esys.escript.pdetools Package

Classes

class esys.escript.pdetools.ArithmeticTuple(*args)

Bases: object

Tuple supporting inplace update x+=y and scaling x=a*y where x,y is an ArithmeticTuple and a is a float.

Example of usage:

from esys.escript import Data
from numpy import array
a=eData(...)
b=array([1.,4.])
x=ArithmeticTuple(a,b)
y=5.*x
__init__(*args)

Initializes object with elements args.

Parameters:args – tuple of objects that support inplace add (x+=y) and scaling (x=a*y)
class esys.escript.pdetools.CorrectionFailed

Bases: esys.escriptcore.pdetools.SolverSchemeException

Exception thrown if no convergence has been achieved in the solution correction scheme.

__init__()

Initialize self. See help(type(self)) for accurate signature.

args
with_traceback()

Exception.with_traceback(tb) – set self.__traceback__ to tb and return self.

class esys.escript.pdetools.Defect

Bases: object

Defines a non-linear defect F(x) of a variable x. This class includes two functions (bilinearform and eval) that must be overridden by subclassing before use.

__init__()

Initializes defect.

bilinearform(x0, x1)

Returns the inner product of x0 and x1

NOTE: MUST BE OVERRIDDEN BY A SUBCLASS

Parameters:
  • x0 – value for x0
  • x1 – value for x1
Returns:

the inner product of x0 and x1
Return type:

float
derivative(F0, x0, v, v_is_normalised=True)

Returns the directional derivative at x0 in the direction of v.

Parameters:
  • F0 – value of this defect at x0
  • x0 – value at which derivative is calculated
  • v – direction
  • v_is_normalised – True to indicate that v is nomalized (self.norm(v)=0)
Returns:

derivative of this defect at x0 in the direction of v
Note:

by default numerical evaluation (self.eval(x0+eps*v)-F0)/eps is used but this method maybe overwritten to use exact evaluation.
eval(x)

Returns the value F of a given x.

NOTE: MUST BE OVERRIDDEN BY A SUBCLASS

Parameters:x – value for which the defect F is evaluated
Returns:value of the defect at x
getDerivativeIncrementLength()

Returns the relative increment length used to approximate the derivative of the defect. :return: value of the defect at x :rtype: positive float

norm(x)

Returns the norm of argument x.

Parameters:x – a value
Returns:norm of argument x
Return type:float
Note:by default sqrt(self.bilinearform(x,x) is returned.
setDerivativeIncrementLength(inc=1.4901161193847656e-05)

Sets the relative length of the increment used to approximate the derivative of the defect. The increment is inc*norm(x)/norm(v)*v in the direction of v with x as a starting point.

Parameters:inc (positive float) – relative increment length
class esys.escript.pdetools.HomogeneousSaddlePointProblem(**kwargs)

Bases: object

This class provides a framework for solving linear homogeneous saddle point problems of the form:

*Av+B^*p=f*
*Bv     =0*

for the unknowns v and p and given operators A and B and given right hand side f. B^* is the adjoint operator of B. A may depend weakly on v and p.

__init__(**kwargs)

initializes the saddle point problem

Bv(v, tol)

Returns Bv with accuracy tol (overwrite)

Return type:equal to the type of p
Note:boundary conditions on p should be zero!
getAbsoluteTolerance()

Returns the absolute tolerance.

Returns:absolute tolerance
Return type:float
getDV(p, v, tol)

return a correction to the value for a given v and a given p with accuracy tol (overwrite)

Parameters:
  • p – pressure
  • v – pressure
Returns:

dv given as dv= A^{-1} (f-A v-B^*p)
Note:

Only A may depend on v and p
getTolerance()

Returns the relative tolerance.

Returns:relative tolerance
Return type:float
inner_p(p0, p1)

Returns inner product of p0 and p1 (overwrite).

Parameters:
  • p0 – a pressure
  • p1 – a pressure
Returns:

inner product of p0 and p1
Return type:

float
inner_pBv(p, Bv)

Returns inner product of element p and Bv (overwrite).

