A Stabilized Finite Element Dynamic Overset Method for the Navier-Stokes Equations
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Transcript of A Stabilized Finite Element Dynamic Overset Method for the Navier-Stokes Equations
A Stabilized Finite Element Dynamic Overset Method for the Navier-Stokes Equations
Chao LiuSimCenter: Center of Excellence in Applied Computational Science and Engineering
University of Tennessee at Chattanooga
What is Overset?
Motivation
• For complex configuration• Allow changing of individual
component without regenerating whole mesh
• Great for quick prototyping of a range of complex configurations
Motivation• For large relative motion
• Allowing arbitrary motion
Problems (Questions) to be solved (answered)
• What is the overset methodology for stabilized FEM?
• Would overset deteriorate solution accuracy for FEM?
• Would breaking a domain into separate overlapping domain impact solver performance, and how to improve it?
• Automatic and high-quality overset grid assembly (A.K.A hole-cutting)?
• Parallel execution of overset grid assembly and flow solver on the fly
Problems (Questions) to be solved (answered)
• What is the overset methodology for stabilized FEM?
• Would overset deteriorate solution accuracy for FEM?
• Would breaking a domain into separate overlapping domain impact solver performance, and how to improve it?
• Automatic and high-quality overset grid assembly (A.K.A hole-cutting)?
• Parallel execution of overset grid assembly and flow solver on the fly
Governing Equations• SUPG formulation of compressible Navier-Stokes equations in
conservative form
• Test function
• Stabilization matrix
𝜑 = 𝑁𝑖 𝐼 + 𝛻𝑁𝑖 ∙ 𝐴 𝜏
𝜏 −1 =
𝑖=1
𝑛
𝛻𝑁𝑖 ∙ 𝐴 + 𝑉 𝛻𝑁𝑖 ∙ 𝐴 = 𝑇 Λ 𝑇 −1
න
Ω 𝑡
𝜑𝜕𝑸 𝑥, 𝑡
𝜕𝑡+ 𝛻 ∙ 𝐹𝑒 𝑸 − 𝐹𝑣 𝑸,𝛻𝑸 − 𝑺 𝑸, 𝛻𝑸 𝑑Ω 𝑡 = 0
Erwin, J. T. "Stabilized Finite Elements for Compressible Turbulent Navier-Stokes." Ph.D. Dissertation,
University of Tennessee at Chattanooga, Chattanooga, TN, 2013.
Governing Equations
00 0
11 111 1
xyxx xz
yyxy yz
x y yyzxz zzv v v
xy yy yz xz yz zzxx xy xz
F F FT TT
u v w u v wu v wy zx
zx y
0
0
0
0
0
TS
S
e e
t t t
t t t
t t tx y z
e e e
t t t
t t t t t t
t t t
F F
u x v y w z
u u x p u v y u w z
v u x v v y p v w zF F F
w u x w v y w w z p
E p u x px E p v y py E p w z pz
u x v y w z
gQV
Governing Equations
• Utilizing integration by parts the weak form becomes
Boundary terms
𝜕
𝜕𝑡න෩Ω
𝑁𝑖𝑸 𝐽 𝑑෩Ω − න෩Ω
𝐽−1𝛻𝝃𝑁𝑖 ∙ 𝐹𝑒 − 𝐹𝑣 𝐽 𝑑෩Ω + ර
෩Γ
𝑁𝑖 𝐹𝑒 − 𝐹𝑣 ∙ ෝ𝒏 𝐽∗ 𝑑 ෨Γ
− න෩Ω
𝑁𝑖𝑺 𝐽 𝑑෩Ω +𝜕
𝜕𝑡න෩Ω
𝑃 𝑸 𝐽 𝑑෩Ω + න෩Ω
𝑃 𝐽−1𝛻𝝃 ∙ 𝐹𝑒 − 𝐹𝑣 − 𝑺 𝐽 𝑑Ω = 0
Overset Boundary Condition
Example of overset problem
Overset Boundary Condition
Convective term treated as Riemann problem
are obtained locallyare interpolated from donor element
,
,
L L
R R
Q Q
Q Q
ഥ𝐹𝑒 = ഥ𝐹𝑒+𝑸𝐿 + ഥ𝐹𝑒
−𝑸𝑅
𝐹𝑣 =1
2𝐹𝑣 𝑸𝐿, 𝛻𝑸𝐿 + 𝐹𝑣 𝑸𝑅, 𝛻𝑸𝑅
Solution Procedure
• Implicit local time marching for steady simulation
• Discrete-Newton relaxation for unsteady simulation
• GMRES with ILU(k) preconditioning to solve linear system
• Linearization (Jacobian matrix) of overset boundary flux is calculated w.r.t. left and right stage
Problems (Questions) to be solved (answered)
• What is the overset methodology for stabilized FEM?
• Would overset deteriorate solution accuracy for FEM?
• Would breaking a domain into separate overlapping domain impact solver performance, and how to improve it?
• Automatic and high-quality overset grid assembly (A.K.A hole-cutting)?
• Parallel execution of overset grid assembly and flow solver on the fly
Manufactured Solutions: 2D
• Inviscid and laminar (Re=100)
• P1, P2, P3 triangular element
• Forcing functions is added to the governing equation to drive solution to:
1.0cos2.0)]1.0(cos[2.01
1.0cos2.0)]1.0(cos[2.01
1.0cos2.0)]1.0(cos[2.01
cos2.0)](cos[2.01
4444
3333
2222
1111
ysxcysxcTT
ysxcysxcvv
ysxcysxcuu
ysxcysxc
o
o
o
o
Manufactured Solutions: 2D
Temperature on coarsest meshes, laminar, P3 elements
Manufactured Solutions: 2D
Temperature, invisicd Temperature, laminar
Order of accuracy for inviscid and laminar flow
Manufactured Solutions: 3D
• Inviscid
• P1 and P2 tetrahedral and hexahedral element
• Forcing functions is added to the governing equation to drive solution to:
1.0* 1.0 sin cos sin cos sin cos
0.5* 1.0 sin 1.5 cos 1.5 sin 1.5 cos 1.5 sin 1.5 cos 1.5
0.5* 1.0 sin 1.5 cos 1.5 sin 1.5 cos 1.5 sin 1.5 cos 1.5
0.1* 1.0 sin 1.5 cos 1.5
x x y y z z
u x v x y y z z
v x v x y y z z
w x v x
2 2 2
sin 1.5 cos 1.5 sin 1.5 cos 1.5
3.0* 1.0 sin sin sin
y y z z
E x y z
Manufactured Solutions: 3D
coarsest meshes 𝑃1 solutions of temperature and z-velocity on second finest mesh
Manufactured Solutions: 3D
(a) P1 elements (b) P2 elements
Observed order of accuracy
Steady Turbulent Flow
Single grid Zero-layer non-matched overset grid Multi-layer overlapping overset grid
60.2, 2 ,Re 10M
Steady Turbulent Flow
P1 elements P2 elements P3 elements
X-velocity profile at x=0.24 and 0.32
Sinusoidally Oscillating Wing
Single-grid Overset-grid, pre-cut Overset-grid, dynamic cut
• 𝑀∞ = 0.6, 𝛼∞ = 0°
• ONERA M6 wing pitch about its 0.6 chord
0
( ) sin(2 )
where 2.89 , 2.41 , 0.0808
m o
m
t kM t
k
Sinusoidally Oscillating Wing
Time history of lift coefficient of the sinusoidally
oscillating ONERA M6 wing
Steady-State Turbulent WPFS
• 𝑅𝑒 = 106, 𝛼 = 0°,𝑀𝑎 = 0.6• Modified Spalart-Allmaras turbulent model• 𝑃2element, 𝑄1 mesh• Number of nodes:
• Single-grid: 2,073,761 • Overset-grid: 2,102,028
• 500 CPU cores
Steady-State Turbulent WPFS
Steady-State Turbulent WPFSSi
ngl
e-gr
idO
vers
et-g
rid
Steady-State Turbulent WPFSSi
ngl
e-gr
idO
vers
et-g
rid
Steady-State Turbulent WPFSSi
ngl
e-gr
idO
vers
et-g
rid
Steady-State Turbulent WPFS
𝐶𝑝 plot at various azimuthal locations on the store
0° 30° 60°
90° 120° 150°
180° 210° 240°
270° 300°330°
Steady-State Turbulent WPFS
𝐶𝑝 plots on inboard/outboard sides of the pylon
y = -0.45 y = -0.8
y = -1.15
Steady-State Turbulent WPFS
𝐶𝑝 plots at various span-wise locations on the wing
z = 6.2 z = 6.8
Question answered
• Would overset deteriorate solution accuracy for FEM?• Same order of accuracy as the non-overset FEM can be achieved
• In real world simulations, attention to grid quality is needed, especially on overset boundary
Problems (Questions) to be solved (answered)
• What is the overset methodology for stabilized FEM?
