Lonny L. Thompson

A review of finite-element methods for time-harmonic acoustics

State-of-the-art finite-element methods for time-harmonic acoustics governed by the Helmholtz equation are reviewed. Four major current challenges in the field are specifically addressed: the effective treatment of acoustic scattering in unbounded domains, including local and nonlocal absorbing boundary conditions, infinite elements, and absorbing layers; numerical dispersion errors that arise in the approximation of short unresolved waves, polluting resolved scales, and requiring a large computational effort; efficient algebraic equation solving methods for the resulting complex-symmetric (non-Hermitian) matrix systems including sparse iterative and domain decomposition methods; and a posteriori error estimates for the Helmholtz operator required for adaptive methods. Mesh resolution to control phase error and bound dispersion or pollution errors measured in global norms for large wave numbers in finite-element methods are described. Stabilized, multiscale, and other wave-based discretization methods dev...

Accurate radiation boundary conditions for the time‐dependent wave equation on unbounded domains

Asymptotic and exact local radiation boundary conditions (RBC) for the scalar time-dependent wave equation, rst derived by Hagstrom and Hariharan, are reformulated as an auxiliary Cauchy problem for each radial harmonic on a spherical boundary. The reformulation is based on the hierarchy of local boundary operators used by Bayliss and Turkel which satisfy truncations of an asymptotic expansion for each radial harmonic. The residuals of the local operators are determined from the solution of parallel systems of linear rstorder temporal equations. A decomposition into orthogonal transverse modes on the spherical boundary is used so that the residual functions may be computed e ciently and concurrently without altering the local character of the nite element equations. Since the auxiliary functions are based on residuals of an asymptotic expansion, the proposed method has the ability to vary separately the radial and transverse modal orders of the RBC. With the number of equations in the auxiliary Cauchy problem equal to the transverse mode number, this reformulation is exact. In this form, the equivalence with the closely related non-re ecting boundary condition of Grote and Keller is shown. If fewer equations are used, then the boundary conditions form highorder accurate asymptotic approximations to the exact condition, with corresponding reduction in work and memory. Numerical studies are performed to assess the accuracy and convergence properties of the exact and asymptotic versions of the RBC. The results demonstrate that the asymptotic formulation has dramatically improved accuracy for time domain simulations compared to standard boundary treatments and improved e ciency over the exact condition. Copyright ? 2000 John Wiley & Sons, Ltd.

Implementation of exact non-reflecting boundary conditions in the finite element method for the time-dependent wave equation

When solving the wave equation in infinite regions using finite element methods, the domain must be truncated. We investigate the accuracy of time-dependent non-reflecting boundary conditions (NRBC) derived in Grote, Keller (1995), when implemented in the finite element method. The NRBC annihilate the first N wave harmonics on a spherical truncation boundary. High-order temporal derivatives are formulated as a system of first-order ordinary diAerential equations. Several versions of implicit and explicit multi-step, time-integration schemes are presented for solution of the finite element equations concurrently with the first-order system appearing in the NRBC. An alternative scaling of the boundary variables is introduced which leads to a well-conditioned coeAcient matrix. Although the boundary conditions are global over the boundary, when implemented in the finite element method, they only require inner products of spherical harmonics within the force vector, and as a result, they are easy to implement and do not disturb the banded/ sparse structure of the matrix equations. Several numerical examples are presented which demonstrate the improvement in accuracy over standard finite element methods. ” 2000 Elsevier Science S.A. All rights reserved.

Local high‐order radiation boundary conditions for the two‐dimensional time‐dependent structural acoustics problem

The time‐dependent structural acoustics problem involving solution of the coupled wave equation over an infinite fluid domain is posed as a coupled problem over a finite fluid domain with local time‐dependent radiation boundary conditions applied to the fluid truncation boundary. The proposed radiation boundary conditions are based on an asymptotic approximation to the exact solution in the frequency domain expressed in negative powers of a nondimensional wave number. A sequence of differential operators that match the leading terms of the asymptotic expansion provide boundary conditions that are of progressively higher order and increasing accuracy. Time‐dependent boundary conditions are obtained through an inverse Fourier transform. The relationship of these approximate local operators to the exact nonlocal Dirichlet‐to‐Neumann map is examined. To illustrate their effectiveness, the boundary conditions are employed in a finite element formulation for the time‐dependent structural acoustics problem. In c...

