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Assume the time-dependent linear elasticity equation. Using a finite element discretization we obtain

$$M\ddot{u}=Ku+F_\text{ext}$$

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

$$ M\dot{u}_{n+1}=dt(Ku_n+F_\text{ext})+M\dot{u}_n \tag{1} \label{1} $$

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.

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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.

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