Surrogate Management Framework Research Articles

Optimizing Gas Turbine Performance Using the Surrogate Management Framework and High-Fidelity Flow Modeling

This work couples high-fidelity moving-domain finite element compressible flow modeling with a Surrogate Management Framework (SMF) for optimization to effectively design a variable speed gas turbine stage. The superior accuracy of high-fidelity modeling, however, comes with relatively high computational costs, which are further amplified in the iterative design process that relies on parametric sweeps. An innovative approach is developed to reduce the number of iterations needed for optimal design, leading to a significant reduction in the computational cost without sacrificing the high fidelity of the analysis. The proposed design optimization approach is applied to a novel incidence-tolerant turbomachinery blade technology that articulates the stator- and rotor-blade positions of an annular single-stage high pressure turbine to achieve peak performance. This work also extends our understanding of rotor–stator interactions by simulating complex internal flows occurring during multi-speed turbine operation. Potential variable-speed gas turbine stage designs and the proposed optimization approach are presented to provide valuable insight into this new turbomachinery technology that can positively impact future propulsion systems.

A concurrent implementation of the surrogate management framework with application to cardiovascular shape optimization

The surrogate management framework (SMF) is an effective approach for derivative-free optimization of expensive objective functions. The SMF is typically comprised of surrogate-based infill methods (SEARCH step) coupled to pattern search optimization (POLL step). Although the latter is easy to parallelize, parallelization of the SEARCH step requires surrogate-based strategies that generate multiple candidates at each iteration. The impact of such SEARCH methods on SMF performance remains poorly explored. In this paper, we extend the SMF to incorporate concurrent evaluations at the SEARCH step by comparing two different infill approaches: single search multiple error sampling and expected improvement constant liar approaches. These variants are generalized to address non-linearly constrained problems by the filter method. The proposed methods are benchmarked for different infill sizes, while accounting for the variability in initialization. We then demonstrate the proposed methods on two shape optimization problems motivated by hemodynamically-driven surgical design. Surrogate-based multiple-infill strategies outperform their single-infill counterparts for a fixed computational time budget on bound constrained problems. Insights drawn from this study have implications not only on future instances of the SMF, but also for other surrogate-based and hybrid parallel infill methods for derivative-free optimization.

Optimization of Tissue-Engineered Vascular Graft Design Using Computational Modeling.

Tissue-engineered vascular grafts hold great promise in many clinical applications, especially in pediatrics wherein growth potential is critical. A continuing challenge, however, is identification of optimal scaffold parameters for promoting favorable neovessel development. In particular, given the countless design parameters available, including those related to polymeric microstructure, material behavior, and degradation kinetics, the number of possible scaffold designs is almost limitless. Advances in computationally modeling the growth and remodeling of native blood vessels suggest that similar simulations could help reduce the search space for candidate scaffold designs in tissue engineering. In this study, we meld a computational model of in vivo neovessel formation with a surrogate management framework to identify optimal scaffold designs for use in the extracardiac Fontan circulation while comparing the utility of different objective functions. We show that evolving luminal radius and graft compliance can be matched to that of the native vein by the end of the simulation period with judicious combinations of scaffold parameters, although the inability to match these metrics at all times reveals constraints engendered by current materials. We emphasize further that there is yet a need to examine additional metrics, and combinations thereof, when seeking to optimize functionality and reduce the potential for adverse outcomes. Impact Statement Tissue-engineered vascular grafts have considerable promise for treating myriad conditions, and multiple designs are now in FDA-approved trials. Nevertheless, the search continues for the optimal design of the underlying polymeric scaffold. We present a novel melding of a computational model of vascular adaptation and a formal method of optimization that can aid in identifying optimal design parameters, with potential to save development time and costs while improving clinical outcomes.

