esys.downunder.dcresistivityforwardmodeling Package

Classes

class esys.downunder.dcresistivityforwardmodeling.DcResistivityForward

This class allows for the solution of dc resistivity forward problems via the calculation of a primary and secondary potential. Conductivity values are to be provided for the primary problem which is a homogeneous half space of a chosen conductivity and for the secondary problem which typically varies it conductivity spatially across the domain. The primary potential acts as a reference point typically based of some know reference conductivity however any value will suffice. The primary potential is implemented to avoid the use of dirac delta functions.

__init__()

This is a skeleton class for all the other forward modeling classes.

checkBounds()
getApparentResistivity()
getElectrodes()

retuns the list of electrodes with locations

getPotential()
class esys.downunder.dcresistivityforwardmodeling.DipoleDipoleSurvey(domain, primaryConductivity, secondaryConductivity, current, a, n, midPoint, directionVector, numElectrodes)

DipoleDipoleSurvey forward modeling

__init__(domain, primaryConductivity, secondaryConductivity, current, a, n, midPoint, directionVector, numElectrodes)

This is a skeleton class for all the other forward modeling classes.

getApparentResistivityPrimary()
getApparentResistivitySecondary()
getApparentResistivityTotal()
getPotential()

Returns 3 list each made up of a number of list containing primary, secondary and total potentials diferences. Each of the lists contain a list for each value of n.

class esys.downunder.dcresistivityforwardmodeling.LinearPDE(domain, numEquations=None, numSolutions=None, isComplex=False, debug=False)

This class is used to define a general linear, steady, second order PDE for an unknown function u on a given domain defined through a Domain object.

For a single PDE having a solution with a single component the linear PDE is defined in the following form:

-(grad(A[j,l]+A_reduced[j,l])*grad(u)[l]+(B[j]+B_reduced[j])u)[j]+(C[l]+C_reduced[l])*grad(u)[l]+(D+D_reduced)=-grad(X+X_reduced)[j,j]+(Y+Y_reduced)

where grad(F) denotes the spatial derivative of F. Einstein’s summation convention, ie. summation over indexes appearing twice in a term of a sum performed, is used. The coefficients A, B, C, D, X and Y have to be specified through Data objects in Function and the coefficients A_reduced, B_reduced, C_reduced, D_reduced, X_reduced and Y_reduced have to be specified through Data objects in ReducedFunction. It is also allowed to use objects that can be converted into such Data objects. A and A_reduced are rank two, B, C, X, B_reduced, C_reduced and X_reduced are rank one and D, D_reduced, Y and Y_reduced are scalar.

The following natural boundary conditions are considered:

n[j]*((A[i,j]+A_reduced[i,j])*grad(u)[l]+(B+B_reduced)[j]*u)+(d+d_reduced)*u=n[j]*(X[j]+X_reduced[j])+y

where n is the outer normal field. Notice that the coefficients A, A_reduced, B, B_reduced, X and X_reduced are defined in the PDE. The coefficients d and y are each a scalar in FunctionOnBoundary and the coefficients d_reduced and y_reduced are each a scalar in ReducedFunctionOnBoundary.

Constraints for the solution prescribe the value of the solution at certain locations in the domain. They have the form

u=r where q>0

r and q are each scalar where q is the characteristic function defining where the constraint is applied. The constraints override any other condition set by the PDE or the boundary condition.

The PDE is symmetrical if

A[i,j]=A[j,i] and B[j]=C[j] and A_reduced[i,j]=A_reduced[j,i] and B_reduced[j]=C_reduced[j]

For a system of PDEs and a solution with several components the PDE has the form

-grad((A[i,j,k,l]+A_reduced[i,j,k,l])*grad(u[k])[l]+(B[i,j,k]+B_reduced[i,j,k])*u[k])[j]+(C[i,k,l]+C_reduced[i,k,l])*grad(u[k])[l]+(D[i,k]+D_reduced[i,k]*u[k] =-grad(X[i,j]+X_reduced[i,j])[j]+Y[i]+Y_reduced[i]

A and A_reduced are of rank four, B, B_reduced, C and C_reduced are each of rank three, D, D_reduced, X_reduced and X are each of rank two and Y and Y_reduced are of rank one. The natural boundary conditions take the form:

n[j]*((A[i,j,k,l]+A_reduced[i,j,k,l])*grad(u[k])[l]+(B[i,j,k]+B_reduced[i,j,k])*u[k])+(d[i,k]+d_reduced[i,k])*u[k]=n[j]*(X[i,j]+X_reduced[i,j])+y[i]+y_reduced[i]

The coefficient d is of rank two and y is of rank one both in FunctionOnBoundary. The coefficients d_reduced is of rank two and y_reduced is of rank one both in ReducedFunctionOnBoundary.

