Abstract
In this article, the element free Galerkin (EFG) method is applied to carry out the topology optimization of the geometrically non-linear continuum structures. In EFG method, the moving least squares shape function is used to approximate the displacements. The 2D geometrically non-linear formulation is presented based on the EFG method. The penalty method is explored to enforce the essential boundary conditions. Considering the relative density of nodes as design variables, the minimization of compliance as an objective function, the mathematical formulation of the topology optimization is developed using the solid isotropic microstructures with penalization interpolation scheme. Sensitivity of the objective function is derived based on the adjoint method. Numerical examples show that the proposed approach is feasible and effective for the topology optimization of the geometrically non-linear continuum structures.
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References
Belytschko, T., Lu, Y.Y., Gu, L.: Element-free Galerkin method. Int. J. Numer. Methods Eng. 37, 229–256 (1994a)
Belytschko, T., Gu, L., Lu, Y.Y.: Fracture and crack growth by element free Galerkin methods. Modell. Simul. Mater. Sci. Eng. 2, 519–534 (1994b)
Belytschko, T., Krysl, P., Krongauz, Y.: A three-dimensional explicit element-free Galerkin method. Int. J. Numer. Methods Fluids 24(12), 1253–1270 (1997)
Belytschko, T., Liu, W.K., Moran, B.: Nonlinear finite elements for continua and structures. Wiley, New York (2000)
Bendsoe, M.P., Kikuchi, N.: Generating optimal topology in structural design using a homogenization method. Comput. Methods Appl. Mech. Eng. 71, 197–224 (1989)
Bendsoe, M.P., Sigmund, O.: Material interpolation schemes in topology optimization. Arch. Appl. Mech. 69, 635–654 (1999)
Bendsoe, M.P., Sigmund, O.: Topology Optimization: Theory, Methods, and Applications. Springer, Berlin, Heidelberg, New York (2003)
Bruns, T.E., Tortorelli, D.A.: Topology optimization of non-linear elastic structures and compliant mechanisms. Comput. Methods Appl. Mech. Eng. 190, 3443–3459 (2001)
Bruns, T.E., Sigmund, O., Tortorelli, D.A.: Numerical methods for the topology optimization of structures that exhibit snap-through. Int. J. Numer. Methods Eng. 55(10), 1215–1237 (2002)
Buhl, T., Pedersen, C.B.W., Sigmund, O.: Stiffness design of geometrically nonlinear structures using topology optimization. Struct. Multidiscipl. Optim. 19, 93–104 (2000)
Cho, S., Kwak, J.: Topology design optimization of geometrically non-linear structures using meshfree method. Comput. Methods Appl. Mech. Eng. 195, 5909–5925 (2006)
Du, Y.X., Chen, L.P., Luo, Z.: A meshless Galerkin approach for topology optimization of monolithic compliant mechanisms using optimality criteria method. Acta Mech. Solida Sin. 28(1), 102–108 (2007)
Jung, D., Gea, H.C.: Topology optimization of nonlinear structures. Finite Elem. Anal. Des. 40, 1417–1427 (2004)
Kang, Z., Wang, Y.Q.: Structural topology optimization based on non-local Shepard interpolation of density field. Comput. Methods Appl. Mech. Eng. 200, 3515–3525 (2011)
Kang, Z., Wang, Y.Q.: A nodal variable method of structural topology optimization based on Shepard interplant. Int. J. Numer. Meth. Eng. 90(3), 329–342 (2012)
Li, S., Atluri, S.N.: The MLPG mixed collocation method for material orientation and topology optimization of anisotropic solids and structures. Comput. Model. Eng. Sci. 30(1), 37–56 (2008)
Liu, G.R., Chen, X.L.: A mesh-free method for static and free vibration analyses of thin plates of complicated shape. J. Sound Vib. 241(5), 839–855 (2001)
Luo, Z., Zhang, N., Gao, W., Ma, H.: Structural shape and topology optimization using a meshless Galerkin level set method. Int. J. Numer. Methods Eng. 90, 369–389 (2012)
Luo, Z., Zhang, N., Wang, Y., Gao, W.: Topology optimization of structures using meshless density variable approximants. Int. J. Numer. Methods Eng. 93, 443–464 (2013)
Matsui, K., Terada, K.: Continuous approximation of material distribution for topology optimization. Int. J. Numer. Methods Eng. 59, 1925–1944 (2004)
Rahmatalla, S.F., Swan, C.C.: A Q4/Q4 continuum structural topology optimization implementation. Struct. Multidiscipl. Optim. 27, 130–135 (2004)
Rozvany, G., Kirch, U., Bendsøe, M.P., Sigmund, O.: Layout optimization of structures. Appl. Mech. Rev. 48, 41–119 (1995)
Sigmund, O., Petersson, J.: Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct. Optim. 16(1), 68–75 (1998)
Svanberg, K.: The method of moving asymptotes: a new method for structural optimization. Int. J. Numer. Methods Eng. 24, 359–373 (1987)
Wang, M.Y., Wang, X.M., Guo, D.M.: A level set method for structural topology optimization. Comput. Methods Appl. Mech. Eng. 192(1–2), 227–246 (2003)
Xie, Y.M., Steven, G.P.: A simple evolutionary procedure for structural optimization. Comput. Struct. 49, 885–896 (1993)
Zheng, J., Long, S.Y., Li, G.Y.: The topology optimization design for continuum structure based on the element free Galerkin method. Eng. Anal. Boundary Elem. 34(7), 666–672 (2010)
Zhou, J.X., Zou, W.: Meshless approximation combined with implicit topology description for optimization of continua. Struct. Multidiscipl. Optim. 36(4), 347–353 (2008)
Zhou, M., Shyy, Y.K., Thomas, H.L.: Checkerboard and minimum member size control in topology optimization. Struct. Multidiscipl. Optim. 21(2), 152–158 (2001)
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This work was supported by National Natural Science Foundation of China (No. 11202075, No. 11372106), Research Fund for the Doctoral Program of Higher Education of China (No. 20120161120006).
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Zheng, J., Yang, X. & Long, S. Topology optimization with geometrically non-linear based on the element free Galerkin method. Int J Mech Mater Des 11, 231–241 (2015). https://doi.org/10.1007/s10999-014-9257-y
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DOI: https://doi.org/10.1007/s10999-014-9257-y