Discrete Time

Discretization in time links $A$ to our degrees of freedom (DOF) $x$ . In the continuous setting, $A (X, t) = \frac{\partial ^{2} x}{\partial t ^{2}} (X, t)$ . Now, let us divide time into small intervals, $t^{0}, t^{2}, \dots, t^{n}, \dots$ , as discussed in the first chapter. Using the finite difference formula, we can conveniently approximate $A$ in terms of $x$ .

For example, with backward Euler: $A^{n} (X) V^{n} (X) = \frac{V ^{n} ( X ) - V ^{n - 1} ( X )}{t ^{n} - t ^{n - 1}}, = \frac{x ^{n} ( X ) - x ^{n - 1} ( X )}{t ^{n} - t ^{n - 1}},$ which gives us: $A^{n} (X) = \frac{x ^{n} ( X ) - ( x ^{n - 1} ( X ) + h V ^{n - 1} ( X ))}{Δ t ^{2}},$ where $Δ t = t^{n} - t^{n - 1}$ . Applying this relation at the sample points into Equation (17.1.3), we obtain: $M_{\overset{a}{^} b} \frac{x _{b ∣ \hat{i}}^{n} - ( x _{b ∣ \hat{i}}^{n - 1} + h V _{b ∣ \hat{i}}^{n - 1} )}{Δ t ^{2}} = \int_{\partial Ω^{0}} N_{\overset{a}{^}} (X) T_{\hat{i}} (X, t^{n}) d s (X) - \int_{Ω^{0}} N_{\overset{a}{^}, j} (X) P_{\hat{i} j} (X, t^{n}) d X . (17.2.1)$

Then, by applying mass lumping and zero traction boundary conditions, i.e., $T (X, t) = 0$ , we finally see that Equation (17.2.1) is the $(\overset{a}{^} d + \hat{i})$ -th row of the discrete form of backward Euler time integration in the first lecture: $M (x^{n + 1} - (x^{n} + Δ t v^{n})) - Δ t^{2} f (x^{n + 1}) = 0,$ where the elasticity force $f (x)$ is obtained by evaluating: $- \int_{Ω^{0}} N_{\overset{a}{^}, j} (X) P_{\hat{i} j} (X, t) d X,$ which will be discussed in the next chapter.