板壳截面优化的理论与基于MSC/NASTRAN的软件二次开发
|
摘要
在给定了板壳的材料常数、板壳结构的边界形状和边界条件的前提下，确定表征板壳厚度的设计变量，在满足约束条件下实现结构重量最小：
1.对于尺寸和应力约束的问题，使用满应力方法求解。在每一步迭代过程建立优化模型之前，采用射线步技术，调整了结构的性态，保证了收敛的稳定性。
2.对于尺寸、应力和位移约束的问题，将应力约束化为动态下限，用单位虚荷载方法将位移约束近似显式化，构造优化问题的数学规划模型，将其对偶规划处理为二次规划问题，采用LEMKE算法进行求解，得到满足尺寸、应力和位移约束条件的截面最优解。
在MSC/NASTRAN基础上，利用MSC/PATRAN提供的PCL语言，对板壳结构的截面优化的优化模块进行了二次开发。将建立优化模型的菜单融合到PATRAN界面，用户可以通过人机界面的交互功能确定设计变量、约束条件及目标函数三要素，实现程序的可视化。
多个算例验证了程序具有合理性、精确性和收敛速度快等特点。
On the premise of a given set of material parameter, structural boundary shape and condition, design variables-thickness of plate and shell, is designed to minimize the structural weight subjected to the constraint conditions.
1.For the problem with size and stress constraints, full stress design method is used to solve the sectional optimization of plate and shell structures. Before the optimization modeling, the scaling step technique is used to adjust the structural behavior, and the stability of computational convergence is ensured.
2.For the problem with size, stress and displacement constraints, the stress constraint is transformed into movable lower bounds of sizes, the displacement constraint is transformed into an approximate function which explicitly includes design variables by using Mohr integral theory. A mathematical programming model of the optimization problem is set up. The dual programming of the model is approached into a quadratic programming model. Finally, the Lemke algorithm is used to get the design result which satisfies the size, stress and displacement constraints.
Based on MSC/NASTRAN software, by using of PCL language (PATRAN Command Language), new software of optimum design module was secondly developed for sectional optimization of plate and shell structures. Visualization is realized by setting the menu into PATRAN. Users can define design variables, constraint conditions and objective function by the iterative function of computer interface.
Some numerical examples of optimum design verified the credibility, validity and speed convergence of the program.
1.对于尺寸和应力约束的问题，使用满应力方法求解。在每一步迭代过程建立优化模型之前，采用射线步技术，调整了结构的性态，保证了收敛的稳定性。
2.对于尺寸、应力和位移约束的问题，将应力约束化为动态下限，用单位虚荷载方法将位移约束近似显式化，构造优化问题的数学规划模型，将其对偶规划处理为二次规划问题，采用LEMKE算法进行求解，得到满足尺寸、应力和位移约束条件的截面最优解。
在MSC/NASTRAN基础上，利用MSC/PATRAN提供的PCL语言，对板壳结构的截面优化的优化模块进行了二次开发。将建立优化模型的菜单融合到PATRAN界面，用户可以通过人机界面的交互功能确定设计变量、约束条件及目标函数三要素，实现程序的可视化。
多个算例验证了程序具有合理性、精确性和收敛速度快等特点。
On the premise of a given set of material parameter, structural boundary shape and condition, design variables-thickness of plate and shell, is designed to minimize the structural weight subjected to the constraint conditions.
1.For the problem with size and stress constraints, full stress design method is used to solve the sectional optimization of plate and shell structures. Before the optimization modeling, the scaling step technique is used to adjust the structural behavior, and the stability of computational convergence is ensured.
2.For the problem with size, stress and displacement constraints, the stress constraint is transformed into movable lower bounds of sizes, the displacement constraint is transformed into an approximate function which explicitly includes design variables by using Mohr integral theory. A mathematical programming model of the optimization problem is set up. The dual programming of the model is approached into a quadratic programming model. Finally, the Lemke algorithm is used to get the design result which satisfies the size, stress and displacement constraints.
Based on MSC/NASTRAN software, by using of PCL language (PATRAN Command Language), new software of optimum design module was secondly developed for sectional optimization of plate and shell structures. Visualization is realized by setting the menu into PATRAN. Users can define design variables, constraint conditions and objective function by the iterative function of computer interface.
Some numerical examples of optimum design verified the credibility, validity and speed convergence of the program.
