Goal-adaptive Isogeometric Analysis with hierarchical splines

G. Kurua,∗, C.V. Verhooselb, K.G. van der Zeeb, E.H. van Brummelenb

aONERA, The French Aerospace Lab, CFD and Aeroacoustics Department, BP72 - 29 avenue de la Division Leclerc, FR-92322 Châtillon, FrancebMultiscale Engineering Fluid Dynamics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

Abstract

In this work, a method of goal-adaptive Isogeometric Analysis is proposed. We combine goal-oriented error estimation and adap-tivity with hierarchical B-splines for local h-refinement. The goal-oriented error estimate is computed with a p-refined discretedual space, which is adaptively refined alongside the primal space. This discrete dual space is proven to be a strict superset of theprimal space. Hierarchical refinements are introduced in marked regions that are formed as the union of chosen coarse-level splinesupports from the primal basis. We present two ways of extracting localized refinement indicators suitable for the hierarchicalrefinement procedure: one based on a partitioning of the dual-weighted residual into contributions of basis function supports andone based on the combination of element indicators within a basis function support. The proposed goal-oriented adaptive strategy isexemplified for the Poisson problem and a free-surface flow problem. Numerical experiments on these problems show convergenceof the adaptive method with optimal rates. Furthermore, the corresponding goal-oriented error estimators are shown to be accurate,with effectivity indices in the range of 0.7-1.1.

Keywords: Isogeometric Analysis, Goal-oriented error estimation, Adaptive refinement, Hierarchical splines

1. Introduction

Since the usage of computers has become widespread in en-gineering design, performing analysis on geometries designedusing Computer Aided Design (CAD) software has been a com-mon task. Isogeometric Analysis [1, 2], is a framework forsolving (partial) differential equations on domains generated byCAD software. It aims to eliminate or significantly reduce thetime required for preparing the designed geometry for analy-sis, by directly using the CAD representation of the geometryin the analysis step. In addition to the benefits from an en-gineering management perspective, the exact representation ofthe design geometries in the analysis step has benefits for ap-plications where the smoothness of the boundary of the domainplays a role, such as flow boundary layers and sliding contactof surfaces [1]. Possibility of having higher order differentiablesolutions in Isogeometric Analysis has also proven to be verydesirable for many engineering applications where continuity ofthe solution is a requirement due to the employed formulation,such as binary phase separation [3], gradient damage models[4] and analysis of shell structures [5, 6]. Due to its positiveattributes, the interest in Isogeometric Analysis has grown veryrapidly after its initial introduction, with applications to manycomplicated engineering problems in material fracture [7, 8],contact mechanics [9, 10], turbulence computation [11, 12],free-surface flows [13], fluid-structure interaction [14], shapeoptimization [15, 16] and many others.

Although Isogeometric Analysis aims to overcome someof the problems often encountered in the engineering design-

∗Corresponding authorEmail address: gokturk.kuru@onera.fr (G. Kuru )

through-analysis process, the fact that the objects used for ge-ometry representation are often not adequate for attaining a cer-tain accuracy in analysis means that there has to be some sort ofa preprocessing step. If this step is done heuristically, it againrequires manual labor. To have a truly smooth work flow fromdesign to analysis, adapting the initial design representationsbased on an automated procedure is essential. Adaptive Isoge-ometric Analysis methods based on error estimators are vitalin performing this task with effectively no intervention by theengineer. Achieving this requires two issues to be addressed:Enriching the approximation space through local refinement ofthe basis, and finding reliable refinement indicators to assess ifand where refinement should take place. The aim of this workis to address these issues by applying goal-oriented error esti-mation principles to develop an adaptive Isogeometric Analy-sis method using hierarchical B-splines as the local refinementtechnique.

In assessing CAD technologies in the context of adaptive Iso-geometric Analysis, various qualities are important. Especiallylocality of the refinement procedure plays an important role,along with the requirement of linear independence and com-patibility with the state of the art CAD technologies that are inuse in the industry. Hierarchical splines [17] fulfill all of theserequirements, yet they are constructed in a simple way. Thehierarchical refinement method has been applied to B-splines[17, 18] and NURBS [19]. In the field of engineering design,NURBS is the industry standard technology [2, 20, 21]. How-ever, B-splines have many properties in common with NURBSand in terms of refinement techniques, virtually every conceptcan be extended from B-splines to NURBS with relative ease.Therefore, hierarchical B-splines are chosen to be used in this

Preprint submitted to Computer Methods in Applied Mechanics and Engineering November 22, 2013

study.Adaptive Isogeometric Analysis with hierarchical splines has

already been studied using refinement indicators based on socalled bubble functions [18] and gradients of the approximatesolution [22]. In addition to these, hierarchically refined basesare used for Galerkin discretizations for several applications[23, 24]. Adaptive Isogeometric Analysis has also been studiedwith T-splines [25, 26], PHT-splines [27, 28] and using com-mon Finite Element Method (FEM)-type basis functions withspline parametrizations [29, 30]. Adaptivity with a method ofconstructing locally refined B-spline elements ensuring up toC1-conformity through a matching procedure had been investi-gated [31] even before the inception of Isogeometric Analysis.

Next to the techniques of constructing locally refined ap-proximation spaces, an integral element of adaptive refinementis the refinement indicator that is used in deciding where andhow a mesh needs to be refined. Refinement indicators areusually based on an a-posteriori estimate of the error. Amongthe most prominent a-posteriori error estimation methods forGalerkin discretizations are recovery-based methods, residualbased methods and goal-oriented methods. For a review of var-ious a-posteriori error estimation techniques, see [32, 33]. Thechoice of goal-oriented error estimation is motivated by its verynature, i.e. estimating the error in a quantity that is of practicalrelevance for the application. In mesh adaptivity using estima-tors that do not take the quantity of interest into account, therefinement aims to resolve features of the solution with equalaccuracy over the whole computational domain. In many prob-lems the complicated features of the solution on some parts ofthe domain might not influence the quantity of interest. Hence,improving the approximate solution based on goal-oriented er-ror estimators is a very natural choice, given that there is a clearquantity of interest.

This work considers the local h-refinement of IsogeometricAnalysis spaces using hierarchical B-splines, with refinementindicators derived from goal-oriented error estimators. Themain contributions of this work are a goal-oriented error esti-mator using adapted hierarchical B-spline primal spaces that arestrictly nested within the proposed dual spaces, the constructionof refinement indicators that are suitable for hierarchical refine-ment, the numerical investigation of these indicators and actualgoal-adaptive Isogeometric discretizations of the two exampleproblems.

The proposed methods are applied to the solution of exem-plary elliptic problems of Poisson’s equation and a prototypi-cal free-surface flow problem. Both problems are posed on 2Ddomains, yet the methodology is formulated in general dimen-sions and in an operator-independent way wherever possible.Hence, extension to other elliptic problems should require min-imal modifications to the proposed methodology. Furthermore,for problems in 3D, the benefits of local refinement in terms ofcomputational cost would be even more relevant.

Starting with the a brief introduction to B-splines, the em-ployed refinement technique using hierarchical B-splines is re-viewed in Section 2 in the light of adaptive Isogeometric Anal-ysis. The method of goal-oriented error estimation is intro-duced in a general setting and an error estimator suitable for

adaptive-refinement is proposed in Section 3. Furthermore, inSection 3 two refinement indicators are proposed using this es-timator. In Section 4, the error estimator and refinement indica-tors are studied numerically and the adaptive procedure is testedfor two problems. Conclusions and future recommendations arepresented in Section 5.

