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Optimization problems with multiple optima (local or global) are called multimodal problems. Many real world problems are multimodal and challenging to solve as compared to unimodal problems due to the possibility of premature convergence to a local optimum solution and possibly requiring a higher number of function evaluations. A multimodal optimization algorithm provides multiple solutions, thus a better understanding of the design space at minimal additional computational cost. The goal of multimodal particle swarm optimizer (MPSO) is to converge to multiple local and global optima with a reasonable number of function evaluations. The modifications to PSO include reduction in the personal best weight and an additional step to replace the global best with a group best in the PSO procedure. The modifications only allow a user defined number of particles (m ) to converge to a solution and relocate the particles if more than m particles converge to a solution. The relocation is active or inactive based on a predefined set of rules. MPSO is demonstrated on several optimization problems such as benchmark problems from the literature, spring design, and sequential sampling.

The use of Surrogate-based optimization has become increasingly prevalent in the design of engineering systems. Each optimization cycle consists of fitting a surrogate through an initial number of designs and performing optimization based on the surrogate to get a new design, where an exact simulation is performed. Algorithms like Efficient Global Optimization use uncertainty estimates available with the Kriging surrogate to guide the selection of new point(s). With access to parallel computation, adding multiple points per optimization cycle has become increasingly attractive. With EGO, Expected Improvement is commonly used as the criterion for selection of new point, but it is difficult and computationally expensive to add multiple points using this criterion. It is much easier to select multiple points when Probability of targeted Improvement is used as the selection criterion, but it suffers from the issue of target setting due to lack of knowledge of the true function. An adaptive target setting method is proposed in this paper, which changes the target to reflect the improvement achieved in each optimization cycle. This method is demonstrated to be highly efficient for three analytic examples. For these examples the proposed method is compared to a constant target setting and is shown to give better results. It is also seen that when multiple points are added per cycle using this method, the convergence rate becomes faster and also, the confidence in the final result becomes much higher.

Reliability analysis performed with high-delity simulations is usually expensive. As a result, a popular strategy is to replace the costly computer models by surrogates built using few simulations (sampled prefentially near the limit state). Unfortunately, estimation based on surrogates introduces uncertainty in the probability of failure. In this paper, we use kriging to study the uncertainty in probability of failure estimation. Kriging models the response as a Gaussian process. For given set of kriging parameters, the prediction at any point follows a Gaussian distribution. This makes the failure region uncertain and consequently the estimated probability of failure. Additionally, the choice of kriging parameters adds another layer of uncertainty. We studied both sources of uncertainty and found that: (a) with a single set of parameters (e.g., maximum likelihood estimate), only a portion of the probability of failure uncertainty is accounted for, and (b) uncertainty in probability of failure correlates very well with kriging accuracy.

Modeling fatigue crack growth is a challenging computational fracture mechanics problem because the crack growth rate can only be evaluated at the current crack size. The forward Euler method has been a common choice in integrating fatigue crack growth, whose accuracy can only be guaranteed with a very small size of increment. This hinders failure investigation of systems with complex geometry, which would require expensive finite element simulations. Higher-order integration methods, such as the midpoint method, might allow larger increment size but require additional evaluation of crack growth rate at crack sizes larger than the current one. In arbitrary geometry, this is not an easy task because the direction of crack growth is unknown in advance, and additional simulations are often prohibitive. In this paper, two surrogate models are generated for the prior crack growth direction and stress intensity factor data and are used to predict the crack growth rate at future locations without the need for additional finite element simulations. The step size for the numerical integration is chosen based on the prior accuracy of the extrapolated data for the crack growth direction and stress intensity factor. Several examples were tested in which crack growth follows linear and curved paths under a range of boundary conditions leading to different relationships between stress intensity factor and crack size. Results showed that a large increase in the allowable step size may be used with increased accuracy over the Euler method without the need for additional simulations.