引文
[1] 隋允康．建模·变换·优化—结构综合方法新进展.大连理工大学出版社.1996.
[2] Schmit, L.A.,Structural design by systematic synthesis. Proc. of Conf. On Electronic Computation, ASCE. New York(1960):105~122.
[3] Venkayya, V.B., Khot, N.S., Reddyu, V.S., Optimization of structures based on the study of endrgy distribution. Proc. Of 2nd Conf.on Matrix Methods in Struct. Mech., WPAFB, AFFDLTR-68-150, U.S.A.F(Dec.1969): 111~153.
[4] Gellatly, R.A., Berke, L., Gibson, W., The use of optimality criteria in automated structural design. Paper Presented at the 3rd Conf. On Matrix Methods in Struct. Mech., WPAFB. (Oct.1971).
[5] Schmit, L.A., Miura, H., Approximation concepts for efficient structural synthesis. NASA CR-2552(March 1976).
[6] Fleury C. Structural weight optimization by dual method of convex programming. International Journal for Numerical methods in Engineering, 1979,14:1761~1783.
[7] Schmit.L.A., Fleury.C., Structurla synthesis by combining approximation concepts and dual methods. AIAA, 18(1980):1252~1260.
[8] Khan.M.r. Willmert. K.D., Thornton. W.A., An optimality criterion method for large-scale structures. AIAA.J. 17.7(1979):753~761.
[9] Dorn WS, Gomory RE, Greenoerg HJ. Automatic design of optimal structures. Journal de Mechanics.1964.3:25~52.
[10] 钱令希.工程结构优化设计.北京.水力电力出版社.1983.
[11] 钱令希、钟万勰.结构优化设计的一个方法.
[12] 钱令希、钟万勰、程耿东、隋允康.工程结构优化设计的一个新途径— —序列二次规划SQP.计算结构力学及其应用.1984,1
[13] 王光远、周正源、霍达.结构优化设计的两相优化法.力学学报.1983,4
[14] 霍达.多工况结构优化的两相优化法.哈尔滨建筑工程学院学报.1984,3
[15] 夏人伟、周明.结构设计的统一优化准则法. 计算结构力学及其应用.1985,11.Vol.2.No.4:29
[16] 许素强、夏人伟.杆系结构几何优化中的广义变量近似方法.计算结构力学及其应用.1995,2.Vol.12.No.1:17~25
[17] 程耿东.关于桁架结构拓扑优化中的奇异最优解.大连理工大学学报.2000,7.
[18] 柴山、石连栓、孙焕纯.包含两类变量的离散变量的桁架结构拓扑优化设计.力学学报. 1999,9.Vol.31.No.5:574~583.
[19] 石连栓、孙焕纯.离散变量结构形状优化设计的综合算法.力学学报.1999,11.
[20] 隋允康、阳志光.应用两点有理逼近改进的牛顿法和对偶法.大连理工大学学报.1994,2.Vol.34.No.1:1~9.
[21] 隋允康、刑誉峰、张桂明.基于两点累积信息的原/倒变量展开的对偶优化方法.力学学报.1994,11. Vol.26.No.6: 699~710.
[22] 隋允康、于新.曲线寻优的收敛性和近似解析解法。大连理工大学学报.1995,35.3.
[23] 隋允康.结构优化的曲线寻优理论及其逼近论和常微分方程组解法.中国学术期刊文摘.1995,1.2.
[24] 隋允康、林龙富.求解非线性规划的序列有理规划SRP方法.计算结构力学及其应用.1994,5.Vol.11.No.2:200~212
[25] 隋允康、叶宝瑞.一种方便实用的有理逼近及其对于大量优化方法的改进.运筹学杂志.1993,6.Vol.12.No.4:52~66.
[26] 隋允康、由衷.具有两类变量的空间桁架分层优化方法.计算结构力学及其应用.1998.
[27] 隋允康、杨德庆.桁架形状优化设计最优目标敏度法.计算力学学报. 1997,14（增刊）
[28] 隋允康、杨德庆.最优目标敏度紧系的桁架几何两步优化方法.力学学报.1997,7
[29] 隋允康、杨德庆.桁架结构拓扑优化设计密度变量的两种模式.力学与实践.1997,5
[30] Yun Kang Sui,Yang Deqing.A New Method for Structural Topological Optimization Based on the Concept of Independent Continuous Variables and Smooth Model. ACTA MECHANICA SINICA.1998.5 Vol.14.No.2:179~185.
