Active Subspaces of Airfoil Shape Parameterizations

Global sensitivity analysis and uncertainty quantification of soot formation in an n-dodecane spray flame

Aerodynamic shape optimization plays a fundamental role in aircraft design. However, useful parameterizations of shapes for engineering models often result in high-dimensional design spaces which can create challenges for both local and global optimizers. In this paper, we employ an active subspace method (ASM) to discover and exploit low-dimensional, monotonic trends in the quantity of interest as a function of the design variables. The trend enables us to eciently and eectively nd an optimal design in appropriate areas of the design space. We apply this approach to the ONERA-M6 transonic wing, parameterized with 50 Free-Form Deformation (FFD) design variables. Given an initial set of 300 designs, the ASM discovered a low-dimensional linear subspace of the input space that explained the majority of the variability in the drag and lift coecients. This revealed a global trend that we exploited to nd an optimal design with reduced computational cost.

The problems of interest are where the objective function must be evaluated using sampling techniques. The sampled function values are used to build an approximation to the response surface over the product space of uncertainties and design variables. This approximation is successively used in the integration over the space of uncertainties and in the optimization (minimization) over the space of design variables. A surrogate-based trust region method is used to determine a succession of design points for optimization

Derivative-informed projected neural networks for high-dimensional parametric maps governed by PDEs

Design candidates obtained from optimization techniques may have meaningful information, which provides not only the best solution, but also a relationship between object functions and design variables. In particular, trade-off studies for optimum airfoil shape design involving various objectives and design variables require the effective analysis tool to take into account a complexity between objectives and design variables. In this study, for the multiple-conflicting objectives that need to be simultaneously fulfilled, the real-coded Adaptive Range Multi-Objective Genetic Algorithm code, which represents the global and stochastic multi-objective evolutionary algorithm, was developed for an airfoil shape design. Furthermore, the PARSEC method reflecting geometrical properties of airfoil is adopted to generate airfoil shapes. In addition, the Self-Organizing Maps, based on the neural network, are used to visualize trade-offs of a relationship between the objective function space and the design variable space obtained by evolutionary computation. The Self-Organizing Maps that can be considered as data mining of the engineering design generate clusters of object functions and design variables as an essential role of trade-off studies. The aerodynamic data for all candidate airfoils is obtained through Computational Fluid Dynamics. Lastly, the relationship between the maximum lift coefficient and maximum lift-to-drag ratio as object functions and 12 airfoil design parameters based on the PARSEC method is investigated using the Self-Organizing Maps method.

Preliminary design information is characteristically imprecise or fuzzy: specifications and requirements are subject to change, and the design description is vague and incomplete. The Method of Imprecision uses the mathematics of fuzzy sets to explicitly represent and manipulate imprecise preliminary design information, enabling the designer to better understand the full range of designs and performances that satisfy an imprecise set of specifications and requirements. This paper discusses the foundations of this methodology, and introduces pragmatic extensions that provide computationally tractable methods to map design imprecision from multiple design variables onto multiple performance variables. These methods attempt to minimize the number of function evaluations required while retaining an appropriate level of accuracy. This is achieved by using optimization to obtain extremal points for each performance variable and selectively applying a linear approximation for the mapping from the design variable space (DVS) to the performance variable space (PVS) to interpolate between extremal points. This linear approximation is constructed using regression techniques adapted from experiment design.

The structural synthesis problem, which is currently viewed as an inequality constrained minimization problem in design variable space, is transformed into an unconstrained minimization problem in a space such that each point represents a set of values for the design variables as well as for each behavior variable in each load condition. Using penalty functions and a trial value of the optimum weight (W0)^ rather than slack variables and Lagrange multipliers, a positive function i/' is constructed such that any point where \[/ = 0 represents an acceptable design of weight W0 or less. This integrated formulation is used to seek optimum designs by systematically reducing W0, thus generating a sequence of acceptable designs with decreasing weight. Numerical examples of optimum designs, based on a nonlinear analysis for a one node m-bar planar truss, are presented together with computer running times. These results are obtained using a steep-descent-type procedure in conjunction with the \//function formulation. The results indicate that the integrated approach offers the prospect of making substantial improvements in the efficiency of the structural synthesis process, particularly when linearization of the structural analysis is inappropriate.