2. Hierarchical refinement of B-splines

Given a bounded open interval Ω ⊂ R, named the parameterdomain, B-splines are piecewise polynomials Ni,p : Ω → R ofdegree p ∈ Z≥0, that form a basis of a globally Ck-continuouspolynomial space, with −1 ≤ k < p. The space formed by theB-splines can be characterized by a non-decreasing sequence ofknots, named the knot vector. Knots are points in Ω, at whichthe B-splines have reduced continuity, i.e. locations at whichthe basis functions are not C∞-continuous. A B-spline curveC : Ω → Ω is a parametrization from the parameter domainΩ ⊂ R to the physical domain Ω ⊂ Rd′ , with d′ ≥ 1, using theB-spline basis. The coefficients of the B-splines Bi ∈ Rd′ arereferred to as the control points. The B-spline curve is describedas:

C(ξ) =

n∑i=1

Ni,p(ξ)Bi

where the B-splines Ni,p are defined by the Cox-de Boor rela-tion [34, 35], which states for p = 0:

Ni,0(ξ) =

1 if ξi ≤ ξ < ξi+1,

0 otherwise(1a)

and recursively for p > 0:

Ni,p(ξ) =ξ − ξi

ξi+p − ξiNi,p−1(ξ) +

ξi+p+1 − ξ

ξi+p+1 − ξi+1Ni+1,p−1(ξ) (1b)

For multivariate objects, commonly the tensor product ofmultiple univariate spline bases is used. For example, the bi-variate B-spline surface is constructed as:

C(ξ, ζ) =

n∑i=0

m∑j=0

Ni,p(ξ)N j,p(ζ)Bi j

In the context of Isogeometric Analysis, global h-,p- and k-refinements of the B-spline bases are investigated in [2]. Theglobal h-refinement of the B-spline basis corresponds to knotinsertion. The p-refinement is achieved by order elevation andknot insertions at all knot positions. k-refinement is a possibilityunique to spline bases, where the order is elevated along withthe continuity order, which results in a minimal increase in thenumber of degrees of freedom to obtain a certain polynomialdegree.

Traditional h-refinement of B-spline bases via knot insertioncan also be done locally in 1D, however in multiple dimensions,the tensor product structure of the basis causes creation of unde-sired elements, resulting in increased computational cost witharguably little gain in accuracy. This is illustrated graphically

2

0 0.25 0.5 0.75 1ξ0

ξ11

0.75

0.5

0.25

0

0 0.25 0.5 0.75 1ξ0

ξ11

0.75

0.5

0.25

0

Figure 1: Tensor product refinement (left) and hierarchical re-finement (right) of a bivariate linear Bézier element. Differentcolors for the hierarchically refined mesh represent mesh parti-tions belonging to different levels.

in Figure 1, along with a more desirable alternative using hi-erarchical splines. The meshes plotted in Figure 1 are referredto as the Bézier meshes and the lines represent the boundariesof the Bézier elements, across which the polynomial space canadmit non-C∞ solutions.

In the sequel we will introduce hierarchical splines and pro-vide some properties that are relevant to isogeometric analy-sis, viz. linear independence and nestedness of hierarchicallyrefined spline spaces. The basic results outlined here can befound in slightly different forms in the literature [36, 18, 37],but we extend the nestedness result to strict nestedness (Propo-sition 7) which will be used later for the proof of an importantresult (Theorem 8) that is of specific interest for goal-orientederror estimation with hierarchical splines.

2.1. Hierarchical B-splines

Hierarchical B-splines aim to construct bases on non-tensorproduct meshes consisting of locally refined portions. Thiscan be seen as an extension of C0-continuity preserving localh-refinement of elements as seen in Adaptive Finite ElementMethods [38] to Ck-continuity preserving local h-refinement ofdegree p > k B-spline patches.

The idea of hierarchical refinement is based on combiningcoarse level B-splines and fine level B-splines at different re-gions of the domain. The fine level B-splines are generatedfrom coarse level B-splines by global h- or p-refinement oper-ations and only a subset of the fine level functions are includedbased on a selected refinement region.

In the definition of hierarchical splines and throughout thistext, the open support of a function f : A → R will be denotedas supp f :

supp f := x ∈ A : f (x) , 0

where A is an open set. Furthermore, supp f will be used todenote the closure of supp f .

Definition 1 (Hierarchical B-spline basis). Let (N r)0≤r<R be asequence of R B-spline bases such that span(N r+1) ⊃ span(N r).Let (Ωr

#)0≤r≤R be a sequence of R + 1 open sets such that Ωr+1# ⊆

Ωr# with Ω0

# = Ω and ΩR# = ∅. The hierarchical B-spline basis

00.20.40.60.8

1

N0

00.20.40.60.8

1

N1

00.20.40.60.8

1

0 1 2 3 4 5

H2

ξ

Figure 2: Local hierarchical h-refinement of the B-spline basiswith p = 2, Ξ = 0, 0, 0, 1, 2, 3, 4, 5, 5, 5 and Ω1

# = (3, 5)

00.20.40.60.8

1

H1

00.20.40.60.8

1N

2

00.20.40.60.8

1

0 1 2 3 4 5

H2

ξ

Figure 3: Further application of hierarchical refinement to thebasis in row 3 of Figure 2 with a too small selection Ω2

# =

(4, 4.5)

at level R is defined as:

HR :=R−1⋃r=0

Nr ∈ N r : supp Nr ⊆ Ωr# ∧ supp Nr * Ωr+1

# (2)

Definition 1 is equivalent to the definition provided in [18].It essentially states that the hierarchical B-spline basis is con-structed by excluding coarse-level functions whose supports arecontained within the refinement region and replacing them bythe fine-level functions whose supports are contained withinthe same region. Although only h-refined B-spline spaces areconsidered here, the properties outlined in the following aregenerally valid for any sequence of B-spline spaces such thatspan(N r+1) ⊃ span(N r).

A hierarchically h-refined basis H2 is exemplified for a uni-variate case in the third row of Figure 2 with Ω1

# = (3, 5) asthe marked region of refinement. The B-splines N0

i ∈ N0 are

plotted in the first row and the h-refined B-splines N1i ∈ N

1

are plotted in the second row. To construct H2, functionsfrom N0 whose supports are not contained within (3, 5) andfunctions from N1 whose supports are contained within (3, 5)are included. A further application of the refinement withthe selection region corresponding to a single Bézier elementΩ2

# = (4, 4.5) is depicted in Figure Figure 3. This operation

3

results in no refinement, since the marked refinement region istoo small to fully contain the support of a function from N2.This illustrates how element-wise marking common to adaptivefinite elements would not be suitable with hierarchical splines.

Remark 1. The way the hierarchical B-spline basis is definedmakes its properties equally valid for arbitrary-dimensionalmultivariate spline objects.

B-splines have the property of local linear independence.(see e.g. [39]) This is an important property that will be used insome proofs later on, and is defined as follows:

Definition 2 (Local linear independence). A set of functionsM, withM 3 Ni : Ω → R, is said to be locally linearly inde-pendent if for any open subset A ⊂ Ω there do not exist nonzerocoefficients di such that ∑

i:Ni∈M, supp Ni∩A,∅diNi|A = 0

where Ni|A are the restrictions of Ni to A.

Global linear independence is defined very similarly, exceptA is not taken as an arbitrary open subset, but A = Ω. Usinglocal linear independence property of B-splines, global linearindependence of the hierarchical B-splines is demonstrated bythe following theorem:

Theorem 3 (Global linear independence of HR). The hierar-chically constructed set of splines as defined in Definition 1 arelinearly independent on Ω.

Proof. See [18].

Remark 2. In general, hierarchical B-splines do not have theproperty of local linear independence, which is evident from in-specting the number of quadratic polynomials over the interval(4, 5) on the last row of Figure 2. The practical implication ofthis is that applying hierarchical refinement before a trimmingoperation can make the mesh unsuitable for analysis: Introduc-ing a cut to the hierarchical splines of Figure 2 at point ξ = 4results in linearly dependent functions to the right of the cut.The restriction of the remaining coarse level spline to (4, 5) canbe exactly represented by the restrictions of the remaining finelevel splines, implying linear dependence for the trimmed ge-ometry. However, it is worth stressing that any trimming wouldlikely be introduced during the design of the object, and subse-quent hierarchical refinements on trimmed objects do not giverise to this problem.

In the following proposition, we show that hierarchical re-finement produces enriched spaces in which the initial spaceis nested, which is an important property in view of adaptivealgorithms.

Proposition 4 (Nestedness of hierarchical B-spline spaces).The space spanned by a hierarchical B-spline basis HR isnested within the space spanned by HR+1 that is obtained bya further refinement ofHR.

