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	0.02772026	0.1697782	0.02959305
1	1	-0.730191	-0.05513265	1.536219
2	2	-0.4083213	0.6251446	2.557532
3	3	-0.03512366	-1.551634	5.408803
4	4	0.06106403	0.5456437	4.301456
5	5	1.012743	0.3416257	6.142357
6	6	1.0148	1.861681	10.49568
7	7	-1.89551	0.68089	11.05657
8	8	0.5093272	-0.3883912	8.410262
9	9	-0.878559	0.1473901	9.79359

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

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Vertex value function¶