Sirisha Rangavajhala

A probabilistic approach for representation of interval uncertainty

In this paper, we propose a probabilistic approach to represent interval data for input variables in reliability and uncertainty analysis problems, using flexible families of continuous Johnson distributions. Such a probabilistic representation of interval data facilitates a unified framework for handling aleatory and epistemic uncertainty. For fitting probability distributions, methods such as moment matching are commonly used in the literature. However, unlike point data where single estimates for the moments of data can be calculated, moments of interval data can only be computed in terms of upper and lower bounds. Finding bounds on the moments of interval data has been generally considered an NP-hard problem because it includes a search among the combinations of multiple values of the variables, including interval endpoints. In this paper, we present efficient algorithms based on continuous optimization to find the bounds on second and higher moments of interval data. With numerical examples, we show that the proposed bounding algorithms are scalable in polynomial time with respect to increasing number of intervals. Using the bounds on moments computed using the proposed approach, we fit a family of Johnson distributions to interval data. Furthermore, using an optimization approach based on percentiles, we find the bounding envelopes of the family of distributions, termed as a Johnson p-box. The idea of bounding envelopes for the family of Johnson distributions is analogous to the notion of empirical p-box in the literature. Several sets of interval data with different numbers of intervals and type of overlap are presented to demonstrate the proposed methods. As against the computationally expensive nested analysis that is typically required in the presence of interval variables, the proposed probabilistic representation enables inexpensive optimization-based strategies to estimate bounds on an output quantity of interest.

Discretization Error Estimation in Multidisciplinary Simulations

This paper proposesmethods to estimate the discretization error in the system output of coupledmultidisciplinary simulations. In such systems, the governing equations for each discipline are numerically solved by a different computational code, and each discipline has different mesh size parameters. A classic example of multidisciplinary analysis involves fluid–structure interaction, where the element sizes in fluid and structure meshes are typically different. The general case of three-dimensional steady-state problems is considered in the current paper, where mesh refinement is possible in all three spatial directions for each discipline. Two aspects of discretization error, which are of interest in multidisciplinary analysis, are considered: disciplinary mesh sizes and the mismatch of disciplinary meshes at the interface at which boundary conditions are exchanged. Two alternate representations for the discretization error for the previously specified generic case are presented: 1) ignoring mesh mismatch at the interface and 2) considering mesh mismatch at the interface. Polynomial, rational function, and Gaussian process error models are used to represent the discretization error. The proposed error models are illustrated using a threedimensional fluid–structure interaction problem of an aircraft wing.

Joint Probability Formulation for Multiobjective Optimization Under Uncertainty

This paper presents a new approach to solve multiobjective optimization problems under uncertainty. Unlike the existing state-of-the-art, where means/variances of the objectives are used to ensure optimality, we employ a distributional formulation. The proposed formulations are based on joint probability, i.e., probability that all objectives are simultaneously bound by certain design thresholds under uncertainty. For minimization problems, these thresholds can be viewed as the desired upper bounds on the individual objectives. The tradeoffs are illustrated using the so-called decision surface, which is the surface of maximized joint probabilities for a set of design thresholds. Two optimization formulations to generate the decision surface are proposed, which provide the designer with the distinguishing capability that is not available in the existing literature, namely, decision making under uncertainty, while ensuring joint objective performance: (1) Maximum probability design: Given a set of thresholds (preferences within each objective), find a design that maximizes the joint probability while using a probabilistic aggregation as against an ambiguous weight-based method. (2) Optimum threshold design: Given a designer-specified joint probability, find a set of thresholds that satisfy the joint probability specification while allowing for a specification of preferences among the objectives.

Design Optimization under Aleatory and Epistemic Uncertainties

This paper presents a design optimization methodology under three sources of uncertainty: physical variability (aleatory); data uncertainty (epistemic) due to sparse or imprecise data; and model uncertainty (epistemic) due to modeling errors/approximations. A likelihood-based method is use to fuse multiple formats of information, and a non-parametric probability density function (PDF) is constructed. Two types of model errors are considered: model form error and numerical solution error, each of which is a function of the design variables that are changing at each iteration of the optimization. Gaussian process (GP) surrogate models are constructed for efficient computation of model errors in the optimization. The treatment in this paper yields a distribution of the output that accounts for various sources of uncertainty. The use of a probabilistic approach to include both aleatory and epistemic uncertainties allows for their efficient integration into the optimization framework. The proposed methods are illustrated using a three-dimensional wing design problem involving fluid-structure interaction analysis.

