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Journal articles on the topic 'Boundary integral method'

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Published: 4 June 2021

Last updated: 31 January 2023

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1

Sládek, V., and J. Sládek. "Boundary integral method in magnetoelasticity." International Journal of Engineering Science 26, no. 5 (January 1988): 401–18. http://dx.doi.org/10.1016/0020-7225(88)90001-8.

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2

Benedetti, Ivano, Vincenzo Gulizzi, and Vincenzo Mallardo. "Boundary Element Crystal Plasticity Method." Journal of Multiscale Modelling 08, no. 03n04 (September 2017): 1740003. http://dx.doi.org/10.1142/s1756973717400030.

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A three-dimensional (3D) boundary element method for small strains crystal plasticity is described. The method, developed for polycrystalline aggregates, makes use of a set of boundary integral equations for modeling the individual grains, which are represented as anisotropic elasto-plastic domains. Crystal plasticity is modeled using an initial strains boundary integral approach. The integration of strongly singular volume integrals in the anisotropic elasto-plastic grain-boundary equations are discussed. Voronoi-tessellation micro-morphologies are discretized using nonstructured boundary and volume meshes. A grain-boundary incremental/iterative algorithm, with rate-dependent flow and hardening rules, is developed and discussed. The method has been assessed through several numerical simulations, which confirm robustness and accuracy.

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3

Setukha, A. V. "Method of Boundary Integral Equations with Hypersingular Integrals in Boundary-Value Problems." Journal of Mathematical Sciences 257, no. 1 (July 29, 2021): 114–26. http://dx.doi.org/10.1007/s10958-021-05475-3.

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4

Yu, Dehao, and Longhua Zhao. "Natural boundary integral method and related numerical methods." Engineering Analysis with Boundary Elements 28, no. 8 (August 2004): 937–44. http://dx.doi.org/10.1016/s0955-7997(03)00120-6.

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5

Cong, Wenxiang, and Ge Wang. "Boundary integral method for bioluminescence tomography." Journal of Biomedical Optics 11, no. 2 (2006): 020503. http://dx.doi.org/10.1117/1.2191790.

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6

Zakerdoost, Hassan, Hassan Ghassemi, and Mehdi Iranmanesh. "Solution of Boundary Value Problems Using Dual Reciprocity Boundary Element Method." Advances in Applied Mathematics and Mechanics 9, no. 3 (January 17, 2017): 680–97. http://dx.doi.org/10.4208/aamm.2014.m783.

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AbstractIn this work we utilize the boundary integral equation and the Dual Reciprocity Boundary Element Method (DRBEM) for the solution of the steady state convection-diffusion-reaction equations with variable convective coefficients in two-dimension. The DRBEM is a numerical method to transform the domain integrals into the boundary only integrals by using the fundamental solution of Helmholtz equation. Some examples are calculated to confirm the accuracy of the approach. The results obtained by the analytic solutions are in good agreement with ones provided by the DRBEM technique.

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7

Wen, Pi Hua, and M. H. Aliabadi. "Dynamic Crack Problems Using Meshless Method." Key Engineering Materials 525-526 (November 2012): 601–4. http://dx.doi.org/10.4028/www.scientific.net/kem.525-526.601.

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A Meshless Approximation Based on the Radial Basis Function (RBF) Is Developed for Analysis of Dynamic Crack Problems. A Weak Form for a Set of Governing Equations with a Unit Test Function Is Transformed into Local Integral Equations. A Completed Set of Closed Forms of the Local Boundary Integrals Are Obtained. as the Closed Forms of the Local Boundary Integrals Are Obtained, there Are No any Domain or Boundary Integrals to Be Calculated Numerically in this Approach.

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8

Ying, Wenjun, and Wei-Cheng Wang. "A Kernel-Free Boundary Integral Method for Variable Coefficients Elliptic PDEs." Communications in Computational Physics 15, no. 4 (April 2014): 1108–40. http://dx.doi.org/10.4208/cicp.170313.071113s.