Proof. Here we repeat the proof of Vuong et al. [18] in a modi-fied form that is organized it in two parts, since we will refer tothe individual parts later on:

1. When going from level R to R + 1, the discarded functionsare denoted NR

# := N ∈ NR : supp N ⊆ ΩR+1# and

the newly added ones are denoted NR+1# := N ∈ NR+1 :

supp N ⊆ ΩR+1# . If NR

# = ∅, for any other basis NR+1#

it holds that span(NR+1# ) ⊇ span(NR

# ). Otherwise, for allN ∈ NR

# it holds that supp N ⊆ Ω# due to the definitionof NR

# . Also N ∈ span(NR+1) since N ∈ span(NR) andspan(NR) ⊂ span(NR+1). Therefore,

∃di ∈ R : N =∑

i:N1i ∈N

R+1

diN1i

which can be split up as:

∃ci, d j ∈ R : N =∑

i:N1i ∈N

R+1\NR+1#

ciN1i +

∑j:N1

j ∈NR+1#

d jN1j

If NR+1 \ NR+1# = ∅, the first summation drops. Other-

wise, ∀N1i ∈ N

R+1 \ NR+1# , supp N1

i * supp N sincesupp N1

i * Ω# and by the local linear independence ofNR+1 \ NR+1

# on Ω \ Ω#, all coefficients ci are necessarilyzero as N(ξ) = 0,∀ξ ∈ Ω \ Ω#. This too means that thesummation over NR+1 \ NR+1

# disappears, hence

∃d j ∈ R : N =∑

j:N1j ∈N

R+1#

d jN1j

which implies that N ∈ span(NR+1# ). Since this holds for

all N ∈ NR# , it has to hold that

span(NR+1# ) ⊇ span(NR

# ) (3)

2. In the refinement operation from level HR to HR+1, thediscarded basis functions of HR are HR \ HR+1 = NR

#and the newly added basis functions of HR+1 are HR+1 \

HR = NR+1# . By (3), we have span(HR+1 \ HR) ⊇

span(HR\HR+1), from which it follows that span(HR+1) ⊇span(HR \ HR+1). Furthermore, we have the trivial rela-tion span(HR+1) ⊇ span(HR+1∩HR) for the functions thatremain unchanged by the refinement. Combining the two,we have span(HR+1) ⊇ span(HR).

Remark 3. Nestedness of at least the initial space within thesubsequent spaces is required for performing IsogeometricAnalysis, due to isoparametric representation of geometry.

2.2. Hierarchical refinement strategyIn order to use hierarchical B-spline bases in adaptive Isoge-

ometric Analysis, some decisions have to be made in terms ofthe refinement strategy; most notably on the choice of B-splinebases of each level, the refinement domain Ωr+1

# and how furtherrefinements can be applied to an already refined basisHR.

4

The hierarchical B-spline bases use a sequence of globallyrefined B-spline bases N r for their construction. The mainproperty assumed of these bases is that span(N r) ⊂ span(N r+1).This can be satisfied in multiple ways; both h-refined and p-refined B-spline bases can be used, possibly with anisotropicrefinement. In this work, the elementary case of isotropic h-refinement is considered.

Remark 4. Globally k-refined B-splines do not satisfy the as-sumption span(N r) ⊂ span(N r+1), hence are not directly usablein hierarchical refinement.

In Finite Element Methods, local refinement is generallydone element-wise based on a chosen element indicator. How-ever, in the hierarchical refinement procedure, selection ofΩr+1

# = Ki, where Ki ∈ Tr is a single element of the Béziermesh T at level r, could possibly cause no change in the basis,as no basis function support might be contained within a singleelement. To guarantee refinement, Ωr+1

# must at least containthe support of a single basis function supp Nr+1

i , Nr+1i ∈ N r+1

from the higher-level basis. Due to the typically smaller sup-ports of finer level functions, having this as a minimal selec-tion criterion could lead to the addition of fine level functionswithout excluding any coarse level functions, leading to exces-sive overlap of functions from different levels. This choice hassome disadvantages with respect to e.g. possibility of havinga partition of unity basis [18] and the bandwidth of the result-ing stiffness matrices. A study of the bandwidth of matricesobtained from Galerkin discretizations employing hierarchicalB-spline bases is performed in [22].

Remark 5. For the univariate case, the hierarchical B-splinespaces can be spanned by ordinary B-splines that are refinedlocally by knot insertion. In Appendix A, these two bases arecompared in terms of the condition numbers for the solution ofan adaptively discretized Poisson problem, with results slightlyfavoring hierarchical bases. See [22, 40] for studies of differentbases for span(HR).

In this work, the support condition proposed by Vuong et al.[18] is employed for the selection of refinement regions. Thismeans Ωr+1

# ⊇ supp Nri , for a basis function Nr

i ∈ Nr. With this

requirement it is chosen to select functionsN r# and construct the

refinement region by Ωr+1# = int

⋃Ni∈N

r#

supp Ni, where int A :=A \ ∂A denotes the interior of a set A.

The following definition of a globally refined basis is nec-essary for proving strict nestedness for the chosen hierarchicalrefinement strategy:

Definition 5 (Globally refined basis). Given a basis M0, thebasis M1 is said to be globally refined with respect to M0 ifand only if for all N ∈ M0

#N1i ∈ M

1 : supp N1i ⊆ supp N

> #Ni ∈ M0 : supp Ni ⊆ supp N

Remark 6. Definition 5 does not imply that span(M1) ⊆span(M0). However, for the globally h-refined sequence of B-splines employed in this work, this is indeed the case.

Lemma 6. Let M0 and M1 be sets of locally linearly inde-pendent functions on Ω such that span(M1) ⊃ span(M0) andM1 be globally refined with respect toM0. Let Ω# be an opensubset of Ω such that ∃N ∈ M0 : supp N ⊆ Ω#. For the setsM0

# := N ∈ M0 : supp N ⊆ Ω# and M1# := N ∈ M1 :

supp N ⊆ Ω# it holds that

span(M1#) ) span(M0

#)

Proof. The same approach as the first part of the proof forProposition 4 is followed, except with #M1

# , 0 due to therestriction on the selection of the refinement region Ω#. Fur-thermore, by Definition 5, dim(M1

#) > dim(M0#), whereby

span(M1#) , span(M0

#). Hence it holds that span(M1#) )

span(M0#).

Proposition 7 (Strict nestedness of hierarchical B-splinespaces). Let (N r)0≤r<R be a sequence of globally refined B-spline bases and let the refinement regions be chosen with therestriction that ∃Nr

i ∈ Nr : Ωr+1

# ⊇ supp Nri . The space

spanned by the resulting hierarchical B-spline basis HR isstrictly nested within the space spanned by HR+1 that is ob-tained by a further refinement ofHR.

Proof. Again, the proof is similar to that of Proposition 4.The first part is replaced with Lemma 6 to conclude thatspan(HR+1) ) span(HR \ HR+1). In the second part, we havespan(HR+1) ) span(HR+1 ∩ HR) thanks to the restriction onΩ# that guarantees refinement. Combining the two proves thatspan(HR+1) ) span(HR).

Remark 7. Limiting refinement regions Ωr# to unions of basis

function supports makes the adaptive algorithm basis depen-dent. However, in the case of B-splines, this is not an arbitrarychoice and relates directly to the spanned polynomial space,since B-splines form a least support basis due to their local lin-ear independence [41].

Refinement of an already refined basisHR requires assigningthe region Ω# ⊆ Ω, which is marked based on some indicator, toa level i ≤ R. Due to the requirement that Ωr

# ⊇ Ωr+1# when the

marked region Ω# is added to Ωi# all lower level selections 0 <

r < i also have to be adjusted such that Ωr# ⊇ Ωi

#. The selectionof level i is done through inspecting which level the functionN i ∈ HR

# ∩ Ni# that contributes to the refinement region Ωr

#belongs to. The process of selecting a refinement level from acertain selection of functionsN# ⊆ H

R# and the readjustment of

the refinement regions based on this is outlined in Algorithm 1

Remark 8. It is also possible to have coarsening with hierarchi-cal B-splines by choosing a coarsening region and subtractingit from the refinement regions.

3. Goal-oriented error estimation and adaptivity

In the adaptive refinement of discretizations, local refinementindicators are used to determine the regions in the computa-tional domain that need a finer discretization. Most often, these

5

Algorithm 1 Readjustment of selected regions for further re-finement

function readjust(Ωi#

Ri=1, N#)

for all Nri ∈ N# do

Ωr+1# ← int

(Ωr+1

# ∪ supp Nri

)for l← r, 1 do

Ωl# ← Ωl

# ∪ Ωl+1#

end forend for

end function

indicators are based on a method of error estimation and the par-titioning of the estimator into localized contributions. There-fore, for an efficient adaptive method it is important to havea good error estimator and reliable local refinement indicatorsbased on this estimator. Many concepts on goal-oriented errorestimation from the finite element literature apply directly toIsogeometric Analysis. In this work, the originality in terms ofgoal-adaptivity is the construction of an error estimator for hier-archical B-spline discretizations, the extraction of suitable localrefinement indicators from it, and the study of their properties.The application of goal-oriented error estimation to Isogeomet-ric Analysis can be found in [42] for free boundary problemsand in [43] for non-linear rods, where in the latter k-refinementis used for the estimator. Nonetheless, no attempts have beenmade on goal-adaptivity, for which additional nontrivial ingre-dients are required. In the rest of this section, the concept ofgoal-adaptivity is outlined, an error estimator for hierarchicalB-spline discretizations is introduced and two methods of ex-tracting suitable refinement indicators from the error estimatorare proposed.

3.1. Goal-oriented error estimation

The aim in goal-oriented error estimation is having a quan-titative estimate of the influence of the discretization error ona chosen quantity of interest. Here, we recall the derivation ofthe goal-oriented error estimation for linear problems and linearquantities of interest. Extension to non-linear cases is possible(see e.g. [44, 38, 45, 46]).

Let a linear problem be given in the abstract form:

Find u ∈ V such that:

B(u, v) = L(v) ∀v ∈ V (4)

with continuous linear form L(·) and continuous bilinear formB(·, ·) coercive on V , where V is a Hilbert space, typically aSobolev space. The corresponding conforming Galerkin ap-proximation writes:

Find uh ∈ Vh ⊂ V such that:

B(uh, vh) = L(vh) ∀vh ∈ Vh (5)

The discretization error is defined as:

e := u − uh (6)

where e ∈ V , with u ∈ V being the exact solution and uh ∈ Vh

being its Galerkin approximation. For Galerkin approximationsof linear problems, the following holds for the error:

B(e, ψh) = 0 ∀ψh ∈ Vh (7)

which is called Galerkin orthogonality.The residual functional Rh(·) : V → R for the Galerkin ap-

proximation uh is defined as follows:

Rh(v) := L(v) − B(uh, v) = B(u, v) − B(uh, v) = B(e, v) (8)

Our quantity of interest is represented by a linear goal func-tional Q(·) : V → R of the solution, whereby the error in thequantity of interest is expressed as:

Q(u) − Q(uh) = Q(u − uh) = Q(e) (9)

For finding the error in terms of the goal functional, the follow-ing dual problem is introduced:

Find z ∈ V such that:

B(v, z) = Q(v) ∀v ∈ V (10)

Noticing that e ∈ V , equation (10) is also valid for v = e. Hencefor e it holds that B(e, z) = Q(e). Using this and (9), the exacterror in the quantity of interest can be expressed as:

Q(u) − Q(uh) = B(e, z) (11)

Due to Galerkin orthogonality (7) it holds that ∀ψh ∈

Vh, B(e, z) = B(e, z−ψh). Using this, expression in (11) can berewritten as:

Q(u) − Q(uh) = B(e, z − ψh) = L(z − ψh) − B(uh, z − ψh)

= Rh(z − ψh)

∀ψh ∈ Vh, which can be directly evaluated if an approxima-tion z ≈ z is found such that B(v, z) ≈ Q(v), ∀v ∈ V . Usingthe approximate dual solution a goal-oriented error estimatoris obtained by evaluating the dual-weighted residual for anyψh ∈ Vh:

Q(u) − Q(uh) ≈ Rh(z − ψh) =: Est (12)

In FEM, a common way of obtaining z is through Galerkindiscretization of the dual problem (10) with z ∈ V ⊂ V . Thediscretized dual problem is then stated as:

Find z ∈ V such that:

B(v, z) = L(v) ∀v ∈ V (13)

6

Remark 9. The solution space for the dual problem, V , shouldnot be in Vh, i.e. V * Vh. Otherwise B(e, z) = 0 by (7) andhence Est = 0. V is often taken such that V ) Vh.

With the strategy of refining regions composed of functionsupports introduced in Section 2, methods of marking regionsthat comply with this marking restriction are needed. With thisingredient to be addressed later on, the goal-adaptive procedureis outlined in Algorithm 2

Algorithm 2 Goal-adaptive algorithm with hierarchical splines

i← 0,H i ← N0p , H

i ← N0p+1

while i ≤ imax douh ←solve_primal(H i), z←solve_dual(H i)Ωi+1

# ←mark_supports(uh, z)H i+1 ←refine(H i, Ωi+1

# ), H i+1 ←refine(H i, Ωi+1# ),

i← i + 1end while

3.2. The discrete dual problem

The dual problem is also approximated using Galerkin meth-ods with suitable discrete spaces. In the context of Isogeomet-ric Analysis, there are multiple options for the discretizationof the dual problem. In classical FEM the goal-oriented esti-mators are often obtained using discrete dual spaces that resultfrom uniform h- or p-refinement of the primal basis. In Iso-geometric discretizations, due to the additional freedom in ad-justing smoothness across elements, there are many more pos-sibilities for the choice of the dual space V , even with tensorproduct spaces. In [47], an overview of the choices is providedwith dimensions of the dual spaces. An interesting approachthat is unique to Isogeometric Analysis is the application ofk-refinement for the dual discretization, which reduces com-putational cost compared to the more conventional h- and p-refinement. Error estimation with k-refined dual discretizationshas been investigated on uniform meshes in [43].

In this work, we limit ourselves to the rather safe choice ofp-refined dual discretizations: For an h, p, k approximation ofthe primal solution, the proposed estimator will be based on anh, p + 1, k approximation of the dual solution. This means thatelement sizes h, inter-element continuities k and geometries ofelements match for the dual and primal meshes, but the discretedual solution will have a degree p + 1 polynomial approxima-tion, compared to degree p for the primal solutions. With thischoice V ⊃ Vh and the dual basis functions have similar sup-ports to the primal basis functions. This is relevant because theadaptive algorithm uses function supports as refinement regionsand the refinement of primal and dual meshes are performed inconjunction. The dual basis corresponding to the primal basisof Figure 2 is given in Figure 4.

For the chosen estimator, based on an h, p + 1, k approxi-mation of the dual solution, the dual space has the propertythat V ) Vh on an unrefined B-spline mesh. However, thisproperty is not necessarily valid once Vh = span(H i) has been

00.20.40.60.8

1

N0

00.20.40.60.8

1

N1

00.20.40.60.8

1

0 1 2 3 4 5

H2

ξ

Figure 4: The h, p + 1, k dual basis H2, corresponding to theprimal basis in Figure 2, along with its globally refined con-stituents N0 and N1

refined. To have an estimator that behaves similarly after adap-tive refinements, it is proposed that dual space V = span(H i) isadapted alongside Vh using the same refinement regions Ωi

#.The choice of having H i and H i refined simultaneously, at

least guarantees that V ) Vh, which is a property of the initialB-spline spaces:

Theorem 8. LetHR be a hierarchically refined basis that usesglobally h-refined B-spline bases (N r)0≤r<R of degree p with asequence of refinement regions (Ωr

#)0≤r≤R such that Ωr# contains

the support of at least one function fromN r and let HR be a hi-erarchically refined basis that uses the same refinement regions(Ωr

#)0≤r≤R with globally h-refined B-spline bases (N r)0≤r<R ofdegree p + 1 with elements matching those of (N r)0≤r<R. Thefollowing holds for these bases:

span(HR) ) span(HR)