Uncertainty Visualization in Multiobjective Robust Design Optimization

Visualization of the solution of multiobjective problems can be challenging because of high problem dimensionality. In multiobjective robust design optimization problems, additional objective functions, e.g., those related to objective robustness and constraint satisfaction, are typically introduced into the problem. These additional objectives add to the existing high dimensionality of the original problem, making visualization more challenging than deterministic problems. An effective visualization of the relevant information in such problems can significantly aid decision making under uncertainty. We observe that each Pareto optimal solution in multiobjective RDO problems has three uncertainty attributes that can aid the designer in decision making: (1) mean objective performance, (2) variation in performance, and (3) constraint satisfaction. In this paper, we propose a visualization scheme that presents uncertainty information in terms of the above three attributes graphically. The multiobjective problem under uncertainty is first solved to obtain a set of Pareto solutions. Using designer requirements in the above three attributes, a filtering scheme is used to extract relevant data, which is then plotted in the mean objective space. Based on this filtering, desirable regions of the mean objective space from an uncertainty perspective are identified. The proposed visualization scheme is illustrated with the help of a weld assembly design example.

Concurrent Optimization of Mesh Refinement and Design Parameters in Multidisciplinary Design

accuracy in terms of discretization error. Further discussed are the challenges that a design-optimization setting poses to the estimation of discretization error and how the ‘optimum’ mesh-refinement assessment is, in fact, nested within the design-optimization problem. The paper puts forth two significant contributions for multidisciplinary design-optimization formulations: 1) investigation of the impact of the so-called design inputs to discretization error in multidisciplinary design optimization, and 2) development of a concurrent optimization framework for simultaneous mesh refinement and design parameter optimization for multidisciplinary systems. The proposed method is illustrated using a simplified aircraft wing-design problem. I. Introduction I

A new approach to estimate discretization error for multidisciplinary and multidirectional mesh refinement

Discretization error estimation in the system output of multidisciplinary simulations, where each disciplinary simulation has multidirectional mesh refinement, is considered in this paper. In such systems, the governing equations for each discipline are numerically solved by a different computational code, and each discipline has different mesh size parameters. The general case of three-dimensional steady state problems is considered in the current paper. Two aspects of discretization error, that are of interest in multidisciplinary analysis, are considered: disciplinary mesh sizes, and the mismatch of disciplinary meshes at the interface at which boundary conditions are exchanged. Two alternate representations for discretization error for the above specified generic case are presented: (1) ignoring mesh mismatch at the interface, and (2) considering mesh mismatch at the interface. Polynomial, rational function, and Gaussian process error models are used to represent the discretization error. The proposed error models are illustrated using a three-dimensional fluid-structure interaction problem of an aircraft wing using ANSYS multifield module.

Representation and Propagation of both Probabilistic and Interval Uncertainty

This paper develops and illustrates a probabilistic approach for uncertainty representation and propagation in system analysis, when the information on the uncertain input variables and/or their distribution parameters may be available as either probability distributions or simply intervals (single or multiple). The uncertainty described by interval data is represented through a flexible family of probability distributions. Conversion of interval data to a probabilistic format enables the use of computationally efficient methods for probabilistic uncertainty propagation. Two methods are explored for the implementation of the proposed approach, based on: (1) sampling and (2) optimization. The sampling based strategy is more expensive and tends to underestimate the output bounds. The optimization based methodology improves both aspects.

Decision Making in Product Family Optimization Under Uncertainty

In this paper, we study the impact of design variable uncertainty on product family optimization. One of the important challenges in product family optimization is the selection of platform and scaling variables. This variable selection in deterministic problems is generally based on the designer’s relative preferences between two conflicting quantities of interest: commonality between products in the product family, and performance of the individual products. When uncertainty is considered in the design variables, and is propagated through the product family optimization problem, additional important considerations arise. These considerations affect not only the selection of platform and scaling variables, but also the feasibility of constraints under uncertainty and robustness of performance of the products. In this context, the proposed formulation can be used as an effective product family design tool under uncertainty. We demonstrate the proposed approach using a ten bar truss optimization problem.

Towards a Better Understanding of the Equality Constraints in Robust Design Optimization

Equality constraints in deterministic problems pose strict limitations on design feasibility because of the exactitude associated with such constraints. Equality constraints in robust design optimization (RDO) problems can be classified into two types: (1) those that must be satisfied regardless of uncertainty, examples include physics-based constraints, such as F = ma, and (2) those that cannot be satisfied because of uncertainty, which are typically designer-imposed, such as dimensional constraints. Our goal is to maintain design feasibility under uncertain conditions – to exactly satisfy physics based equality constraints, and to satisfy designer-imposed constraints exactly or as closely as possible. Whether or not a particular equality constraint can be exactly satisfied depends on the nature of the design variables that exist in the constraint. In this context, the contribution of this paper is two-fold. First, we present a rank-based matrix approach to interactively classify equality constraints into the above two types. Second, we present an approach to incorporate designer’s intra-constraint and inter-constraint preferences for designer-imposed constraints into the RDO formulation. Intra-constraint preference expresses how closely a designer wishes to satisfy a particular constraint, in terms of its mean and standard deviation. A designer may express inter-constraint preference if satisfaction of a particular designer-imposed constraint is more important than that of another. In other words, a designer might desire higher constraint satisfaction for some equality constraints, even if it is at the expense of lower constraint satisfaction for other equality constraints. The above discussed constraint satisfaction preferences give the designer the means to explore design space possibilities; and entail interesting implications in terms of decision making. An example is provided to illustrate the proposed approach.