Structural and Multidisciplinary Optimization

Metamaterial systems have opened new, unexpected, and exciting paths for the design of acoustic devices that only few years ago were considered completely out of reach. However, the development of an efficient design methodology still remains challenging due to highly intensive search in the design space required by the conventional optimization-based approaches. To address this issue, this study develops two machine learning (ML)-based approaches for the design of one-dimensional periodic and non-periodic metamaterial systems. For periodic metamaterials, a reinforcement learning (RL)-based approach is proposed to design a metamaterial that can achieve user-defined frequency band gaps. This RL-based approach surpasses conventional optimization-based methods in the reduction of computation cost when a near-optimal solution is acceptable. Leveraging the capability of exploration in RL, the proposed approach does not require any training datasets generation and therefore can be deployed for online metamaterial design. For non-periodic metamaterials, a neural network (NN)-based approach capable of learning the behavior of individual material units is presented. By assembling the NN representation of individual material units, a surrogate model of the whole metamaterial is employed to determine the properties of the resulting assembly. Interestingly, the proposed approach is capable of modeling different metamaterial assemblies satisfying user-defined properties while requiring only a one-time network training procedure. Also, the NN-based approach does not need a pre-defined number of material unit cells, and it works when the physical model of the unit cell is not well understood, or the situation where only the sensor measurements of the unit cell are available. The robustness of the proposed two approaches is validated through numerical simulations and design examples.

The unique structure of negative Poisson’s ratio (NPR) materials provides a good energy absorption capacity that has found wide application in automotive and aeronautics industries. The present work proposes a novel NPR energy absorption structure installed between the bumper and anti-collision beam of automobiles for improving pedestrian lower leg protection. The performance of the proposed NPR structure based on tibia acceleration, knee bending angle, and knee shear displacement is evaluated by comparison with the performance of a conventional energy absorption structure. The performance of the proposed structure is further improved by conducting multi-objective design optimization with consideration for the perturbation induced by parameter uncertainties. For optimization, a parametric model of the NPR structure is first established based on the relationships between parameters. To promote computational efficiency, an optimal Latin hypercube sampling technique and the dual response surface method are combined to build surrogate models between responses and inputs. A multi-objective particle swarm optimization algorithm and six sigma criteria are then applied to obtain the optimal design parameters of the structure. The optimization results are validated by comparisons with the results of multi-objective deterministic optimization. The comparison results show that the proposed NPR energy absorption structure improves pedestrian lower leg protection significantly, and its performance is further improved by the proposed multi-objective robust design optimization. This study serves as a good example of the safety promotion provided by applying NPR structures with enhanced energy absorption capacity in vehicles.

Reducing the cost of calculation without compromising the accuracy of the solution is a recognized challenge for optimizing the reliability analysis, which became possible using surrogate models trained with robust techniques, such as active learning Kriging (AK) reliability methods. In the AK reliability method, a Kriging predictor is built with a small size of design of experiments (DoE) and becomes more accurate in the vicinity of the limit state function (LSF) in a stepwise manner, called the learning process, until a stopping criterion is met. The motivation of the current study is to enhance the accuracy and efficiency of AK reliability analysis by developing new learning functions, new stopping criteria, and a new method of selection of the next candidate for updating the DoE in the learning process. In this paper, two new learning functions named Kriging occurrence (KO) and weighted KO (WKO) are proposed based on a probability-based approach. A hybrid selection for the next candidate is introduced which simultaneously considers the probability of improvement and the density of DoE and a new stopping criterion is recommended based on the relative mean of the learning functions. A thorough study of the literature is conducted where 12 learning functions are summarized and their performances are compared to that of newly developed learning functions through five comparative examples. The result of the study shows that the new learning function can enhance the accuracy and efficiency of the learning process.

Performing synthesis during conceptual design provides substantial cost savings by selecting an efficient design topology and geometry, in addition to selecting the structural member properties. A new evolutionary-based representation, which combines redundancy and implicit fitness constraints, is introduced to represent and search for design solutions in an unstructured, multi-objective structural frame problem. The implicit redundant representation genetic algorithm, in tandem with the unstructured problem domain definition, allows the evaluation of diverse frame topologies and geometries. The IRR GA allows the representation of a variable number of location independent parameters, which overcomes the fixed parameter limitations of standard GAs. The novel frame designs evolved by the IRR GA synthesis design method compare favourably with traditional frame design solutions calculated by trial and error.

