Cartesian Grid Embedded Boundary Methods for Partial Differential Equations. APDEC ISIC: Phil Colella, Dan Graves, Terry Ligocki, Brian van Straalen (LBNL); Caroline Bono, Bjorn Sjogreen, David Trebotich (LLNL); Marsha Berger (NYU) UC Davis: Mike Barad (DOE CSGF Program), Greg Miller
Cartesian Grid Embedded Boundary Methods for Partial Differential Equations
APDEC ISIC: Phil Colella, Dan Graves, Terry Ligocki, Brian van Straalen (LBNL); Caroline Bono, Bjorn Sjogreen, David Trebotich (LLNL); Marsha Berger (NYU)
UC Davis: Mike Barad (DOE CSGF Program), Greg Miller
LBNL: Cameron Geddes, Eric Esarey, Wim Leemans (AFRD); Peter Schwartz, Thomas Deschamps (CRD); Adam Arkin, Matt Onsum (PBD).
UCSF: David Saloner
Univ. of North Carolina: David Adalsteinsson
Geometric quantities required for discretization:
All quantities other than must be computed to second-order accuracy.
Aftosmis, Berger, and Melton (1998): generate geometric quantities directly from intersections with surface triangulation of boundary.
Moment equations are derived using the divergence theorem:
. Implicit function grid generator provides a general and flexible tool for analytic representations, image data, geophysical data.
If the fluxes at centroids are computed to second-order accuracy, then the truncation error \ satisfies
Modified equation analysis indicates the expected relationship between the truncation error and the solution error.
Embedded Boundary Methods for Elliptic Equations
Graphical depiction of redistribution
Shock diffraction over an ellipsiod
Convergence results in L1 for a simple wave in a 3D circular tube.
EB Chombo generalizes Chombo: rectangular grids become more general graphs that map into rectangular grids. Nodes of the graph correspond to control volumes, while arcs of the graph correspond to faces that connect adjacent control volumes.
The Chombo parallel infrastructure is sufficiently general to support patch-based parallelism for data defined over unions of rectangles.
Multigrid convergence history for EB discretization of Poisson’s equation on an N3 grid for N=64,128,256.
AMR calculation of shock diffraction over an ellipsoid.
Embedded boundary method to compute the unsteady propagation of a jet into a vacuum chamber.
We solve the Incompressible Navier-Stokes equations using a projection method, splitting the equations into three parts:
Each of these equations are solved using the EB algorithms and software described above, and coupled using a second-order accurate predictor-corrector method.
Vortex shedding past a cylinder, Re = 200
Diffusion on a surface
Can be represented as diffusion in the annular region surrounding the surface
and solved using embedded boundary methods.
and can be combined with implicit function grid generation methods on biological image data.
The resulting method is second-order accurate
Convergence study for diffusion on a sphere
300 mm x 60 mm channel
300 mm x 60 mm channel
Microfluidic MEMS (LBNL, LLNL, UCB)
Air flow in the trachea (LBNL, LLNL, UCSF):
Level set description
Embedded boundary calculation
Results using 1D algorithm for a tracked shock overtaking an expansion fan.
Image of tracked-front data defined on AMR hierarchy.