85 research outputs found

    New hybrid quadrature schemes for weakly singular kernels applied to isogeometric boundary elements for 3D Stokes flow

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    This work proposes four novel hybrid quadrature schemes for the efficient and accurate evaluation of weakly singular boundary integrals (1/r kernel) on arbitrary smooth surfaces. Such integrals appear in boundary element analysis for several partial differential equations including the Stokes equation for viscous flow and the Helmholtz equation for acoustics. The proposed quadrature schemes apply a Duffy transform-based quadrature rule to surface elements containing the singularity and classical Gaussian quadrature to the remaining elements. Two of the four schemes additionally consider a special treatment for elements near to the singularity, where refined Gaussian quadrature and a new moment-fitting quadrature rule are used. The hybrid quadrature schemes are systematically studied on flat B-spline patches and on NURBS spheres considering two different sphere discretizations: An exact single-patch sphere with degenerate control points at the poles and an approximate discretization that consist of six patches with regular elements. The efficiency of the quadrature schemes is further demonstrated in boundary element analysis for Stokes flow, where steady problems with rotating and translating curved objects are investigated in convergence studies for both, mesh and quadrature refinement. Much higher convergence rates are observed for the proposed new schemes in comparison to classical schemes

    International Workshop on Finite Elements for Microwave Engineering

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    When Courant prepared the text of his 1942 address to the American Mathematical Society for publication, he added a two-page Appendix to illustrate how the variational methods first described by Lord Rayleigh could be put to wider use in potential theory. Choosing piecewise-linear approximants on a set of triangles which he called elements, he dashed off a couple of two-dimensional examples and the finite element method was born. … Finite element activity in electrical engineering began in earnest about 1968-1969. A paper on waveguide analysis was published in Alta Frequenza in early 1969, giving the details of a finite element formulation of the classical hollow waveguide problem. It was followed by a rapid succession of papers on magnetic fields in saturable materials, dielectric loaded waveguides, and other well-known boundary value problems of electromagnetics. … In the decade of the eighties, finite element methods spread quickly. In several technical areas, they assumed a dominant role in field problems. P.P. Silvester, San Miniato (PI), Italy, 1992 Early in the nineties the International Workshop on Finite Elements for Microwave Engineering started. This volume contains the history of the Workshop and the Proceedings of the 13th edition, Florence (Italy), 2016 . The 14th Workshop will be in Cartagena (Colombia), 2018

    Windowed Green function method for wave scattering by periodic arrays of 2D obstacles

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    This paper introduces a novel boundary integral equation (BIE) method for the numerical solution of problems of planewave scattering by periodic line arrays of two-dimensional penetrable obstacles. Our approach is built upon a direct BIE formulation that leverages the simplicity of the free-space Green function but in turn entails evaluation of integrals over the unit-cell boundaries. Such integrals are here treated via the window Green function method. The windowing approximation together with a finite-rank operator correction -- used to properly impose the Rayleigh radiation condition -- yield a robust second-kind BIE that produces super-algebraically convergent solutions throughout the spectrum, including at the challenging Rayleigh-Wood anomalies. The corrected windowed BIE can be discretized by means of off-the-shelf Nystr\"om and boundary element methods, and it leads to linear systems suitable for iterative linear-algebra solvers as well as standard fast matrix-vector product algorithms. A variety of numerical examples demonstrate the accuracy and robustness of the proposed methodolog

    On a Calder\'on preconditioner for the symmetric formulation of the electroencephalography forward problem without barycentric refinements

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    We present a Calder\'on preconditioning scheme for the symmetric formulation of the forward electroencephalographic (EEG) problem that cures both the dense discretization and the high-contrast breakdown. Unlike existing Calder\'on schemes presented for the EEG problem, it is refinement-free, that is, the electrostatic integral operators are not discretized with basis functions defined on the barycentrically-refined dual mesh. In fact, in the preconditioner, we reuse the original system matrix thus reducing computational burden. Moreover, the proposed formulation gives rise to a symmetric, positive-definite system of linear equations, which allows the application of the conjugate gradient method, an iterative method that exhibits a smaller computational cost compared to other Krylov subspace methods applicable to non-symmetric problems. Numerical results corroborate the theoretical analysis and attest of the efficacy of the proposed preconditioning technique on both canonical and realistic scenarios

