hp-convergence study: smooth star interface

The next example is another more difficult problem, referred to in Section 4.4.1.4 as a smooth

star interface reinitialisation problem. That is the level set function is initialised as the L2

projection of ˜ φ0(x) = x2+ y2−1 − 0.4 sin6 arctany x  , (6.4)

in the domain Ω = (−2, 2)2_{. This is similar to the smooth star interface problem from Section}

4.4.1.4, however, this time chosen such that the level set interface has a slightly larger curvature. The shape of such an interface and its position relative to the domain boundaries can be seen in Figure 6.6. The signed distance function to this smooth star interface can be computed

0 50 100 150 200 250 300 ndof1/2 10-6 10-5 10-4 10-3 10-2 10-1 100 Error L2 Error DG Error Error Estimate

Figure 6.4: Error against the square root of the number of degrees of freedom (ndof), for the solution of the square interface reinitialisation 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.

numerically by solving the minimisation problem ˜ φ(x, ϑ) = min ϑ ψ = min ϑ q

(x −p1 − 0.4 sin(6ϑ) cos(ϑ))2_{+ (y −}p1 − 0.4 sin(6ϑ) sin(ϑ))2

 .

(6.5)

That is finding the roots to the equation, dψ dϑ = 2 sin (ϑ)p1 − 0.4 sin (6ϑ) + (1.2 cos (ϑ) cos (6ϑ)) p1 − 0.4 sin (6ϑ) !  x −p1 − 0.4 sin (6ϑ) cos (ϑ) 

+ 2y −sin (ϑ)p1 − 0.4 sin (6ϑ) −p1 − 0.4 sin (6ϑ) cos (ϑ) + (1.2 sin (ϑ) cos (6ϑ))

p1 − 0.4 sin (6ϑ) !

= 0, (6.6)

using a bisection method with a tolerance of |ϑ| < 10−15. Using the numerical solution above

to find the value of the solution at a point x, the gradient of the signed distance function to the smooth star interface can be computed using a first-order finite difference method as follows

       ∂ ˜φ(x, y) ∂x ∂ ˜φ(x, y) ∂y        =        ˜ φ(x, y) − ˜φ(x + ∆x, y) ∆x ˜ φ(x, y) − ˜φ(x, y + ∆x) ∆x        , (6.7)

with spatial step size, ∆x = 10−12.

(a) Entire narrow banded adaptive mesh over the domain.

(b) Mesh near to one of the singular regions.

Figure 6.5: Final computed mesh configuration for the square interface reinitialisation problem on an hp-adaptively refined mesh where the colour of the element denotes the polynomial order of that element. The thick black line denotes the computed interface position.

Figure 6.6: Relative position of the initial interface and the domain boundaries for the smooth

star interface reinitialisation problem, defined by Equation (6.4), on the domain Ω = (−2, 2)2_.

tion is initialised on a Cartesian mesh with square elements of size, h = 0.4, of uniform polyno- mial order, p = 2, which is then, after each reinitialisation, adaptively refined and the initial level set function reprojected onto the new mesh, which continues in a loop until one of the stopping criteria is satisfied. The variation in error with mesh density during this study for the smooth star interface is presented in Figure 6.7. As with the previous two examples it can be seen that there is roughly exponential convergence, with initially a slower rate of convergence where the singularities remain inside the narrow band and then a faster rate of convergence once the singularities are narrow banded out. Also it can again be seen that the error estimate roughly follows the pattern of the DG norm, bounding it from below, and providing an upper bound

on the L2 _{norm. Towards the end of the hp-convergence study, however, it can be seen that}

the rate of convergence slows again and almost begins to stagnate. One reason for this is that the error tolerance is not satisfied by the time those elements requiring refinement have reached the maximum allowed polynomial order of p = 8. As these elements are still flagged for refine- ment, the enforceRefinementLimits function switches the flag from p to h, and h-refinement is known to be less efficient than p for smooth problems, and therefore this ultimately limits the rate of convergence. This can be seen to be the case by noticing where the error tolerance shrinks as a result of shrinking narrow band (see the red line on Figure 6.7), which occurs when the elements near to the interface become small as a result of h-refinement which coincides with the stagnating error.

