Trust Region Model Management Research Articles

Aerostructural Design Optimization Using a Multifidelity Quasi-Newton Method

The traditional aircraft design process relies upon low-fidelity models for expedience and resource savings. However, the reduced accuracy and reliability of low-fidelity tools often lead to the discovery of design defects or inadequacies late in the design process. These deficiencies result in either costly changes or the acceptance of a configuration that does not meet expectations. Multifidelity methods attempt to blend the increased accuracy and reliability of high-fidelity models with the reduced cost of low-fidelity models. A new multifidelity algorithm has been proposed, combining elements from typical trust region model management and classical quasi-Newton methods. In this paper, the algorithm is compared with a single-fidelity quasi-Newton method for complex aeroelastic simulations. The vehicle design problem includes variables for planform shape, structural sizing, and cruise condition with constraints on trim and structural stresses. Considering the objective function reduction versus computational expenditure, the multifidelity process performs better in three of four cases in early iterations compared with a single-fidelity approach. A contracting trust region is found to slow the progress of the multifidelity optimizer. However, by leveraging the approximate inverse Hessian, the optimization can be seamlessly continued using high-fidelity data alone. Ultimately, the proposed new algorithm produces better designs in the cases considered. Investigating the return on investment confirms that the multifidelity advantage is greatest in early iterations, and managing the transition to pure high-fidelity optimization is critical.

Multifidelity Quasi-Newton Method for Design Optimization

Multifidelity approaches are frequently used in design when high-fidelity models are too expensive to use directly and lower-fidelity models of reasonable accuracy exist. In optimization, corrected low-fidelity data are typically used in a sequence of independent, approximate optimizations bounded by trust regions. A new, unified, multifidelity quasi-Newton approach is presented that preserves an approximate inverse Hessian between iterations, determines search directions from high-fidelity data, and uses low-fidelity models for line searches. The proposed algorithm produces better search directions, maintains larger step sizes, and requires significantly fewer low-fidelity function evaluations than Trust Region Model Management. The multifidelity quasi-Newton method also provides an expected optimal point that is forward looking and is useful in building superior low-fidelity corrections. The new approach is compared with Trust Region Model Management and the BFGS quasi-Newton method on several analytic test problems using polynomial and kriging corrections. For comparison, a technique is demonstrated to initialize high-fidelity optimization when transition away from approximate models is deemed fruitful. In summary, the unified multifidelity quasi-Newton approach required fewer or equal high-fidelity function evaluations than Trust Region Model Management in about two-thirds of the test cases, and similarly reduced cost in more than half of cases compared with BFGS.

Bayesian-Enhanced Low-Fidelity Correction Approach to Multifidelity Aerospace Design

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.

Multi-Fidelity Design Optimization: Challenges In Complex Physics-based Computational Mechanics

Engineering systems development is currently pushing the envelope of traditional multidisciplinary design capabilities. Bringing multiple physics into the design loop earlier in the design process has shown promise in handling the strict requirements constantly being placed on various areas of computational mechanics, such as the design of next generation military aircraft. The goal in multi-fidelity design is to aid in this process to expand traditional design capabilities through the implementation of techniques developed to mitigate inadequacies and/or obstacles associated with various levels of complex physics in a single design process. Achieving a desired level of accuracy while maintaining a low computational cost may very well be the greatest obstacle combating computational design. However, other hindrances exist such as determining the appropriate physics (i.e. acoustic, thermal, structural), level of physics (i.e. Potential Flow, Euler, Navier Stokes), and mesh refinement to utilize in any given computational model. This work focuses on leveraging higher fidelity information to correct lower fidelity models so as to take advantage of the speed associated with the latter without compromising accuracy. Corrections are implemented via a custom Hybrid Bridge Function (HBF) while the design aspect is governed through the implementation of a special Trust Region Model Management (TRMM) methodology. Multifidelity design optimization is demonstrated on a thermal plate demonstration problem consisting of four differing levels of fidelity. Results show that employment of the described methodology succeeds in obtaining a design at a lower cost while maintaining a necessary level of accuracy.

Multi-objective ceramic matrix composite material design using the variable fidelity model management optimization framework

The increasing computational requirements of advanced numerical tools for simulating material behaviour can prohibit direct integration of these tools in a design optimization procedure where multiple iterations are required. Therefore, a design approach is needed that can incorporate multiple simulations (multi-physics with different input variables) of varying fidelity in an iterative model management framework that can significantly reduce design cycle times. In this research, a material design tool based on a variable fidelity model management framework is applied to obtain the optimal size of a second phase, consisting of silicon carbide (SiC) fibres, in a silicon-nitride (Si3N4) matrix to obtain continuous fibre SiC-Si3N4 ceramic composites (CFCCs) with maximum high temperature strength and high temperature creep resistance. This investigation shows how models with different dimensions and input design variables can be handled and integrated efficiently by the trust region model management framework, while significantly reducing design cycle times in application to the design of multiphase composite materials.

