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The first center is chosen from a uniform distribution over the input space.This method places the next kernel center at the location of minimum density; hence, the resulting sequence of kernel centers fills the space approximately uniformly. The input space is searched for the minimum using multi-start sequential quadratic programming (fmincon in (13)(14)(15)Multidimensional, Multimodal Kernel Based FunctionsTest Functions and Experimental Design MATLAB®), stopping if more than one second elapses or less than 1% improvement occurs for 5 consecutive sequential quadratic programming iteration.Although this procedure is not guaranteed to find the global minimum, finding the global minimum is not imperative; we only need to find a point in a low density region. As for the choice of smoothing parameter, setting it too small will result in a surface with spiky peaks at each kernel center, while setting it too large creates humps with maxima that are not necessarily located at the kernel center. However, for the case of the triweight kernel function, one can guarantee an N-modal function by setting the smoothing parameter to 95% of the minimum Euclidean distance between any two kernel centers.The KDE method is used to create three multimodal functions that are scaled from one through ten dimensions (i.e., one through ten independent variables). In the first scenario, there are N = 2 kernels (modes) regardless of the dimensionality of the problem. In the second scenario, there are N = D kernels, i.e. the number of modes is equal to the number of independent variables. Lastly, in the third scenario, there are N = 2D kernels and the number of modes is always equal to twice the number of independent variables. Posing the problem in this way creates a unique challenge for the metamodeling methods. Specifically, the effect of scaling the number of independent variables can be investigated directly for different levels of modality. Two Stream Counter-Flow Heat ExchangerIn addition to the functions generated using the KDE method, a two stream, counter-flow shell and tube heat exchanger, such as that used in a shipboard freshwater cooling loop, is used as an example function. A schematic of this type of heat exchanger is shown in Fig. 1a. The heat exchanger features a bundle of conductive tubes inside a cylindrical shell. The hot and cold streams flow in opposite directions resulting in temperature profiles similar to those shown in Fig. 1b. The working fluid in the hot and cold streams is assumed to be fresh liquid water. The temperature dependence of the fluid properties is not ignored, and the outer surface of the shell is assumed to be adiabatic. at a time in the order listed in Table 1. The variables whose main effects have the most curvature are added last in an effort to exploit weaknesses of any particular method in high dimensional space.Training and Test Point Sampling StrategyThe method for sampling training points from the base model can have a significant effect on the accuracy of the resulting metamodel. In contrast to physical experiments, which are stochastic in nature, deterministic computer models are not subjected to repeated sampling because their predictions typically do not vary unless the input variables change. Therefore, sampling strategies for computer experiments aim to fill the design space as uniformlyas possible (Koehler and Owens, 1996). There are several so-called space filling designs such as Latin hypercube designs (Mckayet al., 1979), Hammersley sequence sampling (Hammersley, 1960), orthogonal arrays (Owen, 1992), and uniform designs (Fang et al., 2000). Simpson et al. (Simpson et al., 2002) find that uniform designs and Hammersley sequence The output of this model is the overall heat transfer rate from the hot stream to the cold stream. This response, given by (16), is obtained by multiplying the maximum theoretical heat transfer rate by the heat exchanger effectiveness ɛ:whereandIn (17), ṁh and ṁc are the mass flow rates of the hot and cold streams in the heat exchanger, respectively. The Cp,h and Cp,c terms refer to the specific heats of the hot and cold streams, respectively. In (18), UA is the overall heat transfer coefficient between the two fluids and is calculated using a combination of conductive and convective heat transfer equations (Mills, 1998). Lastly, the maximum theoretical heat transfer rate is given by (19):where TH,in and TC,in are the hot stream and cold stream inlet temperatures, respectively. A thorough explanation and derivation of this heat exchanger model is provided in (Mills, 1998).Ten independent variables from the above heat exchanger model are selected for this study. The variables and response are listed in Table 1, along with their respective symbols and units. The variables listed in Table 1 are self explanatory with the exception of the flow area ratio.