Proof. Due to the property that a B-spline space is nested in itsp-refined space, it holds that:

span(N r) ) span(N r) (14)

For the hierarchically refined basis at a level r+1 > 1, it followsfrom Definition 1 that:

span(H r+1) =(span(H r) \ span(N r−1

#r))∪ span(N r

#r) (15)

where N r#i

= N ∈ N r : supp N ⊆ Ωi#. Both N r−1

#rand N r

#rare

nonempty due to the support restriction on the selection of Ωr#

Similar to (15), it also holds that span(H r+1) =(span(H r) \ span(N r−1

#r))∪ span(N r

#r), which implies

span(H r+1) ⊇ span(N r#r

). Comparing the newly addedprimal and dual basis functions, by Lemma 6 it should holdthat span(N r

#r) ) span(N r

#r) for a marked region Ωr

# ⊆ Ω,since N r is globally p-refined with respect to N r andspan(N r) ) span(N r), whereby it is proven that

span(H r+1) ) span(N r#r

) (16)

7

For the initial spaces, (14) is valid as they have not undergoneany refinement. Thus:

span(H1)(14)( span(H1)

Prp.7( span(H2) (17)

and for the next step

span(H2)(15)=

(span(H1) \ span(N0

#1))∪ span(N1

#1)

span(H2) ⊂ span(H1) ∪ span(N1#1

)

span(H2)(16),(17)( span(H2)

Prp.7( span(H3) (18)

Using (16) and the result from the previous step, the same pro-cess as in going from (17) to (18) can be repeated to yieldspan(HR) ) span(HR).

3.3. Refinement indicators for hierarchical B-spline bases

The chosen refinement strategy with hierarchical B-splinesis based on marking function supports for refinement, as ex-plained in Section 2. This is somewhat different from the adap-tive methods based on marking of elements. In this section, twomethods are introduced for obtaining refinement indicators lo-calized to function supports from the global goal-oriented errorestimator proposed earlier. The employed marking strategy willalso be explained for the two indicators.

3.3.1. A function support refinement indicatorA natural way of obtaining local indicators from error esti-

mation is by partitioning the estimator into local contributions.Since the indicators are meant for function supports, the er-ror is partitioned into basis function contributions. Differentlyfrom element partitions, basis function supports in general donot form a disjoint partition of the computational domain, i.e.∀Ni ∈ H

R,∃N j ∈ HR : supp Ni ∩ supp N j , ∅ if p , 0, with

HR : Vh = span(HR) being the primal basis.Considering B-splines of degree p > 0, it holds that Ω =⋃Ni∈H

R supp Ni, and with the assumption that the domainparametrization T : Ω → Ω is a continuous map, it also holdsthat Ω =

⋃Ni∈H

R supp N′i , where N′i = Ni T−1. This allowsfor splitting up any Cl(Ω)-continuous function f in the follow-ing way:

f =∑

i:Ni∈HR

fi (19)

where fi are Cl(Ω)-continuous partitions of f local to the sup-port of the basis function Ni; namely, fi|supp N′i ∈ Ck(supp N′i )and fi|Ω\supp N′i = 0. It can be required that fi|∂ supp N′i \∂Ω = 0thanks to the overlap of supports; i.e. all points on ∂ supp Ni \

∂Ω are interior points of the support of some other basis func-tion N ∈ HR. This is the approach that is taken in splitting thedual solution, z ∈ Ck(Ω) with 0 ≤ k < p, and its approximationin Vh, ψh, that are in the argument of the dual-weighted residual(12). For approximating z with ψh ∈ Vh, L2-orthogonal projec-tion is selected for convenience. Since L(·) and B(uh, ·) consistonly of integrals over the whole domain Ω and the boundary Γ,the partitioning of z − ψh in this way results in integrals over

basis function supports supp N′i . With this approach, the errorestimator can be rewritten as:

Est = Rh(z − ψh) =∑

i:Ni∈HR

Rh(zi − ψhi ) =

∑i:Ni∈H

R

ηsi

with support contributions:

ηsi = Rh(zi − ψ

hi ) (20)

|ηsi | is the support refinement indicator for basis function Ni.

With a marking ratio λ ∈]0, 1], the marked region using thefunction support indicator is given as:

Ωs# = int

⋃N∈HR

#

supp N HR# = Ni ∈ H

R : |ηsi | ≥ λ max

N∈HR|ηs

N |

(21)which is essentially a maximum marking strategy for supports.Remark 10. The function support partitioning of the global es-timator results in refinement indicators that are basis dependent.

For the partitioning of z − ψh, the basis HR of the dual spaceV can be conveniently made use of. For the proposed indicator,we choose to separate z−ψh into locally supported functions inthe following way:

zi − ψhi =

∑j:N j∈H

Ri

z j − ψhj

#HRj

N′j (22)

where HRi = N ∈ HR : supp N ⊆ supp Ni for a primal basis

function Ni ∈ HR and HR

j = N ∈ HR : supp N ⊇ supp N j

for a dual basis function N j ∈ HR. The constants z j and ψh

j arethe coefficients of the dual basis function N j. The expression(22) essentially distributes the dual basis function contributionsz jN′j and ψh

j N′j equally to all primal basis functions N′i whose

supports contain supp N′j.Remark 11. By Theorem 8 and the local linear independence ofsingle-level B-spline bases, it can be shown that both sets HR

iandHR

j seen in (22) are nonempty.Remark 12. For this particular partition, it is necessary thatspan(HR) = V ⊃ Vh, whereby it is possible to express anyψh ∈ Vh in terms of the dual basis as in ψh =

∑j:N j∈H

R ψhj N′j.

3.3.2. An element-based refinement indicatorApart from creating refinement indicators specifically meant

for function supports, it is also possible to use element refine-ment indicators for marking and subsequently to extend themarked elements to function supports by including neighbor-ing elements.

Given a partial differential equation expressed in the formLu = f in Ω, with a differential operator L of order at most k +

1, appropriate boundary conditions on ∂Ω and an approximatesolution uh ∈ Vh ⊂ Ck(Ω); the element indicators are definedas:

ηi =

∫Ki

( f − Luh)(z − ψh) dx +

∫∂Ω∩∂Ki

G(z − ψh) ds (23)

with G(·) denoting the integrand of any integrals over ∂Ω

present in the expression for Rh(·).

8

Remark 13. Indicators of the form (23) can be obtained by el-ementwise integration by parts of the dual-weighted residual(12). The absence of interior boundary terms is due to thediscrete solution having continuous kth derivatives; the bound-ary terms resulting from the elementwise integration by partscancel across the interior element edges, affirming that Est =∑

i:Ki∈TR ηi also when they are discarded from the element con-tributions. In Section 4.1, this is exemplified for the C1 dis-cretizations of the Laplace operator.

It must be noted that the indicator ηi given in (23) is the re-finement indicator of an individual element. In the proposedelement-based method, the marking decision is done based onthe element indicator ηi

Te# = Ki ∈ TR : |ηKi | ≥ λmax

K∈TR|ηK |

using a marking ratio λ ∈ (0, 1]. Afterwards, a marked elementKi ∈ Te

# is extended to a function support supp Ni ⊇ Ki bytaking the combinations of indicators for neighboring elementswhich yield the highest refinement indicator:

Ωei# = supp Ni with Ni = arg max

N∈HRKi

∣∣∣∣∣∣∣∣∑

n:Kn∈SN

ηn

∣∣∣∣∣∣∣∣ (24)

where HRKi

= N ∈ HR : supp N ⊇ Ki is the set of basisfunctions that have some support over Ki and SN = K ∈ TR :K ⊆ supp N is the set of elements that are within the supportof basis function N. The refinement region is then the union ofthese regions for every marked element:

Ωe# = int

⋃i:Ki∈Te

#

Ωei# (25)

Remark 14. This process of combining elements to obtain suit-able refinement regions as required by the hierarchical refine-ment algorithm (Algorithm 2) is analogous to the completionprocedures [48, 49] in traditional adaptive methods, which areused for obtaining conforming discrete spaces after a bisectionoperation.

