Efficient FMM accelerated vortex methods in three dimensions via the Lamb-Helmholtz decomposition

Vortex element methods are often used to efficiently simulate incompressible flows using Lagrangian techniques. Use of the FMM (Fast Multipole Method) allows considerable speed up of both velocity evaluation and vorticity evolution terms in these methods. Both equations require field evaluation of constrained (divergence free) vector valued quantities (velocity, vorticity) and cross terms from these. These are usually evaluated by performing several FMM accelerated sums of scalar harmonic functions. We present a formulation of the vortex methods based on the Lamb-Helmholtz decomposition of the velocity in terms of two scalar potentials. In its original form, this decomposition is not invariant with respect to translation, violating a key requirement for the FMM. One of the key contributions of this paper is a theory for translation for this representation. The translation theory is developed by introducing "conversion" operators, which enable the representation to be restored in an arbitrary reference frame. Using this form, extremely efficient vortex element computations can be made, which need evaluation of just two scalar harmonic FMM sums for evaluating the velocity and vorticity evolution terms. Details of the decomposition, translation and conversion formulae, and sample numerical results are presented

Scalable Fast Multipole Methods on Heterogeneous Architecture

The N-body problem appears in many computational physics simulations. At each time step the computation involves an all-pairs sum whose complexity is quadratic, followed by an update of particle positions. This cost means that it is not practical to solve such dynamic N-body problems on large scale. To improve this situation, we use both algorithmic and hardware approaches. Our algorithmic approach is to use the Fast Multipole Method (FMM), which is a divide-and-conquer algorithm that performs a fast N-body sum using a spatial decomposition and is often used in a time-stepping or iterative loop, to reduce such quadratic complexity to linear with guaranteed accuracy. Our hardware approach is to use heterogeneous clusters, which comprised of nodes that contain multi-core CPUs tightly coupled with accelerators, such as graphics processors unit (GPU) as our underline parallel processing hardware, on which efficient implementations require highly non-trivial re-designed algorithms. In this dissertation, we fundamentally reconsider the FMM algorithms on heterogeneous architectures to achieve a significant improvement over recent/previous implementations in literature and to make the algorithm ready for use as a workhorse simulation tool for both time-dependent vortex flow problems and for boundary element methods. Our major contributions include: 1. Novel FMM data structures using parallel construction algorithms for dynamic problems. 2. A fast hetegenenous FMM algorithm for both single and multiple computing nodes. 3. An efficient inter-node communication management using fast parallel data structures. 4. A scalable FMM algorithm using novel Helmholz decomposition for Vortex Methods (VM). The proposed algorithms can handle non-uniform distributions with irregular partition shapes to achieve workload balance and their MPI-CUDA implementations are highly tuned up and demonstrate the state of the art performances

Efficient Fast Multipole Accelerated Boundary Elements via Recursive Computation of Multipole Expansions of Integrals

In boundary element methods (BEM) in $\mathbb{R}^3$, matrix elements and right hand sides are typically computed via analytical or numerical quadrature of the layer potential multiplied by some function over line, triangle and tetrahedral volume elements. When the problem size gets large, the resulting linear systems are often solved iteratively via Krylov subspace methods, with fast multipole methods (FMM) used to accelerate the matrix vector products needed. When FMM acceleration is used, most entries of the matrix never need be computed explicitly - {\em they are only needed in terms of their contribution to the multipole expansion coefficients.} We propose a new fast method for the analytical generation of the multipole expansion coefficients produced by the integral expressions for single and double layers on surface triangles; charge distributions over line segments and over tetrahedra in the volume; so that the overall method is well integrated into the FMM, with controlled error. The method is based on the $O(1)$ per moment cost recursive computation of the moments. The method is developed for boundary element methods involving the Laplace Green's function in ${\mathbb R}^3$. The derived recursions are first compared against classical quadrature algorithms, and then integrated into FMM accelerated boundary element and vortex element methods. Numerical tests are presented and discussed.Comment: 6 figures, preprin