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AbstractThis work proposes a generalized boundary integral method for variable coefficients elliptic partial differential equations (PDEs), including both boundary value and interface problems. The method is kernel-free in the sense that there is no need to know analytical expressions for kernels of the boundary and volume integrals in the solution of boundary integral equations. Evaluation of a boundary or volume integral is replaced with interpolation of a Cartesian grid based solution, which satisfies an equivalent discrete interface problem, while the interface problem is solved by a fast solver in the Cartesian grid. The computational work involved with the generalized boundary integral method is essentially linearly proportional to the number of grid nodes in the domain. This paper gives implementation details for a second-order version of the kernel-free boundary integral method in two space dimensions and presents numerical experiments to demonstrate the efficiency and accuracy of the method for both boundary value and interface problems. The interface problems demonstrated include those with piecewise constant and large-ratio coefficients and the heterogeneous interface problem, where the elliptic PDEs on two sides of the interface are of different types.

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9

Koumoutsakos, P., and A. Leonard. "Improved boundary integral method for inviscid boundary condition applications." AIAA Journal 31, no. 2 (February 1993): 401–4. http://dx.doi.org/10.2514/3.11682.

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10

Moshaiov, A., and C. Vitooraporn. "Application of Boundary Integral Method to Continuous Plate Structures." Journal of Ship Research 34, no. 03 (September 1, 1990): 212–17. http://dx.doi.org/10.5957/jsr.1990.34.3.212.

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Ship hulls are composed of complex plate structures. Such structures can be analyzed by analytical as well as numerical methods. Here it is demonstrated that boundary integrals can be used to analyze such structures by a seminumerical method. Boundary integrals have been used for analyzing single plate bending problems, and only recently have attempts been made to extend their use to complex plate structures. In this paper the Direct Boundary Integral Method for bending of thin elastic plates is extended to analysis of the continuous plate structures. The concepts of compatibility and equilibrium conditions along the common boundary have been imposed in conjunction with a boundary element formulation for each panel. Several examples of loading conditions are presented and compared with analytical solutions. The excellent agreement achieved illustrates the effectiveness of the extended formulation.

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11

Hriberšek, Matjaž, Polde Škerget, and Herbert Mang. "Preconditioned conjugate gradient methods for boundary-domain integral method." Engineering Analysis with Boundary Elements 12, no. 2 (January 1993): 111–18. http://dx.doi.org/10.1016/0955-7997(93)90005-6.

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12

Bodin, Anthony, Jianfeng Ma, X. J. Xin, and Prakash Krishnaswami. "A meshless integral method based on regularized boundary integral equation." Computer Methods in Applied Mechanics and Engineering 195, no. 44-47 (September 2006): 6258–86. http://dx.doi.org/10.1016/j.cma.2005.12.005.

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13

Petrov, Andrey, Sergey Aizikovich, and Leonid A. Igumnov. "Modeling of Wave Propagation in the Unsaturated Soils Using Boundary Element Method." Key Engineering Materials 743 (July 2017): 158–61. http://dx.doi.org/10.4028/www.scientific.net/kem.743.158.

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Problems of wave propagation in poroelastic bodies and media are considered. The behavior of the poroelastic medium is described by Biot theory for partially saturated material. Mathematical model is written in term of five basic functions – elastic skeleton displacements, pore water pressure and pore air pressure. Boundary element method (BEM) is used with step method of numerical inversion of Laplace transform to obtain the solution. Research is based on direct boundary integral equation of three-dimensional isotropic linear theory of poroelasticity. Green’s matrices and, based on it, boundary integral equations are written for basic differential equations in partial derivatives. Discrete analogue are obtained by applying the collocation method to a regularized boundary integral equation. To approximate the boundary consider its decomposition to a set of quadrangular and triangular 8-node biquadratic elements, where triangular elements are treated as singular quadrangular. Every element is mapped to a reference one. Interpolation nodes for boundary unknowns are a subset of geometrical boundary-element grid nodes. Local approximation follows the Goldshteyn’s generalized displacement-stress matched model: generalized boundary displacements are approximated by bilinear elements whereas generalized tractions are approximated by constant. Integrals in discretized boundary integral equations are calculated using Gaussian quadrature in combination with singularity decreasing and eliminating algorithms.

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14

KISHIDA, Michiya, Kazuaki SASAKI, and Shiroh MACHINO. "Accuracy of numerical surface integrals in the indirect boundary integral method." Transactions of the Japan Society of Mechanical Engineers Series A 57, no. 533 (1991): 181–87. http://dx.doi.org/10.1299/kikaia.57.181.

