A Sequential Linear Programming Coordination Algorithm for Analytical Target Cascading

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.

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.

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.

Design engineers and decision-makers across various fields are constantly working to make optimal design decisions for multidisciplinary engineering systems in an effort to improve performance and reduce costs. The multiple disciplines that decision-makers are forced to consider can range from different physical components of a system, to competing physical phenomena influencing a component (e.g. flow forces and structural strength), to completely separate models of interest to a system (e.g. engineering performance and lifecycle cost). The common element that all these decision-making scenarios share is the presence of couplings between the considered disciplines (or subsystems). How the values for these coupling parameters are determined within a decision-making or optimization framework is the subject of countless research efforts. At present the multidisciplinary design optimization (MDO) community has settled on a few proven techniques such Collaborative Optimization (CO) and Analytic Target Cascading (ATC). However, current MDO techniques have issues that limit their effectiveness in solving various MDO problems. Many of these strategies require close coordination between subsystem optimization solvers, require significant effort by decision-makers to pose their problems in a suitable format, and/or can have large computational efficiency problems due to the fact that they involve solving nested optimization problems. In an effort to alleviate some of these issues and make MDO easier to implement and more computationally efficient, a new sequential MDO algorithm called Cooperative Design Optimization (CDO) is proposed. The CDO approach functions through a series of subsystem optimizations using a successively smaller cooperation space. The cooperation space is analogous to the design space of a traditional optimization problem, but includes only the coupling parameters that are a factor in multiple subsystems. A single iteration of the approach can be thought of as a negotiation between all of the subsystems regarding the boundaries of the cooperation space. To facilitate this cooperation, each subsystem optimization problem is restructured as a multi-objective optimization problem so that a Pareto set of optimal solutions, or agreeable design alternatives, are produced. As a result, after all subsystem optimizations are accomplished, each subsystem’s obtained Pareto optimal solution sets are compared with respect to the coupling parameters and new bounds in the cooperation space are determined for the next iteration. This process allows each subsystem to be optimized completely independently within the boundaries of the agreed upon cooperation space while coordination is achieved through an analysis of obtained optimal solutions for each subsystem. Iterations, or negotiations, are repeated until an acceptable solution or set of solutions is obtained for all included subsystems through this cooperative process.

†‡ § ** In traditional structural optimization, the geometric properties of the product are optimized for a specific set of material properties. This paper seeks to extend the domain of product optimization to also include the material modeling parameters by treating the product and material as a hierarchical multilevel system that can be decomposed and solved using the analytical target cascading (ATC) method. Two material models are considered here. The low-fidelity model is based on a simple elastic-plastic constitutive law with linear hardening whereas the high-fidelity model is based on an internal-state-variable representation that is influenced by the microstructural features of the material. The ATC method is applied to product-material design optimization of a multi-cell, thin-walled tube for impact energy absorption and durability. The integrated product-material optimization problem is solved using both single and multilevel approaches with results compared.

Design of a successful product requires evaluating many design alternatives in a limited design cycle time. This can be achieved through leveraging design space exploration tools and available computing resources on the network. This paper presents a parallel trade study architecture to integrate trade study clients and computing resources on a network using Web services. The parallel trade study solution is demonstrated to accelerate design of experiments, genetic algorithm optimization, and a cost as an independent variable (CAIV) study for a space system application.

This paper presents a novel adaptive-sparse polynomial dimensional decomposition (PDD) method for stochastic design optimization of complex systems. The method entails an adaptive-sparse PDD approximation of a high-dimensional stochastic response for statistical moment and reliability analyses; a novel integration of the adaptive-sparse PDD approximation and score functions for estimating the first-order design sensitivities of the statistical moments and failure probability; and standard gradient-based optimization algorithms. New analytical formulae are presented for the design sensitivities that are simultaneously determined along with the moments or the failure probability. Numerical results stemming from mathematical functions indicate that the new method provides more computationally efficient design solutions than the existing methods. Finally, stochastic shape optimization of a jet engine bracket with 79 variables was performed, demonstrating the power of the new method to tackle practical engineering problems.

The Analytic Target Cascading (ATC) is an effective method for solving hierarchical Multidisciplinary Design Optimization (MDO) problems. At the same time, this method suffers from poor convergence and low accuracy, which is caused by the inconsistency of system constraints. In this paper, a novel ATC method based on dynamic relaxation factor is proposed. The dynamic relaxation factor of consistency constraint is added in the system level and is adjusted by the deviation of the linking variables between the levels to ensure the feasible region of the design space. The effectiveness and accuracy of this method are verified by a mathematical example. This method is used to solve the lightweight problem of the trussed front part of the vehicle body frame based on active and passive safety to achieve the collaborative optimization of lightweight trussed frame, crash safety, and aerodynamic characteristics. The important value of the novel ATC method based on dynamic relaxation factor in engineering applications is proven.

Effective electrification of automotive vehicles requires designing the powertrain's configuration along with sizing its components for a particular vehicle type. Employing planetary gear (PG) systems in hybrid electric vehicle (HEV) powertrain architectures allows various architecture alternatives to be explored, including single-mode architectures that are based on a fixed configuration and multimode architectures that allow switching power flow configuration during vehicle operation. Previous studies have addressed the configuration and sizing problems separately. However, the two problems are coupled and must be optimized together to achieve system optimality. An all-in-one (AIO) system solution approach to the combined problem is not viable due to the high complexity of the resulting optimization problem. This paper presents a partitioning and coordination strategy based on analytical target cascading (ATC) for simultaneous design of powertrain configuration and sizing for given vehicle applications. The capability of the proposed design framework is demonstrated by designing powertrains with one and two PGs for a midsize passenger vehicle.

Analytical Target Cascading (ATC) is a product development tool that computes component design specifications so that the final system design is consistent and meets design targets. It is a promising methodology for decomposition of complex system design and optimization. However, ATC has some shortcomings inherently. In this paper, after analyzing the theory and model of ATC, a more promising method entitled Analytical Target Cascading Based on Physical Programming (ATC-PP) is proposed. This new approach is intended to substantially place the design process into a more flexible and practical framework by adding the advantages of Physical Programming (PP) into ATC. And the preference function which is proposed in PP is used to definite the deviation between response and linking variables in ATC in the new method. It can completely eliminate the need for assigning weight coefficient, which is the object of the typical computational bottleneck in large design optimization problems. And it could also give the designers' preferences in practical design. Finally, the validity of ATC-PP is approved by studying an anchor design problem.

This paper presents a decomposed co-design optimization framework to jointly design the geometry and the controller of a Continuously Variable Transmission (CVT), accounting for its low-level dynamics. Specifically, we first devise a model of the CVT and the feedback controller, and use it to formulate the optimal co-design problem with the goal of minimizing the transmission's mass as well as the losses that occur in the system, including the lower-level actuation. Second, we divide the resulting nonlinear multi-objective optimization problem into separate hierarchical optimization subproblems and we leverage the concept of Analytical Target Cascading (ATC) to solve the separate optimization subproblems using an interior-point optimization algorithm. Finally, we showcase our framework on a representative drive cycle. Our results demonstrate that the presented co-design method can achieve a more compact CVT design without compromising the desired ratio trajectory and reducing the overall losses by up to 14%.

In this article, we propose an algorithm which solves unconstrained optimization problems by combining the trust region method with the quasi-Newton line search method. During consecutive trust region steps, when the quasi-Newton matrix B k approximates the Hessian matrix of the objective function at x k well, the algorithm tries a full quasi-Newton step and, if this step is successful, continues trying quasi-Newton steps. On the other hand, when employing the line search method, if the quasi-Newton direction is nearly orthogonal to the gradient of the objective function or the step size becomes too small, the algorithm switches back to trust region steps. Convergence properties of the algorithm are proved, and numerical results are presented.

Summary Using the conventional trust-region (C-TR) method to solve multiphase equilibrium calculations involving water, CO2, and heavy hydrocarbons might fail to converge to correct solutions. To address this issue, we develop a new four-phase equilibrium calculation algorithm based on two trust-region methods: a scaled trust-region (S-TR) method and a nonmonotonic trust-region (N-TR) method. We test the robustness and efficiency of the modified algorithm by generating P-x and P-T phase diagrams of six benchmark fluid mixtures that involve four-phase equilibria. Comparison against the C-TR algorithm indicates that the modified trust-region (M-TR)-based algorithm effectively resolves most nonconvergence problems and leads to improved computational efficiency as well.

An exponential penalty function (EPF) formulation based on method of multipliers is presented for solving multilevel optimization problems within the framework of analytical target cascading. The original all-at-once constrained optimization problem is decomposed into a hierarchical system with consistency constraints enforcing the target-response coupling in the connected elements. The objective function is combined with the consistency constraints in each element to formulate an augmented Lagrangian with EPF. The EPF formulation is implemented using double-loop (EPF I) and single-loop (EPF II) coordination strategies and two penalty-parameter-updating schemes. Four benchmark problems representing nonlinear convex and non-convex optimization problems with different number of design variables and design constraints are used to evaluate the computational characteristics of the proposed approaches. The same problems are also solved using four other approaches suggested in the literature, and the overall computational efficiency characteristics are compared and discussed.

Simulation optimization refers to the iterative procedure in search of the optimal parameter when the objective function can only be evaluated by stochastic simulation. STRONG (Stochastic Trust Region Response Surface Convergent Method) is a newly developed design-of-experiments based simulation optimization method. It incorporates the idea of trust region method (TRM) for deterministic optimization into the traditional response surface methodology (RSM) to eliminate the human intervention required by RSM and to achieve the desired convergence. In the earlier paper, we proved the convergence of STRONG and demonstrated its computational efficiency. The original STRONG assumes the stochastic response follows a normal distribution. This paper relaxes the normal assumption and develops a framework called STRONG-X which is applicable for generally distributed additive noise with bounded variance. The convergence of STRONG-X can be proved and the generality of STRONG-X makes it an appealing method.