The development of mathematical modeling techniques and computer science over the past decades has greatly increased the possibility to perform detailed simulations of complex systems. However, detailed simulations requires significant, sometimes impractical, computational resources, even when using state of the art hardware. The conventional strategy has therefore been to focus on modeling a particular scale of a system. Consequently, it is common to assume that effects from other scales, e.g. molecular dynamics, can be neglected by invoking assumptions, such as pseudo steady state, about the behaviour or by using constitutive relations.
Multi-scale modeling, on the other hand, involves simultaneous consideration of the different scales of a system, as well as the combination of these scales into an overall model for the system. The objective is to achieve the efficiency of a higher scale model combined with the accuracy of a lower scale. In this context, the interactions between different scales play a significant role in determining the quality of the overall model. Surrogate models provide an inexpensive and efficient way of linking the different models in a multi-scale system. As long as the error of the surrogate model is kept within an acceptable range, the surrogate approximates the behaviour of the complex model on the lower scale well enough in a given design space.
The MoDeNa [1] framework aims at facilitating interactions between the models on across the different scales of the system by extensive use of surrogate models. The respective surrogate models are derived by the experts responsible for the corresponding complex models, but the surrogate models contain parameters, i.e. degrees of freedom, that are used for optimization. The parameters are determined using Design of Experiments (DoE), i.e. formulating experimental designs within a design space such that it leads to a good enough surrogate model, in combination with fitting techniques. Based on the argument that computer simulation are deterministic [3], the statistically optimal points in which the complex simulations should be performed are chosen using DoE methods such as latin hypercube sampling. Finally, the parameters of the surrogate model are fitted to results of these experiments using nonlinear least square optimisation. The procedure, as outlined in [2], is called model-based design of experiments.
However, surrogate models in MoDeNa are 'dynamic', meaning that the parameters of the model can be updated depending on where in the design space the user wants to use the model. In context of MoDeNA, the DoE and parameter fitting are defined as strategies specific to a surrogate model. This means that the model itself can employ these strategies whenever required - depending upon its structure, desired range of validity and application.
The feature is particularly useful because it allows the the user to use the surrogate model outside the design space for which the model was originally validated. MoDeNa handles this situation by expanding the design space beyond the user-specified input, in order to ensure that the input is within the range of the surrogate model, before performing new experiments in the extended space. The parameter fitting procedure is applied afterwards in order to extend the global range of the surrogate model by updating its parameters.
However, it is important to ensure that the physical boundaries of the design variables are not crossed when expanding the design space. Moreover, by studying the trajectory of the user specified inputs in previous runs of the simulation, a predictive algorithm for expanding the design space can be formulated. This helps in reducing the computational time as the surrogate model is updated in a new design space even before the user intends to compute it in that space. The accuracy of the surrogate model is ensured by validating it agains the points from the DoE that was not used in order to fit the parameters.
[1] | MoDeNa. The modena project. [ www.modenaproject.eu ] |
[2] | Gaia Franceschini and Sandro Macchietto. Model-based design of experiments for parameter precision: State of the art. Chemical Engineering Science, 63(19):4846 - 4872, 2008. Model-Based Experimental Analysis. [ DOI | http ] |
[3] | Anthony A. Giunta, Steven F. Wojtkiewicz Jr., and Michael S. Eldred. Overview of modern design of experiments methods for computational simulations. In 41st AIAA aerospace sciences meeting and exhibit, 2003. |
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