Vertex value function¶

A vertex value function $f_{vertexvalue}: \mathcal{D} \times \mathbb{R}^d \rightarrow \mathcal{D} \times \mathbb{R}^q$ is a particular field function that lets invariant the mesh of a field and defined by a function $h : \mathbb{R}^n \times \mathbb{R}^d \rightarrow \mathbb{R}^q$ such that:

$\begin{aligned} f_{vertexvalue}(\underline{t}, \underline{x})=(\underline{t}, h(\underline{t},\underline{x}))\end{aligned}$

Let’s note that the input dimension of $f_{vertexvalue}$ still design the dimension of $\underline{x}$ : $d$ . Its output dimension is equal to $q$ .

The creation of the VertexValueFunction object requires the function $h$ and the integer $n$ : the dimension of the vertices of the mesh $\mathcal{M}$ .

This example illustrates the creation of a field from the function $h:\mathbb{R}\times\mathbb{R}^2$ such as:

$\begin{aligned} h(\underline{t}, \underline{x})=(t+x_1^2+x_2^2) \end{aligned}$

import openturns as ot

ot.Log.Show(ot.Log.NONE)

Create a mesh

N = 100
mesh = ot.RegularGrid(0.0, 1.0, N)

Create the function that acts the values of the mesh

h = ot.SymbolicFunction(["t", "x1", "x2"], ["t+x1^2+x2^2"])

Create the field function

f = ot.VertexValueFunction(h, mesh)

Evaluate f

inF = ot.Normal(2).getSample(N)
outF = f(inF)

# print input/output at first 10 mesh nodes
txy = mesh.getVertices()
txy.stack(inF)
txy.stack(outF)
txy[:10]

	t	X0	X1	y0
0	0	0.3727055	0.2187344	0.1867541
1	1	0.03266357	1.970728	4.884836
2	2	-0.6898897	-1.818467	5.782771
3	3	0.8556508	-0.9339485	4.604398
4	4	0.4625881	-1.57769	6.703093
5	5	-0.3676633	0.6516417	5.559813
6	6	-2.466355	-0.2502447	12.14553
7	7	-0.6671274	1.009165	8.463472
8	8	-0.05370702	-0.4977871	8.250676
9	9	-0.7666144	0.5144517	9.852358

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An Open source initiative for the Treatment of Uncertainties, Risks'N Statistics

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Vertex value function¶