harmonic polynomial cell (hpc) method with immersed boundaries · harmonic polynomial cell (hpc)...

Harmonic Polynomial Cell (HPC) Method with Immersed Boundaries

Yanlin Shao, Department of Mechanical Engineering, DTU

Finn-Christian W. Hanssen, Department of Marine Technology, NTNU

17/04/2008Presentation name2 DTU Mechanical Engineering, Technical University of Denmark

About Yanlin Shao

• Assistant Professor, Technical University of Denmark

• PhD in Marine Hydrodynamics (2010), NTNU. Supervisor: Odd M. Faltinsen

• Previous jobs

Senior Engineer, Marine Analysis, Sevan Marine (2014-2016)

Senior Engineer, Hydrodynamics, DNV GL (2012-2014)

Postdoc, NTNU (2010-2012)

About Finn-Christian W. Hanssen

•Final-year PhD student in Department of Marine Technology, NTNU•Head of Department, Naval Technology, Moss Maritime ASA


Contents

• Short overview of numerical methods in marine hydrodynamics

• Fully nonlinear potential-flow (FNPF) model

• Harmonic polynomials

• Harmonic polynomial cell (HPC) method

• Some applications of the original HPC method

• Immersed boundaries strategy


Short overview of numerical methods in marine hydrodynamics


Numerical methods in marine hydrodynamics

Linear (frequency-domain & time-domain)

Weakly-nonlinear

2nd order: mean, sum-frequency, slow-drift wave forces3rd order: Triple-frequencyHigher order: Impractical

Fully-nonlinear potential-flow

Fully nonlinear free surface and body boundary conditions

Capable of describe higher-order harmonics

Navier-Stokes Equations

Viscous flow separation, local wave breaking…

Com

puta

tional Eff

ort

(C

PU

tim

e)

Industry Academic


Navier-Stokes Equations are still considered too slow to impact early design stage

6


Fully nonlinear potential-flow (FNPF) model

• Governed by Laplace equation (potential flow assumption)

• Boundary conditions are ‘fully-nonlinear’, satisfied at instantaneous position

Surface capturing using markers

Free surface conditions

Body boundary conditions


Challenges in fully nonlinear potential flow model

• Efficiency & accuracy of Laplace equation solver

• Free surface tracking (Lagrangian, Semi- Lagrangian, ALE etc.)

• Instability due to quadratic and convective terms in free surface conditions

• Capturing of wave breaking ?


Harmonic polynomials


In 2D space, they are defined by Re and Im parts of complex polynomials

Examples up to 4th order:

1,

x, y,

x2-y2 , 2xy,

x3-3xy2 , 3x2y-y3 ,

x4-6x2y2+y4 , 4x3y-4xy3

In 3D space, either follow Euler (1756–1757) or spherical harmonics

Examples up to 3rd order:

1

x, y, z

x2-(y2+z2)/2, xy, xz, yz, y2-z2

x3-3(xy2+xz2)/2; x2y-y(y2+z2)/4; x(y2-z2), y(y2-3z2); x2z-z(y2+z2)/4, xyz; z(3y2-z2)

Harmonic polynomials satisfy Laplace equation exactly


Euler’s memoir ‘Principia motus fluidorum’ (Euler, 1756–1757)

English translation available at www.oca.eu/etc7/EE250/texts/euler1761eng.pdf

Leonhard Euler (1707-1783)

Where are Harmonic Polynomials from ?


Use polynomials to represent velocity potential

Constraints on coefficients in order to satisfy Laplace equation

, ,x y z

2 0


Harmonic polynomial cell (HPC) method


2D data interpolation

1

2

3

i

M

M-1

i+1


An example of using harmonic polynomials (2D):

A Dirichlet boundary-value problem

1 2 3

4 5

6 7 8

2 0

8

1

,, j

j

j f x yx by

Unknown coefficients

Harmonic polynomials

Applying boundary conditions at the nodes gives

8

1

, 1,, ,8j ii j

j

if x x y y ib

8

,

1

, 1, ,8i i j j

j

c ib

8 8

,

1 1

, ,j i j i

i j

x y c f x y

Linear combination of , 1, ,8.i i ( , )id x y


• The HPC method for a general potential-flow problem

(i,j) (i+1,j) (i+2,j)(i-1,j)

(i-1,j-1) (i,j-1) (i+1,j-1) (i+2,j-1)

(i,j+1) (i+1,j+1) (i+2,j+1)(i-1,j+1)

1. Discretize by quadrilateral elements

2. Operate with cells that contain 4neighboring quadrilateral elements and 9 grid points

1 2 3

4 5

6 7 8

9

3. Consider a sub-Dirichlet problem in each cell

8 8

,

1 1

, ,j i j i

i j

x y c f x y

9

In fluid:

On Neumann boundaries:

8 8

,

1 1

, , ( , )j i j i

i j

x y c f x y n x yn

8

9 9 1,

1

0, 0 i i

i

x x y y c

Sparse matrix with at most 9 nonzeros in each row. close to 4th order accuracy.


