The shape of the elements suggests the continuous formation of new elements during steady-state deformation.
The proposed method is very flexible: it is formulated for unstructured grids, regardless the shape of the elements and the number of spatial dimensions.
Lemma 3.4 For the Raviart-Thomas elements, there is a positive constant C, which only depends on A, Ω, and the shape of the elements and their maximal polynomial degree k, such that ∥ q ( u h ) − q h ∥ div + ∥ z ( u h ) − z h ∥ 0 ≤ C ( η 2 + η 3 ), (3.37).
Lemma 3.2 For the Raviart-Thomas elements, there is a positive constant C, which only depends on A, Ω, and the shape of the elements and their maximal polynomial degree k, such that ∥ p ( u h ) − p h ∥ div + ∥ y ( u h ) − y h ∥ 0 ≤ C η 2, (3.15).
Lemma 3.3 For the Raviart-Thomas elements, there is a positive constant C, which only depends on A, Ω, and the shape of the elements and their maximal polynomial degree k, such that C η 2 ≤ ∥ p ( u h ) − p h ∥ div + ∥ y ( u h ) − y h ∥ 0. (3.26).
Then, for the Raviart-Thomas elements, there is a positive constant C, which only depends on A, Ω, and the shape of the elements and their maximal polynomial degree k, such that ∥ p ( u h ) − p h ∥ div 2 + ∥ y ( u h ) − y h ∥ 0 2 + ∥ q ( u h ) − q h ∥ div 2 + ∥ z ( u h ) − z h ∥ 0 2 + ∥ u − u h ∥ U 2 ≤ C ∑ i = 1 3 η i 2, (3.39).
In such a case the shape of the element cannot persist; for, if it did, the contained mass would not be in continuous contact with the containing body; while, if its shape is changed, it will cease to be water, since the distinctive quality is shape.
These operations are realised by a series of operators that uses the Minkowski sum of a Solid with a structuring element (a cube or dodecahedron in this case); as shown in Boeters et al. [6] and Donkers [10] the shape of the element will influence the resulting shape and thus the results.
None is complete, but the preserved jugal fragments (Figs. 1B E, 4S,T, 6A,B) collectively indicate the shape of the element.
In the subtectorial space model, the intervals of mesh in the cross section are varied to avoid severe distortion of the cubic shape of the element.
Scaling arguments prove that the constants involved depend only on the shape of the element patch ω T. The mesh-refinement chosen guarantees that only finitely many patches occur so that these constants depend, in fact, only on the boundary Γ and the mesh-refinement strategy.
