# dolfinX : How to calculate a gradient/derivative and get it as a numpy.array?

**URL:** <https://fenicsproject.discourse.group/t/dolfinx-how-to-calculate-a-gradient-derivative-and-get-it-as-a-numpy-array/17929>\
**Category:** dolfinx\
**Tags:** dolfinx, variational-form\
**Created:** [July 9, 2025, 1:02pm UTC](https://fenicsproject.discourse.group/t/dolfinx-how-to-calculate-a-gradient-derivative-and-get-it-as-a-numpy-array/17929 "2025-07-09T13:02:45Z")\
**Posts on this page:** 3\
**Page:** 1

<div class="post-metadata">

**Author:** ![Romai2m](https://avatars.discourse-cdn.com/v4/letter/r/839c29/32.png) [@Romai2m](https://fenicsproject.discourse.group/u/Romai2m)\
**Post date:** [July 9, 2025, 1:02pm UTC](https://fenicsproject.discourse.group/t/dolfinx-how-to-calculate-a-gradient-derivative-and-get-it-as-a-numpy-array/17929/1 "2025-07-09T13:02:45Z")

</div>

Hello everyone,

Dolfinx version is **0.7.3** and have been installed with **conda**.  
Other version I need in the minimal code (at the bottom of the post) are :

```
numpy : 1.26.4
mpi4py : 3.1.6
ufl : 2.23.2.0

```

**Context :** I’m solving an acoustic equation with Neumann condition with an iteratived domain decomposition method (no overlapping). I have 2 subdomains named \Omega \_1 and \Omega\_2. The common boundary (interface) between the two subdomains is named \Gamma .

\begin{equation} \mathcal{L}\_{1}[\mathbf{u}\_{1}^{m}] = 0 \; \; \; \text{on} \; \; \Omega \_{1} \\ \mathcal{L}\_{2}[\mathbf{u}\_{2}^{m}] = 0 \; \; \; \text{on} \; \; \Omega \_{2} \\ \mathbf{t}\_{1}^{m} + \alpha \, \mathbf{u}\_1^{m} = -\mathbf{t}\_{2}^{m-1} + \alpha \, \mathbf{u}\_2^{m-1} \; \; \; \text{ sur } \Gamma \\ \mathbf{t}\_{2}^{m} + \alpha \, \mathbf{u}\_2^{m} = -\mathbf{t}\_{1}^{m-1} + \alpha \, \mathbf{u}\_1^{m-1} \; \; \; \text{ sur } \Gamma \\ \text{ } \\ \text{where } \; \mathbf{t}\_{i}^{m} = \nabla [\mathbf{u}\_{i}^{m}]\cdot \mathbf{n}\_{i}^{ } \end{equation} \; \text{ and } \; \mathbf{n}\_{i}^{ } \; \text{ a normal vector to } \; \Gamma .

For example, the weak formulation is the domain 2 is :

\mathcal{A}\_{2}^{m} (\mathbf{u}\_{2}^{m},\mathbf{v}\_{2}^{}) = \displaystyle {\int \_ {\Gamma } } \left( - \mathbf{t} \_1 ^{m-1} + \alpha \mathbf{u} \_1 ^{m-1} \right) \cdot \mathbf{v}\_2

**Problem :**  
I must calculate the gradient at each iteration and get it with a **numpy.array type** to initialise the next calculation.

**What i find to this purpose :**

```auto
# Define a problem
problem = LinearProblem(a_form, L_form, bcs = [])

# Solution
uh = problem.solve()

```

It is possible to use _.x.array_ method on the solution _uh_ of a probleme :

```auto
array = uh.x.array

```

Then, the gradient and the traction are defined by :

```auto
def sigma(u):
    return ufl.nabla_grad(u)

traction = ufl.dot(sigma(uh),ufl.as_vector([1.0,0.0,0.0]))

```

Type of these three objets are :

```auto
 type(uh) : <class 'dolfinx.fem.function.Function'>
 type(sigma(uh)) : <class 'ufl.differentiation.NablaGrad'>
 type(traction) : <class 'ufl.tensoralgebra.Dot'>

```

This is to say i must **convert** these two last types in **dolfinx.fem.function.Function** to use the _.x.array_ method.

