Journal articles: 'Fast Boundary Element Methods'

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Kravčenko, Michal, Michal Merta, and Jan Zapletal. "Distributed fast boundary element methods for Helmholtz problems." Applied Mathematics and Computation 362 (December 2019): 124503. http://dx.doi.org/10.1016/j.amc.2019.06.017.

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Gumerov, Nail A., and Ramani Duraiswami. "Fast multipole accelerated boundary element methods for room acoustics." Journal of the Acoustical Society of America 150, no. 3 (September 2021): 1707–20. http://dx.doi.org/10.1121/10.0006102.

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Of, G., O. Steinbach, and P. Urthaler. "Fast Evaluation of Volume Potentials in Boundary Element Methods." SIAM Journal on Scientific Computing 32, no. 2 (January 2010): 585–602. http://dx.doi.org/10.1137/080744359.

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Harbrecht, H., and M. Peters. "Comparison of fast boundary element methods on parametric surfaces." Computer Methods in Applied Mechanics and Engineering 261-262 (July 2013): 39–55. http://dx.doi.org/10.1016/j.cma.2013.03.022.

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Gumerov, Nail, and Ramani Duraiswami. "Simulations of room acoustics using fast multipole boundary element methods." Journal of the Acoustical Society of America 148, no. 4 (October 2020): 2693–94. http://dx.doi.org/10.1121/1.5147458.

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MUKHERJEE, SUBRATA, and YIJUN LIU. "THE BOUNDARY ELEMENT METHOD." International Journal of Computational Methods 10, no. 06 (May 2, 2013): 1350037. http://dx.doi.org/10.1142/s0219876213500370.

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The boundary element method (BEM), along with the finite element and finite difference methods, is commonly used to carry out numerical simulations in a wide variety of subjects in science and engineering. The BEM, rooted in classical mathematics of integral equations, started becoming a useful computational tool around 50 years ago. Many researchers have worked on computational aspects of this method during this time.This paper presents an overview of the BEM and related methods. It has three sections. The first, relatively short section, presents the governing equations for classical applications of the BEM in potential theory, linear elasticity and acoustics. The second describes specialized applications in bodies with thin features including micro-electro-mechanical systems (MEMS). The final section addresses current research. It has three subsections that present the boundary contour, boundary node and fast multipole methods (BCM, BNM and FMM), respectively. Several numerical examples are included in the second and third sections of this paper.

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Dargush, G. F., and M. M. Grigoriev. "Fast and Accurate Solutions of Steady Stokes Flows Using Multilevel Boundary Element Methods." Journal of Fluids Engineering 127, no. 4 (February 23, 2005): 640–46. http://dx.doi.org/10.1115/1.1949648.

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Most recently, we have developed a novel multilevel boundary element method (MLBEM) for steady Stokes flows in irregular two-dimensional domains (Grigoriev, M.M., and Dargush, G.F., Comput. Methods. Appl. Mech. Eng., 2005). The multilevel algorithm permitted boundary element solutions with slightly over 16,000 degrees of freedom, for which approximately 40-fold speedups were demonstrated for the fast MLBEM algorithm compared to a conventional Gauss elimination approach. Meanwhile, the sevenfold memory savings were attained for the fast algorithm. This paper extends the MLBEM methodology to dramatically improve the performance of the original multilevel formulation for the steady Stokes flows. For a model problem in an irregular pentagon, we demonstrate that the new MLBEM formulation reduces the CPU times by a factor of nearly 700,000. Meanwhile, the memory requirements are reduced more than 16,000 times. These superior run-time and memory reductions compared to regular boundary element methods are achieved while preserving the accuracy of the boundary element solution.

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van 't Wout, Elwin, Reza Haqshenas, Pierre Gélat, and Nader Saffari. "Fast and accurate boundary element methods for large-scale computational acoustics." Journal of the Acoustical Society of America 154, no. 4_supplement (October 1, 2023): A179. http://dx.doi.org/10.1121/10.0023190.

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The boundary element method (BEM) is a powerful algorithm to solve the Helmholtz equation for harmonic acoustic waves. The explicit use of Green’s functions avoids domain truncation of unbounded regions and accurately models wave propagation through homogeneous materials. Furthermore, fast multipole and hierarchical compression techniques provide efficient computations for dense matrix multiplications. However, the convergence of the iterative linear solvers deteriorates significantly when frequencies are high or materials have large contrasts in density or speed of sound. This talk presents several algorithmic improvements of the BEM. First, a preconditioner based on on-surface radiation conditions drastically reduces the iteration count of linear solvers at high frequencies. Second, anovel boundary integral formulation remains well-conditioned for high-contrast transmission problems. We used our fast and accurate BEM implementation to simulate focused ultrasound propagation in the human body, which can be translated to important biomedical applications such as the non-invasive treatment of liver cancer and neuromodulation of the brain. We validated the methodology within the benchmarking exercise of the International Transcranial Ultrasonic Stimulation Safety and Standards (ITRUSST) consortium. As a second application, we simulated the collective resonances of water-entrained arrays of air bubbles. Finally, we implemented all functionality in our open-source Python library, OptimUS.

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Newman, J. N., and C. H. Lee. "Boundary-Element Methods In Offshore Structure Analysis." Journal of Offshore Mechanics and Arctic Engineering 124, no. 2 (April 11, 2002): 81–89. http://dx.doi.org/10.1115/1.1464561.

