esys.modellib.mechanics Package¶
Classes¶
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class
esys.modellib.mechanics.
DruckerPrager
(**kwargs)¶ -
__init__
(**kwargs)¶ set up model
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doInitialization
()¶
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doStep
(dt)¶
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doStepPostprocessing
(dt)¶ accept all the values:
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doStepPreprocessing
(dt)¶
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setStress
()¶
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setTangentialTensor
()¶
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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.
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__init__
(*args, **kwargs)¶ Initialize self. See help(type(self)) for accurate signature.
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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 inFunction
and the coefficients A_reduced, B_reduced, C_reduced, D_reduced, X_reduced and Y_reduced have to be specified throughData
objects inReducedFunction
. It is also allowed to use objects that can be converted into suchData
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 inReducedFunctionOnBoundary
.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 inReducedFunctionOnBoundary
.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 asJ[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
ofFunctionOnContactZero
. For a system of PDEs the contact condition takes the formn[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 inFunctionOnContactZero
orFunctionOnContactOne
. The coefficient d_contact_reduced is of rank two and y_contact_reduced is of rank one both inReducedFunctionOnContactZero
orReducedFunctionOnContactOne
. In case of a single PDE and a single component solution the contact condition takes the formn[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
orFunctionOnContactOne
and the coefficient d_contact_reduced and y_contact_reduced are each scalar both inReducedFunctionOnContactZero
orReducedFunctionOnContactOne
.Typical usage:
p = LinearPDE(dom) p.setValue(A=kronecker(dom), D=1, Y=0.5) u = p.getSolution()
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__init__
(domain, numEquations=None, numSolutions=None, isComplex=False, debug=False)¶ Initializes a new linear PDE.
- Parameters
domain (
Domain
) – domain of the PDEnumEquations – 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
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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.
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createOperator
()¶ Returns an instance of a new operator.
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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 equalsNone
the current solution is used.- Returns
flux
- Return type
Data
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getRequiredOperatorType
()¶ Returns the system type which needs to be used by the current set up.
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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 inself.getFunctionSpaceForSolution()
. If u is not present or equalsNone
the current solution is used.- Returns
residual of u
- Return type
Data
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getSolution
()¶ Returns the solution of the PDE.
- Returns
the solution
- Return type
Data
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getSystem
()¶ Returns the operator and right hand side of the PDE.
- Returns
the discrete version of the PDE
- Return type
tuple
ofOperator
andData
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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
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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 onFunction
) – value for coefficientA
A_reduced (any type that can be cast to a
Data
object onReducedFunction
) – value for coefficientA_reduced
B (any type that can be cast to a
Data
object onFunction
) – value for coefficientB
B_reduced (any type that can be cast to a
Data
object onReducedFunction
) – value for coefficientB_reduced
C (any type that can be cast to a
Data
object onFunction
) – value for coefficientC
C_reduced (any type that can be cast to a
Data
object onReducedFunction
) – value for coefficientC_reduced
D (any type that can be cast to a
Data
object onFunction
) – value for coefficientD
D_reduced (any type that can be cast to a
Data
object onReducedFunction
) – value for coefficientD_reduced
X (any type that can be cast to a
Data
object onFunction
) – value for coefficientX
X_reduced (any type that can be cast to a
Data
object onReducedFunction
) – value for coefficientX_reduced
Y (any type that can be cast to a
Data
object onFunction
) – value for coefficientY
Y_reduced (any type that can be cast to a
Data
object onReducedFunction
) – value for coefficientY_reduced
d (any type that can be cast to a
Data
object onFunctionOnBoundary
) – value for coefficientd
d_reduced (any type that can be cast to a
Data
object onReducedFunctionOnBoundary
) – value for coefficientd_reduced
y (any type that can be cast to a
Data
object onFunctionOnBoundary
) – value for coefficienty
d_contact (any type that can be cast to a
Data
object onFunctionOnContactOne
orFunctionOnContactZero
) – value for coefficientd_contact
d_contact_reduced (any type that can be cast to a
Data
object onReducedFunctionOnContactOne
orReducedFunctionOnContactZero
) – value for coefficientd_contact_reduced
y_contact (any type that can be cast to a
Data
object onFunctionOnContactOne
orFunctionOnContactZero
) – value for coefficienty_contact
y_contact_reduced (any type that can be cast to a
Data
object onReducedFunctionOnContactOne
orReducedFunctionOnContactZero
) – value for coefficienty_contact_reduced
d_dirac (any type that can be cast to a
Data
object onDiracDeltaFunctions
) – value for coefficientd_dirac
y_dirac (any type that can be cast to a
Data
object onDiracDeltaFunctions
) – value for coefficienty_dirac
r (any type that can be cast to a
Data
object onSolution
orReducedSolution
depending on whether reduced order is used for the solution) – values prescribed to the solution at the locations of constraintsq (any type that can be cast to a
Data
object onSolution
orReducedSolution
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
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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
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__init__
(**kwargs)¶ set up the model
- Parameters
debug (
bool
) – debug flag
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SAFTY_FACTOR_ITERATION
= 0.01¶
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doInitialization
()¶ initialize model
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doStep
(dt)¶
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doStepPostprocessing
(dt)¶ accept all the values:
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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!!!!!!!)
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getSafeTimeStepSize
(dt)¶ returns new step size
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terminateIteration
()¶ iteration is terminateIterationd if relative pressure change is less than rel_tol
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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.
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UNDEF_DT
= 1e+300¶
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doFinalization
()¶ Finalizes the time stepping.
This function may be overwritten.
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doInitialPostprocessing
()¶ Finalises the initialization iteration process. This method is not called in case of a restart.
This function may be overwritten.
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doInitialStep
()¶ Performs an iteration step in the initialization phase. This method is not called in case of a restart.
This function may be overwritten.
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doInitialization
()¶ Initializes the time stepping scheme. This method is not called in case of a restart.
This function may be overwritten.
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doStep
(dt)¶ Executes an iteration step at a time step.
dt
is the currently used time step size.This function may be overwritten.
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doStepPostprocessing
(dt)¶ Finalises the time step.
dt is the currently used time step size.
This function may be overwritten.
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doStepPreprocessing
(dt)¶ Sets up a time step of step size dt.
This function may be overwritten.
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finalize
()¶ Returns False if the time stepping is finalized.
This function may be overwritten.
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getSafeTimeStepSize
(dt)¶ Returns a time step size which can be safely used.
dt
gives the previously used step size.This function may be overwritten.
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setUp
()¶ Sets up the model.
This function may be overwritten.
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terminateInitialIteration
()¶ Returns True if iteration at the inital phase is terminated.
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terminateIteration
()¶ Returns True if iteration on a time step is terminated.
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toDom
(esysxml, node)¶ toDom
method of Model class.
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