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15

Ghassemi, Hassan, and Ahmad Reza Kohansal. "Solutions of the Acoustic Problem in the 3D Form of the Helmholtz Equation Using DRBEM." International Journal of Advance Research and Innovation 4, no. 1 (2016): 272–77. http://dx.doi.org/10.51976/ijari.411639.

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This paper presents to the solutions of the acoustic problem using dual reciprocity boundary element method (DRBEM). The acoustic propagates inside the silencer and its sound can be formulated by the three dimensional (3D) Helmholtz equation. The form of the equation is which can be solved by the integral form on the boundary. The conventional boundary element method (BEM) is not suitable for solving the acoustic problems of silencers with higher Mach number subsonic flow, due to the presence of domain integral. The dual reciprocity method (DRM) is a method that converts the domain integral into the boundary integral. Mathematical formulations discretisation form and evaluation of the integrals are described and discussed. Algorithm of the method is also presented.

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16

Kristalinskii, V. R. "ABOUT THE APPROXIMATE SOLUTION OF THE USUAL AND GENERALIZED HILBERT BOUNDARY VALUE PROBLEMS FOR ANALYTICAL FUNCTIONS." Mathematical Modelling and Analysis 5, no. 1 (December 15, 2000): 119–26. http://dx.doi.org/10.3846/13926292.2000.9637134.

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In this article the methods for obtaining the approximate solution of usual and generalized Hilbert boundary value problems are proposed. The method of solution of usual Hilbert boundary value problem is based on the theorem about the representation of the kernel of the corresponding integral equation by τ = t and on the earlier proposed method for the computation of the Cauchy‐type integrals. The method for approximate solution of the generalized boundary value problem is based on the method for computation of singular integral of the formproposed by the author. All methods are implemented with the Mathcad and Maple.

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17

DAI, Baodong. "Improved Meshless Local Boundary Integral Equation Method." Chinese Journal of Mechanical Engineering 44, no. 10 (2008): 108. http://dx.doi.org/10.3901/jme.2008.10.108.

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18

Hornberger, Klaus, and Uzy Smilansky. "The boundary integral method for magnetic billiards." Journal of Physics A: Mathematical and General 33, no. 14 (March 31, 2000): 2829–55. http://dx.doi.org/10.1088/0305-4470/33/14/315.

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19

Yu,, De-hao, and D. Givoli,. "Natural Boundary Integral Method and Its Applications." Applied Mechanics Reviews 56, no. 5 (August 29, 2003): B65. http://dx.doi.org/10.1115/1.1584411.

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20

Kot, V. A. "Integral Method of Boundary Characteristics: Neumann Condition." Journal of Engineering Physics and Thermophysics 91, no. 2 (March 2018): 445–70. http://dx.doi.org/10.1007/s10891-018-1765-4.

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21

Cerrolaza, M. "The p-adaptive boundary integral equation method." Advances in Engineering Software 15, no. 3-4 (January 1992): 261–67. http://dx.doi.org/10.1016/0965-9978(92)90108-r.

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22

Hao, Wenrui, Bei Hu, Shuwang Li, and Lingyu Song. "Convergence of boundary integral method for a free boundary system." Journal of Computational and Applied Mathematics 334 (May 2018): 128–57. http://dx.doi.org/10.1016/j.cam.2017.11.016.

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23

Hu, Jin-Xiu, Hai-Feng Peng, and Xiao-Wei Gao. "Numerical Evaluation of Arbitrary Singular Domain Integrals Using Third-Degree B-Spline Basis Functions." Mathematical Problems in Engineering 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/284106.

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A new approach is presented for the numerical evaluation of arbitrary singular domain integrals. In this method, singular domain integrals are transformed into a boundary integral and a radial integral which contains singularities by using the radial integration method. The analytical elimination of singularities condensed in the radial integral formulas can be accomplished by expressing the nonsingular part of the integration kernels as a series of cubic B-spline basis functions of the distancerand using the intrinsic features of the radial integral. In the proposed method, singularities involved in the domain integrals are explicitly transformed to the boundary integrals, so no singularities exist at internal points. A few numerical examples are provided to verify the correctness and robustness of the presented method.

