underworld.systems module

This module contains routines relating to differential system.

Module Summary

functions:

underworld.systems.Solver This method simply returns a necessary solver for the provided system.

classes:

underworld.systems.Stokes This class provides functionality for a discrete representation of the Stokes flow equations.
underworld.systems.SteadyStateHeat This class provides functionality for a discrete representation of the steady state heat equation.
underworld.systems.SteadyStateDarcyFlow This class provides functionality for a discrete representation of the steady state darcy flow equation.
underworld.systems.TimeIntegration Abstract class for integrating numerical objects (fields, swarms, etc.) in time.
underworld.systems.AdvectionDiffusion This class provides functionality for a discrete representation of an advection-diffusion equation.
underworld.systems.SwarmAdvector Objects of this class advect a swarm through time using the provided velocity field.

Module Details

functions:

underworld.systems.Solver(eqs, type='BSSCR', *args, **kwargs)[source]

This method simply returns a necessary solver for the provided system.

classes:

class underworld.systems.Stokes(velocityField, pressureField, fn_viscosity, fn_bodyforce=None, fn_one_on_lambda=None, fn_lambda=None, fn_source=None, voronoi_swarm=None, conditions=[], _removeBCs=True, _fn_viscosity2=None, _fn_director=None, fn_stresshistory=None, _fn_stresshistory=None, _fn_v0=None, _fn_p0=None, _callback_post_solve=None, **kwargs)[source]

Bases: underworld._stgermain.StgCompoundComponent

This class provides functionality for a discrete representation of the Stokes flow equations.

Specifically, the class uses a mixed finite element method to construct a system of linear equations which may then be solved using an object of the underworld.system.Solver class.

The underlying element types are determined by the supporting mesh used for the ‘velocityField’ and ‘pressureField’ parameters.

The strong form of the given boundary value problem, for \(f\), \(g\) and \(h\) given, is

\[\begin{split}\begin{align} \sigma_{ij,j} + f_i =& \: 0 & \text{ in } \Omega \\ u_{k,k} + \frac{p}{\lambda} =& \: H & \text{ in } \Omega \\ u_i =& \: g_i & \text{ on } \Gamma_{g_i} \\ \sigma_{ij}n_j =& \: h_i & \text{ on } \Gamma_{h_i} \\ \end{align}\end{split}\]

where,

  • \(\sigma_{i,j}\) is the stress tensor
  • \(u_i\) is the velocity,
  • \(p\) is the pressure,
  • \(f_i\) is a body force,
  • \(\lambda\) is a bulk viscosity,
  • \(H\) is the compressible equation source term,
  • \(g_i\) are the velocity boundary conditions (DirichletCondition)
  • \(h_i\) are the traction boundary conditions (NeumannCondition).

The problem boundary, \(\Gamma\), admits the decompositions \(\Gamma=\Gamma_{g_i}\cup\Gamma_{h_i}\) where \(\emptyset=\Gamma_{g_i}\cap\Gamma_{h_i}\). The equivalent weak form is:

\[\int_{\Omega} w_{(i,j)} \sigma_{ij} \, d \Omega = \int_{\Omega} w_i \, f_i \, d\Omega + \sum_{j=1}^{n_{sd}} \int_{\Gamma_{h_j}} w_i \, h_i \, d \Gamma\]

where we must find \(u\) which satisfies the above for all \(w\) in some variational space.

Parameters:
  • velocityField (underworld.mesh.MeshVariable) – Variable used to record system velocity.
  • pressureField (underworld.mesh.MeshVariable) – Variable used to record system pressure.
  • fn_viscosity (underworld.function.Function) – Function which reports a viscosity value. Function must return scalar float values.
  • fn_bodyforce (underworld.function.Function, Default = None) – Function which reports a body force for the system. Function must return float values of identical dimensionality to the provided velocity variable.
  • fn_minus_one_on_lambda (underworld.function.Function, Default = None) – Function which defines a non solenoidal velocity field via the relationship div(velocityField) = -fn_minus_one_on_lambda * pressurefield + fn_source When this is left as None a incompressible formulation of the stokes equation is formed, ie, div(velocityField) = 0. fn_minus_one_on_lambda is incompatible with the ‘penalty’ stokes solver, ensure a ‘penalty’ equal to 0 is used when fn_minus_one_on_lambda is used. By default this is the case.
  • fn_source (underworld.function.Function, Default = None) – Function which defines a non solenoidal velocity field via the relationship div(velocityField) = -fn_minus_one_on_lambda * pressurefield + fn_source. fn_minus_one_on_lambda is incompatible with the ‘penalty’ stokes solver, ensure the ‘penalty’ of 0, is used when fn_lambda is used. By default this is the case.
  • fn_stresshistory (underworld.function.Function, Default = None) – Function which defines the stress history term used for viscoelasticity. Function is a vector of size 3 (dim=2) or 6 (dim=3) representing a symetric tensor.
  • voronoi_swarm (underworld.swarm.Swarm) – If a voronoi_swarm is provided, voronoi type numerical integration is utilised. The provided swarm is used as the basis for the voronoi integration. If no voronoi_swarm is provided, Gauss integration is used.
  • conditions (underworld.conditions.SystemCondition) – Numerical conditions to impose on the system. This should be supplied as the condition itself, or a list object containing the conditions.

