Simulation-based Design Optimization (SBDO)

To design products, the designer often needs a mathematical description of the system behavior as a function of a set of input (or design) variables. Systematic use of these mathematical models allows the designer to gain knowledge about the influence of different design variables on the responses of the system, and to search for the best solution to the design problem in terms of a suitable set of performance measures.

In practice, however, the designer rarely has a model of the system in the initial stages of the design process and, if such a model is available, it is usually expensive to evaluate and thus unsuited for systematic use in preliminary design. Creation of approximate models, or metamodels, of system behavior based on a limited set of data is a common approach used in practice to circumvent these problems. Metamodels, also known as surrogate models, or simply "surrogates", are mathematical approximations of system's behavior, built using data collected by experimentation with the system itself, a prototype, or with a computationally-expensive mathematical model of it. Assuming that the cost of performing one experiment, either physical or computational, is relatively large, it can be argued that metamodels allow significant savings by reducing the resources devoted to modeling during the design process.

Simulation-based Design Optimization relates to the development, implementation, verification, validation and dissemination of methods to assist decision making and optimization during the design process. Although it is a research area with wide applicability, developments have focused especially in the design of complex systems with deterministic behavior that can be modeled with computationally-expensive computer codes. Particular attention has been paid to methods based on metamodels to reduce the cost of the design process, an area known as Metamodel-based Design Optimization (MBDO).

To this research area, we have developed and are developing contributions in the following subfields:

Multiple Responses

In applications in which several response variables (e.g. performance metrics) of the system are obtained from the same computer simulation, we are working under the hypothesis that the observed correlation between the responses can be used to improve the prediction capabilities of the metamodels beyond what is possible by modeling each response variable independently, hence achieving better models with smaller data sets.

However, in order to realize this potential for improved prediction capabilities in the context of deterministic computer experiments, there is the need for multi-response covariance/correlation functions that are able to represent the observed correlation among the response variables, and to do so with the fewest parameters possible. We address this issue by proposing a family of multi-response covariance functions and studying the predictive performance of the resulting metamodels in an empirical simulation study with a set of correlated test functions.

Let us illustrate the potential for application of multi-response modeling techniques with an example. Figure 1 shows a section of a cross-flow heat exchanger. Fluid "A" flows in the direction pointed to by the velocity vectors and exchanges heat with fluid "B", which is flowing inside the tubes in the direction perpendicular to the plane of the figure. Also shown in the figure are some design variables for this system, such as the distance between tubes (p), the diameter of the tubes (D), and the eccentricity between consecutive tube rows (e). The performance of the system is measured by the overall heat transfer coefficient between the two fluids and by the pressure drop caused in fluid "A", which obviously depend on the settings of the design variables.

Figure 1. Schematic representation of the cross-flow heat exchanger problem.

However, note that due to the underlying physics behind this phenomenon, the behavior of the two performance metrics of interest is somehow correlated, as design configurations with the largest heat transfer coefficient (desirable) will most likely be those that cause the largest pressure drops in the fluid (undesirable). This might lead the designer to wonder, for example, to what extent knowing the heat transfer coefficient of a particular design configuration helps determine the corresponding pressure drop, or if there is any sound approach for using such information to help in decision making.

In addition, note that in the context of Simulation-Based Design and Optimization (SBDO) of this system, both performance metrics would be typically obtained from a single CFD simulation or physical experiment. These facts raise some interesting questions regarding, for example, whether there is anything to be gained by metamodeling both responses simultaneously or, alternatively, if by metamodeling each response variable individually we are disregarding available knowledge about the system behavior. Reflecting on these issues in more general terms, we may question ourselves regarding the suitability of state-of-the-art metamodeling techniques and experimental designs for deterministic computer experiments in a multi-response context. Answering these questions is the underlying goal of our research.

Integration of Information Sources

In principle, available information about system responses, such as legacy data from previous analysis, expert opinions, back-of-the-envelope calculations, monotonicity and sensitivity information, as well as designer intuition, could be used to improve predictions without requiring more samples. This information could be used (a) indirectly, by influencing metamodeling decisions, such as experimental design, metamodel type, fitting method and design of the validation set, or (b) directly, by influencing metamodel predictions. Moreover, it is our opinion that the Bayesian paradigm provides a suitable framework for these applications, as it enables consideration of prior information about system responses and model parameters, albeit in the form of prior distributions. Since rigorous Bayesian methods are usually difficult and time-consuming to implement, pseudo-Bayesian methods could provide a compromise solution for the integration of information sources into metamodels.

