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.

class ShapeFunction(*args, **kwargs)¶

Base class for approximation functions with compact support.

When a continuous variable of e.g. space and time $x(z,t)$ is decomposed in a series $\tilde x = \sum\limits_{i=1}^{\infty} \varphi_i(z) c_i(t)$ the $\varphi_i(z)$ denote the shape functions.

classmethod cure_interval(cls, interval, **kwargs)¶

Create a network or set of functions from this class and return an approximation base (Base) on the given interval.

The kwargs may hold the order of approximation or the amount of functions to use. Use them in your child class as needed.

If you don’t need to now from which class this method is called, overwrite the @classmethod decorator in the child class with the @staticmethod decorator.

Short reference: Inside a @staticmethod you know nothing about the class from which it is called and you can just play with the given parameters. Inside a @classmethod you can additionally operate on the class, since the first parameter is always the class itself.

Parameters

interval (Domain) – Interval to cure.
**kwargs – Various arguments, depending on the implementation.

Returns

Approximation base, generated by the created shape functions.

Return type

Base

Shapefunction Types¶

class LagrangeFirstOrder(start, top, end, **kwargs)¶

Bases: pyinduct.shapefunctions.ShapeFunction

Lagrangian shape functions of order 1.

Parameters

start – Start node
top – Top node, where $f(x) = 1$
end – End node

Keyword Arguments

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_interval(domain, **kwargs)¶

Cure the given interval with LagrangeFirstOrder shape functions.

Parameters: domain (Domain) – Domain to be cured, the points specify the nodes which will be used.
Returns: Base, generated by a set of LagrangeFirstOrder shapefunctions.
Return type: pi.Base

class LagrangeSecondOrder(start, mid, end, **kwargs)¶

Bases: pyinduct.shapefunctions.ShapeFunction

Lagrangian shape functions of order 2.

Parameters

start – start node
mid – middle node, where $f(x) = 1$
end – end node

Keyword Arguments

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_interval(domain, **kwargs)¶

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=- np.inf, np.inf)¶

Bases: pyinduct.shapefunctions.ShapeFunction

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_interval(domain, **kwargs)¶

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

Base, generated by the created shapefunctions.

Return type

Base