This issue, however, is perhaps more insidious than simply enforcing an upper limit on the amount of allowed p-refinement. For example, an important question one could ask at this point, is ‘why is the method unable to satisfy the error tolerance?’; especially given that the problem is smooth, the problem is discretised using what would be considered by many in the finite element community as very high-order polynomial bases, and the Elliptic Reinitialisation method solving this type of problem on Cartesian meshes of fixed polynomial order were shown

0 50 100 150 200 250 300 350 400 450 500 ndof1/2 10-8 10-6 10-4 10-2 100 102 Error L2 Error DG Error Error Estimate

Figure 6.7: Error against the square root of the number of degrees of freedom (ndof), for the solution of the smooth star interface reinitialisation problem, initialised as in Equation (6.4), 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.

in Section 4.4.1.4 to converge optimally. One reason for this is that at each refinement step, when the reinitialisation routine is called, the required number of iterations for the reinitialisation to be stopped as a result of satisfying the convergence criterion, would, if allowed to continue until that point, dwarf the chosen maximum number of iterations. In other words the errors presented in Figure 6.7 are always after 100 iterations, as this is the maximum allowed number of iterations, which on the increasingly well refined meshes seems to not be large enough for the new mesh to outperform the previous mesh. This was not observed in similar earlier examples, as there was no limit on the maximum number of iterations.

Investigating the relative change in the solution, as well as the error in the solution, for this (or any) example problem, using the Elliptic Reinitialisation method (using Picard’s method as the iterative solver), one can see that both of these quantities decrease monotonically when plotted against the number of computed iterations. Furthermore, the numerical examples in Section 4.4.1, present evidence suggesting that when the Elliptic Reinitialisation method is allowed to reach this convergence criterion, the method will converge optimally in h for fixed p. In other words, without consulting practical limitations on time and memory, the Elliptic Reinitialisation method is capable of returning a suitable signed distance function. For this specific example however, the convergence rate of the Picard iterative method is so slow that it almost stagnant, and thus placing a limit on the number of iterations, for reasons of practicality, seems to necessarily mean that the convergence of the refinement will similarly be limited.

It was attempting to overcome this issue which lead investigations into methods such as Anderson acceleration (see Section 5.3) and even Newton schemes (results not presented) for the Elliptic Reinitialisation method, as well as the various temporal discretisations of the Parabolic Reinitialisation method. What was found was that attempts at increasing the rate of convergence

Figure 6.8: Final computed mesh configuration for the smooth star interface reinitialisation problem, initialised as in Equation (6.4), on an hp-adaptively refined mesh where the colour of the element denotes the polynomial order of that element. The thick black line denotes the computed interface position.

would invariably lead to instability. One of the advantages of Anderson acceleration in this respect, and why it is used here, is that when instabilities become apparent it can simply revert to Picard, which has demonstrated stability when solving these problems. It seems ultimately then, that in order to overcome this issue one must either; allow more time for reinitialisation, that is choosing a larger maximum allowed number of iterations, and/or use more powerful hardware and therefore allow for higher levels of refinement and attempt to save lost time using by parallelising the computation. Discussions relating to this issue follow in Sections 6.4.4 and 6.6.4, however, beyond that stated here a suitable solution was unfortunately not found during this research period.

The final mesh configuration for the adaptively refined smooth star reinitialisation problem is shown in Figure 6.8. As a result of the stagnation in the error, the solution never reaches the desired tolerance on the signed distance error. Instead the mesh is configured such that it reaches the upper limit on first p and then eventually h, and is therefore deemed to be optimally refined given these limits. This explains why the mesh looks how it does with the majority of the elements being of order, p = 8, and forming a much narrower narrow band near to the interface than either of the previous examples.