Accurate radiation boundary conditions for the two-dimensional wave equation on unbounded domains☆

A recursive sequence of radiation boundary conditions first given by Hagstrom and Hariharan [Appl. Numer. Math. 27 (1998) 403] for the time-dependent wave equation in a two-dimensional exterior region are re-derived based on direct application of the hierarchy of local boundary operators of Bayliss and Turkel [Commun. Pure Appl. Math. 33 (1980) 707] and a recursion relation for the expansion coefficients appearing in an asymptotic wave expansion. By introducing a decomposition into tangential Fourier modes on a circle we reformulate the sequence of local boundary conditions in integro-differential form involving systems of first-order temporal equations for auxiliary functions associated with each mode and the Fourier transform of the solution evaluated on the boundary. The auxiliary functions are recognized as residuals of the local boundary operators acting on the asymptotic wave expansion. Direct finite element implementations for the original local sequence of boundary conditions are compared to implementations of the Fourier transformed auxiliary functions. We show that both implementations easily fit into a standard finite element discretization provided that independent time integration algorithms are used for the interior and boundary equations with coupling through the boundary force vectors at each time step. For both of our direct and modal finite element implementations, the amount of work and storage is less than that required for the finite element calculation in the interior region within the boundary. One advantage of the tangential modal implementation is that far-field solutions may be computed separately for each Fourier mode without saving lengthy time-history data at interior points. Numerical studies confirm the progressive improvement in accuracy with increasing number of auxiliary functions included.

Design of Honeycomb Mesostructures for Crushing Energy Absorption

This paper presents the energy absorption properties of hexagonal honeycomb structures of varying cellular geometries under high speed in-plane crushing. While the crushing responses in terms of energy absorption and densification strains have been extensively researched and reported, a gap is identified in the generalization of honeycombs with contr’olled and varying geometric parameters. This paper addresses this gap through a series of finite element (FE) simulations where the cell angle and the inclined wall thickness, are varied while maintaining a constant mass of the honeycomb structure. A randomly filled, nonrepeating design of experiments (DOEs) is generated to determine the effects of these geometric parameters on the output of energy absorbed and a statistical sensitivity analysis is used to determine the parameters significant for the crushing energy absorption of honeycombs. It is found that while an increase in the inclined wall thickness enhances the energy absorption of the structure, increases in either the cell angle or ratio of cell angle to inclined wall thickness have adverse effects on the output. Finally, the optimization results suggest that a cellular geometry with a positive cell angle and a high inclined wall thickness provides for maximum energy absorption, which is verified with a 6% error when compared to a FE simulation. [DOI: 10.1115/1.4006739]

Finite element formulation of exact non‐reflecting boundary conditions for the time‐dependent wave equation

A modi ed version of an exact Non-re ecting Boundary Condition (NRBC) rst derived by Grote and Keller is implemented in a nite element formulation for the scalar wave equation. The NRBC annihilate the rst N wave harmonics on a spherical truncation boundary, and may be viewed as an extension of the secondorder local boundary condition derived by Bayliss and Turkel. Two alternative nite element formulations are given. In the rst, the boundary operator is implemented directly as a ‘natural’ boundary condition in the weak form of the initial–boundary value problem. In the second, the operator is implemented indirectly by introducing auxiliary variables on the truncation boundary. Several versions of implicit and explicit timeintegration schemes are presented for solution of the nite element semidiscrete equations concurrently with the rst-order di erential equations associated with the NRBC and an auxiliary variable. Numerical studies are performed to assess the accuracy and convergence properties of the NRBC when implemented in the nite element method. The results demonstrate that the nite element formulation of the (modi ed) NRBC is remarkably robust, and highly accurate. Copyright ? 1999 John Wiley & Sons, Ltd.