[31] N.C.Huang.Optimal design of elastic structures for maximum stiffness.Int.J.Solids struck.4-7(1968).P689
[32] N.V.Banichuk.Optimiging Hole Shape in Plates Working in Bending.Mekhanika Tverdogo Jela.1977.Vol.12.No.3. P81
[33] N.V.Banichuk.Optimiging Elastic Plates Shape in Bending Problems.Mekhanika Tverdogo Jela.1975.Vol.10.No.5.P180
[34] J.Kiusalsos and G.B.Reddy.DESAP1-A Structural Design Program With Stress and Displacement Constraints.NASACR-2794 to 2796.Feb.1977
[35] M.N.K.Rao and H.Tottenham.Fully Stressed Design of Variable Thickness Slabs and Column Footings by Finites Element Methods.J.of Structural Eng.1973.Vol.1.No.1. P9
[36] F.ERBATUR and Y.MENGI.Optimal Design of Plates under the Influnence of Dead Weigft and Surface Loading.J.Struct.Mech.5-4(1977).P345
[37] D.Lamblin.C.Cinquini and G.Guerlement.Application of linear programing to the optimal plastic design of circular plates subject to technological constraints.Comp.Meth.Appl.Mech.Eng.13(1978).P233
[38] J-L.Armand and B.Lodier.Optimal Design of Bending Elements.Int.J.Num.Meth.Eng.13-2(1978).P373
[39] 隋允康、王希诚．适合复杂工程实际的结构优化程序DDDU-2F版本.计算结构力学及其应用. 1985.Vol. 2.No 3:73~81.
[40] 钱令希、隋允康、王希诚、凌克明．壳体结构优化及其DDDU(W)新版本.计算结构力学及其应用. 1992.Vol.9.No.1:79~89.
[41] 隋允康、钟万勰、钱令希．杆－膜－梁组合结构优化的DDDU－2程序系统. 大连工学院学报. 1983.Vol.22.No.1:21~36.
[42] 隋允康.组合结构离散断面优化方法和程序系统DDDU—4.上海力学.1987,4
[43] Sui Yunkang, Yang deqing. Modified Newton Method and Dual Method Through a Rational Approximation at Two Expanded Points.OPA.1995,10 Vol.24:1~17.
[44] Schmit LA Farshi B. Some approximation concepts for structural synthesis. AIAA J. 1974,12(2):231~233.
[45] Sui Yunkang, Yu Xin. The Topological optimization for Truss Structures with Stress Constraints Based on the Exist-Null Conmbined Model.ACTA MECHANICA SINICA.1998,11.Vol.14.No.1:363~370
[46] Yang R.J multidiscipline topology optimization, computers & structures, 1997.Vol.63.No.6:1205~1212
[47] 顾元宪、程耿东.计算机辅助结构优化设计软件MCADS的开发与应用.计算结构力学及其应用.1995,8.Vol.12.No.3:304~307
[48] 隋允康.函数变换下的满应力设计方法.应用力学学报.1987,4.2
[49] 钟万勰、程耿东.跨世纪的中国计算力学.力学与实践.1999,1:11~16
[50] 程耿东.计算力学学科的发展. 计算结构力学及其应用.1993,5.Vol.10.No.2.p187
[51] 隋允康、耿树森.广义几何规划二阶缩并的原算法及其在结构优化中的应用. 大连工学院学报. 1985.Vol.2.No.1:39~47.
[52] 于新、隋允康.二维连续体结构优化中常用单元应力约束的有效处理.力学与实践.1999.Vol.21:48~49
[53] 郑承沛.材料力学.北京工业大学出版社.1994
[54] 刘鸿文.板壳理论.浙江大学出版社.1987
[55] 建筑结构静力计算手册编写组.建筑结构静力计算手册.1975,6
[56] Esping, The OASIS Structural Optimization System, Computers and Structures, 1984, Vol.23:365~377
[57] Braibant and Fleury. Shape Optimal Design Using B-splines, Computers Methods in Applied Mechanics and Engineering, 1984, Vol.44:247~267
[58] Bennett and Botkin, Structural Shape Optimization with Geometric Description and Adaptive Mesh Refinement, AIAA Journal, 1985, Vol.23:458~464
[59] Botkin, Yang and Bennett, Shape Optimization of Three Dimensional Stamped and Solid Automotive Components, The Optimum Shape. Automated Structural Design, Bennett and Bokin, Plenum Press, New York, 1986.
