Isogeometric mortar methods

In this section, we set the geometry decomposition with the functional frame-work. Then we discuss the requirements on the approximation spaces in order to have an optimal method.

2.1.2.1 Description of the computational domain

Let a decomposition of the domain Ω into K non-overlapping domains Ωk be given:

Ω = ^[K

k=1

Ωk, and Ωi∩Ωj =∅, i6=j.

For 1 ≤ k1, k2 ≤ K, k1 6= k2, we define the interface as the interior of the intersection of the boundaries, i.e., γ_k1k2 = ∂Ωk1 ∩∂Ωk2, where γ_k1k2 is open.

Let the non-empty interfaces be enumerated by γ_l, l = 1, . . . , L. We define the skeleton Γ =^SL_l=1γlas the union of all interfaces. As with classical mortar methods, for each interface, one of the adjacent subdomains is chosen as the master side and one as the slave side. The slave side is used in the following to define the Lagrange multiplier space that enforces the coupling between the master and the slave side. We denote the index of the former by m(l), the index of the latter one by s(l), and thus γ_l =∂Ωm(l)∩∂Ωs(l). Note that one subdomain can at the same time be classified as a master domain for one

interface and as a slave domain for another interface, see Figure 2.1. On the interface γ_l, we define the outward unit normal nl of the master side ∂Ωm(l)

and denote by ∂u/∂nl the normal derivative on γ_l from the master side.

n₁

Figure 2.1: Geometrical conforming case (left) and slave conforming case (right).

Each subdomain Ωkis given as the image of the parametric spaceΩ = (0^b ,1)^d by one single NURBS parametrization Fk: Ω^b →Ωk, see Section 2.1.1.3, which satisfies the Assumption 2.1.1. We assume that the decomposition represents the Dirichlet boundary in the sense, that the pull-back of ∂Ωk ∩ ΓD is ei-ther empty or the union of whole faces of the unit d-cube. The h-refinement procedure, see Sections 2.1.1.2 and 2.1.1.3, yields a family of meshes denoted M_k,h, with each mesh being a refinement of the initial one, where we require Assumption 2.1.2. Under these assumptions, the family of meshes is shape regular.

We furthermore assume that for each interface, the pull-back with respect to the slave domain is a whole face of the unit d-cube in the parametric space.

Under these assumptions, we are not necessarily in a geometrically conforming situation, but in a slave conforming situation, see Figure 2.1 (right). We are in a fully geometrically conforming situation, if we also assume that the pull-back with respect to the master domain is a whole face of the unit d-cube, see Figure 2.1 (left).

2.1.2.2 The variational problem

For each subdomain Ωk, we consider the space H_D¹(Ωk) and globally define the broken Sobolev spaces V = Π^K_k=1H_D¹(Ωk), endowed with the broken norm kvk²_V =^PK_k=1kvk²_H1(Ωk), and M = Π^L_l=1H^−1/2(γl).

For the scalar model problem (1.2), we define the broken bilinear and linear formsa: V ×V →Rand f: V →R, such that

We assume that jumps of α and β are solely located at the skeleton, which is important for a reasonable approximation of the solution by the smooth splines.

2.1.2.3 Isogeometric mortar discretization

In the following, we set our discrete approximation spaces used in the mortar context. We introduceV_k,h as the approximation space on Ωk by

V_k,h ={v_k =vb_k◦F⁻¹_k ∈H_D¹(Ωk): vb_k ∈N^pk(Ξk)},

which is defined on the knot vectorΞkof degreep_k. We denote byh_k the mesh size of Vk,h but note that we use the maximal mesh size h = maxkhk as the mesh parameter. In the following lemma, we recall the optimal approximation properties of NURBS spaces, see, e.g., [21, 23, 195].

Lemma 2.1.3. Given a quasi-uniform mesh and let r, s be such that they satisfy 0 ≤ r ≤ s ≤ p_k + 1. Then, there exists a constant c depending only on p_k, θ_k, Fk and W^c_weight,k, such that for any v ∈ H^s(Ωk) there exists an approximation vh ∈Vk,h, such that

kv−v_hk_H^r_(Ωk) ≤ch^s−rkvk_H^s_(Ωk).

On Ω, we define the product space V_h = Π^K_k=1V_k,h ⊂ V, which forms an H¹(Ω) non-conforming space as it is discontinuous over the interfaces.

