4.6.5 pde.tools.expressions module
Handling mathematical expressions with sympy
This module provides classes representing expressions that can be provided as
human-readable strings and are converted to numpy
and numba
representations using sympy
.
return a number compiled from an expression |
|
describes a mathematical expression of a scalar quantity |
|
describes a mathematical expression of a tensorial quantity |
|
evaluate an expression involving fields |
- class ExpressionBase(expression, signature=None, *, user_funcs=None, consts=None, repl=None)[source]
Bases:
object
abstract base class for handling expressions
Warning
This implementation uses
exec()
and should therefore not be used in a context where malicious input could occur.- Parameters:
expression (
sympy.core.basic.Basic
) – A sympy expression or array. This could for instance be an instance ofExpr
orNDimArray
.signature (list of str, optional) – The signature defines which variables are expected in the expression. This is typically a list of strings identifying the variable names. Individual names can be specified as list, in which case any of these names can be used. The first item in such a list is the definite name and if another name of the list is used, the associated variable is renamed to the definite name. If signature is None, all variables in expressions are allowed.
user_funcs (dict, optional) – A dictionary with user defined functions that 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
.repl (dict, optional) – Replacements that are applied to symbols before turning the expression into a python equivalent.
- get_compiled(single_arg=False)[source]
return numba function evaluating expression
- Parameters:
single_arg (bool) – Determines whether the function takes all variables in a single argument as an array or whether all variables need to be supplied separately.
- Returns:
the compiled function
- Return type:
function
- class ScalarExpression(expression=0, signature=None, *, user_funcs=None, consts=None, repl=None, explicit_symbols=None, allow_indexed=False)[source]
Bases:
ExpressionBase
describes a mathematical expression of a scalar quantity
Warning
This implementation uses
exec()
and should therefore not be used in a context where malicious input could occur.- Parameters:
expression (str or float) – The expression, either a number or a string that sympy can parse.
signature (list of str) – The signature defines which variables are expected in the expression. This is typically a list of strings identifying the variable names. Individual names can be specified as lists, in which case any of these names can be used. The first item in such a list is the definite name and if another name of the list is used, the associated variable is renamed to the definite name. If signature is None, all variables in expressions are allowed.
user_funcs (dict, optional) – A dictionary with user defined functions that 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
.repl (dict, optional) – Replacements that are applied to symbols before turning the expression into a python equivalent.
explicit_symbols (list of str) – List of symbols that need to be interpreted as general sympy symbols
allow_indexed (bool) – Whether to allow indexing of variables. If enabled, array variables are allowed to be indexed using square bracket notation.
- derivatives
differentiate the expression with respect to all variables
- class TensorExpression(expression, signature=None, *, user_funcs=None, consts=None, repl=None, explicit_symbols=None)[source]
Bases:
ExpressionBase
describes a mathematical expression of a tensorial quantity
Warning
This implementation uses
exec()
and should therefore not be used in a context where malicious input could occur.- Parameters:
expression (str or float) – The expression, either a number or a string that sympy can parse.
signature (list of str) – The signature defines which variables are expected in the expression. This is typically a list of strings identifying the variable names. Individual names can be specified as list, in which case any of these names can be used. The first item in such a list is the definite name and if another name of the list is used, the associated variable is renamed to the definite name. If signature is None, all variables in expressions are allowed.
user_funcs (dict, optional) – A dictionary with user defined functions that 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
.repl (dict, optional) – Replacements that are applied to symbols before turning the expression into a python equivalent.
explicit_symbols (list of str) – List of symbols that need to be interpreted as general sympy symbols
- derivatives
differentiate the expression with respect to all variables
- differentiate(var)[source]
return the expression differentiated with respect to var
- Parameters:
var (str) –
- Return type:
- get_compiled_array(single_arg=True)[source]
compile the tensor expression such that a numpy array is returned
- property value
the value for a constant expression
- evaluate(expression, fields, *, bc='auto_periodic_neumann', bc_ops=None, user_funcs=None, consts=None, label=None)[source]
evaluate an expression involving fields
Warning
This implementation uses
exec()
and should therefore not be used in a context where malicious input could occur.- Parameters:
expression (str) – The expression, which is parsed by
sympy
. The expression may contain variables (i.e., fields and spatial coordinates of the grid), standard local mathematical operators defined by sympy, and the operators defined in thepde
package. Note that operators need to be specified with their full name, i.e., laplace for a scalar Laplacian and vector_laplace for a Laplacian operating on a vector field. Moreover, the dot product between two vector fields can be denoted by using dot(field1, field2) in the expression, and outer(field1, field2) calculates an outer product. More information can be found in the expression documentation.fields (dict or
FieldCollection
) – Dictionary of the fields involved in the expression. The dictionary keys specify the field names allowed in expression. Alternatively, fields can be aFieldCollection
with unique labels.bc (BoundariesData) – Boundary conditions for the operators used in the expression. The conditions here are applied to all operators that do not have a specialized condition given in bc_ops. Boundary conditions are generally given as a list with one condition for each axis. 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 a tuple of two conditions). 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 ‘natural’ imposes periodic boundary conditions for periodic axis and a vanishing derivative otherwise. More information can be found in the boundaries documentation.
bc_ops (dict) – Special boundary conditions for some operators. The keys in this dictionary specify the operator to which the boundary condition will be applied.
user_funcs (dict, optional) – A dictionary with user defined functions that can be used in the expressions in rhs.
consts (dict, optional) – A dictionary with user defined constants that can be used in the expression. These can be either scalar numbers or fields defined on the same grid as the actual simulation.
label (str) – Name of the field that is returned.
- Returns:
The resulting field. The rank of the returned field (and thus the precise class) is determined automatically.
- Return type: