Multidisciplinary Optimization Problems Research Articles

Surrogate-assisted differential evolution using manifold learning-based sampling for high dimensional expensive constrained optimization problems

To address the challenges of high-dimensional constrained optimization problems with expensive simulation models, a Surrogate-Assisted Differential Evolution using Manifold Learning-based Sampling (SADE-MLS) is proposed. In SADE-MLS, differential evolution operators are executed to generate numerous high-dimensional candidate points. To alleviate the curse of dimensionality, a Manifold Learning-based Sampling (MLS) mechanism is developed to explore the high-dimensional design space effectively. In MLS, the intrinsic dimensionality of the candidate points is determined by a maximum likelihood estimator. Then, the candidate points are mapped into a low-dimensional space using the dimensionality reduction technique, which can avoid significant information loss during dimensionality reduction. Thus, Kriging surrogates are constructed in the low-dimensional space to predict the responses of the mapped candidate points. The candidate points with high constrained expected improvement values are selected for global exploration. Moreover, the local search process assisted by radial basis function and differential evolution is performed to exploit the design space efficiently. Several numerical benchmarks are tested to compare SADE-MLS with other algorithms. Finally, SADE-MLS is successfully applied to a solid rocket motor multidisciplinary optimization problem and a re-entry vehicle aerodynamic optimization problem, with the total impulse and lift to drag ratio being increased by 32.7% and 35.5%, respectively. The optimization results demonstrate the practicality and effectiveness of the proposed method in real engineering practices.

Design-space adaptation method for multiobjective and multidisciplinary optimization

This paper developed a new method that adaptively adjusts a design space by considering the actual solution distribution of a problem to overcome the conventional design-space adaptation method that assumes the solutions distribution to be a normal distribution because the distributions of solutions are rarely normal distributions for real-world problems. The developed method was applied to nineteen multiobjective test functions that are widely used to evaluate the characteristics and performance of optimization approaches. The results showed that this method adapted the design space to an appropriate design space where the solution existence probability was high. The optimization performance achieved using the developed method was higher than that of the conventional methods. Furthermore, the developed method was applied to the conceptual design of an unmanned spacecraft to confirm its validity in real-world design and multidisciplinary-optimization problems. The results showed that the Pareto solutions of the developed method were superior to those of conventional methods. Additionally, the optimization efficiency with the developed method was improved by more than 1.4 times over that of the conventional methods. In this regard, the developed method has the potential to be applied to complicated real-world optimization problems to achieve better performance and efficiency.

BIO19: Patient-Specific Econotherapeutic Closed-Clinician/Engineer-in-the-loop Framework for VAD Physiological Control

Background: Mechanical circulatory support therapy essentially restricts patients to the hospital. Numerous clinical publications and national/international presentations indicate the need for early patient discharge, both from a quality of life and economic viewpoint. An alternative strategy is to look at providing time-appropriate VAD support to temporize patients for transplantation or other definitive therapy. However, current VAD systems do not possess high-level feedback control targeting therapy-specific objectives. Current Controllers (i) operate in open-loop fixed-average-speed mode (do not adjust to match physiological demand or cardiac function) and (ii) have minimal ability to assess and respond to adverse events. Therefore, our group is embarking on developing an Intelligent VAD ControllerTM (iVC™) with advanced, modern features that address the deficits of prevailing technologies. Methods: As an initial step towards the design of an Intelligent VAD ControllerTM (iVC™), we developed an Econotherapeutic framework based on concepts of modern control theory, inter-operator communication (Kanban Method), and industrial supply-demand management (Tanpin-Kanri System). The design inputs included considerations for 1. collaboration of clinician and engineer; 2. patient-specific therapeutic goals; 3. fusion of multiple control objectives; 4. non-invasive monitoring of the patient and hardware, and 5. considerations of the financial/resource cost. From the perspective of a control problem, we extend the engineering theory behind its implementation by including, in our framework, the therapy and healthcare protocols to the control problem as constraints. Results: The resulting Econotherapeutic framework is depicted schematically in Figures 1 and 2 below. Figure 1 outlines the framework’s architecture, which includes two sub-frameworks: Therapeutic and Economical. The outlines of the framework’s functions are provided to optimize patient-specific therapy and follow-up and the economic burden of care and control. Figure 2 outlines the inter-operator communication (inspired by Kanban Methods) protocols that form the basis for the framework in terms of roles and responsibilities, design aspects of the control, and implementation steps (including support). The framework addresses the design inputs with the outlines mentioned in Figure 1 as to what the framework addresses, with the implementation of the framework specific to a patient carried out by the protocols outlined in Figure 2. Conclusion: A VAD closed-loop physiological control is, in essence, a multi-disciplinary optimization problem driven by both medical and engineering considerations. The econotherapeutic framework derived above provides a framework for developing an iVC. We will now proceed to develop, in conjunction with clinicians, the possible details of the framework and, using The PediaFlowTM Pediatric VAD, conduct feasibility studies of the iVC that tests the framework.Figure 1. Architecture of the Econotherapeutic FrameworkFigure 2. The Design, Management and Monitoring Communication protocols of the Econotherapeutic Framework for iVC for every individual patient.

Decision making for multi‐objective problems: Mean and median metrics

AbstractWhen dealing with problems with more than two objectives, sophisticated multi‐objective optimization algorithms might be needed. Pareto optimization, which is based on the concept of dominated and non‐dominated solutions, is the most widely utilized method when comparing solutions within a multi‐objective setting. However, in the context of optimization, where three or more objectives are involved, the effectiveness of Pareto dominance approaches to drive the solutions to convergence is significantly compromised as more and more solutions tend to be non‐dominated by each other. This in turn reduces the selection pressure, especially for algorithms that rely on evolving a population of solutions such as evolutionary algorithms, particle swarm optimization, differential evolution, etc. The size of the non‐dominated set of trade‐off solutions can be quite large, rendering the decision‐making process difficult if not impossible. The size of the non‐dominated solution set increases exponentially with an increase in the number of objectives. This paper aims to expand a framework for coping with many/multi‐objective and multidisciplinary optimization problems through the introduction of a min‐max metric that behaves like a median measure that can locate the center of a data set. We compare this metric to the Chebyshev norm L_∞ metric that behaves like a mean measure in locating the center of a data set. The median metric is introduced in this paper for the first time, and unlike the mean metric is independent of the data normalization method. These metrics advocate balanced, natural, and minimum compromise solutions about all objectives. We also demonstrate and compare the behavior of the two metrics for a Tradespace case study involving more than 1200 CubeSat design alternatives identifying a manageable set of potential solutions for decision‐makers.

Multidisciplinary Design Optimization of a Re-Entry Spacecraft via Radau Pseudospectral Method

The design and optimization of re-entry spacecraft or its subsystems is a multidisciplinary or multiobjective optimization problem by nature. Multidisciplinary design optimization (MDO) focuses on using numerical optimization in designing systems with several subsystems or disciplines that have interactions and independent actions. In the present paper, the system-level optimizer, trajectory, geometry and shape, aerodynamics, and aerothermodynamics differential equations, are converted to algebraic equations using the Radau pseudospectral method (RPM) since a spacecraft is a nonlinear, extensive, and sparse system. The solution to the problem with the help of MDO is reached by iterating all the disciplines together; one can simultaneously enhance the design, decrease the time and cost of the entire design cycle, and minimize the structural mass of a re-entry spacecraft. Considering various methods presented in earlier research works, a combined and innovative all-at-once (AAO), RPM-based MDO method, including the key subsystems in the design process of a re-entry capsule-shape spacecraft with a low lift-to-drag ratio (L/D), is presented. Considering the applicable state and control variables, various constraints, and parameters applied to several geometric shapes of a blunt capsule and using Apollo’s aerodynamic and aerothermodynamic coefficients, the optimized dimensions for a re-entry spacecraft are presented. The introduced optimization scheme led to a 17% mass reduction compared to the original mass of the Apollo vehicle. Fast computing and simplified models are used together in this method to analyze a wide range of vehicle shapes and entry types during conceptual design.