Optimization of Biomimetic Propulsive Kinematics of a Flexible Foil Using Integrated Computational Fluid Dynamics–Computational Structural Dynamics Simulations

A computational methodology, which combines a computational fluid dynamics (CFD) technique and a computational structural dynamics (CSD) technique, is employed to design a deformable foil whose kinematics is inspired by the propulsive motion of the fin or the tail of a fish or a cetacean. The unsteady incompressible Navier–Stokes equations are solved using a second-order accurate finite difference method and an immersed-boundary method to effectively impose boundary conditions on complex moving boundaries. A finite element-based structural dynamics solver is employed to compute the deformation of the foil due to interaction with fluid. The integrated CFD–CSD simulation capability is coupled with a surrogate management framework (SMF) for nongradient-based multivariable optimization in order to optimize flapping kinematics and flexibility of the foil. The flapping kinematics is manipulated for a rigid nondeforming foil through the pitching amplitude and the phase angle between heaving and pitching motions. The flexibility is additionally controlled for a flexible deforming foil through the selection of material with a range of Young's modulus. A parametric analysis with respect to pitching amplitude, phase angle, and Young's modulus on propulsion efficiency is presented at Reynolds number of 1100 for the NACA 0012 airfoil.

Optimizing fluid–structure interaction systems with immersogeometric analysis and surrogate modeling: Application to a hydraulic arresting gear

This work describes a fluid–structure interaction (FSI) design optimization framework and applies it to improving the structural performance of a water brake used to stop aircraft landing on short runways. Inside the water brake, a dissipative torque is exerted on a rotor through interactions between rotor blades and a surrounding fluid. We seek to optimize blade shape over a parameterized design space, to prevent potentially-damaging stress concentrations without compromising performance. To avoid excessive numbers of costly simulations while exploring the design space, we use a surrogate management framework that combines derivative-free pattern search optimization with automated construction of a low-fidelity surrogate model, requiring only a handful of high-fidelity FSI simulations. We avoid the difficult problem of generating fluid and structure meshes at new points in the design space by using immersogeometric FSI analysis. The structure is analyzed isogeometrically: its design geometry also serves as a computational mesh. This geometry is then immersed in an unfitted fluid mesh that does not depend on the structure’s design parameters. We use this framework to make significant improvements to a baseline design found in the literature. Specifically, there is a 35% reduction of von Mises stress variance and a 25% reduction of maximum of stress, while the resisting torque and mass of the optimized blades remain uncompromised.

Statistical Surrogate Formulations for Simulation-Based Design Optimization

Typical challenges of simulation-based design optimization include unavailable gradients and unreliable approximations thereof, expensive function evaluations, numerical noise, multiple local optima, and the failure of the analysis to return a value to the optimizer. One possible remedy to alleviate these issues is to use surrogate models in lieu of the computational models or simulations and derivative-free optimization algorithms. In this work, we use the R dynaTree package to build statistical surrogates of the blackboxes and the direct search method for derivative-free optimization. We present different formulations for the surrogate problem (SP) considered at each search step of the mesh adaptive direct search (MADS) algorithm using a surrogate management framework. The proposed formulations are tested on 20 analytical benchmark problems and two simulation-based multidisciplinary design optimization (MDO) problems. Numerical results confirm that the use of statistical surrogates in MADS improves the efficiency of the optimization algorithm.

Shape optimization of pulsatile ventricular assist devices using FSI to minimize thrombotic risk

In this paper we perform shape optimization of a pediatric pulsatile ventricular assist device (PVAD). The device simulation is carried out using fluid---structure interaction (FSI) modeling techniques within a computational framework that combines FEM for fluid mechanics and isogeometric analysis for structural mechanics modeling. The PVAD FSI simulations are performed under realistic conditions (i.e., flow speeds, pressure levels, boundary conditions, etc.), and account for the interaction of air, blood, and a thin structural membrane separating the two fluid subdomains. The shape optimization study is designed to reduce thrombotic risk, a major clinical problem in PVADs. Thrombotic risk is quantified in terms of particle residence time in the device blood chamber. Methods to compute particle residence time in the context of moving spatial domains are presented in a companion paper published in the same issue (Comput Mech, doi: 10.1007/s00466-013-0931-y , 2013). The surrogate management framework, a derivative-free pattern search optimization method that relies on surrogates for increased efficiency, is employed in this work. For the optimization study shown here, particle residence time is used to define a suitable cost or objective function, while four adjustable design optimization parameters are used to define the device geometry. The FSI-based optimization framework is implemented in a parallel computing environment, and deployed with minimal user intervention. Using five SEARCH/POLL steps the optimization scheme identifies a PVAD design with significantly better throughput efficiency than the original device.

