote, respectively, its restrictions to Ω(f) and Ω(s).
We start by introducing the spaces of kinematically admissible displacements
Figure 1.2 A submerged elastic structure.
Next, we define the following Hermitian forms

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The space X, when endowed with the following inner product,
is a Hilbert space. In addition, we define space H as the completion of X with inner product (u, w)H, and, finally, we define space D(Ã) as follows
Now, we consider the following weak form of the fluid-solid interaction problem (1.2), (1.8), (1.9), (1.10):
Find
such that
and
It is easily seen that (1.19) is a classical virtual work formulation of (1.2), (1.8), (1.9), (1.10). In fact, (1.19) is valid for any test functions of appropriate regularity, not only for those of the form of gradient.
Let , where denotes the space of test functions. Then (1.18) implies that (1.2) and (1.8) are satisfied in a distributional sense. Next, define a as
Green's formula applied to Ω(l) and Ω(s) gives

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which gives transition conditions (1.10). Here n = (ni) is the unit outward normal to the elastic body and τ = (τi) is the corresponding unit outward tangent.
3. Within the above formalism, the initial boundary value problem of structural acoustics can be reinterpreted as an abstract second order Cauchy problem:
where Ã satisfies
u = u(t) is an H- valued function of time, and u0 and v0 are initial conditions.
To proceed further, we reinterpret (1.23) as a first-order system in time. To this end, we introduce:
• the group variable
• the Hilbert space
• the operator A
Then, (1.22) becomes:

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Here U0 specifies initial conditions, .
To show the well-posedness of (1.28) we consider an operator
and we show that
(i) is accretive, i.e.,
and,
(ii) is surjective, i.e., the range of is the whole space
Indeed,
and, hence,
To prove that is surjective, we need to show that for each there exists a solution of (A + 2I) = F, or equivalently, of the following system
System (1.34) can be reduced to the following equation

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which, by virtue of the Lax-Milgram Theorem, possesses a unique solution . This, together with (1.35) and the definition of D(Ã), implies that . Then (1.34)2 implies that .
Summarizing, is accretive, is surjective, and D(A) is dense in , which implies that the Cauchy problem for operator is well posed (cf., the Lummer-Philips Theorem in Yosida, 1973). Finally, by using the ansatz
we get the well posedness of (1.28). Thus, given , there exists a unique such that (1.28) is satisfied. Moreover,
Problem (1.28) is to be solved numerically by using high-order Taylor-Galerkin methods and hp-adaptive 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). Thus, we solve the following first-order system. (We label it the ''velocity-stress'' formulation.)
• equations of linear acoustics:
• equations of elastodynamics:
• contact conditions:

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together with appropriate boundary and initial conditions. Here: , and sij are the components of the stress tensors.
As before, (1.38), (1.39), (1.40) can be cast in the form of a Cauchy problem
where U is the group variable U = (vT,wT)T, i is the imaginary unit, and A is an appropriately defined operator. Again, (1.41) is solved numerically by using high-order Taylor-Galerkin methods and hp-adaptive finite element methods.
HIGH-ORDER TAYLOR-GALERKIN (TG) METHODS
The Taylor-Galerkin (TG) schemes represent Galerkin approximations of temporal Taylor expansions of the solution. The first TG scheme was presented by Oden (1974) in connection with a Lax-Wendroff approximation of nonlinear waves by finite element methods. The TG schemes were made popular by a series of important papers by Donea and collaborators (Donea, 1984; Selmin et al., 1985), but were not applied to schemes of higher "order" than three. By a scheme of "order s" we shall understand that the one-step truncation error measured in L2 -norm is bounded by CΔts+1, where At is a time-step size and C is a constant independent of Δt. The idea of developing multi-stage high-order TG schemes was presented in the dissertation by Safjan (1993) and in subsequent papers by Safjan and Oden (1993, 1994, 1995a,b). A brief summary of the algorithm is presented below.
Let H be a Hilbert space with inner product (·,·) and corresponding norm , and be a self-adjoint operator in H. Being self-adjoint, A admits the spectral decomposition of the form (Yosida, 1973)
where Eλ is a uniquely defined spectral family. In this section we introduce TG schemes for solving abstract Cauchy problems of the form
Any approximation to (1.44) must involve discretization both in space and time variables. We will adopt the assumption here that the final approximation is obtained by using finite differences in time and

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Figure 1.14 Elastic scattering problem. Finite element mesh at time t = 0.5. Different shadings correspond to different element spectral orders p.
Figure 1.15 Elastic scattering problem. Finite element mesh at time t = 1.0. Different shadings correspond to different element spectral orders p.

