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

**Strong Form**:

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.

Governing equation:

Boundary condition (traction force):

Boundary condition (end-point displacement):

Reminder:

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”.

Sources:

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