A surrogate management framework using rigorous trust-region steps

Surrogate models are frequently used in the optimization engineering community as convenient approaches to deal with functions for which evaluations are expensive or noisy, or lack convexity. These methodologies do not typically guarantee any type of convergence under reasonable assumptions. In this article, we will show how to incorporate the use of surrogate models, heuristics, or any other process of attempting a function value decrease in trust-region algorithms for unconstrained derivative-free optimization, in a way that global convergence of the latter algorithms to stationary points is retained. Our approach follows the lines of search/poll direct-search methods and corresponding surrogate management frameworks, both in algorithmic design and in the form of organizing the convergence theory.

New horizons in sphere-packing theory, part II: lattice-based derivative-free optimization via global surrogates

Derivative-free algorithms are frequently required for the optimization of nonsmooth scalar functions in n dimensions resulting, for example, from physical experiments or from the statistical averaging of numerical simulations of chaotic systems such as turbulent flows. The core idea of all efficient algorithms for problems of this type is to keep function evaluations far apart until convergence is approached. Generalized pattern search (GPS) algorithms, a modern class of methods particularly well suited to such problems, accomplish this by coordinating the search with an underlying grid which is refined, and coarsened, as appropriate. One of the most efficient subclasses of GPS algorithms, known as the surrogate management framework (SMF; see Booker et al. in Struct Multidiscip Optim 17:1---13, 1999), alternates between an exploratory search over an interpolating function which summarizes the trends exhibited by existing function evaluations, and an exhaustive poll which checks the function on neighboring points to confirm or confute the local optimality of any given candidate minimum point (CMP) on the underlying grid. The original SMF algorithm implemented a GPS step on an underlying Cartesian grid, augmented with a Kriging-based surrogate search. Rather than using the n-dimensional Cartesian grid (the typical choice), the present work introduces for this purpose the use of lattices derived from n-dimensional sphere packings. As reviewed and analyzed extensively in Part I of this series (see Belitz, PhD dissertation, University of California, San Diego, 2011, Chap. 2), such lattices are significantly more uniform and have many more nearest neighbors than their Cartesian counterparts. Both of these facts make them far better suited for coordinating GPS algorithms, as demonstrated here in a variety of numerical tests.

The impact of uncertainty on shape optimization of idealized bypass graft models in unsteady flow

It is well known that the fluid mechanics of bypass grafts impacts biomechanical responses and is linked to intimal thickening and plaque deposition on the vessel wall. In spite of this, quantitative information about the fluid mechanics is not currently incorporated into surgical planning and bypass graft design. In this work, we use a derivative-free optimization technique for performing systematic design of bypass grafts. The optimization method is coupled to a three-dimensional pulsatile Navier–Stokes solver. We systematically account for inevitable uncertainties that arise in cardiovascular simulations, owing to noise in medical image data, variable physiologic conditions, and surgical implementation. Uncertainties in the simulation input parameters as well as shape design variables are accounted for using the adaptive stochastic collocation technique. The derivative-free optimization framework is coupled with a stochastic response surface technique to make the problem computationally tractable. Two idealized numerical examples, an end-to-side anastomosis, and a bypass graft around a stenosis, demonstrate that accounting for uncertainty significantly changes the optimal graft design. Results show that small changes in the design variables from their optimal values should be accounted for in surgical planning. Changes in the downstream (distal) graft angle resulted in greater sensitivity of the wall-shear stress compared to changes in the upstream (proximal) angle. The impact of cost function choice on the optimal solution was explored. Additionally, this work represents the first use of the stochastic surrogate management framework method for robust shape optimization in a fully three-dimensional unsteady Navier–Stokes design problem.