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Boundary-element methods, also known as panel methods, have been widely used for computations of wave loads and other hydrodynamic characteristics associated with the interactions of offshore structures with waves. In the conventional approach, based on the low-order panel method, the submerged surface of the structure is represented by a large number of small quadrilateral plane elements, and the solution for the velocity potential or source strength is approximated by a constant value on each element. In this paper, we describe two recent developments of the panel method. One is a higher-order method where the submerged surface can be represented exactly, or approximated to a high degree of accuracy by B-splines, and the velocity potential is also approximated by B-splines. This technique, which was first used in the research code HIPAN, has now been extended and implemented in WAMIT. In many cases of practical importance, it is now possible to represent the geometry exactly to avoid the extra work required previously to develop panel input files for each structure. It is also possible to combine the same or different structures which are represented in this manner, to analyze multiple-body hydrodynamic interactions. Also described is the pre-corrected Fast Fourier Transform method (pFFT) which can reduce the computational time and required memory of the low-order method by an order of magnitude. In addition to descriptions of the two methods, several different applications are presented.

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Chen, Leilei, Steffen Marburg, Wenchang Zhao, Cheng Liu, and Haibo Chen. "Implementation of Isogeometric Fast Multipole Boundary Element Methods for 2D Half-Space Acoustic Scattering Problems with Absorbing Boundary Condition." Journal of Theoretical and Computational Acoustics 27, no. 02 (June 2019): 1850024. http://dx.doi.org/10.1142/s259172851850024x.

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Isogeometric Analysis (IGA), which tries to bridge the gap between Computer Aided Engineering (CAE) and Computer Aided Design (CAD), has been widely proposed in recent research. According to the concept of IGA, this work develops a boundary element method (BEM) using non-Uniform Rational B-Splines (NURBS) as basis functions for the 2D half-space acoustic problems with absorbing boundary condition. Fast multipole method (FMM) is applied to accelerate the solution of an isogeometric BEM (IGA-BEM). Several examples are tested and it is shown that this advancement on isogeometric fast multipole boundary element method improves the accuracy of simulations.

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Proskurov, S., R. Ewert, M. Lummer, M. Mößner, and J. W. Delfs. "Sound shielding simulation by coupled discontinuous Galerkin and fast boundary element methods." Engineering Applications of Computational Fluid Mechanics 16, no. 1 (August 16, 2022): 1690–705. http://dx.doi.org/10.1080/19942060.2022.2098827.

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Buchau, Andre, and Wolfgang M. Rucker. "Meshfree Computation of Field Lines Across Multiple Domains Using Fast Boundary Element Methods." IEEE Transactions on Magnetics 51, no. 3 (March 2015): 1–4. http://dx.doi.org/10.1109/tmag.2014.2359520.

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Chaillat, Stéphanie, Marion Darbas, and Frédérique Le Louër. "Fast iterative boundary element methods for high-frequency scattering problems in 3D elastodynamics." Journal of Computational Physics 341 (July 2017): 429–46. http://dx.doi.org/10.1016/j.jcp.2017.04.020.

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Gumerov, Nail A., and Ramani Duraiswami. "Wideband fast multipole accelerated boundary element methods for the three‐dimensional Helmholtz equation." Journal of the Acoustical Society of America 125, no. 4 (April 2009): 2566. http://dx.doi.org/10.1121/1.4808753.

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Rodgers, Michael J., Shuangbiao Liu, Q. Jane Wang, and Leon M. Keer. "Boundary Element Methods for Steady-State Thermal-Mechanical Problems of Counterformal Contact." Journal of Tribology 126, no. 3 (June 28, 2004): 443–49. http://dx.doi.org/10.1115/1.1757492.

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This paper presents a concise boundary integral equation framework for relating the thermal-mechanical surface load (the three traction components and the normal heat flux) to the thermal-mechanical response (the three quasi-static displacement components and the steady-state temperature). This uncoupled thermoelastic framework allows the simultaneous calculation of displacement and temperature—without subsurface discretization—because it is based on classical Green’s functions for displacement and for temperature and on newly derived Green’s functions for thermoelastic displacement. In general, the boundary element method (BEM) can be applied with this framework to finite geometry problems of steady-state thermal-mechanical contact. Here, example calculations are performed for counterformal contact problems, which can be modeled as contact on a halfspace. A linear element BEM is developed and compared with the constant element BEM for speed and accuracy. The linear element BEM uses newly derived influence coefficients for constant loads over an arbitrary triangular element, and these closed form expressions are used to improve the accuracy of the numerical algorithm. The constant element BEM uses the discrete convolution fast Fourier transform (DC-FFT) algorithm, which is based on influence coefficients for constant loads over rectangular elements. The quasi-static surface displacements and the steady-state surface temperature are calculated from an applied semi-ellipsoidal pressure with accompanying frictional heating effects. The surface thermal-mechanical behavior of the counterformal contact is shown in graphs vs. the radius, and the deviations from axisymmetry are highlighted.

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Pozrikidis, C. "Boundary Element Grid Optimization for Stokes Flow With Corner Singularities." Journal of Fluids Engineering 124, no. 1 (November 13, 2001): 22–28. http://dx.doi.org/10.1115/1.1436091.

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The accuracy of boundary-element methods for computing Stokes flow past boundaries with sharp corners where singularities occur is discussed. To resolve the singular behavior, a graded mesh of boundary elements whose length increases in a geometrical fashion with respect to distance from the corners according to a prescribed stretch ratio is used. Numerical results for two-dimensional Stokes flow past bodies with polygonal shapes reveal the existence of an optimal value of the stretch ratio for best accuracy in the computation of the force and torque. When the optimal value is used, fast convergence is achieved with respect to the number of boundary elements.

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Preuss, Simone. "A fast multipole boundary element method for acoustics in viscothermal fluids." INTER-NOISE and NOISE-CON Congress and Conference Proceedings 267, no. 1 (November 5, 2023): 326–29. http://dx.doi.org/10.3397/no_2023_0071.

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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.

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TAKAHASHI, Toru, and Toshikazu EBISUZAKI. "Fast Computing of Boundary Element Methods using the Special-Purpose Computer for Molecular Dynamics." Proceedings of The Computational Mechanics Conference 2002.15 (2002): 823–24. http://dx.doi.org/10.1299/jsmecmd.2002.15.823.

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19

Wang, Haitao, Zhenhan Yao, and Pengbo Wang. "On the preconditioners for fast multipole boundary element methods for 2D multi-domain elastostatics." Engineering Analysis with Boundary Elements 29, no. 7 (July 2005): 673–88. http://dx.doi.org/10.1016/j.enganabound.2005.03.002.

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Chen, Zejun, and Hong Xiao. "The fast multipole boundary element methods (FMBEM) and its applications in rolling engineering analysis." Computational Mechanics 50, no. 5 (February 28, 2012): 513–31. http://dx.doi.org/10.1007/s00466-012-0692-z.

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Wang, Qiao, Wei Zhou, Yonggang Cheng, Gang Ma, and Xiaolin Chang. "Fast multipole cell-based domain integration method for treatment of volume potentials in 3D elasticity problems." Engineering Computations 34, no. 6 (August 7, 2017): 1849–73. http://dx.doi.org/10.1108/ec-03-2016-0111.

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Purpose Domain integrals, known as volume potentials in 3D elasticity problems, exist in many boundary-type methods, such as the boundary element method (BEM) for inhomogeneous partial differential equations. The purpose of this paper is to develop an accurate and reliable technique to effectively evaluate the volume potentials in 3D elasticity problems. Design/methodology/approach An adaptive background cell-based domain integration method is proposed for treatment of volume potentials in 3D elasticity problems. The background cells are constructed from the information of the boundary elements based on an oct-tree structure, and the domain integrals are evaluated over the cells rather than volume elements. The cells that contain the boundary elements can be subdivided into smaller sub-cells adaptively according to the sizes and levels of the boundary elements. The fast multipole method (FMM) is further applied in the proposed method to reduce the time complexity of large-scale computation. Findings The method is a boundary-only discretization method, and it can be applied in the BEM easily. Much computational time is saved by coupling with the FMM. Numerical examples demonstrate the accuracy and efficiency of the proposed method.. Originality/value Boundary elements are used to create adaptive background cells, and domain integrals are evaluated over the cells rather than volume elements. Large-scale computation is made possible by coupling with the FMM.

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Fan, Hong Ling. "Fast Algorithm for the Inverse Matrices of Periodic Adding Element Tridiagonal Matrices." Advanced Materials Research 159 (December 2010): 464–68. http://dx.doi.org/10.4028/www.scientific.net/amr.159.464.

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Adding element tridiagonal periodic matrices have an important effect for the algorithms of solving linear systems，computing the inverses, the triangular factorization，the boundary value problems by finite difference methods, interpolation by cubic splines, three-term difference equations and so on. In this paper, we give a fast algorithm for the Inverse Matrices of periodic adding element tridiagonal matrices.

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Nadimi, Sadegh, Ali Ghanbarzadeh, Anne Neville, and Mojtaba Ghadiri. "Effect of particle roughness on the bulk deformation using coupled boundary element and discrete element methods." Computational Particle Mechanics 7, no. 3 (October 11, 2019): 603–13. http://dx.doi.org/10.1007/s40571-019-00288-3.

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Abstract Particles slide and roll on each other when a granular medium is sheared. Consequently, the tribological properties, such as inter-particle friction and adhesion, play a major role in influencing their bulk failure and rheology. Although the influence of roughness on adhesion and friction of contacting surfaces is known, the incorporation of the surface roughness in the numerical modelling of granular materials has received little attention. In this study, the boundary element method (BEM), which is widely used for simulating the mechanics of interacting surfaces, is coupled with discrete element method (DEM) and the bulk deformation of granular materials is analysed. A BEM code, developed in-house, is employed to calculate the normal force–displacement behaviour for rough contact deformations, based on which a contact model is proposed. This is an efficient and relatively fast method of calculating the contact mechanics of rough surfaces. The resulting model is then implemented in the simulations by DEM to determine the effect of micro-scale surface roughness on the bulk compression of granular materials. This study highlights the importance of the effect of surface characteristics on contact behaviour of particles for the case of shallow footing and provides an efficient approach for modelling the flow behaviour of a large number of rough particles.

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MAGOULÈS, FRÉDÉRIC, and ROMAN PUTANOWICZ. "OPTIMAL CONVERGENCE OF NON-OVERLAPPING SCHWARZ METHODS FOR THE HELMHOLTZ EQUATION." Journal of Computational Acoustics 13, no. 03 (September 2005): 525–45. http://dx.doi.org/10.1142/s0218396x05002748.

