# zbMATH — the first resource for mathematics

Topology optimization of steady-state heat conduction structures using meshless generalized finite difference method. (English) Zbl 07268580
Summary: This paper proposes the topology optimization for steady-state heat conduction structures by incorporating the meshless-based generalized finite difference method (GFDM) and the solid isotropic microstructures with penalization interpolation model. In the meshless GFDM numerical scheme, the explicit formulae of the partial differential equation are expressed by the Taylor series expansions and the moving-least squares approximations to address the required partial derivatives of unknown nodal variables. With the relative density of meshless GFDM node as the design variable, the implementation of the topology optimization is formulated involving the minimization of heat potential capacity as the objective function under node number constraint. Moreover, sensitivity of the objective function is derived based on the adjoint method, and sensitivity filtering subsequently suppresses the checkerboard pattern. Next, the update of design variables at each iteration is solved by the optimality criteria method. At last, several numerical examples are illustrated to demonstrate the validity and feasibility of the proposed method.

##### MSC:
 74S20 Finite difference methods applied to problems in solid mechanics 65N06 Finite difference methods for boundary value problems involving PDEs 49Q12 Sensitivity analysis for optimization problems on manifolds 74P15 Topological methods for optimization problems in solid mechanics 80A19 Diffusive and convective heat and mass transfer, heat flow
Full Text:
##### References:
 [1] Bendsøe, M. P.; Kikuchi, N., Generating optimal topologies in structural design using a homogenization method, Comput Meth Appl Mech Eng, 71, 2, 197-224 (1988) · Zbl 0671.73065 [2] Bendsøe, M. P., Optimal shape design as a material distribution problem, Struct Optim, 1, 4, 193-202 (1989) [3] Zhou, M.; Rozvany, G. I.N, The COC algorithm, part II: topological, geometry and generalized shape optimization, Comput Meth Appl Mech Eng, 89, 1-3, 197-224 (1991) [4] Stolpe, M.; Svanberg, K., An alternative interpolation scheme for minimum compliance topology optimization, Struct Multidiscip Optim, 22, 2, 116-124 (2001) [5] Xie, Y.; Steven, G. P., A simple evolutionary procedure for structural optimization, Comput Struct, 49, 5, 885-896 (1993) [6] Huang, X.; Xie, Y. M.; Burry, M. C., A new algorithm for bi-directional evolutionary structural optimization, Int J C-Mech Sy, 49, 4, 1091-1099 (2007) [7] Huang, X.; Xie, Y. M., Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method, Finite Elem Anal Des, 43, 14, 1039-1049 (2007) [8] Sui, Y. K.; Peng, X. R., The ICM method with objective function transformed by variable discrete condition for continuum structure, Acta Mech Sin, 22, 1, 68-75 (2006) · Zbl 1200.74117 [9] Wang, M. Y.; Wang, X.; Guo, D., A level set method for structural topology optimization, Comput Meth Appl Mech Eng, 192, 1-2, 227-246 (2003) · Zbl 1083.74573 [10] Wang, M. Y.; Zhou, S., Phase field: a variational method for structural topology optimization, Comp Model Eng, 6, 6, 547-566 (2004) · Zbl 1152.74382 [11] Norato, J. A.; Bendsoe, M. P.; Haber, R. B.; Tortorelli, D. A., A topological derivative method for topology optimization, Struct Multidiscip Optim, 33, 4-5, 375-386 (2007) · Zbl 1245.74074 [12] Guo, X.; Zhang, W. S.; Zhong, W. L., Doing topology optimization explicitly and geometrically-a new moving morphable components based framework, J Appl Mech Trans, 81, 8, Article 081009 pp. (2014) [13] Rozvany, G. I.N, A critical review of established methods of structural topology optimization, Struct Multidiscip Optim, 7, 3, 217-237 (2009) · Zbl 1274.74004 [14] Sigmund, O.; Maute, K., Topology optimization approaches A comparative review, Struct Multidiscip Optim, 48, 6, 1031-1055 (2013) [15] Deaton, J. D.; Grandhi, R. V., A survey of structural and multidisciplinary continuum topology optimization: post 2000, Struct Multidiscip Optim, 49, 1, 1-38 (2014) [16] Benz, W., Applications of smooth particle hydrodynamics (SPH) to astrophysical problems, Comput Phys Commun, 48, 1, 97-105 (1988) [17] Belytschko, T.; Lu, Y. Y.; Gu, L., Element-free Galerkin methods, Int J Numer Methods Eng, 37, 2, 229-256 (1994) · Zbl 0796.73077 [18] Liu, W. K.; Jun, S.; Zhang, Y. F., Reproducing kernel particle methods, Int J Numer Methods Fluids, 20, 1081-1106 (1995) · Zbl 0881.76072 [19] Atluri, S. N.; Zhu, T., A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput Mech, 22, 2, 117-127 (1998) · Zbl 0932.76067 [20] Wang, F. J.; Fan, C. M.; Hua, Q. S.; Gu, Y., Localized MFS for the inverse Cauchy problems of two-dimensional Laplace and biharmonic equations, Appl Math Comput, 364, Article 124658 pp. (2020) · Zbl 1433.65338 [21] Qu, W. Z.; Fan, C. M.; Li, X. L., Analysis of an augmented moving least squares approximation and the associated localized method of fundamental solutions, Comput Math Appl, 80, 1, 13-30 (2020) · Zbl 1446.65195 [22] Wang, F. J.; Gu, Y.; Qu, W. Z.; Zhang, C. Z., Localized boundary knot method and its application to large-scale acoustic problems, Comput Meth Appl Mech Eng, 361, Article 112729 pp. (2020) · Zbl 1442.76071 [23] Wang, F. J.; Fan, C. M.; Zhang, C. Z.; Lin, J., A localized space-time method of fundamental 3 solutions for diffusion and convection-diffusion problems, Adv Appl Math Mech, 12, 940-958 (2020) [24] Kim, J., Topology optimization for two-dimensional continuum using element free galerkin method (1997), The University of Texas: The University of Texas Arlington [25] Zheng, J.; Long, S. Y.; Li, G. Y., The topology optimization design for continuum structures based on the element free Galerkin method, Eng Anal Bound Elem, 34, 7, 666-672 (2010) · Zbl 1267.74094 [26] Yang, X. J.; Zheng, J.; Long, S. Y., Topology optimization of continuum structures with displacement constraints based on meshless method, Int J Mech Mater Des, 13, 2, 311-320 (2017) [27] Wu, Y.; Ma, Y. Q.; Feng, W.; Cheng, Y. M., Topology optimization using the improved element-free Galerkin method for elasticity, Chin Phys B, 26, 8, Article 080203 pp. (2017) [28] Shobeiri, V., Topology optimization using bi-directional evolutionary structural optimization based on the element-free Galerkin method, Eng Optimiz, 48, 3, 380-396 (2016) [29] He, Q. Z.; Kang, Z.; Wang, Y. Q., A topology optimization method for geometrically nonlinear structures with meshless analysis and independent density field interpolation, Comput Mech, 54, 3, 629-644 (2014) · Zbl 1311.74098 [30] Wang, Y.; Luo, Z.; Wu, J. L.; Zhang, N., Topology optimization of compliant mechanisms using element-free Galerkin method, Adv Eng Softw, 85, 61-72 (2015) [31] Cui, M. T.; Chen, H. F.; Zhou, J. L.; Wang, F. L., A meshless method for multi-material topology optimization based on the alternating active-phase algorithm, Eng Comput, 33, 4, 871-884 (2017) [32] Zhang, J. P.; Wang, S. S.; Zhou, G. Q.; Gong, S. G.; Yin, S. H., Topology