包含两类变量的离散变量桁架结构拓扑优化设计

建立了包含截面和拓扑两类变量的离散变量结构拓扑优化设计的数学模型，该模型考虑了截面变量与拓扑变量间的耦合关系，反映了拓扑优化问题的组合优化本质，可以较好地解决"极限应力"、"最优解的奇异性"等困扰结构拓外优化设计的问题．同时采用相对差商法进行离散变量桁架结构拓扑优化，直接求解包含两类变量的离散变量结构拓扑优化设计数学模型，收到了比较满意的效果．

TOPOLOGY OPTIMIZATION OF TRUSS STRUCTURES WITH DISCRETE VARIABLES INCLUDING TWO KINDS OF VARIABLES

Because the topology of structure and the sizes of cross sections must be designedin the topology optimization, two kinds of variables, topology variables al and size variables ofcross sections Al are included in the mathematics model of topology optimization. The model isa mixed discrete programming or a pure discrete one including two kinds of variables. It is morecomplex and difficult to be solved than sizing optimization of structures, so it is a challengingproblem in the field of structural optimization, and it is a focus in the optimization of structures.Comparing with the topology optimization of continuous variables, the development of topologyoptimization of discrete variables is slower because of the limitation of development in sectionalsizing optimization of structures with discrete variables.Received 20 October 1997, revised 16 April 19981) The project supported by the Natural Science Foundation of Shandong Province.The basic method for topology optimization is based on ground structure. A set of nodes isconstructed according to the given supporting conditions, load cases and other conditions, and thenodes are linked by members to form the initial ground structure. The all topological forms ofstructure can be obtained by the combinations of members in the initial ground structure. Thisproblem is expressed as the combination of programming P1 of sectional sizing optimization withthe lower limitation of zero and other programming P2 only including the topological variables.After obtained the optimum solution of P1, the topological optimization is done by cancelingsome members of zero sectional area and some other members according to some canceling orresuming rules. In fact, this is an optimization method of separating variables. The sectionalvariables and topological variables are separated and solved respectively. Its advantages are relativeless computational efforts and higher computational efficiency. But there are stronger couplingrelations between topological variables and sectional variables of actual structures. If the couplingrelations are neglected, the optimum solution (or local optimum solution) may not be obtained.In addition, some difficult problems, such as "the limiting stress" and "singular of optimizationsolution" must be faced. TO overcome the problems caused by separating the variables, two kindsof variables should be treated together through the total procedure of optimization.In this paper, a mathematical model of topology optimization of truss structures with discretevariables including two kinds of variables is developed. The model has considered the couplingrelations between cross section variables and topological variables, so that it reflects the innatecharacters of topology optimization as a combinatorial optimization problem. And the problemssuCh as the "limit stress" and "the singular solution of structural optimization" can be solved byusing this model. The model of topology optimization of truss structures with discrete variablesincluding two kinds of variables are solved directly by using the relative difference quotient algorithm. The computational results are satisfactory and some new topologies and better solution areobtained.