Shapefunctions¶

The shapefunctions module contains generic shapefunctions that can be used to approximate distributed systems without giving any information about the systems themselves. This is achieved by projecting them on generic, piecewise smooth functions.

Shapefunction Types¶

class LagrangeFirstOrder(start, top, end, **kwargs)[source]¶

Bases: pyinduct.core.Function

Lagrangian shape functions of order 1.

Keyword Arguments:
Parameters:	start – start node top – top node, where $f(x) = 1$ end – end node
	half – right_border – left_border –

Example plot of the functions funcs generated with

>>> nodes, funcs = cure_interval(LagrangeFirstOrder, (0, 1), node_count=7)

static cure_hint(domain, **kwargs)[source]¶

Hint function that will cure the given interval with LagrangeFirstOrder.

Parameters:	domain (`Domain`) – domain to be cured
Returns:	(domain, funcs), where funcs is set of `LagrangeFirstOrder` shapefunctions.
Return type:	tuple

class LagrangeSecondOrder(start, mid, end, **kwargs)[source]¶

Bases: pyinduct.core.Function

Lagrangian shape functions of order 2.

Keyword Arguments:
Parameters:	start – start node mid – middle node, where $f(x) = 1$ end – end node
	curvature (str) – “concave” or “convex” half (str) – Generate only “left” or “right” half. domain (tuple) – Domain on which the function is defined.

Example plot of the functions funcs generated with

>>> nodes, funcs = cure_interval(LagrangeSecondOrder, (0, 1), node_count=7)

static cure_hint(domain, **kwargs)[source]¶

Hint function that will cure the given interval with LagrangeSecondOrder.

Parameters:	domain (`Domain`) – domain to be cured
Returns:	(domain, funcs), where funcs is set of `LagrangeSecondOrder` shapefunctions.
Return type:	tuple

class LagrangeNthOrder(order, nodes, left=False, right=False, mid_num=None, boundary=None, domain=(-inf, inf))[source]¶

Bases: pyinduct.core.Function

Lagrangian shape functions of order $n$ .

Note

The polynomials between the boundary-polynomials and the peak-polynomials, respectively between peak-polynomials and peak-polynomials, are called mid-polynomials.

Parameters:

order (int) – Order of the lagrangian polynomials.
nodes (numpy.array) – Nodes on which the piecewise defined functions have to be one/zero. Length of nodes must be either $order * 2 + 1$ (for peak-polynomials, see notes) or ‘order +1’ (for boundary- and mid-polynomials).
left (bool) – State the first node (nodes[0]) to be the left boundary of the considered domain.
right (bool) – State the last node (nodes[-1]) to be the right boundary of the considered domain.
mid_num (int) – Local number of mid-polynomials (see notes) to use (only used for order >= 2). $\text{mid\_num} \in \{ 1, ..., \text{order} - 1 \}$
boundary (str) – provide “left” or “right” to instantiate the according boundary-polynomial.
domain (tuple) – Domain of the function.

Example plot of the functions funcs generated with

>>> nodes, funcs = pi.cure_interval(sh.LagrangeNthOrder, (0, 1), node_count=9, order=4)

static cure_hint(domain, **kwargs)[source]¶

Hint function that will cure the given interval with LagrangeNthOrder. Length of the domain argument $L$ must satisfy the condition

$L = 1 + (1 + n) order \quad \forall n \in \mathbb N.$

E.g. n - order = 1 -> $L \in \{2, 3, 4, 5, ...\}$ - order = 2 -> $L \in \{3, 5, 7, 9, ...\}$ - order = 3 -> $L \in \{4, 7, 10, 13, ...\}$ - and so on.

Parameters:	domain (`Domain`) – Domain to be cured. order (int) – Order of the lagrange polynomials.
Returns:	(domain, funcs), where funcs is a set of `LagrangeNthOrder` shapefunctions.
Return type:	tuple

Curing an Interval¶

All classes contained in this module can easily be used to cure a given interval. For example let’s approximate the interval from $z=0$ to $z=1$ with 3 piecewise linear functions:

>>> nodes, funcs = cure_interval(LagrangeFirstOrder, (0, 1), node_count=3)

The approximation nodes of the functions are chosen automatically:

>>> list(nodes)
[0.0, 0.5, 1.0]

cure_interval(shapefunction_class, interval, node_count=None, node_distance=None, **kwargs)[source]¶

Use shape functions to cure an interval with either node_count nodes or nodes with node_distance.

Parameters:	shapefunction_class – Class to cure the interval (e.g. `LagrangeFirstOrder`). The given class has to provide a `cure_hint()`. interval (tuple) – Limits that constrain the interval. node_count (int) – Amount of nodes to use. node_distance (numbers.Number) – Distance of nodes.

Note

Either node_count or node_node_distance can be specified. If both are given and are not consistent, an exception will be raised by Domain.

Raises:	`TypeError` – If given class does not provide a static `cure_hint()` method.
Returns:	`(domain, funcs)`: Where `domain` is a `Domain` instance and `funcs` is a list of (e.g. `LagrangeFirstOrder`) shapefunctions.
Return type:	tuple