Reliability-based multi-scale design optimization of composite frames considering structural compliance and manufacturing constraints

Abstract

The paper proposes an efficient methodology for concurrent reliability-based multi-scale design optimization (RBMDO) of composite frames to minimize structural cost subjecting to compliance constraint. Two types of variables are systematically considered in RBMDO, which are deterministic design variables of the frame components, the discrete fiber winding angles at the two geometrical scales, and random parameters of material properties and loading conditions in both magnitude and direction. To overcome the difficulty of highly nonlinear compliance constraint when using fiber winding angles as design variables and improve efficiency and accuracy of RBMDO of composite frames, the improved single loop and single vector (SLSV) approach based on modified chaos control (MCC) scheme, which is abbreviated hereafter as SLSV-MCC, is proposed, and sensitivities at the current design point are utilized to further increase accuracy of the proposed SLSV-MCC. Six types of specific manufacturing constraints are explicitly considered in the proposed RBMDO to reduce the risk of local failure in the laminated composite. The deterministic multi-scale design optimization (DMDO) model is also presented and utilized for comparison to distinguish differences between deterministic and reliability-based optimization results. Efficiency and accuracy of the proposed SLSV-MCC are compared with the first-order reliability method (FORM) and conventional SLSV approach. Meanwhile, the Monte Carlo simulation (MCS) method is further utilized to validate the accuracy of the proposed RBMDO. The discrete material optimization (DMO) approach is utilized to couple two geometrical scales: macroscopic topology and microscopic material selection. Capabilities of the proposed RBMDO are demonstrated by optimization of 2D and 3D composite frames. Numerical study reveals that the uncertainties in material properties and loading conditions will lead to different macroscopic sizing and topology configurations for deterministic and reliability-based solutions.

Appendix A: Evaluation of convergence

The convergence measure given in Stegmann and Lund (2005) is adopted to describe whether the optimization has converged to a satisfactory result, i.e., a single candidate material has been chosen in a specified element and all other materials have been discarded. For each layer, the following inequality is evaluated as

$$ {\omega}_{i,j,c}\ge \upvarepsilon \sqrt{\omega_{i,j,1}^2+{\omega}_{i,j,2}^2+\cdots +{\omega}_{i,j,{\mathrm{N}}^{\mathrm{cand}}}^2} $$

(A-1)

where ε is the tolerance, typically, ε ∈ [0.95~0.99]. If the inequality in (A-1) is satisfied for any ω_{i, j, c} in the j-th layer, then the layer is flagged as converged. The convergence assessment criterion H_ε is defined as the ratio between the number of converged layers ${N}_c^{l, tot}$ and the total number of layers N^{l, tot}. N^lay is the number of layers in each tube, and it is assumed that each tube has the same number of layers in the paper. N^tub is the number of tubes in the frame structure. Thus, N^{l, tot} can be expressed as the number of tubes multiplied by the number of layers in a tube, that is, N^{l, tot} = N^tub ∙ N^lay. Then, the convergence assessment H_ε can be expressed as

$$ {H}_{\varepsilon }=\frac{N_c^{l, tot}}{N^{l, tot}} $$

(A-2)

If the tolerance is 0.95 and the optimization is fully converged, i.e., H_{ε = 0.95} = 1, all layers have a single weight factor that contributes more than 95% to the Euclidian norm of the weight factors. More discussion about convergence criteria can be found in the references (Xu et al. 2019; Duan et al. 2015).