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Transient thermal analysis

Force Matrix for Transient Thermal Analysis

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I have developed a 2D transient thermal analysis FE code which doesn't
seem to be correct. I am sure there is a conceptual problem in the
formulation itself which I have not been able to understand and could
not find concrete information in books.

I am putting up the question in a very simplified form so as to address the correctness of the formulation.

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