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	0.511256	0.1665579	0.2891242
1	1	-0.6232318	-1.515787	3.686029
2	2	0.2845683	-0.8480072	2.800095
3	3	1.907478	0.3817661	6.784218
4	4	-0.1213199	-0.01847693	4.01506
5	5	-0.4192256	0.9502552	6.078735
6	6	-0.1123913	1.578882	8.505499
7	7	0.5726894	1.402128	9.293936
8	8	1.011314	1.542861	11.40318
9	9	0.4464708	-1.067271	10.3384

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

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Vertex value function¶