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24

Al-Swat, Kawther K., and Said G. Ahmed. "A New Iterative Method for Multi-Moving Boundary Problems Based Boundary Integral Method." Journal of Applied Mathematics and Physics 03, no. 09 (2015): 1126–37. http://dx.doi.org/10.4236/jamp.2015.39140.

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25

Yanishevskyi, V. S., and L. S. Nodzhak. "The path integral method in interest rate models." Mathematical Modeling and Computing 8, no. 1 (2020): 125–36. http://dx.doi.org/10.23939/mmc2021.01.125.

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An application of path integral method to Merton and Vasicek stochastic models of interest rate is considered. Two approaches to a path integral construction are shown. The first approach consists in using Wieners measure with the following substitution of solutions of stochastic equations into the models. The second approach is realised by using transformation from Wieners measure to the integral measure related to the stochastic variables of Merton and Vasicek equations. The introduction of boundary conditions is considered in the second approach in order to remove incorrect time asymptotes from the classic Merton and Vasicek models of interest rates. By the example of Merton model with zero drift, a Dirichlet boundary condition is considered. A path integral representation of term structure of interest rate is obtained. The estimate of the obtained path integrals is performed, where it is shown that the time asymptote is limited.

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26

Sulym, Heorhiy, Iaroslav Pasternak, Mariia Smal, and Andrii Vasylyshyn. "Mixed Boundary Value Problem for an Anisotropic Thermoelastic Half-Space Containing Thin Inhomogeneities." Acta Mechanica et Automatica 13, no. 4 (December 1, 2019): 238–44. http://dx.doi.org/10.2478/ama-2019-0032.

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Abstract The paper presents a rigorous and straightforward approach for obtaining the 2D boundary integral equations for a thermoelastic half-space containing holes, cracks and thin foreign inclusions. It starts from the Cauchy integral formula and the extended Stroh formalism which allows writing the general solution of thermoelastic problems in terms of certain analytic functions. In addition, with the help of it, it is possible to convert the volume integrals included in the equation into contour integrals, which, in turn, will allow the use of the method of boundary elements. For modelling of solids with thin inhomogeneities, a coupling principle for continua of different dimensions is used. Applying the theory of complex variable functions, in particular, Cauchy integral formula and Sokhotski–Plemelj formula, the Somigliana type boundary integral equations are constructed for thermoelastic anisotropic half-space. The obtained integral equations are introduced into the modified boundary element method. A numerical analysis of the influence of boundary conditions on the half-space boundary and relative rigidity of the thin inhomogeneity on the intensity of stresses at the inclusions is carried out.

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27

Zhou, H. L., B. X. Bian, Y. Tian, B. Yu, and Z. R. Niu. "The Calculation of Potential Derivatives by Using NBIE for Anisotropic Potential Problems." Journal of Mechanics 33, no. 2 (July 15, 2016): 183–91. http://dx.doi.org/10.1017/jmech.2016.66.

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AbstractThe natural boundary integral equation (NBIE) is developed to calculate potential derivatives for potential problems with anisotropic media. Firstly, the governing equation of the two-dimensional anisotropic potential problem is transformed into standard Laplace equation by a coordinate transformation method. Then a potential derivative boundary integral equation named as NBIE is extended to solve the anisotropic potential problem. The most important virtue of the NBIE is that the singularity of the integral kernel function is reduced by one order in comparison with the conventional potential derivative boundary integral equation(CDBIE). Therefore the new potential derivative boundary integral equation only contains strongly singular integrals rather than hyper-singular integrals. Thus the NBIE can calculate more accurate potential derivative results for both boundary nodes and interior points. Moreover, in combination with the analytical integral regularization algorithm of nearly singular integrals, the NBIE can obtain more accurate potential derivatives of interior points very close to the boundary than the CDBIE. Numerical examples on heat conduction in anisotropic media demonstrate the accuracy and efficiency of the NBIE.

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28

Jiang, Xin Rong, and Xin Fu. "Virtual Boundary Integral Method for Anisotropic Potential Problems." Applied Mechanics and Materials 155-156 (February 2012): 370–74. http://dx.doi.org/10.4028/www.scientific.net/amm.155-156.370.

