基于扩展等几何分析和混沌离子运动算法的带孔结构形状优化设计
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摘要
为了解决带孔结构形状优化问题,提出了一种将扩展等几何分析方法和混沌离子运动算法相结合的优化求解模式。针对带孔结构的力学计算,采用扩展等几何分析方法,以几何体外轮廓划分背景网格,利用非均匀有理B样条描述带孔边界,其中在劲度矩阵组装过程中,孔内区域不做积分。另外,为获得高精度的积分计算,与孔边界相关的单元采用自适应四叉树细化规则。在优化模型中,以描述结构形状的控制点作为设计变量,以结构质量最小作为优化目标;利用离子运动优化算法代替传统的敏感性移动渐进法对优化模型进行求解。带孔无限平板算例的扩展等几何分析计算结果和转矩臂结构优化算例的计算结果证明了本文方法的有效性。
To solve the problem of the shape optimization of holed structure, a method that integrates extended isogeometric analysis and a chaotic ion motion algorithm is proposed. For mechanical calculations of holed structure, extended isogeometric analysis is used to divide the background meshes in geometric shape, and to describe the boundary of the holes with non-uniform rational B-splines where there is no integral in the area of the holes in the assembly of the stiffness matrix. In addition, to obtain high precision integral calculations, refinement by the adaptive four-forked tree is performed in the element related to the hole boundary. In the optimization model, the control points for describing the structure are set to be the design variables, and the optimization objective is to minimize the mass of the structure. The optimization model is then solved by using the ion motion optimization algorithm instead of the traditional asymptotic method based on sensitivity analysis. The calculation results of the infinite plate with a circular hole by extended isogeometric analysis and the optimization result of a torque arm structure proved the validity of this method.
To solve the problem of the shape optimization of holed structure, a method that integrates extended isogeometric analysis and a chaotic ion motion algorithm is proposed. For mechanical calculations of holed structure, extended isogeometric analysis is used to divide the background meshes in geometric shape, and to describe the boundary of the holes with non-uniform rational B-splines where there is no integral in the area of the holes in the assembly of the stiffness matrix. In addition, to obtain high precision integral calculations, refinement by the adaptive four-forked tree is performed in the element related to the hole boundary. In the optimization model, the control points for describing the structure are set to be the design variables, and the optimization objective is to minimize the mass of the structure. The optimization model is then solved by using the ion motion optimization algorithm instead of the traditional asymptotic method based on sensitivity analysis. The calculation results of the infinite plate with a circular hole by extended isogeometric analysis and the optimization result of a torque arm structure proved the validity of this method.
引文
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[30]Alatas B,Akin E,Ozer A B.Chaos embedded particle swarm optimization algorithm[J].Chaos Solitons and Fractals,2009,40(4):1715―1734.
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[34]Bennett J,Botkin M.Structural shape optimization with geometric description and adaptive mesh refinement[J].AIAA Journal,1985,23(3):458―464.
[35]蔡守宇,张卫红,李杨.基于面片删减的带孔结构等几何形状优化方法[J].机械工程学报,2013,49(13):150―157.Cai Shouyu,Zhang Weihong,Li Yang.Isogeometric shape optimization method with patch removal for holed structures[J].Journal of Mechanical Engineering,2013,49(13):150―157.(in Chinese)
[2]Sun S H,Yu T T,Nguyen T T,et al.Structural shape optimization by IGABEM and particle swarm optimization algorithm[J].Engineering Analysis with Boundary Elements,2018,88:26―40.
[3]Belytschko T,Black T.Elastic crack growth in finite elements with minimal remeshing[J].International Journal for Numerical Methods in Engineering,1999,45(5):601―620.
[4]Asareh I,Yoon Y C,Song J H.A numerical method for dynamic fracture using the extended finite element method with non-nodal enrichment parameters[J].International Journal of Impact Engineering,2018,121:63―76.
[5]Sukumar N,Chopp D L,Mo?s N,et al.Modeling holes and inclusions by level sets in the extended finite-element method[J].Computer Methods in Applied Mechanics and Engineering,2001,190(46/47):6183―6200.
[6]Duysinx P,Miegroet L V,Jacobs T,et al.Generalized shape optimization using X-FEM and level set methods[C]//IUTAM Symposium on Topological Design Optimization of Structures,Machines and Materials.Dordrecht:Springer,2006:23―32.
[7]Miegroet L V,Jacobs T,Duysinx P.Recent developments in fixed mesh optimization with XFEM and level set description[J].Journal of Physics C:Solid State Physics,2007,12(7):1239―1244.
[8]Miegroet L V,Duysinx P.Stress concentration minimization of 2D Filets using X-FEM and level set description[J].Structural and Multidisciplinary Optimization,2007,33(4/5):425―438.
[9]Wei P,Wang M Y,Xing X H.A study on X-FEM in continuum structural optimization using level set model[J].Computer Aided Design,2010,42(8):708―719.
[10]Hughes T J R,Cottrell J A,Bazilevs Y.Isogeometric analysis:CAD,finite elements,NURBS,exact geometry and mesh refinement[J].Computer Methods in Applied Mechanics and Engineering,2005,194(39/41):4135―4195.
[11]薛冰寒,林皋,胡志强,等.求解摩擦接触问题的IGA-B可微方程组方法[J].工程力学,2016,33(10):35―43.Xue Binghan,Lin Gao,Hu Zhiqiang,et al.Analysis of frictional contact mechanics problems by IGA-Bdifferential equation method[J].Engineering Mechanics,2016,33(10):35―43.(in Chinese)
[12]Lieu Q X,Lee J,Lee D,et al.Shape and size optimization of functionally graded sandwich plates using isogeometric analysis and adaptive hybrid evolutionary firefly algorithm[J].Thin-Walled Structures,2018,124:588―604.
[13]Benson D J,Bazilevs Y,Luycker D E,et al.Ageneralized finite element formulation for arbitrary basis functions:From isogeometric analysis to XFEM[J].International Journal for Numerical Methods in Engineering,2010,83(6):765―785.
[14]Haasemann G,Kastner M,Pruger S,et al.Development of a quadratic finite element formulation based on the XFEM and NURBS[J].International Journal for Numerical Methods in Engineering,2011,86(4/5):598―617.
[15]Luycker D E,Benson D J,Belytschko T,et al.X-FEM in isogeometric analysis for linear fracture mechanics[J].International Journal for Numerical Methods in Engineering,2011,87(6):541―565.
[16]Ghorashi S S,Valizadeh N,Mohammadi S.Extended isogeometric analysis for simulation of stationary and propagating cracks[J].International Journal for Numerical Methods in Engineering,2011,89(9):1069―1101.
[17]Svanberg K.The method of moving asymptotes-a new method for structural optimization[J].International Journal for Numerical Methods in Engineering,1987,24(2):359―373.
[18]Kennedy J,Eberhart R C.Particle swarm optimization[C]//IEEE International Conference on Neural Networks,Piscataway:IEEE Press,1995,4:1942―1948.
[19]颜欣桐,徐龙河.基于遗传算法的钢筋混凝土框架-剪力墙结构失效模式多目标优化[J].工程力学,2018,35(4):69―77.Yan Xintong,Xu Longhe.Multi-objective optimization of genetic algorithm-based failure mode for reinforced concrete frame-shear wall structures[J].Engineering Mechanics,2018,35(4):69―77.(in Chinese)
[20]Dorigo M,Maniezzo V,Colorni A.The ant system:optimization by a colony of cooperating agents[J].IEEETransactions on Systems,Man,and Cybernetics,1996,26(1):29―41.
[21]Schnetzler B.Optimization by simulated annealing[J].Science,1992,220(4598):671―680.
[22]Karaboga D,Akay B.A comparative study of artificial bee colony algorithm[J].Applied Mathematics and Computation,2009,214(1):108―132.
[23]Krishnanand K N,Ghose D.Glowworm swarm optimization for simultaneous capture of multiple local optima of multimodal functions[J].Swarm Intelligence,2009,3(2):87―124.
[24]Yang X S.Firefly algorithms for multimodal optimization[J].International Conference on Stochastic Algorithms:Foundations and Applications,2009,5792:169―178.
[25]Yang X S,Cui Z,Xiao R,et al.Swarm intelligence and bio-inspired computation[M].Netherlands:Elsevier,2013:3―23.
[26]Javidy B,Hatamlou A,Mirjalili S.Ions motion algorithm for solving optimization problems[J].Applied Soft Computing,2015,32(3):72―79.
[27]过斌,葛建立,杨国来,等.三维实体结构NURBS等几何分析[J].工程力学,2015,32(9):42―48.Guo Bin,Ge Jianli,Yang Guolai,et al.NURBS-based isogeometric analysis of three-dimensional solid structures[J].Engineering Mechanics,2015,32(9):42―48.(in Chinese)
[28]Gu J M,Yu T T,Lich L V,et al.Multi-inclusions modeling by adaptive XIGA based on LR B-splines and multiple level sets[J].Finite Elements in Analysis and Design,2018,148(1):48―66.
[29]Cai S Y,Zhang W H,Zhu J H,et al.Stress constrained shape and topology optimization with fixed mesh:AB-spline finite cell method combined with level set function[J].Computer Methods in Applied Mechanics and Engineering,2014,278(7):361―387.
[30]Alatas B,Akin E,Ozer A B.Chaos embedded particle swarm optimization algorithm[J].Chaos Solitons and Fractals,2009,40(4):1715―1734.
[31]Xiang T,Liao X,Wong K W.An improved particle swarm optimization algorithm combined with piecewise linear chaotic map[J].Applied Mathematics and Computation,2007,190(2):1637―1645.
[32]Coelho L D S,Mariani V C.A novel chaotic particle swarm optimization approach using Hénon map and implicit filtering local search for economic load dispatch[J].Chaos Solitons and Fractals,2009,39(2):510―518.
[33]Zaslavskii G M.The simplest case of a strange attractor[J].Physics Letters A,1978,69(3):145―147.
[34]Bennett J,Botkin M.Structural shape optimization with geometric description and adaptive mesh refinement[J].AIAA Journal,1985,23(3):458―464.
[35]蔡守宇,张卫红,李杨.基于面片删减的带孔结构等几何形状优化方法[J].机械工程学报,2013,49(13):150―157.Cai Shouyu,Zhang Weihong,Li Yang.Isogeometric shape optimization method with patch removal for holed structures[J].Journal of Mechanical Engineering,2013,49(13):150―157.(in Chinese)