4. Numerical simulations

In this section we first consider the Poisson problem to studythe quality of the proposed error estimators under uniform meshrefinement and its adaptive discretization on an L-shaped do-main. The effectivity of the refinement indicators in com-bination with the proposed adaptivity approach is then stud-ied. Finally the adaptive discretization of a prototypical free-surface flow problem is demonstrated. For both problems two-dimensional domains are considered.

4.1. The Poisson problem

We consider the Poisson problem over a domain Ω with ho-mogeneous Dirichlet conditions over the boundary ΓD, and gen-eral Neumann boundary conditions over the boundary ΓN . Note

that ΓD ∪ ΓN = ∂Ω and int(ΓD) ∩ int(ΓN) = ∅. The weak formproblem is given by Problem (4) with

B(u, v) =

∫Ω

∇u · ∇v dx, L(v) =

∫Ω

f v dx +

∫ΓN

gv ds (26)

andV = v ∈ H1(Ω) : v|ΓD = 0 (27)

where f and g represent the source term and Neumann data,respectively. The corresponding dual problem is given by prob-lem (10) using the same bilinear form B(·, ·), a linear goal func-tional Q(·), and the dual space V = H1

0(Ω).For this problem, the function support indicator (20) as intro-

duced in Section 3.3.1 substantiates as

|ηsi | =

∣∣∣∣∣∣∫

ΓN∩∂ supp N′ig(z − ψh) ds

−

∫supp N′i

∇ uh · ∇(z − ψh) − (z − ψh) f dx∣∣∣∣∣∣

(28)

The element-based refinement indicator (23) as introduced inSection 3.3.2 follows as

|ηi| =

∣∣∣∣∣∣∫

ΓN∩∂Ki

(g − ∇ uh · n)(z − ψh) ds

+

∫Ki

(∆ uh + f )(z − ψh) dx∣∣∣∣∣∣

(29)

For both indicators, ψh ∈ Vh is taken as the orthogonal L2-projection of z onto Vh.

4.1.1. Effectivity of the error estimatorsEffectivity of an estimator is defined as the ratio of the actual

error and the estimated error:

Ieff =Est

Q(u) − Q(uh)

To assess the effectivity of the proposed error indicators, wefirst consider the convergence of their corresponding error es-timators under uniform refinement. We consider the domainΩ = (−0.5, 0.5) × (−0.5, 0.5) with homogeneous Dirichlet con-ditions over ΓD = ∂Ω and the source term that satisfies (26)when the exact solution is given by u(x, y) = (x − 1/2)(x +

1/2)(y − 1/2)(y + 1/2)/ ((x + 2/3)(y + 3/4)). The quantity ofinterest is taken as the average value of the solution over thewhole domain, i.e. Q(u) =

∫Ω

u dx.The primal and dual problems are discretized on uniformly

h-refined meshes with Cp−1-continuous B-splines of degree pand p + 1, respectively. In Figure 5, the actual and estimatederrors are plotted against the dimension of the discrete primalspace, n, along with their effectivity indices. The observed op-timal convergence rate for the errors, i.e. −2p/d expected ofthe exact error by the theory of ordinary adaptive FEM [48],and the effectivity indices approaching 1 demonstrate that theerror estimator adequately reflects the error in the quantity ofinterest.

9

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

10 100 1000 10000

Err

orinQ

(u)

n

slope−2

slope−3

Est, p = 2, p = 3Est, p = 3, p = 4

Exact, p = 2Exact, p = 3

(a) Convergence in the quantity of interest

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

0 200 400 600 800 1000 1200 1400

I eff

n

p = 2, p = 3p = 3, p = 4

(b) The effectivity indices

Figure 5: Study of the estimator under uniform refinement

In Figure 6 the sum of the absolute value of refinement in-dicators is plotted for both localizations of the estimated error.The observed optimal rates of convergence for both indicatorssuggest that the proposed refinement indicators are asymptoti-cally equivalent to the actual error in the quantity of interest.

4.1.2. Adaptive discretization on an L-shaped domain

We next consider the adaptive discretization of the homoge-neous Poisson problem (i.e. f = 0) on the L-shaped domain asshown in Figure 7. Homogeneous Dirichlet conditions are as-sumed over the boundary ΓD and the Neumann data over ΓN isgiven by

g(x, y) =

g1(x, y) x > y and (x, y) ∈ ΓN2

g2(x, y) x ≤ y and (x, y) ∈ ΓN2

−g2(x, y) x > y and (x, y) ∈ ΓN1

−g1(x, y) x ≤ y and (x, y) ∈ ΓN1

(30)

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

10 100 1000 10000

∑ i:Ni∈H

R|η

s i|

n

slope−2

slope−3

p = 2, p = 3p = 3, p = 4

(a) Support indicators

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

10 100 1000 10000

∑ Ki∈TR|η

Ki|

n

slope−2

slope−3

p = 2, p = 3p = 3, p = 4p = 2, p = 3p = 3, p = 4

(b) Element-based indicators

Figure 6: Convergence of the refinement indicators under uni-form mesh refinement

10

x

y

ΓD

ΓD

ΓN1

ΓN2

ΓN2

ΓN1

(0, 0)(−1, 0)

(−1, 1) (1, 1)

(1,−1)(0,−1)

Ω

Figure 7: Description of the L-shaped domain

with ΓN1 and ΓN2 as described in Figure 7 and

g1(x, y) =2[x sin

(2θ(x,y)+π

3

)− y cos

(2θ(x,y)+π

3

)]3(x2 + y2)2/3

g2(x, y) =2[y sin

(2θ(x,y)+π

3

)+ x cos

(2θ(x,y)+π

3

)]3(x2 + y2)2/3

where θ(x, y) is the clockwise angle between the x-axis and theline connecting points (0, 0) and (x, y). We choose the quantityof interest as:

Q(u) = u(1,−1) =

∫Ω

u(x, y)δ(1 − x)δ(−1 − y) dx (31)

where δ is the Dirac distribution. For the exact solution Q(u) =3√

2/2. It is to be noted that in this case both the primal problemand the dual problem incur singularities, near (0, 0) and (1,−1),respectively.

For the discretization of the primal problem a series of hier-archically refined B-spline spaces Vh = span(H ′) ∩ V ⊂ C1(Ω)constructed from a hierarchy of continuously differentiable h-refined second order B-spline bases N r. For the discretiza-tion of the dual problem, a series of discrete spaces V =

span(H ′) ∩ V constructed from a hierarchy of C1-continuoush-refined third order B-spline bases N r is chosen. To ensure C1-continuity throughout the interior of the domain, the L-shapedgeometry is parametrized using repeating control points at thecorners (0, 0) and (1, 1), as also been done e.g. in [1, Fig. 17].The refinement indicators used for this problem are expressedin (28) and (29).

For the marking strategy three ratios λ ∈ 0.95, 0.5, 0.25are used, meaning all regions with indicator |η| ≥ λ|ηmax| aremarked for refinement. The solutions of the primal and the dualproblems after 50 refinement steps with λ = 0.95 are plottedin Figure 8. The Bézier meshes after 15, 30 and 60 refinementsteps using both indicators are plotted in Figure 9, for the caseλ = 0.95. As would be expected, refinement with both indi-cators is generally concentrated around the singularities of theprimal and the dual solutions, namely near the re-entrant cornerat the origin and the location of the quantity of interest, (1,−1).Overall, the element-based indicator appears to mark less perrefinement step. This behavior can be attributed to the fact that,with the support indicator, a single element that has a large er-

Figure 8: Solutions of the primal (left) and the dual (right) prob-lems of Poisson problem on the L-shaped domain, λ = 0.95

(a) Support indicator

(b) Element-based indicator

Figure 9: Bézier meshes for the Poisson problem; after 15, 30and 60 refinements, with λ = 0.95

11

1e-06

1e-05

0.0001

0.001

0.01

0.1

10 100 1000 10000

Err

orinQ

(u)

n

slope −3/2

slope−2

UniformElem. ind., exact

Elem. ind., EstSupp. ind., exact

Supp. ind., Est

(a) Timid marking (λ = 0.95)

1e-06

1e-05

0.0001

0.001

0.01

0.1

10 100 1000 10000

Err

orinQ

(u)

n

slope −3/2

slope−2

UniformElem. ind., exact

Elem. ind., EstSupp. ind., exact

Supp. ind., Est

(b) Balanced marking (λ = 0.5)

1e-06

1e-05

0.0001

0.001

0.01

0.1

10 100 1000 10000

Err

orinQ

(u)

n

slope −3/2

slope−2

UniformElem. ind., exact

Elem. ind., EstSupp. ind., exact

Supp. ind., Est

(c) Liberal marking (λ = 0.25)

Figure 10: Errors in the quantity of interest under adaptive dis-cretization of the Poisson problem on the L-shaped domain,with various marking ratios

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 200 400 600 800 1000 1200 1400 1600 1800

I eff

n

Elem. ind.Supp. ind.

