Construction with higher order approximation property . 59

We restrict ourselves to the one-dimensional construction, since multivariate biorthogonal basis functions can be constructed by a tensor product structure.

However it should be noted that the biorthogonality of tensor product splines only holds on the parametric domain and with respect to the standard L² scalar product. To use the biorthogonal basis for three-dimensional problems, we can formulate the mortar coupling with respect to the parametric space, instead of the exact geometry as before.

Without loss of generality, we consider the unit interval γ = (0,1). Let W_h be a B-spline space of degree p on γ with the break points 0 = ζ1 < ζ2 <

. . . < ζ_E = 1 and the basis (B^bp_i)i=1,...,n, where n = dimW_h. NURBS and the weight of a geometry transformation can be included in the consideredL² scalar product.

We use the construction described in [169], which only slightly needs to be adapted to splines. Instead of the purely algebraic construction introduced there, we consider an equivalent functional setting.

In addition to W_h, we consider a broken polynomial space:

W_h⁻¹ ={v ∈L²(0,1) : v|_(ζ

i,ζi+1)∈Pp}

of dimension N = dimW_h⁻¹ = (E−1)(p+ 1). We note that, besides possible Dirichlet boundary conditions, the broken space W_h⁻¹ is obtained by breaking apart the spline space W_h. See Figure 2.15 for an illustration of the broken space.

We seek dual basis functionsψ_j ∈W_h⁻¹ satisfying biorthogonality:

Z

γ

Bb_i^pψ_j dx=δ_ij,

as well as polynomial reconstruction of degree q. Polynomial reproduction holds exactly if the quasi-interpolation

Qf =^Xn

i=1

f,B^b_i^p

0ψ_i (2.3)

is invariant for polynomials p∈P^q, i.e., Qp=p.

Since W_h ⊂ W_h⁻¹, we can extend the B-spline basis (B^b_i^p)i=1,...,n of W_h to a basis (ϕi)i=1,...,N of W_h⁻¹, such that ϕ_i = B^b_i^p for i ≤ n. To retain a local support, we suggest the following construction:

The basis is build using an auxiliary decomposition ofW_h⁻¹ intonsubspaces, related to the primal basis functions. Each basis function B^b_i^p is supported on n_i ≤p+1 elements. RestrictingB^b_i^pto each of these elements yields then_iparts Bb_i,j^p ∈ W_h⁻¹ (each supported on a single element), such that B^b_i^p = ^Pnj=1ⁱ Bb_i,j^p . We collect the local contributions W_h,i⁻¹ = span{B^b_i,j^p : j = 1, . . . , ni} and note that W_h⁻¹ =^Ln_i=1W_h,i⁻¹. The collection of B^b_i,j^p forms a basis of W_h⁻¹, but the basis does not include the original B-splines as desired. Therefore we construct another basis as follows.

Fori= 1, . . . , n, we extendB^b_i^pbyni−1 basis functionsφi,j to a basis ofW_h,i⁻¹, which guarantees suppφi,j ⊂suppB^bp_i. We note that we do not require the basis to be orthogonal but a good condition number of the basis is advantageous for the numerical computations. Combining these local basis functions yields the desired basis of W_h⁻¹:

ϕ_i

i=1,...,N =B^b₁^p, . . . ,B^b_n^p,(φ_1,j)j=1,...,n1−1, . . . ,(φ_n,j)j=1,...,nn−1

.

See Figure 2.16 for an illustration of a possible realization of the local basis (B^b_i^p,(φ_i,j)j=1,...,ni−1) for a fixed i ∈ {1, . . . , n}. In the case of Dirichlet condi-tions on the space W_h, the corresponding basis functions must additionally be reincluded in the basis of W_h⁻¹.

By a local inversion of the mass matrix, we can construct a dual basisψ^e_i on W_h⁻¹, biorthogonal to the recently constructed primal basis (ϕi)i=1,...,N:

Z

γ

ψe_iϕ_j dx=δ_ij.

See Figure 2.17 for an illustration of ψ^e_i fori= 1, . . . , n, which is biorthogonal

ζ_l ζ_l+1 ζ_l+2 ζ_l+3 -1

0

1 B_i^p

φ_i,1 φ_i,2

Figure 2.16: Local basis contribution based on the B-splineBb_i^p forp= 2.

toB^b_i^p and ψ^ej for j > n, which is orthogonal to all B^b_i^p.

0 1 2 3 4 5

-1 -0.5 0 0.5 1 1.5 2

0 1 2 3 4 5

-1 -0.5 0 0.5 1

Figure 2.17: Orthogonal basis function with same support (left) and basis functions orthog-onal to allNi (right) forp= 2.

Clearly for any choice of (z_ki)k,i, ψ_i =ψ^e_i+ ^XN

k=n+1

z_kiψ^e_k, for i= 1, . . . , n, (2.4) is a biorthogonal basis to (B^b_i^p)i=1,...,n.

We can choosez_ki fork =n+1, . . . , N and i= 1, . . . nsuch that polynomial reconstruction holds and a local support is preserved.

Polynomial reconstruction of degree q holds if the quasi-interpolation (2.3) is invariant for polynomialsp∈P^q, i.e.,

(p, ϕ_j)0 =^Xn

i=1

p,B^b_i^p

0(ψ_i, ϕ_j)₀, for any j = 1, . . . , N, p∈Pq.