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The non-overlapping Schwarz method with absorbing boundary conditions instead of the Dirichlet boundary conditions is an efficient variant of the overlapping Schwarz method for the Helmholtz equation. These absorbing boundary conditions defined on the interface between the subdomains are the key ingredients to obtain a fast convergence of the iterative Schwarz algorithm. In a one-way subdomains splitting, non-local optimal absorbing boundary conditions can be obtained and leads to the convergence of the Schwarz algorithm in a number of iterations equal to the number of subdomains minus one. This paper investigates different local approximations of these optimal absorbing boundary conditions for finite element computations in acoustics. Different approaches are presented both in the continuous and in the discrete analysis, including high-order optimized continuous absorbing boundary conditions, and discrete absorbing boundary conditions based on algebraic approximation. A wide range of new numerical experiments performed on unbounded acoustics problems demonstrate the comparative performance and the robustness of the proposed methods on general unstructured mesh partitioning.

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Achdou, Y., and O. Pironneau. "A Fast Solver for Navier–Stokes Equations in the Laminar Regime Using Mortar Finite Element and Boundary Element Methods." SIAM Journal on Numerical Analysis 32, no. 4 (August 1995): 985–1016. http://dx.doi.org/10.1137/0732046.

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Fedelinski, Piotr. "Computer Modelling of Dynamic Fracture Experiments." Key Engineering Materials 454 (December 2010): 113–25. http://dx.doi.org/10.4028/www.scientific.net/kem.454.113.

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In this work the time-domain boundary element method (BEM) is applied to simulate dynamic fracture experiments. The fast fracture is modelled by adding new boundary elements at the crack tip. The direction of crack growth is perpendicular to the direction of maximum circumferencial stress. The time dependent loading of specimens and velocities of crack growth are taken from experiments as input data for computer simulations. The method is used to analyze: a short beam specimen, a special mixed-mode specimen and a three-point bend specimen subjected to impact loads. The dynamic stress intensity factors (DSIF) and the crack paths are compared with the results obtained by other authors who used the finite element method (FEM) and experimental methods.

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Zhao, Yan Lei, and Xue Ting Liu. "Fast Algorithm for the Inverse Matrices of Adding Element Tridiagonal Periodic Matrices in Signal Processing." Advanced Materials Research 121-122 (June 2010): 682–86. http://dx.doi.org/10.4028/www.scientific.net/amr.121-122.682.

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Adding element tridiagonal matrices play a very important role in the theory and practical applications, such as the boundary value problems by finite difference methods, interpolation by cubic splines, three-term difference equations and so on. In this paper, we give a fast algorithm for the Inverse Matrices of periodic adding element tridiagonal matrices.

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Dölz, Jürgen, Stefan Kurz, Sebastian Schöps, and Felix Wolf. "Isogeometric Boundary Elements in Electromagnetism: Rigorous Analysis, Fast Methods, and Examples." SIAM Journal on Scientific Computing 41, no. 5 (January 2019): B983—B1010. http://dx.doi.org/10.1137/18m1227251.

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Ping, Xuecheng, Mengcheng Chen, Wei Zhu, Yihua Xiao, and Weixing Wu. "Computations of Singular Stresses Along Three-Dimensional Corner Fronts by a Super Singular Element Method." International Journal of Computational Methods 14, no. 06 (August 2017): 1750065. http://dx.doi.org/10.1142/s0219876217500657.

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In order to consider corner configurations with straight corner fronts in three-dimensional (3D) solids, a super polygonal prismatic element containing a straight corner front is established by using the numerical eigensolutions of singular stress fields and the Hellinger–Reissner variational principle. Singular stresses near the corner front subject to far-field boundary conditions can be obtained by incorporating the super singular element with conventional 3D brick elements. The numerical studies are conducted to demonstrate the simplicity of the proposed technique in handling fracture problems of 3D corner configurations and cracks. The usage of the super singular element can avoid mesh refinement near the corner front domain that is necessary for conventional and enriched finite element methods, and lead to high accuracy and fast convergence. Compared with the conventional finite element methods and existing analytical methods, the present method is more suitable for dealing with complicated problems of stress singularity in elasticity including multiple defects.

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YASUDA, Y., S. SAKAMOTO, Y. KOSAKA, T. SAKUMA, N. OKAMOTO, and T. OSHIMA. "NUMERICAL ANALYSIS OF LARGE-SCALE SOUND FIELDS USING ITERATIVE METHODS PART I: APPLICATION OF KRYLOV SUBSPACE METHODS TO BOUNDARY ELEMENT ANALYSIS." Journal of Computational Acoustics 15, no. 04 (December 2007): 449–71. http://dx.doi.org/10.1142/s0218396x07003470.

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The convergence behavior of the Krylov subspace iterative solvers towards the systems with the 3D acoustical BEM is investigated through numerical experiments. The fast multipole BEM, which is an efficient BEM based on the fast multipole method, is used for solving problems with up to about 100,000 DOF. It is verified that the convergence behavior of solvers is much affected by the formulation of the BEM (singular, hypersingular, and Burton-Miller formulation), the complexity of the shape of the problem, and the sound absorption property of the boundaries. In BiCG-like solvers, GPBiCG and BiCGStab2 have more stable convergence than others, and these solvers are useful when solving interior problems in basic singular formulation. When solving exterior problems with greatly complex shape in Burton-Miller formulation, all solvers hardly converge without preconditioning, whereas the convergence behavior is much improved with ILU-type preconditioning. In these cases GMRes is the fastest, whereas CGS is one of the good choices, when taken into account the difficulty of determining the timing of restart for GMRes. As for calculation for rigid thin objects in hypersingular formulation, much more rapid convergence is observed than ordinary interior/exterior problems, especially using BiCG-like solvers.