Constrained optimization of an idealized Y-shaped baffle for the Fontan surgery at rest and exercise

Coupling shape optimization to three-dimensional unsteady cardiovascular simulations poses several key challenges, including high computational cost, a need to handle constraints, and a need for automatic generation of parameterized vessel geometry. In this work we extend our previous framework for cardiovascular optimization to include constraints, pulsatile flow under both rest and exercise conditions, and multiple geometric parameters. Optimization is performed using a derivative-free algorithm called the surrogate management framework, with constraints enforced using a filter method. In this work, we examine a specific surgery called the Fontan, which is performed to treat single-ventricle heart patients. These patients typically undergo a series of three surgeries, the last of which connects the inferior vena cava to the pulmonary arteries. Our group and others have recently proposed and evaluated a new Y-graft modification of the Fontan operation that replaces the current tube shaped baffle. Preliminary simulations have shown that the Y-graft modification is a promising design that increases energy efficiency and improves flow distribution to the pulmonary arteries. In this work, we perform optimization on a model Y-graft design problem. This work represents the first use of formal design optimization methods for the Fontan surgery, and also demonstrates the applicability of the optimization framework on a pulsatile flow problem with multiple design parameters and constraints. The idealized Y-graft model was parameterized with six geometric parameters including graft diameter and anastomosis locations, and the optimization procedure, including model construction, meshing, and simulation, was executed automatically. Energy efficiency was chosen as the objective function. A constraint on the wall shear stress (WSS), a presumed correlate to thrombosis risk, was added to the problem using a filter method, which allowed for exploration of the trade-offs between WSS and energy efficiency. Optimization was performed at two exercise levels with effects of respiration incorporated, and differences in optimal solutions were examined. It was shown that optimal shapes differed between rest and exercise, as well as steady and pulsatile flow conditions, with wide-span branches and decreasing graft branch size favored with increasing exercise level. The optimization method was found to be robust for different polling strategies, and computationally efficient both with and without constraints.

A method for stochastic constrained optimization using derivative-free surrogate pattern search and collocation

Recent advances in coupling novel optimization methods to large-scale computing problems have opened the door to tackling a diverse set of physically realistic engineering design problems. A large computational overhead is associated with computing the cost function for most practical problems involving complex physical phenomena. Such problems are also plagued with uncertainties in a diverse set of parameters. We present a novel stochastic derivative-free optimization approach for tackling such problems. Our method extends the previously developed surrogate management framework (SMF) to allow for uncertainties in both simulation parameters and design variables. The stochastic collocation scheme is employed for stochastic variables whereas Kriging based surrogate functions are employed for the cost function. This approach is tested on four numerical optimization problems and is shown to have significant improvement in efficiency over traditional Monte-Carlo schemes. Problems with multiple probabilistic constraints are also discussed.

Optimization of Active Flow Control over an Airfoil Using a Surrogate-Management Framework

An efficient method based on the surrogate-management framework has been excised to optimize the actuation parameters of active flow control over an airfoil via a synthetic jet. In this approach, sample points are chosen by the design of experiments method, and approximation models are built based on the sampled data obtained from unsteady Reynolds-averaged Navier―Stokes simulations. The accuracy of these approximation models is evaluated at some test points by comparing the approximated values with the accurate values obtained from unsteady Reynolds-averaged Navier―Stokes simulations. Three types of approximation models (quadratic response-surface model, kriging model, and radial-basis-function neutral network) are built from the same data set. The model with highest accuracy is chosen as the surrogate model to be used to replace the unsteady Reynolds-averaged Navier― Stokes analysis during optimization. The optimization objective is to maximize the lift coefficient of a NACA 0015 airfoil at given angles of attack (14 to 22°), with the jet momentum coefficient, nondimensional frequency, and jet angle being the design variables. The surrogate model is coupled with a simulated annealing genetic algorithm optimizer to efficiently obtain the global optimum. As a result of the optimization process, the lift coefficient at an angle of attack of 16° is increased by 16.9% and the corresponding drag is decreased by 13.4% with respect to the initial controlled flow. It is preliminarily shown that the presented method is efficient and applicable for optimization of active flow control via a synthetic jet.

