Interdisciplinary Applied Mathematics

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In Chapter 14 we discuss representative numerical methods for continuum-based simulations. The significant geometric complexity of flows in microsystems suggests that finite elements are more suitable than finite differences, while high-order accuracy is required for efficient discretization. To this end, we focus on spectral element and meshless methods in stationary and moving domains. We also discuss methods for modeling particulate microflows and focus on the force coupling method, a particularly fast approach suitable for three-dimensional simulations. These methods represent three different classes of discretization philosophies and have been used with success in diverse applications of microsystems.


In Chapter 15 we discuss theory and numerical methodologies for simulating gas flows at the mesoscopic and atomistic levels. Such a description is necessary for gases in the transition and free-molecular regimes. First, we present the Direct Simulation Monte Carlo (DSMC) method, a stochastic approach suitable for gases. We discuss limitations and errors in the steady version of DSMC and subsequently present a similar analysis for the unsteady DSMC. In order to bridge scales between the continuum and atomistic scales we present the Schwarz iterative coupling algorithm and apply it to modeling microfilters. We then give an overview of the Boltzmann equation, describing in some detail gas-surface interactions, and include benchmark solutions for validation of numerical codes and of macromodels. A main result relevant to accurately bridging microdynamics and macrodynamics is the Boltzmann inequality, which we also discuss in the last section on lattice Boltzmann methods (LBM). These methods represent a “minimal” discrete form of the Boltzmann equation, and they are applicable to both compressible and incompressible flows; in fact, the majority of LBM applications focuses on incompressible flows.

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