Adaptive time stepping in elastoplasticity

Figure 1.  Numerical approximation of the solution for Example 5.1 at times $ t = 8, 10, 13, 15 $. The grey shading corresponds to the norm of plastic strain $ p $. For visibility purposes, we only applied 4 red refinements as opposed to our actual tests, where 6 red refinements were used. The radially symmetric nature of the problem allows us to save computation time by only computing the solution for a quarter of the ring

Figure 2.  Results of the tests using Example 5.1. Logarithmic plot of the error term $ \eta_{{\rm tot}}^2 $ as a function of $ K^{-1} \sim \tau $ for the method with constant time steps (star) and the adaptive algorithm with $ \theta = 1 $ (cross). The remaining graphs (circle, diamond and square) consider the total number of computed steps $ K_{{\rm tot}} $ instead of the number of time steps $ K $ and show the results for different values of $ \theta $ (see Algorithm 4.1). In the case of constant time steps, we have $ K = K_{{\rm tot}} $. The triangle in the bottom right references the slope of a quadratic function. An error reduction by the adaptive method is observed. The difference is less significant when considering $ K_{{\rm tot}} $, but increases for smaller values of $ \theta $

Figure 3.  Illustration of the geometry used in Example 5.2

Figure 4.  Numerical approximation of the solution for Example 5.2 at time $ T = 10 $. The original triangulation of three halved squares was red refined 6 times and the grey shading corresponds to the norm of plastic strain $ p $. The increasing force causes a curved fissure to develop across the domain

Figure 5.  Results of the tests for Examplek 5.2 using notation from Figure 2. The adaptive time stepping leads to an error reduction

Figure 6.  Results of the tests for Example 5.3 using notation from Figure 2. A stronger improvement of adaptivity is observed than in Examples 5.1 and 5.2

Figure 7.  Development of error function $ \varepsilon_{\tau} $ in time. The algorithm with constant time steps was used on Examples 5.1 (top-left), 5.2 (top-right) and 5.3 (bottom-left). The plot on the bottom-right shows the adaptive method for Example 5.1 and with $ {\varepsilon_{{\rm max}}} = 3.16\cdot 10^{-5} $. The graphs start at $ t > 0 $, because the error in the left out parts is almost zero. Similar results are observed for Examples 5.1 and 5.2, where after a certain amount of time, $ \varepsilon_{\tau} $ starts to monotonically increase. In Example 5.3, a single peak occurs during the step containing the time $ t_p \approx 138.32 $, when plastic strain first emerges