Concurrent Design of Product-Material Systems Using Multilevel Optimization

The area of Multiparametric Optimization (MPO) solves problems that contain unknown problem data represented by parameters. The solutions map parameter values to optimal design and objective function values. In this paper, for the first time, MPO techniques are applied to improve and advance Multidisciplinary Design Optimization (MDO) to solve engineering problems with parameters. A multiparametric subgradient algorithm is proposed and applied to two MDO methods: Analytical Target Cascading (ATC) and Network Target Coordination (NTC). Numerical results on test problems show the proposed parametric ATC and NTC methods effectively solve parametric MDO problems and provide useful insights to designers. In addition, a novel Two-Stage ATC method is proposed to solve nonparametric MDO problems. In this new approach elements of the subproblems are treated as parameters and optimal design functions are constructed for each one. When the ATC loop is engaged, steps involving the lengthy optimization of subproblems are replaced with simple function evaluations.

ATC provides a systematic approach in solving decomposed large scale systems that has solvable subsystems. However， complex engineering system usually has a high computational cost , which result in limiting real-life applications of ATC based on high-fidelity simulation models. To address these problems, this paper aims to develop an efficient approximation model building techniques under the analytical target cascading (ATC) framework, to reduce computational cost associated with multidisciplinary design optimization problems based on high-fidelity simulations. An approximation model building techniques is proposed: approximations in the subsystem level are based on variable-fidelity modeling (interaction of low- and high-fidelity models). The variable-fidelity modeling consists of computationally efficient simplified models (low-fidelity) and expensive detailed (high-fidelity) models. The effectiveness of the method for modeling under the ATC framework using variable-fidelity models is studied. Overall results show the methods introduced in this paper provide an effective way of improving computational efficiency of the ATC method based on variable-fidelity simulation models.

Under dynamic conditions, the dynamic force between the suspended equipment and the car body is substantially increased. This increase not only affects the ride comfort but also substantially raises the likelihood of fatigue damage to the suspended equipment. A multiobjective analytical target cascading (ATC) optimization is proposed to improve the ride comfort of high-speed trains and reduce the vibration of the suspended equipment. A mathematical simulation model of the vehicle equipment system is established, and the acceleration frequency-response function expression of the car body and the suspension system is obtained. The comfort index of the car body and the acceleration root mean square (RMS) of the suspended equipment are calculated by combining the German vertical irregular track spectrum and the comfort filter function. Meanwhile, the ATC method is used to optimize the car body comfort index and the acceleration RMS of the underframe suspended equipment. Then, the availability of the optimization method is evaluated via numerical simulation. Compared with the original suspension scheme of the underframe equipment, when the running speed of the vehicle is 300 km/h, the RMS value of the vibrational acceleration of the underframe equipment after optimization decreases by 19.9 %, and the ride comfort indexes at the car body center and above the front and rear bogies are improved by 6.4 %, 0.1 %, and 1.1 %, respectively. Simulation results demonstrate that ATC optimization can improve the railway vehicle ride comfort and reduce the vibration of the suspended equipment. This paper provides a new approach to the suspension parameter design of equipment.

Analytical Target Cascading method has been widely developed to solve hierarchical design optimization problems. In the Analytical Target Cascading method, a weighted-sum formulation has been commonly used to coordinate the inconsistency between design points and assigned targets in each level while minimizing the cost function. However, the choice of the weighting coefficients is very problem dependent and improper selections of the weights will lead to incorrect solutions. To avoid the problems associated with the weights, single objective functions in the hierarchical design optimization are formulated by a new Bounded Target Cascading method. Instead of point targets assigned for design variables in the Analytical Target Cascading method, bounded targets are introduced in the new method. The target bounds are obtained from the optimal solutions in each level while the response bounds are updated back to the system level. If the common variables exist, they are coordinated based on their sensitivities with respect to design variables. Finally, comparisons of the results from the proposed method and the weighted-sum Analytical Target Cascading are presented and discussed.

