User:Eml5526.s11.team04.premchand/HW4

= Problem 4.2 - Weak form for heat conduction in one dimension =

Problem Statement
Obtain the weak form for the equations of heat conduction with the boundary conditions $$T(0)=100$$ and $$\bar{q}=hT$$ .The condition on the right is a convection condition.

Solution


w= weighted function

integrating by part yields

The weak form is

Find $$T(x)\epsilon \upsilon $$ such that

= Problem 4.6 - Strong and weak form of elastic bar problem with variable distributed spring =

Problem Statement
Consider an elastic bar with a variable distributed spring p(x) along its length .The distributed spring imposes an axial force on the bar in proportion to the displacement. Consider a bar of length l,cross-sectional area A(x),young's modulus E(x)with body force b(x) and boundary conditions as follows

a. Construct the strong form

b. Construct the weak form

Solution
First consider an element of this elastic bar.



From the force equilibrium i.e D'Alembert's principle,

taking the limit as $$\Delta x\rightarrow 0$$

for the governing ODE

Boundary conditions are

So the strong form is

The weak form derivation:

integrating by parts for the first term yields

which can be expanded as follows

Let w (0)=0 be the essential boundary condition.

The the weak form is as follows

Find u(x) smooth enough and satisfy u(0)=0,such that