[60] 张新竹.板壳梁组合结构的有限元分析及其优化.太原工业大学研究生毕业论文.1990.
[61] 孙宪学、孙侠生、段世慧.优化设计系统COM-PASS的发展及其工程应用［J］.计算力学学报.1997,14（增刊）:653~656
[62] 金桂根、张益群.结构动态优化设计在微机上的实现.昆明理工大学学报.1998,10.Vol.23.No.5: 114~124
[63] 叶元烈、秦东晨、陈丽华.薄壁覆盖件的结构优化设计.机械强度.1999,9.Vol.21.No.3:18~185
[64] 杜群贵、迟永滨.通用结构优化设计系统的研究与实现.机械.2001,Vol.28.No.6:22~24
[65] 李芳、徐兴、凌道盛.薄板优化设计的二次形态模型法.工程设计.2001,3:139~142
[66] 乔志宏.桁架结构截面优化的理论与软件—用PCL对MSC/NASTRAN进行优化模块的二次开发.北京工业大学硕士学位论文.2002
[67] 夏利娟、郑靖明、金咸定.工程结构的优化设计方法与应用.上海交通大学学报.2002,11.Vol.36.No.11:1572~1575
[68] 傅潮江、由敬舜.坝体优化设计的序列二次规划法. 江西水利科技.2000,6.Vol.26.No.2: 67~70
[69] 陈刚、孙靖明.二次规划迭代法及其在结构设计中的应用. 计算结构力学及其应用.1987,2.Vol.4.No.1.P83
[70] 藤召波、张世永.陈华富、何光中.非线性规划一般约束条件的SQP方法. 电子科技大学学报.2001,2.Vol.30.No.1.p103
[71] 张菊亮、章祥荪.一个等式约束问题的SQP方法及其收敛性.应用数学学报.2001,9.Vol.24.No.1.p1
[72] 刘萍、凌晓东.一种求解二次规划的简便方法.数值计算与计算机应用.1996,9.第三期
[73] 禹智涛、王红.工程结构优化的序列模糊二次规划方法及其解法.广东工业大学学报.2002,6.Vol.19.No.2.p82
[74] 李师正、李刚.非线性规划的对偶问题.山东科学.1999,6.Vol.12.No.2.p1
[75] 宣兆成、李兴斯.结构优化的内点序列二次规划方法.计算力学学报.1997,8.Vol.14.No.3.p263
[76] 王周宏、王能超、钟毅芳.求解一般半正定二次规划的数值稳定方法.华中科技大学学报（自然科学版）.2002,4.Vol.30.No.4.p110
[77] 孙秦、范建华、徐晓飞.有限元大型二次规划解的一种新算法.数值计算与计算机应用.1999,3.第一期
[78] 隋允康、兆文忠.非线性方程组的二次规划解法和应用.计算力学学报. 2002,5.Vol.19.No.2:245~246
[79] 高自友、吴方、赖炎连.非线性最优化中超线性收敛的序列线性方程组方法.北方交通大学学报. 1995,12.Vol.19.No.4.p423
[80] 段宝岩、张劲柏.基于可视化的工程结构优化设计.计算力学学报. 2001.Vol.18 .No.4.435~441.
[81] 席少霖.非线性最优化方法.高等教育出版社.1992
[82] 薛毅.最优化原理与方法.北京工业大学出版社.2001,2
[83] 张向、许晶月等.面向对象的有限元程序设计.计算力学学报.1999,5
[84] 高自友、吴方.非线性约束条件下的SQP可行方法. 应用数学学报.1995,10.Vol.18.No.4.p579
[85] 张向、许晶月等.面向对象的有限元程序设计.计算力学学报.1999,5
[86] Walther F, Baumgartner A, Mattheck C. Finding new designs for pressure tools using optimization methods which are based om the simulation of biological growth. Computer Aided Optimization Design of Structures, 1993,3:601~622.
[87] Tutorial Manual Handy MSC/NSTRAN
[88] Example Problem Manual Handy MSC/NSTRAN
[89] MSC.Patran 2000(r2) PCL Reference Manual Volume 1:Function Descriptions
MSC.Patran 2000(r2) PCL Reference Manual Volume 1:Function
[90] Descriptions
[91] MSC.Patran 2000(r2) PCL Reference Manual Volume 2:Code Example
[92] MSC.Patran 2000(r2) PCL and Customization
[2] Schmit, L.A.,Structural design by systematic synthesis. Proc. of Conf. On Electronic Computation, ASCE. New York(1960):105~122.