In the development of complex engineering systems, many engineers from different disciplines collaborate to identify feasible designs that satisfy all system requirements. Analytical target cascading (ATC) is a method for the design optimization of hierarchical, multilevel systems and has been successfully employed in the design of complex engineering systems. In this paper, we propose a novel data‐driven set‐based ATC (SBATC) method for hierarchical design optimization problems using machine learning techniques. The proposed SBATC offers two advantages in engineering processes. First, it decomposes hierarchical design optimization problems into sets of suboptimization problems. The feasible regions of the suboptimization problems are explored and cascaded to lower‐level optimization problems. Using the set‐based approach, couplings between two adjacent levels in the optimization process are not required. Second, an efficient strategy is employed to determine feasible regions based on Bayesian active learning (BAL). In BAL, the Gaussian process (GP) of all cost functions is trained. An acquisition function that combines the probability of feasibility and entropy search is evaluated using posterior distributions of the trained GP. The acquisition function is maximized to generate new sampling points around the feasible regions by balancing the exploitation and exploration of the design space. To verify the effectiveness of the proposed method, numerical examples of hierarchical optimization problems are evaluated.

Multi-objective optimization of multi-axle heavy duty vehicle ride comfort and handling stability based on analytical target cascading method

Analytical target cascading (ATC) is a multidisciplinary design optimization method for multilevel hierarchical systems. To improve computational efficiency, especially for problems under uncertainty or with strong monotonicity, a sequential linear programming (SLP) algorithm was previously employed as an alternate coordination strategy to solve ATC and probabilistic ATC problems. The SLP implementation utilizes L∞ norms to maintain the linearity of SLP subsequences. This note offers a proof that there exists a set of weights such that the ATC algorithm converges when L∞ norms are used. Examples are also provided to illustrate the effectiveness of using L∞ norms as a penalty function to maintain the formulation linear and differentiable. The examples show that the proposed method provides more robust results for linearized ATC problems due to the robustness of the linear programming solver.

The scope of this paper is to apply a model-correction-based strategy for efficient reliability analysis of uncertain dynamical systems based on a low-fidelity (LF) model whose outcomes are corrected in a probabilistic sense to represent the more realistic outcomes of a high-fidelity (HF) model. In the model-correction approach utilized, the LF model is calibrated to the HF model close to the so-called most probable point in standard normal space, which allows a more realistic assessment of the considered complex dynamical system. Since only few expensive limit state function evaluations of the HF model are required, an efficient reliability analysis is enabled. In an application example, the LF model describes an existing single-span railway bridge modelled as simply supported Euler–Bernoulli beam subjected to moving single forces representing the axle loads of a moving train. The HF modelling approach accounts for the bridge–train interaction by modelling the passing train as mass-spring-damper system, however increasing the computational effort of the limit state function evaluations. Failure probabilities evaluated with the model-correction approach are contrasted and discussed with failure probabilities of the sophisticated bridge–train interaction model evaluated with the first-order reliability method (FORM). It is demonstrated that the efficiency of the method depends on the correlation between the LF and the HF model. A comparison of the results of FORM and the model-correction-based approach shows that the latter provides reliable failure probability prediction of the HF model while leading to a significant reduction in computational effort.

Distributed optimization for EVs integrated power system considering flexible ramping requirement