• Would overset deteriorate solution accuracy for FEM?
• Would breaking a domain into separate overlapping domain impact solver performance, and how to improve it?
• Automatic and high-quality overset grid assembly (A.K.A hole-cutting)?
• Parallel execution of overset grid assembly and flow solver on the fly
Preconditioner for Overset
Illustration of mesh and Jacobian matrix for the 16-airfoil overset-grid case
Inter-grid (overset) Jacobian matrix is ignored by ILU(k) preconditionerUp to 2x - 3x GMRES search directions needed by observationSame problem also exists for parallel non-overset simulations using ILU(k) preconditioner. We can kill two birds at the same time
Jacobi Inter-grid ILU
Applying preconditioner 𝒙 = (𝐿𝑈 + 𝑂)−1𝒃,
by solving 𝐿𝑈 + 𝑂 𝒙 = 𝒃 for x using Jacobi iteration
𝒙0 = 0
do 𝑖 = 1, 𝑛
solve 𝐿𝑈𝒙𝑖 = 𝑏 − 𝑂𝒙𝑖−1 for 𝒙𝑖
end do
𝒙 = 𝒙𝑛
𝐴𝑝𝑟𝑒 = 𝐿𝑈 + 𝑂 ≈ 𝐴
Preconditioner for Overset• Ma = 0.2, 𝛼 = 2°, steady, inviscid, 𝑃1 element
• CFL ramps up from 1 to 2000 in 100 iterations
• ILU(k) filling level = 1
• Residual of linear system drops 10 orders in magnitude
Question answered
• Would breaking a domain into separate overlapping domain impact solver performance?
• Yes, if overset Jacobian matrix is ignored in LHS or in preconditioner like ILU
• How to improve it?• ILU(k) can be modified to incorporate overset Jacobian. The idea can be
extended to general parallel simulation regardless of whether using overset
• Other preconditioner consider intra-grid Jacobian matrix also exist, but has not been tested:• Line Gauss-Seidel solver
• Schur complement
Problems (Questions) to be solved (answered)
• What is the overset methodology for stabilized FEM?
• Would overset deteriorate solution accuracy for FEM?
• Would breaking a domain into separate overlapping domain impact solver performance, and how to improve it?
• Automatic and high-quality overset grid assembly (A.K.A hole-cutting)?
• Parallel execution of overset grid assembly and flow solver on the fly
Cell Selection Based on Mesh Metric
Cell selection based on cell volume
Cell selection based on cell volume
Cell Selection Based on Mesh Metric
Cell Selection Based on Distance Function
Selection based on distance function
Elliptic Hole Cutting
Airfoil 1
2-airfoil overset grids
Grid-1 Grid-2
Airfoil-1
Airfoil-2Airfoil-2
Airfoil-1
Elliptic Hole Cutting
Grid 1 Grid 2
Solution of Poisson problems
Elliptic Hole Cutting
Final mesh3D view of Poisson solution
Elliptic Hole Cutting
Problems (Questions) to be solved (answered)
• What is the overset methodology for stabilized FEM?
• Would overset deteriorate solution accuracy for FEM?
• Would breaking a domain into separate overlapping domain impact solver performance, and how to improve it?
• Automatic and high-quality overset grid assembly (A.K.A hole-cutting)?
• Parallel execution of overset grid assembly and flow solver on the fly
Parallel Overset Grid Assembly:Grid Partition
Grid 2Grid 1
Mesh
Block 1 Block 2
Grid
Parallel Overset Grid Assembly: Grid Profiling and Collision Detection
Parallel Overset Grid Assembly:Point Search Algorithm
Stencil walking in a “block”, starting from nearest surface
Decomposition Surface
Geometric boundary
Non-geometric boundary
Build an Octree
Parallel Overset Grid Assembly:Nearest Neighbor search and Distance Function
Laminar Wing/Finned Store Separation
• 𝑅𝑒 = 103, 𝛼 = 0°,𝑀𝑎 = 0.6• Prescribed store motion• 𝑃2 tetrahedral element, mesh has straight
edges/surfaces• DOF: 506,030 • 68 CPU cores
Laminar Wing/Finned Store Separation
Laminar Wing/Finned Store Separation
Summary and Contributions• First to develop overset grid method for stabilized finite
element scheme (SUPG).
