John E. Renaud

Uncertainty quantification using evidence theory in multidisciplinary design optimization

Abstract Advances in computational performance have led to the development of large-scale simulation tools for design. Systems generated using such simulation tools can fail in service if the uncertainty of the simulation tools performance predictions is not accounted for. In this research an investigation of how uncertainty can be quantified in multidisciplinary systems analysis subject to epistemic uncertainty associated with the disciplinary design tools and input parameters is undertaken. Evidence theory is used to quantify uncertainty in terms of the uncertain measures of belief and plausibility. To illustrate the methodology, multidisciplinary analysis problems are introduced as an extension to the epistemic uncertainty challenge problems identified by Sandia National Laboratories. After uncertainty has been characterized mathematically the designer seeks the optimum design under uncertainty. The measures of uncertainty provided by evidence theory are discontinuous functions. Such non-smooth functions cannot be used in traditional gradient-based optimizers because the sensitivities of the uncertain measures are not properly defined. In this research surrogate models are used to represent the uncertain measures as continuous functions. A sequential approximate optimization approach is used to drive the optimization process. The methodology is illustrated in application to multidisciplinary example problems.

Trust Region Augmented Lagrangian Methods for Sequential Response Surface Approximation and Optimization

A common engineering practice is the use of approximation models in place of expensive computer simulations to drive a multidisciplinary design process based on nonlinear programming techniques. The use of approximation strategies is designed to reduce the number of detailed, costly computer simulations required during optimization while maintaining the pertinent features of the design problem. To date the primary focus of most approximate optimization strategies is that application of the method should lead to improved designs. This is a laudable attribute and certainly relevant for practicing designers. However to date few researchers have focused on the development of approximate optimization strategies that are assured of converging to a solution of the original problem. Recent works based on trust region model management strategies have shown promise in managing convergence in unconstrained approximate minimization. In this research we extend these well established notions from the literature on trust-region methods to manage the convergence of the more general approximate optimization problem where equality, inequality and variable bound constraints are present. The primary concern addressed in this study is how to manage the interaction between the optimization and the fidelity of the approximation models to ensure that the process converges to a solution of the original constrained design problem. Using a trust-region model management strategy, coupled with an augmented Lagrangian approach for constrained approximate optimization, one can show that the optimization process converges to a solution of the original problem. In this research an approximate optimization strategy is developed in which a cumulative response surface approximation of the augmented Lagrangian is sequentially optimized subject to a trust region constraint. Results for several test problems are presented in which convergence to a Karush-Kuhn-Tucker (KKT) point is observed.

Concurrent Subspace Optimization Using Design Variable Sharing in a Distributed Computing Environment

This paper reviews recent implementation advances and modifications in the continued development of a Concurrent Subspace Op timization (CSSO) algorithm for Multidisciplinary Design Optimization (MDO) The CSSO MDO algorithm implemented in this research incor porates a Coordination Procedure of System Approximation (CP-SA) for design updates This study also details the use of a new discipline based decomposition strategy which provides for design variable sharing across discipline design regimes (i e subspaces) A graphical user interface is developed which provides for menu driven execution of MDO algorithms and results display this new programming environment highlights the modularity of the CSSO algorithm The algorithm is implemented in a distributed computing environment using the graphical user interface providing for truly concurrent discipline design Implementation studies introduce two new multidisciplinary design test problems the optimal design of a high performance, low cost structural system and the preliminary sizing of a general aviation aircraft concept for optimal perfor mance Significant time savings are observed when using distributed computing for concurrent design across disciplines The use of design vari able sharing across disciplines does not introduce any difficulties in implementation as the design update in the CSSO MDO algorithm is gener ated in the CP-SA Application of the CSSO algorithm results in a considerable decrease in the number of system analyses required for optimization in both test problems More importantly for the fully coupled aircraft concept sizing problem a significant reduction in the number of individual contributing analyses is observed

Mechanical behavior of acrylonitrile butadiene styrene fused deposition materials modeling

Analytical/Computational models for the fused deposition (FD) material stiffness and strength as a function of mesostructural parameters are developed. Effective elastic moduli are obtained using the strength of materials approach and an elasticity approach based on the asymptotic theory of homogenization. Theoretical predictions for unidirectional FD‐acrylonitrile butadiene styrene materials are validated with experimentally determined values of moduli and strength. For moduli predictions, the results were found to be satisfactory with difference between experimental and theoretical values of less than 10 percent in most cases.

Decision-Based Collaborative Optimization

In this research a Collaborative Optimization (CO) approach for multidisciplinary systems design is used to develop a decision based design framework for non-deterministic optimization. To date CO strategies have been developed for use in application to deterministic systems design problems. In this research the decision based design (DBD) framework proposed by Hazelrigg [1,2] is modified for use in a collaborative optimization framework. The Hazelrigg framework as originally proposed provides a single level optimization strategy that combines engineering decisions with business decisions in a single level optimization. By transforming this framework for use in collaborative optimization one can decompose the business and engineering decision making processes. In the new multilevel framework of Decision Based Collaborative Optimization (DBCO) the business decisions are made at the system level. These business decisions result in a set of engineering performance targets that disciplinary engineering design teams seek to satisfy as part of subspace optimizations. The Decision Based Collaborative Optimization framework more accurately models the existing relationship between business and engineering in multidisciplinary systems design.