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Heat conduction in anisotropic materials has important applications in science and engineering. In this paper the virtual boundary element method (VBEM) is applied to solve these problems. Due to the fact of a virtual boundary outside the real boundary, the VBEM does not need to treat the singular boundary integrals, and thus, is more accurate and convenient than the traditional one. Numerical examples are presented, to demonstrate the efficiency and accuracy of this method.

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29

Gao, X. W. "A Boundary Element Method Without Internal Cells for Two-Dimensional and Three-Dimensional Elastoplastic Problems." Journal of Applied Mechanics 69, no. 2 (October 25, 2001): 154–60. http://dx.doi.org/10.1115/1.1433478.

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In this paper, a new and simple boundary element method without internal cells is presented for the analysis of elastoplastic problems, based on an effective transformation technique from domain integrals to boundary integrals. The strong singularities appearing in internal stress integral equations are removed by transforming the domain integrals to the boundary. Other weakly singular domain integrals are transformed to the boundary by approximating the initial stresses with radial basis functions combined with polynomials in global coordinates. Three numerical examples are presented to demonstrate the validity and effectiveness of the proposed method.

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30

Mužík, Juraj, and Roman Bulko. "Regularized singular boundary method for groundwater flow in a cofferdam." MATEC Web of Conferences 196 (2018): 03025. http://dx.doi.org/10.1051/matecconf/201819603025.

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The boundary integral methods form one group of methods for solving differential equations. The boundary element method (BEM) is the basic method of this group. However, it requires the boundary mesh of elements and the evaluation of improper singular integrals, that arise due to fundamental solution singularity. Therefore, boundary meshless methods have recently have come into focus to remove these shortcomings. One of the most promising boundary collocation numerical schemes is the singular boundary method (SBM). To tackle the singularity of the fundamental solution, this method adopts the concept of original intensity factors (OIFs). The application of the proposed SBM scheme to groundwater flow in the cofferdam structure is presented and compared to the finite element method (FEM) solution.

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31

Kosztin, Ioan, and Klaus Schulten. "Boundary Integral Method for Stationary States of Two-Dimensional Quantum Systems." International Journal of Modern Physics C 08, no. 02 (April 1997): 293–325. http://dx.doi.org/10.1142/s0129183197000278.

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The boundary integral method for calculating the stationary states of a quantum particle in nano-devices and quantum billiards is presented in detail at an elementary level. According to the method, wave functions inside the domain of the device or billiard are expressed in terms of line integrals of the wavefunction and its normal derivative along the domain's boundary; the respective energy eigenvalues are obtained as the roots of Fredholm determinants. Numerical implementations of the method are described and applied to determine the energy level statistics of billiards with circular and stadium shapes and demonstrate the quantum mechanical characteristics of chaotic motion. The treatment of other examples as well as the advantages and limitations of the boundary integral method are discussed.

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32

Sládek, V. "Boundary element method in micropolar thermoelasticity. Part I: Boundary integral equations." Engineering Analysis with Boundary Elements 2, no. 1 (March 1985): 40–50. http://dx.doi.org/10.1016/0955-7997(85)90041-4.

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33

Ying, Wenjun, and Craig S. Henriquez. "A kernel-free boundary integral method for elliptic boundary value problems." Journal of Computational Physics 227, no. 2 (December 2007): 1046–74. http://dx.doi.org/10.1016/j.jcp.2007.08.021.

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34

Sládek, V., and J. Sládek. "Boundary element method in micropolar thermoelasticity. Part I: Boundary integral equations." Engineering Analysis 2, no. 1 (March 1985): 40–50. http://dx.doi.org/10.1016/0264-682x(85)90050-4.

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35

Sharp, S., and S. L. Crouch. "Boundary Integral Methods for Thermoelasticity Problems." Journal of Applied Mechanics 53, no. 2 (June 1, 1986): 298–302. http://dx.doi.org/10.1115/1.3171755.

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The boundary integral method for solving transient heat flow problems is extended to calculate thermally induced stresses and displacements. These results are then corrected by means of an elastostatic solution to satisfy the boundary conditions.

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36

Wu, Kuang-Chong. "A New Boundary Integral Equation Method for Cracked Piezoelectric Bodies." Key Engineering Materials 306-308 (March 2006): 465–70. http://dx.doi.org/10.4028/www.scientific.net/kem.306-308.465.

