4.1.5 pde.fields.tensorial module
Defines a tensorial field of rank 2 over a grid.
- class Tensor2Field(grid, data='zeros', *, label=None, dtype=None, with_ghost_cells=False)[source]
Bases:
DataFieldBaseTensor field of rank 2 discretized on a grid.
Warning
Components of the tensor field are given in the local basis. While the local basis is identical to the global basis in Cartesian coordinates, the local basis depends on position in curvilinear coordinate systems. Moreover, the field always contains all components, even if the underlying grid assumes symmetries.
- Parameters:
grid (
GridBase) – Grid defining the space on which this field is defined.data (Number or
ndarray, optional) – Field values at the support points of the grid. The flag with_ghost_cells determines whether this data array contains values for the ghost cells, too. The resulting field will contain real data unless the data argument contains complex values. Special values are “zeros” or None, initializing the field with zeros, and “empty”, just allocating memory with unspecified values.label (str, optional) – Name of the field
dtype (numpy dtype) – The data type of the field. If omitted, it will be determined from data automatically.
with_ghost_cells (bool) – Indicates whether the ghost cells are included in data
- divergence(bc, out=None, **kwargs)[source]
Apply tensor divergence and return result as a field.
The tensor divergence is a vector field \(v_\alpha\) resulting from a contracting of the derivative of the tensor field \(t_{\alpha\beta}\):
\[v_\alpha = \sum_\beta \frac{\partial t_{\alpha\beta}}{\partial x_\beta}\]- Parameters:
bc (BoundariesData | None) – The boundary conditions applied to the field. Boundary conditions are generally given as a dictionary with one condition for each axis side. For periodic axes, only periodic boundary conditions are allowed (indicated by ‘periodic’ and ‘anti-periodic’). For non-periodic axes, different boundary conditions can be specified for the lower and upper end (using specific identifiers, like x- and y+). For instance, Dirichlet conditions enforcing a value NUM (specified by {‘value’: NUM}) and Neumann conditions enforcing the value DERIV for the derivative in the normal direction (specified by {‘derivative’: DERIV}) are supported. Note that the special value ‘auto_periodic_neumann’ imposes periodic boundary conditions for periodic axis and a vanishing derivative otherwise. More information can be found in the boundaries documentation. If the special value None is given, no boundary conditions are enforced. The user then needs to ensure that the ghost cells are set accordingly.
out (VectorField, optional) – Optional scalar field to which the result is written.
label (str, optional) – Name of the returned field
**kwargs – Additional arguments affecting how the operator behaves.
- Returns:
result of applying the operator
- Return type:
- dot(other, out=None, *, conjugate=True, label='dot product')[source]
Calculate the dot product involving a tensor field.
This supports the dot product between two tensor fields as well as the product between a tensor and a vector. The resulting fields will be a tensor or vector, respectively.
- Parameters:
other (VectorField or Tensor2Field) – the second field
out (VectorField or Tensor2Field, optional) – Optional field to which the result is written.
conjugate (bool) – Whether to use the complex conjugate for the second operand
label (str, optional) – Name of the returned field
- Returns:
VectorFieldorTensor2Field: result of applying the dot operator- Return type:
- classmethod from_expression(grid, expressions, *, user_funcs=None, consts=None, label=None, dtype=None)[source]
Create a tensor field on a grid from given expressions.
Warning
This implementation uses
exec()and should therefore not be used in a context where malicious input could occur.- Parameters:
grid (
GridBase) – Grid defining the space on which this field is definedexpressions (list of str) – A 2d list of mathematical expression, one for each component of the tensor field. The expressions determine the values as a function of the position on the grid. The expressions may contain standard mathematical functions and they may depend on the axes labels of the grid. More information can be found in the expression documentation.
user_funcs (dict, optional) – A dictionary with user defined functions that can be used in the expression
consts (dict, optional) – A dictionary with user defined constants that can be used in the expression. The values of these constants should either be numbers or
ndarray.label (str, optional) – Name of the field
dtype (numpy dtype) – The data type of the field. If omitted, it will be determined from data automatically.
- Return type:
- plot_components(kind='auto', *args, title=None, constrained_layout=True, filename=None, action='auto', fig_style=None, fig=None, **kwargs)[source]
Visualize all the components of this tensor field.
- Parameters:
kind (str or list of str) – Determines the kind of the visualizations. Supported values are image or line. Alternatively, auto determines the best visualization based on the grid.
title (str) – Title of the plot. If omitted, the title might be chosen automatically. This is shown above all panels.
constrained_layout (bool) – Whether to use constrained_layout in
matplotlib.pyplot.figure()call to create a figure. This affects the layout of all plot elements. Generally, spacing might be better with this flag enabled, but it can also lead to problems when plotting multiple plots successively, e.g., when creating a movie.filename (str, optional) – If given, the figure is written to the specified file.
action (str) – Decides what to do with the final figure. If the argument is set to show,
matplotlib.pyplot.show()will be called to show the plot. If the value is none, the figure will be created, but not necessarily shown. The value close closes the figure, after saving it to a file when filename is given. The default value auto implies that the plot is shown if it is not a nested plot call.fig_style (dict) – Dictionary with properties that will be changed on the figure after the plot has been drawn by calling
matplotlib.pyplot.setp(). For instance, using fig_style={‘dpi’: 200} increases the resolution of the figure.fig (
matplotlib.figures.Figure) – Figure that is used for plotting. If omitted, a new figure is created.**kwargs – All additional keyword arguments are forwarded to the actual plotting function of all subplots.
- Returns:
Instances that contain information to update all the plots with new data later.
- Return type:
2d list of
PlotReference
- symmetrize(make_traceless=False, inplace=False)[source]
Symmetrize the tensor field in place.
- Parameters:
- Returns:
result of the operation
- Return type:
- to_scalar(scalar='auto', *, label='scalar `{scalar}`')[source]
Return scalar variant of the field.
The invariants of the tensor field \(\boldsymbol{A}\) are
\[\begin{split}I_1 &= \mathrm{tr}(\boldsymbol{A}) \\ I_2 &= \frac12 \left[ (\mathrm{tr}(\boldsymbol{A})^2 - \mathrm{tr}(\boldsymbol{A}^2) \right] \\ I_3 &= \det(A)\end{split}\]where tr denotes the trace and det denotes the determinant. Note that the three invariants can only be distinct and non-zero in three dimensions. In two dimensional spaces, we have the identity \(2 I_2 = I_3\) and in one-dimensional spaces, we have \(I_1 = I_3\) as well as \(I_2 = 0\).
- Parameters:
- Returns:
the scalar field after applying the operation
- Return type:
- trace(label='trace')[source]
Return the trace of the tensor field as a scalar field.
- Parameters:
label (str, optional) – Name of the returned field
- Returns:
scalar field of traces
- Return type: