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# Nonlinear heat transfer FEA

## Nonlinear heat transfer FEA

(OP)
Hello everyone.

I'm trying to describe Abaqus solver for my masters thesis, more exactly, for problem of heat conduction in solid, with temperature dependent thermal conductivity (k=k(T)).

I would like to derive the formulas from the ground up (see bottom picture), with example of a simple truss (1D). For start, I would work with steady-state problem and later upgrade to transient. I realize, that this is a non-linear problem, and am aware how Abaqus solves plastic deformation (Newton-Raphson alghoritm).

I have basically turned around google and google scholar, but all i find are books that are too general (FE), have to do with plasticity or articles that are too specific.

I am almost desperate at this point, so I'm asking this community to please give me some hint or maybe some literature for solving heat conduction with k = k(T).

### RE: Nonlinear heat transfer FEA

The best source of information about the theory behind Abaqus analysis procedure is the documentation of this software. There’s a whole Theory Guide in which equations and algorithms used by Abaqus for various simulation types are described. However, books about general FEA for thermomechanical applications should also be useful, especially that derivations in Abaqus documentation are reduced to minimum.

### RE: Nonlinear heat transfer FEA

Steady-state should be straightforward enough: use the standard methodology to derive the weak form (e.g., using Galerkin method of weighted residuals), determine the expression for k(T) (analogous to e.g., determining a displacement-dependent Young´s modulus for a solid mechanics rod problem), develop the discretized system "K(T)u=q" and apply an element interpolant (e.g., linear in a 1D bar), and then create the system of equations and solve it using e.g., direct iteration (more straightforward method) or Newton-Raphson.

You will find this type of problem in many books dealing with normal FE and nonlinear FE. Start with your course notes, and then move to linear FE (to get acquainted with the process of formulating the problem when k is not temperature dependent), and then find a book detailing the nonlinear FE for 1D elements - preferably for the problem at hand, but do note that the nonlinear elastic rod problem is essentially analogous to what you are solving.

The transient problem is much more complicated. It will require discretization also for the time domain and a numerical time integration procedure.

I would recommend "The Finite Element Method" by Thomas J.R. Hughes; nonlinear FE by Reddy; FE for solids by Zienkiewicz & Taylor; 1D FE by A. Öchsner & M. Merkel for a start.

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