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IGERT: SPH and Free Surface Flows. Robert A. Dalrymple Civil 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. - PowerPoint PPT Presentation
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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
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