Non-convex Design Space Research Articles

Robust design modeling and optimization for dealing with a non-convex design space

Classic experimental designs effectively generate experimental design points under standard design spaces such as a circle, a cube, or a sphere. However, for non-convex design spaces, traditional experimental designs are not appropriate due to nonstandard experimental design conditions. No attention has been given to developing an algorithm to generate design points for a non-convex design space in the literature. This paper is three-fold. One, this paper presents a novel exchange algorithm based on an inner approximation approach in order to construct optimal design points where the design space is non-convex. Besides, the proposed algorithm efficiently produces experimental design points for non-convex design space. In addition, it is provided a fair comparison of the proposed algorithm performance. Two, the fitted response functions of the mean, standard deviation, and variance are found using the proposed regression model construction technique where the design space is non-convex. Three, a selected optimal design concept-based robust design optimization model is developed to find Pareto optimal solution settings of chosen design factors for the non-convex design space. Finally, an Industry 4.0 concept-based numerical example is provided to illustrate how to apply the proposed methodology effectively for the non-convex design space.

Bayesian topology optimization for efficient design of origami folding structures

Bayesian optimization (BO) is a popular method for solving optimization problems involving expensive objective functions. Although BO has been applied across various fields, its use in structural optimization area is in its early stages. Origami folding structures provide a complex design space where the use of an efficient optimizer is critical. In this work for the first time we demonstrate the ability of BO to solve origami-inspired design problems. We use a Gaussian process (GP) as the surrogate model that is trained to mimic the response of the expensive finite element (FE) objective function. The ability of this BO-FE framework to find optimal designs is verified by applying it to well-known origami design problems. We compare the performance of the proposed approach to traditional gradient-based optimization techniques and genetic algorithm methods in terms of ability to discover designs and computational efficiency. BO has many user-defined components/parameters and intuitions for these for structural optimization are currently limited. In this work, we study the role of hyperparameter tuning and the sensitivity of Bayesian optimization to the quality and size of the initial training set. Taking a holistic view of the computational expense, we propose various heuristic approaches to reduce the overall cost of optimization. Our results show that Bayesian optimization is an efficient alternative to traditional methods. It allows for the discovery of optimal designs using fewer finite element solutions, which makes it an attractive choice for the non-convex design space of origami fold mechanics.

Rules for trajectory updating in decision based design

Optimization is an indispensable tool in decision based design (DBD). There is an emerging consensus in DBD that the only correct objective to maximize is the decision maker's utility from a design. However, a utility function assessed a priori may not capture the preferences of the decision maker over the entire design space. As a result, when the optimizer searches for the optimal design, it may traverse (or end up) in regions in the design space where the preference order among different solutions is different from that used to build the utility function. For a highly non-convex design space, this can lead to convergence to a grossly suboptimal design depending on where we start the solution search. In this article, we propose two approaches to address this issue. First, we track the trajectory of the solution as generated by the optimizer and generate ranking questions that are presented to the designer to verify the correctness of the utility function, and second we propose backtracking rules if a local utility function is very different from the initially assessed function. We demonstrate our methodology using a mathematical example and a welded-beam design problem.

On the Existence of Non-Convexities in the Design Space of Isolated Wind-Battery Systems

A set of all feasible design options depicted graphically on a diagram with system design variables is known as the design space. For an isolated wind-battery system, the design space is depicted on a rotor diameter vs. generator rating diagram with battery bank size as a parameter. Design space representation enables the distinction between the feasible and infeasible combinations of design variables. It forms a basis to understand the inter-relation between different design variables outlining the tradeoffs in design. In certain cases, it is observed that the feasible solution set may lead to a non-convex design space. Non-convexities in the design space imply that intuitive selection or arbitrary over sizing of design parameter(s) may lead to non-compliance with respect to overall reliability requirement. Through an illustrative example, it is shown that the primary reasons leading to design space non-convexity are the uneven distribution of wind speed standard deviation in the time horizon, high overall system reliability requirement and the wind regime of the site under consideration. From a graph of system reliability vs. coefficient of variation of wind speed (COV), it is observed that non-convexity in design space is most likely to occur for particular combinations of system reliability requirement and COV of wind speed. This representation facilitates predicting the occurrence of non-convexity in the design envelope prior to detailed system design and optimization; thus providing valuable insight to the system designer and preventing a possible failure of performance and loss of investment.

A Hybrid Particle Swarm-Gradient Algorithm for Global Structural Optimization

: The particle swarm optimization (PSO) method is an instance of a successful application of the philosophy of bounded rationality and decentralized decision making for solving global optimization problems. A number of advantages with respect to other evolutionary algorithms are attributed to PSO making it a prospective candidate for optimum structural design. The PSO-based algorithm is robust and well suited to handle nonlinear, nonconvex design spaces with discontinuities, exhibiting fast convergence characteristics. Furthermore, hybrid algorithms can exploit the advantages of the PSO and gradient methods. This article presents in detail the basic concepts and implementation of an enhanced PSO algorithm combined with a gradient-based quasi-Newton sequential quadratic programming (SQP) method for handling structural optimization problems. The proposed PSO is shown to explore the design space thoroughly and to detect the neighborhood of the global optimum. Then the mathematical optimizer, starting from the best estimate of the PSO and using gradient information, accelerates convergence toward the global optimum. A nonlinear weight update rule for PSO and a simple, yet effective, constraint handling technique for structural optimization are also proposed. The performance, the functionality, and the effect of different setting parameters are studied. The effectiveness of the approach is illustrated in some benchmark structural optimization problems. The numerical results confirm the ability of the proposed methodology to find better optimal solutions for structural optimization problems than other optimization algorithms.

