Function¶
The function module defines the Evaluable
class and derived objects,
commonly referred to as nutils functions. They represent mappings from a
nutils.topology
onto Python space. The notabe class of Array
objects map onto the space of Numpy arrays of predefined dimension and shape.
Most functions used in nutils applicatons are of this latter type, including the
geometry and function bases for analysis.
Nutils functions are essentially postponed python functions, stored in a tree
structure of input/output dependencies. Many Array
objects have
directly recognizable numpy equivalents, such as Sin
or
Inverse
. By not evaluating directly but merely stacking operations,
complex operations can be defined prior to entering a quadrature loop, allowing
for a higher level style programming. It also allows for automatic
differentiation and code optimization.
It is important to realize that nutils functions do not map for a physical xy-domain but from a topology, where a point is characterized by the combination of an element and its local coordinate. This is a natural fit for typical finite element operations such as quadrature. Evaluation from physical coordinates is possible only via inverting of the geometry function, which is a fundamentally expensive and currently unsupported operation.
-
exception
nutils.function.
EvaluationError
(etype, evalue, evaluable, values)[source]¶ evaluation error
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class
nutils.function.
Array
(args: tuple, shape: tuple, dtype: <function <lambda> at 0x7fa7565db950>)[source]¶ array function
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class
nutils.function.
Interpolate
(x: <function <lambda> at 0x7fa7565db9d8>, xp: nutils.numeric.const, fp: nutils.numeric.const, left=None, right=None)[source]¶ interpolate uniformly spaced data; stepwise for now
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class
nutils.function.
BlockAdd
(funcs: nutils.util.frozenmultiset)[source]¶ block addition (used for DG)
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class
nutils.function.
Sampled
(data: nutils.util.frozendict, trans=<nutils.function.SelectChain object>)[source]¶ sampled
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class
nutils.function.
Zeros
(shape: tuple, dtype: <function <lambda> at 0x7fa7565db950>)[source]¶ zero
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class
nutils.function.
Guard
(fun: <function <lambda> at 0x7fa7565db9d8>)[source]¶ bar all simplifications
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class
nutils.function.
Find
(where: <function <lambda> at 0x7fa7565db9d8>)[source]¶ indices of boolean index vector
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class
nutils.function.
DerivativeTargetBase
(args: tuple, shape: tuple, dtype: <function <lambda> at 0x7fa7565db950>)[source]¶ base class for derivative targets
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class
nutils.function.
Argument
(name, shape: tuple, nderiv: int = 0)[source]¶ Array argument, to be substituted before evaluation.
The
Argument
is anArray
with a known shape, but whose values are to be defined later, before evaluation, e.g. usingreplace_arguments()
.It is possible to take the derivative of an
Array
to anArgument
:>>> from nutils import function >>> a = function.Argument('x', []) >>> b = function.Argument('y', []) >>> f = a**3 + b**2 >>> function.derivative(f, a).simplified == (3.*a**2).simplified True
Furthermore, derivatives to the local cooardinates are remembered and applied to the replacement when using
replace_arguments()
:>>> from nutils import mesh >>> domain, x = mesh.rectilinear([2,2]) >>> basis = domain.basis('spline', degree=2) >>> c = function.Argument('c', basis.shape) >>> replace_arguments(c.grad(x), dict(c=basis)) == basis.grad(x) True
Parameters:
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class
nutils.function.
Polyval
(coeffs: <function <lambda> at 0x7fa7565db9d8>, points: <function <lambda> at 0x7fa7565db9d8>, ngrad: int = 0)[source]¶ Computes the \(k\)-dimensional array
\[\begin{split}j_0,\dots,j_{k-1} \mapsto \sum_{\substack{i_0,\dots,i_{n-1}\in\mathbb{N}\\i_0+\cdots+i_{n-1}\le d}} p_0^{i_0} \cdots p_{n-1}^{i_{n-1}} c_{j_0,\dots,j_{k-1},i_0,\dots,i_{n-1}},\end{split}\]where \(p\) are the \(n\)-dimensional local coordinates and \(c\) is the argument
coeffs
and \(d\) is the degree of the polynomial, where \(d\) is the length of the last \(n\) axes ofcoeffs
.Warning
All coefficients with a (combined) degree larger than \(d\) should be zero. Failing to do so won’t raise an
Exception
, but might give incorrect results.
-
nutils.function.
matmat
(arg0, *args)[source]¶ helper function, contracts last axis of arg0 with first axis of arg1, etc
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nutils.function.
polyfunc
(coeffs, dofs, ndofs, transforms, *, issorted=True)[source]¶ Create an inflated
Polyval
with coefficientscoeffs
and corresponding dofsdofs
. The argumentscoeffs
,dofs
andtransforms
are assumed to have matching order. In addition, ifissorted
is true, thetransforms
argument is assumed to be sorted.
-
nutils.function.
replace_arguments
(value, arguments)[source]¶ Replace
Argument
objects invalue
.Replace
Argument
objects invalue
according to thearguments
map, taking into account derivatives to the local coordinates.Parameters: - value (
Array
) – Array to be edited. - arguments (
collections.abc.Mapping
withArray
s as values) –Argument
s replacements. The key correspond to thename
passed to anArgument
and the value is the replacement.
Returns: The edited
value
.Return type: - value (
-
class
nutils.function.
Namespace
(*, default_geometry_name='x')[source]¶ Namespace for
Array
objects supporting assignments with tensor expressions.The
Namespace
object is used to storeArray
objects.>>> from nutils import function >>> ns = function.Namespace() >>> ns.A = function.zeros([3, 3]) >>> ns.x = function.zeros([3]) >>> ns.c = 2
In addition to the assignment of
Array
objects, it is also possible to specify an array using a tensor expression string — seenutils.expression.parse()
for the syntax. All attributes defined in this namespace are available as variables in the expression. If the array defined by the expression has one or more dimensions the indices of the axes should be appended to the attribute name. Examples:>>> ns.cAx_i = 'c A_ij x_j' >>> ns.xAx = 'x_i A_ij x_j'
It is also possible to simply evaluate an expression without storing its value in the namespace by passing the expression to the method
eval_
suffixed with appropriate indices:>>> ns.eval_('2 c') Array<> >>> ns.eval_i('c A_ij x_j') Array<3> >>> ns.eval_ij('A_ij + A_ji') Array<3,3>
For zero and one dimensional expressions the following shorthand can be used:
>>> '2 c' @ ns Array<> >>> 'A_ij x_j' @ ns Array<3>
When evaluating an expression through this namespace the following functions are available:
opposite
,sin
,cos
,tan
,sinh
,cosh
,tanh
,arcsin
,arccos
,arctan2
,arctanh
,exp
,abs
,ln
,log
,log2
,log10
,sqrt
andsign
.Parameters: default_geometry_name ( str
) – The name of the default geometry. This argument is passed tonutils.expression.parse()
. Default:'x'
.-
arg_shapes
¶ types.MappingProxyType
– A readonly map of argument names and shapes.
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default_geometry
¶ nutils.function.Array
– The default geometry, shorthand forgetattr(ns, ns.default_geometry_name)
.
-