A Modified Helmholtz Equation Least Squares Method for Reconstructing Vibroacoustic Quantities on an Arbitrarily Shaped Vibrating Structure

A modified Helmholtz equation least-square (HELS) method is developed to reconstruct vibroacoustic quantities on an arbitrarily shaped vibrating structure. Unlike the traditional nearfield acoustical holography that relies on the acoustic pressures collected on a hologram surface at a short stand-off distance to a target structure, this modified HELS method takes the partial normal surface velocities and partial acoustic pressures as the input data. The advantages of this approach include but not limited to: (1) The normal surface velocities that represent the nearfield effects are collected directly, which lead to a more accurate reconstruction of the normal surface velocity distribution; (2) The field acoustic pressures are also measured, which leads to a more accurate reconstruction of the acoustic pressure on the source surface as well as in the field; and (3) There is no need to measure the normal surface velocities over the entire surface, which makes this approach quite appealing in practice because most vibrating structures do not allow for measuring the normal surface velocities over the entire source surface as there are always obstacles or constrains around a target structure. Needless to say, regularization is necessary in reconstruction process since all inverse problems are mathematically ill-posed. To validate this approach, both numerical simulations and experimental results are presented. An optimal reconstruction scheme is developed via numerical simulations to achieve the most cost-effective reconstruction results for practical applications.

Dual Reciprocity Boundary Element Method untuk Menyelesaikan Masalah Infiltrasi Stasioner pada Saluran Datar Periodik

Abstrak. Penelitian ini membahas tentang penyelesaian masalah infiltrasi stasioner dari saluran datar dengan Dual Reciprocity Boundary Element Method (DRBEM). Persamaan pembangun untuk masalah ini adalah persamaan Richard. Menggunakan transformasi Kirchhoff dan relasi eksponensial konduktifitas hidrolik, persamaan Richard ditransformasi ke dalam persamaan infiltrasi stasioner dalam Matric Flux Potential (MFP). Persamaan infiltrasi dalam MFP selanjutnya diubah ke dalam persamaan Helmholtz termodifikasi. Model matematika infiltrasi stasioner pada saluran datar berbentuk Masalah Syarat batas Helmholtz termodifikasi Solusi numerik diperoleh dengan menyelesaikan persamaan Helmholtz termodifikasi menggunakan Dual Reciprocity Boundary Element Method (DRBEM) dengan pengambilan jumlah titik kolokasi eksterior dan interior yang bervariasi. Lebih lanjut, solusi numerik dan solusi analitik dibandingkan..Kata Kunci: Infiltrasi, saluran datar, persamaan helmholtz termodifikasi, DRBEM.Abstract. This research discusses about the problem solving of steady infiltration problem from flat channel with Dual Reciprocity Boundary Element Method (DRBEM). The governing equation for this problem is Richard’s equation. Using Kirchhoff transformation and exponential hydraulic conductivity relation, Richard’s equation is transformed into steady infiltration equation in the form of MFP. Infiltration equation in the form of MFP is then transformed to modified Helmholtz equation. A mathematical model of steady infiltration from flat channel in the form of boundary condition problem of modified Helmholtz EQUATION. Numerical solution is obtained by solving modified Helmholtz equation by using Dual Reciprocity Boundary Element Method (DRBEM) with various number of exterior and interior collocation points. Moreover, numerical and analytic solution are then compared.Keywords: infiltration, flat channel, modified Helmholtz equation, DRBEM

PENENTUAN JENIS FUNGSI BASIS RADIAL DALAM DUAL RECIPROCITY BOUNDARY ELEMENT METHOD

Problems involving modified Helmholtz equation are considered in this paper. To solve the problem numerically, dual reciprocity boundary element method (DRBEM) is employed. Some stage have been passed, using reciprocal relation to approximate boundary integral and domain integral in modified Helmholtz equation . Until, linear equation system are obtained in matrix form. MATLAB is used to calculate the solutions of Solutions of are compared between the exact solution and the numerical solution of modified Helmholtz equation. The numerical results are based on the using of three types of radial basis function to approximate domain integral, such as polinomial, poliharmonik spline and linear types. The solutions show that the polinomial and poliharmonik spline types are more stable and approach to exact solution than linear types.

Solution to the Modified Helmholtz Equation for Arbitrary Periodic Charge Densities

We present a general method for solving the modified Helmholtz equation without shape approximation for an arbitrary periodic charge distribution, whose solution is known as the Yukawa potential or the screened Coulomb potential. The method is an extension of Weinert’s pseudo-charge method [Weinert M, J Math Phys, 1981, 22:2433–2439] for solving the Poisson equation for the same class of charge density distributions. The inherent differences between the Poisson and the modified Helmholtz equation are in their respective radial solutions. These are polynomial functions, for the Poisson equation, and modified spherical Bessel functions, for the modified Helmholtz equation. This leads to a definition of a modified pseudo-charge density and modified multipole moments. We have shown that Weinert’s convergence analysis of an absolutely and uniformly convergent Fourier series of the pseudo-charge density is transferred to the modified pseudo-charge density. We conclude by illustrating the algorithmic changes necessary to turn an available implementation of the Poisson solver into a solver for the modified Helmholtz equation.