7.13 Package FL.PROBLEM

The FL.PROBLEM package introduces the general class <problem> and some subclasses. The most interesting subclass is <pde-problem>. A <pde-problem> is defined on a domain and provides a table mapping the domain patches to property lists containing the coefficient functions.

Several subclasses of <pde-problem> are defined in own packages, e.g. <cdr-problem> in FL.CDR, <elasticity-problem> in FL.ELASTICITY and <navier-stokes-problem in FL.NAVIER-STOKES.

Class: <COEFFICIENT>

The coefficient class.

Superclasses: PROPERTY-MIXIN

Direct slots:

• NAME: Self-explanatory.
• DIMENSION: The dimension of the cell on which this coefficient is active. The value T means that it is active on all cells lying on the patch. The default value NIL means that it is active on cells with the same dimension as the patch.
• DEMANDS: A list indicating which information the evaluation function needs. Possible choices depend on problem and discretization, e.g. :local, :global, :fe, :cell are possible choices. One element can also be a list starting with the keyword :fe-parameters and followed by symbols indicating names of finite element functions on the discretization blackboard.
• EVAL: The evaluation funtion. It accepts a list of keyword parameters which should correspond to the list in DEMANDS.
• ASSEMBLY-TYPE: A list of two symbols. If :matrix is a member, it means evaluation that the coefficient contributes to the Jacobian, if :residual is a member, it contributes to assembling the residual. As noted in the documentation of *assembly-type*, this functionality is not yet used.

Class: <DOMAIN-PROBLEM>

An instance of this class describes a problem posed on domain with coefficients given on each patch of the domain. The slot multiplicity is a positive integer which denotes the number of right-hand sides and solutions (e.g. when computing several eigenvectors at once).

Superclasses: <PROBLEM>

Direct slots:

• DOMAIN: Self-explanatory.
• COMPONENTS: A list whose elements are lists of the form (symbol dim) or (symbol dim type) or (symbol subcomponents) describing the components occuring in the pde. Alternatively, this slot can contain a function/dictionary mapping patches to such lists.
• MULTIPLICITY: Multiplicity of the right-hand side.
• COEFFICIENTS: A function mapping patches to coefficient lists.

Class: <EVP-MIXIN>

A mixin for eigenvalue problems.

Direct slots:

• MULTIPLICITY: The multiplicity of the eigenspace.
• EIGENVALUES: The current approximation of the eigenvalues.
• MU: The multiplier for the system matrix.

Class: <EVP>

Standard class for discrete eigenvalue problems.

Superclasses: <EVP-MIXIN> <NONLINEAR-PROBLEM>

Class: <INTERPOLATION-PROBLEM>

Interpolation problem on a domain. The function which is to be interpolated is given as a coefficient with key FUNCTION in the coefficient list.

Superclasses: <DOMAIN-PROBLEM>

Class: <LS-EVP>

Generalized eigenvalue problem for matrices.

Superclasses: <EVP>

Direct slots:

• STIFFNESS-MATRIX: Self-explanatory.
• MASS-MATRIX: Self-explanatory.

Class: <LSE>

Standard form of a linear system of equations.

Superclasses: <PROBLEM>

Direct slots:

• MATRIX: Self-explanatory.
• RHS: Self-explanatory.

Class: <NLSE>

Class for nonlinear system of equations.

Superclasses: <NONLINEAR-PROBLEM>

Class: <NONLINEAR-PROBLEM>

Class for nonlinear problems. The linearization contains a function returning a linear problem.

Superclasses: <PROBLEM>

Direct slots:

• LINEARIZATION: A function linearizing the problem.
• SOLUTION: An approximation to the solution.

Class: <PDE-PROBLEM>

Base-class for a pde-problem.

Superclasses: <DOMAIN-PROBLEM>

Class: <PROBLEM>

Base class for all problems.

Superclasses: PROPERTY-MIXIN

Class: <TIME-DEPENDENT-PROBLEM>

A mixin which should be used together with a <PDE-PROBLEM> in a call to MAKE-PROGRAMMATIC-INSTANCE.

Direct slots:

• START-TIME: The start time of the problem (NIL=unspecified).
• END-TIME: The end time of the problem (NIL=unspecified).

Function: ADD-FE-PARAMETERS-DEMAND DEMANDS NEW-PARAS

Adds a list of fe-functions to the demands.

Macro: COEFF NAME ARGS &BODY BODY

A local coeff defines a coefficient function inside setup-coefficients. It is defined here at the toplevel such that the Lisp editor indents the definitions correctly.

Function: COEFFICIENTS-OF-CELL CELL MESH PROBLEM

Returns the coefficients of problem on cell.

Function: COEFFICIENTS-OF-PATCH PATCH PROBLEM

Reader for the coefficients of patch for problem.

Function: COMPONENT-POSITION COMPONENTS COMP

Translates a symbol denoting a component to a position.

Function: COMPONENTS-OF-CELL CELL MESH PROBLEM

Returns the components of problem on cell.

Function: COMPONENTS-OF-PATCH PATCH PROBLEM

Reader for the components of problem on patch.

Function: CONSTANT-COEFFICIENT NAME VALUE &REST OTHER-VALUES

Returns a coefficient which takes the given value. Several values can be passed which is needed, for example, for returning also the type of a boundary condition.

