A problem with different thermal conductivities: a general question about the implementation

[amnchacon]

Hello everyone. Imagine that I have a wall composed of different materials, so we have different thermal conductivities. Is it possible to implement a heat conduction problem in Firedrake with this fact? because those thermal conductivities would affect our stiffness matrix $\int_\Omega k \nabla \theta \nabla v \hspace{1mm} d\Omega$ ....

[colinjcotter]

This is a maths question rather than a Firedrake question. But you just integrate by parts to derive the formula in the usual way. On 7 Feb 2024, at 22:49, Andrés Mauricio Nieves Chacón ***@***.***> wrote:  Hello everyone. Imagine that I have a wall composed of different materials, so we have different thermal conductivities. Is it possible to implement a heat conduction issue in Firedrake with this fact? because those thermal conductivities would affect our stiffness matrix $\int_\Omega k \nabla \theta \nabla v \hspace{1mm} d\Omega$ .... Captura.de.tela.de.2024-02-07.16-45-01.png (view on web)<https://github.com/firedrakeproject/firedrake/assets/146268309/ff1d71ef-590b-4a8b-a11c-0a77df5009ec> — Reply to this email directly, view it on GitHub<#3396>, or unsubscribe<https://github.com/notifications/unsubscribe-auth/ABOSV4T5OBRPC53LTHCAEC3YSPZHPAVCNFSM6AAAAABC6SONDGVHI2DSMVQWIX3LMV43ERDJONRXK43TNFXW4OZWGE4TGMJSHA>. You are receiving this because you are subscribed to this thread.Message ID: ***@***.***>

[amnchacon]

But I would have to denote which parts of my mesh will have an specific thermal conductivity, how to specify that in Firedrake? how to translate it in the variational formulation? if I have a Unit Interval Mesh, how could I specify that the first thermal conductivity is applied from one specific point to another, and the same with follow thermal conductivy... and the follow, and that way sequentially. If I have a square mesh for example, where the changes of thermal conductivity are not only on the boundaries but within the domain, how would I specify it? is possible specify the integration limits within the variational formulation written in Firedrake, where govern an specific thermal conductivity?

[ksagiyam]

Details depend on your math problem, but the followings probably help.

You could use conditional to represent thermal conductivity; see, e.g., https://github.com/firedrakeproject/firedrake/blob/master/tests/regression/test_conditional.py for examples.
If the surface of discontinuity coincides with the cell boundaries, you could interpolate the thermal conductivity into the DG0 space.

You could specify subdomain of cell integration as dx(subdomain_label) just as you do for ds. If subdomains are not labeled, you could label them using mark_entities() method of the mesh object: https://www.firedrakeproject.org/firedrake.html#firedrake.mesh.MeshGeometry.mark_entities.

[amnchacon]

I see. That's what I needed to know. I'll check it out carefully. Thanks for this information Mr.