A more detailed tutorial¶

In the previous tutorial (Learning the ropes) we provided a high level overview of the PySPH framework. No details were provided on equations, integrators and solvers. This tutorial assumes that you have read the previous one.

Recall that in the previous tutorial, a circular patch of fluid with a given initial velocity field was simulated using a weaky-compressible SPH scheme. In that example, a WCSPHScheme object was created in the create_scheme method. The details of what exactly the scheme does was not discussed. This tutorial explains some of those details by solving the same problem using a lower-level approach where the actual SPH equations, the integrator, and the solver are created manually. This should help a user write their own schemes or modify an existing scheme. The full code for this example can be seen in elliptical_drop_no_scheme.py.

Imports¶

This example requires a few more imports than the previous case.

the first several lines are imports of various modules:

import os
from numpy import array, ones_like, mgrid, sqrt

# PySPH base and carray imports
from pysph.base.utils import get_particle_array_wcsph
from pysph.base.kernels import Gaussian

# PySPH solver and integrator
from pysph.solver.application import Application
from pysph.solver.solver import Solver
from pysph.sph.integrator import EPECIntegrator
from pysph.sph.integrator_step import WCSPHStep

# PySPH sph imports
from pysph.sph.equation import Group
from pysph.sph.basic_equations import XSPHCorrection, ContinuityEquation
from pysph.sph.wc.basic import TaitEOS, MomentumEquation


Note

This is common for all examples that do not use a scheme and it is worth noting the pattern of the PySPH imports. Fundamental SPH constructs like the kernel and particle containers are imported from the base subpackage. The framework related objects like the solver and integrator are imported from the solver subpackage. Finally, we import from the sph subpackage, the physics related part for this problem.

The methods defined for creating the particles are the same as in the previous tutorial with the exception of the call to self.scheme.setup_properties([pa]). In this example, we do not create a scheme, we instead create all the required PySPH objects from the application. We do not override the create_scheme method but instead have two other methods called create_solver and create_equations which handle this.

Setting up the PySPH framework¶

As we move on, we encounter instantiations of the PySPH framework objects. These are the pysph.solver.application.Application, pysph.sph.integrator.TVDRK3Integrator and pysph.solver.solver.Solver objects. The create_solver method constructs a Solver instance and returns it as seen below:

def create_solver(self):
kernel = Gaussian(dim=2)

integrator = EPECIntegrator( fluid=WCSPHStep() )

dt = 5e-6; tf = 0.0076
solver = Solver(kernel=kernel, dim=2, integrator=integrator,
cfl=0.05, n_damp=50,
output_at_times=[0.0008, 0.0038])

return solver


As can be seen, various options are configured for the solver, including initial damping etc.

Intuitively, in an SPH simulation, the role of the EPECIntegrator should be obvious. In the code, we see that we ask for the “fluid” to be stepped using a WCSPHStep object. Taking a look at the create_particles method once more, we notice that the ParticleArray representing the circular patch was named as fluid. So we’re essentially asking the PySPH framework to step or integrate the properties of the ParticleArray fluid using WCSPHStep. It is safe to assume that the framework takes the responsibility to call this integrator at the appropriate time during a time-step.

The Solver is the main driver for the problem. It marshals a simulation and takes the responsibility (through appropriate calls to the integrator) to update the solution to the next time step. It also handles input/output and computing global quantities (such as minimum time step) in parallel.

Specifying the interactions¶

At this stage, we have the particles (represented by the fluid ParticleArray) and the framework to integrate the solution and marshall the simulation. What remains is to define how to actually go about updating properties within a time step. That is, for each particle we must “do something”. This is where the physics for the particular problem comes in.

For SPH, this would be the pairwise interactions between particles. In PySPH, we provide a specific way to define the sequence of interactions which is a list of Equation objects (see SPH equations). For the circular patch test, the sequence of interactions is relatively straightforward:

• Compute pressure from the Equation of State (EOS): $$p = f(\rho)$$
• Compute the rate of change of density: $$\frac{d\rho}{dt}$$
• Compute the rate of change of velocity (accelerations): $$\frac{d\boldsymbol{v}}{dt}$$
• Compute corrections for the velocity (XSPH): $$\frac{d\boldsymbol{x}}{dt}$$

Care must be taken that the EOS equation should be evaluated for all the particles before the other equations are evaluated.

We request this in PySPH by creating a list of Equation instances in the create_equations method:

def create_equations(self):
equations = [
Group(equations=[
TaitEOS(dest='fluid', sources=None, rho0=self.ro,
c0=self.co, gamma=7.0),
], real=False),

Group(equations=[
ContinuityEquation(dest='fluid',  sources=['fluid',]),

MomentumEquation(dest='fluid', sources=['fluid'],
alpha=self.alpha, beta=0.0, c0=self.co),

XSPHCorrection(dest='fluid', sources=['fluid']),

]),
]
return equations


Each Group instance is completed before the next is taken up. Each group contains a list of Equation objects. Each interaction is specified through an Equation object, which is instantiated with the general syntax:

Equation(dest='array_name', sources, **kwargs)


The dest argument specifies the target or destination ParticleArray on which this interaction is going to operate on. Similarly, the sources argument specifies a list of ParticleArrays from which the contributions are sought. For some equations like the EOS, it doesn’t make sense to define a list of sources and a None suffices. The specification basically tells PySPH that for one time step of the calculation:

• Use the Tait’s EOS to update the properties of the fluid array
• Compute $$\frac{d\rho}{dt}$$ for the fluid from the fluid
• Compute accelerations for the fluid from the fluid
• Compute the XSPH corrections for the fluid, using fluid as the source

Note

Notice the use of the ParticleArray name “fluid”. It is the responsibility of the user to ensure that the equation specification is done in a manner consistent with the creation of the particles.

With the list of equations, our problem is completely defined. PySPH now knows what to do with the particles within a time step. More importantly, this information is enough to generate code to carry out a complete SPH simulation. For more details on how new equations can be written please read The PySPH framework.

The example may be run the same way as the previous example:

\$ pysph run elliptical_drop_no_scheme


The resulting output can be analyzed or viewed the same way as in the previous example.

In the previous example (Learning the ropes), the equations and solver are created automatically by the WCSPHScheme. If the create_scheme is overwritten and returns a scheme, the create_equations and create_solver need not be implemented. For more details on the various application methods, please see pysph.solver.application.Application. Implementing other schemes can be done by either implementing the equations directly as done in this example or one could implement a new pysph.sph.scheme.Scheme.