For j = 1, . . . , n it holdsϕ_j =B^b_j^p and we may use the biorthogonality of ψ_i

and ϕ_j: modification of the basis ψ:

n To sum up, it remains to solve

n z (with nq+ 1 on a sufficiently fine mesh), we have some flexibility in the choice of the unknown z. This flexibility allows us to choose z sparsely, such that the resulting basis functions have a local support.

For a fixed ˆ=n+1, . . . , N, let us choose an index setI(ˆ) with|I(ˆ)|=q+1, Although the equation system is split into parts, we do not solve for one dual basis function with each system. Instead, we can choose which dual basis functions shall be influenced by ψ^e_ˆ. We note this already by looking at the dimensions. We solve N −n equation systems of dimension (q+ 1)×(q+ 1), each one defining n values of z.

Leaving apart the special case, wherez_ˆi = 0 appears in the solution of (2.5), we can see the support of ψ_i by the choice of the index sets. With (2.4), anyj with i∈ I(j) yields suppψ^e_j ⊂ suppψ_i and we can estimate the support of ψ_i indirectly.

As an example, let us consider the reproduction order q = p and for sim-plicity no repeated knots. For any ψ_ˆ, we consider the mid-element of its support. If the function is supported on an even number of elements, we arbi-trarily choose the mid-element closer to the boundary. The indices of allp+ 1 basis functions B_i^p that are non-zero on this element are considered in I(ˆ).

This way the support of each biorthogonal basis functions contains maximal 2p+ 1 elements, compared to p+ 1 elements of the primal function. Assump-tion 2.1.6 is fulfilled by construcAssump-tion, yielding optimal convergence rates by

0 1 2 3 4 5

−2

−1 0 1 2

ζ

B−Spline basis function Biorthogonal function

Figure 2.18: A quadratic basis function and its corresponding (rescaled) biorthogonal basis function with a local support and optimal approximation properties.

Theorem 2.1.7. See Figure 2.18 for an example with p= 2 and compare with the straightforward construction of Figure 2.14.

2.3.2 Numerical results

We test the newly constructed biorthogonal basis in a systematical convergence test. We consider two geometries, a square domain and a curved annulus, each one being divided into two subdomains. Two analytically known solutions are defined, one solving the Poisson problem, the other one the equation of linear elasticity and we consider spline spaces of degree 2,3 and 4. Additionally, the ratio of the master and slave mesh sizes is investigated by considering three cases. In two cases the ratio between the mesh sizes on the two subdomains is 3 : 2, in case (a) the slave mesh is finer, in case (b) the master mesh. In the third case, (c), the mesh size ratio is approximately one. See Figure 2.19 for the initial mesh of all cases.

0 1 0 1

1 0

0 1

0 1 2

1

0 2

0 1 2

1

0 2

Figure 2.19: Initial meshes. Left: Square with mesh size ratio ≈1 : 1 (c) and 2 : 3 (a,b).

Right: Annulus with mesh size ratio≈1 : 1 (c) and 2 : 3 (a,b).

The manufactured solution for the Poisson problem is u(x, y) = sin (2π(x−0.33)) cos(2πy),

with f =−∆u and the exact trace as Dirichlet boundary conditions.

The solution for elasticity is manufactured, based on the solution for the infinite plate with hole. To avoid a radial symmetric solution, we slightly shift the function. With r² = (x+ 0.5)²+y² and θ = arctan(y/(x+ 0.5)):

u rcos(θ) rsinθ)

!

= cos(θ)ur(r, θ)−sin(θ)uθ(r, θ) sin(θ)u_r(r, θ) + cos(θ)u_θ(r, θ)

!

,

withur, uθ as in (1.6). See Figure 2.20 for the stress distribution of the solution on the annulus domain.

|σ|

Figure 2.20: Stress magnitude of the elasticity equation on the annulus.

In all cases, the result of a biorthogonal dual basis is compared to a standard equal order pairing as a reference computation. We consider convergence in theH¹(Ω) error, but note that theL²(Ω) errors behave similar in all considered cases. For the square geometry, the convergence rates are shown in Figure 2.21.

We see optimal convergence rates in all considered cases. However, for the biorthogonal basis, we observe a slightly larger error in the pre-asymptotics of the Poisson equation, in the case where the slave mesh is coarse in comparison to the master mesh. The effect gets larger with increasing polynomial degree, but the difference between the error curves vanishes with more mesh refine-ments. For the example of elasticity, no difference in the six shown curves can be noticed.

1 2 3 4 5 6 7

Figure 2.21:H¹ error for the square geometry. From left to right: p= 2,3,4. Top: Poisson equation. Bottom: Elasticity equation.

The results on the annulus, shown in Figure 2.22, show a similar effect.

However, compared to the square domain, the pre-asymptotics is seen more intense, now including the case of linear elasticity. As on the square mesh, the difference in the error vanishes with more mesh refinements. In contrast to the previous example also a difference for the equal order can be seen. The error on the mesh with a size ratio 2 : 3 is smaller compared to the mesh size ratio 1 : 1, independent of the master-slave choice, which could be due to the local refinement of the outer ring. The difference is about the same magnitude as the error increase due to the pre-asymptotics of the biorthogonal basis. For the example of elasticity, this effect cannot be seen.

This indicates how small issues can increase the error by a similar value as the observed pre-asymptotics and again shows the promising behavior of the novel biorthogonal basis.

Im Dokument Hybrid Finite Element Methods for Non-linear and Non-smooth Problems in Solid Mechanics  (Seite 59-65)