Acoustic shape sensitivity analysis using the boundary integral equation

Abstract

Boundary integral equations are formulated for the shape sensitivity analysis of the acoustic problems. The concept of the material derivative is employed in deriving the sensitivity equations. Since the equation is derived by the direct differentiation of the boundary integrals containing the field values, it is expected that the sensitivity would be computed more effectively and accurately than the conventional finite difference method. In addition, the equation has the potential to be applied to many complex acoustic problems, because the derived equation is regularized by using the integral identity that incorporates the one-dimensional propagating wave and its material derivative. The validity of the formulations is demonstrated through examples having regular shapes such as the three-dimensional pulsating sphere and the one-dimensional duct, for which the analytical solutions are available. As an example for an irregular domain, the two-dimensional model of an automotive interior cavity is dealt with in the view point of the noise level at the passenger’s ear position. The results show that the present method can be an effective tool for the shape optimization in designing the desired sound field. It is noted that the present method permits accurate sensitivities of the acoustic pressure on the boundary as well as at the field points. The present method is thought to be an alternative to the previous finite difference techniques for computing the shape sensitivity using the boundary element method and the formal derivative method using the finite element method.

A BEM sensitivity formulation for three‐dimensional active noise control

“…The derivative of the nodal coordinates can be obtained by superposition of the source centre coordinates C j and the source local coordinates X j , i.e., (25) x j,m = (…”

“…The proposed BEM is coupled with the Adaptive Cross Approximation (ACA), the Hierarchical matrix (H-matrix) format and the GMRES [21] to allow fast solutions for large scale problems. Previous work on BEM sensitivities can be found in [22][23][24][25][26][27][28].…”

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

“…The above discussion indicates that (23) is the non-singular representation of the normal derivative of Green's formula (18). Analytically evaluating hypersingular integrals along curved boundaries can be carried out only in a few special cases.…”

“…The preceding equation indicates that (18) actually is a Fredholm integral equation of the ÿrst kind for both the Dirichlet and Neumann problems. Fredholm integral equations of the ÿrst kind with non-singular kernels are known to be ill-conditioned; special methods are therefore necessitated for solving such integral equations.…”

“…Let us examine next the non-singular representation (23) of the normal derivative equation (18), with the same elliptic cylinder and surface functions and @ =@n given in the previous case. The series coe cients a m in (21) and b n in (22) were again obtained from (31), with the last coe cients being less than 10 −6 ; all integrals in the right-hand side of (23) were evaluated in terms of the 30-point Gauss-Legendre quadrature, with one exception when the nearly singular kernel appeared.…”