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A novel integral equation method is developed in this paper for the analysis of two-dimensional general piezoelectric cracked bodies. In contrast to the conventional boundary integral methods based on reciprocal work theorem, the present method is derived from Stroh’s formalism for anisotropic elasticity in conjunction with Cauchy’s integral formula. The proposed boundary integral equations contain generalized boundary displacement (displacements and electric potential) gradients and generalized tractions (tractions and electric displacement) on the non-crack boundary, and the generalized dislocations on the crack lines. The boundary integral equations can be solved using Gaussian-type integration formulas without dividing the boundary into discrete elements. The crack-tip singularity is explicitly incorporated and the generalized intensity factors can be computed directly. Numerical examples of generalized stress intensity factors are given to illustrate the effectiveness and accuracy of the present method.

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37

Du, Liang Liang, and Xiong Hua Wu. "Natural Boundary Integral Method for Irregular Plate Problems." Applied Mechanics and Materials 138-139 (November 2011): 693–98. http://dx.doi.org/10.4028/www.scientific.net/amm.138-139.693.

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Natural boundary integral method is applied to deal with plate problems defined in irregular domains. We divide the solution into two parts, a particular solution for inhomogeneous biharmonic equation and the general solution for homogeneous biharmonic equation. For the former, the direct expansion method of boundary conditions is used to treat the arbitrary domains, and the processes of natural boundary integral method coupling with finite element method are omitted. Numerical experiments show that the method is very simple and of high accuracy.

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38

Guiggiani, M., G. Krishnasamy, T. J. Rudolphi, and F. J. Rizzo. "A General Algorithm for the Numerical Solution of Hypersingular Boundary Integral Equations." Journal of Applied Mechanics 59, no. 3 (September 1, 1992): 604–14. http://dx.doi.org/10.1115/1.2893766.

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The limiting process that leads to the formulation of hypersingular boundary integral equations is first discussed in detail. It is shown that boundary integral equations with hypersingular kernels are perfectly meaningful even at non-smooth boundary points, and that special interpretations of the integrals involved are not necessary. Careful analysis of the limiting process has also strong relevance for the development of an appropriate numerical algorithm. In the second part, a new general method for the evaluation of hypersingular surface integrals in the boundary element method (BEM) is presented. The proposed method can be systematically applied in any BEM analysis, either with open or closed surfaces, and with curved boundary elements of any kind and order (of course, provided the density function meets necessary regularity requirements at each collocation point). The algorithm operates in the parameter plane of intrinsic coordinates and allows any hypersingular integral in the BEM to be directly transformed into a sum of a double and a one-dimensional regular integrals. Since all singular integrations are performed analytically, standard quadrature formulae can be used. For the first time, numerical results are presented for hypersingular integrals on curved (distorted) elements for three-dimensional problems.

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39

Landman, K. A. "A boundary integral method for contaminant transport in two adjacent porous media." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 30, no. 3 (January 1989): 251–67. http://dx.doi.org/10.1017/s0334270000006214.

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AbstractThe problem of transient two-dimensional transport by diffusion and advection of a decaying contaminant in two adjacent porous media is solved using a boundary-integral method. The method requires the construction of appropriate Green's functions. Application of Green's theorem in the plane then yields representations for the contaminant concentration in both regions in terms of an integral of the initial concentration over the region's interior and integrals along the boundaries of known quantities and the unknown interfacial flux between the two adjacent media. This flux is given by a first-kind integral equation, which can be solved numerically by a discretisation technique. Examples of contaminant transport in fractured porous media systems are presented.

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40

Lavrova, Olga, and Viktor Polevikov. "APPLICATION OF COLLOCATION BEM FOR AXISYMMETRIC TRANSMISSION PROBLEMS IN ELECTRO- AND MAGNETOSTATICS." Mathematical Modelling and Analysis 21, no. 1 (January 26, 2016): 16–34. http://dx.doi.org/10.3846/13926292.2016.1128488.

