Solving Elasticity Problems with Finite Element Method

[Lojzek]

I am trying to make a program that solves elasticity problems with finite element method and
I don't understand how to bring in boundary conditions.

Constant displacement boundary conditions seem simple: replace variables that represent the displacements at surface nodes with the prescribed constants and drop corresponding Euler-Lagrange equations for this variables.

But what if boundary conditions define pressures on the boundaries instead of displacements?
And how do we deal with the problem with both types of boundary conditions?

 

[Mapes]

How about replacing pressures on boundaries with equivalent forces on nodes? Then when a node moves, a work term is generated. The nodes will collectively displace to minimize the sum of work terms and strain energy in the body.

(I haven't tried this personally, but it may give you some ideas.)

 

[Lojzek]

I think I got the solution now. The unknown pressures on element surfaces should be left as unknown variables in the Lagrange equations together with unknown displacements and a sistem of linear equations can be obtained, where the unknown vector contains both unknown displacements and pressures.
Replacing pressures on boundaries with equivivalent forces on nodes would probably work in a similar way. Then unknown displacements and unknown forces would be determined by the linear system.