How to apply Dirichlet boundary conditions to time-dependent PDEs?

Question

Assume the time-dependent linear elasticity equation. Using a finite element discretization we obtain
$$Mddot{u}=Ku+F_text{ext}$$
where $M$ is the mass matrix,$K$ is the stiffness matrix, and $F_text{ext}$ is the external load vector. Further using a time discretization scheme(e.g. Forward Euler), we obtain
$$
Mdot{u}_{n+1}=dt(Ku_n+F_text{ext})+Mdot{u}_n
tag{1}
label{1}
$$
for $N$ time steps. How can I apply the Dirichlet BC in $eqref{1}$? Consider the case of a 2D rectangle with the bottom edge fixed and a distributed load on the left edge.

knl · Answer

You simply make sure that the initial condition satisfies the boundary conditions and that you don't add anything to the elements of $u_{n+1}$ corresponding to the boundary. In other words, you drop the rows and columns of $M$ that correspond to the boundary nodes and solve only in the interior nodes.
Let me add that inhomogeneous Dirichlet boundary conditions are more tricky, especially if they are time-dependent.

How to apply Dirichlet boundary conditions to time-dependent PDEs?

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