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This paper considers the numerical solution of boundary integral equations for an exterior transmission problem in a three-dimensional axisymmetric domain. The resulting potential problem is formulated in a meridian plane as the second kind integral equation for a boundary potential and the first kind integral equation for a boundary flux. The numerical method is an axisymmetric collocation with equal order approximations of the boundary unknowns on a polygonal boundary. The complete elliptic integrals of the kernels are approximated by polynomials. An asymptotic kernels behavior is analyzed for accurate numerical evaluation of integrals. A piecewise-constant midpoint collocation and a piecewise-linear nodal collocation on a circular arc and on its polygonal interpolation are used for test computations on uniform meshes. We analyze empirically the influence of the polygonal boundary interpolation to the accuracy and the convergence of the presented method. We have found that the polygonal boundary interpolation does not change the convergence behavior on the smooth boundary for the piecewise-constant and the piecewise-linear collocation.

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41

Nishimura, N. "Fast multipole accelerated boundary integral equation methods." Applied Mechanics Reviews 55, no. 4 (July 1, 2002): 299–324. http://dx.doi.org/10.1115/1.1482087.

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Fundamentals of Fast Multipole Method (FMM) and FMM accelerated Boundary Integral Equation Method (BIEM) are presented. Developments of FMM accelerated BIEM in the Laplace and Helmholtz equations, wave equation, and heat equation are reviewed. Applications of these methods in computational mechanics are surveyed. This review article contains 173 references.

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42

Hamdin, H. A. M. Ben, and G. Tanner. "Multi-Component BEM for the Helmholtz Equation: A Normal Derivative Method." Shock and Vibration 19, no. 5 (2012): 957–67. http://dx.doi.org/10.1155/2012/451785.

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We describe a multi-component boundary element method for predicting wave energy distributions in a complex built-up system with material properties changing discontinuously at boundaries between sub-components. We point out that depending on the boundary conditions and the number of interfaces between sub-components, it may be advantageous to use a normal derivative method to set up the integral kernels. We describe how the resulting hypersingular integral kernels can be regularised. The method can be used to minimise the number of weakly singular integrals thus leading to BEM formulations which are easier to handle.

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43

Duan, Jun-Sheng, Li-Xia Jing, and Ming Li. "The Mixed Boundary Value Problems and Chebyshev Collocation Method for Caputo-Type Fractional Ordinary Differential Equations." Fractal and Fractional 6, no. 3 (March 9, 2022): 148. http://dx.doi.org/10.3390/fractalfract6030148.

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The boundary value problem (BVP) for the varying coefficient linear Caputo-type fractional differential equation subject to the mixed boundary conditions on the interval 0≤x≤1 was considered. First, the BVP was converted into an equivalent differential–integral equation merging the boundary conditions. Then, the shifted Chebyshev polynomials and the collocation method were used to solve the differential–integral equation. Varying coefficients were also decomposed into the truncated shifted Chebyshev series such that calculations of integrals were only for polynomials and can be carried out exactly. Finally, numerical examples were examined and effectiveness of the proposed method was verified.

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Constanda, Christian. "The boundary integral equation method in plane elasticity." Proceedings of the American Mathematical Society 123, no. 11 (November 1, 1995): 3385. http://dx.doi.org/10.1090/s0002-9939-1995-1301017-3.

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Christoph, G. H. "Roughness Effects in Head’s Integral Boundary-Layer Method." Journal of Fluids Engineering 107, no. 3 (September 1, 1985): 428–30. http://dx.doi.org/10.1115/1.3242504.

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Yang, Yajun. "A Discrete Collocation Method for Boundary Integral Equations." Journal of Integral Equations and Applications 7, no. 2 (June 1995): 233–61. http://dx.doi.org/10.1216/jiea/1181075870.

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Jakobsen, Per. "Calculating optical forces using the boundary integral method." Physica Scripta 80, no. 3 (August 25, 2009): 035401. http://dx.doi.org/10.1088/0031-8949/80/03/035401.

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Yu, Pei, B. Dietz, and L. Huang. "Quantizing neutrino billiards: an expanded boundary integral method." New Journal of Physics 21, no. 7 (July 23, 2019): 073039. http://dx.doi.org/10.1088/1367-2630/ab2fde.

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Nicolas, A., and L. Nicolas. "3D automatic mesh generation for boundary integral method." IEEE Transactions on Magnetics 26, no. 2 (March 1990): 767–70. http://dx.doi.org/10.1109/20.106430.

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Sucec, James. "Modern Integral Method Calculation of Turbulent Boundary Layers." Journal of Thermophysics and Heat Transfer 20, no. 3 (July 2006): 552–57. http://dx.doi.org/10.2514/1.16397.

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