• Differentiable multi-fidelity inference: BMFIA enables gradient-based Bayesian inference for high-dimensional, non-differentiable multi-physics problems using low-fidelity model gradients. • Efficiency with minimal HF data: Accurate posterior estimates are achieved from only 100–300 HF–LF simulations with orders-of-magnitude speed-ups over HF-only inference. • Accuracy despite crude LF models: Robust posterior reconstruction is maintained even with weakly correlated or simplified low-fidelity models. High-dimensional Bayesian inverse analysis ( dim ≫ 100 ) is mostly unfeasible for computationally demanding, nonlinear physics-based high-fidelity (HF) models. Usually, the use of more efficient gradient-based inference schemes is impeded if the multi-physics models are provided by complex legacy codes. Adjoint-based derivatives are either exceedingly cumbersome to derive or nonexistent for practically relevant large-scale nonlinear and coupled multi-physics problems. Similarly, holistic automated differentiation w. r. t. primary variables of multi-physics codes is usually not yet an option and requires extensive code restructuring if not considered from the outset in the software design. This absence of differentiability further exacerbates the already present computational challenges. To overcome the existing limitations, we propose a novel inference approach called Bayesian multi-fidelity inverse analysis (BMFIA) , which leverages simpler and computationally cheaper lower-fidelity (LF) models that are designed to provide model derivatives. BMFIA learns a simple, probabilistic dependence of the LF and HF models, which is then employed in an altered likelihood formulation to statistically correct the inaccurate LF response. From a Bayesian viewpoint, this dependence represents a multi-fidelity (MF) conditional density (discriminative model). We demonstrate how this MF conditional density can be learned robustly in the small data regime from only a few HF and LF simulations (50 to 300), which would not be sufficient for naive surrogate approaches. The formulation is fully differentiable and allows the flexible design of a wide range of LF models. We demonstrate that BMFIA solves Bayesian inverse problems for scenarios that used to be prohibitive, such as finely-resolved and hence high-dimensional spatial reconstruction problems in two-dimensional Euclidean domains with static posteriors, given nonlinear and transient coupled poro-elastic media physics. We show that the resulting static MF posteriors are in excellent agreement with the (usually inaccessible) HF posteriors or ground-truth data and note that extending the framework to arbitrary three-dimensional domains is a natural and important direction for future work.

What is Transferred and How Much is Retained? A Simulation Study of Complex Surgical Skills

History matching is a critical step within the reservoir management process to synchronize the simulation model with the production data. The history-matched model can be used for planning optimum field development and performing optimization and uncertainty quantifications. We present a novel history matching workflow based on a Bayesian framework that accommodates subsurface uncertainties. Our workflow involves three different model resolutions within the Bayesian framework: 1) a coarse low-fidelity model to update the prior range, 2) a fine low-fidelity model to represent the high-fidelity model, and 3) a high-fidelity model to re-construct the real response. The low-fidelity model is constructed by a multivariate polynomial function, while the high-fidelity model is based on the reservoir simulation model. We firstly develop a coarse low-fidelity model using a two-level Design of Experiment (DoE), which aims to provide a better prior. We secondly use Latin Hypercube Sampling (LHS) to construct the fine low-fidelity model to be deployed in the Bayesian runs, where we use the Metropolis-Hastings algorithm. Finally, the posterior is fed into the high-fidelity model to evaluate the matching quality. This work demonstrates the importance of including uncertainties in history matching. Bayesian provides a robust framework to allow uncertainty quantification within the reservoir history matching. Under uniform prior, the convergence of the Bayesian is very sensitive to the parameter ranges. When the solution is far from the mean of the parameter ranges, the Bayesian introduces bios and deviates from the observed data. Our results show that updating the prior from the coarse low-fidelity model accelerates the Bayesian convergence and improves the matching convergence. Bayesian requires a huge number of runs to produce an accurate posterior. Running the high-fidelity model multiple times is expensive. Our workflow tackles this problem by deploying a fine low-fidelity model to represent the high-fidelity model in the main runs. This fine low-fidelity model is fast to run, while it honors the physics and accuracy of the high-fidelity model. We also use ANOVA sensitivity analysis to measure the importance of each parameter. The ranking gives awareness to the significant ones that may contribute to the matching accuracy. We demonstrate our workflow for a geothermal reservoir with static and operational uncertainties. Our workflow produces accurate matching of thermal recovery factor and produced-enthalpy rate with physically-consistent posteriors. We present a novel workflow to account for uncertainty in reservoir history matching involving multi-resolution interaction. The proposed method is generic and can be readily applied within existing history-matching workflows in reservoir simulation.

