Half Space Acoustic Problems Analysis by Fast Multipole Boundary Face Method

Eddy current nondestructive evaluation (NDE) of airframe structures involves the detection of electromagnetic field irregularities due to non-conducting inhomogeneities in an electrically conducting material. Usually, the eddy current NDE problem can be formulated by the boundary integral equations (BIE) and discretized into matrix equations by the method of moments (MoM) or the boundary element method (BEM). The fast multipole method (FMM) is a well-established and effective method for accelerating numerical solutions of the matrix equations. Accelerated by the FMM, the BIE method can now solve large-scale electromagnetic wave propagation and diffusion problems. The traditional BIE method requires O(N) operations to compute the system of equations and another O(N) operations to solve the system using direct solvers, with N being the number of unknowns; in contrast, the BIE method accelerated by the two-level FMM can potentially reduce the operations and memory requirement to O(N). Moreover, several approaches have been proposed for the field calculation in the presence of flaws in three dimensional NDE; however, seldom work has been done in applying efficient methods to seek rapid solution in eddy current NDE simulation. As elaborated in the dissertation, we introduce a fast multipole BIE method for two-dimensional diffusive scalar problem and an efficient BIE method for three-dimensional eddy current NDE. Firstly, we work with the two-dimensional Helmholtz equation with a complex wave number for non-trivial boundary geometry. We describe the FMM acceleration procedure of the BIE method and its features briefly, explaining that the FMM is not only efficient in meshing complicated geometries, accurate for solving singular fields or

Standard numerical models in acoustics rely on the isentropic Helmholtz equation. Its derivation assumes adiabatic and reversible, i.e., dissipation-free, wave propagation. Sound waves in fluids are, however, subject to viscous and thermal losses. These losses originate from viscous friction and heat conduction, leading to the formation of acoustic boundary layers. Considering these effects becomes significant in setups with acoustic cavities of similar dimension as the boundary layers. Recently, boundary element methods (BEM) accounting for the viscothermal dissipation have been proposed. These methods limit the discretization to the surface of the fluid domain and require significantly fewer degrees of freedom than comparable finite element models. However, the BEM coefficient matrices are fully populated, resulting in high computational costs and storage requirements. This study develops a new BEM formulation for acoustics in viscothermal fluids that uses the fast multipole method - a low-rank approximation technique based on a hierarchical subdivision of the computational domain - to alleviate this shortcoming. It is shown that the fast viscothermal BEM improves the algorithmic complexity over the conventional formulation, reducing both the memory requirements and solution time. The results indicate good scalability of the method making it feasible for larger applications such as acoustic metamaterials.

Fast multipole singular boundary method for Stokes flow problems

Numerical simulations related to human hearing, architectural and underwater acoustics, and multiple scattering require computational solution of the Helmholtz equation in three dimensions with complex shaped boundaries. Boundary element methods (BEM) are among the most accurate and efficient methods used for this purpose. However, solution of high frequency/large domain problems is challenging due to poor scaling of conventional solvers with the frequency and domain size. The use of the fast multipole methods (FMM) resolves many problems related to the scalability [N.A. Gumerov and R. Duraiswami, J. Acoust. Soc. Am. 125(1), 191205, 2009]. Additional accelerations are needed to be able to solve practical problems over the entire range of human audible frequencies and can be provided using graphics processors (GPUs) with multicore CPUs. In this work, we report development and demonstration of the FMM/GPU accelerated BEM for the Helmholtz equation in 3D designed for hybrid CPU/GPU architectures. Innovations related to choices of preconditioners, parallelization strategies, and choice of optimal parameters will be presented. A single PC version of the algorithm shows accelerations of the order of 10 times compared to the BEM accelerated with the FMM alone. Results for computing head related transfer functions and other standard calculations will be provided.

A numerical solution for the diffraction of linear waves around a vertical cylinder is presented in this paper using the fast multipole boundary element method (FMBEM). The fast multipole method (FMM) is employed to accelerate the traditional boundary element method. The accuracy and efficiency of the FMBEM for 2D Helmholtz equation is demonstrated, and it’s proved that this approach is suitable for solving large-scaled problems. In addition, numerical results of water dynamics coefficients, which are related to the diffraction problems of linear waves around a cylinder, are supplied in this paper.