Regarding the use of available information directly in metamodel predictions, previous work by our group focused on the integration of information sources into global metamodels. These studies identified two categories of information that might be available to the designer, namely a priori and a posteriori information. An example of the former are "physical priors", i.e. physically-based, simplified models that typically provide upper or lower bounds on system behavior due to their best-case/worst-case nature. We have developed and are currently testing a framework for including this upper/lower bound information into the metamodels.

As a further development of this idea, our group is currently developing a formulation for multi-stage, multi-response Bayesian surrogate models (MSMR-BSM). This formulation enables the integration of different sources of information about multiple system responses, with different levels of accuracy, into a single, global model of the system. Our preliminary results have been very encouraging, showing that the use of bound information results in significant improvements in metamodel accuracy without requiring extra samples.

Rational Design based on Variable Fidelity Models

Computer models and simulations are essential system design tools that can reduce the cost of the design process and allow for improved decision making during all phases of the process. However, the most accurate models tend to be computationally expensive and can only be used sporadically. Consequently, designers are forced to choose between exploring many alternatives with less accurate, inexpensive models and evaluating fewer alternatives with the most accurate models.

To achieve both broad exploration of the design space and accurate determination of the best alternatives, surrogate and variable accuracy modeling is gaining in popularity. A surrogate model is a mathematically tractable approximation of a more expensive model based on a limited sampling of that model. Variable accuracy modeling involves a collection of different models of the same system with different accuracies and computational costs. We hypothesize that designers can determine the best solutions more efficiently using surrogate and variable accuracy models. This hypothesis is based on the observation that very poor solutions can be eliminated using only less accurate models. The most accurate models are then reserved for discerning the best solution from the set of good solutions.

In this research project, headed by Prof. Christiaan Paredis (Mechanical Engineering, Georgia Institute of Technology, USA) and developed in collaboration with our group, a new approach for global optimization is introduced, which combines variable accuracy models with kriging surrogate models and a sequential sampling strategy based on a Value of Information (VOI) metric.

There are two main contributions. The first is a novel surrogate modeling method that accommodates data from any number of different models of varying accuracy and cost. The proposed surrogate model is based on Gaussian process, much like classic kriging modeling approaches. However, in this new approach, the error between the model output and the unknown truth (the real world process) is explicitly accounted for. When variable accuracy data is used, the resulting response surface does not interpolate the data points but provides an approximate fit giving the most weight to the most accurate data. The second contribution is a new method for sequential sampling. Information from the current surrogate model is combined with the underlying variable accuracy models' cost and accuracy to determine where best to sample next using the VOI metric. This metric is used to mathematically determine where next to sample and with which model. In this manner, the cost of further analysis is explicitly taken into account during the optimization process.

Optimization of Wind Farm Layouts

In recent years, there has been a growing interest on sustainable energy resources, such as solar, geothermal, wave, and wind energy. The renewable energy sector's share of the energy supply is expected to grow to 18.6% by 2030. Moreover, the International Energy Agency (IEA) predicts that wind energy will have a 12% share of the global energy supply by 2050. To reach this target, the wind energy production capacity will have to increase at an average rate of 47 GW/year, resulting in massive investments and a dynamic, growing market for wind-related technology and services.

The overall vision for this research program is to develop a state-of-the-art methodology for optimal design of wind farms, with simultaneous consideration of power generation, environmental impact with respect to noise, and life-cycle cost of the wind farm, including its decommissioning. This goal will be achieved by joining efforts with an industrial partner, a global leader in the energy infrastructure sector with extensive expertise and experience in the design, construction and commissioning of wind farms.

During the first stage of this project, covering the first 6 months, the research team will evaluate the limitations of existing methods for the design of the wind farm layout under consideration of energy output and noise generation only, and will tailor existing multi-objective optimization methods with the goal of facilitating compliance with the Renewable Energy Approval process (see Figure).

Figure 2. The iterative wind farm design process.