A non-probabilistic reliability topology optimization method based on aggregation function and matrix multiplication considering buckling response constraints

A non-probabilistic reliability topology optimization method is proposed based on the aggregation function and matrix multiplication.The expression of the geometric stiffness matrix is derived,the finite element linear buckling analysis is conducted,and the sensitivity solution of the linear buckling factor is achieved.For a specific problem in linear buckling topology optimization,a Heaviside projection function based on the exponential smooth growth is developed to eliminate the gray cells.The aggregation function method is used to consider the high-order eigenvalues,so as to obtain continuous sensitivity information and refined structural design.With cyclic matrix programming,a fast topology optimization method that can be used to efficiently obtain the unit assembly and sensitivity solution is conducted.To maximize the buckling load,under the constraint of the given buckling load,two types of topological optimization columns are constructed.The variable density method is used to achieve the topology optimization solution along with the moving asymptote optimization algorithm.The vertex method and the matching point method are used to carry out an uncertainty propagation analysis,and the non-probability reliability topology optimization method considering buckling responses is developed based on the transformation of non-probability reliability indices based on the characteristic distance.Finally,the differences in the structural topology optimization under different reliability degrees are illustrated by examples.

Reliability-Based Topology Optimization of Fail-Safe Structures Using Moving Morphable Bars

This paper proposes an effective reliability design optimizationmethod for fail-safe topology optimization(FSTO)considering uncertainty based on the moving morphable bars method to establish the ideal balance between cost and robustness,reliability and structural safety.To this end,a performancemeasure approach(PMA)-based doubleloop optimization algorithmis developed tominimize the relative volume percentage while achieving the reliability criterion.To ensure the compliance value of the worst failure case can better approximate the quantified design requirement,a p-norm constraint approach with correction parameter is introduced.Finally,the significance of accounting for uncertainty in the fail-safe design is illustrated by contrasting the findings of the proposed reliabilitybased topology optimization(RBTO)method with those of the deterministic design method in three typical examples.Monte Carlo simulation shows that the relative error of the reliability index of the optimized structure does not exceed 3%.

An Efficient Strategy for Non-probabilistic Reliability-Based Multi-material Topology Optimization with Evidence Theory

It is essential to consider the effects of incomplete measurement,inaccurate information and inadequate cognition on structural topology optimization.For the multi-material structural topology optimization with non-probability uncertainty,the multi-material interpolation model is represented by the ordered rational approximation of mat erial properties(ordered RAMP).Combined with structural compliance minimization,the multi-material topology optimization with reliability constraints is established.The corresponding non-probability uncertainties are described by the evidence theory,and the uniformity processing method is introduced to convert the evidence variables into random variables.The first-order reliability method is employed to search the most probable point under the reliability index constraint,and then the random variables are equivalent to the deterministic variables according to the geometric meaning of the reliability index and sensitivity information.Therefore,the non-probabilistic reliability-based multi-material topology optimization is transformed into the conventional deterministic optimization format,followed by the ordered RAMP method to solve the optimization problem.Finally,through numerical examples of 2D and 3D structures,the feasibility and effectiveness of the proposed method are verified to consider the geometrical dimensions and external loading uncertainties.

A Comparison of Deterministic, Reliability-Based Topology Optimization under Uncertainties

Reliability and optimization are two key elements for structural design. The reliability~ based topology optimization （RBTO） is a powerful and promising methodology for finding the optimum topologies with the uncertainties being explicitly considered, typically manifested by the use of reliability constraints. Generally, a direct integration of reliability concept and topol- ogy optimization may lead to computational difficulties. In view of this fact, three methodologies have been presented in this study, including the double-loop approach （the performance measure approach, PMA） and the decoupled approaches （the so-called Hybrid method and the sequential optimization and reliability assessment, SORA）. For reliability analysis, the stochastic response surface method （SRSM） was applied, combining with the design of experiments generated by the sparse grid method, which has been proven as an effective and special discretization technique. The methodologies were investigated with three numerical examples considering the uncertainties including material properties and external loads. The optimal topologies obtained using the de- terministic, RBTOs were compared with one another; and useful conclusions regarding validity, accuracy and efficiency were drawn.