OPEN ACCESS
This paper presents a numerical method for topology optimisation for two-dimensional elastodynamics based on the level set method and the boundary element method (BEM) accelerated by the H-matrix method and its application to identifications of defects in an infinite elastic medium. Gradient-based topology optimisation methods require design sensitivity, which is obtained by solving some boundary value problems. The BEM is employed for this sensitivity analysis because the BEM can deal with infinite domains rigorously without any approximation. However, the computational cost in the BEM is expensive, and this is a serious drawback since we need to repeat sensitivity analysis even for a single optimisation process. In this study, the H-matrix method is used as an acceleration method of the BEM for the reduction of the computational cost of the sensitivity analysis. Also proposed is a method to improve the efficiency of the H-matrix method by exploiting a property of the kernel function of the elastodynamic fundamental solution. Some numerical examples are demonstrated, and the effectiveness of the proposed method is confirmed.
boundary element method, defect identification, elastic wave, level set method, topological derivative, topology optimisation, H-matrix method
[1] Reissner, E. & Stavsky, Y., Bending and stretching of certain types of heterogeneous aeolotropic elastic plates. Transactions of the ASME, Journal of Applied Mechanics, 28(3), pp. 402–408, 1961.https://doi.org/10.1115/1.3641719
[2] Whitney, J.M. & Leissa, A.W., Analysis of heterogeneous, anisotropic plates. Transac-tions of the ASME, Journal of Applied Mechanics, 36, pp. 261–266, 1969.https://doi.org/10.1115/1.3564618
[3] Whitney, J.M. & Leissa, A.W., Analysis of a simply supported laminated anisotropic rectangular plate. AIAA Journal, 8(1), pp. 28–33, 1970.https://doi.org/10.2514/3.5601
[4] Whitney, J.M., The effect of boundary conditions on the response of laminated compos-ites. Journal of Composite Materials, 4(2), pp. 192–203, 1970.https://doi.org/10.1177/002199837000400205
[5] Harris, G.Z., The buckling and post-buckling behaviour of composite plates under biaxial loading. International Journal of Mechanical Sciences, 17, pp. 187–202, 1975. https://doi.org/10.1016/0020-7403(75)90052-1
[6] Noor, A.K., Mathers, M.D. & Anderson, M.S., Exploiting symmetries for efficient post-buckling analysis of composite plates. AIAA Journal, 15(1), pp. 24–32, 1977.https://doi.org/10.2514/3.60601
[7] Syngellakis, S., A boundary element approach to buckling of general laminates. WIT Transactions on Modelling and Simulation, 53, pp. 145–155, 2012.https://doi.org/10.2495/be120131
[8] Hwu, C., Green’s function for the composite laminates with bending extension coupling. Composite Structures, 63, pp. 283–292, 2004.https://doi.org/10.1016/s0263-8223(03)00175-2
[9] Hwu, C., Boundary integral equations for general laminated plates with coupled stretching-bending deformation. The Royal Society, Proceedings: Mathematical, Physical and Engineering Sciences, 466(2116), pp. 1027–1054, 2010.https://doi.org/10.1098/rspa.2009.0432
[10] Syngellakis, S., Fundamental solutions for the coupled extension-flexure laminate problem. WIT Transactions on Modelling and Simulation, 61, pp. 235–246, 2015. https://doi.org/10.2495/bem380191
[11] Courant, R. & Hilbert, D., Methods of Mathematical Physics vol. II: Partial Differential Equations, Interscience Publishers: New York, 1953.
[12] Hildebrand, F.B., Advanced Calculus for Applications, Prentice-Hall, Inc.: Englewood Cliffs, N.J., 1962.
[13] Cheng, Z.-Q. & Reddy, J.N., Octet formalism for Kirchhoff anisotropic plates. The Royal Society, Proceedings: Mathematical, Physical and Engineering Sciences, 458(2022), pp. 1499–1517, 2002.https://doi.org/10.1098/rspa.2001.0934