Research On Topology Optimization Of Uncertain Continuum Structure

Topology optimization has currently become an important tool for innovative design of key components in the fields(such as aerospace,automobile and heavy machinery)for its advantages such as large design freedom,providing high-quality new deigns with different performance.However,most of existing topology optimization methods need to obtain sensitivity information of the objective function and constraints in the optimization model when they are employed to perform the structural optimization design,which will not only bring difficulties in the numerical derivation,but also probably increase the computational burden and result in some difficulties which are adverse to the implementation of optimization approaches.Moreover,in engineering structure design,various uncertain factors are common and run through the entire life cycle of the structure.Although the values of these uncertainties are relatively small in most cases,their coupling effect will cause large deviations in the response of the structural system,and the uncertainty can only be reduced as much as possible.Based on the latest topology optimization theories and methods,this paper studied the deterministic topology optimization problem of continuum structure,and proposed the improved proportional topology optimization(IPTO)algorithms that can avoid obtaining sensitivity information of the objective function and constraints in the optimization problem.Subsequently,combined with the uncertain analysis,the reliability topology optimization,robust topology optimization and topology optimization avoiding unsupported materials of continuum structure were studied in turn.The main work in this paper is as follows:(1)The topology optimization problem of continuum structure with a wide application background was studied,where the objective is to minimize the structural compliance while considering the volume constraint.Under the premise of avoiding obtaining sensitivity information of the objective function and constraints in the optimization problem,the solid isotropic microstructures with penalization was used as the material interpolation model.Combined with the advantages of density filtering and fully considering the intermediate variables obtained during performing the topology optimization design for the structure,four IPTO algorithms(called respectively as IPTO＿A、IPTO＿B、IPTO＿C and IPTO＿D)were proposed on the basis of the proportional topology optimization(PTO)algorithm by designing new compliance proportional filtering function and new density variable increment update scheme and by modifying the density variable update schemes.Subsequently,on the basis of giving evaluation indicators,the numerical example was used and other topology optimization methods were compared to test the effect of the proposed strategies and to verify the effectiveness and superiority of the new algorithms.At the same time,it was found that the IPTO＿A algorithm had the best performance.In addition,the effect of control parameters(λ and α)in the new algorithms on structural optimization results was also discussed in depth.(2)The topology optimization problem of continuum structure with minimum volume was investigated under stress constraints.Taking avoiding the acquisition of sensitivity information of the objective function and constraints as the premise and combined with the advantages of density filtering,a new improved proportional topology optimization(IPTOs)algorithm was proposed on the basis of the PTO and IPTO＿A algorithms by introducing the Heaviside operator and the weighting function considering the Gaussian distribution and by designing the new target material volume update scheme and new density variable increment update scheme.Subsequently,the numerical examples were employed and the PTO algorithm was compared to validate the effect of the proposed strategies and the new algorithm.At the same time,the effect of control parameters(λ and α)in the new algorithm on structural optimization results was analyzed.(3)The reliability topology optimization problem of continuum structure was studied while considering the uncertainties of structural geometrical dimensions,material volume and external load.Under the premise of simplifying the calculation process as much as possible,a new reliability-based topology optimization(RBTO)method was proposed by effectively combining the IPTO＿A algorithm with multiple methods such as probability method,hybrid(or concurrent)method and first-order reliability method(FORM).On the basis of describing the uncertainties of design variables with the probability method,the mathematical model of the optimization problem was established,where the objective was to minimize the structural compliance while the material volume constraint and reliability constraint were comprehensively considered.To improve the computational efficiency,the hybrid method was used to decouple the reliability topology optimization model into two independent sub-design stages including the reliability analysis and the equivalent deterministic topology optimization.In the reliability analysis stage,the modified random variables satisfying the reliability constraint were obtained with the help of the FORM.In the equivalent deterministic topology optimization stage,the modified random variables were taken as design parameters,and the IPTO＿A algorithm was employed to perform the topology optimization design for the structure.Finally,the effectiveness of the new method was confirmed with numerical examples,and the effect of the reliability index on structural optimization results was also discussed.(4)The reliability topology optimization problem of continuum structure with stress constraints was studied while considering the uncertainties of the external load and the thickness of a structure.By effectively combining the IPTOs algorithm with multiple methods(such as probability method,sequential optimization and reliability assessment(SORA),FORM,response surface methodology(RSM)and Kriging method),two new RBTO methods were proposed.Based on the probability method to describe the uncertainties of design variables,the mathematical model of the optimization problem was established,and then the SORA was employed to decouple the optimization problem into two independent sub-design stages comprising the equivalent deterministic topology optimization and the reliability analysis.In the equivalent deterministic topology optimization stage,the IPTOs algorithm was used to perform the structural optimization design.In the reliability analysis stage,the RSM and the Kriging method were respectively combined with the central composite design,FORM and iterative solution technique to approximate the limit state function of structural stress with respect to random variables,and subsequently the inverse reliability analysis and the inverse transformation were carried out to obtain the most probable point which was introduced as design parameters into the next equivalent deterministic topology optimization.Finally,the effectiveness of two new RBTO methods and the accuracy of topology designs obtained in the reliability topology optimization were validated by using numerical examples and the Monte Carlo method.At the same time,it was concluded that the RSM and the Kriging method were suitable for constructing the approximated limit state function of structural stress.(5)The robust topology optimization of continuum structure was studied while considering loading uncertainty.Under the premise of simplifying the calculation process as much as possible,a new robust topology optimization(RTO)method was proposed under the combination of the IPTO＿A algorithm and multiple methods(such as probability method,weighted combination method,displacement superposition principle of linear body,and Monte Carlo method).Among these,the probability method was used to describe the uncertainties of loading magnitude and direction,the weighted combination method was applied to constructing the objective function of the mathematical model of the optimization problem,the calculation method of the objective function was given by using multiple strategies such as displacement superposition principle of linear body and Monte Carlo method,and the core functions of the IPTO＿A algorithm were redesigned to ensure that the optimization problem was able to be tacked.Finally,the effectiveness and superiority of the new RTO method was confirmed by applying numerical examples and by comparing with other RTO methods.At the same time,it was found that different values of the standard deviation of the angle between the acting direction of the load and the horizontal direction in the rectangular coordinate would affect structural optimization results.(6)The topology optimization problem of continuum structure avoiding unsupported materials was studied under symmetrical loads.Taking the 2D cantilever beam as the research object,the necessity of tackling the structural topology optimization problem that can avoid the unsupported material was expounded with the help of the optimized results of the cantilever beam obtained by the IPTO＿A algorithm,and on this basis,a scheme of applying the RTO method under the condition of considering loading uncertainty was proposed to address the studied problem.Finally,the effectiveness of the scheme was demonstrated by the numerical example.The improved proportional topology optimization algorithms proposed in this paper can not only effectively tackle the optimization design problems of continuum structure,but also break through the bottleneck of obtaining sensitivity information that was required in most of existing topology optimization methods.Under the premise of simplifying the calculation process as much as possible and reducing the acquisition of sensitivity information,the research on topology optimization of continuum structure considering the uncertainty can respectively obtain the optimized structures with high reliability and strong robustness and avoiding unsupported material.This study enriches the theory and method of topology optimization of continuum structure,and has important theoretical research significance and engineering application value.