Figure 11: Effectivity indices under adaptive discretization ofthe Poisson problem on the L-shaped domain, with λ = 0.95

ror can result in the marking of multiple basis functions whosesupports contain that element.

The exact and estimated errors are plotted versus the numberof degrees of freedom in Figure 10 for the three marking ratios.Due to the singularities in the primal and dual solutions, theuniform-refinement approximation converges at a suboptimalrate of −3/2. The adaptive approximation recovers the opti-mal convergence rate of −2, for both the support indicator andthe element-based indicator. Furthermore, the accuracy of theadaptive discretizations is not very sensitive to how liberallymarking is performed, as described by the parameter λ. This isbeneficial in limiting the number of intermediate solves in theadaptive algorithm. Both indicators show similar behavior interms of the improvement of accuracy per added degree of free-dom, with the support indicator performing only slightly better.

The effectivities of the error estimators (Figure 11) stay insimilar regions for adaptive discretizations using the two indi-cators. Although the actual error is underestimated by the es-timator in both cases, they are is in close agreement with theerror for most part, at a level above or around 70% of the actualerror. The adaptive discretizations using the element indicatorslightly outperforms those with the support indicator in termsof the accuracy of the estimated error.

In Table 1 we give a general indication of the total computa-tional cost of the proposed adaptive algorithm, for various val-ues of the marking parameter λ. In all cases investigated hereexcept for the adaptive refinement based on the element indi-cator with the impractical marking ratio of λ = 0.95, the to-tal cost of the adaptive algorithm is smaller than a single solveof the uniformly refined case. For the adaptive discretizations,n denotes the number of degrees of freedom after the last re-finement step. Cholesky (LDLT) decomposition is used for alllinear solves. The underperformance of the element-based in-dicator in terms of the total CPU time can be attributed to thefact that for a given λ, the support indicator marks much moreliberally; whereby the same error level is reached with a muchsmaller number of refinement steps. Lastly, it is worth noting

12

Table 1: CPU times with various values of the marking param-eter λ

Discretization n |Q(u) − Q(uh)| CPU timeUniform 2244 5.72972 · 10−5 155 sSupp. ind., λ = 0.25 294 5.47364 · 10−5 13 sSupp. ind., λ = 0.5 325 4.91581 · 10−5 19 sSupp. ind., λ = 0.95 315 5.48413 · 10−5 44 sElem. ind., λ = 0.25 314 5.63982 · 10−5 135 sElem. ind., λ = 0.5 301 6.19853 · 10−5 150 sElem. ind., λ = 0.95 301 6.19853 · 10−5 232 s

Γ

ΓDw

ΓD fΓD f

(0, 0) (1, 0)

(1.5, 0.5)

(2, 0) (4, 0)

(4, 1)(0, 1)

y

x

Ω

αθΓ0

Figure 12: Description of the notched channel domain for thefree-surface flow problem

that the CPU times are obtained with no effort on optimizingthe implementation of the adaptive algorithm.

4.2. Free-surface flow

We consider the Bernoulli free-boundary problem [42] overthe domain depicted in Figure 12 as a prototypical free-surfaceproblem for a flow through a notched channel. This problem isparticularly of interest to B-spline discretizations, since the dualproblem contains the curvature of the free surface for whichhigher-continuity discrete geometries are preferred. Laplace’sequation, i.e. ∆u = 0, is solved over the domain Ω while Dirich-let boundary conditions are prescribed over the fixed part of theboundary, homogeneous over ΓDw and nonhomogeneous overΓD f according to u = uD = y. Over the free boundary Γ, bothNeumann and Dirichlet conditions are prescribed as u = uD = 1and ∂u

∂xn=: g = 1.

In order to satisfy both boundary conditions, the boundary Γ

has to attain a specific shape, which makes the Bernoulli prob-lem a free-boundary problem. Since the solution to the bound-ary value problem depends on the position of the free-surfaceand vice versa, this problem is inherently non-linear. We em-ploy the explicit Neumann update scheme [42, 50] to solve thefree boundary problem. Alternatively, a Newton scheme can befound in [51].

As a reference configuration, a free surface Γ0 is assumed(i.c. a horizontal surface at y = 1). The displacement of the freeboundary is parametrized with respect to this reference configu-ration using the displacement field θ : Γ0 → R2. The boundaryin the current configuration is denoted by Γθ. The correspond-ing domain Ωθ is defined with respect to the initial domain Ω0,where the deformation is found through the solution of an auxil-iary vector Laplacian problem with θ as the Dirichlet conditionon Γ0 and homogeneous Dirichlet conditions on ΓD.

In the Neumann scheme, based on the iteratively approxi-mated free-boundary θh and the approximate domain Ωθh , theweak problem (4) is solved with

Bθh (u, v) =

∫Ωθh

∇ u · ∇ v dx Lθh (v) =

∫Γθh

gv ds (32)

and

Uθh = u ∈ H1(Ωθh ) : u|ΓD = uD Vθh = v ∈ H1(Ωθh ) : v|ΓD = 0

For the discretization of this problem C1-continuous, secondorder hierarchically refined spline spaces are constructed onan approximate current configuration Ωθh . To ensure C1-continuity in the interior and the free boundary, the kinks inthe geometry of the notch are parametrized by repeating con-trol points.

Once the free-surface problem is solved, a dual problem lin-earized around the approximate free boundary Γθh can be solvedover the deformed domain Ωθh . The derivation of the shape-linearized dual problem can be found in [52]. Here we considerthe elevation αθ(x) of the free boundary Γθ at point x = 2 +

√2

as the goal functional:

Qθ(u) =

∫Γ0

qbdαθ ds and qbd(x) = δ(x−2−√

2) (33)

where the elevation is measured from Γ0. A reference value forthis quantity of interest is obtained from [42] as 0.02269.

The weak form dual problem is given by (10) with

Bθh (z, v) =

∫Ωθh

∇ v · ∇ z dx +

∫Γθh

∂ng + κhgg

zv ds (34)

Lθh (v) = −

∫Γθh

qbd

gv ds (35)

where κh : Γθh → R is the approximate additive curvatureof the boundary. Due to the curvature term, a C1 B-splineparametrization of the free boundary is instrumental for thegoal-adaptive discretization of this problem.

The dual solution is computed on the approximate domainΩθh with a discrete space constructed using C1-continuous, cu-bic hierarchical B-spline basis H ′r. For this problem the pro-posed refinement indicators have the same form as in (28) and(29), but they are computed on the approximate domain Ωθh andthe approximate boundary Γθh

The goal-adaptive discretization of the Bernoulli problem us-ing C1-continuous second order hierarchically refined splinesfor the primal problem and C1-continuous third order splinesfor the dual problem after 60 refinement steps yields the solu-tions plotted in Figure 13. The Bézier meshes for the two indi-cators are given in Figure 14. As expected, most refinement isperformed in the regions of the tip of the notch and the vicinityof the quantity of interest. The two indicators produce largelysimilar meshes, with the element-based indicator again mark-ing less densely than the support indicator per refinement step.Furthermore, the change in mesh density seems to be faster withthe element-based indicator.

