Hybridized Isogeometric Method for Elliptic Problems on CAD Surfaces with Gaps Tobias Jonsson Mats G. Larson Karl Larsson

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Hybridized Isogeometric Method for Elliptic

Problems on CAD Surfaces with Gaps

Tobias Jonsson Mats G. Larson Karl Larsson

March 8, 2023

Abstract

We develop a method for solving elliptic partial diﬀerential equations on surfaces

described by CAD patches that may have gaps/overlaps. The method is based on

hybridization using a three-dimensional mesh that covers the gap/overlap between

patches. Thus, the hybrid variable is deﬁned on a three-dimensional mesh, and we

need to add appropriate normal stabilization to obtain an accurate solution, which

we show can be done by adding a suitable term to the weak form. In practical

applications, the hybrid mesh may be conveniently constructed using an octree to

eﬃciently compute the necessary geometric information. We prove error estimates

and present several numerical examples illustrating the application of the method

to diﬀerent problems, including a realistic CAD model.

1 Introduction

CAD models describe surfaces using a collection of patches that meet in curves and

points. Ideally, the CAD surface is watertight, but in practice, there are often gaps or

overlaps between neighboring patches. These gaps/overlaps may cause serious meshing

and ﬁnite element analysis problems and in practical applications the CAD model often

needs to be corrected before meshing is possible. This paper develops a robust isogeo-

metric method [8] for handling CAD surfaces with gaps/overlaps. The main idea is to

cover the gaps/overlaps with a three-dimensional mesh and then use a hybrid variable on

this mesh together with a Nitsche-type formulation. The hybrid variable transfers data

between neighboring patches, and there is no direct communication between the patches.

To obtain a convergent method, the hybrid variable must be given enough stiﬀness in the

directions normal to the interface. We show that this can be done by adding a suitable

term to the weak statement. We allow trimmed patches and add appropriate stabiliza-

tion terms to control the behavior of the ﬁnite element functions in the vicinity of the

trimmed boundaries using techniques from CutFEM, see [3]. In practice, we suggest an

octree structure for setting up the hybrid mesh to facilitate eﬃcient computation of the

involved terms. We allow standard conforming ﬁnite element spaces as well as spline

spaces with higher regularity. We derive error estimates and present several numerical

examples illustrating the method’s convergence and application to a realistic CAD model.

Related Work. A framework that is also based on a patchwise parametrically de-

scribed geometry combined with a Nitsche type method to couple the solution over patch

arXiv:2210.01617v2 [math.NA] 7 Mar 2023

interfaces is the discontinuous Galerkin isogeometric analysis [16, 17], which considers

gaps/overlaps in [12, 13]. One major diﬀerence to the present work is that the method

involves the explicit construction of a parametric map between corresponding points over

interfaces with gaps, which in our method is implicit through the stabilization of the

hybrid variable. In our view the hybridized approach leads to a considerably more con-

venient and robust implementation that also has the beneﬁt of supporting interfaces

coupling more than two patches, cf. [10]. Our usage of the hybrid variable resembles the

bending strip method for Kirchhoﬀ plates [15], in which strips of ﬁctitious material with

unidirectional bending stiﬀness and zero membrane stiﬀness are placed to cover the gaps

and are used for coupling the solution over the patch interfaces. The coupling of solu-

tions over imperfect interfaces is also addressed in overlapping mesh problems where the

solution is deﬁned on two separate meshes whose boundaries do not match, but rather

intersect each other’s meshes. This was extended to gaps in [1, 9] where elements close

to the interface were modiﬁed to cover the gap, eliminating the gap regions and creating

an overlapping mesh situation instead. However, it is not clear how overlapping mesh

techniques could be utilized to couple solutions on surfaces since the patch meshes do not

necessarily lie on the same smooth surface.

Outline. The paper is organized as follows: In Section 2 we present the method, in

Section 3 we show stability and error estimates, and in Section 4 we present numerical

experiments and examples.

2 Model Problem and Method

The main contribution of this paper is the robust coupling of solutions over patch in-

terfaces with gaps/overlaps. To simplify the derivation and analysis of the method, we

consider a simpliﬁed model problem that allows us to focus on the central issue and

avoid complicated notation and unrelated technical arguments. We include remarks and

references on how the method is extended to more general problems on CAD surfaces.

2.1 Model Problem

We introduce a two-dimensional model problem with a gap at an internal interface, derive

a hybridized formulation and the corresponding ﬁnite element method, together with the

necessary notation to proceed with the analysis.

Model for a Domain with Gap. We introduce the following set-up and notation,

illustrated in Figure 1:

•Consider a domain Ω ⊂R2and let Ω1and Ω2be a partition of Ω into two subsets

separated by a smooth interface Γ, such that Ω1is the exterior domain and Ω2is the

interior domain. Let Uδ(Γ) ⊂R3be the open three-dimensional tubular neighborhood

of Γ with thickness 2δ. Then there is δ0>0 such that the closest point mapping

pΓ:Uδ0(Γ) →Γ is well deﬁned.

•Let Ωi,δ be obtained by perturbing Γ in the normal direction by a function γi∈C(Γ)

such that

kγikL∞(Γ) .δ≤δ0(2.1)

Ω1

Ω2

nΓ

(a) Two patch domain

Ω1,δ

Ω2,δ

nΓ

(b) Perturbed patches

Ω1,δ

Ω2,δ

2δ

Uδ(Γ)

(d) Three-dimensional hybrid mesh covering the interface

Figure 1: Model problem with gap/overlap. Top: In the derivation and anal-

ysis of the method we use this conceptual construction of a two-dimensional

two-patch domain with gaps/overlaps stemming from perturbation of the

patch boundaries facing the interface. Bottom: While the perturbed two-

patch domain entirely lives in the two-dimensional plane, the hybrid variable

for increased generality will live on a three-dimensional mesh covering the

imperfect interface.

More precisely

∂Ωi,δ =[

x∈Γ

x+γi(x)nΓ(x) (2.2)

where nΓ(x) is the unit normal to Γ exterior to Ω2. Note that the functions γ1and

γ2are diﬀerent and therefore the domains Ω1,δ and Ω2,δ do not perfectly match at the

interface, instead there may be a gap or an overlap but in view of (2.1) we will have

∂Ω1,δ ∪∂Ω2,δ ⊂Uδ(Γ) ⊂Uδ0(Γ) (2.3)

Exact Model Problem. Consider the following model interface problem on the exact

partition of Ω (without a gap/overlap): Find ufulﬁlling

−∆ui=fiin Ωi, i = 1,2 (2.4)

with interface conditions

u1=u2,∇n1u1+∇n2u2= 0 on Γ (2.5)

and a homogeneous Dirichlet boundary condition u= 0 on ∂Ω. Here uiindicates the

solution on the patch Ωi, and we let u0denote the solution on the interface Γ. We assume

a regularity of the weak solution on each patch ui∈Hs(Ωi)∩H1

0(Ω)|Ωi, where s > 3/2.

Further, for the solution on the interface we assume u0∈Hs(Γ), which is likely 1/2 more

regularity than is required since this is essentially the trace along Γ but we maintain this

assumption for simplicity. In summary, we assume a weak solution with the following

decomposition into three ﬁelds

u= (u0;u1;u2)∈W=V0⊗V1⊗V2=Hs(Γ) ⊗Hs(Ω1)⊗Hs(Ω2)∩H1

0(Ω) (2.6)

Extended Solution. We will next derive a weak formulation on the perturbed patches

Ωi,δ instead of on the exact patches Ωi. To make sense of the exact solution uin such a

formulation we must ﬁrst extend uto the perturbed domains. We recall that there is an

extension operator Ei:Hs(Ωi)→Hs(R2), independent of s, such that

kEivkHs(R2).kvkHs(Ωi)(2.7)

and Eiv=von Ωi, see [23]. For the derivation of the hybridized formulation we introduce

ﬁelds u0, v0deﬁned on a domain Ω0⊂R3fulﬁlling

∂Ω1,δ ∪∂Ω2,δ ∪Γ⊂Ω0⊂Uδ0(Γ) (2.8)

and hence we must also extend the exact solution uon Γ to Ω0. To this end we deﬁne

an extension E0:Hs(Γ) →Hs(Uδ0(Γ)) such that (E0v)|x=v◦pΓ(x). Clearly, E0v=v

on Γ. We then have

kE0vkHs(Uδ0(Γ)) .δ0kvkHs(Γ) (2.9)

see [7]. For compactness we introduce the notation

ue= (ue

0;ue

1;ue

2) = (E0u0;E1u1;E2u2) (2.10)

where it is implied by the subscript of the ﬁeld which extension operator is used. We also

apply this notation to spaces such that, for instance, We={v=we:w∈W}.