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Figure 1.16 Elastic scattering problem. Finite element mesh at time t = 1.5. Different shadings correspond to different element spectral orders p.
Figure 1.17 The vibrating cylinder problem.

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Figure 1.18 The vibrating cylinder problem. Mesh of elements of fourth order.
Figure 1.19 The vibrating cylinder problem. Operator . Pressure distribution at time t = 1.0.

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Figure 1.20 The vibrating cylinder problem. Operator . Pressure distribution at time t = 1.5.
Figure 1.21 The vibrating cylinder problem. Operator . Pressure distribution at time t = 2.0.

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Figure 1.22 The vibrating cylinder problem. Operator B2. Pressure distribution at time t = 2.5.
Figure 1.23 The vibrating cylinder problem. Operator B1. Pressure distribution at time t = 2.5.

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Example 3:The Coupled FE/BE Methods versus the Time-Domain Method
The Galerkin-Burton-Miller formulation developed here is intended to be used to solve problems at very high frequencies. To verify this, we consider an example of a harmonic plane incident wave
impinging on a cylinder with a boundary condition on the surface of the cylinder (as shown in Figure 1.24). The analytic solution is available in the case of β being a constant (Demkowicz et al., 1991b). Clearly, for β = 0, we have the problem of acoustical wave scattering on a rigid body. 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 . 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
is around 2 percent in this example where Pe represents the exact solution and ph is the numerical solution.
Figure 1.24 The test problem: a plane harmonic wave impinging on a cylinder.
Equations (1.105)-(1.108) define a problem of an elastic structure surrounded by an inviscid fluid field. Because of the absence of damping, the problem is not uniquely solvable at certain frequencies (resonant frequencies) and the stiffness matrix becomes singular or ill-conditioned when the frequency ω is at or close to one of these resonant frequencies.

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Figure 1.25 The comparison between numerical solution and analytic solution around the cylindrical boundary with ka = 400 and . 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) and the wave number here varies from k = 0.02 to 14 with Δk = 0.02. A uniform mesh of 256 quadratic finite elements and 128 quadratic boundary elements is used in the computation. Figure 1.26 plots the relationship between the amplitude of total pressure at a far field point (1,0) and wave number k. The experimental results show that the test problem possesses several resonant frequencies in the range of . As a comparison, the analytic solution based on elastic thin shell theory (see Junger and Feit, 1972) is also given in Figure 1.26, and for the small wave number, the elastic solution and thin shell solution are close to each other.
To understand the influence of resonant frequencies on the numerical solution, the test problem given in Example 1 is solved again by the frequency-domain approach. The following incident wave is used

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Figure 1.26 The resonant frequencies of the coupled elastic cylinder and fluid model. The plain harmonic incident wave is used and the problem is solved from k = 0.02 to k = 14 with Δk = 0.02.
where τ = x - ct and the time unit is 1/c. As shown in (1.126), the incident wave pulse must be supposed as a periodical function and T here must be large enough so that the solution at the time period in question is not affected by the solution wave from other periods. In this example, we are interested in the solution in the time period and T = 5.2 is selected. Then, the incident wave in (1.126) is decomposed into the sum of a series of sine functions, i.e.,
and N = 2,048. At most frequencies, pi is in fact almost zero; here, we solve the problem only at a frequency when its corresponding amplitude satisfies
Then, instead of 2,048 different frequencies, we need to solve the problem only in 124 different frequencies.
Figures 1.27 and 1.28 present the results at t = 0 and t = 2/c for the rigid scattering (, upper half part) and elastic scattering (lower half part), respectively. For the rigid scattering, because the

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Helmholtz equation with Neumann boundary condition is well defined on the exterior domain (no resonant frequencies), the frequency-domain method produces essentially the same result obtained by the time-domain method. However, the frequency-domain method does not provide a satisfactory result for elastic scattering, because the method falls at the frequencies close to the resonant frequencies of the coupled solid-fluid system. Figure 1.27 shows spurious scattering waves produced by aliasing in the frequency-domain approach. These nonexistent waves are produced ahead of the plane wave that actually excites the structure. It is likely that these types of erroneous excitations could be controlled, if viscous and material damping effects were added to the model.
Figure 1.27 The acoustic pressure contours for the rigid scattering (upper half) and elastic scattering (lower half) at t = 0. The frequency-domain approach is used and the problem is solved in 124 different frequencies. The elastic cylinder produces spurious waves before the incident plane wave excites it.