optimization of thermal structure for isotropic and anisotropic materials using the element-free Galerkin method, Eng Optimiz (2019) [33] Cho, S. H.; Kwak, J., Topology design optimization of geometrically non-linear structures using meshfree method, Comput Meth Appl Mech Eng, 195, 44-47, 5909-5925 (2006) · Zbl 1124.74039 [34] Zhou, J. X.; Zou, W., Meshless approximation combined with implicit topology description for optimization of continua, Struct Multidiscip Optim, 36, 4, 347-353 (2008) [35] Li, S.; Atluri, S. N., The MLPG mixed collocation method for material orientation and topology optimization of anisotropic solids and structures, CMES-Comp Model Eng Sci, 30, 1, 37-56 (2008) [36] Li, S. L.; Long, S. Y.; Li, G. Y., A topology optimization of moderately thick plates based on the meshless numerical method, CMES-Comp Model Eng Sci, 60, 1, 73-94 (2010) · Zbl 1231.74361 [37] Zheng, J.; Long, S. Y.; Xiong, Y. B.; Li, G. Y., A topology optimization design for the continuum structure based on the meshless numerical technique, CMES-Comp Model Eng Sci, 34, 2, 137-154 (2008) · Zbl 1232.74126 [38] Lin, J.; Guan, Y. J.; Zhao, G. Q.; Naceur, H.; Lu, P., Topology optimization of plane structures using smoothed particle hydrodynamics method, Int J Numer Methods Eng, 110, 8, 726-744 (2017) [39] Jaskowiec, J.; Milewski, S., The effective interface approach for coupling of the FE and meshless FD methods and applying essential boundary conditions, Comput Math Appl, 70, 5, 962-979 (2015) · Zbl 1443.65342 [40] Jaworska, I.; Orkisz, J., On nonlinear analysis by the multipoint meshless FDM, Eng Anal Bound Elem, 92, 231-243 (2018) · Zbl 1403.65112 [41] Cecot, W.; Milewski, S.; Orkisz, J., Determination of overhead power line cables configuration by FEM and meshless FDM, Int J Comput Methods, 15, 2, Article 1850004 pp. (2018) · Zbl 1404.74156 [42] Tolstykh, A. I.; Shirobokov, D. A., On using radial basis functions in a ‘‘finite difference mode’’ with applications to elasticity problems, Comput Mech, 33, 1, 68-79 (2003) · Zbl 1063.74104 [43] Mishra, P. K.; Fasshauer, G. E.; Sen, M. K.; Ling, L., A stabilized radial basis-finite difference (RBF-FD) method with hybrid kernels, Comput Math Appl, 77, 9, 2354-2368 (2019) · Zbl 1442.65324 [44] Hidayat, M. I.P, Meshless local B-spline collocation method for heterogeneous heat conduction problems, Eng Anal Bound Elem, 101, 76-88 (2019) · Zbl 07034716 [45] Hidayat, M. I.P.; Ariwahjoedi, B.; Parman, S.; Rao, T. V.V. L.N, Meshless local B-spline collocation method for two-dimensional heat conduction problems with nonhomogenous and time-dependent heat sources, J Heat Trans-T ASME, 139, 7, Article 071302 pp. (2017) [46] Cole, J. B., A high-accuracy realization of the Yee algorithm using non-standard finite differences, IEEE Trans Microw Theory Tech, 45, 6, 991-996 (1997) [47] Liszka, T., An interpolation method for an irregular net of nodes, Int J Numer Methods Eng, 20, 9, 1599-1612 (1984) · Zbl 0544.65006 [48] Liszka, T.; Orkisz, J., The finite difference method at arbitrary irregular grids and its application in applied mechanics, Comput Struct, 11, 1-2, 83-95 (1980) · Zbl 0427.73077 [49] Payre, G. M.J., Influence graphs and the generalized finite difference method, Comput Meth Appl Mech Eng, 196, 1933-1945 (2007) · Zbl 1173.76376 [50] Ureña, F.; Benito, J. J.; Salete, E.; Gavete, L., A note on the application of the generalized finite difference method to seismic wave propagation in 2D, J Comput Appl Math, 236, 12, 3016-3025 (2012) · Zbl 1236.86011 [51] Benito, J. J.; Urena, F.; Gavete, L., Influence of several factors in the generalized finite difference method, Appl Math Model, 25, 12, 1039-1053 (2001) · Zbl 0994.65111 [52] Qu, W. Z.; Fan, C. M.; Zhang, Y. M., Analysis of three-dimensional heat conduction in functionally graded materials by using a hybrid numerical method, Int J Heat Mass Transf, 145, Article 118771 pp. (2019) [53] Li, P. W.; Fu, Z. J.; Gu, Y.; Song, L. N., The generalized finite difference method for the inverse Cauchy problem in two-dimensional isotropic linear elasticity, Int J Solids Struct, 174, 69-84 (2019) [54] Gavete, L.; Urena, F.; Benito, J. J.; Salete, E., A note on the dynamic analysis using the generalized finite difference method[J], J Comput Appl Math, 252, 132-147 (2013) · Zbl 1290.74043 [55] Jaskowiec, J.; Milewski, S., Coupling finite element method with meshless finite difference method in thermomechanical problems, Comput Math Appl, 72, 9, 2259-2279 (2016) · Zbl 1368.74061 [56] Gu, Y.; Qu, W. Z.; Chen, W.; Song, L. N.; Zhang, C. Z., The generalized finite difference method for long-time dynamic modeling of three-dimensional coupled thermoelasticity problems, J Comput Phys, 384, 42-59 (2019) [57] Qu, W. Z., A high accuracy method for long-time evolution of acoustic wave equation, Appl Math Lett, 98, 135-141 (2019) · Zbl 1448.35325 [58] Fu, Z. J.; Xie, Z. Y.; Ji, S. Y.; Tsai, C. C.; Li, A. L., Meshless generalized finite difference method for water wave interactions with multiple-bottom-seated-cylinder-array structures, Ocean Eng, 195, Article 106736 pp. (2020) [59] Gu, Y.; Hua, Q. S.; Zhang, C. Z.; He, X. Q., The generalized finite difference method for long-time transient heat conduction in 3D anisotropic composite materials, Appl Math Model, 71, 316-330 (2019) · Zbl 07186619 [60] Qu, W. Z.; Gu, Y.; Zhang, Y. M.; Fan, C. M.; Zhang, C. Z., A combined scheme of generalized finite difference method and Krylov deferred correction technique for highly accurate solution of transient heat conduction problems, Int J Numer Methods Eng, 117, 1, 63-83 (2019) [61] Fan, C. M.; Huang, Y. K.; Li, P. W.; Chiu, C. L., Application of the generalized finite-difference method to inverse biharmonic boundary-value problems, Numer Heat Tranf B-Fundam, 65, 2, 129-154 (2014) [62] Gu, Y.; Wang, L.; Chen, W.; Zhang, C. Z.; He, X. Q., Application of the meshless generalized finite difference method to inverse heat source problems, Int J Heat Mass Transf, 108, 721-729 (2017) [63] Jaworska, I.; Milewski, S., On two-scale analysis of heterogeneous materials by means of the meshless finite difference method, Int J Multiscale Comput Eng, 14, 2, 113-134 (2016) [64] Jaworska, I., On some aspects of the meshless FDM application for the heterogeneous materials, Int J Multiscale Comput Eng, 15, 4, 359-378 (2017) [65] Gersborg-Hansen, A.; Bendsoe, M. P.; Sigmund, O., Topology optimization of heat conduction problems using the finite volume method, Struct Multidiscip Optim, 31, 4, 251-259 (2006) · Zbl 1245.80011 [66] Tang, L.; Gao, T.; Song, L. L.; Meng, L.; Zhang, C. Q.; Zhang, W. H., Topology optimization of nonlinear heat conduction problems involving large temperature gradient, Comput Meth Appl Mech Eng, 357, Article 112600 pp. (2019) · Zbl 1442.74180 [67] Page, L. G.; Dirker, J.; Meyer, J. P., Topology optimization for the conduction cooling of a heat-generating volume with orthotropicmaterial, Int J Heat Mass Transf, 103, 1075-1083 (2016)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.