A Review on Techniques of Domain Integrals Computation in Boundary Elements Method

Document Type : Review Article

Authors

Abstract

In this article, a review of the evaluation methods of the domain integrals in the boundary element method will be presented. The emergence of domain integrals in the formulation of the boundary element method mainly originates from the inertia term in dynamic problems, body forces in static problems or the effects of material heterogeneity. There are several approaches to calculate boundary and domain integrals in boundary element methods. Choosing the type of integration method has a prominent effect on the accuracy of the numerical solution. In this research, a comprehensive review on the techniques of domain integrals computation will be presented based on two approaches, i.e. domain splitting, and converting the domain integrals to boundary ones. The review focuses primarily on the formulation of approaches without requiring domain splitting, because of their popularity. Among them, the dual reciprocity method and the radial integration method have been described as the most efficient. At the end, the details of the modified radial integration method for calculating the integrals within non-convex domains will be stated.  

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References

  1. 1. Katsikadelis, J. T., “Boundary Elements: Theory and Applications”, Vol. 1, UK: Elsevier Science Ltd, 2002.

    1. Balaš, J., Sládek, J., and Sládek V., “Stress Analysis by Boundary Element Methods”, Elsevier, 2013.
    2. Alatawi, I. A., and Trevelyan, J., “A Direct Evaluation of Stress Intensity Factors Using the Extended Dual Boundary Element Method”, Engineering Analysis with Boundary Elements, Vol. 52, pp. 56-63, 2015.
    3. Aliabadi, M. H., “Boundary Element Formulations in Fracture Mechanics”, 1997.
    4. Aliabadi, M.H., “A New Generation of Boundary Element Methods in Fracture Mechanics”, International Journal of Fracture, Vol. 86, No. 1, pp. 91-125, 1997.
    5. Fedeliński, P., “Boundary Element Method in Dynamic Analysis of Structures with Cracks”, Engineering Analysis with Boundary Elements, Vol. 28, No. 9, pp. 1135-1147, 2004.
    6. Andrade, H. C., and Leonel, E.D., “An Enriched Dual Boundary Element Method Formulation for Linear Elastic Crack Propagation”, Engineering Analysis with Boundary Elements, Vol. 121, pp.158-179, 2020.
    7. Katsikadelis, J. T., “Α Powerful BEM-Based Solution Technique for Solving Linear and Nonlinear Engineering Problems, in Boundary Element Method XVI”, 16th International Boundary Element Method Conference (BEM XVI), Computational Mechanics Publications: Southampton, UK., pp. 167–182, 1994.
    8. Katsikadelis, J. T., “The Boundary Element Method for Plate Analysis”, Vol. 1, UK: Elsevier Inc, 2014.
    9. Riveiro, M.A., and Gallego, R., “Boundary Elements and the Analog Equation Method for the Solution of Elastic Problems in 3-D Non-Homogeneous Bodies”, Computer Methods in Applied Mechanics and Engineering, Vol. 263, pp. 12-19, 2013.
    10. Ishiguro, S., Nakajima, H., and Tanaka, M., “Analysis of Two-Dimensional Steady-State Heat Conduction in Anisotropic Solids by Boundary Element Method Using Analog Equation Method and Green's Theorem”, Journal of Computational Science and Technology, Vol. 3, No. 1, pp. 66-76, 2009.
    11. Carrer, J. A.M., and Telles, J.C.F., “A Boundary Element Formulation to Solve Transient Dynamic Elastoplastic Problems”, Computers & Structures, Vol. 45, No. 4, pp. 707-713, 1992.
    12. Carrer, J.A.M., and Mansur, W.J., “Alternative Time-Marching Schemes for Elastodynamic Analysis with the Domain Boundary Element Method Formulation”, Computational Mechanics, Vol. 34, No. 5, pp. 387-399, 2004.
    13. Katsikadelis, J. T., “Boundary Elements: Theory and Applications”, 2002.
    14. Beskos, D.E., “Boundary Element Analysis of Plates and Shells”, Vol. 1, Berlin: Springer-Verlag, 1991.
    15. Beer, G., Smith, I. M., and Duenser, C., “The Boundary Element Method with Programming”, Vienna, Austria: Springer, 2008.
    16. Tan, C. L., “Three-Dimensional Boundary Integral Equation Stress Analysis of Cracked Components”, Imperial College, University of London, 1979.
    17. Najarzadeh, L., Movahedian, B. and Azhari, M., “Free Vibration and Buckling Analysis of Thin Plates Subjected to High Gradients Stresses Using the Combination of Finite Strip and Boundary Element Methods”, Thin-Walled Structures, Vol. 123, pp. 36-47, 2018.
    18. Feng, W.Z., Gao X.W., Liu, J., and Yang, K., “A New BEM for Solving 2D and 3D Elastoplastic Problems without Initial Stresses/Strains”, Engineering Analysis with Boundary Elements, Vol. 61, pp. 134-144, 2015.
    19. Danson, D. J., “Boundary Element Formulation for

    Problems in Linear Isotropic Elasticity with Body Forces”, Springer Verlag, 1981.

    1. De Mey, G., Suciu, R., Munteanu, C., and Matthys, L., “BEM Solution of Poisson's Equation in Two Dimensions with Polynomial Interpolation of the Source Function”, Engineering Analysis with Boundary Elements, Vol. 18, pp. 175-178, 1997.
    2. Suciu, R., De Mey, G., and De Baetselier, E., “BEM Solution of Poisson's Equation with Source Function Satisfying ∆ρ=constant”, Engineering Analysis with Boundary Elements, Vol. 25, No. 2, pp. 141-145, 2001.
    3. Nowak, A. J., and Brebbia, C. A., “The Multiple Reciprocity Method: A New Approach for Transforming BEM Domain Integrals to the Boundary”, Engineering Analysis, Vol. 6, pp. 164-167, 1989.
    4. Al-Jawary, M. A., “The Radial Integration Boundary Integral and Integro-differential Equation Methods for Numerical Solution of Problems with Variable Coefficients”, Brunel University: London, 2012.
    5. Neves, A. C., and Brebbia, C. A., “The Multiple Reciprocity Boundary Element Method in Elasticity: A New Approach for Transforming Domain Integrals to the Boundary”, International Journal for Numerical Methods in Engineering, Vol. 31, No. 4, pp. 709-727. 1991.
    6. Shiah, Y. C., and Lin, Y. J., “Multiple Reciprocity Boundary Element Analysis of Two-Dimensional Anisotropic Thermoelasticity Involving an Internal Arbitrary Non-Uniform Volume Heat Source”, International Journal of Solids and Structures, Vol. 40, No. 24, pp. 6593-6612, 2003.
    7. Chen, W., Fu, Z. J., and Chen, C. S., “Boundary-Type RBF Collocation Methods, in Recent Advances in Radial Basis Function Collocation Methods”, Springer Berlin Heidelberg: Berlin, Heidelberg. pp. 51-87, 2014.
    8. Gao, X. W., “The Radial Integration Method for Evaluation of Domain Integrals with Boundary-Only Discretization”, Engineering Analysis with Boundary Elements, Vol. 26, No. 10, pp. 905-916, 2002.
    9. Chen, W., Fu, Z. J., and Jin, B. T., “A truly Boundary-only Meshfree Method for Inhomogeneous Problems Based on Recursive Composite Multiple Reciprocity Technique”, Engineering Analysis with Boundary Elements, Vol. 34, No. 3, pp. 196-205, 2010.
    10. Nardini, D. and Brebbia, C., “A New Approach to Free Vibration Analysis Using Boundary Elements”, Applied Mathematical Modelling, Vol. 7, No. 3, pp. 157-162, 1983.
    11. Partridge P. W., Brebbia, C. A., and Wrobel, L.C, “The Dual Reciprocity Boundary Element Method”, Computational Mechanics Publications: Southampton, 1992.
    12. Loeffler, C. F., Mansur, W. J., de Melo Barcelos, H., and Bulcão, A., “Solving Helmholtz Problems with the Boundary Element Method Using Direct Radial Basis Function Interpolation”, Engineering Analysis with Boundary Elements, Vol. 61, pp. 218-225, 2015.
    13. Hamzeh Javaran, S., Khaji, N., and Noorzad, A., “First Kind Bessel Function (J-Bessel) as Radial Basis Function for Plane Dynamic Analysis Using Dual Reciprocity Boundary Element Method”, Acta Mechanica, Vol. 218, pp. 247–258, 2011.
    14. Hamzeh Javaran, S., and Khaji, N., “Complex Fourier Element Shape Functions for Analysis of 2D Static and Transient Dynamic Problems Using Dual Reciprocity Boundary Element Method”, Engineering Analysis with Boundary Elements, Vol. 95, pp. 222–237, 2018
    15. Wen, P.H., Aliabadi, M.H., and Young, A., “Transformation of Domain Integrals to Boundary Integrals in BEM Analysis of Shear Deformable Plate Bending Problems”, Computational Mechanics Vol. 24, No. 4, pp. 304-309, 1999.
    16. Uğurlu, B., “A Dual Reciprocity Boundary Element Solution Method for the Free Vibration Analysis of Fluid-Coupled Kirchhoff Plates”, Journal of Sound and Vibration, Vol. 340, pp. 190-210, 2015.
    17. Useche, J., and Albuquerque, E. L., “Dynamic Analysis of Shear Deformable Plates Using the Dual Reciprocity Method”, Engineering Analysis with Boundary Elements, Vol. 36, No. 5, pp. 627-632, 2012.
    18. Davies, T.W., and Moslehy, F.A., “Modal Analysis of Plates Using the Dual Reciprocity Boundary Element Method”, Engineering Analysis with Boundary Elements, Vol. 14, No. 4, pp. 357-362, 1994.
    19. Soares Jr, R. A., Palermo Jr, L., and Wrobel, L. C., “Application of the Dual Reciprocity Method for the Buckling Analysis of Plates with Shear Deformation”, Engineering Analysis with Boundary Elements, Vol. 106, pp. 427-439, 2019.
    20. Chen, C.S., Brebbia, C.A. and Power, H., “Dual Reciprocity Method Using Compactly Supported Radial Basis Functions”, Communications in Numerical Methods in Engineering, Vol. 15, No. 2, pp. 137-150, 1999.
    21. Li, J., Hon, Y. C., and Chen, C. S., “Numerical Comparisons of Two Meshless Methods Using Radial Basis Functions”, Engineering Analysis with Boundary Elements, Vol. 26, No. 3, pp. 205-225, 2002.
    22. Chanthawara, K., Kaennakham, S., and Toutip, W., “The Dual Reciprocity Boundary Element Method (DRBEM) with Multiquadric Radial Basis Function for Coupled Burger’s Equations”, The International Journal of Multiphysics, Vol. 8., No. 2, pp. 123-144, 2014.
    23. Gao, X.W., “A Boundary Element Method Without Internal Cells for Two-Dimensional and Three-Dimensional Elastoplastic Problems”, Journal of Applied Mechanics, Vol. 69, No. 2, pp. 154-160, 2001.
    24. Deng, Q., Li, C. G., Wang, S. L., Zheng, H., and Ge, X. R., “A Nonlinear Complementarity Approach for Elastoplastic Problems by BEM without Onternal Cells”, Engineering Analysis with Boundary Elements, Vol. 35, No.3, pp. 313-318, 2011.
    25. Gao, X.W., “Boundary Element Analysis in Thermoelasticity with and without Internal Cells”, International Journal for Numerical Methods in Engineering, Vol. 57, No. 7, pp. 975-990, 2003.
    26. Gao, X. W., Zheng, B. J., Yang, K., and Zhang, C., “Radial Integration BEM for Dynamic Coupled Thermoelastic Analysis Under Thermal Shock Loading”, Computers & Structures, Vol. 158, pp. 140-147, 2015.
    27. Yao, W. A., Yao, H. X., and Yu, B., “Radial Integration BEM for Solving Non-Fourier Heat Conduction Problems”, Engineering Analysis with Boundary Elements, Vol. 60, pp. 18-26, 2015.
    28. Feng, W. Z., Yang, K., Cui, M., and Gao, X. W., “Analytically-Integrated Radial Integration BEM for Solving Three-dimensional Transient Heat Conduction Problems”, International Communications in Heat and Mass Transfer, Vol. 79, pp. 21-30, 2016.
    29. Yang, K., Peng, H. F., Wang, J., Xing, C. H., and Gao, X. W., “Radial Integration BEM for Solving Transient Nonlinear Heat Conduction with Temperature-dependent Conductivity”,  International Journal of Heat and Mass Transfer, Vol. 108, pp. 1551-1559, 2017.
    30. Cui, M., Xu, B. B., Feng, W. Z., Zhang, Y., Gao, X. W., and Peng, H. F., “A Radial Integration Boundary Element Method for Solving Transient Heat Conduction Problems with Heat Sources and Variable Thermal Conductivity”,  Numerical Heat Transfer, Part B: Fundamentals, Vol. 73, No. 1, pp. 1-18, 2018.
    31. Chen, L., Li, K., Peng, X., Lian, H., Lin, X., and Fu, Z., “Isogeometric Boundary Element Analysis for 2D Transient Heat Conduction Problem with Radial Integration Method”, Computer Modeling in Engineering & Sciences, Vol. 126, No. 1, pp. 125-146, 2021.
    32. Albuquerque, E. L., Sollero, P., and Portilho de Paiva, W., “The Radial Integration Method Applied to Dynamic Problems of Anisotropic Plates”, Communications in Numerical Methods in Engineering, Vol. 23, No. 9, pp. 805-818, 2007
    33. Soares Jr, R. A., Palermo Jr, L., and Wrobel, L. C., “Application of the Radial Integration Method for the Buckling Analysis of Plates with Shear Deformation”, Engineering Analysis with Boundary Elements,Vol. 118, pp. 250-264, 2020.
    34. Schoenberg, I.J., “Metric Spaces and Positive Definite Functions”, Transactions of the American Mathematical Society, Vol. 44, No. 3, pp. 522-536, 1938.
    35. Micchelli, C. A., “Interpolation of Scattered Data: Distance Matrices and Conditionally Positive Definite Functions, in Approximation Theory and Spline Functions”, Springer, pp. 143-145, 1984.
    36. M. A. Golberg, Chen, C. S., and Bowman, H., “Some Recent Results and Proposals for the Use of Radial Basis Functions in the BEM”, Engineering Analysis with Boundary Elements, Vol. 23, pp. 285–296, 1999.
    37. Golberg, M. A., Chen, C. S. and Karur, S. R., “Improved Multiquadric Approximation for Partial Differential Equations”, Engineering Analysis with Boundary Elements, Vol. 18, No. 1, pp. 9-17, 1996.
    38. Najarzadeh, L., Movahedian, B., and Azhari, M., “Stability Analysis of the Thin plates with Arbitrary Shapes Subjected to Non-Uniform Stress Fields Using Boundary Element and Radial Integration Methods”, Engineering Analysis with Boundary Elements,Vol. 87: pp. 111-121, 2018.
    39. Najarzadeh, L., Movahedian, B., and Azhari, M., “Numerical Solution of Scalar Wave Equation by the Modified Radial Integration Boundary Element Method”, Engineering Analysis with Boundary Elements,Vol. 105: pp. 267-278, 2019.
    40. Najarzadeh, L., Movahedian, B., and Azhari, M., “Numerical Solution of Water Wave Propagation Problems Over Variable Bathymetries Using the Modified Radial Integration Boundary Element Method”, Ocean Engineering, Vol. 257, p. 111613, 2022.
    41. Zonoubi, K., Movahedian, B., and Azhari, M., ‘The Solution of Elastodynamic Problems Using Modified Radial Integration Boundary Element Method (MRIBEM)”, Engineering Analysis with Boundary Elements, Vol. 144, pp. 482-491, 2022.
Journal of Computational Methods in Engineering
Volume 42, Issue 1 - Serial Number 25
September 2023
Pages 77-88

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