Design and Optimization of Complex Systems

Truely optimal solutions to system design can only be obtained if the entire system is considered. In this research we consider design of commercial aircraft, but we expand the system to include a family of planes. A multidisciplinary design optimization framework is developed in which multiple aircraft, each with different missions, can be optimized simultaneously. Results are presented for a two-member family whose individual missions differ significantly. We show that both missions can be satisfied with common designs, and that by optimizing both planes simultaneously rather than following the traditional baseline plus derivative approach, the common solution is vastly improved. The new framework is also used to gain insight to the effect of design variable scaling on the optimization algorithm.Singapore-MIT Alliance (SMA

mfEGRA: Multifidelity Efficient Global Reliability Analysis through Active Learning for Failure Boundary Location

This paper develops mfEGRA, a multifidelity active learning method using data-driven adaptively refined surrogates for failure boundary location in reliability analysis. This work addresses the issue of prohibitive cost of reliability analysis using Monte Carlo sampling for expensive-to-evaluate high-fidelity models by using cheaper-to-evaluate approximations of the high-fidelity model. The method builds on the Efficient Global Reliability Analysis (EGRA) method, which is a surrogate-based method that uses adaptive sampling for refining Gaussian process surrogates for failure boundary location using a single-fidelity model. Our method introduces a two-stage adaptive sampling criterion that uses a multifidelity Gaussian process surrogate to leverage multiple information sources with different fidelities. The method combines expected feasibility criterion from EGRA with one-step lookahead information gain to refine the surrogate around the failure boundary. The computational savings from mfEGRA depends on the discrepancy between the different models, and the relative cost of evaluating the different models as compared to the high-fidelity model. We show that accurate estimation of reliability using mfEGRA leads to computational savings of $\sim$46% for an analytic multimodal test problem and 24% for a three-dimensional acoustic horn problem, when compared to single-fidelity EGRA. We also show the effect of using a priori drawn Monte Carlo samples in the implementation for the acoustic horn problem, where mfEGRA leads to computational savings of 45% for the three-dimensional case and 48% for a rarer event four-dimensional case as compared to single-fidelity EGRA

Data-Driven Model Reduction for the Bayesian Solution of Inverse Problems

One of the major challenges in the Bayesian solution of inverse problems governed by partial differential equations (PDEs) is the computational cost of repeatedly evaluating numerical PDE models, as required by Markov chain Monte Carlo (MCMC) methods for posterior sampling. This paper proposes a data-driven projection-based model reduction technique to reduce this computational cost. The proposed technique has two distinctive features. First, the model reduction strategy is tailored to inverse problems: the snapshots used to construct the reduced-order model are computed adaptively from the posterior distribution. Posterior exploration and model reduction are thus pursued simultaneously. Second, to avoid repeated evaluations of the full-scale numerical model as in a standard MCMC method, we couple the full-scale model and the reduced-order model together in the MCMC algorithm. This maintains accurate inference while reducing its overall computational cost. In numerical experiments considering steady-state flow in a porous medium, the data-driven reduced-order model achieves better accuracy than a reduced-order model constructed using the classical approach. It also improves posterior sampling efficiency by several orders of magnitude compared to a standard MCMC method

An Accelerated Greedy Missing Point Estimation Procedure

Model reduction via Galerkin projection fails to provide considerable computational savings if applied to general nonlinear systems. This is because the reduced representation of the state vector appears as an argument to the nonlinear function, whose evaluation remains as costly as for the full model. Masked projection approaches, such as the missing point estimation and the (discrete) empirical interpolation method, alleviate this effect by evaluating only a small subset of the components of a given nonlinear term; however, the selection of the evaluated components is a combinatorial problem and is computationally intractable even for systems of small size. This has been addressed through greedy point selection algorithms, which minimize an error indicator by sequentially looping over all components. While doable, this is suboptimal and still costly. This paper introduces an approach to accelerate and improve the greedy search. The method is based on the observation that the greedy algorithm requires solving a sequence of symmetric rank-one modifications to an eigenvalue problem. For doing so, we develop fast approximations that sort the set of candidate vectors that induce the rank-one modifications, without requiring solution of the modified eigenvalue problem. Based on theoretical insights into symmetric rank-one eigenvalue modifications, we derive a variation of the greedy method that is faster than the standard approach and yields better results for the cases studied. The proposed approach is illustrated by numerical experiments, where we observe a speed-up by two orders of magnitude when compared to the standard greedy method while arriving at a better quality reduced model.German Research Foundation (grant Zi 1250/2-1)United States. Department of Energy (award DE-FG02-08ER258, as part of the DiaMonD Multifaceted Mathematics Integrated Capability Center)United States. Department of Energy (award DE-SC000929, as part of the DiaMonD Multifaceted Mathematics Integrated Capability Center