Quantifying uncertainty with ensembles of surrogates for blackbox optimization

Blackbox optimization tackles problems where the functions are expensive to evaluate and where no analytical information is available. In this context, a tried and tested technique is to build surrogates of the objective and the constraints in order to conduct the optimization at a cheaper computational cost. This work introduces an extension to a specific type of surrogates: ensembles of surrogates, enabling them to quantify the uncertainty on the predictions they produce. The resulting extended ensembles of surrogates behave as stochastic models and allow the use of efficient Bayesian optimization tools. The method is incorporated in the search step of the mesh adaptive direct search (MADS) algorithm to improve the exploration of the search space. Computational experiments are conducted on seven analytical problems, two multi-disciplinary optimization problems and two simulation problems. The results show that the proposed approach solves expensive simulation-based problems at a greater precision and with a lower computational effort than stochastic models.

Multi-objective, Multidisciplinary Optimization of Low-Boom Supersonic Transports Using Multifidelity Models

A multidisciplinary optimization (MDO) method has been developed to design a computational fluid dynamics (CFD)-based low-boom configuration that can be obtained from a Pareto solution of a low-fidelity multi-objective MDO problem with mission constraints. This paper refines the developed MDO method by using multifidelity models for CFD-based multi-objective MDO and a better method for the system-level trade between the target low-boom ground noise level and the overland range. The refined MDO method can generate a low-boom configuration that satisfies the mission requirements, has the lowest takeoff gross weight and the longest range for the low-boom mission, trims the low-boom cruise flight with fuel redistributions, and has a reversed equivalent area distribution closely matching a low-boom target with ground noise level below 70 perceived level in decibels. The validity of the refined MDO method is demonstrated by a design study of a low-boom supersonic transport that carries 40 passengers, flies a low-boom mission with cruise Mach of 1.7 and range of 3500 n mile, and cruises overwater at Mach 1.8 with range of 3882 n mile. Moreover, the refined MDO method eliminates the difference between the assumed cruise weight for CFD-based low-boom inverse design optimization and the estimated cruise weight of the optimal inverse design solution with respect to the mission requirements.

A hybrid ITLHHO algorithm for numerical and engineering optimization problems

Harris hawks optimization (HHO) is one of the newest metaheuristic algorithms (MHAs) which mimic the interdependent behaviour and hunting style of Harris hawks in nature. It is an efficient swarm optimization technique that has been used to solve various kinds of optimization problems. However, for some optimization cases, it has a tendency to be trapped into local search space and it endures an improper balance between exploitation and exploration. To get rid of this situation and to explore the global searching ability of HHO, an effective hybrid method improved teaching–learning HHO (ITLHHO) has been developed using improved teaching–learning-based optimization for solving different kinds of engineering design and numerical optimization problems. The performance of ITLHHO has been demonstrated by 33 well-known benchmark functions, including IEEE Congress of Evolutionary Computation Benchmark Test Functions (CEC-C06, 2019 Competition) and 10 multidisciplinary challenging engineering optimization problems. After illustration, the outcomes of the proposed ITLHHO are compared with several recently developed competitive MHAs. Additionally, the ITLHHO results are statistically investigated with the Wilcoxon rank-sum test and multiple comparison test to show the significance of the results. The experimental results suggest that ITLHHO significantly outperforms other algorithms and becomes a remarkable and promising tool for solving various kinds of optimization problems.

Integration of Low-Fidelity MDO and CFD-Based Redesign of Low-Boom Supersonic Transports

A mixed-fidelity low-boom multidisciplinary optimization (MDO) problem is formulated for integration of low-fidelity MDO and computational fluid dynamics (CFD)-based low-boom inverse design optimization. The mixed-fidelity low-boom MDO problem aims to enforce the weight consistency for CFD-based low-boom inverse design optimization: the optimum low-boom configuration is designed for the weight at the start of cruise of the same configuration for the low-boom overland mission. Moreover, it also seeks the optimum trades between the maximum takeoff gross weight (MTOGW), cruise Mach, and range for the low-boom overland mission while meeting the requirement of flying a transatlantic overwater mission. A block coordinate optimization (BCO) method is developed to find an approximate solution of the mixed-fidelity low-boom MDO problem. The BCO method is successfully applied to generate two CFD-based low-boom configurations that closely match two reversed equivalent area targets with ground noise levels below 70 PLdB, respectively. Moreover, these CFD-based low-boom configurations can be obtained with minor wing modifications from the solutions of the low-fidelity MDO problem. The low-fidelity MDO solutions are Pareto points for constrained multi-objective optimization of MTOGW, low-boom cruise Mach, and low-boom range. The best low-fidelity MDO concept flies at cruise Mach 1.8 for a low-boom overland mission with range of 2950 nm and an overwater mission with range of 3600 nm.

