Boundary Conditions (BCs) vs. Displacement BCs in FEM

This post aims to address the question that arises when one cannot distinguish between boundary conditions (BCs) and displacement BCs in the flowchart of FEM process.

Since FEM is all about solving the FE equation in matrix form, we approach this question using the classic FE equation of a linear elastic problem, KU = f. Let us assume it expands to look like Figure 1:

Figure 1: FE Equation of a linear elastic problem, KU = f

We can now interpret the difference between BCs and displacement BCs from the physical and mathematical perspectives:

1. Physically

• BCs are the applied displacement load, point load, pressure load, initial strains etc.
• Displacement BCs are fixed ends, rollers, fix x and y degree of freedom etc.

2. Mathematically we can think of

• BCs as necessary for assembling the global force (f) matrix (i.e. f31 = Fx, f32 = Fy)
• Displacement BCs as a way to remove known degrees of freedom (i.e. U21 = U22 = U31 = U32 = 0) from the system of equations (Figure 2).

Figure 2: Applying displacement BCs to the FE equation. This is equivalent to removing known degrees of freedom from the system of equations

By doing so, the stiffness matrix is no longer singular (2 equations, 2 unknowns) and hence can now be inverted to solve for U (Figure 3).

Figure 3: The FE equation can now be inverted

One last point to make. Although the BCs and displacement BCs have a subtle mathematical difference in FEM, during the practical applications of FEM (i.e. FEA), FEA programs usually group both together under the "Boundary Conditions" section and allow user to define them as one step through graphical interface.

Also read
The role of boundary conditions in FEM