SPH researchat National University of Ireland, GalwayNathan Quinlan, Marty Lastiwka, Mihai Basa

10 October, 2005

SPH SIG, 10 October 2005

Background and Motivation

Making CFD more accessibleCan we do without mesh generation?

Began working on SPH in 2001

Funding awarded by Irish Research Council for Science, Engineering and Technology for 4-year project starting 2003

Biomedical flows

Moving geometries (artery walls, heart valves)

Complex, unique geometries from 3D and “4D” medical imaging

SPH SIG, 10 October 2005

Activities to date

• Theoretical study of accuracy

• Adaptive particle distribution

• Viscous flow

• Incompressible flow

SPH SIG, 10 October 2005

Accuracy of SPH

SPH does not exactly reproduce a constant-valued function – it is not zero-order consistent

0.25 0.3 0.35 0.4 0.45

0.9999

1

1.0001

1.0002

1.0003

1.0004

x

test

fun

ctio

n

exact particle valuesSPH estimate

Consistency-corrected SPH methods (like RKPM) guarantee exact reproduction of polynomials of order 0, 1, …

SPH SIG, 10 October 2005

Truncation error analysis of SPH in 1D

22 3

3 43

2

2

2 3

ˆ

6

1ˆ ˆ

1 1ˆ ˆ ˆ2 ...24

ij

j ij

i

j j jj j j j

j j

j j jj j j j j j

j j

W AA x

x x

h A Ws ds O h

x s

x x xAW AW

h x h

x x xhAW AW AW

h x h

discretisation error

smoothing error

xj = particle location

xj = particle volume

xj = centre of particle volumeA(x) = data function

SPH SIG, 10 October 2005

Numerical experiments in 3D

standard kernel

corrected kernel

SPH SIG, 10 October 2005

The need for adaptive SPH

flow

inlet

outlet

particles

inserted at inlet

shoc

k

SPH SIG, 10 October 2005

Test case: quasi-3D shock tube flow

instantaneous density field

x

z

y

Location of discontinuity

at t=0

flow

SPH SIG, 10 October 2005

Results – adaptive particle distribution

SPH SIG, 10 October 2005

Method 1: mixed finite-difference / SPHMonaghan (1992), Morris et al. (1996)

Method 2: Direct second derivatives of kernelSuccessfully used by Takeda et al. (1994), with Gaussian kernels.

Method 3: Two passes of standard SPH with WIntroduced by Flebbe et al. (1994) and Watkins et al. (1996)

Evaluation of second derivatives for viscous flow

SPH SIG, 10 October 2005

Evaluation of second derivatives for viscous flow

finite difference / SPH

2-pass

SPH SIG, 10 October 2005

Incompressible flow

Similar to pressure projection technique of Cummins and Rudman

New method based on Clebsch-Weber decomposition

SPH SIG, 10 October 2005

Incompressible flow

0 200 400 600 800 10000

0.5

1

1.5

2

2.5

3x 10

-4

Clebsch-WeberPressure Projection

time step

.u

, norm

alis

ed

SPH SIG, 10 October 2005

Current and future work

• Boundary conditions

• Turbulence modelling

• Parallelisation

• Application to mechanical heart valves