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- Transformation,
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Numerical Studies of Wave Propagation in Polycrystalline Shape ...

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 pr**of**ile 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 th**in** l**in**e 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 smear**in**g effect can be elim**in**ated by decreas**in**g 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** discont**in**uities **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 l**in**ear 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 convergence**in** 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 mesh**in**gresults **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 (m**in**) Elements Time (m**in**)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 load**in**g problemA more realistic **in**itial-boundary value problem is one for which, **in**stead **of** step load**in**g, 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 regard**in**g theuniqueness **of** the weak solution for times t > t pulse when unload**in**g takes place.The stress level used for the numerical simulation is σ 0 = 800MP a and the **in**itial 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 temperaturepr**of**ile (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 pr**of**ile at different **in**stances **of** time for a square pulse **in** adiabatic load**in**g

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