Accuracy & efficiency (1)

Neumann

2 0

Lh

Length = L, Height = h, L = 40h

Uniform rectangular grids x y

Analytical velocity potential cosh ( ) sink y h kx

Mixed Dirichlet-Neumann boundary value problem

GMRES solver used for all the 5 methods

Dirichlet surface

NeumannNeumann


Comparison based on mixed Dirichlet-Neumann problem on a 2D shoe box

BEM: Constant Boundary Element Method

FMM-BEM: Fast Multipole Accelerated BEM

HPC: Harmonic Polynomial Cell method

Required CPU time to achieve 10-4 accuracy

FMM-BEM: > 1 sec

BEM: much much longer time

HPC: 0.06 sec

CPU time

L2

errors

N N

103

2x103

3x103

0.1

1

10

CP

U ti

me

(s)

Number of unknowns

BEM

FMM-BEM

HPC

103

2x103

3x103

10-9

10-7

10-5

10-3

10-1

101

HPC

L2 e

rror

s

Number of unkowns

BEM, Dirichlet surface

BEM, Neumann surface


Accuracy & efficiency (2)X

Y

Z

F ram e 0 0 1 1 7 Apr 2 0 1 2 pane l on episode solid

Dirichlet surface

sin exp( )x y zk x k y k z

2 20.5, 0.5,x y z x yk k k k k

yh x z

X Y

Z

F ram e 0 0 1 1 7 Apr 2 0 1 2 pane l on episode solid

Neumann surface


𝑆𝐷: Dirichlet surface

𝑆𝑁: Neumann surface

QBEM = Quadratic boundary element method

FMM-QBEM = Fast Multiple Method accelerated QBEM

L2

errors CPU time

∆ℎ 𝑈𝑛𝑘𝑛𝑜𝑤𝑛𝑠 𝑜𝑛 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦

N=8448 in QBEM, 79,507unknowns in HPC~ 20s for HPC-3D; ~ 300s for FMA-QBEM (p=12)~ 2000s for QBEM

Based on Intel 2.0GHz CPU


Some applications of the original HPC method


Wave focusing by tuning the phase angles for each wave component

Solitary wave against wall

Wave height = 2mWater depth = 4m

The original HPC method usesBoundary-fitted grid


20 21 22 23 24 25 26-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

(

m)

t (s)

Num. Exp.

26 27 28 29 30 31 32-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

(

m)

t (s)

Num. Exp.

28 29 30 31 32 33 34-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

t (s)

(

m)

Num. Exp.

35 36 37 38 39 40 41-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

Num. Exp.

t (s)

(

m)

0.4m

0.3m1:

20

1:1

0

6m

6m

2m

3m

13m

12.5m

14.5m

17.3m

21m

Nonlinear waves over submerged trapezoidal bar

Experiments results available from :Beji & Battjes (1993), Luth et al. (1994)

x = 12.5 m x = 14.5 m

x = 17.3 m x = 21 m


XY

ZF ram e 0 0 1 0 2 M ay 2 0 1 2 3 d m eshes | 2 d m esh

Wave focusing due to uneven seafloor

Water surface

Sea floor

HPC results agree well with experiments

0 5 10 15 20 25 30 350.000

0.004

0.008

0.012

0.016

0.020

0.024

x (m)

wa

ve

am

plit

ud

e (

m)

1st harmonic,exp.

2nd

harmonic,exp.

3rd harmonic,exp.

1st harmonic,num.

2nd

harmonic,num.

3rd harmonic,num.


0.00 0.05 0.10 0.15 0.20 0.256.0

6.2

6.4

6.6

6.8

7.0

7.2

kA

Present

Ferrant

|F1| /g

AR

2

Analytical

Experiment

0.00 0.05 0.10 0.15 0.20 0.250.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

kA

Analytical

Experiment

Present

Ferrant

|F2| /gA

2R

0.00 0.05 0.10 0.15 0.20 0.250.0

0.1

0.2

0.3

0.4

kA|F

3| /gA

3

Present

Ferrant

A

Analytical

Experiment

0.00 0.05 0.10 0.15 0.20 0.250.0

0.1

0.2

0.3

0.4

0.5

kA

|F4| /gA

4R

-1 Experiment

Ferrant

Present

LinearForce

Sum-FrequencyForce

Triple-FrequencyForce

Quadruple-FrequencyForce

Wave slope

Wave slope

Wave slope

Wave slope

20.245, /g=wave number, radius, wave amplitudekR k R A F ra m e 0 0 1 2 2 J a n 2 0 1 3 4 -n o d e s F E M p a n e ls | F E - V o lu m e B ric k D a ta


• Weakly-nonlinear theory (e.g. Faltinsen, Newman & Vinje model) is popular in offshore industry. It should be only used in estimation of non-impulsive loads.