Analytical target cascading (ATC) method has been widely applied to solve multilevel decomposed system design optimization problems. In the ATC method, concurrent design is achieved by target cascading. However, due to the complexity and the presence of uncertainty, it is a challenging task to set proper targets. In this article, instead of using point value targets, interval targets are analyzed and propagated through the multilevel system with the goal of reducing the effects of uncertainty while providing more flexibility to a design process. In the proposed method, the design of a hierarchical system at each level is taken as a single-objective optimization problem, by minimizing the degree of deviation between the target response interval and the achievable response interval. Not only the optimal design performance is considered in this method, but also the acceptable variation range of the performance is analyzed. When the present target for a lower level system and the achievable response from a lower level system are not point values, but rather intervals, their probability distributions are not available. Therefore, these variables are treated as interval variables. When the random and interval variables are present, the most probable point-based first-order reliability and the interval analysis methods are used to calculate the reliability bounds. The proposed method for flexibility under uncertainty provides more degree of freedom to the design of lower level systems, while also keeping the performance of the upper systems stable within a tolerable range. The accuracy of the proposed method is demonstrated via comparing results from both the proposed and traditional methods.

This work discusses a novel approach, Bayesian-inspired multifidelity optimization, for solving complex optimization requiring computationally expensive simulations. Bayesian-inspired multifidelity optimization aims to minimize computational cost in achieving accurate high-fidelity results through the use of low-fidelity (computationally cheaper) models in combination with a surrogate correction model. A novel Bayesian hybrid bridge function was developed to serve as the low-fidelity correction technique. This Bayesian hybrid bridge function is a Bayesian weighted average of two standard bridge functions, additive and multiplicative. The correction technique is implemented in parallel with a modified trust region model management optimization scheme. It is shown that optimization on the corrected low-fidelity model converges to the same local optimum as optimization on the high-fidelity model in fewer high-fidelity model. This technique is developed to reach that optimum in fewer high-fidelity function evaluations than traditional optimization, ultimately reducing computational cost. This work also extends the low-fidelity correction optimization beyond the traditional bifidelity optimization to that of optimization with multiple-fidelity objective and constraint functions. The proposed solution technique allows for use of commercial optimizers and is demonstrated on three separate problems. The first demonstration is on a one-dimensional analytical test case in which the Bayesian correction is shown. Then, the ability to handle multiple fidelities in both the objective and constraint functions is presented and discussed using a two-dimensional analytical test case. An airfoil shape optimization problem is then used to illustrate the effectiveness of this approach in an engineering design scenario in which the various fidelities arise as a difference in computational physics. It is shown via these demonstrations that implementation of this Bayesian low-fidelity correction optimization approach results in convergence to a high-fidelity optimum at a reduced computational cost in comparison to traditional optimization techniques.

The interaction of trailing vortices with lifting surfaces is investigated using two levels of modeling fidelity. An overset mesh-based Reynolds-averaged Navier–Stokes solver is considered as the high-fidelity computational model. A lower-fidelity model is developed by combining a vortex panel method with a propeller aerodynamic model and slipstream theory. The high-fidelity model is first validated against available experimental data obtained from the interaction of a trailing vortex generated by an upstream wing with a downstream wing. The ability of both models to represent the development of the vortex wake and integrated loads is assessed for a number of parametric configurations, including a case in which the vortex core directly impinges on the wing surface. Following this, isolated propeller and wing-mounted propeller configurations are studied. In all of these cases, the high-fidelity model is effective in predicting the details of the flow and integrated airloads. The low-fidelity model, although less accurate, is shown to accurately predict interactional air loads and performance at orders of magnitude less cost than the high-fidelity model, justifying its role as a viable tool in design and trajectory-planning applications.