Notes

Constructor must be called by collectively all processes.

eqResiduals

Returns the stokes flow equations’ residuals from the latest solve. Residual calculations use the matrices and vectors of the discretised problem. The residuals correspond to the momentum equation and the continuity equation.

Returns:r1 is the momentum equation residual r2 is the continuity equation residual
Return type:(r1, r2) - 2 tuple of doubles

Notes

This method must be called collectively by all processes.

fn_bodyforce

The body force function. You may change this function directly via this property.

fn_one_on_lambda

A bulk viscosity parameter

fn_source

The volumetric source term function. You may change this function directly via this property.

fn_viscosity

The viscosity function. You may change this function directly via this property.

stokes_callback

Return the callback function used by this system

velocity_rms()[source]

Calculates RMS velocity as follows

\[v_{rms} = \sqrt{ \frac{ \int_V (\mathbf{v}.\mathbf{v}) \, \mathrm{d}V } {\int_V \, \mathrm{d}V} }\]
class underworld.systems.SteadyStateHeat(temperatureField, fn_diffusivity, fn_heating=0.0, voronoi_swarm=None, conditions=[], _removeBCs=True, **kwargs)[source]

Bases: underworld._stgermain.StgCompoundComponent

This class provides functionality for a discrete representation of the steady state heat equation.

The class uses a standard Galerkin finite element method to construct a system of linear equations which may then be solved using an object of the underworld.system.Solver class.

The underlying element types are determined by the supporting mesh used for the ‘temperatureField’.

The strong form of the given boundary value problem, for \(f\), \(h\) and \(h\) given, is

\[\begin{split}\begin{align} q_i =& - \alpha \, u_{,i} & \\ q_{i,i} =& \: f & \text{ in } \Omega \\ u =& \: g & \text{ on } \Gamma_g \\ -q_i n_i =& \: h & \text{ on } \Gamma_h \\ \end{align}\end{split}\]

where, \(\alpha\) is the diffusivity, \(u\) is the temperature, \(f\) is a source term, \(g\) is the Dirichlet condition, and \(h\) is a Neumann condition. The problem boundary, \(\Gamma\), admits the decomposition \(\Gamma=\Gamma_g\cup\Gamma_h\) where \(\emptyset=\Gamma_g\cap\Gamma_h\). The equivalent weak form is:

\[-\int_{\Omega} w_{,i} \, q_i \, d \Omega = \int_{\Omega} w \, f \, d\Omega + \int_{\Gamma_h} w \, h \, d \Gamma\]

where we must find \(u\) which satisfies the above for all \(w\) in some variational space.

Parameters:
  • temperatureField (underworld.mesh.MeshVariable) – The solution field for temperature.
  • fn_diffusivity (underworld.function.Function) – The function that defines the diffusivity across the domain.
  • fn_heating (underworld.function.Function) – A function that defines the heating across the domain. Optional.
  • voronoi_swarm (underworld.swarm.Swarm) – If a voronoi_swarm is provided, voronoi type numerical integration is utilised. The provided swarm is used as the basis for the voronoi integration. If no voronoi_swarm is provided, Gauss integration is used.
  • conditions (underworld.conditions.SystemCondition) – Numerical conditions to impose on the system. This should be supplied as the condition itself, or a list object containing the conditions.

Notes

Constructor must be called collectively by all processes.

