An Efficient Numerical Method for Calculation of Elastic and Thermo-Elastic Fields in a Homogeneous Medium with Several Heterogeneous Inclusions

Author(s)
Sergey Kanaun

ABSTRACT

The work is devoted to calculation of static elastic and thermo-elastic fields in a homogeneous medium with a finite number of isolated heterogeneous inclusions. Firstly, the problem is reduced to the solution of inte-gral equations for strain and stress fields in the medium with inclusions. Then, Gaussian approximating func-tions are used for discretization of these equations. For such functions, the elements of the matrix of the dis-cretized problem are calculated in explicit analytical forms. The method is mesh free, and only the coordi-nates of the approximating nodes are the geometrical information required in the method. If such nodes compose a regular grid, the matrix of the discretized problem obtains the Toeplitz properties. By the calcula-tion of matrix-vector products with such matrices, the Fast Fourier Transform technique may be used. The latter accelerates essentially the process of the iterative solution of the disretized problem. The results of calculations of elastic fields in 3D-medium with an isolated spherical heterogeneous inclusion are compared with exact solutions. Examples of the calculation of elastic and thermo-elastic fields in the medium with several inclusions are presented.

The work is devoted to calculation of static elastic and thermo-elastic fields in a homogeneous medium with a finite number of isolated heterogeneous inclusions. Firstly, the problem is reduced to the solution of inte-gral equations for strain and stress fields in the medium with inclusions. Then, Gaussian approximating func-tions are used for discretization of these equations. For such functions, the elements of the matrix of the dis-cretized problem are calculated in explicit analytical forms. The method is mesh free, and only the coordi-nates of the approximating nodes are the geometrical information required in the method. If such nodes compose a regular grid, the matrix of the discretized problem obtains the Toeplitz properties. By the calcula-tion of matrix-vector products with such matrices, the Fast Fourier Transform technique may be used. The latter accelerates essentially the process of the iterative solution of the disretized problem. The results of calculations of elastic fields in 3D-medium with an isolated spherical heterogeneous inclusion are compared with exact solutions. Examples of the calculation of elastic and thermo-elastic fields in the medium with several inclusions are presented.

KEYWORDS

Elasticity, Heterogeneous Medium, Integral Equations, Gaussian Approximating Functions, Toeplitz Matrix, Fast Fourier Transform

Elasticity, Heterogeneous Medium, Integral Equations, Gaussian Approximating Functions, Toeplitz Matrix, Fast Fourier Transform

Cite this paper

nullS. Kanaun, "An Efficient Numerical Method for Calculation of Elastic and Thermo-Elastic Fields in a Homogeneous Medium with Several Heterogeneous Inclusions,"*World Journal of Mechanics*, Vol. 1 No. 2, 2011, pp. 31-43. doi: 10.4236/wjm.2011.12005.

nullS. Kanaun, "An Efficient Numerical Method for Calculation of Elastic and Thermo-Elastic Fields in a Homogeneous Medium with Several Heterogeneous Inclusions,"

References

[1] I. Kunin, “Elastic Media with Microstructure II,” Springer, Berlin, Toronto and New York, 1983.

[2] W. Chew, “Waves and Fields in Inhomogeneous Media,” Van Nostrand Reinhold, Amsterdam, 1990.

[3] A. Peterson, S. Ray and R. Mittra, “Computational Methods for Electromagnetics,” IEEE Press, New York, 1997. doi:10.1109/9780470544303

[4] C. Y. Dong, S. H. Lo and Y. K. Cheung, “Numerical Solution of 3d-Elastostatic Inclusion Problems Using the Volume Integral Equation Method,” Computer Methods in Applied Mechanics and Engineering, Vol. 192, No. 1-3, 2003, pp. 95-106. doi:10.1016/S0045-7825(02)00534-0

[5] C. Y. Dong, S. H. Lo and Y. K. Cheung, “Numerical Solution for Elastic Inclusion Problems by Domain Integral Equation with Integration by Means of Radial Basis Functions,” Engineering Analysis with Boundary Elements, Vol. 28, No. 6, 2004, pp. 623-632. doi:10.1016/j.enganabound.2003.06.001

[6] V. Maz’ya and G. Schmidt G, “Approximate Approximation,” American Mathematical Society, Mathematical Surveys and Monographs, Vol. 141, 2007, pp. 1-348.

[7] S. Kanaun, “Fast Calculation of Elastic Fields in A Homogeneous Medium with Isolated Heterogeneous Inclusions,” International Journal of Multiscale Computational Engineering, Vol. 7, No. 4, 2009, pp. 263-276. doi:10.1615/IntJMultCompEng.v7.i4.30

[8] S. Kanaun and V. Levin, “Self-Consistent Methods for Composites, Vol. 1: Static Problems (Solids Mechanics and Its Applications),” Springer, Dordecht, Vol. 148, 2008.

[9] S. Mikhlin, “Multidimensional Singular Integrals and In- tegral Equations,” Pergamon Press, Oxford & New York, 1965.

[10] W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, “Numerical Recipes in FORTRAN: The Art of Scientific Computing,” 2nd Edition, Cambridge University Press, New York, 1992.

[11] G. Golub and C. Van Loan, “Matrix Computations,” Johns Hopkins University Press, Baltimore, 1993.

[12] I. Kunin, “Methods of Tensor Analysis in the Theory of Dislocations,” US Department of Commerce, Clearinghouse for Federal Scientific Technology and Information, Springfield, VA 221151, 1965.

[13] S. Kanaun, “A Method for the Solution of the Diffraction Problem on Perfectly Conducting Screens,” Journal of Computational Physics, Vol. 176, No. 1, 2002, pp. 170- 190. doi:10.1006/jcph.2001.6974

[14] S. Kanaun, “Scattering of Monochromatic Electromagnetic Waves on 3D-Dielectric Bodies of Arbitrary Shapes,” Progress in Electromagnetics Research B, Vol. 21, No. 1, 2010, pp. 129-150.

[1] I. Kunin, “Elastic Media with Microstructure II,” Springer, Berlin, Toronto and New York, 1983.

[2] W. Chew, “Waves and Fields in Inhomogeneous Media,” Van Nostrand Reinhold, Amsterdam, 1990.

[3] A. Peterson, S. Ray and R. Mittra, “Computational Methods for Electromagnetics,” IEEE Press, New York, 1997. doi:10.1109/9780470544303

[4] C. Y. Dong, S. H. Lo and Y. K. Cheung, “Numerical Solution of 3d-Elastostatic Inclusion Problems Using the Volume Integral Equation Method,” Computer Methods in Applied Mechanics and Engineering, Vol. 192, No. 1-3, 2003, pp. 95-106. doi:10.1016/S0045-7825(02)00534-0

[5] C. Y. Dong, S. H. Lo and Y. K. Cheung, “Numerical Solution for Elastic Inclusion Problems by Domain Integral Equation with Integration by Means of Radial Basis Functions,” Engineering Analysis with Boundary Elements, Vol. 28, No. 6, 2004, pp. 623-632. doi:10.1016/j.enganabound.2003.06.001

[6] V. Maz’ya and G. Schmidt G, “Approximate Approximation,” American Mathematical Society, Mathematical Surveys and Monographs, Vol. 141, 2007, pp. 1-348.

[7] S. Kanaun, “Fast Calculation of Elastic Fields in A Homogeneous Medium with Isolated Heterogeneous Inclusions,” International Journal of Multiscale Computational Engineering, Vol. 7, No. 4, 2009, pp. 263-276. doi:10.1615/IntJMultCompEng.v7.i4.30

[8] S. Kanaun and V. Levin, “Self-Consistent Methods for Composites, Vol. 1: Static Problems (Solids Mechanics and Its Applications),” Springer, Dordecht, Vol. 148, 2008.

[9] S. Mikhlin, “Multidimensional Singular Integrals and In- tegral Equations,” Pergamon Press, Oxford & New York, 1965.

[10] W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, “Numerical Recipes in FORTRAN: The Art of Scientific Computing,” 2nd Edition, Cambridge University Press, New York, 1992.

[11] G. Golub and C. Van Loan, “Matrix Computations,” Johns Hopkins University Press, Baltimore, 1993.

[12] I. Kunin, “Methods of Tensor Analysis in the Theory of Dislocations,” US Department of Commerce, Clearinghouse for Federal Scientific Technology and Information, Springfield, VA 221151, 1965.

[13] S. Kanaun, “A Method for the Solution of the Diffraction Problem on Perfectly Conducting Screens,” Journal of Computational Physics, Vol. 176, No. 1, 2002, pp. 170- 190. doi:10.1006/jcph.2001.6974

[14] S. Kanaun, “Scattering of Monochromatic Electromagnetic Waves on 3D-Dielectric Bodies of Arbitrary Shapes,” Progress in Electromagnetics Research B, Vol. 21, No. 1, 2010, pp. 129-150.