Horsetail Matching for Optimization Under Probabilistic, Interval and Mixed Uncertainties

The importance of including uncertainties in the design process of aerospace systems is becoming increasingly recognized, leading to the recent development of many techniques for optimization under uncertainty. Most existing methods represent uncertainties in the problem probabilistically; however, in many real life design applications it is often difficult to assign probability distributions to uncertainties without making strong assumptions. Existing approaches for optimization under different types of uncertainty mostly rely on treating combinations of statistical moments as separate objectives, but this can give rise to stochastically dominated designs. Horsetail matching is a flexible approach to optimization under any mix of probabilistic and interval uncertainties that overcomes some of the limitations of existing approaches. The formulation delivers a single, differentiable metric as the objective function for optimization. It is demonstrated on algebraic test problems and the design of a flying wing using a coupled aero-structural analysis code.Engineering and Physical Sciences Research CouncilUnited States. Office of Naval Research. Multidisciplinary University Research Initiative (Award Number FA9550-15-1-0038

Dynamic data-driven model reduction: adapting reduced models from incomplete data

This work presents a data-driven online adaptive model reduction approach for systems that undergo dynamic changes. Classical model reduction constructs a reduced model of a large-scale system in an offline phase and then keeps the reduced model unchanged during the evaluations in an online phase; however, if the system changes online, the reduced model may fail to predict the behavior of the changed system. Rebuilding the reduced model from scratch is often too expensive in time-critical and real-time environments. We introduce a dynamic data-driven adaptation approach that adapts the reduced model from incomplete sensor data obtained from the system during the online computations. The updates to the reduced models are derived directly from the incomplete data, without recourse to the full model. Our adaptivity approach approximates the missing values in the incomplete sensor data with gappy proper orthogonal decomposition. These approximate data are then used to derive low-rank updates to the reduced basis and the reduced operators. In our numerical examples, incomplete data with 30–40 % known values are sufficient to recover the reduced model that would be obtained via rebuilding from scratch.United States. Air Force Office of Scientific Research (AFOSR MURI on multi-information sources of multi-physics systems, Award Number FA9550-15-1-0038)United States. Dept. of Energy (Applied Mathematics Program, Award DE-FG02 08ER2585)United States. Dept. of Energy (Applied Mathematics Program, Award DE-SC0009297

A MATHEMATICAL AND COMPUTATIONAL FRAMEWORK FOR MULTIFIDELITY DESIGN AND ANALYSIS WITH COMPUTER MODELS

A multifidelity approach to design and analysis for complex systems seeks to exploit optimally all available models and data. Existing multifidelity approaches generally attempt to calibrate low-fidelity models or replace low-fidelity analysis results using data from higher fidelity analyses. This paper proposes a fundamentally different approach that uses the tools of estimation theory to fuse together information from multifidelity analyses, resulting in a Bayesian-based approach to mitigating risk in complex system design and analysis. This approach is combined with maximum entropy characterizations of model discrepancy to represent epistemic uncertainties due to modeling limitations and model assumptions. Mathematical interrogation of the uncertainty in system output quantities of interest is achieved via a variance-based global sensitivity analysis, which identifies the primary contributors to output uncertainty and thus provides guidance for adaptation of model fidelity. The methodology is applied to multidisciplinary design optimization and demonstrated on a wing-sizing problem for a high altitude, long endurance vehicle.United States. Air Force Office of Scientific Research. Small Business Technology Transfer Program (Contract FA9550-09-C-0128