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Winckelmans, G. S., J. K. Salmon, M. S. Warren, A. Leonard, and B. Jodoin. "Application of fast parallel and sequential tree codes to computing three-dimensional flows with the vortex element and boundary element methods." ESAIM: Proceedings 1 (1996): 225–40. http://dx.doi.org/10.1051/proc:1996039.

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Zhou, H., Y. Zhang, J. Wen, and S. Cui. "Mould cooling simulation for injection moulding using a fast boundary element method approach." Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture 224, no. 4 (October 26, 2009): 653–62. http://dx.doi.org/10.1243/09544054jem1407.

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The existing cooling simulations for injection moulding are mostly based on the boundary element method (BEM). In this paper, a fast BEM approach for mould cooling analysis is developed. The actual problem is decoupled into a one-dimensional transient heat conduction problem within the thin part and a cycle-averaged steady state three-dimensional heat conduction problem of the mould. The BEM is formulated for the solution of the mould heat transfer problem. A dynamic allocation strategy of integral points is proposed when using the Gaussian integral formula to generate the BEM matrix. Considering that the full and unsymmetrical influence matrix of the BEM may lead to great storage space and solution time, this matrix is transformed into a sparse matrix by two methods: the direct rounding method or the combination method. This approximated sparsification approach can reduce the storage memory and solution time significantly. For validation, six typical cases with different element numbers are presented. The results show that the error of the direct rounding method is too large while that of the combination method is acceptable.

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Ma, F., J. Chatterjee, and P. K. Banerjee. "New Fast Convolution Algorithm in Boundary-Element Methods for Two- and Three-Dimensional Linear Soil Consolidation Analysis." International Journal of Geomechanics 7, no. 3 (May 2007): 236–49. http://dx.doi.org/10.1061/(asce)1532-3641(2007)7:3(236).

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TAKAHASHI, Toru, Hiroshi ISAKARI, and Toshiro MATSUMOTO. "An implementation and benchmark of a fast-multipole-type direct-solver for two-dimensional boundary element methods." Proceedings of The Computational Mechanics Conference 2014.27 (2014): 296–98. http://dx.doi.org/10.1299/jsmecmd.2014.27.296.

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Ochmann, Martin. "Two fast methods for the calculation of sound radiation: Multipol radiator synthesis and boundary element multigrid method." Journal of the Acoustical Society of America 87, S1 (May 1990): S74. http://dx.doi.org/10.1121/1.2028356.

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Song, Haiming, Kai Zhang, and Yutian Li. "Finite Element and Discontinuous Galerkin Methods with Perfect Matched Layers for American Options." Numerical Mathematics: Theory, Methods and Applications 10, no. 4 (September 12, 2017): 829–51. http://dx.doi.org/10.4208/nmtma.2017.0020.

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AbstractThis paper is devoted to the American option pricing problem governed by the Black-Scholes equation. The existence of an optimal exercise policy makes the problem a free boundary value problem of a parabolic equation on an unbounded domain. The optimal exercise boundary satisfies a nonlinear Volterra integral equation and is solved by a high-order collocation method based on graded meshes. This free boundary is then deformed to a fixed boundary by the front-fixing transformation. The boundary condition at infinity (due to the fact that the underlying asset's price could be arbitrarily large in theory), is treated by the perfectly matched layer technique. Finally, the resulting initial-boundary value problems for the option price and some of the Greeks on a bounded rectangular space-time domain are solved by a finite element method. In particular, for Delta, one of the Greeks, we propose a discontinuous Galerkin method to treat the discontinuity in its initial condition. Convergence results for these two methods are analyzed and several numerical simulations are provided to verify these theoretical results.

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Ebrahimnejad, Latif, and Reza Attarnejad. "A novel way of using fast wavelet transforms to solve dense linear systems arising from boundary element methods." Engineering Computations 26, no. 5 (July 17, 2009): 483–99. http://dx.doi.org/10.1108/02644400910970167.

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Amlani, Faisal, Stéphanie Chaillat, and Adrien Loseille. "An efficient preconditioner for adaptive Fast Multipole accelerated Boundary Element Methods to model time-harmonic 3D wave propagation." Computer Methods in Applied Mechanics and Engineering 352 (August 2019): 189–210. http://dx.doi.org/10.1016/j.cma.2019.04.026.

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Pan, Jie, Jingwei Huang, Yunli Wang, Gengdong Cheng, and Yong Zeng. "A self-learning finite element extraction system based on reinforcement learning." Artificial Intelligence for Engineering Design, Analysis and Manufacturing 35, no. 2 (April 21, 2021): 180–208. http://dx.doi.org/10.1017/s089006042100007x.

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AbstractAutomatic generation of high-quality meshes is a base of CAD/CAE systems. The element extraction is a major mesh generation method for its capabilities to generate high-quality meshes around the domain boundary and to control local mesh densities. However, its widespread applications have been inhibited by the difficulties in generating satisfactory meshes in the interior of a domain or even in generating a complete mesh. The element extraction method's primary challenge is to define element extraction rules for achieving high-quality meshes in both the boundary and the interior of a geometric domain with complex shapes. This paper presents a self-learning element extraction system, FreeMesh-S, that can automatically acquire robust and high-quality element extraction rules. Two central components enable the FreeMesh-S: (1) three primitive structures of element extraction rules, which are constructed according to boundary patterns of any geometric boundary shapes; (2) a novel self-learning schema, which is used to automatically define and refine the relationships between the parameters included in the element extraction rules, by combining an Advantage Actor-Critic (A2C) reinforcement learning network and a Feedforward Neural Network (FNN). The A2C network learns the mesh generation process through random mesh element extraction actions using element quality as a reward signal and produces high-quality elements over time. The FNN takes the mesh generated from the A2C as samples to train itself for the fast generation of high-quality elements. FreeMesh-S is demonstrated by its application to two-dimensional quad mesh generation. The meshing performance of FreeMesh-S is compared with three existing popular approaches on ten pre-defined domain boundaries. The experimental results show that even with much less domain knowledge required to develop the algorithm, FreeMesh-S outperforms those three approaches in essential indices. FreeMesh-S significantly reduces the time and expertise needed to create high-quality mesh generation algorithms.

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Chatzipantelidis, Panagiotis, Zoltan Horváth, and Vidar Thomée. "On Preservation of Positivity in Some Finite Element Methods for the Heat Equation." Computational Methods in Applied Mathematics 15, no. 4 (October 1, 2015): 417–37. http://dx.doi.org/10.1515/cmam-2015-0018.

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AbstractWe consider the initial boundary value problem for the homogeneous heat equation, with homogeneous Dirichlet boundary conditions. By the maximum principle the solution is nonnegative for positive time if the initial data are nonnegative. We complement in a number of ways earlier studies of the possible extension of this fact to spatially semidiscrete and fully discrete piecewise linear finite element discretizations, based on the standard Galerkin method, the lumped mass method, and the finite volume element method. We also provide numerical examples that illustrate our findings.

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Illyashenko, Ludmila, and Alexander Nerukh. "APPLICATION OF SPECTRAL METHODS OF BOUNDARY INTEGRAL EQUATIONS FOR MODELING OF NANOOPTICAL DEVICES." Bulletin of the National Technical University "KhPI". Series: Mathematical modeling in engineering and technologies, no. 1 (August 1, 2023): 122–27. http://dx.doi.org/10.20998/2222-0631.2023.01.18.

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Efficiency of modeling of optical nanostructures depends not only on the accuracy of the description of the new physical processes which appear in the new configurations of resonantly scattering and resonantly absorbing structures, but also on the selection of appropriate algorithms for solving the corresponding mathematical problem and numerical parameters of modeling depending on parameters of elements. This is why solving complicated problems of modeling complex resonantly scattering and resonantly absorbing electrodynamic nanostructures involves deep learning of all groups of new unknown effects. In this work a semi-analytical algorithm is developed based on parametrized by conformal mapping techniques spectral method of boundary integral equations with analytical regularization based on singularity subtraction enhanced by Fast Fourier transform, that contrary to the classical schemes, which are based on finite difference and finite element methods, allows to take into account the complex-valued functional dependence of dielectric permittivity of plasmonic materials on the wavelength (even when its value is tabulated) and to solve the problems with static and dynamic singularities in integral equations. Due to sensibility of plasmon resonances to changes in external medium such nanostructures are used in modern medicine, pharmacy and also as chemical and biological sensors. In this work main efforts are directed on generation of algorithm for investigation of dielectric nanostructures with static singularities.

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Wang, Junpeng, Jinyou Xiao, and Lihua Wen. "A Numerical Method for Estimating the Nonlinear Eigenvalue Numbers of Boundary Element." Xibei Gongye Daxue Xuebao/Journal of Northwestern Polytechnical University 37, no. 1 (February 2019): 28–34. http://dx.doi.org/10.1051/jnwpu/20193710028.

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Recently, some new proposed methods for solving nonlinear eigenvalue problems (NEPs) have promoted the development of large-scale modal analysis using BEM. However, the efficiency and robustness of such methods are generally still dependent on input parameters, especially on the parameters related to the number of eigenvalues to be solved. This limitation obviously restricts the popularization of the practical engineering application of modal analysis using BEM. Therefore, this paper develops a numerical method for estimating the number of nonlinear eigenvalues of the boundary element method. Firstly, the interpolation method based on the discretized Cauchy integral formula of analytic function is used for obtaining the BEM matrix's derivative with regard to frequency, and this method is easily combined with the mainstream fast algorithm libraries of BEM. Secondly, the method for evaluating the eigenvalue number of BEM under various boundary conditions is obtained by combining the interpolation method with the analytic formula to obtain the eigenvalue number, while the unbiased estimation is used to determine the trace of matrix. Finally, a series of typical examples are used to explore the principle for selecting optimal input parameters in this method, and then a set of optimal input parameters are determined. The overall excellent performance of this method is verified by a complex large-scale example.

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Li, Chengxi, and Jijian Lian. "Development and Application of a Pre-Corrected Fast Fourier Transform Accelerated Multi-Layer Boundary Element Method for the Simulation of Shallow Water Acoustic Propagation." Applied Sciences 10, no. 7 (April 1, 2020): 2393. http://dx.doi.org/10.3390/app10072393.