Parameters:
  • p – a pressure increment
  • Bv – a residual
Returns:

inner product of element p and Bv
Return type:

float
Note:

used if PCG is applied.
norm_Bv(Bv)

Returns the norm of Bv (overwrite).

Return type:equal to the type of p
Note:boundary conditions on p should be zero!
norm_p(p)

calculates the norm of p

Parameters:p – a pressure
Returns:the norm of p using the inner product for pressure
Return type:float
norm_v(v)

Returns the norm of v (overwrite).

Parameters:v – a velovity
Returns:norm of v
Return type:non-negative float
resetControlParameters(K_p=1.0, K_v=1.0, rtol_max=0.01, rtol_min=1e-07, chi_max=0.5, reduction_factor=0.3, theta=0.1)

sets a control parameter

Parameters:
  • K_p (float) – initial value for constant to adjust pressure tolerance
  • K_v (float) – initial value for constant to adjust velocity tolerance
  • rtol_max (float) – maximuim relative tolerance used to calculate presssure and velocity increment.
  • chi_max (float) – maximum tolerable converegence rate.
  • reduction_factor (float) – reduction factor for adjustment factors.
setAbsoluteTolerance(tolerance=0.0)

Sets the absolute tolerance.

Parameters:tolerance (non-negative float) – tolerance to be used
setControlParameter(K_p=None, K_v=None, rtol_max=None, rtol_min=None, chi_max=None, reduction_factor=None, theta=None)

sets a control parameter

Parameters:
  • K_p (float) – initial value for constant to adjust pressure tolerance
  • K_v (float) – initial value for constant to adjust velocity tolerance
  • rtol_max (float) – maximuim relative tolerance used to calculate presssure and velocity increment.
  • chi_max (float) – maximum tolerable converegence rate.
setTolerance(tolerance=0.0001)

Sets the relative tolerance for (v,p).

Parameters:tolerance (non-negative float) – tolerance to be used
solve(v, p, max_iter=20, verbose=False, usePCG=True, iter_restart=20, max_correction_steps=10)

Solves the saddle point problem using initial guesses v and p.

Parameters:
  • v (Data) – initial guess for velocity
  • p (Data) – initial guess for pressure
  • usePCG (bool) – indicates the usage of the PCG rather than GMRES scheme.
  • max_iter (int) – maximum number of iteration steps per correction attempt
  • verbose (bool) – if True, shows information on the progress of the saddlepoint problem solver.
  • iter_restart (int) – restart the iteration after iter_restart steps (only used if useUzaw=False)
Return type:

tuple of Data objects
Note:

typically this method is overwritten by a subclass. It provides a wrapper for the _solve method.
solve_AinvBt(dp, tol)

Solves A dv=B^*dp with accuracy tol

Parameters:dp – a pressure increment
Returns:the solution of A dv=B^*dp
Note:boundary conditions on dv should be zero! A is the operator used in getDV and must not be altered.
solve_prec(Bv, tol)

Provides a preconditioner for (BA^{-1}B^ * ) applied to Bv with accuracy tol

Return type:equal to the type of p
Note:boundary conditions on p should be zero!
class esys.escript.pdetools.IndefinitePreconditioner

Bases: esys.escriptcore.pdetools.SolverSchemeException

Exception thrown if the preconditioner is not positive definite.

__init__()

Initialize self. See help(type(self)) for accurate signature.

args
with_traceback()

Exception.with_traceback(tb) – set self.__traceback__ to tb and return self.

class esys.escript.pdetools.IterationBreakDown

Bases: esys.escriptcore.pdetools.SolverSchemeException

Exception thrown if the iteration scheme encountered an incurable breakdown.

__init__()

Initialize self. See help(type(self)) for accurate signature.

args
with_traceback()

Exception.with_traceback(tb) – set self.__traceback__ to tb and return self.

class esys.escript.pdetools.Locator(where, x=array([0., 0., 0.]))

Bases: object

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.escript.pdetools.MaxIterReached

Bases: esys.escriptcore.pdetools.SolverSchemeException

Exception thrown if the maximum number of iteration steps is reached.