• Breaking-wave induced impulsive loads needs to be considered additionally by a slamming model

Higher velocity

at wave crest

Wave Impact

ImpactforceImpact

Inducedvibration


Matched multi-block structure meshes

Fluid domain divided into sub-blocks

For points shared by more than 3 blocks, a least-square fitting was used construct harmonic polynomial cell


Extended to consider singularity at sharp corners

Extended to consider hydro-foils


Immersed boundaries strategy


Immersed boundaries

Immersed boundaries in Cartesian background grid

Boundary-fitted grid

Easier to deal with complex structural boundaries

Optimum accuracy can be achieved using rectangular cells

Avoid distortion of meshes in case of large boundary motions

Motivation of using immersed boundaries in HPC method


Immersed free surface (Dirichlet)

• Position and velocity potential on free-surface markers are updated each time step

• Continuity enforced at fluid point

• Artificial layers above free surface

• Extension of solution from water domain to artificial layers using Dirichletboundary condition and harmonic polynomials

9 8

9 9 1,

1

0, 0 i i

i

x x y y c

Free-surface marker, free to move in vertical direction

Fluid node

Ghost node above free surface

Unused node outside domain


Piston wave maker

How does it look like ?


Immersed body surface (Neumann)

Standard ghost-node approach


• Neumann boundary condition for a point on the body surface

Example of immersed body surface (Neumann)

Fluid node

Ghost node above free surface

Unused node outside domain

Ghost node inside body


Oscillating cylinder in infinite fluid


It seems to be a common issue with ghost-node method.

It was later solved by

Use of immersed boundaries + overlapping grids

Solving additional boundary value problem for 𝜙𝑡


Immersed boundaries + overlapping grids

• Use a fixed background grid

• Use body-fixed Cartesian grids

• Couple solution in local and background grids

• Free surface is an immersed boundary

• Body surfaces are immersed boudnaries


Illustration by forced oscillation of a horizontal circular cylinder

• A challenging case where local wave nonlinearities develop near wave-body intersection points due to high oscillation frequency

● Free surface markers● Fluid nodes● Communication nodes

◻ Free-surface ghost nodes◆ Body-boundary ghost nodes○ Voided nodes


• Animation with part of Cartesian background grid and the local grid


Results for heaving cylinder

• Heave force (Fourier components)



Wave elevation, high-frequency case



• Fluid pressure force in heave, high-frequency case


Wave-body interaction: Lewis form in regular waves

Location of wave probes

Damping of reflected waves Damping of transmitted wave

Flap wavemaker

Breadth 0.50

Draft 0.25

Displacement/length 125.0

Lewis form parameter 1.0

1.0

Vertical center of gravity 0.135

Surge restoring coefficient 197.58

Surge damping coefficient 19.80

Variable Dimensional Non-dimensional

Time

Sway (surge) force

Heave force

Roll (pitch) moment

Sway (surge) drift force


Lewis form in regular waves: Fixed body

Sway force

Heave force

Roll moment

Sway drift force

Wave reflection coeff.

Wave transmission coeff.


Lewis form in regular waves: Floating body

Sway RAO Heave RAO Roll RAO

Work in progress


People who are contributing/have contributed to the development/extension of HPC method

Professors in NTNU: Odd M. Faltinsen, Marilena Greco,

Professor in CNR-INSEAN:Claudio Lugni

Post doc:Yanlin Shao (2010-2012)

Phd students: Arnt Gunvald Fredriksen (finished 2015, NTNU)Hui Liang (finished 2015, DUT & NTNU ) Finn-Christian Hanssen (expected 2017, NTNU) Mohd Atif Siddiqui (NTNU)Shaojun Ma (NTNU)Ida Marlen Strand (NTNU)Adrea Bardazzi (CNR-INSEAN)

Master students:Wenbo Zhu (NTNU, finished 2015)Chao Tong (SJTU, 2016-)

Shao & Faltinsen, OMAE 2012Shao & Faltinsen, JCP 2014Shao & Faltinsen, JOMAE 2014Bardazzi et al., JCP, 2015Liang, Faltinsen & Shao, APOR 2015Hanssen, Greco & Shao, OMAE 2015Zhu, Greco & Shao, IJNAOE, 2017Hanssen, Greco & Faltinsen, IWWWFB 2017


Thank you

harmonic polynomial cell (hpc) method with immersed boundaries · harmonic polynomial cell (hpc)...

Documents