Function: CONSTRAINT-COEFFICIENT COMPONENTS MULTIPLICITY

Returns a coefficient function which sets Dirichlet zero boundary conditions for all components of a PDE system.

Macro: CREATE-PROBLEM TYPE (&KEY DOMAIN COMPONENTS (MULTIPLICITY) PROPERTIES) &BODY BODY

Creates a PDE problem. type is the type of the problem which can be the name of a problem class or a list of class names. domain is the domain for this problem, multiplicity is the multiplicity of the solution, e.g. the number of eigenvectors we search for. In body, patch-dependent coefficients should be defined with setup-coefficients. It is also possible to define patch-dependent components with setup-components.

Function: DUAL-PROBLEM PROBLEM FUNCTIONAL

Returns the dual problem for problem with the right-hand side given by functional. The solution of this problem measures the sensitivity of functional applied to the solution of problem with respect to errors in the solution.

Function: ENERGY EVP X

Evaluates the energy bilinear form for a generalized eigenvalue problem.

Function: ENSURE-COEFFICIENT NAME OBJ

Returns obj if it is a coefficient, converts obj into a coefficient depending on the space variable if obj is a function; otherwise, obj is made into a constant coefficient.

Function: ENSURE-RESIDUAL PROBLEM BLACKBOARD

Ensures that the field :RESIDUAL is computed and that the flag :RESIDUAL-P is set on the blackboard.

Function: ENSURE-SOLUTION PROBLEM BLACKBOARD

Ensures that the field :SOLUTION is set on the blackboard.

Function: EXTRACT-FROM SOLUTION FROM NCOMPS &OPTIONAL SCALAR-P INDEX

Extracts numbers or subvectors from the solution vector.

Function: EXTRACTION-INFORMATION COMPONENTS COMPONENT

If component is in components, a triple consisting of position, length, and a flag is returned. The flag is true, if the component is a scalar.

Function: FILTER-APPLICABLE-COEFFICIENTS COEFFS CELL PATCH &KEY (CONSTRAINTS T)

Filters out the applicable coefficients for the respective cell with the given patch.

Function: FU->COEFFICIENT NAME FUNC

The function argument func is transformed into a coefficient depending on the solution.

Function: FX->COEFFICIENT NAME FUNC &KEY (OFFSET 0)

The function argument func is transformed into a coefficient depending on global coordinates.

Function: FXU->COEFFICIENT NAME FUNC &KEY (K 0)

The function argument func is transformed into a coefficient depending on position and solution. If k is different from 0 then the k-jet of f is returned as arguments.

Function: GET-COEFFICIENT COEFFS NAME

Get coefficient name from the list coeffs.

Function: GET-PROPERTY OBJECT PROPERTY

Gets property for object. Returns NIL also if property is not available.

Function: IDENTIFICATION-COEFFICIENT MASTER MAPPING

A special coefficient used for identifying parts of the domain. The coefficient evaluation returns the master coordinates.

Function: LINEAR-P PROBLEM

Predicate determining if a problem is linear or nonlinear.

Function: LINEARIZE PROBLEM SOLUTION

Linearize the nonlinear problem PROBLEM at the point SOLUTION. The result should be a linear problem.

Function: LSE &REST ARGS

Constructs a standard LSE.

Function: MAKE-COEFFICIENTS-FOR PROBLEM NAME PATCH DEMANDS EVALUATOR

Generates a coefficient while dispatching on problem and coefficient name. May return a single coefficient or a list of several coefficients.

Function: MASS EVP X

Evaluates the mass bilinear form for a generalized eigenvalue problem.

Function: MULTIPLICITY VEC

We allow multiple vectors, for solving linear problems in parallel.

Function: NLSE &REST ARGS

Constructs a standard NLSE.

Function: NR-OF-COMPONENTS PROBLEM

Returns the number of components for problem.

Function: PREPARE-COEFFICIENT-ARGUMENTS COMPONENTS ARGS

Prepares arguments for the given coefficient function.

Function: REQUIRED-FE-FUNCTIONS COEFFS

Returns a list of finite element functions required by the coefficients in the property list coeffs.

Function: SELECT-LINEAR-SOLVER OBJECT BLACKBOARD

Selects a linear solver for OBJECT. OBJECT is usually a matrix or a linear problem with certain characteristics.

Function: SELECT-SOLVER OBJECT BLACKBOARD

Selects a solver for OBJECT. OBJECT is usually a problem with certain characteristics.

Function: SELF-ADJOINT-P PROBLEM

Returns two values. The first says if problem is self-adjoint, the second says if that value has really been checked.

Macro: SETUP-COEFFICIENTS (&OPTIONAL (PATCH)) &BODY PATCH-DEFINITIONS

Defines coefficients dispatching on patch.

Macro: SETUP-COMPONENTS (&OPTIONAL (PATCH)) &BODY PATCH-DEFINITIONS

Defines components dispatching on patch.

Function: STATIONARY-PROBLEM-CLASS TDP

Finds the stationary pde problem for the time-dependent problem TDP.

Function: ZERO-CONSTRAINTS PROBLEM

Returns a coefficient function which constrains all system components to zero.