Constraints take the form

u[i]=r[i] where q[i]>0

r and q are each rank one. Notice that at some locations not necessarily all components must have a constraint.

The system of PDEs is symmetrical if

  • A[i,j,k,l]=A[k,l,i,j]

  • A_reduced[i,j,k,l]=A_reduced[k,l,i,j]

  • B[i,j,k]=C[k,i,j]

  • B_reduced[i,j,k]=C_reduced[k,i,j]

  • D[i,k]=D[i,k]

  • D_reduced[i,k]=D_reduced[i,k]

  • d[i,k]=d[k,i]

  • d_reduced[i,k]=d_reduced[k,i]

LinearPDE also supports solution discontinuities over a contact region in the domain. To specify the conditions across the discontinuity we are using the generalised flux J which, in the case of a system of PDEs and several components of the solution, is defined as

J[i,j]=(A[i,j,k,l]+A_reduced[[i,j,k,l])*grad(u[k])[l]+(B[i,j,k]+B_reduced[i,j,k])*u[k]-X[i,j]-X_reduced[i,j]

For the case of single solution component and single PDE J is defined as

J[j]=(A[i,j]+A_reduced[i,j])*grad(u)[j]+(B[i]+B_reduced[i])*u-X[i]-X_reduced[i]

In the context of discontinuities n denotes the normal on the discontinuity pointing from side 0 towards side 1 calculated from FunctionSpace.getNormal of FunctionOnContactZero. For a system of PDEs the contact condition takes the form

n[j]*J0[i,j]=n[j]*J1[i,j]=(y_contact[i]+y_contact_reduced[i])- (d_contact[i,k]+d_contact_reduced[i,k])*jump(u)[k]

where J0 and J1 are the fluxes on side 0 and side 1 of the discontinuity, respectively. jump(u), which is the difference of the solution at side 1 and at side 0, denotes the jump of u across discontinuity along the normal calculated by jump. The coefficient d_contact is of rank two and y_contact is of rank one both in FunctionOnContactZero or FunctionOnContactOne. The coefficient d_contact_reduced is of rank two and y_contact_reduced is of rank one both in ReducedFunctionOnContactZero or ReducedFunctionOnContactOne. In case of a single PDE and a single component solution the contact condition takes the form

n[j]*J0_{j}=n[j]*J1_{j}=(y_contact+y_contact_reduced)-(d_contact+y_contact_reduced)*jump(u)

In this case the coefficient d_contact and y_contact are each scalar both in FunctionOnContactZero or FunctionOnContactOne and the coefficient d_contact_reduced and y_contact_reduced are each scalar both in ReducedFunctionOnContactZero or ReducedFunctionOnContactOne.

Typical usage:

p = LinearPDE(dom)
p.setValue(A=kronecker(dom), D=1, Y=0.5)
u = p.getSolution()
__init__(domain, numEquations=None, numSolutions=None, isComplex=False, debug=False)

Initializes a new linear PDE.

Parameters

  • domain (Domain) – domain of the PDE

  • numEquations – number of equations. If None the number of equations is extracted from the PDE coefficients.

  • numSolutions – number of solution components. If None the number of solution components is extracted from the PDE coefficients.

  • debug – if True debug information is printed

checkSymmetry(verbose=True)

Tests the PDE for symmetry.

Parameters

verbose (bool) – if set to True or not present a report on coefficients which break the symmetry is printed.

Returns

True if the PDE is symmetric

Return type

bool

Note

This is a very expensive operation. It should be used for degugging only! The symmetry flag is not altered.

createOperator()

Returns an instance of a new operator.

getFlux(u=None)

Returns the flux J for a given u.

J[i,j]=(A[i,j,k,l]+A_reduced[A[i,j,k,l]]*grad(u[k])[l]+(B[i,j,k]+B_reduced[i,j,k])u[k]-X[i,j]-X_reduced[i,j]

or

J[j]=(A[i,j]+A_reduced[i,j])*grad(u)[l]+(B[j]+B_reduced[j])u-X[j]-X_reduced[j]

Parameters

u (Data or None) – argument in the flux. If u is not present or equals None the current solution is used.

Returns

flux

Return type

Data

getRequiredOperatorType()

Returns the system type which needs to be used by the current set up.

getResidual(u=None)

Returns the residual of u or the current solution if u is not present.

Parameters

u (Data or None) – argument in the residual calculation. It must be representable in self.getFunctionSpaceForSolution(). If u is not present or equals None the current solution is used.

Returns

residual of u

Return type

Data

getSolution()

Returns the solution of the PDE.

