Interdisciplinary Applied Mathematics

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the cloud). When the kernel is centered at xnj, the interpolation functions, Nj,(xnj), i =1, 2,…, NCj, are computed. Once the interpolation functions



FIGURE 15.6.    Placement    of    the    kernel    and    the    definition    of    the    cloud    for    (a)


DSMC-to-continuum interpolation and for (b) continuum-to-DSMC interpolation.


are computed, u(xnj) is computed by


NCj


u(xnj)    3 ^ Ni(xnj )uci•    (15.3)


i=1


Continuum-to-DSMC Interpolation


Let ni    denote the    nodes in    the    continuum    domain, and    let    Cj    denote    the


DSMC cells. Let xni and xcj denote the position of the continuum nodes and the DSMC    cells,    respectively.    Let    uni    denote    the    nodal    parameter    for


the solution    at    the    continuum    node    with    location    xni,    and    let    u(xcj) be


the interpolated solution at the DSMC cell center with location xcj. To compute u(xcj), a kernel or a weighting function is centered at the position xcj. The kernel centered at xcj defines    the    cloud    Oj    and    the number of


continuum nodes, NPj, that lie within    the    cloud    Oj    (see    Figure 15.6(b)


for the definition of the cloud and the continuum nodes that fall within the cloud). When the kernel is centered at xcj, the interpolation functions, Ni (xcj ),i    1, 2,… , NPj, are computed. Once the interpolations functions


are computed, u(xcj) is computed by


NPj


u(xcj) =    Ni(xcj)uni.    (15.4)


i=1

15.3 Multiscale Analysis of Microfilters


In this section we present results on multiscale analysis of microfilters using two coupling schemes. For a discussion on the physics of microfilters, see Section 6.5. The first scheme couples DSMC with the Stokes equations and the second scheme couples DSMC with the Navier-Stokes equations. The


differences in the two coupling schemes are highlighted, and the dependence of the convergence of the multiscale method on various parameters is discussed.

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