Hybridized Weak Formulation. Since an extended function coincides with the orig-

inal function on its original domain, we may replace the ﬁelds in the continuous problem

(2.4)–(2.5) by their extensions. We then, patchwise, multiply (2.4) by a test function

i∈Ve

i, integrate over the perturbed patch Ωi,δ, and apply a Green’s formula to obtain

i=1

(fe

i, ve

i)Ωi,δ =

i=1

(−∆ue

i, ve

i)Ωi,δ (2.11)

i=1

(∇ue

i,∇ve

i)Ωi,δ −(∇nue, ve

i)∂Ωi,δ (2.12)

i=1

(∇ue

i,∇ve

i)Ωi,δ −(∇nue

i, ve

i−ve

0)∂Ωi,δ −(∇nue

i, ve

0)∂Ωi,δ (2.13)

≈

i=1

(∇ue

i,∇ve

i)Ωi,δ −(∇nue

i, ve

i−ve

0)∂Ωi,δ −(ue

i−ue

0,∇nve

i)∂Ωi,δ (2.14)

+βh−1(ue

i−ue

0, ve

i−ve

0)∂Ωδ,i −(∇nue

i, ve

0)∂Ωi,δ (2.15)

where we added and subtracted functions ue

0=u0◦pΓ=u|Γ◦pΓand ve

0, and in the last

step we added terms involving ue−ue

0that are not exactly zero since they are evaluated

on the perturbed curves ∂Ωi,δ, which diﬀer from Γ. The functions ue

0and ve

0will, due

to the construction of E0using the closest point mapping pΓ, in the continuous problem

be constant in the directions orthogonal to Γ. In the discrete setting, this property will

instead be imposed weakly since it is not straightforward to implement strongly.

Application to Surfaces. The model problem can be directly extended to a setting

with a surface built up by a set of patches, O={Ωi:i∈I}with Ian index set, and

interfaces {Γij =∂Ωi∩∂Ωj}. The patches are deﬁned by a mapping Fi:R2⊃b

Ωi→

Ωi⊂R3, and a set of trim curves b

Γij. In the model problem (2.4) the Laplace operator is

replaced by the Laplace-Beltrami operator ∆Ω, the gradients are replaced by tangential

gradients ∇Ω, and the interface conditions are

ui=uj,∇νiui+∇νjuj= 0 on Γij (2.16)

where and ∇νi=νi· ∇Ωare the tangential derivatives along the exterior unit co-normals

νito ∂Ωi,δ. Note that here νimay be diﬀerent from −νjand thus Γij may be a sharp

edge on the surface across which the surface normal is discontinuous. The perturbation

of the surface may be precisely deﬁned by ﬁrst extending Ωito a slightly larger smooth

surface e

Ωiand then assuming that ∂Ωi,δ is smooth curve on e

Ωisuch that

∂Ωi,δ ⊂Uδ(Γ) (2.17)

The surface patches can be further perturbed by the action of a rigid body motion in R3

with norm less than δ. The analysis we present is basically directly applicable to this

setting since the key assumption is (2.17). A further diﬃculty that we do not consider

here is a more general perturbation of the mapping F. We have chosen to present the

method and analysis in the simple setting outlined in the previous paragraph since it

captures the main new challenges and the notation is much simpler.

Implementation. In practice we ﬁrst import a number of patches that do not match

perfectly. These patches {Ωi:i∈I}are each described by the mapping Fitogether with

a set of trim curves {γj:j∈JI}deﬁning the boundary of the patch in the reference

domains. We then compute the intersection with the mapped trim curves F(γi) and

voxels in an octree which allows local reﬁnement. We can then extract a suitable cover of

the gaps between the mapped patches consisting of a face-connected set of voxels which

is the mesh used for the hybrid variable. The precise formulation of such algorithms is

not the focus of this paper and we leave that for future work. Note, in particular, that

no information is passed directly between two patches instead all information is passed

through the hybrid variable.

2.2 Hybridized Finite Element Method

Finite Element Spaces. To deﬁne the ﬁnite element spaces we assume that we have

polygonal domains Ωi⊂e

Ωi⊂R2and families of quasiuniform meshes e

Th,i on e

Ωiwith

mesh parameter h∈(0, h0], for i= 1,2.We deﬁne the active meshes and the correspond-

ing discrete domains by

Th,i ={T∈e

Th,i :T∩Ωi6=∅},Ωh,i =∪T∈Th,i T, i = 1,2 (2.18)

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HybridizedIsogeometricMethodforEllipticProblemsonCADSurfaceswithGapsTobiasJonssonMatsG.LarsonKarlLarssonMarch8,2023AbstractWedevelopamethodforsolvingellipticpartialdierentialequationsonsurfacesdescribedbyCADpatchesthatmayhavegaps/overlaps.Themethodisbasedonhybridizationusingathree-dimensionalmeshth...

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Hybridized Isogeometric Method for Elliptic Problems on CAD Surfaces with Gaps Tobias Jonsson Mats G. Larson Karl Larsson

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