Interdisciplinary Applied Mathematics

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In Chapter 16 we discuss theory and numerical methodologies for simulating liquid flows at the atomistic and mesoscopic levels. The atomistic description is necessary for liquids contained in domains with dimension of fewer than ten molecules. First, we present the Molecular Dynamics (MD) method, a deterministic approach suitable for liquids. We explain details of the algorithm and focus on the various potentials and thermostats that can be used. This selection is crucial for reliable simulations of liquids at the nanoscale. In the next section we consider various approaches in coupling atomistic with mesoscopic and continuum level. Such coupling is quite difficult, and no fully satisfactory coupling algorithms have been developed yet, although significant progress has been made. An alternative method is to embed an MD simulation in a continuum simulation, which we demonstrate in the context of electroosmotic flow in a nanochannel. In the last section we discuss a new method, developed in the late 1990s primarily in Europe: the dissipative particle dynamics (DPD) method. It has features of both LBM and MD algorithms and can be thought of as a coarse-grained version of MD.


In Chapter 17 we turn our attention to simulating full systems across heterogeneous domains, i.e., fluid, thermal, electrical, structural, chemical, etc. To this end, we introduce several reduced-order modeling techniques for analyzing microsystems. Specifically, techniques such as generalized Kirch-hoff networks, black box models, and Galerkin methods are described in detail. In black box models, detailed results from simulations are used to construct simplified and more abstract models. Methods such as nonlinear static models and linear and nonlinear dynamic models are described under the framework of black box models. Finally, Galerkin methods, where the basic idea is to create a set of coupled ordinary differential equations, are described. The advantages and limitations of the various techniques are highlighted.

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