    Nystrom methods for high-order CQ solutions of the wave equation in two dimensions

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    An investigation of high order Convolution Quadratures (CQ) methods for the solution of the wave equation in unbounded domains in two dimensions is presented. These rely on Nystrom discretizations for the solution of the ensemble of associated Laplace domain modified Helmholtz problems. Two classes of CQ discretizations are considered: one based on linear multistep methods and the other based on Runge-Kutta methods. Both are used in conjunction with Nystrom discretizations based on Alpert and QBX quadratures of Boundary Integral Equation (BIE) formulations of the Laplace domain Helmholtz problems with complex wavenumbers. CQ in conjunction with BIE is an excellent candidate to eventually explore numerical homogenization to replace a chaff cloud by a dispersive lossy dielectric that produces the same scattering. To this end, a variety of accuracy tests are presented that showcase the high-order in time convergence (up to and including fifth order) that the Nystrom CQ discretizations are capable of delivering for a variety of two dimensional single and multiple scatterers. Particular emphasis is given to Lipschitz boundaries and open arcs with both Dirichlet and Neumann boundary conditions

    Modelling, Simulation and Data Analysis in Acoustical Problems

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    Modelling and simulation in acoustics is currently gaining importance. In fact, with the development and improvement of innovative computational techniques and with the growing need for predictive models, an impressive boost has been observed in several research and application areas, such as noise control, indoor acoustics, and industrial applications. This led us to the proposal of a special issue about “Modelling, Simulation and Data Analysis in Acoustical Problems”, as we believe in the importance of these topics in modern acoustics’ studies. In total, 81 papers were submitted and 33 of them were published, with an acceptance rate of 37.5%. According to the number of papers submitted, it can be affirmed that this is a trending topic in the scientific and academic community and this special issue will try to provide a future reference for the research that will be developed in coming years

    Fast Solvers and Simulation Data Compression Algorithms for Granular Media and Complex Fluid Flows

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    Granular and particulate flows are common forms of materials used in various physical and industrial applications. For instance, we model the soil as a collection of rigid particles with frictional contact in soil-vehicle simulations, and we simulate bacterial colonies as active rigid particles immersed in a viscous fluid. Due to the complex interactions in-between the particles and/or the particles and the fluid, numerical simulations are often the only way to study these systems apart from typically expensive physical experiments. A standard method for simulating these systems is to apply simple physical laws to each of the particles using the discrete element method (DEM) and evolve the resulting multibody system in time. However, due to the sheer number of particles in even a moderate-scale real-world system, it quickly becomes expensive to timestep these systems unless we exploit fast algorithms and high-performance computing techniques. For instance, a big challenge in granular media simulations is resolving contact between the constituent particles. We use a cone-complementarity formulation of frictional contact to resolve collisions; this approach leads to a quadratic optimization problem whose solution gives us the contact forces between particles at each timestep. In this thesis, we introduce strategies for solving these optimization problems on distributed memory machines. In particular, by imposing a locality-preserving partitioning of the rigid bodies among the computing nodes, we minimize the communication cost and construct a scalable framework for collision detecting and resolution that can be easily scaled to handle hundreds of millions of particles. For robust and efficient simulation of axisymmetric particles in viscous fluids, we introduce a fast method for solving Stokes boundary integral equations (BIEs) on surfaces of revolution. By first transforming the Stokes integral kernels into a rotationally invariant form and then decoupling the transformed kernels using the Fourier series, we reduce the dimensionality of the problem. This approach improves the time complexity of the BIE solvers by an order of magnitude; additionally we can use high-order one-dimensional singular quadrature schemes to construct highly accurate solvers. Finally, coupling our solver framework with the fast multipole method, we construct a fast solver for simulating Stokes flow past a system of axisymmetric bodies. Combining this with our complementarity collision resolution framework, we have the potential to simulate dense particulate suspensions. Physics-based simulations similar to those described above generate large amounts of output data, often in the hundreds of gigabytes range. We introduce data compression techniques based on the tensor-train decomposition for DEM simulation outputs and demonstrate the high compressibility of these large datasets. This allows us to keep a reduced representation of simulated data for post-processing or use in learning tasks. Finally, due to the high cost of physics-based models and limited computational budget, we can typically run only a limited number of simulations when exploring a high-dimensional parameter space. Formally, this can be posed as a matrix/tensor completion problem, and Bayesian inference coupled with a linear factorization model is often used in this setup. We use Markov chain Monte Carlo (MCMC) methods to sample from the unnormalized posteriors in these inference problems. In this thesis, we explore the properties of the posterior in a simple low-rank matrix factorization setup and develop strategies to break its symmetries. This leads to better quality MCMC samples and lowers the reconstruction errors with various synthetic and real-world datasets.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169614/1/saibalde_1.pd

