CutIGA with basis function removal
 Daniel Elfverson^{1},
 Mats G. Larson^{1} and
 Karl Larsson^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s4032301800992
© The Author(s) 2018
Received: 11 January 2018
Accepted: 25 February 2018
Published: 13 March 2018
Abstract
We consider a cut isogeometric method, where the boundary of the domain is allowed to cut through the background mesh in an arbitrary fashion for a second order elliptic model problem. In order to stabilize the method on the cut boundary we remove basis functions which have small intersection with the computational domain. We determine criteria on the intersection which guarantee that the order of convergence in the energy norm is not affected by the removal. The higher order regularity of the Bspline basis functions leads to improved bounds compared to standard Lagrange elements.
Introduction
Background and earlier work

Gradient jump penalties or some related stabilization term, see [3] and [4].

Adding a small amount of extra stiffness to each active element as is done in the finite cell method, see [7] and [12].

Element merging where small elements are associated with a neighbor which has a large intersection. For DG methods see [11] and for CG methods see [1].

Basis function removal where basis functions with support that has a small intersection with the domain are removed. For the case of isogeometric spline spaces see [8].
New contributions
We investigate the basis function removal approach based on simply eliminating basis functions that has a small intersection with the domain in the context of isogeometric analysis, more precisely we employ Bspline spaces of order p with maximal regularity \(C^{p1}\). To this end we need to make the meaning of small intersection precise and our guideline will be that we should not lose order in a given norm. In particular, we consider the error in the energy norm and show that we may remove basis functions with sufficiently small energy norm and still retain optimal order convergence.
We also quantify the meaning of a basis function with sufficiently small energy norm in terms of the size of the intersection between the support of the basis function and the domain. In order to measure the size of the intersection we consider a corner inside the domain and we let \(\delta _i\), \(i=1,\ldots ,d\) with d the dimension, be the distance from the corner to the intersection of edge \(E_i\) with the boundary. If there is no intersection \(\delta _i = h\). We then identify a condition on \(\delta _i\) in terms of the mesh parameter h which guarantees that we have optimal order convergence in the energy norm. The energy norm of the basis functions may be approximated by the diagonal element of the stiffness matrix and we propose a convenient selection procedure based on the diagonal elements in the stiffness matrix which is easy to implement.
We also derive the condition on \(\delta _i\) corresponding to the \(W^1_\infty \) norm, which will be tighter since the norm is stronger and here we also need the continuity of the derivative of the basis functions. We discuss the approach in the context of standard Lagrange basis function where we note that we get much a tighter condition on \(\delta _i\) in the energy norm and in the \(W^1_\infty \) norm we find that it is not possible to remove basis functions.
We impose Dirichlet conditions weakly using nonsymmetric Nitsche, which is coercive by definition. Since the energy norm used in the nonsymmetric Nitsche method does not control the normal gradient on the Dirichlet boundary we do however need to add a standard least squares stabilization term on the elements in the vicinity of the boundary. Note that this term is element wise in contrast to the stabilization terms usually used in CutFEM.
When symmetric Nitsche is used to enforce Dirichlet boundary conditions stabilization appears to be necessary to guarantee that a certain inverse estimate holds. This bound is not improved by the higher regularity of the splines and will not be enforced in a satisfactory manner by basis function removal.
Outline
In “The model problem and method” section we introduce the model problem and the method, in Chapter 3 we derive properties of the bilinear form, define the interpolation operator, define the criteria for basis function removal, derive error bounds, and quantify \(\delta \) in terms of h for various norms, and finally in “Numerical results” section we present some illustrating numerical examples.
The model problem and method
Model problem
The finite element method

Let \(\widetilde{\mathcal {T}}_{h}\), \(h \in (0,h_0]\), be a family of uniform tensor product meshes in \(\mathbb {R}^d\) with mesh parameter h.

Let \(\widetilde{V}_{h}=C^{p1}Q^p(\mathbb {R}^d)\) be the space of \(C^{p1}\) tensor product Bsplines of order p defined on \(\widetilde{\mathcal {T}}_{h}\). Let \(\widetilde{B} = \{\varphi _i\}_{ i\in \widetilde{I}}\) be the standard basis in \(\widetilde{V}_h\), where \(\widetilde{I}\) is an index set.

