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







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







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


Temperature on coarsest meshes, laminar, P3 elements


Temperature, invisicd Temperature, laminar

Order of accuracy for inviscid and laminar flow


• 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


coarsest meshes 𝑃1 solutions of temperature and z-velocity on second finest mesh


(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 WPFSSi

ngl

e-gr

idO

vers

et-g

rid


ngl

e-gr

idO

vers

et-g

rid


ngl

e-gr

idO

vers

et-g

rid


𝐶𝑝 plot at various azimuthal locations on the store

0° 30° 60°

90° 120° 150°

180° 210° 240°

270° 300°330°


𝐶𝑝 plots on inboard/outboard sides of the pylon

y = -0.45 y = -0.8

y = -1.15


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







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







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


Grid 1 Grid 2

Solution of Poisson problems


Final mesh3D view of Poisson solution








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



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