Mechanical behavior of acrylonitrile butadiene styrene (ABS) fused deposition materials. Experimental investigation

An experimental study of the mechanical behavior of fused‐deposition (FD) ABS plastic materials is described. Elastic moduli and strength values are determined for the ABS monofilament feedstock and various unidirectional FD‐ABS materials. The results show a reduction of 11 to 37 per cent in modulus and 22 to 57 per cent in strength for FD‐ABS materials relative to the ABS monofilament. These reductions occur due to the presence of voids and a loss of molecular orientation during the FD extrusion process. The results can be used to benchmark computational models for stiffness and strength as a function of the processing parameters for use in computationally optimizing the mechanical performance of FD‐ABS materials in functional applications.

Hybrid Variable Fidelity Optimization by Using a Kriging-Based Scaling Function

Solving design problems that rely on very complex and computationally expensive calculations using standard optimization methods might not be feasible given design cycle time constraints. Variable fidelity methods address this issue by using lower-fidelity models and a scaling function to approximate the higher-fidelity models in a provably convergent framework. In the past, scaling functions have mainly been either first-order multiplicative or additive corrections. These are being extended to second order. In this investigation variable metric approaches for calculating second-order scaling information are developed. A kriging-based scaling function is introduced to better approximate the high-fidelity response on a more global level. An adaptive hybrid method is also developed in this investigation. The adaptive hybrid method combines the additive and multiplicative approaches so that the designer does not have to determine which is more suitable prior to optimization. The methodologies developed in this research are compared to existing methods using two demonstration problems. The first problem is analytic, whereas the second involves the design of a supercritical high-lift airfoil. The results demonstrate that the krigingbased scaling methods improve computational expense by lowering the number of high-fidelity function calls required for convergence. The results also indicate the hybrid method is both robust and effective.

Implicit Uncertainty Propagation for Robust Collaborative Optimization

In this research we develop a mathematical construct for estimating uncertainties within the bilevel optimization framework of collaborative optimization. The collaborative optimization strategy employs decomposition techniques that decouple analysis tools in order to facilitate disciplinary autonomy and parallel execution. To ensure consistency of the physical artifact being designed, interdisciplinary consistency constraints are introduced at the system level. These constraints implicitly enforce multidisciplinary consistency when satisfied. The decomposition employed in collaborative optimization prevents the use of explicit propagation techniques for estimating uncertainties of system performance. In this investigation we develop and evaluate an implicit method for estimating system performance uncertainties within the collaborative optimization framework. The methodology accounts for both the uncertainty associated with design inputs and the uncertainty of performance predictions from other disciplinary simulation tools. These implicit uncertainty estimates are used as the basis for a new robust collaborative optimization (RCO) framework. The bilevel robust optimization strategy developed in this research provides for disciplinary autonomy in system design, while simultaneously accounting for performance uncertainties to ensure feasible robustness of the resulting system. The method is effective in locating a feasible robust optima in application studies involving a multidisciplinary aircraft concept sizing problem. The system-level consistency constraint formulation used in this investigation avoids the computational difficulties normally associated with convergence in collaborative optimization. The consistency constraints are formulated to have the inherent properties necessary for convergence of

Trust region model management in multidisciplinary design optimization

A common engineering practice is the use of approximation models in place of expensive computer simulations to drive a multidisciplinary design process based on nonlinear programming techniques. The use of approximation strategies is designed to reduce the number of detailed, costly computer simulations required during optimization while maintaining the pertinent features of the design problem. This paper overviews the current state of the art in model management strategies for approximate optimization. Model management strategies coordinate the interaction between the optimization and the fidelity of the approximation models so as to ensure that the process converges to a solution of the original design problem. Approximations play an important role in multidisciplinary design optimization (MDO) by offering system behavior information at a relatively low cost. Most approximate MDO strategies are sequential, in which an optimization of an approximate problem subject to design variable move limits is iteratively repeated until convergence. The move limits or trust region are imposed to restrict the optimization to regions of the design space in which the approximations provide meaningful information. In order to insure convergence of the sequence of approximate optimizations to a Karush–Kuhn–Tucker solution, a trust region model management or move limit strategy is required. In this paper recent developments in approximate MDO strategies and issues of trust region model management in MDO are reviewed.

Topology Optimization Using a Hybrid Cellular Automaton Method With Local Control Rules

The hybrid cellular automaton (HCA) algorithm is a methodology developed to simulate the process of structural adaptation in bones. This methodology incorporates a distributed control loop within a structure in which ideally localized sensor cells activate local processes of the formation and resorption of material. With a proper control strategy, this process drives the overall structure to an optimal configuration. The controllers developed in this investigation include two-position, proportional, integral and derivative strategies. The HCA algorithm combines elements of the cellular automaton (CA) paradigm with finite element analysis (FEA). This methodology has proved to be computationally efficient to solve topology optimization problems. The resulting optimal structures are free of numerical instabilities such as the checkerboarding effect. This investigation presents the main features of the HCA algorithm and the influence of different parameters applied during the iterative optimization process. DOI: 10.1115/1.2336251