13

Figure 13: Solutions of the primal (above) and the dual (below)problems of free-surface flow in a nothced channel

(a) Support indicator

(b) Element-based indicator

Figure 14: Bézier meshes for the free-surface flow problem;after 15, 30 and 60 adaptive refinements, with λ = 0.95

The estimated error and the error with respect to the refer-ence solution are plotted for adaptive refinement using both in-dicators in Figure 15, with marking ratios λ ∈ 0.95, 0.5, 0.25.Adaptive refinement with both indicators result in optimal con-vergence of the error in Q(u), with the support indicator againslightly outperforming overall. For the investigated range ofmarking ratios from λ = 0.25 to λ = 0.95, the adaptivediscretizations produce virtually indistinguishable errors for agiven number of degrees of freedom. The similarity of the adap-tive methods is also seen in their effectivity indices, as plottedin Figure 16. The effectivity indices of the estimator on meshesadapted using both indicators improve rapidly in the first fewadaptivity iterations as the free boundary takes a shape aroundwhich the linearized dual is accurate. The element indicatorseems to reach the point at which the linearization errors arenegligible more rapidly. After this point, both adaptive strate-gies give good estimations of the error, being over 70% of theactual error.

5. Conclusions and recommendations

A goal-adaptive Isogeometric Analysis procedure is pro-posed, where the local refinement of the initial geometry spacesis achieved using hierarchical B-splines. Local h-refinement ofB-spline bases of degree p with refinement indicators derivedfrom a goal-oriented error estimator is considered. For the er-ror estimator, a Galerkin approximation with hierarchically h-refined B-spline bases of degree p + 1 is proposed. The pro-posed discrete dual space is proven to be a strict superset of

1e-06

1e-05

0.0001

0.001

0.01

10 100 1000 10000

Err

orinQ

(u)

n

slope −3/4

slope−2

UniformElem. ind., exact

Elem. ind., EstSupp. ind., exact

Supp. ind., Est

(a) Timid marking (λ = 0.95)

1e-06

1e-05

0.0001

0.001

0.01

10 100 1000 10000

Err

orinQ

(u)

n

slope −3/4

slope−2

UniformElem. ind., exact

Elem. ind., EstSupp. ind., exact

Supp. ind., Est

(b) Balanced marking (λ = 0.5)

1e-06

1e-05

0.0001

0.001

0.01

10 100 1000 10000

Err

orinQ

(u)

n

slope −3/4

slope−2

UniformElem. ind., exact

Elem. ind., EstSupp. ind., exact

Supp. ind., Est

(c) Liberal marking (λ = 0.25)

Figure 15: Errors in the quantity of interest under adaptive dis-cretization of the Bernoulli problem, with various marking ra-tios

14

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 200 400 600 800 1000 1200 1400 1600

I eff

n

Elem. ind.Supp. ind.

Figure 16: Effectivity indices under adaptive refinement of theBernoulli problem, with λ = 0.95

the discrete primal space. To comply with the refinement re-gion requirements of the hierarchical refinement strategy, twonovel refinement indicators are constructed to be used in themarking of function supports for refinement. These indicatorsare the support indicator, which is directly used for the mark-ing of basis function supports, and the element-based indicator,which uses combinations of element indicators to decide whichsupport to refine.

Based on investigations with a model problem, it is con-cluded that the proposed error estimator converges with optimalrates under uniform refinement. Furthermore, the estimated er-ror is assessed to stay sharp under uniform refinement, witheffectivity indices generally in the range of 0.9-1 for the con-sidered case. The local error indicators specifically devised forthe hierarchical refinement procedure are shown to convergewith the same optimal rates under uniform refinement, whichis judged to be a strong indication of sound adaptivity poten-tial.

The studied individual ingredients are combined in an adap-tive procedure whose performance is demonstrated with twosteady example problems: Poisson’s equation on an L-shapeddomain and free-surface flow through a notched channel. Adap-tive refinement using both indicators result in optimal conver-gence rates in terms of the quantity of interest for both exam-ple problems, despite the low regularity of the solutions. Fromthe apparent rates of convergence of the adaptive solutions it isconcluded that both indicators are suitable for the hierarchicalrefinement procedure. Furthermore, the effectivity indices ofthe error estimator on adapted approximation spaces stay in therange 0.7-1 for the most part, showing that the proposed errorestimator remains accurate under adaptive refinement as well,even for problems with significant non-linearities arising fromfree boundaries.

The aim of goal-adaptive refinement of spline spaces isshown to be achievable with some new ideas on the localiza-tion of the error estimator and the flexibility of refinement thatis provided by hierarchical splines. For the ultimate goal of

unifying design and analysis practices in engineering, this re-sult can be seen as a step in eliminating some of the necessitiesof manual interventions that prevent a large scale implementa-tion of the design-through-analysis methodology proposed byIsogeometric Analysis.

It has been remarked that there are possibilities other thanthe ones explored here. In particular, the possibility of localh- and p-refinement offered by hierarchical splines could becombined to permit hp-adaptive Isogeometric Analysis, withsome effort to modify the procedure. Same modifications wouldlikely enable anisotropic refinements as well. Therefore, the ex-tension of the hierarchical refinement procedures to anisotropichp-refinement is a natural direction to follow. Furthermore, itwas suggested that Isogeometric Analysis permits many differ-ent options for the discretization of the dual problem. A studyof error estimators based on other discrete duals may result inindicators suitable for adaptivity purposes that are cheaper tocompute. Lastly, the application of the adaptive procedure tohierarchical NURBS, as already explored in [22] with a dif-ferent error estimator, is an important step for analysis in theindustrial context. As the critical properties of B-splines usedhere are also shared by NURBS, the proposed estimator andthe refinement indicators introduced here are generalizable toadaptivity with hierarchical NURBS.

Acknowledgements

The research of G. Kuru is partially funded by the EuropeanCommision FP7 ANADE (PITN-GA-289428) project. The re-search of K.G. van der Zee and C.V. Verhoosel is funded bythe Netherlands Organisation for Scientific Research (NWO),VENI scheme.

A. Comparison of hierarchical and knot insertion bases in1D

With hierarchical splines, functions from multiple levels canhave overlapping supports. This results in slightly higher band-width for the resulting matrix equations and possibly can affectthe conditioning adversely. In the univariate case, it is possibleto obtain ordinary B-spline bases, locally refined by knot inser-tion, that span the same discrete space as hierarchical splines.In the following, the hierarchical spline bases will be comparedto the knot insertion bases in terms of the condition numberfor the solution of matrix equations arising from a 1D Poissonproblem.

Given the open interval Ω = (−1, 1), the problem in its strongform is:

Find u : Ω→ R such that,

d2udx2 = f in Ω

u = uD on ∂Ω

(36)

15

Figure 17: Adapted bases spanning the same discrete space atstep 15, hierarchical B-spline basis (left) and knot-insertion B-spline basis (right)

with

f (x) =2a3x

(1 + a2x2)2 , uD(x) = arctan(ax)

and a > 0 a scaling coefficient. It can be shown that u =

arctan(ax) is the solution to problem (36).The weak form of problem (36) is discretized by choosing fi-

nite dimensional h-refined quadratic hierarchical spline spacesas the trial and test spaces of the Galerkin method. The hier-archical basis that spans this space is denoted HR and the knotinsertion basis that spans the same space is denoted N i.

Since the exact solution u = arctan(ax) is known, element-wise H1 norms of the true error, ηK = ||uh − u||H1(K), are used aselement indicators to drive the refinement procedure, a thresh-olding-type marking [48], accompanied with a completion stepas with the element-based indicator explained in Section 3.With parameters a = 1000 for the forcing and λ = 0.9 for themarking fraction, the condition numbers σmax/σmin, defined asthe ratio of the maximum and the minimum singular values ofthe stiffness matrix, are plotted against degrees of freedom inFigure 18. An example of the resulting bases at refinement step15 is give in Figure 17, along with the sparsity pattern of the re-sulting stiffness matrix (Figure 19) to give an indication of theinfluence of the increased overlap on the bandwidth.

The results show that having lower and higher level func-tions together results in increased condition numbers for bothbases. However, the condition number is less severe for the hi-erarchically refined basis, with the employed choice of havingrefinement regions defined by the union of basis function sup-ports.

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17