# Time integration for elastodynamics

I'd like to solve the elastodynamics equation for a problem with fast scales from an impact (although no shocks).

$\rho\ddot{\mathbf{u}} - \nabla \cdot \sigma(\mathbf{u}) = \mathbf{f}$

In structural dynamics, this equation is typically solved with the Newmark scheme with Lagrange elements for the spatial domain discretization. The Newmark scheme is implicit, but I believe given the fast scales, it will be cheaper to solve with an explicit scheme. To avoid inverting the mass matrix from the second-order derivative term, I have seen that a common technique is to lump the mass matrix into a diagonal matrix. What's the accuracy of this technique?

Other methods involve using DG for the spatial discretization, which gives a block diagonal matrix that can be easily inverted, could this be cheaper and more accurate than the above approach? What would be the right way to approach this problem? Thanks.

• The Newmark method with $\beta=0$, $\gamma=0.5$, with mass lumping, leads to an explicit scheme. – DanielRch Jul 20 '16 at 18:52
• My question is how accurate the mass lumping is, specially for unstructured meshes. – balborian Jul 20 '16 at 21:18
• The error due to mass lumping is usually smaller than the discretization error. You can lump safely for most usual applications in both standard finite elements and in DG methods. – DanielRch Jul 21 '16 at 19:51
• Could you give a reference about your comment please? – balborian Jul 21 '16 at 22:19
• This is discussed in Hughes TJR, "The Finite Element Method", for example. – DanielRch Jul 23 '16 at 6:36