Algorithms And Engineering Applications Of Topology Optimization Of A Truss-like Structure

With the rapid development of mathematical programming theory,computational mechanics and computer technique,a series of important achievements have been made in the field of structural optimization,which has promoted the sustained development of many fields,such as aerospace,automotive and civil engineering.Compared with structural size and shape optimization,topology optimization has been regarded as the most challenging and rewarding task in the structural optimization field owing to its difficulty,complexity and greater potential.Several approaches for topology optimization have been presented so far.Among them,the topology optimization method based on the truss-like material model is extremely valuable,which overcomes a series of numerical instability problems.At present,truss-like continua are mainly optimized as per the fully stressed criterion which only applies to topology optimization problems with stress constraints under a single load case(SLC).In view of this situation,firstly,an improved mathematical programming method with strong adaptability to multi-constraint optimization problems is developed in this dissertation.Secondly,topology optimization of spatial truss-like structures with stress constraints under a multiple load case(MLC)is more challenging and closer to the actuality.It is worth doing all-round research.In addition,on the basis of the previous researches,a numerical optimization algorithm is proposed that can generate the optimal reinforcement layout of RC structures under an MLC.All in all,this dissertation focuses on algorithms and engineering applications of topology optimization of a truss-like structure.The research contents of this dissertation mainly involve the following three aspects.(1)An improved mathematical programming method to optimize a planar truss-like continuum is presented.The densities and orientations of members at nodes are taken as the design variables.The volume(weight)of the structurematerials is taken as the objective function.Since the orientations of members are independent of the volume(weight)of the structure materials,the densities and orientations of members are optimized in two separate procedures in each iteration.An explicit sub-problem in variable separation form is established at every iteration procedure.At each sub-problem,the stress constraint function is expanded to a Fourier series of member angles.The optimal orientations of members are determined according to extreme condition.The objective function and constraint functions are written as explicit,convex approximation functions of member densities.The method of moving asymptotes(MMA)is used to optimize member densities.(2)A new method for topology optimization of spatial truss-like structures with stress constraints under an MLC is put forward.A spatial truss-like material model with three families of orthotropic members is taken as the research object.An optimality criterion based on the concept of directional stiffness is employed to solve the topology optimization problem of minimum volume(weight)of the structure materials with stress constraints under the MLC.First,the truss-like structure is optimized as per the fully stressed criterion under each SLC.Accordingly,the directional stiffness of the optimal structure under each SLC is obtained.Next,the directional stiffness of the optimal truss-like structure under the MLC,expressed as a closed surface,is determined through the least squares method based on the results above.It has been proved in this dissertation that the eigenvalues and eigenvectors of the coefficient matrix of the closed surface are the optimal densities and orientations,respectively,of the members in the truss-like structures under the MLC.(3)A numerical optimization algorithm is proposed in this dissertation that can automatically generate the optimal reinforcement layout of RC structures under an MLC.It is assumed that concrete is filled with truss-like materials.The truss-like continuum in concrete is used to simulate steel bars.In order to calculate the average stress at any point in the composite materials composed of a truss-like continuum and concrete,the relation between the density components in the local coordinate system and the ones in the global coordinate system is established.Then,to obtainthe average state of stress at any point in the composite materials,the stress components of concrete and steel bars at the point are superimposed according to the proportion of the two materials.Finally,the average principal stress and its principal direction are calculated based on the stress state at one point in the composite materials.The optimal reinforcement layout in concrete is obtained as per the fully stressed criterion under each SLC to guarantee that neither steel bars nor concrete will fail.Similarly,a closed surface(for three-dimensional structures)or a closed curve(for two-dimensional structures)is adopted to fit the maximum directional stiffness under all SLCs.The optimal reinforcement layout of RC structures under the MLC is obtained by solving the eigenvalues problem of the coefficient matrix of the surface or the curve.Due to the limitation of the current design method for RC structures,this method provides a reference for concept design of complex concrete components under stress constraints.