Returns

the solution

Return type

Data

getSystem()

Returns the operator and right hand side of the PDE.

Returns

the discrete version of the PDE

Return type

tuple of Operator and Data

insertConstraint(rhs_only=False)

Applies the constraints defined by q and r to the PDE.

Parameters

rhs_only (bool) – if True only the right hand side is altered by the constraint

setValue(**coefficients)

Sets new values to coefficients.

Parameters

  • coefficients – new values assigned to coefficients

  • A (any type that can be cast to a Data object on Function) – value for coefficient A

  • A_reduced (any type that can be cast to a Data object on ReducedFunction) – value for coefficient A_reduced

  • B (any type that can be cast to a Data object on Function) – value for coefficient B

  • B_reduced (any type that can be cast to a Data object on ReducedFunction) – value for coefficient B_reduced

  • C (any type that can be cast to a Data object on Function) – value for coefficient C

  • C_reduced (any type that can be cast to a Data object on ReducedFunction) – value for coefficient C_reduced

  • D (any type that can be cast to a Data object on Function) – value for coefficient D

  • D_reduced (any type that can be cast to a Data object on ReducedFunction) – value for coefficient D_reduced

  • X (any type that can be cast to a Data object on Function) – value for coefficient X

  • X_reduced (any type that can be cast to a Data object on ReducedFunction) – value for coefficient X_reduced

  • Y (any type that can be cast to a Data object on Function) – value for coefficient Y

  • Y_reduced (any type that can be cast to a Data object on ReducedFunction) – value for coefficient Y_reduced

  • d (any type that can be cast to a Data object on FunctionOnBoundary) – value for coefficient d

  • d_reduced (any type that can be cast to a Data object on ReducedFunctionOnBoundary) – value for coefficient d_reduced

  • y (any type that can be cast to a Data object on FunctionOnBoundary) – value for coefficient y

  • d_contact (any type that can be cast to a Data object on FunctionOnContactOne or FunctionOnContactZero) – value for coefficient d_contact

  • d_contact_reduced (any type that can be cast to a Data object on ReducedFunctionOnContactOne or ReducedFunctionOnContactZero) – value for coefficient d_contact_reduced

  • y_contact (any type that can be cast to a Data object on FunctionOnContactOne or FunctionOnContactZero) – value for coefficient y_contact

  • y_contact_reduced (any type that can be cast to a Data object on ReducedFunctionOnContactOne or ReducedFunctionOnContactZero) – value for coefficient y_contact_reduced

  • d_dirac (any type that can be cast to a Data object on DiracDeltaFunctions) – value for coefficient d_dirac

  • y_dirac (any type that can be cast to a Data object on DiracDeltaFunctions) – value for coefficient y_dirac

  • r (any type that can be cast to a Data object on Solution or ReducedSolution depending on whether reduced order is used for the solution) – values prescribed to the solution at the locations of constraints

  • q (any type that can be cast to a Data object on Solution or ReducedSolution depending on whether reduced order is used for the representation of the equation) – mask for location of constraints

Raises

IllegalCoefficient – if an unknown coefficient keyword is used

class esys.downunder.dcresistivityforwardmodeling.Locator(where, x=array([0., 0., 0.]))

Locator provides access to the values of data objects at a given spatial coordinate x.

In fact, a Locator object finds the sample in the set of samples of a given function space or domain which is closest to the given point x.

__init__(where, x=array([0., 0., 0.]))

Initializes a Locator to access values in Data objects on the Doamin or FunctionSpace for the sample point which is closest to the given point x.

Parameters

  • where (escript.FunctionSpace) – function space

  • x (numpy.ndarray or list of numpy.ndarray) – location(s) of the Locator

getFunctionSpace()

Returns the function space of the Locator.

getId(item=None)

Returns the identifier of the location.

getValue(data)

Returns the value of data at the Locator if data is a Data object otherwise the object is returned.

getX()

Returns the exact coordinates of the Locator.

setValue(data, v)

Sets the value of the data at the Locator.

class esys.downunder.dcresistivityforwardmodeling.PoleDipoleSurvey(domain, primaryConductivity, secondaryConductivity, current, a, n, midPoint, directionVector, numElectrodes)

Forward model class for a poledipole setup

__init__(domain, primaryConductivity, secondaryConductivity, current, a, n, midPoint, directionVector, numElectrodes)
Parameters

  • domain (Domain) – domain of the model

  • primaryConductivity (data) – preset primary conductivity data object

  • secondaryConductivity (data) – preset secondary conductivity data object

  • current (float or int) – amount of current to be injected at the current electrode