Experimental optimization has been used since the early 20th century to help farmers maximize yields (defining inputs such as water and fertilizer). The traditional approach iterates in cycles consisting of fitting a polynomial to samples (that differed in the set of input variables) and optimizing the fitted surrogate. In each cycle, a set of designs is defined and tested. Although engineering design relies mostly on computer experiments, there are cases where simulations are expensive enough and the system is cheap enough to manufacture and test to favor experimental over analytical optimization. In this paper, we use the design of a paper helicopter to illustrate how we can adapt the modern efficient global optimization (EGO) algorithm to handle experimental data. The objective is to maximize the time a simple paper helicopter takes to fall from a specific height. We propose running EGO with multiple surrogates (MSEGO) for generating not only one, but multiple candidate designs per optimization cycle. Here, we use kriging, radial basis neural network, linear Shepard, and support vector regression. We also heavily penalize regions of the design space where designs are predicted to fail, using support vector classification to define the failure region. We found MSEGO reduced the impact of failed designs, allowed for exploration of the design space, and improved the fall time by 10% .

In surrogate-based optimization, each cycle consists of carrying out a number of simulations, fitting a surrogate, performing optimization based on the surrogate, and finally running exact simulation at the candidate solutions. Adaptive sampling algorithms that add one point per cycle are readily available. For example, the efficient global optimization (EGO) algorithm uses the kriging prediction and uncertainty estimate to guide the selection of the next sampling point. However, the addition of one point at a time may not be efficient when it is possible to run simulations in parallel. Additionally, the extension to include multiple points per cycle turns out to be either limited or computationally challenging. We propose an algorithm for adding several points per optimization cycle based on approximated computation of the probability of improvement. We assume that the probabilities at different points of the design space are independent from each other. The approach was tested on three analytic examples. For these examples we compare our approach with traditional sequential optimization based on kriging. We found that indeed our approach was able to deliver better results in a fraction of the optimization cycles needed by the traditional kriging implementation.

Load stiffening ability in the flight load direction and complaint nature in the opposite direction are important structural behaviors of the bendable unmanned air vehicle (UAV) wing, developed at the University of Florida. The complaint nature of the wing enables UAV storage inside smaller packing volumes. The present paper discusses deterministic and probabilistic design optimization of a bendable wing having 24 inch span and 7 inch root chord. In deterministic optimization, the wing shape definition parameters and the layup scheme used to manufacture the wing are treated as design variables. Aerodynamic and structural performances of the wing are studied to determine wing aerodynamic efficiency and a maximum flying velocity that the wing can withstand without buckling or failing under flight loads. The Tsai-Wu criterion is used to check failure of the wing due to rolling stresses to determine minimum safe storage diameter. Multidisciplinary (aerodynamics and structural) shape and layup optimization is performed using an elitist non-dominated sorting genetic algorithm: NSGA-II. Simultaneous maximization of wing aerodynamic efficiency and maximum flying velocity a wing can withstand without in-flight buckling are used as two design objectives. The design points on the deterministic Pareto optimal front achieved are compared with a baseline design to observe some designs with improved performance. A mixed approach is tried during probabilistic optimization. The design variable and random parameter uncertainties are included explicitly and a modeling uncertainty is included through a factor (similar to factor of safety). We found that when uncertainties in modeling, design variables and random parameters are taken into account, deterministic optimization designs show high probability of failure. In the probabilistic design optimization a single constraint of satisfying target probability of failure is used. To reduce computational cost, we use surrogate models for objectives and constraints. After several surrogates were evaluated for each of the objectives and the constraints, the best surrogates based on cross validation or RMSE were selected for the probabilistic optimization work. Finally, the effect of the target probability of failure on the Pareto front was studied. It was observed that as the target probability of failure becomes stringent, all of the Pareto front designs use almost the same leading edge curvature indicating the original storage diameter constraint drives the designs when higher reliability is desired. By relaxing the target probability of failure, stiffer wings can be obtained. Some designs with acceptable probability of failure are identified which have higher aerodynamic efficiency than a baseline wing having higher probability of failure.