A Method to Compute Periodic Sums

In a number of problems in computational physics, a finite sum of kernel functions centered at N particle locations located in a box in three dimensions must be extended by imposing periodic boundary conditions on box boundaries. Even though the finite sum can be efficiently computed via fast summation algorithms, such as the fast multipole method (FMM), the periodized extension, posed as an infinite sum of kernel functions, centered at the particle locations in the box, and their images, is usually treated via a different algorithm, Ewald summation. This is then accelerated via the fast Fourier transform (FFT). A method for computing this periodized sum just using a blackbox finite fast summation algorithm is presented in this paper. The method splits the periodized sum in to two parts. The first, comprising the contribution of all points outside a large sphere enclosing the box, and some of its neighbors, is approximated inside the box by a collection of kernel functions (“sources”) placed on the surface of the sphere. These are approximated within the box using an expansion in terms of spectrally convergent local basis functions. The second part, comprising the part inside the sphere, and including the box and its immediate neighborhood, is treated via the summation algorithm. The coefficients of the sources are determined by least squares collocation of the periodicity condition of the total potential, imposed on a circumspherical surface for the box. While the method is presented in general, details are worked out for the case of evaluating potentials and forces due to electrostatically charged particles in a box. Results show that when used with the FMM, the periodized sum can be computed to any specified accuracy, at a cost of about twice the cost of the free-space FMM with the same accuracy. Several technical details and efficient algorithms for auxiliary computations are also provided, as are numerical comparisons

The double-tree method : An O(n) unsteady aerodynamic lifting surface method

Open Access via Wiley agreement. Acknowledgements Some results in this work were obtained using the Maxwell High Performance Computing Cluster of the University of Aberdeen IT Service (www.abdn.ac.uk/staffnet/research/hpc.php), provided by Dell Inc. and supported by Alces Software. The lead author would also like to thank the University of Aberdeen for their research scholarship funding.Peer reviewedPublisher PD

Método de vórtices discretos e multipolos rápidos aplicados em escoamentos não-estacionários

Orientadores: William Roberto Wolf, Alex Mendonça BimbatoDissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia MecânicaResumo: O método de vórtices discretos (DVM) não necessita de malhas por ser uma descrição lagrangiana da equação do transporte de vorticidade. Esta, por sua vez, é separada em termos difusivos e convectivos. Esta equação é resolvida pela discretização do campo de vorticidade em N vórtices discretos. Diversos métodos podem ser usados para modelar os efeitos da difusão; pode-se citar o método do passo aleatório, método do crescimento do núcleo, método da troca de intensidade, entre outros. Os termos de convecção são resolvidos pela utilização da derivada material para evitar termos não-lineares. Assim, cada vórtice discreto é convectado pelo campo local de velocidade, que é calculado pela contribuição do escoamento livre, superfícies sólidas e pela solução da lei de Biot-Savart que rege a interação entre vórtices. Entretanto, esta última contribuição exige um dispendioso passo de convolução com O(N 2 ) operações, que impõe restrição no uso do método para a solução de problemas típicos de engenharia. Assim, métodos alternativos são necessários para acelerar a solução do DVM. O método de multipolos rápidos, FMM, considerado um dos 10 melhores algoritmos do século 20, foi proposto por Greengard and Rokhlin (1987) para a solução da interação gravitacional entre N corpos. O algoritmo consiste no agrupamento da influência de elementos próximo entre si, e então calcula-se a interação em regiões distantes, como por exemplo o centro de outro agrupamento. Esta operação tem custo computacional de ordem O(N) para um número N suficientemente grande. Assim, a influência entre grupos distantes de elements é calculada mais eficientemente do que a ordem O(N 2 ) para calcular diretamente a lei de Biot-Savart. Neste trabalho, nós usamos um esquema não-adaptativo multi-nível do FMM com melhorias para acelerar o preprocessamento bem como os cálculos de interação no FMM. O acoplamento dos métodos é investigado para três diferentes problemas: a simulação de um cilindro abruptamente acelerado e a evolução temporal de uma esteira de aeronave assim como de uma camada de mistura. Uma comparação do custo computacional do método acelerado é comparado com a solução usando apenas a lei de Biot-Savart. Finalmente, como uma camada de mistura requer condições de contorno periódicas, o estudo de uma série alternativa para o cerne do FMM é feito com a investigação da precisão e do tempo computacionalAbstract: The Discrete Vortex Method (DVM) is a meshless numerical method based on a Lagrangian description of the vorticity transport equation, which is split into diffusive and convective effects. In order to solve this equation, the vorticity field is discretized in N vortex-particles. Several formulations can be used to model the diffusive effects, e.g., the random walk method, the core spreading method, the particle strength exchange, etc. The convection term can be treated using a material derivative to avoid the solution of a non-linear term. Therefore, each vortex is convected with the fluid velocity field, which is evaluated by the contributions from the incident flow, the perturbation due to the body, and the vortex-vortex interactions calculated by the Biot-Savart law. However, the last contribution requires an expensive convolution step of O(N 2 ) calculations, which imposes a heavy limitation on the usage of the method to solve engineering problems. With that in mind, alternative ways are required to accelerate the DVM simulations. The Fast Multipole Method is listed as one of the top 10 algorithms of the twentieth century, and it was developed by Greengard and Rokhlin (1987) for the solution of N -body gravitational problems. The algorithm consists of clustering the influence of elements close to each other, and then evaluating their interaction at distant locations, i.e., the center of far away clusters, with computational cost O(N) for a large number N . This way, the influence among different groups of particles is computed faster than the O(N 2 ) operations required by the direct Biot-Savart law. Here, we use the non-adaptive multi-level FMM with an optimization in the pre-processing steps, along with several techniques to speed up both pre-processing and FMM steps. The coupling of DVM and FMM is investigated in the present work, in three different problems: the simulation of the flow past an impulsively started cylinder and the temporal evolution of both an aircraft wake as well as a mixing layer. For these problems, there is a comparison of the computational time used by both the DVM-FMM and solely by the DVM. Finally, as the temporal evolution of a mixing layer requires periodic boundary conditions, a solution of an alternative kernel for the FMM is also employed in order to solve the problem, followed by the investigation of its precisionMestradoTermica e FluidosMestre em Engenharia Mecânica33003017CAPE

