Smoothed Particle Hydrodynamics

Smoothed Particle Hydrodynamics (SPH, in German: smoothed particle hydrodynamics ) is a numerical method to solve the hydrodynamic equations. It is used among other things in astrophysics, ballistics and tsunami calculations. SPH is a Lagrangian method, ie the coordinates used to swim with the fluid. SPH is a very easy -to-implement, robust method because the calculation of derivatives no mesh is needed.

  • 2.1 symmetrization
  • 2.2 Motion of the fluid
  • 2.3 Continuity Equation in SPH
  • 2.4 Euler equation in SPH
  • 2.5 viscosity
  • 5.1 Films
  • 5.2 code

Method

General

In Smoothed Particle Hydrodynamics to simulate the fluid is divided into elements. The elements are similar to the Monte - Carlo methods, randomly distributed throughout the liquid. This minimizes the expected error. The mean spacing of these elements is represented by the Length Smoothing ( Glättlänge ). It is the most important parameter of the method. Between the particles, the fluid is smoothed by the kernel, thus the name. Any size (e.g., the density ) is calculated by summing over all particles. Each particle receives a share, in the form of a scalar at this size. This is from the partial differential equations of hydrodynamics ordinary differential equations, which greatly simplifies the calculations. SPH is a very empirical method. This means that many things are done because they work, not because there is a strict mathematical derivation.

Derivation

The formal derivation is running either a Lagrangian or a Integralinterpolation. When Integralinterpolation for one size, one starts from an identity, the Dirac delta distribution may refer to:

Then, the distribution is approximated by a core, wherein the smoothing length. In order for the approximation in the limit remains valid, you can ask for normalization and identity with the distribution in the limit as h → 0:

Indeed, this is not the case with most used cores. To get from the division into mass elements, it expanded with the density and leaves greater than 0 for the case of infinitely many, infinitely small particles, the sum becomes the integral. Numerically it is always with finitely many particles must be satisfied:

Here, the mass of the particle and the B density at the position of the particle B:

Thus we have the fundamental equation of Smoothed Particle Hydrodynamics derived ( right panel). The size of A is calculated by a sum of all the particles. You can see that from the dependent of r size a scalar multiplied has become with the kernel. This leads to a great simplification of differential equations as a derivation now no longer affected by the size, but only on the kernel:

Core and smoothing length ( Smoothing Length)

Smoothing length

The most important parameter is the SPH smoothing length. It defines the resolution of the method and thus has a strong influence on accuracy and computational effort for simulations. With an appropriate choice of the kernel (see below) it also defines the number of permanent when calculating with neighbors to be included. Usual up to several tens of particles per size. For good results are based on the average density of the fluid:

With particles dimensions and

In modern codes to choose time-dependent. with

It then uses in areas of large densities higher resolution, while low densities, the smoothing length is greater in areas. This can reduce the computational complexity while maintaining the same accuracy.

Core

The core is probably the most important structure of the SPH method. Different kernels correspond to different difference schemes in grid methods. For the interpretation of SPH equations it is advantageous to use a core in the form of a Gaussian curve:

Numerically this approach, however, is not very appropriate because one often puts on a clear behavior with respect to the reach of the core value in this case. That is to choose a core from a certain zero in order to clearly specify the number of neighbors that are included in the calculation, can. So you can limit the required computational effort. As already mentioned, SPH is a very empirical method, ie for different applications are required very different cores. The exact choice is a matter of experience and is often done after the trial and error approach. Since a nucleus is often implemented in a separate function, the effort has to be replaced or change it to a minimum. Often cores are used on the basis of splines:

With, a normalization constant and the number of dimensions. Here are just particles are included up to next-nearest neighbors in the calculation. Furthermore, the second derivative of this core is not constant, and therefore it is not dependent on the disorder of the particles.

Error estimates

In the derivation above Integralinterpolationsfunktionen two approximations were made. First, it was assumed and the summation is carried out only over a finite number of particles.