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Because of the complexities associated with the domain geometry and environments, accurate prediction of acoustics propagation and scattering in realistic shallow water environments by direct numerical simulation is challenging. Based on the pre-corrected Fast Fourier Transform (PFFT) method, we accelerated the classical boundary element method (BEM) to predict the acoustic propagation in a multi-layer shallow water environment. The classical boundary element method formulate the acoustics propagation problem as a linear equation system in the form of [A]{x}={b}, where [A] is an N×N dense matrix composed of influence coefficients. Solving such linear equation system requires O(N2/N3) computational cost for iterative/direct methods. The developed method, PFFT-BEM, can effectively reduce the computational efforts for direct numerical simulations from O(N2~3) to O(Nlog N), where N is the total number of boundary unknowns. To numerically simulate the sound propagation in a shallow water environment, we applied the first-order non-reflecting boundary condition in the truncated numerical domain boundary to eliminate the errors due to reflected waves. Multi-layer coupled formulation was used to include the environment inhomogeneity in PFFT-BEM. Through multiple convergence tests on the number of layers and elements, we validated and quantified the accuracy of PFFT-BEM. To demonstrate the usefulness and capability of the developed PFFT-BEM, we simulated three-dimensional (3D) underwater sound propagation through 3D geometries to check the efficacy of the established classical method: the 3D Parabolic equation model. Finally, PFFT-BEM was employed to simulate sound propagation through a complex multi-layer shallow water environment with internal waves. The “3D+T” results obtained by PFFT-BEM compared well with the physical test, thereby proving the capability and correctness of this method.

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SHERBAKOV, Sergei S., Mikhail M. POLESTCHUK, and Dzianis E. MARMYSH. "PARALLELIZING OF COMPUTATIONS ON A GRAPHICS PROCESSING UNIT FOR ACCELERATING BOUNDARY ELEMENT CALCULATIONS IN MECHANICS." Mechanics of Machines, Mechanisms and Materials 1, no. 66 (March 2024): 80–85. http://dx.doi.org/10.46864/1995-0470-2024-1-66-80-85.

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In solving problems of computer modeling using various methods, accuracy and computational efficiency questions always arise. This study explores the application of two modifications of the boundary element method to solve the problem of potential distribution within a closed two-dimensional domain with a uniform potential distribution on its boundary. The first modification involves using three nonlinear shape functions instead of one. The second modification applies the Galerkin method to the boundary element approach with three nonlinear shape functions. The essence of this modification lies in the fact that the system of equations is formulated in integral form over the entire boundary element, rather than at collocation points. In addition to this, the paper describes and investigates the advantages and disadvantages of the smoothing modification applied to these approaches. Since the influence matrix consists of independently computable elements, parallelization of calculations using NVIDIA CUDA technology has been proposed to enhance computational efficiency, significantly accelerating the calculation of interaction matrix. The choice of this technology is advantageous due to the prevalence of NVIDIA graphics accelerators in almost every personal computer or laptop, as well as it is easy to use. The study presents an approach to the application of this technology and demonstrates the results, showing the acceleration of parallelized calculations which show the dependence on the number of boundary elements. A comparison of the efficiency of the selected technology when applied to two methods, collocation and Galerkin, is also presented, indicating a significant speedup of up to 22 times by computing the influence matrix of the boundary elements.

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Chernukha, Nikita. "New Numerical Methods for Structural Mechanics Problems in Unbounded Domains." Applied Mechanics and Materials 725-726 (January 2015): 848–53. http://dx.doi.org/10.4028/www.scientific.net/amm.725-726.848.

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The article is devoted to the problem of numerical simulation of unbounded domains in structural mechanics. Nowadays there are many numerical methods to analyze structural mechanics problems in infinite domains. A brief analytical review of existing numerical methods is presented. Among them are finite difference method, boundary element method (BEM), finite element method (FEM) and scaled boundary finite element method (SBFEM). No one suggests general approach for all kinds of problem statements. Vast majority of industrial software realize FEM. Considering this fact it is more reasonable to modify FEM for mechanical problems in unbounded domains. New variational differential method and new FEM modification, based on the approach of quasi-uniform grids modelling in finite difference method, are proposed. New numerical methods enable to solve problems in semi-infinite and infinite domains without introduction of artificial boundaries and setting special non-reflecting conditions. The article shows basic steps of new numerical algorithms for problems in one-dimensional semi-infinite computational domain.

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Fochesato, Christophe, and Frédéric Dias. "A fast method for nonlinear three-dimensional free-surface waves." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 462, no. 2073 (April 5, 2006): 2715–35. http://dx.doi.org/10.1098/rspa.2006.1706.

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An efficient numerical model for solving fully nonlinear potential flow equations with a free surface is presented. Like the code that was developed by Grilli et al . (Grilli et al . 2001 Int. J. Numer. Methods Fluids 35 , 829–867), it uses a high-order three-dimensional boundary-element method combined with mixed Eulerian–Lagrangian time updating, based on second-order explicit Taylor expansions with adaptive time-steps. Such methods are known to be accurate but expensive. The efficiency of the code has been greatly improved by introducing the fast multipole algorithm. By replacing every matrix–vector product of the iterative solver and avoiding the building of the influence matrix, this algorithm reduces the computing complexity from to nearly , where N is the number of nodes on the boundary. The performance of the method is illustrated by the example of the overturning of a solitary wave over a three-dimensional sloping bottom. For this test case, the accelerated method is indeed much faster than the former one, even for quite coarse grids. For instance, a reduction of the complexity by a factor six is obtained for N =6022, for the same global accuracy. The acceleration of the code allows the study of more complex physical problems and several examples are presented.

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Oliveira, Tiago, Wilber Vélez, and Artur Portela. "Formulation of local numerical methods in linear elasticity." Multidiscipline Modeling in Materials and Structures 16, no. 5 (June 4, 2020): 853–86. http://dx.doi.org/10.1108/mmms-05-2018-0094.