Accurate computation of Galerkin double surface integrals in the 3-D boundary element method

Many boundary element integral equation kernels are based on the Green's functions of the Laplace and Helmholtz equations in three dimensions. These include, for example, the Laplace, Helmholtz, elasticity, Stokes, and Maxwell's equations. Integral equation formulations lead to more compact, but dense linear systems. These dense systems are often solved iteratively via Krylov subspace methods, which may be accelerated via the fast multipole method. There are advantages to Galerkin formulations for such integral equations, as they treat problems associated with kernel singularity, and lead to symmetric and better conditioned matrices. However, the Galerkin method requires each entry in the system matrix to be created via the computation of a double surface integral over one or more pairs of triangles. There are a number of semi-analytical methods to treat these integrals, which all have some issues, and are discussed in this paper. We present novel methods to compute all the integrals that arise in Galerkin formulations involving kernels based on the Laplace and Helmholtz Green's functions to any specified accuracy. Integrals involving completely geometrically separated triangles are non-singular and are computed using a technique based on spherical harmonics and multipole expansions and translations, which results in the integration of polynomial functions over the triangles. Integrals involving cases where the triangles have common vertices, edges, or are coincident are treated via scaling and symmetry arguments, combined with automatic recursive geometric decomposition of the integrals. Example results are presented, and the developed software is available as open source

Fast and Accurate Boundary Element Methods in Three Dimensions

The Laplace and Helmholtz equations are two of the most important partial differential equations (PDEs) in science, and govern problems in electromagnetism, acoustics, astrophysics, and aerodynamics. The boundary element method (BEM) is a powerful method for solving these PDEs. The BEM reduces the dimensionality of the problem by one, and treats complex boundary shapes and multi-domain problems well. The BEM also suffers from a few problems. The entries in the system matrices require computing boundary integrals, which can be difficult to do accurately, especially in the Galerkin formulation. These matrices are also dense, requiring O(N^2) to store and O(N^3) to solve using direct matrix decompositions, where N is the number of unknowns. This can effectively restrict the size of a problem. Methods are presented for computing the boundary integrals that arise in the Galerkin formulation to any accuracy. Integrals involving geometrically separated triangles are non-singular, and are computed using a technique based on spherical harmonics and multipole expansions and translations. Integrals involving triangles that have common vertices, edges, or are coincident are treated via scaling and symmetry arguments, combined with recursive geometric decomposition of the integrals. The fast multipole method (FMM) is used to accelerate the BEM. The FMM is usually designed around point sources, not the integral expressions in the BEM. To apply the FMM to these expressions, the internal logic of the FMM must be changed, but this can be difficult. The correction factor matrix method is presented, which approximates the integrals using a quadrature. The quadrature points are treated as point sources, which are plugged directly into current FMM codes. Any inaccuracies are corrected during a correction factor step. This method reduces the quadratic and cubic scalings of the BEM to linear. Software is developed for computing the solutions to acoustic scattering problems involving spheroids and disks. This software uses spheroidal wave functions to analytically build the solutions to these problems. This software is used to verify the accuracy of the BEM for the Helmholtz equation. The product of these contributions is a fast and accurate BEM solver for the Laplace and Helmholtz equations