igert: sph and free surface flows robert a. dalrymple civil engineering
TRANSCRIPT
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IGERT: SPH and Free Surface Flows
Robert A. DalrympleCivil Engineering
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Wave Theories
Flat Bottom Theories
Small Amplitude (Airy Theory, 1845)
Shallow Water (Boussinesq, 1871, KdV, 1895)
Intermediate and Deep Depths (Stokes, 1856, Stokes V)
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Wave Modeling
Numerical modeling began in the 1960’s
Numerically assisted analytical methods
Finite difference modeling (marker and cell)
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Stream Function on the Web
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Finite Difference and FiniteElements in the 70’s
• 2-D to 3-D• Multiple numbers of waves• Shoaling, refraction, diffraction
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Parabolic Modeling (REF/DIF)
Scripps Canyon--NCEXhttp://science.whoi.edu/users/elgar/NCEX/wp-refdif.html
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Boussinesq Model
Empirical breakingWave-induced currents!
Much more computing time
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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)
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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
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Topics
• Meshfree methods• Interpolation methods• SPH modeling • Results• GPU : the future
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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)
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Particle Methods
Discrete Element MethodMolecular DynamicsSPHVortex Methods
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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).
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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)?
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Linear Interpolation
x1 x2
y1
y2
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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
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Polynomial InterpolantForce the interpolant through the four points to get four equations:
Rewriting:
The solution for a is found by inverting p
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ExampleData are: (0,2), (1,0.3975), (2, -0.1126), (3, -0.0986).
Fitting a cubic polynomial through the four points gives:
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Polynomial Fit to Example
Exact: redPolynomial fit: blue
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Beware of Extrapolation
An Nth order polynomial has N roots!
Exact: red
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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:
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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
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Question
Show that the equation above leads to the following expression for the best fit straight line:
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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
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Least Squares InterpolantCubic Least Squares Fit: * is the fitting polynomial o is the given data
Exact
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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
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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
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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)
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Weighting Function
q=x/h
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Moving Least Squares InterpolantMinimizing the weighted squared errors for the coefficients:
,, ,
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Moving Least Squares Interpolant
Solving
The final locally valid interpolant is:
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MLS Fit to (Same) Irregular Data
Given data: circles; MLS: *; exact: line
h=0.51
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.3
.5
1.0
1.5
Varying h Values
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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
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The Kernel (or Weighting Function)
Compact support: 2D-circle of radius h 3D-sphere of radius h
1D-line of length 2h
h
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Fundamental Equation of SPH
where W(s-x,h) is a kernel, which behaves like Dirac function.
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Delta Function (1 D)
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Kernel Requirements (Monaghan)
Monotonically decreasing with distance |s-x|
Symmetric with distance
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Numerical Approximations of the Integrals
The incremental volume: mj/j , where the mass is constant.
which is an interpolant!
Partition of unity
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KernelsGaussian:
Not compactly supported -- extends to infinity.
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KernelsMoving Particle Semi-implicit (MPS):
Quadratic:
(discontinuous slope at q=1)
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Spline Kernel
Same kernel as usedin MLS interpolant.
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Gradients
Given SPH interpolant:
Find the gradient directly and analytically:
O.K., but when uj is a constant, there is a problem.
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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)
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ConsistencyTaylor series expansion of u(x,t) about point s (1-D):
Consistency conditions are then:
Integrals of all higher moments must be zero.
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MLS and Shepard Interpolant
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Governing Equations
Kinematics
Conservation of Mass
Conservation of Momentum
Equation of State: Pressure = f()
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Kinematic Equationfor i=1, np
Mass Conservation
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Conservation of Mass
Integrate both sides by the kernel and integrate over the domain:
Next use Gauss theorem:
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Conservation of Mass (2)
0
Put in discrete form:
j
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Conservation of Mass (3)From before (consistency condition):
The derivative of this expression is
Multiplying by ui and adding to previous conservationequation:
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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.
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Conservation of MomentumEquations of motion:
Multiply by and the kernel and integrate as before:
Subtract the zero sum:
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Conservation of MomentumEquations of motion:
Multiply by and the kernel and integrate as before:
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Conservation of Momentum (2)Another formulation (used in JHU SPH):
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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
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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
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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 ~
*
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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.
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Neighbor ListLink list is reconstructed each time step. Cells are 2h x 2h in size.
from SPHysics Users Manual
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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
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Beeman’s Method
Better velocity estimate (but more storage, and needs an+1)
Predictor-corrector version (for velocity):
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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)
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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
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Model Validation – hydrodynamics model
– Star points are experiment data
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Model Validation – hydrodynamics model
– Star points are experiment data
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Model Validation – hydrodynamics model
– Star points are experiment data
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Bore in a Box
Tsunami represented by a dam break
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Bore in a Box: 2004
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3D Weak Plunger
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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
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Coherent Structures: Q Criterion
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Paradigm Shift: GPU Models
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CPU vs GPU
Floating point operations per second: GPU vs CPU
Nvidia CUDA Programming Guide, 1.1
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Nvidia CUDA Programming Guide, 1.1
CPU/GPU Architectures
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Bore in a Box
Real time: CPU vs GPU on a laptop
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Bore in a Box
677,360 particles
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Nvidia Tesla Computing Card
4 Gb/240 = 16 Mb/ core