Burton Miller Research Articles

Iterative solution of high-order boundary element method for acoustic impedance boundary value problems

A high-order boundary element method for time-harmonic acoustic impedance boundary value problems is presented. The method is based on the Galerkin-type formulation of the Burton–Miller integral equation (BMIE) with high-order polynomial basis and testing functions. This formulation has several important advantages: It is free of the interior resonance problem, the hypersingular integral operator of the traditional BMIE formulation can be avoided, it leads to faster convergence in terms of the number of unknowns than the low-order methods and it shows a good performance with iterative solvers. To avoid the numerical difficulties associated to the implementation of the singular surface integral equations, the singularity extraction technique is applied to evaluate the integrals in the singular and near-singular cases. The resulting matrix equation is solved iteratively with the generalized minimal residual method (GMRES) and a simple preconditioner based on the incomplete LU factorization is applied to expedite the convergence. Numerical results indicate that the BMIE formulation with Galerkin method and high-order basis functions has good convergence properties for various geometries and boundary conditions on a wide frequency range when the GMRES method is applied to solve the matrix equation iteratively. This, in turn, indicates that the formulation is well suited for an efficient application of fast solution procedures, such as the fast multipole method.

Approximate inverse preconditioning for the fast multipole BEM in acoustics

Acoustic radiation from vibrating structures is simulated by a Galerkin boundary element method based on the Burton–Miller approach. The boundary element operators are evaluated by the fast multipole method that allows large-scale computations in the medium frequency range. Two iterative solvers are considered: the generalized minimal residual method and a multigrid solver. Both approaches can be accelerated greatly by the presented approximate inverse preconditioner.

Alternative integral equations for the iterative solution of acoustic scattering problems

This paper addresses the first step of the derivation of well-conditioned integral formulations for the iterative solution of exterior acoustic boundary-value problems. These new formulations are designed to be implemented in a fast multipole method coupled to a Krylov subspace iterative algorithm. Their construction is based on the on-surface radiation condition formalism. Both theoretical developments and numerical aspects are examined in detail. This approach can be viewed as a generalization of the usual Burton–Miller integral equations.

Multipole error estimation of an acoustical boundary-element solver combined with a dual-layer CHIEF approach

The sound radiation from vibrating structures is studied using an iterative boundary-element method. The starting point is a self-adjoint formulation of the Helmholtz integral equation. The generalized minimum residual method (GMRES) is used for solving the corresponding system of equations. Three aspects of the method are investigated. First, the accuracy of the iterative solver is checked by performing multipole error tests: The body is forced to vibrate with the corresponding surface normal velocity of a multipole. Hence, the radiated sound field is analytically known and can be used for determining the exact error caused by the boundary-element solver. Second, instabilities at irregular frequencies are suppressed by combining the iterative solver with a special variant of the CHIEF method, where the total boundary of the body is retracted to an auxiliary surface lying totally within the interior of the structure. CHIEF points can be placed on such an auxiliary surface. The effectiveness of this combined dual-layer GMRES–CHIEF approach and of the Burton–Miller method will be compared. Third, the radiation from a structure with mixed boundary conditions will be investigated. One part of the surface is coated with an absorbing impedance, another part is vibrating with a prescribed normal velocity.

Implementation of CHIEF in MATLAB for prediction of sound radiated from arbitrarily shaped vibrating bodies

The Boundary Element Method can be used to predict sound pressure levels radiated from an arbitrarily shaped vibrating body. Using the direct, indirect, and approximation formulation one can solve such an exterior acoustic problem. In this work, the direct formulation is chosen. The two major methods employed for implementation of this formulation are the Combined Helmholtz Integral Equation Formulation (CHIEF) and the Burton–Miller Method. It was decided that the CHIEF method would be the focus of this research. Using the CHIEF method as a guide, a computer simulation was developed that incorporates quadratic elements, calculates the sound pressure and sound power, is able to import surface velocities, and uses the program ideas as a preprocessor. This program was verified using theoretical models as well as experimental measurements conducted in the Western Michigan University Noise and Vibration Laboratory. This program was written in Matlab with the understanding that it can be processed on a personal computer.

