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.

Temperatures (image/png)

Caption: : 1D Heat Conduction

Diagrama Barra Inicial (binary/octet-stream)

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