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

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	0.6082017	-1.266173	1.973104
1	1	-0.4382656	1.205478	2.645254
2	2	-2.181385	0.3500421	6.880971
3	3	-0.355007	1.437249	5.191716
4	4	0.810668	0.793156	5.286279
5	5	-0.4705256	0.2610179	5.289525
6	6	-2.290062	-1.282885	12.89018
7	7	-1.311781	-0.09078383	8.729011
8	8	0.9957932	-0.1394528	9.011051
9	9	-0.5602056	0.4454897	9.512291

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

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Vertex value function¶