Discrete Time
Discretization in time links to our degrees of freedom (DOF) . In the continuous setting, . Now, let us divide time into small intervals, , as discussed in the first chapter. Using the finite difference formula, we can conveniently approximate in terms of .
For example, with backward Euler: which gives us: where . Applying this relation at the sample points into Equation (17.1.3), we obtain:
Then, by applying mass lumping and zero traction boundary conditions, i.e., , we finally see that Equation (17.2.1) is the -th row of the discrete form of backward Euler time integration in the first lecture: where the elasticity force is obtained by evaluating: which will be discussed in the next chapter.