A Direct Differentiation Boundary Integral Sensitivity Formulation of 3-D Acoustic Problems Oriented to FMBEM

摘要

Analysis of acoustic sensitivity characteristics with respect to design variables is an important step of acoustic design and optimization processes. This paper presents a design sensitivity formulation of boundary integral equation for three-dimensional acoustic problems based on the direct differentiation method. The basic boundary integral equation is a linear combination of standard boundary integral equation and its normal derivative. This type of the boundary integral equation is used to avoid the divergent numerical results for fictitious eigen-frequencies observed for exterior problems [3]. The normal derivative of the standard boundary integral equation is a hyper-singular type, hence in the standard BEM, it is regularized by using the fundamental solution of Laplace's equation [1,2]. However, regularized boundary integral equation is not efficient for fast multipole BEM (FMBEM), because multipole expansion formulas and other translation formulas have to be implemented also for the fundamental solution of Laplace's equation. In this study, we derive a boundary integral sensitivity formula using the hypersingular type boundary Laplace's equation. In this study, we derive a boundary integral sensitivity formula using the hypersingular type boundary integral equation to use FMBEM for numerical analyses. Because the constant elements are assumed to be used in the numerical model, the collocation point can always be assumed to be placed on a smooth part of the boundary. The sound pressure at a point in the domain is given by an integral representation in the form, as follows:p(x)+lrq*(x,y).p(y)dΓ(y)=lrp*(x,y).q(y)dΓ(y) where p*(x,y)is the fundamental solution of three-dimensional Helmholtz equation, and q*(x,y)is itsnormal derivative.The differentiation of Eq. (1) with respect to an arbitrary design variable such as shape design parameter, frequency, mass density and impedance can be obtained by the direct differentiation method, as follows: p(x)+lrq*(x,y).p(y)dΓ(y)+lrq*(x,y).p(y)dΓ(y)+lrq*r (x,y).p(y)dΓ(y)=lrp*(x,y).q(y)dΓ(y)+lrp*(x,y).q(yr )dΓ(y)+lΓp*(x,y).q(y)dΓ(y)where upper dot(.)denotes differentiation with respect to the design variable.It is well-known that, without any care, the solution of an exterior acoustic problem is violated at the eigenfrequencies of the interior problem, when the boundary integral equation (BIE) method is applied to solve the problem directly. To deal with this problem, the Burton-Miller formulation using a linear combination of the boundary integral equation (BIE) and hypersingular BIE (HBIE) is used in this paper as follows, p(x)+lΓq*(x,y).p(y)dΓ(y)+a×lΓq(x,y).p(y)dΓ(y)=r lΓp(x,y).q(y)dΓ(y)+a×l×p(x,y).q(y)d×(y)a×q(x) where is a coupling constant that can be chosen as αi/k[7],k here is the wave number, and (~)denotes the derivative with respect to the normal at the collocation point x. Then, the differentiation of Eq.(3) with respect to a design variable gives p(x)+lΓq*(x,y).p(y)dΓ(y)+a×lΓq.(x,y).p(y)dΓ(y)r +lΓq*(x,y).p(y)dΓ(y)+a×lΓq.(x,y).p(y)dΓ(y)+lΓr q*(x,y).p(y)dΓ(y)+a×lΓq.(x,y).p(y)dΓ(y)=lΓp*(xr,y).q(y)dΓ(y)+a×lΓp*(x,y).q(y)dΓ(y)a×q(x)+lΓpr *(x,y).q(y)dΓ(y)a×lp*(x,y).q(y)dΓ(y)+lΓp*(x,y).r q(y)dΓ(y)+a×lΓp*(x,y).q(y)dΓ(y) By taking the limit of the internal point x of Eq.(4) to the boundary, we obtain the boundary integral equation which relates the sensitivity coefficients of the boundary sound pressure and the particle velocities.