A convex programming optimizer called GMMA (Generalized Method of Moving Asymptotes) is presented in this paper. This method aims at solving engineering design problems including nonlinear equality and inequality constraints. The basic feature of this optimizer is that the efficient dual solution strategy together with the flexible GMMA approximation scheme are used. Especially, nonlinear equality constraints can be exactly satisfied by the intermediate solution of each explicit subproblem because their linearization is updated in an internal loop of the subproblem. This method will be illustrated by a hydrodynamic design application.

In nonlinear optimization, the dual problem is in general not easier to solve than the primal problem. Convex separable optimization problems, frequently arising in electrical and mechanical engineering, constitute a notable exception to the above rule. The dual problem is to optimize the dual objective functionℓ over a non-negative orthant, and the evaluation ofℓ reduces to the execution of independentlinear searches only. To generalize the idea, we also consider partially-separable problems with objective and constraint functions such that the Hessian matrix of the Lagrange function is a block-diagonal matrix with 2*2 blocks. The evaluation of the dual objective function is accordingly reduced to a number of independentplanar searches. Obviously, 3*3 blocks would lead tospatial searches, etc. We compare the performance of a primal and a dual method on a graded set of artificial test problems with increasing size, increasing degree of degeneracy, and increasing ill-conditioning. The observed speed-up by the dual approach varies between 2 and 30. Finally, we consider the potential of the dual approach for execution on parallel computers.

A procedure is developed for the design of reinforced concrete footings subjected to vertical, concentric column loads that satisfies both structural requirements and geotechnical limit states using a hybrid Big Bang-Big Crunch (BB-BC) algorithm. The objectives of the optimization are to minimize cost, CO $_{2}$ emissions, and the weighted aggregate of cost and CO $_{2}$ . Cost is based on the materials and labor required for the construction of reinforced concrete footings and CO $_{2}$ emissions are associated with the extraction and transportation of raw materials; processing, manufacturing, and fabrication of products; and the emissions of equipment involved in the construction process. The cost and CO $_{2}$ objective functions are based on weighted values and are subjected to bending moment, shear force, and reinforcing details specified by the American Concrete Institute (ACI 318-11), as well as soil bearing and displacement limits. Two sets of design examples are presented: low-cost and low-CO $_{2}$ emission designs based solely on geotechnical considerations; and designs that also satisfy the ACI 318-11 code for structural concrete. A multi-objective optimization is applied to cost and CO $_{2}$ emissions. Results are presented that demonstrate the effects of applied load, soil properties, allowable settlement, and concrete strength on designs.

This paper presents a reliability-based topology optimization method under geometrical uncertainties. First, we briefly introduce the concept of topology optimization. Then, we explain how shape uncertainty is modeled in Eulerian description, using an advection equation and a Karhunen-Loève expansion. Based on the shape uncertainty modeling, we formulate a reliability measure for the shape uncertainty, briefly introducing the inverse reliability method. Two optimization problems, a minimum mean compliance problem and an optimum design problem for a compliant mechanism, are then formulated using the proposed shape uncertainty modeling. The design sensitivity analysis for the reliability analysis and optimization procedure, performed using the adjoint variable method, is then explained. A two-level optimization algorithm is constructed next, in which the inner iteration is used for reliability analysis and the outer is used for updating design variables. Finally, three numerical examples are provided to demonstrate the validity and the utility of the proposed method.

An alternative formulation of collaborative optimization (CO) combined with linear approximations (CLA-CO) is recently developed to improve the computational efficiency of CO. However, for optimization problems with nonconvex constraints, conflicting linear approximations may be added into the system level in the CLA-CO iteration process. In this case, CLA-CO is inapplicable because the conflicting constraints lead to a problem that does not have any feasible region. In this paper, a linear approximation filter (LAF) strategy for CLA-CO is proposed to address the application difficulty with nonconvex constraints. In LAF strategy, whether conflict exists is first identified through transforming the identification problem into the existence problem of feasible region of linear programming; then, the conflicting linear approximations are coordinated by eliminating the larger violated linear approximations. Thereafter, the minimum violated linear approximation replaces the accumulative linear approximations as the system-level constraint. To evaluate the violation of linear approximation, a quantification of the violation is introduced based on the CO process. By using the proposed LAF strategy, CLA-CO can solve the optimization problems with nonconvex constraints. The verification of CLA-CO with LAF strategy to three optimizations, a numerical test problem, a speed reducer design problem, and a compound cylinder design problem, illustrates the capabilities of the proposed LAF strategy.