Structural optimization using biological variables to help understand how bones design holes

Bones do not normally fracture near blood vessel holes. This in part may be related to spatial distribution of porosity, mineralization, and microstructure observed near the holes, as these parameters are known to affect bone stiffness and strength. Material parameters representing physical variables are optimized in this paper using a simple two-dimensional model of the bone in an attempt to duplicate the observed material distributions. The objective is to minimize the failure index defined (stress-strength ratio) under a given uniaxial load. In addition to tailoring the elastic properties (by changing porosity and mineral density), and orthotropy (via optimizing material orientation) the study also investigates the effect of the shape of the hole. Of the three variables, shape was the most effective in minimizing the failure index, followed by tailoring of the elastic properties and lastly material orientation. The elastic modulus distribution in the optimum design was similar to that observed around foramen holes in bones. The contributions of the physical variables in minimizing the failure index were almost independent of each other. The optimization was able to reduce the failure index compared to a hole in a homogeneous plate but was not sufficient to prevent failure at the hole. This can be attributed to the limitations of the shape functions used to describe the spatial distribution of mineralization and volume fraction and limitations in models used to relate composition and microstructure to elastic properties. The shape functions used to describe the spatial distribution of mineral mass fraction and volume fraction and the constraints imposed on them resulted in a non-convex design space.

High-Speed Civil Transport Design Space Exploration Using Aerodynamic Response Surface Approximations

A method has been developed to generate and use polynomial approximations to the range and cruise drag components in a highly constrained, multidisciplinary design optimization of a high-speed civil transport (HSCT) cone guration. The method improves optimization performance by eliminating the numerical noise present in the analyses through the use of response surface methodology. In this implementation quadratic polynomials are e t within variable bounds to data gathered from a series of numerical analyses of different aircraft designs. Because the HSCT optimization process contains noise and suffers from a nonconvex design space even when noise is e ltered out, multiple optimization runs are performed from different starting points with and without the response surface models in order to evaluate both their effectiveness as surrogate functions and as a design exploration tool. The alternative method used is variable complexity modeling (VCM). It is shown that response surface methodology facilitates design space exploration, allowing improvements in terms of both convergence performance and computational effort when multiple starting points are required, although using VCM usually produces better e nal designs.

A knowledge-based search framework for designing composite structures

In this paper, a search framework is presented that makes a genetic algorithm (GA) a practical engineering tool for searching discrete and nonconvex design spaces. It is shown that “near optimal” designs can be relatively quickly found if three problems are addressed: (i) the dimensionality of a design space is reasonably reduced, (ii) fast function evaluations are available, and (iii) genetic operators are tailored to the application at hand. In our work, the number of active design variables is reduced by using a rule-based fuzzy logic inference engine to process knowledge about the structure's behaviour. The use of a parallel computing environment for structural analysis in conjunction with data retrieval from binary-like trees effectively reduces the time required to perform genetic searching. Three different selection operators along with a permutation-like operator for the reduced problem are evaluated for their effectiveness in locating the global optimum. As an example, the stacking sequence for maximizing a buckling load for a stiffened square panel with a circular cut-out and subject to thermal and shear loading is designed.

Design of structure/control systems with transient response constraints exhibiting relative minima

Structural optimization problems involving dynamic behaviour constraints often exhibit nonconvex design spaces. The direct application of a global optimization algorithm requires a large number of function evaluations which in turn require a large number of dynamic structural analyses. This work presents a strategy aimed at finding the global optimum for problems with transient dynamic behaviour constraints based on approximation concepts. The method consists of generating and solving a sequence of approximate problems using a global optimizer. The approximations are explicit and capture most of the inherent nonconvexity of the exact functions. A simple example. problem is presented to illustrate the procedure set forth.

A basis for local minima of a symmetric frame topography

Optimization of indeterminate frames involves non-convex stress constraint functions which form a non-convex design space that results in multiple local minima, unknown in number to begin with. The current state-of-the-art in this field recommends employment of different arbitrary initial designs to have a reasonable assurance of reaching every local minimum which, under a particular load, is associated with a specific material distribution. In this study a basis is laid down for (a) determination, at the very outset, of the number of local minima that possibly may exist in a frame of symmetric topography, (b) definition, in terms of relative member stiffnesses, of the material distribution pattern associated with each local minimum, and (c) a guideline for evaluation of member forces for initial proportioning of the pattern, The application of the proposed method for evaluation of local minima is illustrated with an example problem.

Optimum Design With Multimodal SimulatedAnnealing Strategies

Strategies for locating multiple relative optima in a multimodal optimization problem using multiple-point simulated annealing are proposed. Two strategies incorporated with multiple-point simulated annealing are developed to increase the capabilities of locating more relative optima in a nonconvex design space. Balance strategy involves the use of a balance function that evaluates the degree of design spreading over the entire design space and create a corresponding direction bias in the subsequent design-change process. Bounce strategy evaluates the degree of design crowding and aims to push designs away from the crowd center by creating a direction bias in the next design-change process. Both strategies are evaluated on a number of illustrative multimodal problems.

Iterative Design and Structural Optimization

Methods of automatic design of structures are viewed, proposed, and evaluated by a numerical example. Sensitivity coefficients are introduced to help predict the consequences of design changes. Simultaneous linear equations are used to suggest new designs. Linear programming is introduced to generate optimum designs. Non-fully stressed designs and non-convex design spaces are examined. The effect of the choice of the design variables is demonstrated. A number of illustrative examples are included.