Active Manifold and Model-Order Reduction to Accelerate Multidisciplinary Analysis and Optimization

A computational framework is proposed for efficiently solving multidisciplinary analysis and optimization (MDAO) problems in a relatively high-dimensional design parameter space. It relies on projection-based reduced-order models (PROMs) and a new concept of an active manifold (AM) to mitigate the curse of dimensionality during the training of the PROMs. The AM is discovered using a deep convolutional autoencoder for dimensionality reduction: it is proposed as a superior alternative to the concept of active subspace whose capabilities are limited by the associated affine approximation. The computational framework also blends the concept of a global PROM as a surrogate model of a nonlinear partial differential equation (PDE)-based behavior function with that of a database of local, linear PROMs for approximating a linear PDE-based behavior function. It is demonstrated for the solution of a flexible instance of NASA’s Common Research Model for transport aircraft, where the objective function pertains to aerodynamics, the constraint relates to flutter, and the design space contains 58 structural and shape parameters. The obtained results illustrate the potential of the AM concept for mitigating the aforementioned curse of dimensionality. They also demonstrate the feasibility of the proposed computational framework for realistic MDAO problems and its ability to reduce solution time.

Multi-disciplinary Optimization of Wing Sandwich Structure using Proper Orthogonal Decomposition and Automatic Machine Learning

The coupling between different disciplines for multi-disciplinary optimization greatly increases the complexity of a computational framework, while at the same time increasing CPU time and memory usage. To overcome these difficulties, first, proper orthogonal decomposition and radial basis function were used to generate a reduced-order model from the initial experimental points. Second, analysis results for additional experimental points were predicted using the reduced-order model. Third, using automated machine learning, surrogate models for the objective and constraint functions were obtained from the analysis results at the initial and additional experimental points. Last, optimization was performed using the surrogate models for the objective and constraint functions. As an example, the multi-disciplinary optimization problem of determining the thicknesses of the composite lamina and sandwich core when the composite sandwich structure was used as an aircraft wing skin material was analyzed.

Optimal Aircraft Design Decisions Under Uncertainty Using Robust Signomial Programming

Aircraft design benefits from optimization under uncertainty since design feasibility and performance can have large sensitivities to uncertain parameters. Legacy methods of uncertainty protection do not adequately explain the tradeoffs between feasibility and optimality, and they require prior engineering knowledge that may not be available for new system concepts. Furthermore, stochastic optimization over parameter distributions is computationally intractable for solving high-dimensional nonlinear design optimization problems. This paper proposes an efficient solution method for engineering design optimization problems under uncertainty using robust signomial programs (RSPs). Signomial programs (SPs) have demonstrated potential in solving multidisciplinary optimization problems, and the formulation of RSPs enables conceptual design that captures parametric uncertainty with probabilistic guarantees of constraint satisfaction. The proposed method transforms stochastic optimization problems to deterministic problems by considering the worst-case robust counterpart of each design constraint over a parameter uncertainty set, provided that each constraint is SP representable. The RSP formulation extends an existing robust geometric program (RGP) formulation by allowing difference-of-log-convex constraints that appear in many design problems. The RSP is solved efficiently and deterministically using a sequence of local RGP approximations. RSPs are then applied to unmanned aircraft design, and they are used to rigorously explore the tradeoff between robustness and optimality in design decisions.

$$\lambda $$-DNNs and their implementation in conjugate heat transfer shape optimization

