ElementHex2

[gdmcbain]

The next element to be added under #23 is probably the 27-node quadratic three-dimensional ElementHex2.

Quadratic hexahedral elements were recommended over ElementHex1 by Benzley et al (1995), as consulted in #75.

The use of quadratic displacement formulated finite elements significantly improve the
performance of the tetrahedral as well as the hexahedral elements.

ElementHex2 will also be required for Taylor–Hood element for Stokes or Navier–Stokes problems on a MeshHex.

[gdmcbain]

I guess the definition begins like:

class ElementHex2(ElementH1):
    nodal_dofs = 1
    facet_dofs = 1
    interior_dofs = 1
    edge_dofs = 1
    dim = 3
    maxdeg = 6
    dofnames = ['u', 'u', 'u', 'u']

but what is the rule for how the dofs are ordered within each group (8 nodal, 6 facets, 1 interior, 12 edges)? I assume the nodal dofs are the same as ElementHex1 but how are the facets and edges numbered? (And is the order indeed nodal, faces, interior, edges?)

[kinnala]

Tricky question because it depends on how the mesh connectivity is defined in MeshHex._build_mappings(). E.g. by looking at this definition of edges:

        self.edges = np.sort(np.hstack((
            self.t[[0, 1]],
            self.t[[0, 2]],
            self.t[[0, 3]],
            self.t[[1, 4]],
            self.t[[1, 5]],
            self.t[[2, 4]],
            self.t[[2, 6]],
            self.t[[3, 5]],
            self.t[[3, 6]],
            self.t[[4, 7]],
            self.t[[5, 7]],
            self.t[[6, 7]],
        )), axis=0)

Basically you would need to make sure that the first edge basis function would be the one which gets value 1 between the nodes numbered 0 and 1 (rows 0 and 1 in mesh.p[:, mesh.t]), the second basis function gets value 1 between the nodes numbered 0 and 2, etc.

I hope it makes sense. If it doesn't, ask more info and I'll provide.

[kinnala]

I'm now looking into implementing the p-version of the finite element method in scikit-fem, starting with ElementQuadP for arbitrary order. I'll probably change the reference element used for quads and hexes (from (-1, 1)^d to (0, 1)^d) because the book I'm using as a reference uses those and it seems that the notation for the basis functions will be slightly more concise.

Just a remark in case you already started working on this.

[gdmcbain]

Great! As in potentially different p in each cell? Very exciting.

Which book?

+1 for relocating the tensorial elements to the first ray/quadrant/octant. I had briefly lamented when working in one dimension that the canonical line segment element was not both the one-dimensional simplex and the one-dimensional factor from which the quadrilateral and hexahedron are formed as products.

[kinnala]

I'm now using Leszek Demkowicz' books since I got a recommendation from a friend:

https://www.amazon.com/Computing-Hp-Adaptive-Finite-Elements-Vol/dp/1584886714

I'm also simultaneously studying theory from Schwab's book:

https://www.amazon.com/hp-Finite-Element-Methods-Applications/dp/0198503903

[kinnala]

I'm not yet sure how hp-adaptivity would work in scikit-fem, but I'll know better after studying these books. I'm going to start with integrated Legendre polynomial -based basis functions for quads and triangles for arbitrary p.