Transcript Cartesian Grid Embedded Boundary Methods for Partial Differential Equations

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
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).
Grid Generation
Geometric quantities required for discretization:
• Volume fraction
• Nondimensionalized face area
, boundary area
• Face centroids
, boundary centroid
All quantities other than
order accuracy.
must be computed to second-
Aftosmis, Berger, and Melton (1998): generate geometric quantities directly from
intersections with surface triangulation of boundary.
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 lowerdimensional moments - bootstrap up from 1D intersection data and
moments.
• Generalizes to arbitrarily high-order accuracy, any number of dimensions.
Grid Generation from Implicit Function Descriptions
.
Implicit function grid generator provides a general and flexible
tool for analytic representations, image data, geophysical data.
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.
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 Mmatrices (linear interpolation in
3D is unstable, bilinear is stable)
•The smoothing properties of the Green’s function of elliptic operators turn the
singular truncation error into a much smoother solution error:
in max
norm.
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.
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 patchbased parallelism for data defined over unions of rectangles.
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 welldefined discretizations of underresolved geometries.
• Geometric multigrid leads to high-performance, algorithmically scalable
solvers.
AMR calculation of shock diffraction over an
ellipsoid.
Multigrid convergence history for EB discretization of
Poisson’s equation on an N3 grid for N=64,128,256.
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.
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
• Hyperbolic:
solved using the EB algorithms
and software described above,
• Parabolic:
and coupled using a secondorder accurate predictor• Elliptic:
corrector method.
Vortex shedding past a cylinder, Re = 200
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.
The resulting method is second-order
accurate
Convergence study for diffusion on a sphere
and can be combined with implicit
function grid generation methods on
biological image data.
Applications: Non-SciDAC collaborations
Microfluidic MEMS (LBNL, LLNL, UCB)
300
mm
x 60
mm
channel
300
mm
x 60
mm
channel
contraction flow
Embedded
Boundary
100 X
Flow
Experimental channel
X-velocity [cm/s]
Air flow in the trachea (LBNL, LLNL, UCSF):
CT image
Level set description
Pressure [bar]
Geometric detail
Embedded boundary calculation
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.
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Results using 1D algorithm for a tracked
shock overtaking an expansion fan.
Image of tracked-front data defined on AMR hierarchy.
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 > 3, with underlying mapped Cartesian mesh, in support of
FSP edge plasma project.