Topological Optimization Algorithm Based On Discrete Design Variables For Continuum-type Structures And Its Applications

In the field of structural optimization,the topological optimization of continuum structure is increasingly applied to the optimization design of the civil engineering structure.As one of the major topology optimization methods,the evolutionary structural optimization(ESO)method has gradually developed into a mature hard-kill or soft-kill bidirectional ESO(BESO)method,starting from a complete heuristic algorithm.Despite various methods and improved algorithms based on BESO criteria have been developed,design variables take values only at both ends,which usually leads to convergence and stability problems as well as the need for empirical criteria or heuristic algorithms for solving the application of optimization.Therefore,this paper proposes topology optimization algorithms based on discrete variables and BESO criteria and extends them to several variants of optimization problems,and researches these algorithms deeply in the theoretical and practical aspects.Firstly,for extending the discrete size variable-based BESO(DSV-BESO)method to topology optimization design,this paper defines two design variables as discrete element densities and discrete level set functions(DLSFs),and constructs three criteria of bidirectional evolution.The proposed topology optimization method is called the discrete variable-based BESO(DV-BESO)algorithm.Compared with the soft-kill BESO(SBESO)method,the DV-BESO algorithm is of higher precision of sensitivity and better convergence and stability,because of its discrete variables.Secondly,DSV-BESO method may not be able to find the local optimization solution to the multiple displacement constraint-based optimization(MDCO)design,due to the use of empirically-based formula in the Lagrange multiplier(LM)method.Therefore,this paper improves the DV-BESO algorithm by taking the Powell-Hestenes-Rochafellar augmented LM algorithm(‘the PHR algorithm' for short)instead of the empirical formula.In the global displacement control-based optimization(GDCO)design,this study further improves the PHR algorithm on the oscillation of convergence.In GDCO,the DV-BESO algorithm takes the improved PHR algorithm instead of the heuristic algorithm of the SBESO method,so it can define the objection function other than the structural volume,and reduce the difference between the terminal displacement constraint and its limit.Thirdly,the DV-BESO algorithm using the PHR algorithm(‘improved DV-BESO algorithm' for short)is expanded to the stress-constrained optimization(SCO)design.Firstly,this paper introduces two approaches of relaxation and aggregation to construct the relax stress function of each aggregation subregion,and then establishes a Lagrangian function,in which the obtained stress functions are taken as the constraints.Secondly,the true sensitivities are found by solving the derivative of the Lagrangian function with respect to the continuous elemental densities,and then the approximate sensitivities can be found by taking the discrete densities instead of the continuous densities.Finally,the improved DV-BESO algorithm is used to find a local optimal solution.Therefore,it is proved that the improved DV-BESO algorithm can be used to solve optimization designs with large-scale local constraints.The same idea and similar computational process can be adopted to solve the other constraint-based optimization designs.Fourthly,this paper studies the relative parameters of the optimization designs with multiple displacement constraints and stress constraints.The parameter study is used to verify the reasonableness of the DV-BESO algorithm and its improvement,and investigate the influence of each parameter on the final results of topology and objective function.In the MDCO design,the structural volume is treated as the objective function.By using the DV-BESO algorithm with the empirical formula,three kinds of parameters are studies.They are the set of discrete values of elemental density(‘discrete set of density' for short),the coefficient of the improved approach of the sensitivity number using historical information(‘coefficient of sensitivity improvement' for short),and the initial value of evolution rate(‘initial evolution rate' for short).Furthermore,by using the improved DV-BESO algorithm,apart from the above parameters,three kinds of parameters are also studies.They are the coefficient of continuation approach for the filter size,the predefined displacement limit and the coefficient of update formulas of LM,so there are six parameters in total.In the GDCO design,the above six kinds of parameters are studied by the improved DV-BESO algorithm that takes the mean compliance as the objective function.Considering the small difference between the final results of mean compliance,this paper proposes the criteria for selecting the minimum values of mean compliant to find one or three of the local optimal solution.By using the parameter study,this paper obtains the influence range of each parameter on the final volume fraction for the MDCO design and the compliance fraction for the GDCO design respectively,and sorts the selected parameters by influence degree corresponding to each bidirectional evolution criterion.In the SCO design,eight kinds of parameters are studied by the improved DV-BESO algorithm that takes the mean compliance as the objective function.The selected parameters are the discrete set of density,the coefficient of sensitivity improvement,the initial evolution rate,the initial filter size and initial penalty factor and the coefficient of continuation approach for both of them,as well as stress constraints.By using the parameter study,this paper obtains the influence range of each parameter of the part of the eight parameters on the final compliance fraction for each bidirectional evolution criterion,and sorts the selected parameters by influence degree corresponding to each bidirectional evolution criterion and the predefined volume constraint.Finally,the DV-BESO algorithm with the DLSFs is difficult to find the local optimal solution,the shape of the boundary is not smooth enough,and it needs more iterations.The conventional level set methods(LSMs)possess no mechanism to nucleate new holes inside the material domain,and have lower computational efficiency.This paper proposes a novel algorithm by combining both a bi-directional evolutionary algorithm(BEA)and the local level set method(LLSM).A proposed BEA using the DLSFs is implemented according to optimization criteria of the BESO method,until a stable topological solution is found.Then,the LLSM using the local level set function(LLSF)is applied to further improve local details of the topology and shape of the structure.The final DLSFs are transformed into the initial LLSF by iteratively solving a distance-regularized equation(DRE).In the BEM,the topological derivatives and the Shepard interpolation are treated as nodal sensitivities and sensitivity filtering,respectively.The computational efficiency of the LLSM is increased by taking the DRE instead of the re-initialization equation.A conditionally stable difference scheme with reverse diffusion constraints is formulated to ensure the numerical stability of the DRE.In the combined algorithm,the BEA is taken instead of the process of hole nucleation in the LLSM,while the LLSM is able to further improve the convergence of the BEM to obtain at least a local optimal solution.