[3] Venkayya, V.B., Khot, N.S., Reddyu, V.S., Optimization of structures based on the study of endrgy distribution. Proc. Of 2nd Conf.on Matrix Methods in Struct. Mech., WPAFB, AFFDLTR-68-150, U.S.A.F(Dec.1969): 111~153.
[4] Gellatly, R.A., Berke, L., Gibson, W., The use of optimality criteria in automated structural design. Paper Presented at the 3rd Conf. On Matrix Methods in Struct. Mech., WPAFB. (Oct.1971).
[5] Schmit, L.A., Miura, H., Approximation concepts for efficient structural synthesis. NASA CR-2552(March 1976).
[6] Fleury C. Structural weight optimization by dual method of convex programming. International Journal for Numerical methods in Engineering, 1979,14:1761~1783.
[7] Schmit.L.A., Fleury.C., Structurla synthesis by combining approximation concepts and dual methods. AIAA, 18(1980):1252~1260.
[8] Khan.M.r. Willmert. K.D., Thornton. W.A., An optimality criterion method for large-scale structures. AIAA.J. 17.7(1979):753~761.
[9] Dorn WS, Gomory RE, Greenoerg HJ. Automatic design of optimal structures. Journal de Mechanics.1964.3:25~52.
[10] 钱令希.工程结构优化设计.北京.水力电力出版社.1983.
[11] 钱令希、钟万勰.结构优化设计的一个方法.
[12] 钱令希、钟万勰、程耿东、隋允康.工程结构优化设计的一个新途径— —序列二次规划SQP.计算结构力学及其应用.1984,1
[13] 王光远、周正源、霍达.结构优化设计的两相优化法.力学学报.1983,4
[14] 霍达.多工况结构优化的两相优化法.哈尔滨建筑工程学院学报.1984,3
[15] 夏人伟、周明.结构设计的统一优化准则法. 计算结构力学及其应用.1985,11.Vol.2.No.4:29
[16] 许素强、夏人伟.杆系结构几何优化中的广义变量近似方法.计算结构力学及其应用.1995,2.Vol.12.No.1:17~25
[17] 程耿东.关于桁架结构拓扑优化中的奇异最优解.大连理工大学学报.2000,7.
[18] 柴山、石连栓、孙焕纯.包含两类变量的离散变量的桁架结构拓扑优化设计.力学学报. 1999,9.Vol.31.No.5:574~583.
[19] 石连栓、孙焕纯.离散变量结构形状优化设计的综合算法.力学学报.1999,11.
[20] 隋允康、阳志光.应用两点有理逼近改进的牛顿法和对偶法.大连理工大学学报.1994,2.Vol.34.No.1:1~9.
[21] 隋允康、刑誉峰、张桂明.基于两点累积信息的原/倒变量展开的对偶优化方法.力学学报.1994,11. Vol.26.No.6: 699~710.
[22] 隋允康、于新.曲线寻优的收敛性和近似解析解法。大连理工大学学报.1995,35.3.
[23] 隋允康.结构优化的曲线寻优理论及其逼近论和常微分方程组解法.中国学术期刊文摘.1995,1.2.
[24] 隋允康、林龙富.求解非线性规划的序列有理规划SRP方法.计算结构力学及其应用.1994,5.Vol.11.No.2:200~212
[25] 隋允康、叶宝瑞.一种方便实用的有理逼近及其对于大量优化方法的改进.运筹学杂志.1993,6.Vol.12.No.4:52~66.
[26] 隋允康、由衷.具有两类变量的空间桁架分层优化方法.计算结构力学及其应用.1998.
[27] 隋允康、杨德庆.桁架形状优化设计最优目标敏度法.计算力学学报. 1997,14（增刊）
[28] 隋允康、杨德庆.最优目标敏度紧系的桁架几何两步优化方法.力学学报.1997,7
[29] 隋允康、杨德庆.桁架结构拓扑优化设计密度变量的两种模式.力学与实践.1997,5
[30] Yun Kang Sui,Yang Deqing.A New Method for Structural Topological Optimization Based on the Concept of Independent Continuous Variables and Smooth Model. ACTA MECHANICA SINICA.1998.5 Vol.14.No.2:179~185.
