Smoothed Particle Hydrodynamics (SPH) 4. May 2012

Transcription

1 Smoothed Particle Hydrodynamics (SPH) 4. May 2012

2 Calculating density

3 SPH density estimator Weighted summation over nearby particles: ρ(r) = N neigh b=1 m bw (r r b, h) W weight function with dimension of inverse volume. h scale parameter determining fall-off rate of W. Conservation of mass implies W normalisation: V W (r r b, h)dv = 1

4 The smoothing kernel (Weight function) Positive weighting, monotonically decreasing with relative distance, smooth derivatives. Symmetry with respect to (r r ) i.e., W (r r, h) W ( r r, h) Flat central portion (No strong influence by small change in near neighbour position change) A natural choice that satisfies all of the properties is the Gaussian: d number of dimensions. σ normalisation factor. ] W (r r, h) = σ exp [ (r r ) 2 h d h 2

5 The smoothing kernel (Weight function) Problem with Gaussian: Interaction with all particles in the domain necessary, although the contribution of particles quickly becomes negligible with increasing distance. Better choice: Gaussian shaped kernel, but truncated at some finite distance (a few times the scale length h). A popular choice is the Schoenberg (1946) B-Spline: w(q) = σ 1 4 (2 q)3 (1 q) 3, 0 q < 1; 1 4 (2 q)3, 1 q < 2; 0. q 2, There is a large number of alternative Kernels.

6 The smoothing length (scale length) h Specifies the fall-off of the kernel weighting with respect the to particle separation. The number of neighbours (inside the smoothing kernel) can change without changing h (Higher order B-splines) To resolve clustered and sparse regions evenly a natural choice is to relate h to the local number density of particles: h(r) n(r) 1/d n(r) = b W [r r b, h(r)]. For equal mass particles this is equivalent to making h proportional to density itself. Since in turn density itself is a function of h, this leads to the idea of iterative summation to simultaneous obtain density and h. This is done for the two equations: ρ(r a ) = b m bw (r a r b, h a ) h(r a ) = η ( ma ρ a ) 1/d

7 From density to equations of motion How to calculate arbitrary quantities A(r) (which could be velocity, energy, etc.) using the density estimate? First use the identity A(r) = A(r )δ(r r )dr and approximate using the smoothing kernel instead of the Dirac delta function. A(r) = A(r )W (r r, h)dr + O(h 2 ) Now discretise onto a finite set of interpolation points. A(r) = A(r ) ρ(r ) W (r r, h)ρ(r )dr N neigh b=1 m b A b ρ b W (r r b, h)

8 From density to equations of motion The summation interpolant forms the base of all SPH formalisms. Gradient terms can now be straightforwardly calculated. A(r) b m b A b ρ b W (r r b, h) A(r) b m b A b ρ b W (r r b, h) A(r) b m b A b ρ b W (r r b, h) j A i (r) b m b Ai b ρ b j W (r r b, h) In case of a non-constant smoothing length also the derivative of the smoothing length has to be taken into account.

9 From density to equations of motion (Example: continuity equation) Start with the base expression for ρ: Time derivative: ρ a = b m bw (r a r b, h a ) dρ a dt = b m b (v a v b ) a W ab (h). Rewrite: dρ a dt = v a b m b ρ b ρ b a W ab m b b ρ b (ρ b v b ) a W ab Which is an approximate expression for (derivatives from the last slide): v a ρ (ρv) ρ a ( v) a

10 Non constant smoothing length In the case of a non-constant smoothing length, also the derivative of the smoothing length needs to be taken into account. Using the time derivative of the density again: dρ a dt = b m b (v a v b ) a W ab (h). Changes to: With the correction term Ω a : dρ a dt = 1 Ω a b m b (v a v b ) a W ab (h a ) Ω a [ ] 1 ha ρ a b m b W ab(h a) h a

11 Generalised first derivate operators The gradient operators can be generalised using the expressions A = 1 φ [ (φa) A φ] b m b ρ b φ b φ a (A b A a ) a W ab and [ ( )] A = φ A φ + A φ 2 φ ( ) m b φb b ρ b φ a A a + φa φ b A b a W ab where φ is any arbitrary, differentiable scalar quantity defined on the particles. Various alternative SPH formulations are based on different choices for φ. Two examples using the expression for acceleration: (Hernquist, Katz, 1989) dv dt = b m b (Morris, 1996) dv dt = b m b ( 2 PaP b ρ aρ b ) a W ab ( Pa P b ρ aρ b ) a W ab

12 Choice of the derivative The paradox of SPH is that choosing a less accurate derivate estimator can actually be a good decision. Using the accurate derivate estimator of Morris for a constant density box of particles (randomly distributed, which is a bad choice) the particle-particle force is: m a m b [ Pa P b ρ aρ b ] F ab (ˆr a ˆr b ). Here, the force vanishes when the pressure is constant regardless of the particle distribution. Using a less accurate estimate: [ ] m a m P b F ρ 2 ab (ˆr a ˆr b ). a + P ρ 2 b there will be a net (repulsive) force between the particles until a certain arrangement of particles is reached (locally isotropic and regular).

13 Alternative derivation of the SPH formalism Price derives the equations of motion for SPH from first principles using only ( the) Lagrangian L = T V inserted into the Euler-Lagrange equations d L dt v a L r a = 0 From this the whole SPH equations of motion can also be derived. The full equations of motion are (including the treatment of magnetic fields): dv i a dt = b m b [ S ij a Ω aρ 2 a ( ) with S ij P + 1 2µ 0 B 2 δ ij + 1 µ 0 B i B j ] j aw ab (h a ) + Sij b j Ω b ρ aw 2 ab (h b ) b Induction equation for the evolution of the magnetic field: [ db i dt = 1 N [ Ωρ i j=1 m Bi j ( v ij i W i ) v ij ( B i i W i )]]

14 Higher order derivatives With some work one can introduce higher order derivates to calculate quantities such as 2 A or ( A) Those higher order derivatives can be used to discretise additional physics (depending on higher order derivates) like viscosity or thermal conductivity. Taking viscosity as an example, the standard formulation of artificial viscosity is given by: ( dv ) dt diss = b m b α c s,abµ ab +βµ 2 ab ρ ab ˆr ab F ab µ ab = hv ab r ab r ab +ɛ h ab 2 Barred quantities correspond to an average c s is the sound speed α and β are dimensionless parameters (α = 1, β = 2 normally) ɛ 0.01 is a small parameter to prevent divergences.

15 Example for particle rearrangement and (artificial) viscosity

16 Treatment of discontinuities using artificial viscosity and artificial conductivity du dt diss = mb b ρ ab P h 1 2 αvsig (vab i u (u u ) F r ab )2 + αu vsig a b ab

17 SPH Advantage No geometry constraints Auto resolution Perfect advection Energy conservation Easy NBody coupling Disadvantage Shocks broadened Mixing Conduction by hand Derivative estimator Visualisation

18 SPH Codes Gadget Phantom NDSPMHD SPHNG Seren VINE GCSPH SPARTACUS