Development of a new solver for 1D differential equations

[nikoscham]

Overview

This thread will monitor the development of a new FEAScript solver for 1D linear differential equations with the following general form:

,

where u is the unknown quantity we want to find (e.g., temperature, concentration, displacement). x is the position along a 1D line. A, B, C, D are coefficients that the user will provide. By choosing different values for them, the user can model different physical situations.

Tasks

• Review and update the mesh generation and nodal numbering process for 1D meshes
• Create a new generalFormPDEScript solver for 1D differential equations, where the differential equation will be converted into the "weak", integral, form.
• Create a new tutorial and compare the results with the analytical solution
• Integrate the solver with FEAScript's existing utilities

[nikoscham]

This issue has been added to the FEAScript roadmap for the upcoming 0.2.0 release (https://github.com/orgs/FEAScript/discussions/17).

[nikoscham]

Support for time-dependent PDEs is also considered:

,

where we suggest using the Backward Euler time-integration method.

[Stu-ops]

Hey @nikoscham 👋 , could you assign it to me? I’d like to work on fixing it.

[nikoscham]

Hey @nikoscham 👋 , could you assign it to me? I’d like to work on fixing it.

Hi Priyank (@Stu-ops)! Thanks for your interest in contributing to the FEAScript project. Well, actually this one sits on the edge between a “good first issue” and an intermediate task 😃. The methodology is as follows:

1. First you have to derive the weak form of the equation presented here:

Overview

This thread will monitor the development of a new FEAScript solver for 1D linear differential equations with the following general form:

,

where u is the unknown quantity we want to find (e.g., temperature, concentration, displacement). x is the position along a 1D line. A, B, C, D are coefficients that the user will provide. By choosing different values for them, the user can model different physical situations.

I can help with the derivation and will add it as a follow-up comment in the next days. In the meanwhile you can also read some notes I wrote regarding the finite element method: https://blog.feascript.com/tutorial/2023/12/01/finite-element-method-notes.html

2. Next you have to create a new solver generalFormPDEScript in the spirit of the existing solvers https://github.com/FEAScript/FEAScript-core/tree/main/src/solvers. Don't worry though, we can implement it together step by step if you agree.

Best regards and please also consider ⭐️ the library!
Nikos

[Stu-ops]

Thanks @nikoscham for giving me this opportunity. Before I move ahead, I have a small but important request. I’ll be raising the PR after October 1st so that it counts for Hacktoberfest, which is important to me. In the meantime, I’ll continue working on the contribution to be ready by then. I hope that’s not a problem for you.