Download Presentation
## IGERT: SPH and Free Surface Flows

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**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**Numerical modeling began in the 1960’s Numerically assisted analytical methods Finite difference modeling (marker and cell)**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--NCEX http://science.whoi.edu/users/elgar/NCEX/wp-refdif.html**Boussinesq Model**Much more computing time Empirical breaking Wave-induced currents!**HOS/SNOW (MIT, JHU)**Wave Theory & Modeling • Cross validation • Shear instability characteristics • Parameter for simulations Perturbation Theory O(/εn) DNS/LES O() ~ O(10) • Kinematic and dynamic boundary conditions on interface • Mud energy dissipation for wavefield simulation • Cross validation • Interfacial boundary conditions • Parameters for simulations • Dissipation model development Direct Wavefield Simulation O(100) Laboratory Experiments O(10) / Field Measurements O(100)**Meshfree Lagrangian Numerical Method for Wave Breaking**Fluid is described by quantities at discrete nodes Approximated by a summation interpolant; other options: MLS Radius of Kernel function, W Water Particles r 2h Boundary Particles**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-clouds Natural 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 Method Molecular Dynamics SPH Vortex 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**Given two points, (x1,y1), (x2,y2): Fit a straight line between the points. y(x) = a x +b y1= a x1 + b y2= a x2 + b a=(y2-y1)/(x2-x1), b= (y1 x2-y2 x1)/(x2-x1), Use this equation to find y values for any x1 < x < x2 y2 y1 x1x2**Polynomial Interpolants**Given 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 Interpolant**Force the interpolant through the four points to get four equations: Rewriting: The solution for a is found by inverting p**Example**Data 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: red Polynomial fit: blue**Beware of Extrapolation**Exact: red An Nth order polynomial has N roots!**Least Squares Interpolant**For 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 Interpolant**Minimizing 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**Data irregularly spaced x: -0.2 .44 1.0 1.34 1.98 2.50 2.95 3.62 4.13 4.64 4.94 y: 2.75 1.80 -1.52 -2.83 -1.62 1.49 2.98 0.81 -2.14 -2.93 -1.81**Least Squares Interpolant**Cubic Least Squares Fit: * is the fitting polynomial o is the given data Exact**Piecewise Interpolation**Piecewise polynomials: fit all points Linear: continuity in y+, y- (fit pairs of points) Quadratic: +continuity in slope Cubic splines: +continuity in second derivative RBF All 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 ofx**Moving Least Squares Interpolant**Define a weighted mean-squared error: where W(x-xi) is a weighting function that decays with increasing x-xi. Same as previous least squares approach, except for W(x-xi)**Weighting Function**q=x/h**Moving Least Squares Interpolant**Minimizing the weighted squared errors for the coefficients: , , ,**Moving Least Squares Interpolant**Solving The final locally valid interpolant is:**MLS Fit to (Same) Irregular Data**h=0.51 Given data: circles; MLS: *; exact: line**Varying h Values**1.0 .3 1.5 .5**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)**h Compact support: 2D-circle of radius h 3D-sphere of radius h 1D-line of length 2h**Fundamental Equation of SPH**where W(s-x,h) is a kernel,which behaves like Dirac function.**Kernel Requirements (Monaghan)**Monotonically decreasing with distance |s-x| Symmetric with distance**Numerical Approximations of the Integrals**Partition of unity The incremental volume: mj/j , where the mass is constant. which is an interpolant!**Kernels**Gaussian: Not compactly supported -- extends to infinity.**Kernels**Moving Particle Semi-implicit (MPS): Quadratic: (discontinuous slope at q=1)**Spline Kernel**Same kernel as used in 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)**(A) 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)**Consistency**Taylor series expansion of u(x,t) about point s (1-D): Consistency conditions are then: Integrals of all higher moments must be zero.**Governing Equations**Kinematics Conservation of Mass Conservation of Momentum Equation of State: Pressure = f()**Kinematic Equation**for 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