block-structured adaptive mesh renementcalhoun/ · structured amr turbulent v-flame Œ p. 2/21....
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Block-Structured Adaptive MeshRefinement
Lecture 4Geometry
– Embedded Boundary– Software support embedded boundaries
Turbulent V-Flame – p. 1/21
Approaches to geometry
Curvilinear adaptive grids
Over set grid – generalizes curvilinear
Embedded boundary or CartesianGrid methods
Grid generation is tractable –CART3DDiscretization issues arewell-understood away fromboundary
Straightforward coupling tostructured AMR
Turbulent V-Flame – p. 2/21
Approaches to geometry
Curvilinear adaptive grids
Over set grid – generalizes curvilinear
Embedded boundary or CartesianGrid methods
Grid generation is tractable –CART3DDiscretization issues arewell-understood away fromboundary
Straightforward coupling tostructured AMR
-0.60 -0.55 -0.50 -0.45 -0.40 -0.35
-0.10
-0.05
0.00
0.05
0.10
Turbulent V-Flame – p. 2/21
Approaches to geometry
Curvilinear adaptive grids
Over set grid – generalizes curvilinear
Embedded boundary or CartesianGrid methods
Grid generation is tractable –CART3DDiscretization issues arewell-understood away fromboundary
Straightforward coupling tostructured AMR
Turbulent V-Flame – p. 2/21
Approaches to geometry
Curvilinear adaptive grids
Over set grid – generalizes curvilinear
Embedded boundary or CartesianGrid methods
Grid generation is tractable –CART3DDiscretization issues arewell-understood away fromboundary
Straightforward coupling tostructured AMR References
Chern and Colella, 1987
Youngs et al., 1990
Berger and Leveque, 1991
Pember et al., 1994
Johansen and Colella 1998
Colella et al., to appear
Turbulent V-Flame – p. 2/21
PreliminariesPrimary variables defined at cell centers
Λc – Volume fraction of cut cell ≡ Vc/h2
α – aperture ≡ edge length
Solve multiphysics applications using EB &AMR
Develop solvers for classical PDEs
Decompose applications into compo-nent processes
IssuesAccuracy
Stability
h
hαB
αS
Λ
Turbulent V-Flame – p. 3/21
EB – Conservation Laws
Ut + ~F (U) = 0
Finite volume discretizaton
∫ tn+1
tn
∫C
Ut + ~F dx dt = 0
h2ΛcUn+1 = h2ΛcU
n + ∆t(∑
s
αsFs + αBFB)
or
Un+1 = Un +∆t
h2Λc(∑
s
αsFs + αBFB)
where Fs and FB are explicitly computed fluxes
How to compute fluxes
How to handle small-cell stability
C
Turbulent V-Flame – p. 4/21
Fluxes – version 1There are several variations on how to do these things
A simple way to compute fluxes
Extend state to compute fluxes usingGodunov scheme for all edges of a cut cell
Volume weighted sum of values in aneighborhood of point
Modify Godonov scheme to use"essential" stencil for edges with αs = 0
FB computed by solving Riemann problemin local coordinates to boundary
FB
Turbulent V-Flame – p. 5/21
Update
One could update using
Un+1,cu = Un +∆t
h2Λc
∑s
αsFs + αBFB
This defines a conservative update but the timestep for cut cells decreases as Λc decreases.
We would like a conservative update that is sta-ble at full-cell CFL
Define a reference state
Un+1,ref = Un +∆t
h2
∑s
Fs
which represents update as though there wereno boundary in the cut cell
C
Turbulent V-Flame – p. 6/21
Update cont’dDefine
δM = h2Λc(Un+1,cu − Un+1,ref )
Compute stable update
Un+1,p = Un+1,ref +δM
h2
Redistribute (1 − Λc)δM to neighboring cells
Volume weighted
Mass weighted (gas dynamics)
Λ δM 1 − Λ δM
Recover full CFL time stepTurbulent V-Flame – p. 7/21
Enhancements to base algorithm
Extended states (Colella et al., to appear)
Extrapolate along normal direction
Do not use data in adjacent cell
Fluxes (Johansen and Colella, JCP 1998)
Interpolate fluxed to centroid of edges
Higher-order boundary flux in normal direction
Turbulent V-Flame – p. 8/21
AnalysisModified equation
∂Umod
∂t+ ∂ ~F (Umod) = τ
τ localized
O(h2) interior
O(h/Λ) at boundary
Error
O(h2) is boundary is noncharacteristic
O(h) in L∞ and O(h2) in L1 if boundary is characteristic
Turbulent V-Flame – p. 9/21
Poisson equation
Solve elliptic PDE on embedded boundary
∆φ = ρ
Want a cell-centered finite volume discretiza-tion
∇ · ∇φ = ρ