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Figure 1.28 The acoustic pressure contours for the rigid scattering (upper half) and elastic scattering (lower half) at t = 2. The frequency-domain approach is used and the problem is solved in 124 different frequencies.
ACKNOWLEDGMENT
Support of this work by the Office of Naval Research (ONR) under #N00014-89-J-1451 is gratefully acknowledged.
REFERENCES
Babuška, I., and M. Suri, 1987, ''The hp version of the finite element method with quasiuniform meshes,''RAIRO Math. Model. Numer. Anal.21, 199-238.
Burton, A.J., and G.F. Miller, 1971, "The application of integral equation methods to the numerical solution of some exterior boundary-value problems,"Proc. R. Soc. London Ser.A 323,201-210.
Demkowicz, L., J.T. Oden, W. Rachowicz, and O. Hardy, 1989, "Toward a universal hp adaptive finite element strategy. part 1: constrained approximation and data structure,"Comput. Meth. Appl. Mech. Eng.77, 79-112.

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Demkowicz, L., J.T. Oden, M. Ainsworth, and P. Geng,1991 a, "Solution of elastic scattering problems in linear acoustics using hp boundary element methods,"J. Comput. Appl. Math.36, 29-63.
Demkowicz, L., J.T. Oden, W. Rachowicz, and O. Hardy,1991b, "An hp Taylor-Galerkin finite method for compressible Euler equations,"Comput. Meth. Appl. Mech. Eng.88, 363-396.
Demkowicz, L., A. Karafiat, and J.T. Oden,1992, "Solution of elastic scattering problems in linear acoustics using hp boundary element method,"Comput. Meth. Appl. Mech. Eng.101, 251-282.
Donea, J., 1984, "A Taylor-Galerkin method for convective transport problems,"Int. J. Numer. Meth. Eng.88, 101-120.
Enquist, B., and A. Majda,1977, "Radiation boundary conditions for the numerical simulation of waves,"Math. Comp.31, 629-651.
Enquist, B., and A. Majda, 1979, "Radiation boundary conditions for acoustic and elastic wave calculations,"Commun. Pure Appl. Math.32, 313-357.
Hubert, J.H., and E.S. Palencia, 1989, Vibration and Coupling ofContinuous Systems, Berlin, Heidelberg, New York: Springer-Verlag.
Junger, M.C., and D. Feit,1972, Sound, Structures, and Their Interaction , Cambridge, Mass.: MIT Press.
Karafiat, A., J.T. Oden, and P. Geng, 1993, "Variational formulations and hp-boundary element approximation of hypersingular integral equations for Helmholtz-exterior boundary-value problems in two dimensions,"Int. J. Eng. Sci.31, 649-672.
Kutt, H.R., 1975, "On the numerical evaluation of finite part integrals involving an algebraic singularity,"Report WISK 179, Pretoria: The National Research Institute for Mathematical Sciences.
Oden, J.T., 1974, "Formulation and application of certain primal and mixed finite element models of finite deformations of elastic bodies." In: Computing Methods in Applied Sciences and Engineering , R. Glowinski and J.L. Lions (eds.), Berlin, Heidelberg, New York: Springer-Verlag.
Safjan, A., 1993, "High-order Taylor-Galerkin and adaptive hp methods for hyperbolic systems with application to structural acoustics,"Ph.D. Dissertation, University of Texas at Austin.
Safjan, A., and J.T. Oden, 1993, "High-order Taylor-Galerkin and adaptive hp methods for second-order hyperbolic systems: application to elastodynamics,"Comput. Meth. Appl. Mech. Eng.103, 187-230.
Safjan, A., and J.T. Oden, 1994, "High-order Taylor-Galerkin and adaptive hp methods for first-order hyperbolic systems,"TICAM Report.
Safjan, A., and J.T. Oden, 1995a, "High-order Taylor-Galerkin and adaptive hp methods for hyperbolic systems," submitted to J. Comput.Phys.
Safjan, A., and J.T. Oden, 1995b, "Highly accurate nonreflecting boundary conditions for wave propagation problems," to be submitted.
Schenck, A.H., 1967, "Improved integral formation for acoustic radiation problems,"J. Acoust. Soc. Am.44(1), 41-58.
Selmin, V., J. Donea, and L. Quartapelle, 1985, "Finite element methods for nonlinear advection,"Comput. Meth. Appl. Mech. Eng.52, 817-845.
Yosida, K., 1973, Functional Analysis, Fourth edition, Berlin, Heidelberg, New York: Springer-Verlag.