Procedures in deriving element equations (General field problems) Governing equation Identify boundary conditions Discretize the solution domain, Ω For variational approach Define the functional, I(Ø) Minimize the functional. Or For method of weighted residual (MWR) Integrate by parts (Reduce order of integral and introduce BC) .

To derive an element equation for 2-D heat transfer problem. Example To derive an element equation for 2-D heat transfer problem. Governing differential equation (steady state) Boundary conditions Specified T=T(x,y) on s1 *(T-T) convective heat loss q heat loss on boundary due to conduction Discretize the Ωdomain into M number of elements. For each element: For variational approach The functional has only 1st order derivatives, thus T is continuous across the interface and T is constant within element. ň s1 s2 x y Ω

Minimize the functional Terms to be evaluated: .

yield: By method of weight residual (Galerkin’s Method) Integrate by parts Let u=Ni Results: Apply B.C : = -q-h(T(e) –Tα) on s2 yield: