It is revealed that the local optimum is particularly prone to occur in multi-material topology o... more It is revealed that the local optimum is particularly prone to occur in multi-material topology optimization using the conventional SIMP method. To overcome these undesirable phenomena, reciprocal variables are introduced into the formulation of topology optimization for minimization of total weight with the prescribed constraint of various structural responses. The SIMP scheme of multi-phase materials is adopted as the interpolation of the elemental stiffness matrix, mass matrix and weight. The sensitivities of eigenvalue and weight with respect to design variables are derived. Explicit approximations of natural eigenvalue and weight are given with the help of the first and second order Taylor series expansion. Thus, the optimization problem is solved using a sequential quadratic programming approach, by setting up a sub-problem in the form of a quadratic program. The filtering technique by solving the Helmholtz-type partial differential equation is performed to eliminate the checkerboard patterns and mesh dependence. Numerical analysis indicates that it is beneficial to avoid the local optimum by using the reciprocal SIMP formulation. Besides, the structure composed of multi-materials can achieve a lighter design than that made from the exclusive base material. The effectiveness and capability of the proposed method are also verified by nodal displacement constraint and multiple constraints.
We propose an explicit isogeometric topology optimization approach based on Moving Morphable Comp... more We propose an explicit isogeometric topology optimization approach based on Moving Morphable Components (MMCs). The prescribed design domain is discretized using a NURBS patch and NURBS-based Isogeometric Analysis (IGA) method is adopted for structural response analysis and sensitivity analysis. We employ the MMCs to represent the geometries of structural components (a subset of the design domain) with use of explicit design parameters. The central coordinates, half-length, half-width, and inclined angles of MMCs are taken as design variables. The proposed method not only inherits the explicitness of the MMC-based topology optimization, but also embraces the merits of the Isogeometric Analysis (IGA) such as a tighter link with Computer-Aided Design (CAD) and higher-order continuity of the basis functions. Several numerical examples illustrate that the presented method based on IGA is more robust and stable than FEM-based topology optimization using MMCs.
The present work introduces a novel concurrent optimization formulation to meet the requirements ... more The present work introduces a novel concurrent optimization formulation to meet the requirements of lightweight design and various constraints simultaneously. Nodal displacement of macrostructure and effective thermal conductivity of microstructure are regarded as the constraint functions, which means taking into account both the load-carrying capabilities and the thermal insulation properties. The effective properties of porous material derived from numerical homogenization are used for macrostructural analysis. Meanwhile, displacement vectors of macrostructures from original and adjoint load cases are used for sensitivity analysis of the microstructure. Design variables in the form of reciprocal functions of relative densities are introduced and used for linearization of the constraint function. The objective function of total mass is approximately expressed by the second order Taylor series expansion. Then, the proposed concurrent optimization problem is solved using a sequential quadratic programming algorithm, by splitting into a series of sub-problems in the form of the quadratic program. Finally, several numerical examples are presented to validate the effectiveness of the proposed optimization method. The various effects including initial designs, prescribed limits of nodal displacement, and effective thermal conductivity on optimized designs are also investigated. An amount of optimized macrostructures and their corresponding microstructures are achieved.
SUMMARY This paper presents a novel formulation based on Hellinger– Reissner variational principl... more SUMMARY This paper presents a novel formulation based on Hellinger– Reissner variational principle in the framework of quasi-conforming method for static and free vibration analysis of Reissner– Mindlin plates. The formulation starts from polynomial approximation of stresses, which satisfy the equilibrium equations of Reissner– Mindlin plate theory. Then the stress matrix is treated as the weighted function to weaken the strain-displacement equations after the strains are derived by using the constitutive equations. Finally, the string-net functions are introduced to calculate strain integration. As examples, two new plate bending elements, a 4-node quadrilateral element QC-P4-11í µí»½ and a 3-node triangular element QC-P3-7í µí»½, are proposed. Several benchmark examples are demonstrated to show the performance of the elements, and the results obtained are compared with other available ones. Numerical results have proved that both elements possess excellent precision. In particular, the quadrilateral element performs well even when the element shape degenerates into a triangle or concave quadrangle.
In this paper, isogeometric finite element method (IGA) based on Non-Uniform Rational B-splines (... more In this paper, isogeometric finite element method (IGA) based on Non-Uniform Rational B-splines (NURBS) basis function is applied for the buckling analysis of generally laminated composite beam with various boundary conditions. A beam element with four degrees of freedom per control point is investigated , which has considered the bending-torsion deformation. The model for the buckling analysis of laminated composite beam is detailed by the principle of virtual work. Several numerical examples of symmetric and anti-symmetric, cross-ply and angle-ply composite beam are performed. Numerical results of critical buckling loads and mode shapes are presented, and compared with other available results to show the efficiency and accuracy of the present IGA approach. In addition, the impacts of the modulus ratios, slenderness ratios, stacking sequence and the fiber angle, especially the Poisson effect on the critical buckling loads of composite beam are clearly demonstrated. It should be noted the results with the Poisson effect neglected are only suit for the cross-ply composite beams. And the benchmark solutions presented in this work can be used as a reference for the buckling analysis of laminated composite beams in future.
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