Evolution

Once the initialisation has been completed the evolution can begin. The evolution for the GSG circular interface problem will proceed, driven by an advection velocity defined in Equation

(5.20), which can be described as growth at a constant rate in the period te= (0, 0.15], shrinking

at the same constant rate in the period, te= (0.15, 0.45], followed by a period of constant growth

again in the period, te = (0.45, 0.6), such that the interface should arrive finally at its initial

position at time, te = T = 0.6. The function describing the analytical solution over the domain

for all time is defined in Equation (5.21). If at any point during the evolution the signed

distance error grows such that ESD > 10−6×area, this will trigger the refine-reinitialisation

loop (Algorithm 3), which will continue in a while loop until either the tolerance on the error is satisfied, or the mesh becomes maximally refined.

Figure 6.19 shows how the error varies over pseudotime, during the evolution for the GSG circular interface problem on an hp-adaptively refined mesh. As there is no h-refinement away from the origin, the mesh is fairly coarse throughout the evolution. This means that the area of the narrow band remains constant for the majority of the evolution. The two instances where

this is not true are when the interface expands to its largest around te = 0.15, and when the

interface shrinks to its smallest around te= 0.45, whereby the narrow band tracks the evolving

0 0.1 0.2 0.3 0.4 0.5 0.6 Time (s) 10-9 10-8 10-7 10-6 10-5 10-4 Error L2 Error DG Error Error Estimate

Figure 6.19: Error over pseudotime for the GSG circular interface problem on an hp-adaptively refined mesh. The red line denotes the error tolerance defining one of the stopping criteria, which can change as the area of narrow band changes.

periods, the size of the narrow band grows and with it the error; this is most notably apparent

in the signed distance error, but can also be seen in the L2 _{and DG norms too. Once the}

interface begins to shrink again, the area of the interface returns to its original size, and those elements which moved from outside to inside the narrow band once again leave the narrow band and with it the additional error associated with them. During the second change in the shape of the narrow band the area of the narrow band decreases but the elements which move from outside to inside the narrow band in the same instant, are closer to the singularity and have a much higher curvature to capture and thus the error increases once again. Once the interface grows again, these elements eventually leave the narrow band and the additional error associated with them also leaves. As would be expected the cumulative error at each time step increases

over time which is reflected in the increasing error in the L2 _{and DG norms over time, with the}

oscillations explained by the changing shape of the narrow band as noted above.

The red line in Figure 6.19, demonstrates the criterion defining whether refinement is nec- essary. In this example the signed distance error never approaches this line and as such for all time, no further refinement needs to occur. This is to be expected as this is such a simple problem; the level set function evolves as a rigid body in space, and thus the signed distance error is constant over time varying only due to the change in the size of the narrow band.

Comparing the results for the hp-adaptive mesh with the fixed mesh for the same problem in Section 5.4.2.1, it can be observed that the results are similar both in terms of how the error varies over time and the absolute value of the errors. The main difference is that as a coarser mesh of higher-order elements are used here, the evolution in the shape of the narrow band is much less severe, which results in less variation in the error over time. Also using the hp- adaptive mesh for the problem is more efficient both in terms of the number of degrees of freedom required, that is during the evolution, an average of 12769 degrees of freedom are required for the fixed mesh (where the average is computed as the total number of degrees of freedom in

t=0.00 5 6 7 8 Polynomial Order

(a) Mesh and interface position at te= 0.00.

t=0.15 4 5 6 7 8 Polynomial Order

(b) Mesh and interface position at te= 0.15.

t=0.30 5 6 7 8 Polynomial Order

(c) Mesh and interface position at te= 0.30.

t=0.45 5 6 7 8 Polynomial Order

(d) Mesh and interface position at te= 0.45.

t=0.60 5 6 7 8 Polynomial Order

(e) Mesh and interface position at te= 0.60.

Figure 6.20: Configuration of mesh and interface position over time for the GSG circular interface evolution problem, on an hp-adaptively refined mesh. The thick black line denotes the computed interface position.

the narrow band at each time step divided by the total number of time steps), whereas just an average of 5511 are required for the adaptive mesh, as well as the number of time steps required, which is 219 for the fixed mesh versus just 89 for the adaptive mesh. The results for the fixed mesh mentioned here is that mesh where h = 0.05 and p = 3.

Figure 6.20 shows some snapshots of the evolving interface, as well as the evolving adaptive mesh over time for the GSG circular interface problem. As mentioned earlier no additional refinements are required for this example problem after the initialisation of the mesh and as such all that is required during the evolution is for some of the elements outside of the narrow band move inside, and vice versa, to maintain the correct narrow band width based on the position of the evolving interface. Although not plotted, the errors are sufficiently small such that there would be no observable difference between the computed and analytical position of the level set interface.