A data-driven two-branch deep neural network (DNN), to be referred to as $$\lambda $$ -DNN, used to predict scalar fields is presented. The network architecture consists of two separate branches (input layers) connected to the main one towards its output. In multi-disciplinary shape optimization problems, such as those this paper is dealing with, the input to the $$\lambda $$ -DNN contains data relevant to the geometrical shape and the case itself. Herein, the $$\lambda $$ -DNN is used in conjugate heat transfer (CHT) analysis and shape optimization problems, synergistically with codes simulating flows over the fluid domain and solving the heat conduction equations over the solid one. It is used to optimize a duct and an internally cooled turbine blade-airfoil surrounded by hot gas. The $$\lambda $$ -DNNs, after being trained on fields computed using the CHT solver, are used as surrogates for either the heat conduction equation solver of the solid domain, replicating either one out of the two disciplines of the problem or the coupled CHT solver.

Quantifying the Impact of a Tsunami on Data-Driven Earthquake Relief Zone Planning in Los Angeles County via Multivariate Spatial Optimization

Post-earthquake relief zone planning is a multidisciplinary optimization problem, which required delineating zones that seek to minimize the loss of life and property. In this study, we offer an end-to-end workflow to define relief zone suitability and equitable relief service zones for Los Angeles (LA) County. In particular, we address the impact of a tsunami in the study due to LA’s high spatial complexities in terms of clustering of population along the coastline, and a complicated inland fault system. We design data-driven earthquake relief zones with a wide variety of inputs, including geological features, population, and public safety. Data-driven zones were generated by solving the p-median problem with the Teitz–Bart algorithm without any a priori knowledge of optimal relief zones. We define the metrics to determine the optimal number of relief zones as a part of the proposed workflow. Finally, we measure the impacts of a tsunami in LA County by comparing data-driven relief zone maps for a case with a tsunami and a case without a tsunami. Our results show that the impact of the tsunami on the relief zones can extend up to 160 km inland from the study area.

Multidisciplinary free-form optimization of solid structures for mean compliance minimization and time-dependent temperature control

A parameter-free free-form optimization method to solve multidisciplinary shape optimization problems of solid structures is presented in this paper. The design objective is both to minimize the mean compliance of linear elastic structures and to control the temperature at arbitrary locations and time periods for arbitrary loadings and under volume constraint, simultaneously. The multidisciplinary design problem is formulated as a distributed-parameter shape optimization problem based on the variational method, and the global objective function is defined using an original normalized weighted sum method. The shape gradient density functions with respect to the shape variation are derived based on the material derivative and the adjoint variable methods. The shape gradient density functions theoretically derived are applied to the H1 gradient method, a gradient method in the function space, to obtain the optimal shape while maintaining the smoothness of the shape. Numerical examples are considered to demonstrate the validity and the effectiveness of the developed optimization method.

Multidisciplinary collaborative optimization based on relaxation method for solving complex problems

The purpose of the present work is to improve the performance of the standard collaborative optimization (CO) approach based on an existing dynamic relaxation method. This approach may be weakened by starting design points. First, a New Relaxation (NR) method is proposed to solve the difficulties in convergence and low accuracy of CO. The new method is based on the existing dynamic relaxation method and it is achieved by changing the system-level consistency equality constraints into relaxation inequality constraints. Then, a Modified Collaborative Optimization (MCO) approach is proposed to eliminate the impact of the information inconsistency between the system-level and the discipline-level on the feasibility of optimal solutions. In the MCO approach, the impact of the inconsistency is treated by transforming the discipline-level constrained optimization problems into an unconstrained optimization problem using an exact penalty function. Based on the NR method, the performance of the MCO approach carried out by solving two multidisciplinary optimization problems. The obtained results show that the MCO approach has improved the convergence of CO significantly. These results prove that the present MCO succeeds in getting feasible solutions while the CO fails to provide feasible solutions with the used starting design points.

