esys.modellib.mechanics Package

Classes

class esys.modellib.mechanics.DruckerPrager(**kwargs)
__init__(**kwargs)

set up model

doInitialization()
doStep(dt)
doStepPostprocessing(dt)

accept all the values:

doStepPreprocessing(dt)
setStress()
setTangentialTensor()
class esys.modellib.mechanics.IterationDivergenceError

Exception which is thrown if there is no convergence of the iteration process at a time step.

But there is a chance that a smaller step could help to reach convergence.

__init__(*args, **kwargs)

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

class esys.modellib.mechanics.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.modellib.mechanics.Mechanics(**kwargs)

base class for mechanics models in updated lagrangean framework

Note

Instance variable domain - domain (in)

Note

Instance variable internal_force - =Data()

Note

Instance variable external_force - =Data()

Note

Instance variable prescribed_velocity - =Data()

Note

Instance variable location_prescribed_velocity - =Data()

Note

Instance variable temperature - = None

Note

Instance variable expansion_coefficient - = 0.

Note

Instance variable bulk_modulus - =1.

Note

Instance variable shear_modulus - =1.

Note

Instance variable rel_tol - =1.e-3

Note

Instance variable abs_tol - =1.e-15

Note

Instance variable max_iter - =10

Note

Instance variable displacement - =None

Note

Instance variable stress - =None

__init__(**kwargs)

set up the model

Parameters

debug (bool) – debug flag

SAFTY_FACTOR_ITERATION = 0.01
doInitialization()

initialize model

doStep(dt)
doStepPostprocessing(dt)

accept all the values:

doStepPreprocessing(dt)

step up pressure iteration

if run within a time dependend problem extrapolation of pressure from previous time steps is used to get an initial guess (that needs some work!!!!!!!)

getSafeTimeStepSize(dt)

returns new step size

terminateIteration()

iteration is terminateIterationd if relative pressure change is less than rel_tol

class esys.modellib.mechanics.Model(parameters=[], **kwargs)

A Model object represents a process marching over time until a finalizing condition is fulfilled. At each time step an iterative process can be performed and the time step size can be controlled. A Model has the following work flow:

doInitialization()
while not terminateInitialIteration(): doInitialStep()
doInitialPostprocessing()
while not finalize():
    dt=getSafeTimeStepSize(dt)
    doStepPreprocessing(dt)
    while not terminateIteration(): doStep(dt)
    doStepPostprocessing(dt)
doFinalization()

where doInitialization, finalize, getSafeTimeStepSize, doStepPreprocessing, terminateIteration, doStepPostprocessing, doFinalization are methods of the particular instance of a Model. The default implementations of these methods have to be overwritten by the subclass implementing a Model.

__init__(parameters=[], **kwargs)

Creates a model.

Just calls the parent constructor.

UNDEF_DT = 1e+300
doFinalization()

Finalizes the time stepping.

This function may be overwritten.

doInitialPostprocessing()

Finalises the initialization iteration process. This method is not called in case of a restart.

This function may be overwritten.

doInitialStep()

Performs an iteration step in the initialization phase. This method is not called in case of a restart.

This function may be overwritten.

doInitialization()

Initializes the time stepping scheme. This method is not called in case of a restart.

This function may be overwritten.

doStep(dt)

Executes an iteration step at a time step.

dt is the currently used time step size.

This function may be overwritten.

doStepPostprocessing(dt)

Finalises the time step.

dt is the currently used time step size.

This function may be overwritten.

doStepPreprocessing(dt)

Sets up a time step of step size dt.

This function may be overwritten.

finalize()

Returns False if the time stepping is finalized.

This function may be overwritten.

getSafeTimeStepSize(dt)

Returns a time step size which can be safely used.

dt gives the previously used step size.

This function may be overwritten.

setUp()

Sets up the model.

This function may be overwritten.

terminateInitialIteration()

Returns True if iteration at the inital phase is terminated.

terminateIteration()

Returns True if iteration on a time step is terminated.

toDom(esysxml, node)

toDom method of Model class.

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