Implementation of Non-reflecting Boundaries in a Space-time Finite Element Method for Structural Acoustics

This paper examines the development and implementation of second-order accurate non-reflecting boundary conditions in a time-discontinuous Galerkin finite element method for structural acoustics in unbounded domains. The formulation is based on a multi-field space-time variational equation for both the acoustic fluid and elastic solid together with their interaction. This approach to the modeling of the temporal variables allows for the consistent use of high-order accurate adaptive solution strategies for unstructured finite elements in both time and space. An important feature of the method is the incorporation of temporal jump operators which allow for discretizations that are discontinuous in time. Two alternative approaches are examined for implementing non-reflecting boundaries within a timediscontinuous Galerkin finite element method; direct implementation of the exterior acoustic impedance through a weighted variational equation in time and space, and indirectly through a decomposition into two equations involving an auxiliary variable defined on the non-reflecting boundary. The idea for the indirect approach was originally developed in (Kallivokas, 1991) in the context of a standard semi-discrete formulation. Extensions to general convex boundaries are also discussed – numerical results are presented for acoustic scattering from an elongated structure using a first-order accurate boundary condition applied to an elliptical absorbing boundary. INTRODUCTION Recently, a class of high-order accurate space-time finite element methods for transient structural acoustics in unbounded domains has been developed (Thompson, 1996a; Thompson, 1996b; Thompson, 1996c; Thompson, 1996d). An important feature of the space-time formulation is the incorporation of temporal jump operators which allow for finite element basis functions that are discontinuous in time. In (Thompson, 1995a) a multi-field extension is given where independent interpolation functions are used for both the acoustic velocity potential and its time derivative, i.e. the acoustic pressure. The use of a multi-field formulation was first introduced in (Hughes, 1988) in the context of elastodynamics, and allows for greater flexibility over the choice of finite element basis functions used and eliminates a minimum requirement of quadratic basis functions present in a single-field formulation. The multi-field formulation reverts to the single-field formulation with certain choices of basis functions. Time-discontinuous Galerkin space-time finite element methods are capable of delivering very high accuracies for wave propagation simulations, particularly for problems involving sharp gradients in the solution which typically arise in the vicinity of fluid-structure interfaces and near inhomogeneities such as stiffeners, structural joints, and material discontinuities. In these problems, solutions obtained with standard numerical methods may have difficulty resolving the discontinuities in the physical solution – in the case of standard time integrators, large spurious oscillations may appear which pollute the entire 1 Copyright  1997 by ASME solution. In addition, for problems involving the propagation of pulses with broadband frequencies over a large distance, commonly used second-order accurate numerical algorithms may exhibit significant dispersion errors causing misrepresentation of arrival time and directionality at a distant target. When space-time finite element methods are used to solve the structural acoustics problem in unbounded domains, a fluid truncation boundary is introduced where radiation (nonreflecting) boundary conditions are applied to transmit outgoing waves. If accurate non-reflecting boundary conditions are used, fewer fluid elements are needed and considerable cost savings will result. Therefore there is a need for the implementation of high-order accurate non-reflecting (absorbing) boundary conditions which eliminate or minimize reflection of outgoing waves and that also preserve the data structure of the space-time finite element method. In (Thompson, 1996a) we indicated how the time-discontinuous space-time finite element method provides a natural variational setting for the implementation of local in time non-reflecting boundary conditions. In (Thompson, 1996b), a new sequence of high-order accurate and local in time nonreflecting boundary conditions based on an exact representation of the acoustic impedance were developed. In this paper, we clarify the space-time formulation given in (Thompson, 1995a), and focus on the development and consistent implementation of a second-order accurate non-reflecting boundary condition within the multi-field space-time variational equation for the coupled structural acoustics problem in unbounded domains. Two alternative approaches for implementing the non-reflecting boundary condition are examined: in the first, the non-reflecting boundary operator is implemented directly as a ‘natural’ boundary condition in the space-time variational equation, i.e. it is enforced weakly in both space and time; in the second, the non-reflecting boundary operator is implemented indirectly through a decomposition into two equations involving an auxiliary variable. The first equation is implemented as a ‘natural’ boundary condition, while the second equation is incorporated through a consistent Galerkin method. The idea for the indirect approach was originally given in (Kallivokas, 1991; Kallivokas, 1995) in the context of a standard semi-discrete finite element formulation. The advantage of the indirect approach is the resulting symmetry of the formulation in the spatial dimension, at the expense of solving for an additional variable on the nonreflecting boundary. Extensions of the formulation to general convex boundaries is also discussed – as an example, numerical results are presented for acoustic scattering from an elongated structure using a first-order accurate boundary condition applied to an elliptical boundary. Ωs Ωf