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PurposeThis paper is concerned with new formulations of local meshfree and finite element numerical methods, for the solution of two-dimensional problems in linear elasticity.Design/methodology/approachIn the local domain, assigned to each node of a discretization, the work theorem establishes an energy relationship between a statically admissible stress field and an independent kinematically admissible strain field. This relationship, derived as a weighted residual weak form, is expressed as an integral local form. Based on the independence of the stress and strain fields, this local form of the work theorem is kinematically formulated with a simple rigid-body displacement to be applied by local meshfree and finite element numerical methods. The main feature of this paper is the use of a linearly integrated local form that implements a quite simple algorithm with no further integration required.FindingsThe reduced integration, performed by this linearly integrated formulation, plays a key role in the behavior of local numerical methods, since it implies a reduction of the nodal stiffness which, in turn, leads to an increase of the solution accuracy and, which is most important, presents no instabilities, unlike nodal integration methods without stabilization. As a consequence of using such a convenient linearly integrated local form, the derived meshfree and finite element numerical methods become fast and accurate, which is a feature of paramount importance, as far as computational efficiency of numerical methods is concerned. Three benchmark problems were analyzed with these techniques, in order to assess the accuracy and efficiency of the new integrated local formulations of meshfree and finite element numerical methods. The results obtained in this work are in perfect agreement with those of the available analytical solutions and, furthermore, outperform the computational efficiency of other methods. Thus, the accuracy and efficiency of the local numerical methods presented in this paper make this a very reliable and robust formulation.Originality/valuePresentation of a new local mesh-free numerical method. The method, linearly integrated along the boundary of the local domain, implements an algorithm with no further integration required. The method is absolutely reliable, with remarkably-accurate results. The method is quite robust, with extremely-fast computations.

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Yari, Ehsan, and Hassan Ghassemi. "Boundary Element Method Applied to Added Mass Coefficient Calculation of the Skewed Marine Propellers." Polish Maritime Research 23, no. 2 (April 1, 2016): 25–31. http://dx.doi.org/10.1515/pomr-2016-0017.

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AbstractThe paper mainly aims to study computation of added mass coefficients for marine propellers. A three-dimensional boundary element method (BEM) is developed to predict the propeller added mass and moment of inertia coefficients. Actually, only few experimental data sets are available as the validation reference. Here the method is validated with experimental measurements of the B-series marine propeller. The behavior of the added mass coefficients predicted based on variation of geometric and flow parameters of the propeller is calculated and analyzed. BEM is more accurate in obtaining added mass coefficients than other fast numerical methods. All added mass coefficients are nondimensionalized by fluid density, propeller diameter, and rotational velocity. The obtained results reveal that the diameter, expanded area ratio, and thickness have dominant influence on the increase of the added mass coefficients.

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Mora, Jaime, and Leszek Demkowicz. "Fast Integration of DPG Matrices Based on Sum Factorization for all the Energy Spaces." Computational Methods in Applied Mathematics 19, no. 3 (July 1, 2019): 523–55. http://dx.doi.org/10.1515/cmam-2018-0205.

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AbstractNumerical integration of the stiffness matrix in higher-order finite element (FE) methods is recognized as one of the heaviest computational tasks in an FE solver. The problem becomes even more relevant when computing the Gram matrix in the algorithm of the Discontinuous Petrov Galerkin (DPG) FE methodology. Making use of 3D tensor-product shape functions, and the concept of sum factorization, known from standard high-order FE and spectral methods, here we take advantage of this idea for the entire exact sequence of FE spaces defined on the hexahedron. The key piece to the presented algorithms is the exact sequence for the one-dimensional element, and use of hierarchical shape functions. Consistent with existing results, the presented algorithms for the integration of {H^{1}}, {H(\operatorname{curl})}, {H(\operatorname{div})}, and {L^{2}} inner products, have the {\mathcal{O}(p^{7})} computational complexity in contrast to the {\mathcal{O}(p^{9})} cost of conventional integration routines. Use of Legendre polynomials for shape functions is critical in this implementation. Three boundary value problems under different variational formulations, requiring combinations of {H^{1}}, {H(\operatorname{div})} and {H(\operatorname{curl})} test shape functions, were chosen to experimentally assess the computation time for constructing DPG element matrices, showing good correspondence with the expected rates.

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MENG, WENHUI, and JUNZHI CUI. "COMPARATIVE STUDY OF TWO DIFFERENT FMM–BEM METHODS IN SOLVING 2-D ACOUSTIC TRANSMISSION PROBLEMS WITH A MULTILAYERED OBSTACLE." International Journal of Structural Stability and Dynamics 11, no. 01 (February 2011): 197–214. http://dx.doi.org/10.1142/s021945541100404x.

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The fast multipole method (FMM) is an effective approach for accelerating the computation efficiency of the boundary element method (BEM) in solving problems that are computationally intensive. This paper presents two different BEMs, i.e., Kress' and Seydou's methods, for solving two-dimensional (2D) acoustic transmission problems with a multilayered obstacle, along with application of the FMM to solution of the related boundary integral equations. Conventional BEM requires O(MN2) operations to compute the equations for this problem. By using the FMM, both the amount of computation and the memory requirement of the BEM are reduced to order O(MN), where M is the number of layers of the obstacle. The efficiency and accuracy of this approach in dealing with the acoustic transmission problems containing a multilayered obstacle are demonstrated in the numerical examples. It is confirmed that this approach can be applied to solving the acoustic transmission problems for an obstacle with multilayers.

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Journal articles on the topic 'Fast Boundary Element Methods'

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