__init__()

Initialize self. See help(type(self)) for accurate signature.

args
with_traceback()

Exception.with_traceback(tb) – set self.__traceback__ to tb and return self.

class esys.escript.pdetools.NegativeNorm

Bases: esys.escriptcore.pdetools.SolverSchemeException

Exception thrown if a norm calculation returns a negative norm.

__init__()

Initialize self. See help(type(self)) for accurate signature.

args
with_traceback()

Exception.with_traceback(tb) – set self.__traceback__ to tb and return self.

class esys.escript.pdetools.NoPDE(domain, D=None, Y=None, q=None, r=None)

Bases: object

Solves the following problem for u:

kronecker[i,j]*D[j]*u[j]=Y[i]

with constraint

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

where D, Y, r and q are given functions of rank 1.

In the case of scalars this takes the form

D*u=Y

with constraint

u=r where q>0

where D, Y, r and q are given scalar functions.

The constraint overwrites any other condition.

Note:This class is similar to the linearPDEs.LinearPDE class with A=B=C=X=0 but has the intention that all input parameters are given in Solution or ReducedSolution.
__init__(domain, D=None, Y=None, q=None, r=None)

Initializes the problem.

Parameters:
  • domain (Domain) – domain of the PDE
  • D (float, int, numpy.ndarray, Data) – coefficient of the solution
  • Y (float, int, numpy.ndarray, Data) – right hand side
  • q (float, int, numpy.ndarray, Data) – location of constraints
  • r (float, int, numpy.ndarray, Data) – value of solution at locations of constraints
getSolution()

Returns the solution.

Returns:the solution of the problem
Return type:Data object in the FunctionSpace Solution or ReducedSolution
setReducedOff()

Sets the FunctionSpace of the solution to Solution.

setReducedOn()

Sets the FunctionSpace of the solution to ReducedSolution.

setValue(D=None, Y=None, q=None, r=None)

Assigns values to the parameters.

Parameters:
  • D (float, int, numpy.ndarray, Data) – coefficient of the solution
  • Y (float, int, numpy.ndarray, Data) – right hand side
  • q (float, int, numpy.ndarray, Data) – location of constraints
  • r (float, int, numpy.ndarray, Data) – value of solution at locations of constraints
class esys.escript.pdetools.Projector(domain, reduce=True, fast=True)

Bases: object

The Projector is a factory which projects a discontinuous function onto a continuous function on a given domain.

__init__(domain, reduce=True, fast=True)

Creates a continuous function space projector for a domain.

Parameters:
  • domain – Domain of the projection.
  • reduce – Flag to reduce projection order
  • fast – Flag to use a fast method based on matrix lumping
getSolverOptions()

Returns the solver options of the PDE solver.

Return type:linearPDEs.SolverOptions
getValue(input_data)

Projects input_data onto a continuous function.

Parameters:input_data – the data to be projected
class esys.escript.pdetools.SolverSchemeException

Bases: Exception

This is a generic exception thrown by solvers.

__init__()

Initialize self. See help(type(self)) for accurate signature.

args
with_traceback()

Exception.with_traceback(tb) – set self.__traceback__ to tb and return self.

class esys.escript.pdetools.TimeIntegrationManager(*inital_values, **kwargs)

Bases: object

A simple mechanism to manage time dependend values.

Typical usage is:

dt=0.1 # time increment
tm=TimeIntegrationManager(inital_value,p=1)
while t<1.
    v_guess=tm.extrapolate(dt) # extrapolate to t+dt
    v=...
    tm.checkin(dt,v)
    t+=dt
Note:currently only p=1 is supported.
__init__(*inital_values, **kwargs)

Sets up the value manager where inital_values are the initial values and p is the order used for extrapolation.

checkin(dt, *values)

Adds new values to the manager. The p+1 last values are lost.

extrapolate(dt)

Extrapolates to dt forward in time.

getTime()
getValue()

Functions

esys.escript.pdetools.BoundaryValuesFromVolumeTag(domain, **values)

Creates a mask on the Solution(domain) function space where the value is one for samples that touch regions tagged by tags.

