In order to apply Finite Element Method to solve PDEs, we need to convert the PDEs into an equivalent “weak form”.
Example: Axially loaded Elastic Bar
with boundary conditions:
at , traction/force per unit area , is prescribed:
at , displacement is prescribed:
Weak Form: is an arbitrary weight function, multiplied by the governing equation and boundary conditions and integrated over the domain.
Boundary condition (traction force):
Boundary condition (end-point displacement):
Therefore, when we integrate the equation over :
Substitute in :
- From the governing equation, the first term becomes:
- From the relation , the middle term (first term on the right hand side) becomes:
And the equation becomes:
Consider the boundary conditions:
- Boundary condition (traction force):
- Boundary condition (end-point displacement):
Now, let’s say hello to the slimmed and toned equation:
where we only need to recover such that it satisfies .
This form has a weaker continuity requirements than the strong form, hence the name “weak form”.