The mortar method is based on a weak enforcement of continuity across the interfacesγ_l in broken Sobolev spaces. Let a space of discrete Lagrange multi-pliersM_l,h ⊂L²(γ_l) on each interface γ_l be given, which are built on the slave mesh. On the skeleton Γ, we define the discrete product Lagrange multiplier space M_h as M_h = Π^L_l=1M_l,h. Choices of different spaces are discussed in the next section. Furthermore, we define the discrete trace space with additional zero boundary conditions byW_l,h={v|_γl: v ∈V_s(l),h} ∩H₀¹(γ_l).

One possibility for a mortar method is to specify the discrete weak formu-lation as a saddle point problem: Find (uh, λh)∈Vh×Mh, such that

a(u_h, v_h) +b(v_h, λ_h) =f(v_h), v_h ∈V_h, (2.1a) b(u_h, µ_h) = 0, µ_h ∈M_h, (2.1b) whereb(v, µ) = ^PL_l=1^R_γlµ[v]l dγ and [·]l denotes the jump from the master to the slave side overγ_l.

We note, that the Lagrange multiplier λ_h gives an approximation of the normal flux across the skeleton.

Remark 2.1.4. We note that the formulation for linear elasticity follows the same structure. With Vh = (V_h)^d and Mh = (M_h)^d, we consider the saddle point problem with the broken bilinear and linear forms of linear elasticity and a vectorial coupling condition:

a(u,v) = ^XK

k=1

Z

Ωk

σ(u) :ε(v) dx, f(v) = ^XK

k=1

Z

Ωk

bf^>v dx+^Z

ΓN∩∂Ω_k

bt^>v dγ, b(v,µ) =^XL

l=1

Z

γl

µ^>[v]l dγ.

Note that for linear elasticity, the normal stress σ(u)n is the normal flux, which is approximated by the Lagrange multiplier.

For convenience of notation we present the theoretical results for the scalar case, but note that they directly apply to the case of linear elasticity as well.

It is well-known from the theory of mixed and mortar methods, that the following abstract requirements guarantee the method to be well-posed and of optimal order, see [25, 32]. One is a uniform inf-sup stability of the discrete spaces and the second one an approximation requirement of the Lagrange multiplier. We remark that we denote by 0 < c < ∞ a generic constant that is independent of the mesh sizes but possibly depends on p_k.

Although the primal variable of the saddle point problem is in a broken H¹ space, the inf-sup stability can be formulated as an L² stability over each interface. This implies the H₀₀^1/2 −H^−1/2 stability, which can be used in the geometrically conforming situation for d = 2 and in weighted L² norms, for the other cases, see [35].

Assumption 2.1.5. For l = 1, . . . , L and any µ_l ∈M_l,h it holds sup

wl∈W_l,h

R

γlw_lµ_l dγ

kw_lk_L²_(γl) ≥ckµ_lk_L²_(γl).

The second assumption is the approximation order of the dual space. Since for the dual space weaker norms are used, the approximation order of M_l,h with respect to the L² norm can be smaller than the one of Wl,h.

Assumption 2.1.6. For l = 1, . . . , L there exists a fixed η(l), such that for any λ∈H^η(l)(γ_l) it holds

µl∈Minf_l,hkλ−µ_lk_L²_(γl) ≤c h^η(l)kλk_Hη(l)(γl).

The following a priori estimates in the brokenV andM norms can be shown by standard techniques, see [27, 30].

Theorem 2.1.7. Given Assumptions 2.1.5 and 2.1.6, the following conver-gence is given for the primal solution of (2.1). For u ∈ H^σ+1(Ω), with 1/2< σ≤mink,l(p_k, η(l) + 1/2)it holds

1

h²ku−uhk²_L2(Ω)+ku−uhk²_V ≤c

K

X

k=1

h^2σ_k kuk²_Hσ+1(Ωk).

Also an estimate for the dual solution, approximating the normal flux, holds:

L

X

l=1

kα∂u

∂nl

−λ_hk²_H−1/2(γl) ≤c

K

X

k=1

h^2σ_k kuk²_Hσ+1(Ωk).

In the geometrically non-conforming case, as well as for d= 3, the ratio of the mesh sizes on the master and the slave side enters in the a priori estimate, see [138], which does not play a role here due to Assumption 2.1.2.

Optimality of the mortar method holds, when η(l) = p_s(l) − 1/2 can be chosen. Moreover, the dual estimate could still be improved under additional regularity assumptions, see [158].

Im Dokument Hybrid Finite Element Methods for Non-linear and Non-smooth Problems in Solid Mechanics  (Seite 37-41)