Muscat.FE.SymWeakForm module

ArrayToMatrix(array, normalize=False) → MutableDenseMatrix[source]

Convert a list of object into a sympy.Matrix (1D). str objects are converted into a symbols. If normalize is True, the array is rescaled by its L2-norm, if all values are numerical; a warning is issued otherwise.

Parameters:

  • array (arraylike) – a list of objects

  • normalize (bool, optional) – if true and all the values in the array are numerical then the vector is rescaled (norm(array) ==1), by default False

Returns:

a sympy matrix with str converted to symbols

Return type:

Matrix

CheckIntegrity(GUI: bool = False)[source]
Cross(a, b)[source]

Compute the cross product of two 3D vector fields.

Parameters:

  • a (sympy.Matrix) – Vector field of shape (n, 1).

  • b (sympy.Matrix) – Vector field of shape (n, 1).

Returns:

Vector field of shape (n, 1).

Return type:

sympy.Matrix

Diag(arg)[source]

Create a diagonal matrix from symbolic weak forms.

Parameters:

arg (array-like) – Iterable of shape (n,) containing weak forms as SymPy expressions.

Returns:

Diagonal matrix of shape (n, n).

Return type:

sympy.Matrix or numpy.ndarray

Divergence(arg, sdim=3)[source]

Compute the divergence of a vector or matrix field.

Parameters:

  • arg (sympy.Matrix) – Matrix of shape (n, 1) or (n, n) with n equal to 2 or 3.

  • sdim (int, optional) – Dimension of the divergence, either 2 or 3.

Returns:

Expression if arg is a vector, or a matrix of shape (n, 1) if arg is a matrix field.

Return type:

sympy.Expr or sympy.Matrix

FromVoigtSigma(arg)[source]

https://en.wikipedia.org/wiki/Voigt_notation

GetConstant(name, size=1) → Symbol | MutableDenseMatrix[source]

Create a scalar or vector constant symbolic field.

Parameters:

  • name (str) – Label of the constant field.

  • size (int, optional) – Number of components. If 1, a scalar symbol is returned.

Returns:

Scalar symbol when size == 1; otherwise a column matrix of symbols.

Return type:

sympy.Symbol or sympy.Matrix

GetCoordinate(size: int64)[source]

Function to get the coordinate of the current point. The Coordinate symbol will be populated with the proper value when processed by the integrator

Parameters:

size (MuscatIndex) – size: MuscatIndex, dimension of the space. Usually taken equal to the problem dimension

Returns:

sympy.Matrix of shape (size,1) with the special denomination “Coordinate”

Return type:

sympy.Matrix

GetElementDeterminant()[source]

Function to get the determinant of the element jacobian. The ELEMENT_DETERMINANT symbol will be populated with the proper value when processed by the integrator :rtype: sympy.Symbol with the special denomination “ELEMENT_DETERMINANT”

GetField(name: str, size: int64, sdim: int64, star=False, extraCoordinates=[]) → MutableDenseMatrix[source]

Instantiate a scalar, vector or tensor symbolic field.

Parameters:

  • name (str) – Label of the field. If size > 1, each component receives a suffix for the coordinate.

  • size (MuscatIndex or tuple of MuscatIndex) – Dimension of the field. Use 1 for a scalar, 2 for 2D and 3 for 3D. If a tuple is given, a tensor field is created.

  • sdim (MuscatIndex) – Space dimension where the field evolves.

  • star (bool, optional) – Whether the field is a test field.

  • extraCoordinates (list[str], optional) – Extra field parameters, such as time.

Returns:

Matrix containing functions of the spatial coordinates and any extra coordinates.

Return type:

sympy.Matrix

GetNormal(size: int64)[source]

Return the element normal symbolic field.

The Normal symbol is populated with the proper value when processed by the integrator.

Parameters:

size (MuscatIndex) – Dimension of the normal. Usually equal to the problem dimension.

Returns:

Matrix of shape (size, 1) with the special denomination "Normal".

Return type:

sympy.Matrix

GetScalarField(alpha)[source]

Create a constant scalar field from multiple input types.

Parameters:

alpha (object) – Input value used to create the scalar field.

Returns:

sympy.Symbol if alpha is a string, float if alpha is an integer or float, 1.0 if alpha is None, otherwise alpha.

Return type:

sympy.Symbol or float or object

GetTestField(name: str, size: int64, sdim: int64, extraCoordinates=[]) → MutableDenseMatrix[source]

Instantiate a test field.

This is a convenience wrapper around GetField with star=True.

Parameters:

  • name (str) – Label of the field. If size > 1, each component receives a suffix for the coordinate and the test character.

  • size (MuscatIndex or tuple of MuscatIndex) – Dimension of the field. Use 1 for a scalar, 2 for 2D and 3 for 3D. If a tuple is given, a tensor field is created.

  • sdim (MuscatIndex) – Space dimension where the field evolves.

  • extraCoordinates (list[str], optional) – Extra field parameters, such as time.

Returns:

Matrix containing functions of the spatial coordinates and any extra coordinates.

Return type:

sympy.Matrix

GetTestSufixChar() → str[source]
Gradient(arg, sdim=3)[source]

Compute the gradient of a vector or scalar field.

Parameters:

  • arg (sympy.Matrix) – Matrix of shape (1, 1) for a scalar field or (n, 1) for a vector field.

  • sdim (int, optional) – Space dimension on which to differentiate, either 2 or 3.

Returns:

Matrix of shape (sdim, 1) for a scalar field or (sdim, n) for a vector field. The first dimension is the differentiation coordinate and the second is the vector component.

Return type:

sympy.Matrix

Inner(a, b)[source]

Compute the scalar product of two vector fields.

Parameters:

  • a (sympy.Matrix) – Vector field of shape (n, 1).

  • b (sympy.Matrix) – Vector field of shape (n, 1).

Returns:

Scalar product following the first dimension, using the opposite convention to NumPy.

Return type:

sympy.Expr

Notes

Matrix support is not implemented yet.

Laplacian(arg, sdim=3)[source]

Compute the Laplacian of a scalar or vector field.

Parameters:

  • arg (sympy.Matrix) – Matrix of shape (1, 1) for a scalar field or (n, 1) for a vector field.

  • sdim (int, optional) – Space dimension on which to differentiate, either 2 or 3.

Returns:

Expression for a scalar field, or a matrix of shape (n, 1) for a vector field.

Return type:

sympy.Expr or sympy.Matrix

Strain(arg, sdim=3)[source]

Compute the strain matrix of a displacement vector.

Parameters:

  • arg (sympy.Matrix) – Displacement field of shape (n, 1).

  • sdim (int, optional) – Dimension for the derivation, either 2 or 3.

Returns:

Strain matrix of shape (n, n).

Return type:

sympy.Matrix

StrainAxyCol(arg, radius)[source]
ToVoigtEpsilon(arg)[source]

https://en.wikipedia.org/wiki/Voigt_notation

ToVoigtSigma(arg)[source]

https://en.wikipedia.org/wiki/Voigt_notation

Trace(arg)[source]

Compute the trace of a matrix.

Parameters:

arg (sympy.Matrix or numpy.ndarray) – Input matrix.

Returns:

Sum of the diagonal elements.

Return type:

sympy.Expr or numpy scalar