Cartesian Grid Embedded Boundary Methods for Partial Differential Equations

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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

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Embedded Boundary Discretization of Conservation
Laws

• Primary dependent variables approximate values
  at Cartesian cell centers.
• Divergence theorem over each control volume
  leads to finite volume approximation.

• Approximation of fluxes based on finite
  differences of cell-centered data (standard
  conservative differences in regular cells).

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Grid Generation
Geometric quantities required for discretization

• Volume fraction
• Nondimensionalized face area , boundary
  area
• Face centroids , boundary centroid

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.
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Grid Generation from an Implicit Function
Description

• Moment equations are derived using the
  divergence theorem

• Overdetermined system solved using least-squares.
• Right-hand side is obtained from higher-order
  moments or lower-dimensional moments - bootstrap
  up from 1D intersection data and moments.
• Generalizes to arbitrarily high-order accuracy,
  any number of dimensions.

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Grid Generation from Implicit Function
Descriptions

• . Implicit function grid generator provides a
  general and flexible tool for analytic
  representations, image data, geophysical data.

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Numerical Analysis of Embedded Boundary Methods
Formal consistency

• If the fluxes at centroids are computed to
  second-order accuracy, then the truncation error
  \
  satisfies
• at interior cells
• at the boundary

Modified equation analysis indicates the expected
relationship between the truncation error and the
solution error.
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Embedded Boundary Methods for Elliptic Equations

• Fluxes are computed using linear interpolation
  of centered differences in 2D, bilinear
  interpolation in 3D
• Stability is nontrivial matrices are not
  symmetric, nor M-matrices (linear interpolation
  in 3D is unstable, bilinear is stable)

• The smoothing properties of the Greens function
  of elliptic operators turn the singular
  truncation error into a much smoother solution
  error in max norm.

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Embedded Boundary Methods for Hyperbolic Equations

• Small-cell stability hybridize with
  nonconservative stable method, and redistribute
  the missing mass. increment to maintain
  conservation.
• The nonconservative method must be designed
  carefully to maintain stability, robustness, and
  accuracy.
• Modified equation arguments lead us to expect
  second-order accuracy in L1, first-order accuracy
  in max norm.

Graphical depiction of redistribution
Shock diffraction over an ellipsiod
Convergence results in L1 for a simple wave in a
3D circular tube.
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Embedded Boundary Software Infrastructure
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.
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Multigrid and Adaptive Mesh Refinement

• Embedded boundary methods extend naturally to
  nested grid hierarchies.
• Coarsening grid generation is done without
  reference to original geometric description by
  coarsening the graph directly, leading to
  well-defined discretizations of underresolved
  geometries.
• Geometric multigrid leads to high-performance,
  algorithmically scalable solvers.

Multigrid convergence history for EB
discretization of Poissons equation on an N3
grid for N64,128,256.
AMR calculation of shock diffraction over an
ellipsoid.
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Application Gas Jet Simulation for Wakefield
Accelerators

• Embedded boundary method to compute the unsteady
  propagation of a jet into a vacuum chamber.
• Inviscid EB AMR solvers for time-dependent flow
  through a nozzle in 2D (axisymmetric) and 3D,
  including grid generation capabilities.
• Currently implementing parabolic solvers,
  including tensor solvers, for compressible
  viscous terms, heat conduction.

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Application Viscous Incompressible Flow
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.

• Hyperbolic
• Parabolic
• Elliptic

Vortex shedding past a cylinder, Re 200
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Applications Non-SciDAC collaborations
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
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Applications Non-SciDAC collaborations
Microfluidic MEMS (LBNL, LLNL, UCB)
Air flow in the trachea (LBNL, LLNL, UCSF)
CT image
Level set description
Embedded boundary calculation
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Volume-of-Fluid Methods for Free Boundary Problems

• Entension of discretization methods, software to
  the case of sharp fronts.
• Generalizes formally consistent EB
  discretizations to case where solution is defined
  on both sides of a moving boundary.
• Leverages the EB software infrastructure.
• Potential applications tracking flame fronts in
  premixed combustion, type 1A supernovae.

Results using 1D algorithm for a tracked shock
overtaking an expansion fan.
Image of tracked-front data defined on AMR
hierarchy.
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Future Plans

• Complete initial implementations for SciDAC
  applications compressible Navier-Stokes solver
  for plasma-wakefield accelerator project,
  incompressible Navier-Stokes solver for
  combustion (9/30/2005). Continue development of
  these algorithms in response to further
  applications requirements.
• EB software review serial, parallel
  performance, documentation, in preparation for
  initial public release of EB Chombo (12/31/2005).
• Continue algorithm development for formally
  consistent volume-of-fluid front tracking.
• Proposed work development of extension of EB
  infrastructure to dimensions gt 3, with underlying
  mapped Cartesian mesh, in support of FSP edge
  plasma project.