so ∇φ acts like a flux
∑s
αs∂φ
∂n s+ αB
∂φ
∂nB= Λch
2ρ
Turbulent V-Flame – p. 10/21
EB Poisson discretizationEvalute ∂φ/∂n using Johansen–Colella fluxLeads to well-conditioned linear system withapproximately "elliptic" spectral properties
Modified equation gives
∆φh = ρ + τ
where τ is first-order near boundary andsecond-order away from boundary
Smoothing property of inverse operator giveserror, φ − φh = ∆−1τ = O(h2)
However the matrix is notSymmetric
M-Matrix
Turbulent V-Flame – p. 11/21
Extension to three dimensionsTwo possible approaches to extend Johansen–Colella flux to threedimension
Linear interpolation is unstable; but, bilinear is stable
Turbulent V-Flame – p. 12/21
Poisson solution error – 3D
grid ‖ε‖∞
p∞ ‖ε‖2 p2 ‖ε‖1 p1
163 4.80 × 10−4 — 5.17 × 10−5 — 1.83 × 10−5 —323 1.06 × 10−4 2.17 1.25 × 10−5 2.05 4.41 × 10−6 2.05643 2.43 × 10−5 2.13 3.07 × 10−6 2.02 1.09 × 10−6 2.02
Turbulent V-Flame – p. 13/21
Nodal Projection
Projection performs the decomposition
V = Ud + ∇φ
For cut cells, view as extension of finite elementbasis extended to cover all of the cut cell
Projection uses homogeneous Neumann bound-ary conditions at cut cell boundaries
Gives a weak form∫Ω
∇φ · ∇χ dx =
∫Ω
V · ∇χ dx
Youngs et al. – Full potential adaptive transonicflow solver
Turbulent V-Flame – p. 14/21
Multiphysics application
Industrial burnerLow Mach numbercombustion formuationAxisymmetric flow
k − ε turbulence modelLaw of the wall
Discrete ordinates radia-tion
burner axis
.762 m
.381 m
1.651 m
.219 m
.3 m
1.0668 m
burner swirling air
naturalgas
Turbulent V-Flame – p. 15/21
Burner simulation results
Temperature K
CO2 mass frac
CO mass frac
CH4 mass frac
Radial velocity m/sec
Axial velocity m/sec
Turbulent V-Flame – p. 16/21
Burner experimental comparisons
0.0 0.2 0.4
r (m)
-20.0
0.0
20.0
40.0
Axia
l velo
city (
m/s
)
measured
unsteady
steady
0.0 0.2 0.4
r (m)
0.0
5.0
10.0
Tang. velo
city (
m/s
)
measured
unsteady
steady
0.0 0.2 0.4
r (m)
0.0
500.0
1000.0
1500.0
2000.0
Tem
p (
K)
measured
unsteady
steady
0.0 0.2 0.4
r (m)
0.0
2.0
4.0
CO
(%
mol)
measured
unsteady
steady
0.0 0.2 0.4
r (m)
0.0
5.0
10.0
CO
2 (
% m
ol)
measured
unsteady
steady
0.0 0.2 0.4
r (m)
0.0
5.0
10.0
15.0
20.0
O2
(%
mol)
measured
unsteady
steady
Turbulent V-Flame – p. 17/21
AMR considerationsEmbedded boundary + structured AMR is basically straightforward
If coarse / fine boundaries aren’t near the embedded boundary thereis basically nothing to do
When coarse / fine boundaries intersect cut cellsModify hyperbolic redistribution
Follows basic AMR design principlesKeep track of redistributions across coarse / fine boundaryAdjust data to correct errors (analogous to reflux)
Modify Johansen – Colella flux formulaeDrop to first-order for hyperbolic if necessaryUse first-order least squares fit to define boundary flux for ellipticSince these modifications are localized to a co-dimension 2 subset
of the domain they do not effect accuracy
Turbulent V-Flame – p. 18/21
Embedded Boundary SoftwareGrid generation software – Cart3D
Component based approach
Fix-up triangulations
Generate cut cell informationhttp://people.nas.nasa.gov/ aftosmis/cart3d/cart3Dhome.html
Turbulent V-Flame – p. 19/21
Packages supporting EB discretizationsEBChombo – LBNL
BEARCLAW – Univ. of Washington and Univ. of North Carolina
CART3D – NASA Ames
It is beyond the scope of this lecture to discuss EB software issues in detail
We can examine the analogs of some of the data structures discussedbefore
Turbulent V-Flame – p. 20/21
EB Software Design – EBChombo
We generalize rectangular array abstrac-tions to represent more general generalgraphs that map into the rectangular lat-tice Z
D. The nodes of the graph arethe control volumes, while the arcs of thegraph are the faces across which fluxesare defined.
BoxLib EB ChomboZD – EBIndexSpace
Index IntVect VolIndex, FaceIndex
Region of ZD Box EBISBox
Union of rectangles BoxArray EBISLayout
Rectangular array Fab EBCellFAB, EBFaceFAB
Looping construct FabIterator VoFIterator, FaceIterator
Turbulent V-Flame – p. 21/21