    An introduction to operator preconditioning for the fast iterative integral equation solution of time-harmonic scattering problems

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    International audienceThe aim of this paper is to provide an introduction to the improved iterative Krylov solution of boundary integral equations for time-harmonic scattering problems arising in acoustics, electromagnetism and elasticity. From the point of view of computational methods, considering large frequencies is a challenging issue in engineering since it leads to solving highly indefinite large scale complex linear systems which generally implies a convergence breakdown of iterative methods. More specifically, we explain the problematic and some partial solutions through analytical preconditioning for high-frequency acoustic scattering problems and the introduction of new combined field integral equations. We complete the paper with some recent extensions to the case of electromagnetic and elastic waves equations

    Simple and efficient GPU parallelization of existing H-Matrix accelerated BEM code

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    In this paper, we demonstrate how GPU-accelerated BEM routines can be used in a simple black-box fashion to accelerate fast boundary element formulations based on Hierarchical Matrices (H-Matrices) with ACA (Adaptive Cross Approximation). In particular, we focus on the expensive evaluation of the discrete weak form of boundary operators associated with the Laplace and the Helmholtz equation in three space dimensions. The method is based on offloading the CPU assembly of elements during the ACA assembly onto a GPU device and to use threading strategies across ACA blocks to create sufficient workload for the GPU. The proposed GPU strategy is designed such that it can be implemented in existing code with minimal changes to the surrounding application structure. This is in particular interesting for existing legacy code that is not from the ground-up designed with GPU computing in mind. Our benchmark study gives realistic impressions of the benefits of GPU-accelerated BEM simulations by using state-of-the-art multi-threaded computations on modern high-performance CPUs as a reference, rather than drawing synthetic comparisons with single-threaded codes. Speed-up plots illustrate that performance gains up to a factor of 5.5 could be realized with GPU computing under these conditions. This refers to a boundary element model with about 4 million unknowns, whose H-Matrix weak form associated with a real-valued (Laplace) boundary operator is set up in only 100 minutes harnessing the two GPUs instead of 9 hours when using the 20 CPU cores at disposal only. The benchmark study is followed by a particularly demanding real-life application, where we compute the scattered high-frequency sound field of a submarine to demonstrate the increase in overall application performance from moving to a GPU-based ACA assembly

    Nystr\"om methods for high-order CQ solutions of the wave equation in two dimensions

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    We investigate high-order Convolution Quadratures methods for the solution of the wave equation in unbounded domains in two dimensions that rely on Nystr\"om discretizations for the solution of the ensemble of associated Laplace domain modified Helmholtz problems. We consider two classes of CQ discretizations, one based on linear multistep methods and the other based on Runge-Kutta methods, in conjunction with Nystr\"om discretizations based on Alpert and QBX quadratures of Boundary Integral Equation (BIE) formulations of the Laplace domain Helmholtz problems with complex wavenumbers. We present a variety of accuracy tests that showcase the high-order in time convergence (up to and including fifth order) that the Nystr\"om CQ discretizations are capable of delivering for a variety of two dimensional scatterers and types of boundary conditions
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