In this paper, a highly efficient fast boundary element method (BEM) for solving largescale engineering acoustic problems in a broad frequency range is developed and implemented. The acoustic problems are modeled by the Burton-Miller boundary integral equation (BIE), thus the fictitious frequency issue is completely avoided. The BIE is discretized by using the Nyström method based on the curved quadratic elements, leading to simple numerical implementation (no edge or corner problems) and high accuracy in the BEM analysis. The linear systems are solved iteratively and accelerated by using a newly developed kernel-independent wideband fast directional algorithm (FDA) for fast summation of oscillatory kernels. In addition, the computational efficiency of the FDA is further promoted by exploiting the low-rank features of the translation matrices, resulting in two-to three-fold reduction in the computational time of the multipole-tolocal translations. The high accuracy and nearly linear computational complexity of the present method are clearly demonstrated by typical examples. An acoustic scattering problem with dimensionless wave number kD (where k is the wave number and D is the typical length of the obstacle) up to 1000 and the degrees of freedom up to 4 million is successfully solved within 10 hours on a computer with one core and the memory usage is 24 GB.

Purpose -The purpose of this paper is to develop an easy-to-implement and accurate fast boundary element method (BEM) for solving large-scale elastodynamic problems in frequency and time domains. Design/methodology/approach -A newly developed kernel-independent fast multipole method (KIFMM) is applied to accelerating the evaluation of displacements, strains and stresses in frequency domain elastodynamic BEM analysis, in which the far-field interactions are evaluated efficiently utilizing equivalent densities and check potentials. Although there are six boundary integrals with unique kernel functions, by using the elastic theory, the authors managed to accelerate these six boundary integrals by KIFMM with the same kind of equivalent densities and check potentials. The boundary integral equations are discretized by Nyström method with curved quadratic elements. The method is further used to conduct the time-domain analysis by using the frequency-domain approach. Findings -Numerical results show that by the fast BEM, high accuracy can be achieved and the computational complexity is brought down to linear. The performance of the present method is further demonstrated by large-scale simulations with more than two millions of unknowns in the frequency domain and one million of unknowns in the time domain. Besides, the method is applied to the topological derivatives for solving elastodynamic inverse problems. Originality/value -An efficient KIFMM is implemented in the acceleration of the elastodynamic BEM. Combining with the Nyström discretization based on quadratic elements and the frequencydomain approach, an accurate and highly efficient fast BEM is achieved for large-scale elastodynamic frequency domain analysis and time-domain analysis.

SUMMARYWe describe a new equivalent charge formulation (ECF), i.e. the combined ECF (CECF), for electrostatic analysis of structures consisting of conductors and dielectrics. The CECF uses a weighted combination of the single-and the adjoint double-layer operators to account for the potential on the conductor-dielectric surface and is found to have better conditioning than the standard ECF. A perturbation approach is presented to insure that the capacitances are computed accurately even when the permittivity ratios of the dielectrics are vary large. Unlike the original perturbation approach, this new approach uses only one system matrix with different right-hand sides.A wavelet Galerkin boundary element method (WGBEM) for solving the ECF and CECF is developed based on the new variable-order WGBEM introduced in (J. Several numerical examples show that the proposed CECF converges faster than the ECF, and that the WGBEM, using both the ECF and CECF, has almost linear complexity in solving large-scale 3-D electrostatic problems. Moreover, since the truncated non-standard form is computed once and then stored, the WGBEM is very suitable for solving problems with multiple right-hand sides, like the perturbation approach in this paper.

The kernel-independent fast multipole method (KIFMM) proposed in [1] is of almost linear complexity. In the original KIFMM the time-consuming M2L translations are accelerated by FFT. However, when more equivalent points are used to achieve higher accuracy, the efficiency of the FFT approach tends to be lower because more auxiliary volume grid points have to be added. In this paper, all the translations of the KIFMM are accelerated by using the singular value decomposition (SVD) based on the lowrank property of the translating matrices. The acceleration of M2L is realized by first transforming the associated translating matrices into more compact form, and then using low-rank approximations. By using the transform matrices for M2L, the orders of the translating matrices in upward and downward passes are also reduced. The improved KIFMM is then applied to accelerate BEM. The performance of the proposed algorithms are demonstrated by three examples. Numerical results show that, compared with the original KIFMM, the present method can reduce about 40% of the iterating time and 25% of the memory requirement.

A highly efficient fast boundary element method (BEM) for solving large-scale engineering acoustic problems in a broad frequency range is developed and implemented. The acoustic problems are modeled by the Burton-Miller boundary integral equation (BIE), thus the fictitious frequency issue is completely avoided. The BIE is discretized by using the collocation method with piecewise constant elements. The linear systems are solved iteratively and accelerated by using a newly developed kernel-independent wideband fast directional algorithm (FDA) for fast summation of oscillatory kernels. In addition, the computational efficiency of the FDA is further promoted by exploiting the low-rank features of the translation matrices. The high accuracy and nearly linear computational complexity of the present method are clearly demonstrated by typical examples. An acoustic scattering problem with dimensionless wave number kD (where k is the wave number and D is the typical length of the obstacle) up to 1000 and the degrees of freedom up to 4 million is successfully solved within 4 hours on a computer with one core and the memory usage is 24.7 GB.