Computation of far-field solutions based on exact nonreflecting boundary conditions for the time-dependent wave equation

In this work we show how to combine in the exact nonreflecting boundary conditions (NRBC) first derived by Grote and Keller, the calculation of the exterior (far-field) solution for time-dependent radiation and scattering in an unbounded domain. At each discrete time step, radial modes computed on a spherical artificial boundary which drive the exact NRBC for the near-field solution, are imposed as Cauchy data for the radial wave equation in the far-field. Similar to the far-field computation scheme used by Wright, the radial modes in the exterior region are computed using an explicit finite diAerence solver. However, instead of using an ‘infinite grid’, we truncate the exterior radial grid at the far-field point of interest, and for each harmonic, impose the same exact NRBC used for the near-field truncation boundary, here expressed in modal form. Using this approach, two diAerent methods for extrapolating the nearfield solution to the far-field are possible. In the first, the near-field solution is computed using the exact NRBC, then, based on the solution for the radial modes evaluated on the artificial boundary, the exterior solution may be computed as a post-process. In the second, we show how to compute the far-field solution concurrently with the near-field solution and the NRBC. Numerical studies demonstrate that the method is highly accurate and efficient for direct time-domain computations of far-field solutions. ” 2000 Elsevier Science S.A. All rights reserved.

A space‐time finite element method for the exterior acoustics problem

In this paper, the development of a space‐time finite element method for the solution of the transient acoustics problem in exterior domains is discussed. The space‐time formulation for the exterior acoustics problem is obtained from a time‐discontinuous Galerkin variational equation for coupled structural acoustics, specialized to the case of zero normal velocities on the wet surface, i.e., a rigid scatterer. The formulation employs a finite computational acoustic domain surrounding the scatterer and incorporates high‐order time‐dependent nonreflecting (radiation) boundary conditions on the fluid truncation boundary as ‘‘natural’’ boundary conditions in the space‐time variational equation, i.e., they are enforced weakly in both space and time. The result is an algorithm for direct transient analysis of acoustic radiation and scattering with the desired combination of good stability and high accuracy. The method is especially useful for the application of adaptive solution strategies for transient acousti...

The Effects of Chassis Flexibility on Roll Stiffness of a Winston Cup Race Car

Predictable handling of a racecar may be achieved by tailoring chassis stiffness so that roll stiffness between sprung and unsprung masses are due almost entirely to the suspension. In this work, the effects of overall chassis flexibility on roll stiffness and wheel camber response, will be determined using a finite element model (FEM) of a Winston Cup racecar chassis and suspension. The FEM of the chassis/suspension is built from an assembly of beam and shell elements using geometry measured from a typical Winston cup race configuration. Care has been taken to model internal constraints between degrees-of-freedom (DOF) at suspension to chassis connections, e.g. at ball and pin joints and internal releases. To validate the model, the change in wheel loads due to an applied jacking force that rolls the chassis agrees closely with measured data. The roll stiffness predicted from finite element models of the front and rear suspension compared closely to those calculated using a rigidbody kinematics model. To study the effects of chassis flexibility on roll, torsional stiffness is increased by adding strategic members to the chassis structure. Results from the finite element analysis indicate that the effective roll stiffness of the front suspension interacting with the chassis, increased by 7.3 % over a baseline chassis when the chassis torsional stiffness was increased by 130% over a baseline chassis stiffness of 9934 ft-lb/deg. As the chassis stiffness is increased further above this value, the front roll stiffness changed very little. From these results, the minimum torsional stiffness required so that the effective roll stiffness of the front suspension is within 3 % from the roll stiffness with a rigid chassis, is about 23100 ft-lb/deg.