Surrogate-Based Optimization Using Multifidelity Models with Variable Parameterization and Corrected Space Mapping

Surrogate-based-optimization methods provide a means to achieve high-fidelity design optimization at reduced computational cost by using a high-fidelity model in combination with lower-fidelity models that are less expensive to evaluate. This paper presents a provably convergent trust-region model-management methodology for variable-parameterization design models: that is, models for which the design parameters are defined over different spaces. Corrected space mapping is introduced as a method to map between the variable-parameterization design spaces. It is then used with a sequential-quadratic-programming-like trust-region method for two aerospace-related design optimization problems. Results for a wing design problem and a flapping-flight problem show that the method outperforms direct optimization in the high-fidelity space. On the wing design problem, the new method achieves 76% savings in high-fidelity function calls. On a bat-flight design problem, it achieves approximately 45% time savings, although it converges to a different local minimum than did the benchmark.

Sequential approximate optimization using variable fidelity response surface approximations

The dimensionality and complexity of typical multidisciplinary systems hinders the use of formal optimization techniques in application to this class of problems. The use of approximations to represent the system design metrics and constraints has become vital for achieving good performance in many multidisciplinary design optimization (MDO) algorithms. This paper reports recent research efforts on the use of variable fidelity response surface approximations (RSA) to drive the convergence of MDO problems using a trust region model management algorithm. The present study focuses on a comparative study of different response sampling strategies based on design of experiment (DOE) approaches within the disciplines to generate the zero order data to build the RSAs. Two MDO test problems that have complex coupling between disciplines are used to benchmark the performance of each sampling strategy. The results show that these types of variable fidelity RSAs can be effectively managed by the trust region model management strategy to drive convergence of MDO problems. It is observed that the efficiency of the optimization algorithm depends on the sampling strategy used. A comparison of the DOE approaches with those obtained using a optimization based sampling strategy (i.e. concurrent subspace optimization --- CSSO) shows the DOE methodologies to be competitive with the CSSO based sampling methodology in some cases. However, the CSSO based sampling strategy was found to be, in general, more efficient in driving the optimization.

Efficient approximation method for constructing quadratic response surface model

For a large scaled optimization based on response surface methods, an efficient quadratic approximation method is presented in the context of the trust region model management strategy. If the number of design variables inn, the proposed method requires only 2n+1 design points for one approximation, which are a center point and two additional axial points within a systematically adjusted trust region. These design points are used to uniquely determine the main effect terms such as the linear and quadratic regression coefficients. A quasi-Newton formula then uses these linear and quadratic coefficients to progressively update the two-factor interaction effect terms as the sequential approximate optimization progresses. In order to show the numerical performance of the proposed method, a typical unconstrained optimization problem and two dynamic response optimization problems with multiple objective are solved. Finally, their optimization results compared with those of the central composite designs (CCD) or the over-determined D-optimality criterion show that the proposed method gives more efficient results than others.

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.

Convergence of trust region augmented Lagrangian methods using variable fidelity approximation data

To date the primary focus of most constrained approximate optimization strategies is that application of the method should lead to improved designs. Few researchers have focused on the development of constrained approximate optimization strategies that are assured of converging to a Karush-Kuhn-Tucker (KKT) point for the problem. Recent work by the authors based on a trust region model management strategy has shown promise in managing the convergence of constrained approximate optimization in application to a suite of single level optimization test problems. Using a trust-region model management strategy, coupled with an augmented Lagrangian approach for constrained approximate optimization, the authors have shown in application studies that the approximate optimization process converges to a KKT point for the problem. The approximate optimization strategy sequentially builds a cumulative response surface approximation of the augmented Lagrangian which is then optimized subject to a trust region constraint. In this research the authors develop a formal proof of convergence for the response surface approximation based optimization algorithm. Previous application studies were conducted on single level optimization problems for which response surface approximations were developed using conventional statistical response sampling techniques such as central composite design to query a high fidelity model over the design space. In this research the authors extend the scope of application studies to include the class of multidisciplinary design optimization (MDO) test problems. More importantly the authors show that response surface approximations constructed from variable fidelity data generated during concurrent subspace optimizations (CSSOs) can be effectively managed by the trust region model management strategy. Results for two multidisciplinary test problems are presented in which convergence to a KKT point is observed. The formal proof of convergence and the successfull MDO application of the algorithm using variable fidelity data generated by CSSO are original contributions to the growing body of research in MDO.

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.