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A Study on Topology and Shape Optimization of Planar Continuum Structures

Author: LaiYunShan
Tutor: MaHaiTao
School: South China University of Technology
Course: Structural Engineering
Keywords: hybrid stress elements variable thickness topology optimization shape optimization sensitivity analysis
CLC: TB472
Type: Master's thesis
Year: 2011
Downloads: 147
Quote: 1
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Abstract


This dissertation is based on the technologies of shape and topological optimization design for continuum structures, in which plane stress hybrid element for membrane with variable thickness; static topology optimization for continuum structures and the optimal design of 2-D continuum structures using shape optimization approach are mainly discussed.Firstly, a new hybrid stress element formulation is presented for the analysis of membranes with variable thickness based on Hellinger-Reissner variational principle. The general hybrid stress element formulated using independent stress and displacement fields, is improved. Independent membrane force and displacement fields are used in the new formulation. The new hybrid plane stress element for membranes with variable thickness is formulated in details. Numerical examples show the validity and effectiveness of the new element.Secondly, the fundamental theory and models of topology optimization for continuum structures are introduced; the SIMP and RAMP density-stiffness interpolation schemes and numerical instabilities in topology optimization, such as checkerboards and mesh dependencies, are also discussed.For improving the numerical instabilities, using nodal thickness as design variable, the mathematical model of topology optimization based on the hybrid elements of variable nodal thickness with penalization exponent is established. The paper mainly considers the stiffness topology optimizatim of planar continuum structures for minimum compliance as objective and the volume as constraint. Analytical expressions of sensitivity analysis have been derived, and the topology optimization algorithm is presented. The method proposed in the paper was applied to sample problems like MBB beam. The results show that the topology obtained by this new model has clear topology configuration without using any filtering schemes or other technologies. This model avoids checkerboard pattern problems and proves its effectiveness and superiority. Considering the non-smooth final topology boundary, we use B-splines to smooth the boundary of topology optimization, in order to obtain more reasonable topology.Thirdly, a new method of 2-D structural shape optimization based on fixed mesh is presented. Using the coordinates of the boundary control points as design variables, connect through these control points form a straight line, which is called control boundary, to describe the shape of the structural boundary. Two key issues of this new shape optimization method are resolved. The first problem is the calculation of the element stiffness matrices of elements on the control boundary, and the second problem is the sensitivity analysis, i.e., the calculation of partial derivatives of structural response variables to the boundary shape design variables. Through several numerical examples, the effectiveness of the new method is shown.Fourthly, the significant challenges that need to be overcome to effectively solve stress-based optimization problems are discussed. And the topology optimization model of continuum structures considering the global stress constraint based on hybrid stress element with variable nodal thickness is established. And the sensitivity analysis of stress constraint function is derived in detail using adjoint method.Finally, all the work is summarized, and the prospect of the future work is presented.

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