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Numerical Studies of Wave Propagation in Polycrystalline Shape ...

Numerical Studies of Wave Propagation in Polycrystalline Shape ...

10010000Stress,

10010000Stress, MPa-100-200-300Stress, MPa-100-200-300-400-400-5000.00 0.05 0.10 0.15Axial position (meters)(a)-5000.00 0.05 0.10 0.15Axial position (meters)(b)Figure 3. Stress profile at 30µs for a fixed mesh with 2000 elements (a) and an adaptive mesh (b). Both are done for atime step τ = 0.1µs. Mesh nodes in the adaptive mesh (b) are marked with black squares and the thin line at the topshows the density of elements.Table 2. L 1 error in the displacement distribution at T = 100µs.N=2000 N=4000 N=8000 N=16000 N=32000t=0.2 µs 1.98 ∗ 10 −3 1.99 ∗ 10 −3 1.99 ∗ 10 −3t=0.1 µs 1.31 ∗ 10 −2 9.95 ∗ 10 −4 9.97 ∗ 10 −4t=0.05 µs 5.63 ∗ 10 −2 1.26 ∗ 10 −2 4.98 ∗ 10 −4 4.99 ∗ 10 −4t=0.025 µs 8.73 ∗ 10 −2 8.74 ∗ 10 −3 1.24 ∗ 10 −2 2.49 ∗ 10 −4 2.497 ∗ 10 −4Table 3. L 1 error in the stress distribution along the rod at T = 100µs.N=2000 N=4000 N=8000 N=16000 N=32000t=0.2 µs 2.68 ∗ 10 −2 2.75 ∗ 10 −2 2.72 ∗ 10 −2t=0.1 µs 3.42 ∗ 10 −2 1.87 ∗ 10 −2 1.91 ∗ 10 −2t=0.05 µs 8.77 ∗ 10 −2 2.95 ∗ 10 −2 1.32 ∗ 10 −2 1.34 ∗ 10 −2t=0.025 µs 1.26 ∗ 10 −1 5.61 ∗ 10 −2 2.58 ∗ 10 −2 9.41 ∗ 10 −1 9.47 ∗ 10 −1discretization scheme. The smearing effect can be eliminated by decreasing the time step for both the standardand the adaptive version of the FEM. The convergence of the standard FEM solver is shown on Tables 2 and3. Due to the presence of discontinuities in the stress the relative error is measured in the L 1 norm. As seenfrom Table 2 the convergence rate for the displacements are of optimal order 1 for the linear elements used. Apiecewise constant function on the other side has regularity 1 2− δ. That is, for every positive constant δ > 0 thepiecewise constant functions belong to the space H 1 2 −δ0 (0, 1). Therefore one cannot expect optimal convergencein the stresses and as seen from Table 3 the order of convergence is 0.5.The adaptive FEM approach yields the same solutions as the standard FEM but at a much lower computationalcost. For the same time steps for which the convergence of the fixed FEM was tested the adaptive meshingresults in similar accuracy as seen from Table 4. A comparison in the computational performance of the fixedand adaptive FE methods is given in Table 5. The time step used is τ = 0.01µs and the number of elements forthe fixed FEM is 16000. The adaptive solution is chosen so that it has comparable accuracy with the one forthe fixed mesh solution. A comparison of the execution times for the fixed and adaptive methods shows that theadaptive procedure delivers an order of magnitude improvement in performance.

Table 4. L 1 error in the displacement distribution at T = 100µs.τ 0.2 µs 0.1 µs 0.05 µs 0.025 µserror, σ 2.74 ∗ 10 −2 1.93 ∗ 10 −2 1.35 ∗ 10 −2 9.54 ∗ 10 −3error, u 2.68 ∗ 10 −3 1.64 ∗ 10 −3 8.02 ∗ 10 −4 4.34 ∗ 10 −4Table 5. Execution times for fixed and adaptive meshesTime Fixed Mesh Adaptive MeshElements Time (min) Elements Time (min)10 µs 16000 56 161 1:1220 µs 16000 113 199 2:3740 µs 16000 226 256 6:1080 µs 16000 451 301 153.3. Square pulse impact loading problemA more realistic initial-boundary value problem is one for which, instead of step loading, the boundary conditionis a square pulse, that is⎧⎨ 0 for t ≤ 0σ 0 (t) = σ 0 for 0 < t < t pulse(10)⎩0 for t ≥ t pulsewhere t pulse is the duration of the pulse. Due to the complicated constitutive response and boundary conditionsthere is no analytical solution to be compared with. Moreover, there are unresolved questions regarding theuniqueness of the weak solution for times t > t pulse when unloading takes place.The stress level used for the numerical simulation is σ 0 = 800MP a and the initial temperature is T R =320 ◦ K > A of . The same material data as the one from the previous section is used with the exception thatthe value for the difference in the specific entropies is changed to ρ∆s 0 = 3.5 × 10 5 . The stress level is chosenso that the full adiabatic hysteresis loop can be realized. The pulse length is t pulse = 10µs and the time step ist = 0.001µs. The simulation time is 100µs.The evolution of the stress and temperature in the rod up to 90µs is shown in Figures 4 and 6. The two-shocksolution for the stress is clearly visible at the end of the pulse load at t = 10µs (Figure 4). The temperatureprofile (Figure 6) also has two shocks. The maximum temperature T 0 = 378.8 ◦ K is achieved in the region of fullphase transformation. The jump in the elastic shock is T el − T R = 0.66 ◦ K and for this reason it is not clearlyvisible in the figure.100-100Stress, MPa-300-500-700-90010 microseconds30 microseconds60 microseconds90 microseconds0.00 0.10 0.20 0.30 0.40Axial position (meters)Figure 4. Stress profile at different instances of time for a square pulse in adiabatic loading

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