  • For the identity, that is, with and arbitrarily many particles are a Taylor expansion of an error.
  • Also for the Summationsnäherung can calculate an error using the Shoenberg formula, if the particles are ordered distributed in the fluid.
  • In the case of disordered particles, there is no traditional error estimation.

This one is for simulations with SPH always rely on the comparison with other simulations, at least for an error estimate. Some publications mention that the error usually significantly lower than those of a Monte Carlo simulation are, this is also a matter of experience. Generally SPH tends to smearing of discontinuities, is so especially in the case of simulations with few particles locally rather inaccurate. For large numbers of particles, the behavior is but a lot better. However, the global behavior is even at low particle numbers, which corresponds to a low computational cost, very good. That is, global quantities such as the energy are well reproduced. Often can be with SPH a global good simulation program with little effort, which can be expected in a reasonable time on workstations.

Advantages and Disadvantages

Advantages:

  • SPH is a Lagrangian method; the continuity equation is automatically satisfied.
  • The code is very robust, ie almost always produces results
  • The implementation of SPH is relatively simple, just testing different kernels.
  • Using a Gaussian function as the kernel can be theoretical results easy to interpret.
  • In modern code shows a dependence of the computational complexity of the particle.
  • SPH shows good global results at low particle numbers.

Cons:

  • The code is often too robust; Despite a wrong model SPH can deliver results, but are then physically incorrect
  • The error estimate is often a problem and to obtain only in comparison with the results of other methods
  • The method is highly dispersive
  • For good accuracy high particle numbers are needed. This has the advantage low computational cost is not applicable
  • The treatment of discontinuities is often difficult, since structures on scales that are smaller than the smoothing length smoothed.

Hydrodynamic equations in SPH

Symmetrization

To formulate the hydrodynamics in SPH, the seemingly easiest approach is to use the basic equation in the hydrodynamic equations like the Navier -Stokes equation. However, the resulting equations are not symmetrical with respect to Teilchenvertauschung. Therefore, in this case, many apply conservation of energy, angular momentum, etc., not more. Often, however, it is possible to salvage this by writing in the density in the differential operator and uses the product rule:

Often, you can derive so symmetric equations. All this happens not strictly formal, but only because it provides better results.

Movement of the fluid

The simplest way is to use the definition of the speed:

The movement of a particle is not linked to the other, which often can lead to problems. Therefore we developed the XSPH method ( "Extended SPH " ):

With an averaged density:

And a coupling parameter ε. Thus, the order of the particle is better preserved without additional viscosity must be introduced.

Continuity equation in SPH

If we set the density in the basic equation, we obtain

For a particle a From this, calculate the SPH continuity equation

Euler equation in SPH

For the Euler equation yields:

This equation is not symmetric with respect to particle exchange: momentum and torque are not obtained. That's why we use the above-indicated trick for the pressure gradient:

What we obtain the desired balanced equation:

We use a Gaussian function gives a central force equal strongly affects both particles:

Viscosity

Like almost every numerical method also generates SPH by computing inaccuracies viscosity. To model this is often not sufficient. Therefore leads you, similar to the transition from the Euler equation for the Navier -Stokes equation, a Viskositätstensor one. The exact choice of this tensor depends strongly on the model.

Applications

SPH is used in many different fields such as astrophysics application. There are also relativistic and magnetic SPH methods:

  • Gas dynamics
  • Galaxy formation and fusion
  • Binary star systems, accretion disks and stellar collisions
  • Lunar origin
  • Relativistic problems
  • Magnetic problems

Related Publications

  • Monaghan: Smoothed Particle Hydrodynamics; Annu. Rev. Astrophys. 1992
  • Steinmetz, Müller: On the capabilities and limits of sph; Astronomy and Astrophysics 1993
  • Alimi, Courty: Thermodynamic evolution of the cosmological Baryonic gas pt.2; Astronomy and Astrophysics 2005
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