So, my idea was to create a trivial system to get the gradient such that :

\displaystyle{\int \_\Omega} \mathbf{t} \cdot \mathbf{v} = \displaystyle{\int \_\Omega} \mathbf{traction} \cdot \mathbf{v} \;

where \mathbf{t} is a trial function, \mathbf{v} is a test function and \mathbf{traction} the second member.

Then \mathbf{t} will have the \<dolfinx.fem.function.Function\> type and \mathbf{t}_.x.array_ will initialise the next calculation.

**Question** Are there other ways to switch a type to a dolfinx.fem.function.Function or to get an array associated to it ? Especially because this approach **double** my computation time …

**Here is the minimal code :** where I need the traction to be accessible in a _numpy.array_

```auto
## Library

import numpy as np # 1.26.4
from dolfinx import fem, la, mesh, plot # 0.7.3
from mpi4py import MPI # 3.1.6
import ufl # 2.23.2.0
from dolfinx.fem.petsc import LinearProblem # 0.7.3

## Domain parameter

length, height = 10, 3
Nx, Ny = 80, 60
extent = [[0.0, 0.0], [length, height]]
domain = mesh.create_rectangle(MPI.COMM_WORLD, extent, [Nx, Ny], mesh.CellType.quadrilateral)

## Finite Element Space/functions

V = fem.FunctionSpace(domain,("P",2,(3,)))

## Define a Fucntion

def Solution(x):
    X = np.zeros_like(x)
    X[0] = x[0]
    X[1] = np.cos(x[0]+x[2]**2)
    X[2] = 0.0
    return X

sol = fem.Function(V)
sol.interpolate(Solution)

## Traction of sol 

def sigma(u):
    return ufl.nabla_grad(u)

traction = ufl.dot(sigma(uh),ufl.as_vector([1.0,0.0,0.0]))

```

---

<div class="post-metadata">

**Author:** ![dokken](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/dokken/32/1560_2.png) [@dokken](https://fenicsproject.discourse.group/u/dokken)\
**Post date:** [July 9, 2025, 3:50pm UTC](https://fenicsproject.discourse.group/t/dolfinx-how-to-calculate-a-gradient-derivative-and-get-it-as-a-numpy-array/17929/2 "2025-07-09T15:50:12Z")

</div>

Yes,  
You can use:

- `dolfinx.fem.Expression`: see for instance [Approximation with continuous and discontinuous finite elements — FEniCS Workshop](http://jsdokken.com/FEniCS-workshop/src/approximations.html#interpolate-expressions)
- Alternatively, use @a-latyshev’s external operator [dolfinx-external-operator — External operators within FEniCSx/DOLFINx](https://a-latyshev.github.io/dolfinx-external-operator/)

---

<div class="post-metadata">

**Author:** ![Romai2m](https://avatars.discourse-cdn.com/v4/letter/r/839c29/32.png) [@Romai2m](https://fenicsproject.discourse.group/u/Romai2m)\
**Post date:** [July 11, 2025, 3:06pm UTC](https://fenicsproject.discourse.group/t/dolfinx-how-to-calculate-a-gradient-derivative-and-get-it-as-a-numpy-array/17929/3 "2025-07-11T15:06:38Z")

</div>

Dear Dokken,

Thank you for your quick answer.

I have followed the first proposition cause the second looks a bit harder.

Now, it works very well and divide my old computation time by 2. For example (same computation) :

| | Time before (s) | Time now (s) |
| --- | --- | --- |
| Solve the problem : | 106 | 106 |
| Get the traction/gradient : | 106 | 0.6 |

Thanks again.