[31] N.C.Huang.Optimal design of elastic structures for maximum stiffness.Int.J.Solids struck.4-7(1968).P689
[32] N.V.Banichuk.Optimiging Hole Shape in Plates Working in Bending.Mekhanika Tverdogo Jela.1977.Vol.12.No.3. P81
[33] N.V.Banichuk.Optimiging Elastic Plates Shape in Bending Problems.Mekhanika Tverdogo Jela.1975.Vol.10.No.5.P180
[34] J.Kiusalsos and G.B.Reddy.DESAP1-A Structural Design Program With Stress and Displacement Constraints.NASACR-2794 to 2796.Feb.1977
[35] M.N.K.Rao and H.Tottenham.Fully Stressed Design of Variable Thickness Slabs and Column Footings by Finites Element Methods.J.of Structural Eng.1973.Vol.1.No.1. P9
[36] F.ERBATUR and Y.MENGI.Optimal Design of Plates under the Influnence of Dead Weigft and Surface Loading.J.Struct.Mech.5-4(1977).P345
[37] D.Lamblin.C.Cinquini and G.Guerlement.Application of linear programing to the optimal plastic design of circular plates subject to technological constraints.Comp.Meth.Appl.Mech.Eng.13(1978).P233
[38] J-L.Armand and B.Lodier.Optimal Design of Bending Elements.Int.J.Num.Meth.Eng.13-2(1978).P373
[39] 隋允康、王希诚．适合复杂工程实际的结构优化程序DDDU-2F版本.计算结构力学及其应用. 1985.Vol. 2.No 3:73~81.
[40] 钱令希、隋允康、王希诚、凌克明．壳体结构优化及其DDDU(W)新版本.计算结构力学及其应用. 1992.Vol.9.No.1:79~89.
[41] 隋允康、钟万勰、钱令希．杆－膜－梁组合结构优化的DDDU－2程序系统. 大连工学院学报. 1983.Vol.22.No.1:21~36.
[42] 隋允康.组合结构离散断面优化方法和程序系统DDDU—4.上海力学.1987,4
[43] Sui Yunkang, Yang deqing. Modified Newton Method and Dual Method Through a Rational Approximation at Two Expanded Points.OPA.1995,10 Vol.24:1~17.
[44] Schmit LA Farshi B. Some approximation concepts for structural synthesis. AIAA J. 1974,12(2):231~233.
[45] Sui Yunkang, Yu Xin. The Topological optimization for Truss Structures with Stress Constraints Based on the Exist-Null Conmbined Model.ACTA MECHANICA SINICA.1998,11.Vol.14.No.1:363~370
[46] Yang R.J multidiscipline topology optimization, computers & structures, 1997.Vol.63.No.6:1205~1212
[47] 顾元宪、程耿东.计算机辅助结构优化设计软件MCADS的开发与应用.计算结构力学及其应用.1995,8.Vol.12.No.3:304~307
[48] 隋允康.函数变换下的满应力设计方法.应用力学学报.1987,4.2
[49] 钟万勰、程耿东.跨世纪的中国计算力学.力学与实践.1999,1:11~16
[50] 程耿东.计算力学学科的发展. 计算结构力学及其应用.1993,5.Vol.10.No.2.p187
[51] 隋允康、耿树森.广义几何规划二阶缩并的原算法及其在结构优化中的应用. 大连工学院学报. 1985.Vol.2.No.1:39~47.
[52] 于新、隋允康.二维连续体结构优化中常用单元应力约束的有效处理.力学与实践.1999.Vol.21:48~49
[53] 郑承沛.材料力学.北京工业大学出版社.1994
[54] 刘鸿文.板壳理论.浙江大学出版社.1987
[55] 建筑结构静力计算手册编写组.建筑结构静力计算手册.1975,6
[56] Esping, The OASIS Structural Optimization System, Computers and Structures, 1984, Vol.23:365~377
[57] Braibant and Fleury. Shape Optimal Design Using B-splines, Computers Methods in Applied Mechanics and Engineering, 1984, Vol.44:247~267
[58] Bennett and Botkin, Structural Shape Optimization with Geometric Description and Adaptive Mesh Refinement, AIAA Journal, 1985, Vol.23:458~464
[59] Botkin, Yang and Bennett, Shape Optimization of Three Dimensional Stamped and Solid Automotive Components, The Optimum Shape. Automated Structural Design, Bennett and Bokin, Plenum Press, New York, 1986.
