In this section, we introduce a parametric finite element discretization of the contact prob- lem stated in Section 1.2.2. On the one hand, this method uses much more geometric information from a CAD model than standard finite elements; on the other hand, we do not use the same functions for the discrete approximation of the displacement field as for the representation of the geometry, which is done in the so-called “isogeometric analysis” introduced in [114]. This allows for a reasonable multilevel hierarchy in case of low order trial functions to be discussed in the next section.
In the following, the symbols ϕ with some indices stand for certain parameterizations or transformations; this must not be confused with the notation of the deformation in the continuum mechanical model. We denote the (closed) d-simplex by ∆d and its faces
7.2 Parametric finite elements 157
ϕT∆ ,−→
ϕk
,−→
Figure 7.1. From left to right: the reference element bT = ∆3; a mesh of the
simplex ∆3; a parametric mesh (here, K = 1) where each element is an image of an affine element; a tetrahedral decomposition of a cylinder with K = 8.
by ∆dj, j ∈ {1, . . . , d + 1}. To describe the elastic body (here, d = 3) by a practicable parameterization, we consider a non-overlapping simplicial decomposition of Ω ⊂ Rdinto a fixed number of K ≥ 1 subdomains. Formally this reads as
Ω = K [ k=1 Ωk= K [ k=1 ϕk(∆d),
where the notation already indicates that the subdomains (Ωk)k=1,...,Kappear as particular
images of suitable parameterizations (ϕk)k=1,...,K. This is illustrated in Figure 7.1 (right).
Let us assume that the faces of the simplicial cells Ωk, namely the surfaces ϕk(∆dj),
k ∈ {1, . . . , K}, j ∈ {1, . . . , d+1}, are given as B-patches. This way to represent polynomial surfaces is analyzed in [66]. In this case, the author of [158] proposes to construct the full- dimensional mappings ϕk : ∆d → Rd, k ∈ {1, . . . , K}, as transfinite interpolations of the
surface values from the CAD model using certain blending functions. Particularly, the single parameterizations are smooth and they match across these B-patch surfaces if the surfaces themselves match. This gives rise to a consistent global parameterization which we do not write down explicitly. We note that this global mapping is continuous but not necessarily differentiable across the interior interfaces. In addition, one can guarantee that each parameterization ϕk satisfies the regularity assumption
det(∇ϕk) > 0 in ∆d. (7.1)
In fact, this is one of the main results of [158].
In the following, we define the parametric finite element spaces in a rather straightfor- ward way via a lift of standard Lagrange finite elements. For this purpose, let (T`k)`∈N be a family of nested simplicial meshes of ∆d for each k ∈ {1, . . . , K}. To keep the global finite element spaces conforming, we assume that the meshes meeting at the faces of the simpli- cial subdomains Ωk of Ω match at each level ` ∈ N. Let bT be the reference element; here,
b
T = ∆d. Then, for each T∆∈ T`k, there is an affine mapping ϕT∆ such that ϕT∆( bT ) = T∆. Now, we give a concise description of the parametric elements in Ω by employing the special finite element transformations
ϕT := ϕk◦ ϕT∆ : bT → R
158 7 Multigrid methods based on parametric finite elements
which are diffeomorphisms between the reference element bT and the actual elements. That way, the parametric elements at level ` ∈ N are identified as the images of the elements of the meshes (T`k)k=1,...,K; see Figure 7.1. More precisely, a family of parametric meshes
(T`)`∈N of Ω can be defined by T`:= n T = ϕT( bT ) = ϕk(ϕT∆( bT )) | 1 ≤ k ≤ K, T∆∈ T k ` o , ∀ ` ∈ N.
Assume that this family of global meshes is shape regular and quasi-uniform according to the equations (1.16) and (1.17). Note that assumption (7.1), combined with the continuous differentiability of the mappings (ϕk)k=1,...,K, in the compactum ∆d, implies that it is
sufficient to ensure these conditions for each sequence (T`k)`∈N separately as far as we keep K fixed.
Finally, let P := Pr( bT ) be the space of polynomials of degree r in bT . Then, for ` ∈ N,
the parametric finite element space associated with the parametric mesh T` is
X` := v ∈ C0(Ω) | ∀ T ∈ T` ∃ w ∈ P : v(x) = w(ϕ−1T (x)), ∀ x ∈ T
= v ∈ C0(Ω) | v ◦ ϕ
T ∈ P, ∀ T ∈ T` .
(7.3)
Note that, in principle, the above definition makes sense for any reasonable set of finite element transformations (ϕT)T ∈T`. In case the mappings are constructed as in (7.2) via the high order parameterization from [158], this is a “superparametric” concept if the degree r is small. This is in contrast to the subparametric or isoparametric finite elements which are usually considered in the literature; see [56].
From a practical point of view, virtually every kind of parameterization can be em- ployed with the following qualification. For an efficient assembly of the stiffness matrix and the right hand side via sufficiently accurate numerical quadrature, the derivatives of the resulting finite element transformations (7.2) and the mappings themselves must be easy to evaluate; see, e. g., [9].
Let us now apply the above concept. We suppose that the Dirichlet data has been treated appropriately. Then, a discretization of Signorini’s problem described in Sec- tion 1.2.2 is obtained by specifying a suitable set of admissible displacements K` using
the vector-valued parametric finite element space X` := (X`)3 defined by (7.3) with r = 1.
The discrete variational problem reads exactly as before in the standard case of Section 1.3 if, as usual, the non-penetration conditions on the possible contact boundary ΓC are merely
enforced at the nodes N`∩ ΓC. For clarity, we recall the variational inequality: find u`∈ K`
such that
a(u`, v − u`) ≥ F (v − u`), ∀ v ∈ K`:= {v ∈ X`| v(p) · n(p) ≤ g(p), ∀ p ∈ N`∩ ΓC} .
Remark 7.1. Although, from a modeling point of view, as much geometric information as possible should be used for an accurate description of contact phenomena, we remark that a strong pointwise non-penetration condition everywhere on ΓC is usually not suitable for
the variational formulation the (parametric) finite element method relies on. Besides, a decoupled set of constraints is preferable for a variety of reasons. The common remedy is to prescribe the contact constraints with respect to a suitable cone of Lagrange multipliers.