İki boyutlu elastodinamik problemlerin sınır eleman metodu ile formülasyonu

Bu çalışmada, iki boyutlu elastodinamik problemlerin çözümü için süreksiz kuadratik sınır eleman formülasyonu yapılmıştır. Fourier dönüşüm uzayında yapılan formülasyonda, integral denklemlerin çözümü Gauss sayısal integrasyon yöntemi ile yapılmaktadır. P sabit ve Q hareketli noktalarının aynı eleman üzerinde olması halinde ortaya çıkan, 1/r tekilliği kaldırılarak, Ln(1/r) tekilliği ise uygun logaritmik ve standart Gauss yöntemlerinin kullanılmasıyla giderilmiştir. Yapılan formülasyona dayalı, iki boyutlu dinamik analizler için genel amaçlı bilgisayar programı hazırlanmıştır. Fourier dönüşüm uzayında dinamik analiz yapan program ile frekansa küçük değerler verilerek elastostatik analiz de yapılabilmektedir. İki adet dinamik problem çözülmüş ve elde edilen sonuçların, literatürde verilen sonuçlarla uyum içinde olduğu gözlenmiştir.

Formulation of 2D elastodynamic problems with boundary element method

In this study, discontinuous quadratic boundary element formulation is presented for analysis of elastodynamic problems. In the formulation, which is performed in Fourier transform space, the integral equations are solved numerically by Gaussian quadratures. 1/r and Ln(1/r) singularities that exist when source point P and varied point Q are within same element either removed or manipulated by using logarithmic and standard Gaussian quadratures. Based on the formulation presented in this study, a general purpose computer program is developed, for two dimensional dynamic analyses. The program performs the analysis in Fourier transform space and the program which is developed for elastodynamic analysis can also be used for elastostatic analysis by assigning a small value to the frequency. Two dynamic problems are solved and the obtained results are compared with those in the literature.

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1. Beskos, D.E., Boundary Element Methods in Mechanics. Elsevier Science Publishers, Amsterdam, 1987.
2. Brebbia, C.A., and Dominguez, J., Boundary Elements an Introductory Course. Computational Mechanics Publications, Southampton, 1989.
3. Becker, A.A., The Boundary Element Method in Engineering. McGraw-Hill Book Company, London, 1992.
4. Banerjee, P.K., The Boundary Element Methods in Engineering. McGraw-Hill Book Company, London, 1994.
5. Aliabadi, M.H., The Boundary Element Method Applications in Solids and Structures. John Wiley & Sons Limited, West Sussex, 2002.
6. Brebbia, C.A., Telles, J.C.F. and Wrobel, L.C., Boundary Element Techniques. Springer, New York, 1984.
7. Siebrits, E., and Crouch, SL., Two-Dimensional Elastodynamics Displacement Discontinuity Method. Int. J. for Num. Meth. in Eng., 37: 3229–3250, 1994.
8. Shen, S.Y., An Indirect Elastodynamic Boundary Element Method with Analytic Bases. Int. J. for Num. Meth. in Eng., 57: 767–792, 2003.
9. Partridge, P.W., Brebbia, C.A., and Wrobel, L.C., The Dual Reciprocity Boundary Element Method. Computational Mechanics Publications, Southampton and Elsevier Applied Science, London, 1992.
10. Mengi, Y., Tanrıkulu, A.H., and Tanrıkulu, A.K., Boundary Element Method for Elastic Media, An Introduction. ODTÜ Basım İşliği, Ankara, 1994.
11. Frangi, A., Elastodynamics By BEM: A New Direct Formulation. Int. J. for Num. Meth. in Eng., 45: 721–740, 1999.
12. Nedelec, J.C., Integral Equations with Non Integrable Kernels. Integral Equations and Operator Theory, 5: 562-572, 1982.
13. Nishimura, N., and Kobayashi, S., A Regularized Boundary Integral Equation Method for Elastodynamic Crack Problems, Comp. Mech., 4: 319-328, 1989.
14. Manolis, G.D., ve Beskos, D.E., Boundary Element Method in Elastodynamics. Unwin Hyman, London, 1988.
15. Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, V.T., Numerical Recipes, Cambridge University Press, New York, 1986.
16. Chien, C-C., and Wu, T-Y., A Particular Integral BEM/Time-Discontinuous FEM Metodology for Solving 2-D Elastodynamic Problems. Int. J. of Solids and Struc., 38, 289-306, 2001.