The boundary element method is a numerical method that can be advantageously used for a wide range of engineering problems, including the stress concentration problems encountered in fracture mechanics. In linear elastic fracture mechanics (LEFM), the stress intensity factor is an important parameter. Cracks, if present in the region experiencing the modes of deformation, increase the stress amplitude significantly and this high stress may lead to premature failure of the engineering components. If the value of the SIF is known, it is possible to predict whether the crack will propagate or not. As the conventional boundary integral equation (CBIE) degenerates when a mathematical crack is modelled, a previously developed dual boundary integral equation approach has been adopted in the current work. It utilizes the hypersingular boundary integral equation (HBIE) along with the CBIE. A weakly singular form of HBIE is utilized in the current work to eliminate the hypersingularity analytically. Stress intensity factors are evaluated using the crack opening displacement (COD), displacement extrapolation (DE), and the J-integral approaches. A stand-alone code has been developed for calculating the stress intensity factors of general two-dimensional domains with cracks. The code has been validated by evaluating the stress intensity factors for the standard components, for which the stress intensity factor values are available in the literature. Accurate and well-converged results are obtained proving the robustness of the code. A linear combination of the CBIE and HBIE was applied at the crack and a significant (87–97 per cent) reduction in the condition numbers for the system of equations was observed for the examples studied. Again, the results obtained are accurate and well converged.

The boundary element method (BEM) is a numerical method for solving boundary-value or initial-value problems formulated by use of boundary integral equations. In the BEM, only the boundaries - that is, surfaces for three-dimensional problems or curves for two dimensional (2D) problems - of a problem domain need to be discretized. However the boundary element method (BEM) has been limited to solving problems with a few thousand degrees of freedom (DOFs) on a personal computer. This is because the conventional BEM, in general, produces dense and nonsymmetric matrices. The main idea of the fast multipole (FM) BEM is to employ iterative solvers to solve the BEM system of equations. Using this method we can solve models with more than one million equations on a laptop computer. In this paper, the governing equations for elasticity problems are reviewed first. Numerical examples are provided to demonstrate the accuracy and efficiencies of fast multipole method (FMM) for solving 2D elasticity problems.

Though computational molecular dynamics is an effective tool for nano-scale phenomenon analysis, computational costs associated with its computer simulation are extremely high. There are two major computational steps associated with computer simulation of dynamics of molecular structures. They are calculation of interatomic forces and formation and solution of the equations of motion. Currently, these two computational steps are treated separately in most commonly-used methods. For example, Fast Multipole Method (FMM) and Cell Multipole Method (CMM) have been used for calculation of interatomic forces, and Cartesian Coordinate Method (CCM) and Internal Coordinate Molecular Dynamics Method (ICMD) have been created for the formation and solution of equations of motion of an atomistic molecular system. In this paper, a new procedure is presented through a proper integration between multibody molecular algorithms (MMA) and fast multipole methods to improve computational efficiency for computer simulation of the dynamical behaviors of multibody molecular structures in polymers and biopolymers. For the computational costs associated with interatomic forces, a fast multipole method is used to calculate the interatomic forces due to the potentials. For the computational costs associated with formation and solution of equations of motion, a multibody molecular algorithm developed by the author in his previous work will be utilized to integrate with fast multipole methods. The algorithm significantly improves computational efficiency when comparing with its counterpart procedures. The fast multipole method begins by scaling all atoms into a box with coordinate ranges to ensure numerical stability of subsequent operations. The parent box is then divided into half in the direction of each Cartesian axis and each child box is then subdivided to form a computational family tree. The flow of calculations is carried out along the tree structure with five passes. The fast multipole method has been improved and modified to achieve better effectiveness and higher efficiency since it was created. The multibody molecular algorithm starts with numbering subsets, forming bond graph, and developing three computing passes along the tree structure of an atomistic molecular system. Computing data flows in the fast multipole method and the multibody molecular algorithm will properly line up with the parent-child recursive relationship along the configuration of the tree structure due to linear recursive natures of both fast multipole method and multibody molecular algorithm. Then the time spent on the recursive simulation passes in the fast multipole method for computing forces may overlap with the time spent on the three recursive computational passes in the multibody molecular algorithm for forming and solving equations of motion.

