User:Egm5526.s11.team4.zou/HW7

Alternative Solution by Fangfang Zhu:
First derive the equilibrium equation for an arbitrary choose domain showns as:



As showns in the above figure,the equilibrium of heat can be written as:

Using the Taylor series expansion we can write:

So the equilibrium equation can be rewritten as:

Dividing dxdy both side yields:

Written in compact form:

Consider the constitutive equation for 2D in heat transfer. The 2D Fourier Law for 2D space can be written as: $$\displaystyle \vec{q}=-[k]\nabla T$$

Written in cartisan matrix form:

For isotropic material we have $$\displaystyle {{k}_{xy}}={{k}_{yx}}=0$$ and $$\displaystyle {{k}_{xx}}={{k}_{yy}}=k$$ So we can write:

or

Thus the strong form for this problem in index notation can be written as:

Following the routine of deriving weak form we can easily get the weak form for this problem:

Let

The weak form can be written as:

Descritization process: Set the approximated solution as a linear combination of shape functions $$\displaystyle {{N}_{i}}(x)$$ as:

And the approximated weighting function is alinear combination of shape functions $$\displaystyle {{N}_{i}}(x)$$ as:

Where $$\displaystyle \tilde{K}={{\int\limits_{D}{\left[ B \right]}}^{T}}\cdot \left[ K \right]\cdot \left[ B \right]$$

Similoarily:

Where $$\displaystyle \tilde{M}=\int\limits_{D}{{{N}^{T}}\cdot \rho c\cdot N}dD$$

where $$\displaystyle {{F}_{i}}=\left[ \int\limits_{-hTN_{i}^{T}ds}+\int\limits_{{{q}_{0}}N_{i}^{T}ds}+\int\limits_{h{{T}_{a}}{{N}_{i}}ds+\int\limits_{D} – f{{N}_{i}}}dD \right]$$

Thus we can write the discrete weak form:

The temprature distribution