Example

Setup a basic thermal system:

>>> linearMesh = uw.mesh.FeMesh_Cartesian( elementType='Q1/dQ0', elementRes=(4,4), minCoord=(0.,0.), maxCoord=(1.,1.) )
>>> tField = uw.mesh.MeshVariable( linearMesh, 1 )
>>> topNodes = linearMesh.specialSets["MaxJ_VertexSet"]
>>> bottomNodes = linearMesh.specialSets["MinJ_VertexSet"]
>>> tbcs = uw.conditions.DirichletCondition(tField, topNodes + bottomNodes)
>>> tField.data[topNodes.data] = 0.0
>>> tField.data[bottomNodes.data] = 1.0
>>> tSystem = uw.systems.SteadyStateHeat(temperatureField=tField, fn_diffusivity=1.0, conditions=[tbcs])

Example with non diffusivity:

>>> k = tField + 1.0
>>> tSystem = uw.systems.SteadyStateHeat(temperatureField=tField, fn_diffusivity=k, conditions=[tbcs])
>>> solver = uw.systems.Solver(tSystem)
>>> solver.solve()
Traceback (most recent call last):
...
RuntimeError: Nonlinearity detected.
Diffusivity function depends on the temperature field provided to the system.
Please set the 'nonLinearIterate' solve parameter to 'True' or 'False' to continue.
>>> solver.solve(nonLinearIterate=True)
fn_diffusivity

The diffusivity function. You may change this function directly via this property.

fn_heating

The heating function. You may change this function directly via this property.

class underworld.systems.SteadyStateDarcyFlow(pressureField, fn_diffusivity, fn_bodyforce=None, voronoi_swarm=None, conditions=[], velocityField=None, swarmVarVelocity=None, _removeBCs=True, **kwargs)[source]

Bases: underworld._stgermain.StgCompoundComponent

This class provides functionality for a discrete representation of the steady state darcy flow equation.

The class uses a standard Galerkin finite element method to construct a system of linear equations which may then be solved using an object of the underworld.system.Solver class.

The underlying element types are determined by the supporting mesh used for the ‘pressureField’.

The strong form of the given boundary value problem, for \(f\), \(q\) and \(h\) given, is

\[\begin{split}\begin{align} q_i =& \kappa \, ( -u_{,i} + S_i ) & \\ q_{i,i} =& \: f & \text{ in } \Omega \\ u =& \: q & \text{ on } \Gamma_q \\ -q_i n_i =& \: h & \text{ on } \Gamma_h \\ \end{align}\end{split}\]

where,

  • \(\kappa\) is the diffusivity, \(u\) is the pressure,
  • \(S\) is a flow body-source, for example due to gravity,
  • \(f\) is a source term, \(q\) is the Dirichlet condition, and
  • \(h\) is a Neumann condition.

The problem boundary, \(\Gamma\), admits the decomposition \(\Gamma=\Gamma_q\cup\Gamma_h\) where \(\emptyset=\Gamma_q\cap\Gamma_h\). The equivalent weak form is:

\[-\int_{\Omega} w_{,i} \, q_i \, d \Omega = \int_{\Omega} w \, f \, d\Omega + \int_{\Gamma_h} w \, h \, d \Gamma\]

where we must find \(u\) which satisfies the above for all \(w\) in some variational space.

Parameters:
  • pressureField (underworld.mesh.MeshVariable) – The solution field for pressure.
  • fn_diffusivity (underworld.function.Function) – The function that defines the diffusivity across the domain.
  • fn_bodyforce (underworld.function.Function) – A function that defines the flow body-force across the domain, for example gravity. Must be a vector. Optional.
  • voronoi_swarm (underworld.swarm.Swarm) – A swarm with just one particle within each cell should be provided. This avoids the evaluation of the velocity on nodes and inaccuracies arising from diffusivity changes within cells. If a swarm is provided, voronoi type numerical integration is utilised. The provided swarm is used as the basis for the voronoi integration. If no voronoi_swarm is provided, Gauss integration is used.
  • conditions (underworld.conditions.SystemCondition) – Numerical conditions to impose on the system. This should be supplied as the condition itself, or a list object containing the conditions.
  • velocityField (underworld.mesh.MeshVariable) – Solution field for darcy flow velocity. Optional.
  • swarmVarVelocity (undeworld.swarm.SwarmVariable) – If a swarm variable is provided, the velocity calculated on the swarm will be stored. This is the most representative velocity data object, as the velocity calculation occurs on the swarm, away from mesh nodes. Optional.

Notes

Constructor must be called collectively by all processes.

fn_bodyforce

The heating function. You may change this function directly via this property.

fn_diffusivity

The diffusivity function. You may change this function directly via this property.

class underworld.systems.TimeIntegration(order, **kwargs)[source]

Bases: underworld._stgermain.StgCompoundComponent

Abstract class for integrating numerical objects (fields, swarms, etc.) in time.

The integration algorithm is a modified Runge Kutta method that only evaluates midpoint information varying in space - using only the present timestep solution. The order of the integration used can be 1,2,4

Parameters:order (int {1,2,4}) – Defines the numerical order ‘in space’ of the Runge Kutta like integration scheme.
dt

Time integrator timestep size.

time

Time integrator time value.

class underworld.systems.AdvectionDiffusion(phiField, phiDotField, velocityField, fn_diffusivity, fn_sourceTerm=None, conditions=[], _allow_non_q1=False, **kwargs)[source]

Bases: underworld._stgermain.StgCompoundComponent

This class provides functionality for a discrete representation of an advection-diffusion equation.

The class uses the Streamline Upwind Petrov Galerkin SUPG method to integrate through time.

\[\frac{\partial\phi}{\partial t} + {\bf u } \cdot \nabla \phi= \nabla { ( k \nabla \phi ) } + H\]
Parameters:
  • phiField (underworld.mesh.MeshVariable) – The concentration field, typically the temperature field
  • phiDotField (underworld.mesh.MeshVariable) – A MeshVariable that defines the initial time derivative of the phiField. Typically 0 at the beginning of a model, e.g. phiDotField.data[:]=0 When using a phiField loaded from disk one should also load the phiDotField to ensure the solving method has the time derivative information for a smooth restart. No dirichlet conditions are required for this field as the phiField degrees of freedom map exactly to this field’s dirichlet conditions, the value of which ought to be 0 for constant values of phi.
  • velocityField (underworld.mesh.MeshVariable) – The velocity field.
  • fn_diffusivity (underworld.function.Function) – A function that defines the diffusivity within the domain.
  • fn_sourceTerm (underworld.function.Function) – A function that defines the heating within the domain. Optional.
  • conditions (underworld.conditions.SystemCondition) – Numerical conditions to impose on the system. This should be supplied as the condition itself, or a list object containing the conditions.

Notes

Constructor must be called by collectively all processes.

get_max_dt()[source]

Returns a numerically stable timestep size for the current system. Note that as a default, this method returns a value one half the size of the Courant timestep.

Returns:The timestep size.
Return type:float
integrate(dt)[source]

Integrates the advection diffusion system through time, dt Must be called collectively by all processes.

Parameters:dt (float) – The timestep interval to use
class underworld.systems.SwarmAdvector(velocityField, swarm, order=2, **kwargs)[source]

Bases: underworld.systems._timeintegration.TimeIntegration

Objects of this class advect a swarm through time using the provided velocity field.

Parameters:
integrate(dt, update_owners=True)[source]

Integrate the associated swarm in time, by dt, using the velocityfield that is associated with this class

Parameters:
  • dt (double) – The timestep to use in the intergration
  • update_owners (bool) – If this is set to False, particle ownership (which element owns a particular particle) is not updated after advection. This is often necessary when both the mesh and particles are advecting simutaneously.

Example

>>> import underworld as uw
>>> import numpy as np
>>> import numpy.testing as npt
>>> from underworld import function as fn
>>> dim=2;
>>> elementMesh = uw.mesh.FeMesh_Cartesian(elementType="Q1/dQ0", elementRes=(9,9), minCoord=(-1.,-1.), maxCoord=(1.,1.))
>>> velocityField = uw.mesh.MeshVariable( mesh=elementMesh, nodeDofCount=dim )
>>> swarm = uw.swarm.Swarm(mesh=elementMesh)
>>> particle = np.zeros((1,2))
>>> particle[0] = [0.2,-0.2]
>>> swarm.add_particles_with_coordinates(particle)
array([0], dtype=int32)
>>> velocityField.data[:]=[1.0,1.0]
>>> swarmAdvector = uw.systems.SwarmAdvector(velocityField=velocityField, swarm=swarm, order=2)
>>> dt=1.0
>>> swarmAdvector.integrate(dt)
>>> npt.assert_allclose(swarm.particleCoordinates.data[0], [1.2,0.8], rtol=1e-8, verbose=True)