  • a (list) – the spacing between current and potential electrodes

  • n (float or int) – multiple of spacing between electrodes. typicaly interger

  • midPoint – midPoint of the survey, as a list containing x,y coords

  • directionVector – two element list specifying the direction the survey should extend from the midpoint

  • numElectrodes (int) – the number of electrodes to be used in the survey must be a multiple of 2 for polepole survey:

getApparentResistivityPrimary()
getApparentResistivitySecondary()
getApparentResistivityTotal()
getPotential()

Returns 3 list each made up of a number of list containing primary, secondary and total potentials diferences. Each of the lists contain a list for each value of n.

class esys.downunder.dcresistivityforwardmodeling.PolePoleSurvey(domain, primaryConductivity, secondaryConductivity, current, a, midPoint, directionVector, numElectrodes)

Forward model class for a polepole setup

__init__(domain, primaryConductivity, secondaryConductivity, current, a, midPoint, directionVector, numElectrodes)
Parameters

  • domain (Domain) – domain of the model

  • primaryConductivity (data) – preset primary conductivity data object

  • secondaryConductivity (data) – preset secondary conductivity data object

  • current (float or int) – amount of current to be injected at the current electrode

  • a (list) – the spacing between current and potential electrodes

  • midPoint – midPoint of the survey, as a list containing x,y coords

  • directionVector – two element list specifying the direction the survey should extend from the midpoint

  • numElectrodes (int) – the number of electrodes to be used in the survey must be a multiple of 2 for polepole survey:

getApparentResistivityPrimary()
getApparentResistivitySecondary()
getApparentResistivityTotal()
getPotential()

returns a list containing 3 lists one for each the primary, secondary and total potential.

class esys.downunder.dcresistivityforwardmodeling.SchlumbergerSurvey(domain, primaryConductivity, secondaryConductivity, current, a, n, midPoint, directionVector, numElectrodes)

Schlumberger survey forward calculation

__init__(domain, primaryConductivity, secondaryConductivity, current, a, n, midPoint, directionVector, numElectrodes)

This is a skeleton class for all the other forward modeling classes.

getApparentResistivity(delPhiList)
getElectrodeDict()

retuns the electrode dictionary

getPotentialAnalytic()

Returns 3 list each made up of a number of list containing primary, secondary and total potentials diferences. Each of the lists contain a list for each value of n.

getPotentialNumeric()

Returns 3 list each made up of a number of list containing primary, secondary and total potentials diferences. Each of the lists contain a list for each value of n.

getSourcesSamples()

return a list of tuples of sample locations followed by a list of tuples of source locations.

class esys.downunder.dcresistivityforwardmodeling.WennerSurvey(domain, primaryConductivity, secondaryConductivity, current, a, midPoint, directionVector, numElectrodes)

WennerSurvey forward calculation

__init__(domain, primaryConductivity, secondaryConductivity, current, a, midPoint, directionVector, numElectrodes)
Parameters

  • domain (Domain) – domain of the model

  • primaryConductivity (data) – preset primary conductivity data object

  • secondaryConductivity (data) – preset secondary conductivity data object

  • current (float or int) – amount of current to be injected at the current electrode

  • a (list) – the spacing between current and potential electrodes

  • midPoint – midPoint of the survey, as a list containing x,y coords

  • directionVector – two element list specifying the direction the survey should extend from the midpoint

  • numElectrodes (int) – the number of electrodes to be used in the survey must be a multiple of 2 for polepole survey

getApparentResistivityPrimary()
getApparentResistivitySecondary()
getApparentResistivityTotal()
getPotential()

returns a list containing 3 lists one for each the primary, secondary and total potential.

esys.downunder.dcresistivityforwardmodeling.xrange

alias of range

Functions

esys.downunder.dcresistivityforwardmodeling.saveSilo(filename, domain=None, write_meshdata=False, time=0.0, cycle=0, **data)

Writes Data objects and their mesh to a file using the SILO file format.

Example:

temp=Scalar(..)
v=Vector(..)
saveSilo("solution.silo", temperature=temp, velocity=v)

temp and v are written to “solution.silo” where temp is named “temperature” and v is named “velocity”.

Parameters

  • filename (str) – name of the output file (‘.silo’ is added if required)

  • domain (escript.Domain) – domain of the Data objects. If not specified, the domain of the given Data objects is used.

  • write_meshdata (bool) – whether to save mesh-related data such as element identifiers, ownership etc. This is mainly useful for debugging.

  • time (float) – the timestamp to save within the file

  • cycle (int) – the cycle (or timestep) of the data

  • <name> – writes the assigned value to the Silo file using <name> as identifier

Note

All data objects have to be defined on the same domain but they may be defined on separate FunctionSpace s.

Others

  • pi

Packages