• Improved preconditioners for overset simulations that consider inter-grid Jacobian matrix
• Developed novel hole cutting method Elliptic Hole Cutting (EHC)
• Developed a MPI-based parallel 3D dynamics overset grid assembly
• Dynamic viscous moving boundary simulation with high order finite element overset scheme
Recommendations for Future Work
• PDE based Shock-capturing
• Mesh morphing with overset
• Locally-conservative numerical flux integration on overset interface (overset grid with zero-overlap)
• Dynamic load balancing for moving boundary simulation
Products of Current Work• Chao Liu, James C. Newman III and W. Kyle Anderson. “Three-Dimensional Stabilized Finite Elements Dynamic Overset
Method for the Navier-Stokes Equations”. (Journal article in preparation)
• Chao Liu, James C. Newman III and W. Kyle Anderson. “Petrov–Galerkin Overset Grid Scheme for the Navier–Stokes Equations with Moving Domains”, AIAA Journal, 2015, 53(11): 3338~3353. doi: 10.2514/1.J053925 (http://arc.aiaa.org/doi/abs/10.2514/1.J053925)
• Chao Liu, James C. Newman III, W. Kyle Anderson and Behzad R. Ahrabi. “Three-Dimensional Dynamic Overset Method for Stabilized Finite Elements,” 22nd AIAA Computational Fluid Dynamics Conference, Dallas, TX, June 2015, AIAA Paper 2015-3423. (http://arc.aiaa.org/doi/abs/10.2514/6.2015-3423)
• Chao Liu, Behzad R. Ahrabi, James C. Newman III and W. Kyle Anderson. “An Adaptive Streamline/Upwind PetrovGalerkin Overset Grid Scheme for the Navier-Stokes Equations with Moving Domains”, 12th Symposium on Overset Composite Grids and Solution Technology, Atlanta, GA, October 2014. (http://2014.oversetgridsymposium.org/index.php).
• Chao Liu, James C. Newman III and W. Kyle Anderson. “A Streamline/Upwind Petrov Galerkin Overset Grid Scheme for the Navier-Stokes Equations with Moving Domains”, 32nd AIAA Applied Aerodynamics Conference, Atlanta, GA, June 2014, AIAA Paper 2014-2980. (http://arc.aiaa.org/doi/abs/10.2514/6.2014-2980)
Shock Capturing• Modifying SUPG operator
• Adding artificial dissipation
Artificial Dissipation • Arithmetic artificial dissipation: ∈ is given as an explicit function
• PDE Solution based artificial dissipation• Able to providing smooth artificial dissipation, that would improve
solution accuracy
• At extra cost of solving PDE for ∈
Glasby, R. S., and Erwin, J. T. "Introduction to COFFE: The Next-Generation HPCMP CREATE-AV CFD Solver," 54th AIAA Aerospace Sciences Meeting. American Institute of Aeronautics and Astronautics, 2016.
Barter, G. E., and Darmofal, D. L. "Shock capturing with PDE-based artificial viscosity for DGFEM: Part I. Formulation," Journal of Computational Physics Vol. 229, No. 5, 2010, pp. 1810-1827.
Arithmetic Artificial Dissipation• 𝑀𝑎 = 0.5, 𝛼 = 1.25°• P3 element, 31K nodes, curved mesh• No artificial dissipation
• Benchmark solution: FLO82 4096 x 4096 nodes
Arithmetic Artificial Dissipation• 𝑀𝑎 = 0.8, 𝛼 = 0°• P2 element, 13K nodes, curved mesh
• FLO82 4096 x 4096 nodes No artificial dissipation Added artificial dissipation
Arithmetic Artificial Dissipation
• 𝑀𝑎 = 0.8, 𝛼 = 0°• P2 element, 13K nodes, curved mesh
Arithmetic Artificial Dissipation• 𝑀𝑎 = 2, 𝛼 = 0°• P3 element, 31K nodes, curved mesh• Added artificial dissipation
Future of Scientific Computing
• Modern programming method• Modular, generic programming
• Test driving development means less debug
• Parallelization• Dynamic Load balancing for large scale distributed parallel computing
• Dynamic overset, AMR would cause unbalanced work load that need to redistributed
• Task parallelism to explore in-node accelerator (Xeon Phi, GPGPU)
• Multiphysics simulation
• The future of CFD is FEM