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How To Write A Program For The Pd Framework

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Enhanced divergence-free elements for

efficient incompressible flow simulations in

the PDE framework Peano

Tobias Neckel,

Miriam Mehl, Christoph Zenger

Scientific Computing in Computer Science,

Fakultät für Informatik

TU München

Germany

(2)

Outline

• Derivation of Div-free Ansatz Functions

• Enhanced Div-free Elements

• Numerical Results

– Checkerboard Driven Cavity

– Driven Cavity

– Flow around a Cylinder

(3)

Derivation of Div-free Ansatz Functions

Incompressible Navier-Stokes Equations

Discretisation

low-order FEM (Q1Q0, etc.)

Divergence-free elements:

solenoidal velocities in

every

point in a cell

(4)

Derivation of Div-free Ansatz Functions

(5)

Derivation of Div-free Ansatz Functions

du/dx = const = u3 - u0

dv/dy = const = v1 - v2

div(u) = u3 - u0 + v1 - v2 = const = 0

(6)

Derivation of Div-free Ansatz Functions

du/dx = const = u3 - u0

dv/dy = const = v1 - v2

div(u) = u3 - u0 + v1 - v2 = const = 0

(7)

Derivation of Div-free Ansatz Functions

Ansatz functions 45 and Cartesian:

Advantages of 45 representation:

Simplified derivation and representation of elements

Performance: ~20% less runtime for evaluation of

operators D and C

(8)

Enhanced Div-free Elements

• Additional DoF on faces:

– exact representation of fluxes on edges

– no checkerboarding

(9)

Numerical Results – Checkerboard Driven Cavity

enhanced div-free

(steady state)

Q1Q0

(step1,

no convergence

!)

Re=1

(10)

Numerical Results – Driven Cavity

(11)

Numerical Results – Flow around a Cylinder

Re = 20

Re = 100

c

d

c

l

C

d,max

C

l,max

St

88,857

5.68 0.0151 3.225

0.94

0.299

ref.

5.58 0.0107 3.230

1.00

0.298

# DoF

(12)

Outlook

Multigrid

– Peano Framework designed for hierarchical applications

– Speed up computations while keeping low memory requirements

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(14)

The PDE Framework Peano

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The PDE Framework Peano

• Cartesian grids (arbitrary dimensions)

• Plug-in concept for applications

• Space-filling curves, spacetrees, and stack data structures

– Strictly element-wise access

– Low memory demands

– Dynamical load balancing

– Moving geometries, dynamical adaptivity, geometric multigrid

• Software Engineering

– automatic tests, continuous integration, OO, design patterns, ...

• CFD component

– Incompressible flow (FEM, IDO)

– Explicit + implicit time-integration schemes (FE, RK4, BE, (adaptive) TR)

(16)

Backup I

Low memory requirements (FEM + adap.):

0

500

1000

1500

2000

2500

FE

RK

TR adap.

sundance

bytes 2D

bytes 3D

(17)
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Numerical Results - Performance

2D IDO Overhead Peano vs. Aoki (regular):

1.3 – 4.4

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2 level 6 12,676 level 7 116,061 level 8 1,051,253

ratio adaptive vs. regular

2D FEM

(19)

CFD Extensions

• Preparation for turbulent flows

– LES model

– Mesh stretching for regular grids

– Data averaging (2x in space, 1x in time)

– Restart + checkpointing

• Parallelisation

– Regular grids 2D/3D

– Restart + checkpointing

• Moving geometries

– Update of data + grid (regular + adaptive)

– Divergence correction

(20)

CFD Extensions

• Moving geometries

– Update of data + grid (regular + adaptive)

– Divergence correction

References

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