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]
tX0X1y0
000.37270550.21873440.1867541
110.032663571.9707284.884836
22-0.6898897-1.8184675.782771
330.8556508-0.93394854.604398
440.4625881-1.577696.703093
55-0.36766330.65164175.559813
66-2.466355-0.250244712.14553
77-0.66712741.0091658.463472
88-0.05370702-0.49778718.250676
99-0.76661440.51445179.852358