A computational framework for derivative-free optimization of cardiovascular geometries

Predictive simulation and optimization-based design tools have great potential to improve the design of surgeries and interventions used in cardiovascular medicine. The present work builds upon recent advances in blood flow simulation capabilities to develop tools for optimization. A framework for coupling optimal shape design to time-accurate three-dimensional blood flow simulations in idealized cardiovascular geometries is presented. The optimization method employed is a tailored version of the surrogate management framework, a method that was developed previously for expensive functions with little or no gradient information. In this method, we employ a derivative-free approach using surrogates for increased efficiency together with mesh adaptive direct search to guarantee convergence to a local minimum. The optimization procedure has been fully automated to include the generation and parameterization of three-dimensional solid geometries, mesh generation, finite element flow computation, post processing and optimization. This methodology is demonstrated on three model problems that are representative of typical cardiovascular geometries: a stenosis, a vessel bifurcation modeled on Murray’s law, and an end-to-side anastomosis. These problems are used to illustrate the importance of the choice of cost function and the performance of the optimization algorithm. This framework for treatment optimization will be applicable to a wide range of cardiovascular surgical or catheter-based interventions in future work.

Trailing-edge noise reduction using derivative-free optimization and large-eddy simulation

Derivative-free optimization techniques are applied in conjunction with large-eddy simulation (LES) to reduce the noise generated by turbulent flow over a hydrofoil trailing edge. A cost function proportional to the radiated acoustic power is derived based on the Ffowcs Williams and Hall solution to Lighthill's equation. Optimization is performed using the surrogate-management framework with filter-based constraints for lift and drag. To make the optimization more efficient, a novel method has been developed to incorporate Reynolds-averaged Navier–Stokes (RANS) calculations for constraint evaluation. Separation of the constraint and cost-function computations using this method results in fewer expensive LES computations. This work demonstrates the ability to fully couple optimization to large-eddy simulation for time-accurate turbulent flow. The results demonstrate an 89% reduction in noise power, which comes about primarily by the elimination of low-frequency vortex shedding. The higher-frequency broadband noise is reduced as well, by a subtle change in the lower surface near the trailing edge.

Suppression of vortex-shedding noise via derivative-free shape optimization

In this Letter we describe the application of a derivative-free optimization technique, the surrogate management framework (SMF), for designing the shape of an airfoil trailing edge which minimizes the noise of vortex shedding. Constraints on lift and drag are enforced within SMF using a filter. Several optimal shapes have been identified for the case of laminar vortex shedding with reasonable computational cost using several shape parameters, and results show a significant reduction in acoustic power. Physical mechanisms for noise reduction are discussed.

Optimal Aeroacoustic Shape Design Using the Surrogate Management Framework

Shape optimization is applied to time-dependent trailing-edge flow in order to minimize aerodynamic noise. Optimization is performed using the surrogate management framework (SMF), a non-gradient based pattern search method chosen for its efficiency and rigorous convergence properties. Using SMF, design space exploration is performed not with the expensive actual function but with an inexpensive surrogate function. The use of a polling step in the SMF guarantees that the algorithm generates a convergent subsequence of mesh points in the parameter space. Each term of this subsequence is a weak local minimizer of the cost function on the mesh in a sense to be made precise later. We will discuss necessary optimality conditions for the design problem that are satisfied by the limit of this subsequence. Results are presented for an unsteady laminar flow past an acoustically compact airfoil. Constraints on lift and drag are handled within SMF by applying the filter pattern search method of Audet and Dennis, within which a penalty function is used to form and optimize a surrogate function. Optimal shapes that minimize noise have been identified for the trailing-edge problem in constrained and unconstrained cases. Results show a significant reduction (as much as 80%) in acoustic power with reasonable computational cost using several shape parameters. Physical mechanisms for noise reduction are discussed.

Reliability Based Design Optimization Using Design Explorer

Experimental design methods can be applied to engineering design activities to understand which variables affect the system under consideration, how these variables affect the system, and how to select variable settings that will give uniformly long life to the system. The objective of this paper is to demonstrate the use of Design and Analysis of Computer Experiment (DACE) methods (Sacks, J. et al., 1989) and design optimization via the Surrogate Management Framework (Booker, A. J. et al., 1999; Audet, C. et al., 2000) on reliability optimization problems. Reliabilities are calculated using the Probabilistic Structural Analysis Method (Palle Thoft–Christensen and Baker, 1982; Achintya Haldar and Sankaran Mahadevan, 2000), a method for estimation of reliabilities and reliability indices for a structural model given probability distributions for design variables and “environmental” variables such as loads. By maximizing reliability, or minimizing the probability of failure, we attempt to achieve a minimum cost design that is affected minimally by the variability in the design variables.