A modified dual-level fast multipole boundary element method based on the Burton–Miller formulation for large-scale three-dimensional sound field analysis

It is suitable to solve the acoustic problems by using the fast multipole boundary element method (FMBEM), since the FMBEM can accelerate the matrix-vector multiplication dramatically by reducing the CPU time and memory of conventional boundary element method to O(N log2N) and O(N) respectively. We propose a 3D acoustic FMBEM based on Burton-Miller formulation in this paper. A new adaptive algorithm is applied to the diagonal form FMBEM, and a new proposed analytical moment formulation is used in the moment computation. Both of them further improve the efficiency of FMBEM. Acoustic scattering of soft sphere at resonant frequency is investigated to validate the accuracy of solution using Burton-Miller formulation. Comparisons of solution to the multi-radiating spheres problem with the one solved by Bapat's program demonstrate the accuracy and the efficiency of our algorithm in solving large-scale acoustic problems. In the end, we use our algorithm to analyze the inner sound filed of a car and dolphin acoustic scattering.

An adaptive expansion technique in the fast multipole method for 3D acoustics problems at low frequencies

Animportant and computationally demanding part of molecular dynamicssimulations is the calculation of long-range electrostatic interactions.Today, the prevalent method to compute these interactions is particlemesh Ewald (PME). The PME implementation in the GROMACS moleculardynamics package is extremely fast on individual GPU nodes. However,for large scale multinode parallel simulations, PME becomes the mainscaling bottleneck as it requires all-to-all communication betweenthe nodes; as a consequence, the number of exchanged messages scalesquadratically with the number of involved nodes in that communicationstep. To enable efficient and scalable biomolecular simulations onfuture exascale supercomputers, clearly a method with a better scalingproperty is required. The fast multipole method (FMM) is such a method.As a first step on the path to exascale, we have implemented a performance-optimized,highly efficient GPU FMM and integrated it into GROMACS as an alternativeto PME. For a fair performance comparison between FMM and PME, wefirst assessed the accuracies of the methods for various sets of inputparameters. With parameters yielding similar accuracies for both methods,we determined the performance of GROMACS with FMM and compared itto PME for exemplary benchmark systems. We found that FMM with a multipoleorder of 8 yields electrostatic forces that are as accurate as PMEwith standard parameters. Further, for typical mixed-precision simulationsettings, FMM does not lead to an increased energy drift with multipoleorders of 8 or larger. Whereas an ≈50 000 atom simulationsystem with our FMM reaches only about a third of the performancewith PME, for systems with large dimensions and inhomogeneous particledistribution, e.g., aerosol systems with water droplets floating ina vacuum, FMM substantially outperforms PME already on a single node.

The fast multipole method (FMM) is very efficient in solving integral equations. This paper applies the method to solve large solid-solid boundary integral equations for elastic waves in two dimensions. The scattering problem is first formulated with the boundary element method. FMM is then introduced to expedite the solution process. By using the FMM technique, the number of floating-point operations of the matrix-vector multiplication in a standard conjugate gradient algorithm is reduced from O(N 2) to O(N 1.5), where N is the number of unknowns. The matrix-filling time and the memory requirement are also of the order N 1.5. The computational complexity of the algorithm is further reduced to O(N 4/3) by using a ray propagation technique. Numerical results are given to show the accuracy and efficiency of FMM compared to the boundary element method with dense matrix.

This paper discusses a formulation and its applications of the new Fast Multipole Method (FMM) to three-dimensional Boundary Integral Equation Method (BIEM) in elastostatic crack problems. It is shown, through numerical experiments, that the new FMM is more efficient than the original FMM.

The complex variable fast multipole boundary element method for the analysis of strongly inhomogeneous media