Structural and Multidisciplinary Optimization

Uncertainties in deviations of physical properties lead to a probabilistic failure analysis of the composite materials. The proposed optimization model for laminate composites is based on reliability analysis considering the ultimate failure state. To avoid difficulties associated with the complete analysis of the failure modes, bounds are established for the failure probability of the structural system. These bounds are related with theintact and degraded configurations of the structure. Using thefirst ply failure and thelast ply failure theories and a degradation model for the mechanical properties with load sharing rules we obtain the failure probabilities corresponding to the two above configurations. The failure probability of each configuration is obtained using level 2 reliability analysis and the Lind-Hasofer method. The optimization algorithm is developed based on the problem decomposition into three subproblems having as objectives the maximization of the structural efficiency atintact and degraded configurations of the structure and weight minimization subjected to allowable values for the structural reliability. Additionally, the search for the initial design is performed introducing a weight minimization level. It is expected to explore the remaining load capacity of the structures afterfirst ply failure as a function of the anisotropic properties of the composites. The design variables are the ply angles and the thicknesses of the laminates. The structural analysis for the model developed is performed through the finite element method mainly using the isoparametric degenerated shell finite element. The sensitivities are obtained using the discrete approach through the adjoint variable method. In order to show the performance of the analysis two examples are presented.

Evolutionary Structural Optimization (ESO) and its later version bi-directional ESO (BESO) have gained widespread popularity among researchers in structural optimization and practitioners in engineering and architecture. However, there have also been many critical comments on various aspects of ESO/BESO. To address those criticisms, we have carried out extensive work to improve the original ESO/BESO algorithms in recent years. This paper summarizes latest developments in BESO for stiffness optimization problems and compares BESO with other well-established optimization methods. Through a series of numerical examples, this paper provides answers to those critical comments and shows the validity and effectiveness of the evolutionary structural optimization method.

The design of complex engineering systems requires an initial decomposition of the system into subsystems. These systems are linked together by couplings, which represent output data transference from one subsystem to another. Because complex engineering systems can have hundreds or thousands of such couplings, the optimization of these systems is often quite difficult, if not impossible. To reduce the optimization time, it becomes important that a system designer have the ability to select couplings that have little effect on the solution accuracy, and temporarily remove them. Previous coupling strength analysis methods have not related the effect of a coupling’s removal for multiple cycles to solution accuracy. The method presented here identifies weak couplings based on their relationship to the objective function and constraints in the overall system optimization problem. The couplings are then suspended for multiple cycles of the multidisciplinary design optimization process. Discussion of the application of this new method follows, as well as implementation on a decomposed analytical problem. The method significantly reduces the number of subsystem analyses required to optimize the decomposed problem by suspending couplings for multiple design cycles. As a result of the system reduction, considerable computational saving are made without introducing significant error into the results of the optimization. The trade-offs between computational savings and solution accuracy are also shown and discussed.

A geometrical stiffness matrix for a flexibly-connected member is developed for the analysis and design of semirigid framed structures when geometrical non-linearity (P — Δ effects and other second-order effects) must be considered. A “fixity factor” defining the rigidity of a connection relative to the attached member is introduced to model different types of member connectivity. The sensitivities of the geometrical stiffness matrix with respect to the fixity factor are developed with a view to conducting structural optimization. As an example of optimum design, a semi-rigid steel framework is presented along with discussion of the necessity of considering second-order effects.

In this article, a topology optimization method is developed, which is aware of material uncertainties. The uncertainties are handled in a worst-case sense, i.e., the worst possible material distribution over a given uncertainty set is taken into account for each topology. The worst-case approach leads to a minimax problem, which is analyzed throughout the paper. A conservative concave relaxation for the inner maximization problem is suggested, which allows to treat the minimax problem by minimization of an optimal value function. A Tikhonov-type and a barrier regularization scheme are developed, which render the resulting minimization problem continuously differentiable. The barrier regularization scheme turns out to be more suitable for the practical solution of the problem, as it can be closely linked to a highly efficient interior point approach used for the evaluation of the optimal value function and its gradient. Based on this, the outer minimization problem can be approached by a gradient-based optimization solver like the method of moving asymptotes. Examples from additive manufacturing as well as material degradation are examined, demonstrating the efficiency of the suggested method. Finally, the impact of the concave relaxation of the inner problem is investigated. In order to test the conservatism of the latter, a RAMP-type continuation scheme providing a lower bound for the inner problem is suggested and numerically tested.

In this work, we propose to extend the bi-directional evolutionary structural optimization (BESO) method for compliance minimization design subject to both constraints on volume fraction and maximum von Mises stress. To this end, the aggregated p-norm global stress measure is first adopted to approximate the maximum stress. The conventional compliance design objective is augmented with p-norm stress measures by introducing one or multiple Lagrange multipliers. The Lagrange multipliers are employed to yield compromised designs of the compliance and the p-norm stress. An empirical scheme is developed for the determination of the Lagrange multipliers such that the maximum von Mises stress could be effectively constrained through the controlling of the aggregated p-norm stress. To further enforce the satisfaction of stress constraints, the stress norm parameter p is assigned to a higher value after attaining the objective volume. The update of the binary design variables lies in the computationally efficient sensitivity numbers derived using the adjoint method. A series of comparison studies has been conducted to validate the effectiveness of the method on several benchmark design problems.

Active-learning surrogate model–based reliability analysis is widely employed in engineering structural reliability analysis to alleviate the computational burden of the Monte Carlo method. To date, most of these methods are built based on the single-fidelity surrogate model, such as the Kriging model. However, the computational burden of constructing a fine Kriging model may be still expensive if the high-fidelity (HF) simulation is extremely time-consuming. To solve this problem, an active-learning method based on the multi-fidelity (MF) Kriging model for structural reliability analysis (abbreviated as AMK-MCS+AEFF), which is an online data-driven method fusing information from different fidelities, is proposed in this paper. First, an augmented expected feasibility function (AEFF) is defined by considering the cross-correlation, the sampling density, and the cost query between HF and low-fidelity (LF) models. During the active-learning process of AMK-MCS+AEFF, both the location and fidelity level of the updated sample can be determined objectively and adaptively by maximizing the AEFF. Second, a new stopping criterion that associates with the estimated relative error is proposed to ensure that the iterative process terminates in a proper iteration. The proposed method is compared with several state-of-the-art methods through three numerical examples and an engineering case. Results show that the proposed method can provide an accurate failure probability estimation with a less computational cost.