IGERT: SPH and Free Surface Flows

Robert A. DalrympleCivil Engineering

Wave Theories

Flat Bottom Theories

Small Amplitude (Airy Theory, 1845)

Shallow Water (Boussinesq, 1871, KdV, 1895)

Intermediate and Deep Depths (Stokes, 1856, Stokes V)

Wave Modeling

Numerical modeling began in the 1960’s

Numerically assisted analytical methods

Finite difference modeling (marker and cell)

Stream Function on the Web

Finite Difference and FiniteElements in the 70’s

• 2-D to 3-D• Multiple numbers of waves• Shoaling, refraction, diffraction

Parabolic Modeling (REF/DIF)

Scripps Canyon--NCEXhttp://science.whoi.edu/users/elgar/NCEX/wp-refdif.html

Boussinesq Model

Empirical breakingWave-induced currents!

Much more computing time

Perturbation TheoryO(/εn)

Direct Wavefield SimulationO(100)

DNS/LES O() ~ O(10)

Wave Theory & Modeling

Laboratory Experiments O(10) / Field Measurements O(100)

• Kinematic and dynamic boundary conditions on interface

• Mud energy dissipation for wavefield simulation

• Cross validation• Shear instability characteristics• Parameter for simulations

• Cross validation• Interfacial boundary

conditions• Parameters for

simulations• Dissipation model

development

HOS/SNOW (MIT, JHU)

Meshfree Lagrangian Numerical Method for Wave Breaking

WaterParticles

2h

r

Boundary Particles

Radius ofKernel function, W

Fluid is described by quantities at discrete nodes

Approximated by a summation interpolant; other options: MLS

Topics

• Meshfree methods• Interpolation methods• SPH modeling • Results• GPU : the future

Mesh-Free Methods

Smoothed particle hydrodynamics (SPH) (1977)Diffuse element method (DEM) (1992)Element-free Galerkin method (EFG / EFGM) (1994)Reproducing kernel particle method (RKPM) (1995)hp-cloudsNatural element method (NEM)Material point method (MPM)Meshless local Petrov Galerkin (MLPG)Generalized finite difference method (GFDM)Particle-in-cell (PIC)Moving particle finite element method (MPFEM)Finite cloud method (FCM)Boundary node method (BNM)Boundary cloud method (BCM)Method of finite spheres (MFS)Radial Basis Functions (RBF)

Particle Methods

Discrete Element MethodMolecular DynamicsSPHVortex Methods

Why Interpolation?

• For discrete models of continuous systems, we need the ability to interpolate values in between discrete points.

• Half of the SPH technique involves interpolation of values known at particles (or nodes).

Interpolation

• To find the value of a function between known values.

Consider the two pairs of values (x,y): (0.0, 1.0), (1.0, 2.0)

What is y at x = 0.5? That is, what’s (0.5, y)?

Linear Interpolation

x1 x2

y1

y2

Polynomial InterpolantsGiven N (=4) data points,

Find the interpolating function that goes through the points:

If there were N+1 data points, the function would be

with N+1 unknown values, ai, of the Nth order polynomial

Polynomial InterpolantForce the interpolant through the four points to get four equations:

Rewriting:

The solution for a is found by inverting p

ExampleData are: (0,2), (1,0.3975), (2, -0.1126), (3, -0.0986).

Fitting a cubic polynomial through the four points gives:

Polynomial Fit to Example

Exact: redPolynomial fit: blue

Beware of Extrapolation

An Nth order polynomial has N roots!

Exact: red

Least Squares InterpolantFor N points, we will use a lower order fitting polynomial of order m < (N-1).

The least squares fitting polynomial is similar to the exact fit form:

Now p is N x m matrix. Since we have fewer unknown coefficients as data points, the interpolant cannot go through each point. Define the error as the amount of “miss”

Sum of the (errors)2:

Least Squares InterpolantMinimizing the sum with respect to the coefficients a:

Solving,

This can be rewritten in this form,

which introduces a pseudo-inverse.

Reminder:

for cubic fit

Question

Show that the equation above leads to the following expression for the best fit straight line:

Cubic Least Squares Example

x: -0.2 .44 1.0 1.34 1.98 2.50 2.95 3.62 4.13 4.64 4.94y: 2.75 1.80 -1.52 -2.83 -1.62 1.49 2.98 0.81 -2.14 -2.93 -1.81

Data irregularly spaced

Least Squares InterpolantCubic Least Squares Fit: * is the fitting polynomial o is the given data

Exact

Piecewise InterpolationPiecewise polynomials: fit all points

Linear: continuity in y+, y- (fit pairs of points)

Quadratic: +continuity in slope

Cubic splines: +continuity in second derivative

RBFAll of the above, but smoother

Moving Least Squares Interpolant

are monomials in x for 1D (1, x, x2, x3)x,y in 2D, e.g. (1, x, y, x2, xy, y2 ….)

Note aj are functions of x

Moving Least Squares InterpolantDefine a weighted mean-squared error:

where W(x-xi) is a weighting function that decayswith increasing x-xi.

Same as previous least squares approach, except for W(x-xi)

Weighting Function

q=x/h

Moving Least Squares InterpolantMinimizing the weighted squared errors for the coefficients:

,, ,

Moving Least Squares Interpolant

Solving

The final locally valid interpolant is:

MLS Fit to (Same) Irregular Data

Given data: circles; MLS: *; exact: line

h=0.51

.3

.5

1.0

1.5

Varying h Values

Basis for Smoothed Particle Hydrodynamics SPH

From astrophysics (Lucy,1977; Gingold and Monaghan, 1977))

An integral interpolant (an approximation to a Dirac delta function):

kernel

The Kernel (or Weighting Function)

Compact support: 2D-circle of radius h 3D-sphere of radius h

1D-line of length 2h

h

Fundamental Equation of SPH

where W(s-x,h) is a kernel, which behaves like Dirac function.

