ShyyLab :: Research

Optimization/ Surrogate Modeling

Optimization in its purest form seeks the ‘best’ answer given some set of constraints and conditions. This type of answer can indeed be very valuable. However sometimes the question that needs to be answered is, ‘What is happening?’ not ‘What is the optimal solution?’ For both of these questions, the tools available under the optimization framework can help you find the answer.

Quite often, knowledge about practical engineering problems is limited by the fact that analytical solutions to them are difficult or impossible to come by. Information is only known at a finite number of points and these were probably obtained either via complex experiments or time-consuming computer simulations. Particularly helpful in extracting as much meaningful insight from a given set of data are the surrogate modeling and sensitivity analysis tools. Surrogate modeling offers a method of estimating, with varying degrees of sophistication, information away from the known points, whereas sensitivity analysis quantifies the relative importance of the various design variables and their combinations. Together they provide an efficient method for analyzing/simplifying otherwise intractable design spaces and allow us to quantitatively qualify observed correlations.

We are interested in coupling computational fluid dynamics (CFD) and surrogate modeling to address these issues. However, there are uncertainties in predictions using this approach, like empiricism in computational models and surrogate model errors. We develop methods to estimate and to reduce such uncertainties. In our research, we investigate (i) experimental designs using multiple criteria to help data generation, (ii) ensemble of surrogates to reduce modeling uncertainties and sampling strategy, (iii) quantitative measures of prediction errors, (iv) approaches enabling the probing of sensitivity of design variable in a global context, and (v) methodology for supporting multi-objective considerations. Generally, the process of applying the surrogate models to the engineering design problems consists of determining the strategy for exploring the design variable space, constructing the approximate response of the variables on the physical process (objective) and finding the optimal combination of variables that produce the desirable objective. Some key features of the method are as following.

Design of experiments(DoE) : This is the sampling plan of design variables(independent variables) to make the variance of predicted objective functions(dependent variables) from the exact values as small as possible. Since either the resources for the numerical or experimental simulation are limited and bound to various errors, the plan may affect the quality of the models significantly. Popular schemes employed are factorial design, central composite design(CCD), face-centered cube(FCD), Latin hypercube sampling(LHS), D-optimal design, and their various combinations.

Response surface methodology (RSM) : The performance measure of a system or the dependent variable (objective) of a numerical model is called the response. RSM examines the relationships between the input parameters (design variables) and the response to predict its behavior in an unknown region of the design space. Often, this approach is employed to look for optimal response behavior within the design region.

Surrogate models : Instead of doing every experiment or numerical simulation, an explicit numerical description, i.e. a surrogate model can be used to relate the design variables and the response if the model guarantees accuracy. Widely used surrogate models are polynomial response surfaces(PRS), Kriging, artificial neural networks, and combinations of them.

Pareto front : In a multi-objective design problem, it may be possible to point out a response which is more desirable in one objective without sacrificing other objectives. The combination of these points in the objective function space creates a front, i.e. a Pareto front named after the economist, Vilfredo Pareto(1848-1923), which conveys important information on optimality, since the optimal point may be contained in the front.

Global sensitivity analysis(GSA) : GSA is estimating the effect of design variables on the total variability of the objective functions. Since it delivers the domain-averaged effect of the variable which may be correlated to other parameters, the sensitivities of variables can be used as the measure of importance to the model, and thus make the design process more efficient by enabling a designer to choose whether some variables can be ignored or not in the optimization process. Lowering the number of independent design variables can greatly reduce the complexity of the entire exercise.

Surrogate model

Functional relationship between design variables and response y=f(x)

Advantages

Independent of data generation: exp, high/low fidelity comp
Easy coupling with optimization software
Facilitates global sensitivity analysis
Naturally suitable for multi-objective evaluations

Popular surrogates:

Polynomial response surface; Kriging; radial basis neural network

Surrogate based optimization framework:

optimization3_600

Specific examples:

A merit of the present surrogate modeling technique is its capabilities of handling hardware design, as well as modeling and computational issues. We have applied these techniques to design of supersonic turbines, injector flows, radial turbines, diffusers, . We have also adopted the same methodology to help address the modeling issues such as cryogenic cavitation, dielectric barrier discharge(DBD) flow actuator, flapping wing kinematics, and lithium ion intercalation.

Large scale computation-facilitated injector design optimization for liquid rocket propulsion:

Objective: Establish optimization techniques to enable CFD-based design optimization of propulsion components: quantitative cost-benefit analysis, ranked influence of design variables.
Accomplishments: Have conducted single-injector (combined shear coax and impinging) optimization and validated the methodology and framework.
- Design variables: α, ΔHA, ΔOA, OPTT geometry variables for fuel and oxidizer.
- Minimize all objectives: X_cc, TT_max, TW₄, TF_max combustion efficiency and thermal load.
Surrogate model, genetic algorithm & Pareto fronts enable quantitative trade-off consideration
- 10% penalty in TF_max improves TT_max~ 60%
Global analysis ranks ΔHA and ΔOA more influential for X_cc (performance).

References:

U of M \| CoE \| Aerospace
Computational Thermo-Fluids @ U of M, Aerospace Engineering