Finite Element Method for problems in Physics

Linear Elliptic Differential equations in 1D

• 1D Heat Conduction at steady state
• 1D Mass Diffusion at steady state
• 1D Elasticity at steady state.
• 
• Legenda: 1D Heat Conduction

Slide 2

• 
• ug = Specified displacement at x = L.
• t = Specified traction at x = L.
• f = Distributed body force.
• Find \(u(x) : (0,L) ---> R\) given \(u(0)= u_0,u_g,t\) the constitutive relation \(\sigma = E u_,x\) such that \[\frac{d\sigma}{dx}+f=0\].
• with the boundary conditions /( Diff Eq \) , \(u(0)=u_0 \), and either \(u(L)=u_g\) or \(\sigma(L) = t \)

Boundary Conditions

• \(u(0)=U_0 \), \(u(L) = u_g \) - Dirichlet Boundary Conditions - On the primal field
• \(\sigma(L) = t\) - Neumann Boundary Conditions - On the derivative of the primal field.
• For Elasticity : Dirichlet - Displacement
• Neumann - traction
• *( We do not consider neumann at o and L. This would assume that we have a dynamic conditions such as a bar flying.)
• ( We do not have just one answer for this type of problem. (Proof on mooc or notebook.))
• Neumann B.C alone can be specified for the time dependent elasticity problem \(HyperbolicPartialDifferentialEquation\)

The differential Equation

• \[ \frac{d\sigma}{dx} +f(x) = 0 \] \((0,L) \) open interval excluding 0 and L because we have boundary conditions on them.

Constitutive Relation

• \[ \sigma = Eu,_x \] Tell about the constitution of the domain
• \(\sigma\) = Stress
• \( E\) = Young Modulus
• \(u,_x \) = strain -Linearized Elasticity