This paper is concerned with the numerical implementation of the boundary element method for shape optimal design of two-dimensional linear elastic structures. The design objective is to minimize the structural compliance, subject to an area constraint. Sensitivities of objective and constraint functions, derived by means of Lagrangean approach and the material derivative concept with an adjoint variable technique, are computed through analytical expressions that arise from optimality conditions. The boundary element method, used for the discretization of the state problem, applies the stress boundary integral equation for collocation on the design boundary and the displacement boundary integral equation for collocation on other boundaries. The use of the stress boundary integral equation, discretized with discontinuous quadratic elements, allows an efficient and accurate computation of stresses on the design boundary. This discretization strategy not only automatically satisfies the necessary conditions for the existence of the finite-part integrals, which occur naturally in the stress boundary integral equation, but also circumvents the problem of collocation at kinks and corners. The perturbation field is described with linear continuous elements, in which the position of each node is defined by a design variable. Continuity along the design boundary is assured by forcing the end points of each discontinuous boundary element to be coincident with a design node. The optimization problem is solved by the modified method of feasible directions available in the program ADS. Examples of a plate with a hole are analyzed with the present method, for different loading conditions. The accuracy and efficiency of the implementation described herein make this formulation ideal for the study of shape optimal design of structures.

This paper is concerned with the numerical implementation of the dual boundary element method for shape optimal design of two-dimensional linear elastic structures. The design objective is to minimize the structural compliance, subject to an area constraint. Sensitivities of objective and constraint functions, derived by means of Lagrangean approach and the material derivative concept with an adjoint variable technique, are computed through analytical expressions that arise from optimality conditions. The dual boundary element method, used for the discretization of the state problem, applies the stress equation for collocation on the design boundary and the displacement equation for collocation on other boundaries. The use of the stress boundary integral equation, discretized with discontinuous quadratic elements, allows an efficient and accurate computation of stresses on the design boundary. The perturbation field is described with linear continuous elements, in which the position of each node is defined by a design variable. Continuity along the design boundary is assured by forcing the end points of each discontinuous boundary element to be coincident with a design node. The optimization problem is solved by the modified method of feasible directions available in the program ADS. Examples of a plate with a hole are analyzed with the present method, for different loading conditions. The accuracy and efficiency of the implementation described herein make this formulation ideal for the study of shape optimal design of structures.

In this paper, we present a novel approach to geometry parameterization that we apply to the design of mixing elements for single-screw extruders. The approach uses neural networks of a specific architecture to automatically learn an appropriate parameterization. This stands in contrast to the so far common user-defined parameterizations. Geometry parameterization is crucial in enabling efficient shape optimization as it allows for optimizing complex shapes using only a few design variables. Recent approaches often utilize computer-aided design (CAD) data in conjunction with spline-based methods where the spline’s control points serve as design variables. Consequently, these approaches rely on the design variables specified by the human designer. This approach results in a significant amount of manual tuning to define a suitable parameterization. In addition, despite this effort, many times the optimization space is often limited to shapes in close proximity to the initial shape. In particular, topological changes are usually not feasible. In this work, we propose a method that circumvents this dilemma by providing low-dimensional, yet flexible shape parametrization using a neural network, which is independent of any computational mesh or analysis methods. Using the neural network for the geometry parameterization extends state-of-the-art methods in that the resulting design space is not restricted to user-prescribed modifications of certain basis shapes. Instead, within the same optimization space, we can interpolate between and explore seemingly unrelated designs. To show the performance of this new approach, we integrate the developed shape parameterization into our numerical design framework for dynamic mixing elements in plastics’ extrusion. Finally, we challenge the novel method in a competitive setting against current free-form deformation-based approaches and demonstrate the method’s performance even at this early stage.

Global Sensitivity Analysis (GSA) is the study of the influence of any given inputs on the outputs of a model. In the context of engineering design, GSA has been widely used to understand both individual and collective contributions of design variables on the design objectives. So far, global sensitivity studies have often been limited to design spaces with only quantitative (numerical) design variables. However, many engineering systems also contain, if not only, qualitative (categorical) design variables in addition to quantitative design variables. In this paper, we integrate the novel Latent Variable Gaussian Process (LVGP) with Sobol’ analysis to develop the first metamodel-based mixed-variable GSA method. Through two analytical case studies, we first validate and demonstrate the effectiveness of our proposed method for mixed-variable problems. Furthermore, while the new metamodel-based mixed-variable GSA method can benefit various engineering design applications, we employ our method with multi-objective Bayesian optimization (BO) to accelerate the Pareto front design exploration in many-level combinatorial design spaces. Specifically, we implement a sensitivity-aware design framework for metal-organic framework (MOF) materials that are constructed only from qualitative design variables and show the benefits of our method for expediting the exploration of novel MOF candidates from a many-level large combinatorial design space.

To determine the optimal design of the rail profile for a curved segment of a heavy haul railway, an optimization method that considers the extent of rail wear over the entire design cycle was developed. The approach is based on wheel–rail rolling contact theory, vehicle–track dynamic theory and nonlinear programming theory. The changes in the rail profile were analyzed using an updating strategy that considers random irregularities in the railway track. The minimum average metal loss from the vehicle in terms of wear and grinding for the whole design cycle was regarded as the design objective, and discrete point coordinates on the rail were regarded as the design variables. A radial basis function was used to establish an approximate model, which formed an explicit function between design variables and design objectives. The optimization model was established on this basis, and a genetic algorithm was used to solve the optimization model for the optimized profile. The maximum passing gross load for the whole design cycle reached 150.74 x106 t, an increase of 139.73% compared with the standard profile, which achieved the purpose of prolonging the service life of the rail.

To solve structural optimization problems, it is necessary to integrate a structural analysis package and an optimization package. Since most structural analysis packages suffer from closeness of system, it is very difficult to integrate it with an optimization package. To overcome the difficulty, we propose a possible alternative, DAMDO, which integrate Design, Analysis, Modeling, Definition, and Optimization phases into an integration environment as follows. (1) Design first generate many possible structural design alternatives. Each design alternative consists of many design variables X. (2) Analysis employ the structural analysis software to analyze all structural design alternatives to obtain their internal forces and displacements. They are the response variables Y. (3) Modeling employ artificial neural networks to build model Y = f(X) to obtain the relationship functions between the design variables X and the response variables Y. (4) Definition employ the design variables X and the response variables Y to define the objective function and constraint functions. (5) Optimization employ the optimization software to solve the optimization problem consisting of the objective function and the constraint functions to produce the optimum design variables X*. Optimization of truss structures was used to validate the DAMDO approach. The empirical results show that the truss optimization problems can be solved by the DAMDO approach, which employ neural networks to integrate the structural analysis package and optimization package without requiring direct integration of the two packages. This approach is promising in many engineering optimization domains which need to couple an analysis package and an optimization one to obtain the optimum solutions.

Global Sensitivity Analysis (GSA) is the study of the influence of any given input on the outputs of a model. In the context of engineering design, GSA has been widely used to understand both individual and collective contributions of design variables on the design objectives. So far, global sensitivity studies have often been limited to design spaces with only quantitative (numerical) design variables. However, many engineering systems also contain, if not only, qualitative (categorical) design variables in addition to quantitative design variables. In this paper, we integrate Latent Variable Gaussian Process (LVGP) with Sobol’ analysis to develop the first metamodel-based mixed-variable GSA method. Through numerical case studies, we validate and demonstrate the effectiveness of our proposed method for mixed-variable problems. Furthermore, while the proposed GSA method is general enough to benefit various engineering design applications, we integrate it with multi-objective Bayesian optimization (BO) to create a sensitivity-aware design framework in accelerating the Pareto front design exploration for metal-organic framework (MOF) materials with many-level combinatorial design spaces. Although MOFs are constructed only from qualitative variables that are notoriously difficult to design, our method can utilize sensitivity analysis to navigate the optimization in the many-level large combinatorial design space, greatly expediting the exploration of novel MOF candidates.

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.