Let \(B = \{ \varphi \in \widetilde{B} \, : \, {{\mathrm{supp}}}(\varphi ) \cap \Omega \ne \emptyset \}\) be the set of basis functions with support that intersects \(\Omega \). Let I be an index set for B. Let \(V_h = {{\mathrm{span}}}\{B\}\) and let \(\mathcal {T}_h = \{ T \in \widetilde{\mathcal {T}}_h : T \subset \cup _{\varphi \in B } {{\mathrm{supp}}}(\varphi ) \}\). An illustration of the basis functions in 1D is given in Fig. 1.

Let \(B = B_a \cup B_r\) be a partition into a set \(B_a\) of active basis functions which we keep and a set \(B_r\) of basis functions which we remove. Let \(I = I_a \cup I_r\) be the corresponding partition of the index set. Let \(V_{h,a} = {{\mathrm{span}}}\{B_a\}\) be the active finite element space.
Remark 1
The nonsymmetric method
Remark 2
In practice, \(\mathcal {T}_{h,D}\) may be taken as the set of all elements that intersect the Dirichlet boundary \(\partial \Omega _D\) and their neighbors, i.e. \(\mathcal {T}_{h,D} = \mathcal {N}_h(\mathcal {T}_h(\partial \Omega _D))\).
Remark 3
In the symmetric formulation we stabilize to ensure that coercivity holds and this stabilization also implies that the resulting linear system of equations is well conditioned. Therefore, in the symmetric case, we do not employ basis function removal on the Dirichlet boundary.
Error estimates
Basic properties of \(\varvec{A}_{\varvec{h}}\)
Proof
Interpolation error estimates
Lemma 1
Proof

We definedand we have the bound$$\begin{aligned} \delta _{ij} = {\left\{ \begin{array}{ll} 1 &{} \text {if }{{\mathrm{supp}}}(\varphi _i) \cap {{\mathrm{supp}}}(\varphi _j) \ne \emptyset \\ 0 &{} \text {if } {{\mathrm{supp}}}(\varphi _i) \cap {{\mathrm{supp}}}(\varphi _j) = \emptyset \end{array}\right. } \end{aligned}$$(62)$$\begin{aligned} \sum _{j \in I_r} \delta _{ij} \le (2p+1)^d \end{aligned}$$(63)

We used the \(L^\infty (\mathcal {N}_h(\Omega ))\) stability of the interpolant \(\pi _h\) and then the \(L^\infty \) stability of the extension operator and finally the Sobolev embedding theorem$$\begin{aligned} \Vert \pi _h v \Vert _{L^{\infty }(\mathcal {N}_h(\Omega ))} \lesssim \Vert v \Vert _{L^{\infty }(\mathcal {N}_h(\Omega ))} \lesssim \Vert v \Vert _{L^{\infty }(\Omega )} \lesssim \Vert v \Vert _{H^{p+1}(\Omega )} \end{aligned}$$(64)
Error estimate
We have the following error estimate.
Theorem 1
Proof
Remark 4
Bounds in terms of the geometry of the cut elements
In this section we derive a criterion in terms of the geometry of the cut support of the basis function which implies (51). This criterion will in general not be used in practice but it provides insight into the effect of the higher order regularity of the Bsplines.
Numerical results
Linear elasticity
The nonsymmetric method for linear elasticty
A Neumann problem To illustrate the selection of spline basis functions to remove we first consider a pure Neumann problem with the geometry presented in Fig. 5a. The domain is symmetrically pulled from the left and the right using a unitary traction load. We assume a linear isotropic material with an Emodulus of \(E=100\) and a Poisson ratio of \(\nu =0.3\). To ensure the discretized problem is well posed we seek solutions orthogonal to the rigid body modes by using Lagrange multipliers.
Illustration of the selection procedure
Convergence
Conclusion

Basis function removal can be done in a rigorous way which guarantees optimal order of convergence and that the resulting linear system is not arbitrarily close to singular. These results critically depend on the smoothness of the Bspline spaces.

Basis function removal is easy to implement and efficient since there is no fillin in the stiffness matrix as is the case in for instance face based stabilization. Furthermore, basis function removal is consistent in contrast to the finite cell method.
Declarations
Author's contributions
All authors have prepared the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
Not applicable.
Consent for publication
Not applicable.
Ethics approval and consent to participate
Not applicable.
Funding
This research was supported in part by the Swedish Foundation for Strategic Research Grant No. AM130029, the Swedish Research Council Grants Nos. 20134708, 201703911, and the Swedish Research Programme Essence.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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