Usage: m=BoundaryValuesFromVolumeTag(domain, ham=1, f=6)

Parameters:domain (escript.Domain) – domain to be used
Returns:a mask which marks samples that are touching the boundary tagged by any of the given tags
Return type:escript.Data of rank 0
esys.escript.pdetools.GMRES(r, Aprod, x, bilinearform, atol=0, rtol=1e-08, iter_max=100, iter_restart=20, verbose=False, P_R=None)

Solver for

Ax=b

with a general operator A (more details required!). It uses the generalized minimum residual method (GMRES).

The iteration is terminated if

|r| <= atol+rtol*|r0|

where r0 is the initial residual and |.| is the energy norm. In fact

|r| = sqrt( bilinearform(r,r))

Parameters:
  • r (any object supporting inplace add (x+=y) and scaling (x=scalar*y)) – initial residual r=b-Ax. r is altered.
  • x (same like r) – an initial guess for the solution
  • Aprod (function Aprod(x) where x is of the same object like argument x. The returned object needs to be of the same type like argument r.) – returns the value Ax
  • bilinearform (function bilinearform(x,r) where x is of the same type like argument x and r. The returned value is a float.) – inner product <x,r>
  • atol (non-negative float) – absolute tolerance
  • rtol (non-negative float) – relative tolerance
  • iter_max (int) – maximum number of iteration steps
  • iter_restart (int) – in order to save memory the orthogonalization process is terminated after iter_restart steps and the iteration is restarted.
Returns:

the solution approximation and the corresponding residual
Return type:

tuple
Warning:

r and x are altered.
esys.escript.pdetools.MINRES(r, Aprod, x, Msolve, bilinearform, atol=0, rtol=1e-08, iter_max=100)

Solver for

Ax=b

with a symmetric and positive definite operator A (more details required!). It uses the minimum residual method (MINRES) with preconditioner M providing an approximation of A.

The iteration is terminated if

|r| <= atol+rtol*|r0|

where r0 is the initial residual and |.| is the energy norm. In fact

|r| = sqrt( bilinearform(Msolve(r),r))

For details on the preconditioned conjugate gradient method see the book:

“Templates for the Solution of Linear Systems by R. Barrett, M. Berry, T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst”.

Parameters:
  • r (any object supporting inplace add (x+=y) and scaling (x=scalar*y)) – initial residual r=b-Ax. r is altered.
  • x (any object supporting inplace add (x+=y) and scaling (x=scalar*y)) – an initial guess for the solution
  • Aprod (function Aprod(x) where x is of the same object like argument x. The returned object needs to be of the same type like argument r.) – returns the value Ax
  • Msolve (function Msolve(r) where r is of the same type like argument r. The returned object needs to be of the same type like argument x.) – solves Mx=r
  • bilinearform (function bilinearform(x,r) where x is of the same type like argument x and r is. The returned value is a float.) – inner product <x,r>
  • atol (non-negative float) – absolute tolerance
  • rtol (non-negative float) – relative tolerance
  • iter_max (int) – maximum number of iteration steps
Returns:

the solution approximation and the corresponding residual
Return type:

tuple
Warning:

r and x are altered.
esys.escript.pdetools.MaskFromBoundaryTag(domain, *tags)

Creates a mask on the Solution(domain) function space where the value is one for samples that touch the boundary tagged by tags.

Usage: m=MaskFromBoundaryTag(domain, “left”, “right”)

Parameters:
  • domain (escript.Domain) – domain to be used
  • tags (str) – boundary tags
Returns:

a mask which marks samples that are touching the boundary tagged by any of the given tags
Return type:

escript.Data of rank 0
esys.escript.pdetools.MaskFromTag(domain, *tags)

Creates a mask on the Solution(domain) function space where the value is one for samples that touch regions tagged by tags.