Surrogate-based optimization has become popular in the design of complex engineering systems. Each optimization cycle consists of analyzing a number of designs, fitting a surrogate, performing optimization based on the surrogate, and finally performing exact simulation at the design obtained by the optimization. Adaptive sampling algorithms that add one point per cycle are readily available in the literature. They use uncertainty estimators to guide the selection of the next sampling point(s). The addition of one point at a time may not be efficient when it is possible to run simulations in parallel. So we propose an algorithm for adding several points per optimization cycle based on the simultaneous use of multiple surrogates. The need for uncertainty estimates usually limits adaptive sampling algorithms to surrogates such as kriging and polynomial response surface because of the lack of uncertainty estimates in the implementation of other surrogates. We import uncertainty estimates from surrogates having such estimates to use with other surrogates such as support vector regression models. The approach was tested on two analytic examples for nine basic surrogates including kriging, radial basis neural networks, linear Shepard and support vector regression. For these examples we compare our approach with traditional sequential optimization based on kriging. We found that our approach was able to deliver better results in a fraction of the optimization cycles needed by the traditional kriging implementation.

In adaptive sampling and optimization methods, the uncertainty estimators are used to guide the selection of the next sampling point(s). These algorithms often limit themselves to surrogates such as kriging and polynomial response surface because of the lack of uncertainty estimates in the implementation of other surrogates. We propose the importation (borrowing) of uncertainty estimates from one surrogate to another. This would allow the use of support vector regression models together with kriging uncertainty estimates, for example. When multiple surrogates are available, we also propose using crossvalidation to aid in the decision of which surrogate to import from. The approach was tested on two algebraic examples for ten basic surrogates including different instances of kriging, polynomial response surface, radial basis neural networks and support vector regression surrogates. For these examples we found that (i) the statistically based uncertainty estimates do not always correlate well to the errors; (ii) importation of uncertainty structure can offer a reasonable solution; and (iii) cross-validation successfully selects the surrogate according to the quality of the uncertainty structure and therefore it is useful in choosing which surrogate to import the uncertainty estimate from.

The use of metamodeling techniques in the design and analysis of computer experiments has progressed remarkably in the past two decades, but how far have we really come? This is the question that we investigate in this paper, namely, the extent to which the use of metamodeling techniques in multidisciplinary design optimization have evolved in the two decades since the seminal paper on Design and Analysis of Computer Experiments by Sacks et al. As part of this review, we examine the motivation for advancements in metamodeling techniques from both a historical perspective and the research itself. Based on current thrusts in the field, we emphasize multi-level/multi-fidelity approximations and ensembles of metamodels, as well as the availability of metamodels within commercial software and for design space exploration and visualization in this review. Our closing remarks offer insight into future research directions – nearly the same ones that have motivated us in the past.

The dielectric barrier discharge (DBD) plasma actuator offers promising opportunities for flow control because it does not require mass injection and involves no moving mechanical components. In order to gain better understanding of the impact of the materials and operational parameters on the performance of the DBD actuator, and to facilitate design of effective control schemes, the surrogate modeling technique is adopted. The model is established based on three design variables, namely (i) frequency of the applied voltage, (ii) dielectric constant of the insulator, and (iii) polarity (positive/negative) time ratio of the applied waveform, and focuses on two objectives, namely, (i) net force generated, and (ii) power requirement. The 2-species fluid plasma model with helium as a working gas is used in the computational model to generate the data needed by the surrogate model. Multiple surrogate models are compared to enhance the robustness of the surrogate performance. There exist multiple Pareto fronts where the x-directional force is positive with relatively low power and negative with high power respectively. Global sensitivity analysis indicates that the frequency of the applied voltage is important for the actuator performance in one region whereas the time ratio of the applied waveform is in the other, while the dielectric constant is always important. The performance dependency on variables also differs significantly according to the different regions.

This paper presents an inverse procedure for the determination of external loads, given the dynamic responses of the loaded structure and its corresponding finite element model. The influence of the stress-stiffening effect on the dynamic characteristics of structural systems is used to establish a relation between the dynamical responses and the applied external loading. An optimization problem is formulated in which the objective function represents the difference between the measured modal characteristics of the loaded structure and their FE counterparts. The loading parameters (magnitude, position and direction), assumed as being unknown, are considered as design variables. The identification procedure is illustrated by means of numerical simulations, in which the identification problem is solved by using the heuristic named Particle Swarm Optmization together with the Lagrange-Newton SQP (Sequential Quadratic Programming) method.