Model Reduction Framework with a New Take on Active Subspaces for Optimization Problems with Linearized Fluid‐Structure Interaction Constraints

SummaryIn this paper, a new take on the concept of an active subspace for reducing the dimension of the design parameter space in a multidisciplinary analysis and optimization (MDAO) problem is proposed. The new approach is intertwined with the concepts of adaptive parameter sampling, projection‐based model order reduction, and a database of linear, projection‐based reduced‐order models equipped with interpolation on matrix manifolds, in order to construct an efficient computational framework for MDAO. The framework is fully developed for MDAO problems with linearized fluid‐structure interaction constraints. It is applied to the aeroelastic tailoring, under flutter constraints, of two different flight systems: a flexible configuration of NASA's Common Research Model; and NASA's Aeroelastic Research Wing #2 (ARW‐2). The obtained results illustrate the feasibility of the computational framework for realistic MDAO problems and highlight the benefits of the new approach for constructing an active subspace in both terms of solution optimality and wall‐clock time reduction.

An adaptive switchover hybrid particle swarm optimization algorithm with local search strategy for constrained optimization problems

Practical engineering optimization problems are almost constrained optimization problems and difficult to be solved effectively, therefore, how to handle these problems has attracted more and more attention. Particle swarm optimization (PSO) is one of the most popular algorithms in solving the complicated optimization problems due to its relatively strong global optimization capability and low requirement for computing resources. However, PSO is easy to converge prematurely like other swarm intelligence algorithms due to the loss of diversity among particles. This article proposes an adaptive switchover hybrid PSO framework with local search process (ASHPSO), which adaptively switches the optimization searching process between the standard PSO and the differential evolution (DE) modified by a full dimension crossover strategy to avoid the premature convergence problem. Moreover, a local search strategy is employed to improve the boundary search capability of PSO in consideration of the engineering problems characteristics. Experiments on 28 well-known benchmark functions, 5 engineering problems and a full vehicle multi-disciplinary optimization problem demonstrate the effectiveness of the proposed algorithm compared with other hybrid variants.

A global optimization strategy based on the Kriging surrogate model and parallel computing

The computational efficiency and accuracy of the global solution are the main performance indicators of an optimization algorithm to solve the structural and multidisciplinary optimization problems. The Kriging-based optimization algorithm can satisfy certain engineering requirements by applying the single-point sequence sampling method. However, this conventional algorithm does not efficiently apply the parallel performance of high-performance multi-core computers. This study aims to propose a global optimization strategy based on the Kriging surrogate model and parallel computing depending on the multi-peak characteristics of the expect improvement (EI) function. The proposed method searches out the locations of multiple peaks of the EI function by introducing a so-called P-EI function; it then simultaneously includes multiple sampling points located nearby these peaks. Furthermore, an efficient design domain reduction technique is applied to improve the accuracy of the global solution. When compared with the traditional Kriging-based methods, the proposed method can effectively obtain multiple peaks of the EI function without solving complex expressions of the high-dimensional EI function exhibiting a joint probability density distribution. The parallel computation ability, global performance, and solution accuracy of our method are validated via typical test functions and structural optimization problems.

Efficient incorporation of fatigue damage constraints in wind turbine blade optimization

AbstractWind turbine design is a challenging multidisciplinary optimization problem, where the aerodynamic shapes, structural member sizing, and material composition must all be determined and optimized. Some previous blade design methods incorporate static loading with an added safety factor to account for dynamic effects. Others incorporate dynamic loading, but in general limit, the evaluation to a few design cases. By not fully incorporating the dynamic loading of the wind turbine, the final turbine blade design is either too conservative by overemphasizing the dynamic effects or infeasible by failing to adequately account for these effects. We propose an iterative method that estimates fatigue effects during the optimization process while quickly converging to the true solution. We also demonstrate an alternate approach where a surrogate model is trained to efficiently estimate the dynamic loading of the wind turbine in the design process. This surrogate model, once trained, was then incorporated in the optimization loop of the wind turbine blade. In contrast to the iterative method, there is significant upfront computational cost to construct the surrogate model. However, this surrogate model has been generalized to be used for different rated turbines and can predict the fatigue damage of a wind turbine with less than 5% error for baseline wind turbines of the same family. These methods can be used instead of the more computationally expensive method of calculating the dynamic loading of the turbine within the optimization routine.