[60] 张新竹.板壳梁组合结构的有限元分析及其优化.太原工业大学研究生毕业论文.1990.
[61] 孙宪学、孙侠生、段世慧.优化设计系统COM-PASS的发展及其工程应用［J］.计算力学学报.1997,14（增刊）:653~656
[62] 金桂根、张益群.结构动态优化设计在微机上的实现.昆明理工大学学报.1998,10.Vol.23.No.5: 114~124
[63] 叶元烈、秦东晨、陈丽华.薄壁覆盖件的结构优化设计.机械强度.1999,9.Vol.21.No.3:18~185
[64] 杜群贵、迟永滨.通用结构优化设计系统的研究与实现.机械.2001,Vol.28.No.6:22~24
[65] 李芳、徐兴、凌道盛.薄板优化设计的二次形态模型法.工程设计.2001,3:139~142
[66] 乔志宏.桁架结构截面优化的理论与软件—用PCL对MSC/NASTRAN进行优化模块的二次开发.北京工业大学硕士学位论文.2002
[67] 夏利娟、郑靖明、金咸定.工程结构的优化设计方法与应用.上海交通大学学报.2002,11.Vol.36.No.11:1572~1575
[68] 傅潮江、由敬舜.坝体优化设计的序列二次规划法. 江西水利科技.2000,6.Vol.26.No.2: 67~70
[69] 陈刚、孙靖明.二次规划迭代法及其在结构设计中的应用. 计算结构力学及其应用.1987,2.Vol.4.No.1.P83
[70] 藤召波、张世永.陈华富、何光中.非线性规划一般约束条件的SQP方法. 电子科技大学学报.2001,2.Vol.30.No.1.p103
[71] 张菊亮、章祥荪.一个等式约束问题的SQP方法及其收敛性.应用数学学报.2001,9.Vol.24.No.1.p1
[72] 刘萍、凌晓东.一种求解二次规划的简便方法.数值计算与计算机应用.1996,9.第三期
[73] 禹智涛、王红.工程结构优化的序列模糊二次规划方法及其解法.广东工业大学学报.2002,6.Vol.19.No.2.p82
[74] 李师正、李刚.非线性规划的对偶问题.山东科学.1999,6.Vol.12.No.2.p1
[75] 宣兆成、李兴斯.结构优化的内点序列二次规划方法.计算力学学报.1997,8.Vol.14.No.3.p263
[76] 王周宏、王能超、钟毅芳.求解一般半正定二次规划的数值稳定方法.华中科技大学学报（自然科学版）.2002,4.Vol.30.No.4.p110
[77] 孙秦、范建华、徐晓飞.有限元大型二次规划解的一种新算法.数值计算与计算机应用.1999,3.第一期
[78] 隋允康、兆文忠.非线性方程组的二次规划解法和应用.计算力学学报. 2002,5.Vol.19.No.2:245~246
[79] 高自友、吴方、赖炎连.非线性最优化中超线性收敛的序列线性方程组方法.北方交通大学学报. 1995,12.Vol.19.No.4.p423
[80] 段宝岩、张劲柏.基于可视化的工程结构优化设计.计算力学学报. 2001.Vol.18 .No.4.435~441.
[81] 席少霖.非线性最优化方法.高等教育出版社.1992
[82] 薛毅.最优化原理与方法.北京工业大学出版社.2001,2
[83] 张向、许晶月等.面向对象的有限元程序设计.计算力学学报.1999,5
[84] 高自友、吴方.非线性约束条件下的SQP可行方法. 应用数学学报.1995,10.Vol.18.No.4.p579
[85] 张向、许晶月等.面向对象的有限元程序设计.计算力学学报.1999,5
[86] Walther F, Baumgartner A, Mattheck C. Finding new designs for pressure tools using optimization methods which are based om the simulation of biological growth. Computer Aided Optimization Design of Structures, 1993,3:601~622.
[87] Tutorial Manual Handy MSC/NSTRAN
[88] Example Problem Manual Handy MSC/NSTRAN
[89] MSC.Patran 2000(r2) PCL Reference Manual Volume 1:Function Descriptions
MSC.Patran 2000(r2) PCL Reference Manual Volume 1:Function
[90] Descriptions
[91] MSC.Patran 2000(r2) PCL Reference Manual Volume 2:Code Example
[92] MSC.Patran 2000(r2) PCL and Customization