Evaluation of 2‐D Green's boundary formula and its normal derivative using Legendre polynomials, with an application to acoustic scattering problems

AbstractThis paper presents the non‐singular forms, in a global sense, of two‐dimensional Green's boundary formula and its normal derivative. The main advantage of the modified formulations is that they are amenable to solution by directly applying standard quadrature formulas over the entire integration domain; that is, the proposed element‐free method requires only nodal data. The approach includes expressing the unknown function as a truncated Fourier–Legendre series, together with transforming the integration interval [a, b] to [‐1,1] ; the series coefficients are thus to be determined. The hypersingular integral, interpreted in the Hadamard finite‐part sense, and some weakly singular integrals can be evaluated analytically; the remaining integrals are regular with the limiting values of the integrands defined explicitly when a source point coincides with a field point. The effectiveness of the modified formulations is examined by an elliptic cylinder subject to prescribed boundary conditions.The regularization is further applied to acoustic scattering problems. The well‐known Burton–Miller method, using a linear combination of the surface Helmholtz integral equation and its normal derivative, is adopted to overcome the non‐uniqueness problem. A general non‐singular form of the composite equation is derived. Comparisons with analytical solutions for acoustically soft and hard circular cylinders are made. Copyright © 2001 John Wiley & Sons, Ltd.

A robust formulation for solving fluid-structure interaction problems

In this article, a coupled finite element and boundary element approach for the acoustic radiation and scattering from submerged elastic bodies of arbitrary shape is presented. An alternative to the direct boundary element method for acoustics is proposed. By taking an auxiliary source surface inside the radiating boundary and following the usual discretization and integration procedures employed in the boundary element method, both the singularities of the integrands and the nonuniqueness problems do not arise. In addition, the difficulty of slope discontinuity also can be overcome. This procedure is formulated in a similar fashion of wave superposition method, except that the direct boundary integral equations are adopted. The proposed formulation employ the surface Helmholtz integral equation and its normal gradient like that adopted in the Burton–Miller approach, but do not employ any coupling constant. Typical examples are presented that demonstrate the efficiency of the proposed technique.

Analysis of transient wave scattering from rigid bodies using a Burton–Miller approach

Transient scattering from closed rigid bodies can be analyzed using a variety of time domain integral equations, e.g., the Kirchhoff integral equation and its normal derivative. Unfortunately, when the spectrum of the incident field includes one or more of the resonance frequencies of the corresponding interior problem, the solutions to these time domain integral equations become corrupted with spurious interior modes. In this article, this phenomenon is demonstrated via numerical experiments, and a Burton–Miller-type time domain combined field integral equation is proposed as a remedy. To verify that the solutions to this Burton–Miller-type equation are not corrupted by interior modes, various numerical results are presented. It is anticipated that this equation, when used in conjunction with fast time domain integral equation solvers (e.g., plane wave time domain algorithms), will enable the accurate analysis of transient wave scattering from acoustically large bodies.

Numerical solution and a posteriori error estimation of exterior acoustics problems by a boundary element method at high wave numbers

This paper is concerned with the application of boundary element methods to the exterior acoustical problem at high wave numbers. The major issues here are to establish a strong theoretical foundation for the application of the Burton–Miller integral equation and develop a practical way for its numerical implementation. Unlike many conventional approaches, the problem in this study is formulated by the Galerkin method. Through an analysis on its ellipticity, the Burton–Miller equation is proven to be well-posed in the H1/2-Sobolev space and its approximation can attain a quasioptimal convergence rate. The Galerkin method avoids sensitive properties of hypersingular integration, simplifies the numerical implementation, and improves the quality of the numerical solutions, especially at high wave numbers. An L2-norm residual error estimation technique is also implemented for an adaptive scheme for these problems. The numerical implementation is completed on parallel distributed-memory machines as well as conventional sequential machines. The validation of the method at high wave numbers is done through tests on a series of numerical examples.

A user's guide to fortran programs for Wigner and Racah coefficients of SU 3

An accurate numerical solver is developed in this paper for eigenproblems governed by the Helmholtz equation and formulated through the boundary element method. A contour integral method is used to convert the nonlinear eigenproblem into an ordinary eigenproblem, so that eigenvalues can be extracted accurately by solving a set of standard boundary element systems of equations. In order to accelerate the solution procedure, the parameters affecting the accuracy and efficiency of the method are studied and two contour paths are compared. Moreover, a wideband fast multipole method is implemented with a block IDR(s) solver to reduce the overall solution cost of the boundary element systems of equations with multiple right-hand sides. The Burton–Miller formulation is employed to identify the fictitious eigenfrequencies of the interior acoustic problems with multiply connected domains. The actual effect of the Burton–Miller formulation on tackling the fictitious eigenfrequency problem is investigated and the optimal choice of the coupling parameter as α=i/k is confirmed through exterior sphere examples. Furthermore, the numerical eigenvalues obtained by the developed method are compared with the results obtained by the finite element method to show the accuracy and efficiency of the developed method.