Usage: m=MaskFromTag(domain, “ham”)

Parameters:
  • domain (escript.Domain) – domain to be used
  • tags (str) – boundary tags
Returns:

a mask which marks samples that are touching the boundary tagged by any of the given tags
Return type:

escript.Data of rank 0
esys.escript.pdetools.NewtonGMRES(defect, x, iter_max=100, sub_iter_max=20, atol=0, rtol=0.0001, subtol_max=0.5, gamma=0.9, verbose=False)

Solves a non-linear problem F(x)=0 for unknown x using the stopping criterion:

norm(F(x) <= atol + rtol * norm(F(x0)

where x0 is the initial guess.

Parameters:
  • defect (Defect) – object defining the function F. defect.norm defines the norm used in the stopping criterion.
  • x (any object type allowing basic operations such as numpy.ndarray, Data) – initial guess for the solution, x is altered.
  • iter_max (positive int) – maximum number of iteration steps
  • sub_iter_max (positive int) – maximum number of inner iteration steps
  • atol (positive float) – absolute tolerance for the solution
  • rtol (positive float) – relative tolerance for the solution
  • gamma (positive float, less than 1) – tolerance safety factor for inner iteration
  • subtol_max (positive float, less than 1) – upper bound for inner tolerance
Returns:

an approximation of the solution with the desired accuracy
Return type:

same type as the initial guess
esys.escript.pdetools.PCG(r, Aprod, x, Msolve, bilinearform, atol=0, rtol=1e-08, iter_max=100, initial_guess=True, verbose=False)

Solver for

Ax=b

with a symmetric and positive definite operator A (more details required!). It uses the conjugate gradient method with preconditioner M providing an approximation of A.

The iteration is terminated if

|r| <= atol+rtol*|r0|

where r0 is the initial residual and |.| is the energy norm. In fact

|r| = sqrt( bilinearform(Msolve(r),r))

For details on the preconditioned conjugate gradient method see the book:

“Templates for the Solution of Linear Systems by R. Barrett, M. Berry, T.F. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst”.

Parameters:
  • r (any object supporting inplace add (x+=y) and scaling (x=scalar*y)) – initial residual r=b-Ax. r is altered.
  • x (any object supporting inplace add (x+=y) and scaling (x=scalar*y)) – an initial guess for the solution
  • Aprod (function Aprod(x) where x is of the same object like argument x. The returned object needs to be of the same type like argument r.) – returns the value Ax
  • Msolve (function Msolve(r) where r is of the same type like argument r. The returned object needs to be of the same type like argument x.) – solves Mx=r
  • bilinearform (function bilinearform(x,r) where x is of the same type like argument x and r is. The returned value is a float.) – inner product <x,r>
  • atol (non-negative float) – absolute tolerance
  • rtol (non-negative float) – relative tolerance
  • iter_max (int) – maximum number of iteration steps
Returns:

the solution approximation and the corresponding residual
Return type:

tuple
Warning:

r and x are altered.
esys.escript.pdetools.TFQMR(r, Aprod, x, bilinearform, atol=0, rtol=1e-08, iter_max=100)

Solver for

Ax=b

with a general operator A (more details required!). It uses the Transpose-Free Quasi-Minimal Residual method (TFQMR).

The iteration is terminated if

|r| <= atol+rtol*|r0|

where r0 is the initial residual and |.| is the energy norm. In fact

|r| = sqrt( bilinearform(r,r))

Parameters:
  • r (any object supporting inplace add (x+=y) and scaling (x=scalar*y)) – initial residual r=b-Ax. r is altered.
  • x (same like r) – an initial guess for the solution
  • Aprod (function Aprod(x) where x is of the same object like argument x. The returned object needs to be of the same type like argument r.) – returns the value Ax
  • bilinearform (function bilinearform(x,r) where x is of the same type like argument x and r. The returned value is a float.) – inner product <x,r>
  • atol (non-negative float) – absolute tolerance
  • rtol (non-negative float) – relative tolerance
  • iter_max (int) – maximum number of iteration steps
Return type:

tuple
Warning:

r and x are altered.
esys.escript.pdetools.getInfLocator(arg)

Return a Locator for a point with the inf value over all arg.

esys.escript.pdetools.getSupLocator(arg)

Return a Locator for a point with the sup value over all arg.

Others

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