Delta Function (1 D)

Kernel Requirements (Monaghan)

Monotonically decreasing with distance |s-x|

Symmetric with distance

Numerical Approximations of the Integrals

The incremental volume: mj/j , where the mass is constant.

which is an interpolant!

Partition of unity

KernelsGaussian:

Not compactly supported -- extends to infinity.

KernelsMoving Particle Semi-implicit (MPS):

Quadratic:

(discontinuous slope at q=1)

Spline Kernel

Same kernel as usedin MLS interpolant.

Gradients

Given SPH interpolant:

Find the gradient directly and analytically:

O.K., but when uj is a constant, there is a problem.

Gradients (2)

To fix problem, recall partition of unity equation:

The gradient of this equation is zero:

So we can multiply by ui = u(x) and subtract from Eq.(A)

(A)

ConsistencyTaylor series expansion of u(x,t) about point s (1-D):

Consistency conditions are then:

Integrals of all higher moments must be zero.

MLS and Shepard Interpolant

Governing Equations

Kinematics

Conservation of Mass

Conservation of Momentum

Equation of State: Pressure = f()

Kinematic Equationfor i=1, np

Mass Conservation

Conservation of Mass

Integrate both sides by the kernel and integrate over the domain:

Next use Gauss theorem:

Conservation of Mass (2)

0

Put in discrete form:

j

Conservation of Mass (3)From before (consistency condition):

The derivative of this expression is

Multiplying by ui and adding to previous conservationequation:

Conservation of Mass (4)To determine the density at a given point s:

A more accurate approach is:

During computations, the density can be regularly redetermined through this expression, due to Randles and Liberski.

Conservation of MomentumEquations of motion:

Multiply by and the kernel and integrate as before:

Subtract the zero sum:

Conservation of MomentumEquations of motion:

Multiply by and the kernel and integrate as before:

Conservation of Momentum (2)Another formulation (used in JHU SPH):

Equation of State: Weak Compressibility

which implies a speed of sound, CS:

where is usually 7 and B is a constant

No need to solve PDE for pressure

Closure Submodels

Viscosity generally accounted for by an artificial empirical term (Monaghan 1992):

0.

0.

0

ijij

ijij

ij

ijij

ij

c

rv

rv

22

.

ij

ijijij r

h rv

Sub-Particle Scale (SPS) Turbulence Model• Spatial-filter over the governing equations (compressible):

(Favre-averaging)

= SPS stress tensor with elements: ijijkkijtij kSS 3

232 ~~

2

• Eddy viscosity: SlCst2

• Smagorinsky constant: Cs 0.12 See Lo & Shao (2002), Gotoh et al. (2002) for incompressible SPS

2/12 ijijSSS

Sij = strain tensor

ff ~

*

Neighbor Lists

Which particles within 2h of particle i?

In principle, each particle interacts with every other particlein the domain: N x N calculations (“all pair search”)

Linked List:“Cells”: Examine particles in neighboring cells;

only 9 cells to examine.

Tree SearchDevelop a tree. Search geometrically.

Neighbor ListLink list is reconstructed each time step. Cells are 2h x 2h in size.

from SPHysics Users Manual

Time Stepping

Discretize time:

Take an Euler step (first order derivative expression):

Use this to get a value of

Then make a corrector step:

Monaghan; moreaccurate than Euler step

Beeman’s Method

Better velocity estimate (but more storage, and needs an+1)

Predictor-corrector version (for velocity):

Boundary conditions are problematic in SPH due to: the boundary is not well definedkernel sum deficiencies at boundaries, e.g. density

Ghost (or virtual) particles (Takeda et al. 1994)Leonard-Jones forces (Monaghan 1994)Boundary particles with repulsive forces (Monaghan 1999)Rows of fixed particles that masquerade as interior flow particles

(Dalrymple & Knio 2001)

(Can use kernel normalization techniques to reduce interpolation errors at

the boundaries, Bonet and Lok 2001)

Boundary Conditions

b

a f = n R(y) P(x)

y

(slip BC)

DEVELOPED JOINTLY BY RESEARCHERS AT:

Universita di Roma “La Sapienza”

(Italy)

Universidade de Vigo

(Spain)

Johns Hopkins University

(USA)

University of Manchester

(U.K.)

SPHYSICS

Model Validation – hydrodynamics model

– Star points are experiment data

Model Validation – hydrodynamics model

– Star points are experiment data

Model Validation – hydrodynamics model

– Star points are experiment data

Bore in a Box

Tsunami represented by a dam break

Bore in a Box: 2004

3D Weak Plunger

Coherent Structures

The rate of strain tensor and the rotation tensor are

Q criterion

Second invariant of the velocity gradient tensor

Positive values of Q imply vorticity greater than strain

, velocity gradient tensor

Coherent Structures: Q Criterion

Paradigm Shift: GPU Models

CPU vs GPU

Floating point operations per second: GPU vs CPU

Nvidia CUDA Programming Guide, 1.1

Nvidia CUDA Programming Guide, 1.1

CPU/GPU Architectures

Bore in a Box

Real time: CPU vs GPU on a laptop

Bore in a Box

677,360 particles

Nvidia Tesla Computing Card

4 Gb/240 = 16 Mb/ core