The Surface Finite Element Method

4.2.1 Approximation of Geometry and Triangulations

As stated within Section 2.1 under our assumptions for the asymptotic analysis presented in the previous chapter; we restrict to smooth, connected, evolving com- pact hypersurfaces (t) _⇢ Rn+1, where n = 1,2 is the dimension of the surface and t 2 [0, T], with T > 0, and such that @ (t) = ;. We again assume that it is orientable and denote by⌫(_·, t) : (t)_!R3_{,t2[0, T], a spatial unit normal vector}

surface finite element (ESFEM) can, in general, be applied to higher dimensions. We approximate the evolving surface, { (t)}t, by an evolving polyhedral

surface, _{ h(t)}t. For the purpose of generating such an approximation we use a

triangulation of the given hypersurface (t) at timet= 0. That is we approximate (0) by a polyhedral surface h(0) which is formed by taking a given number,

Nh<1, of nodes on the surface (0) and then using n-simplicies with these nodes

as vertices to interpolate between them. The idea is visually represented in Figure 4.1.

Figure 4.1: Approximation of a torus by a polygonal surface induced by a triangu- lation.

The discretely triangulated surface h(0) is thus the union of a finite number

of non-degenerate closed n-simplicies. We denote this set of simplicies Th(0). We

restrict to triangulations such that there is a bijection between (0) and h(0), so

that we have a simple covering. That is for T1, T2 2 Th(0) either T1\T2 = ; or

T1 \T2 is an (n k)-dimensional side simplex (k 2 {1, ..., n}) common to both

elements.

To obtain a triangulated surface for a given time t > 0, we advect the nodes used in the initial triangulation h(0) in the following manner. We denote by

{Xj(t)}j, j = 1, ..., Nh the set of vertices associated with the triangulation at each

time, then the velocity of the nodes is given by ˙Xj(t). It is natural to evolve the

vertices withv as this keeps the nodes on the evolving surface (t) this requires at all times that

˙

Xj(t) =v(Xj(t), t), Xj(0) =Xj0 8j= 1, ..., Nh. (4.1)

Figure 4.2: Example of mesh degeneration due to tangential velocity. Initial uniform triangulation on the left and at later time on right after advecting nodes by the material velocity which is given by v(X(✓, )) = sin(✓)X✓, where X(✓, ) is the

standard parameterisation of a sphere.

although advection by the material velocity is natural, it is possible to see mesh degeneration. An example of this can be seen in Figure 4.2. To overcome this issue one may only require (4.1) to hold in the normal direction and leave the tangential motion arbitrary. Note that we will still have the property thatXj(t)2 (t) 8j =

1, ..., Nh andt2[0, T]. For our procedure we assume there is some intrinsic material

tangential velocity, v⌧ := P v, given as the tangential projection of the material

velocity. We then assume in addition an arbitrary velocity,a⌧, satisfyinga⌧·⌫ = 0

such that nodes are transported according to

PX˙j(t) =a⌧(Xj(t), t) +v⌧(Xj(t), t), Xj(0) =Xj0 8j = 1, ..., Nh. (4.2)

For a surface PDE this leads naturally to an Arbitrary Lagrangian Eulerian (ALE) method, or in our case ALE-ESFEM. The ALE-ESFEM has been introduced in Elliott and Styles [2012] and more rigorously studied in Elliott and Venkataraman [2015]. In most cases we will usea⌧ = 0 and evolve the nodes purely by the surface

tangential velocity. However in some instances the ability to use an arbitrary velocity will enable us to ensure mesh regularity throughout a simulation and thus we will be able to avoid any complications due to re-meshing. In particular in the moving sphere example (Section 4.4.3), since the nodes are all transported towards the south pole, settinga⌧ = v⌧ avoids nodes bunching near the south pole and ensures an

diameter of an n-simplex’s face. In addition to the above we assume that the evolution of the mesh is such that the ratio of the maximal simplex diameter and minimal simplex in-ball radius is uniformly bounded independently of bothhandt. Thus far the procedure for approximating the geometry does not change the topology of our mesh. Since we are interested in producing phase field simulations, the thin moving interfacial layers have to be resolved, this motivates the need for adaptive refinement and coarsening. In this work we only consider ’h’-refinements, as opposed to ’p’-refinements. In ’p’-refinements the polynomial degree of the simplex faces is increased, see Heine [2004], and in ’h’-refinements the maximal diameter is reduced. If we wish to refine the triangulation then we follow the practical point of view by introducing new nodal points on the current triangulation and project these points back onto the smooth surface. An example of the process is seen in Figure 4.3 and the details can be found in Dziuk and Elliott [2007] for a general refinement procedure and Demlow and Dziuk [2007] for adaptive refinement.

Figure 4.3: Example ofh-refinement whereh is reduced due to the introduction of the point ˜X3 which is then projected onto the surface (t) as the point X3. The

red line represents h(t) for one particularh and the green line represents h(t) for

a smallerh.

4.2.2 Finite Element Spaces

Given an appropriate mesh generated by a triangulation, Th(t), we define the fol-

lowing isoparametric finite element spaces at each time t, see Brenner and Scott [2007]:

Sh(t) =

⇢

⌘(_·, t)₂C0( h(t))⌘(·, t)|T(t)2P8T(t)2Th(t) (4.3)

where_Pis the polynomials of degree 1. These finite element spaces are isoparametric in that the polynomial degree of the faces used in the geometry approximation is

the same as the polynomial degree used for the approximating functions. We denote by⌘1(t), ...,⌘Nh(t) the nodal basis of Sh(t), which is characterised by the identity:

⌘i(t, Xj(t)) = ij. (4.4)