higher-order surface treatment for discontinuous galerkin

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDSInt. J. Numer. Meth. Fluids 2015; 00:1–20Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fld

Higher-order surface treatment for discontinuous Galerkinmethods with applications to aerodynamics

A. S. Silveira, R. C. Moura†, A. F. C. Silva, M. A. Ortega

ITA, Technological Institute of Aeronautics, Sao Jose dos Campos, SP, Brazil

SUMMARY

When dealing with high-order numerical methods, an adequate treatment of curved surfaces is required notonly to guarantee that the expected high-order is maintained in the vicinity of surfaces, but also to avoidsteady-state convergence issues. Among the variety of high-order surface treatment techniques that havebeen proposed, the ones employing NURBS (Non-Uniform Rational B-Splines) to describe curved surfacescan be considered superior both in terms of accuracy and compatibility with CAD softwares. The currentstudy describes in detail the integration of NURBS-based geometry description in a high-order solver basedon the discontinuous Galerkin formulation. Particularly, this work also discuss how and why NURBS curvesof very high order can be employed within standard NURBS-based boundary treatment techniques to yieldreduced implementation complexity and computational overhead. Theoretical estimates are provided alongwith numerical experiments in order to support the proposed approach. Minding engineering applicationsin the context of compressible aerodynamics, additional simulations are addressed as numerical examplesto illustrate the advantages of using higher-order NURBS in practical situations. Copyright c© 2015 JohnWiley & Sons, Ltd.

Received . . .

KEY WORDS: Discontinuous Galerkin; Curved boundaries; NURBS; Quadratures; Aerodynamics

1. INTRODUCTION

The rising importance of unstructured high-order methods in recent years has motivated the parallel

development of “support” techniques which are required to be consistent with the high-order

context. Among these techniques, one can mention [1]: sub-cell shock capturing schemes, efficient

time integration methods, robust mesh generation algorithms, as well as high-order boundary

condition approaches to curved surfaces. Regarding this last issue in particular, it is known that

unless an adequate geometry description is employed for the curved surfaces, one can expect not

only loss of accuracy, but also problems concerning residue convergence to machine zero [2].

As a result, the treatment of curved boundaries has been drawing the attention of scientists and

practitioners, and a variety of approaches have been proposed in the literature to tackle this issue.

Probably, the simplest and most intuitive way to deal with curved surfaces in the context of

unstructured high-order methods is to use polynomials to represent these boundaries, employing the

traditional isoparametric finite elements approach, see [3]. A much more recent technique, known

as isogeometric analysis [4], advocates the use of NURBS (Non-Uniform Rational B-Splines –

widely used in CAD softwares [5]) not only when representing surfaces, but also to approximate

the solution itself, in an attempt to unify the processes of geometry description and numerical

simulation. Such approach is however incompatible with general high-order methods because of

†E-mail: [email protected]

Copyright c© 2015 John Wiley & Sons, Ltd.

Prepared using fldauth.cls [Version: 2010/05/13 v2.00]

2 A. S. SILVEIRA, R. C. MOURA, A. F. C. SILVA, M. A. ORTEGA

its obvious basis function restriction. Another modern but less radical technique [6] suggests an

adaptation of the impermeability condition at the boundary so as to make the flow field to “see”

a polygonal low-order surface as a smoothly curved one. A drawback of this approach is the loss

(although arguably small) of conservation at the surface. Finally, there is the technique closely

followed in the present work [7, 8], named NEFEM (Nurbs-Enhanced Finite Element Method), in

which NURBS curves are employed to represent the surfaces, but no restriction is imposed on the

solution’s basis functions.

The NEFEM technique can be considered superior in comparison to other available approaches

for a variety of reasons: (i) its associated results were verified to be at least one order of magnitude

more accurate than the ones obtained with isoparametric finite elements [7, 9], not to mention

the fact that NURBS can be exact when representing conic sections (ellipses, hyperbolas, etc.),

while polynomial-based expansions cannot; (ii) there is no restriction on the numerical formulation

itself, since the use of NURBS curves is required only for surface representation, allowing the

elements not touching the boundaries to be treated according to the adopted numerical formulation;

(iii) the conservation property is guaranteed “by construction” at the surface [10]; (iv) its natural

compatibility with CAD-generated geometries is an important advantage in terms of engineering

applications.

We stress that point (i) above was found in comparisons carried against the standard isoparametric

approach, where the polynomial degree employed for the boundary description and for the solution’s

approximation is the same. In fact, it should be possible to modify this approach to obtain an

improved accuracy by the use of higher-order polynomials for the boundary representation. We

highlight however that the objective of the present study is not to advocate the use of NURBS-based

techniques over other approaches, but to point out a more efficient implementation strategy for those

already using NEFEM, as described in the following.

Within the context of unstructured high-order methods, see [11, 12], boundary conditions are

usually enforced by means of integrations along the element edges defining the boundary of interest.

When employing the NEFEM approach, however, a mismatch between the location of element

vertices and the so-called NURBS breakpoints is commonly observed along curved surfaces. It

happens that a NURBS of degree q has (usually) a continuity of order q − 1 at its breakpoints, while

being infinitely differentiable everywhere else [13]. To account for this piece-wise smooth nature

of NURBS functions, the common practice is to divide integrations along element edges in several

patches so that none of them should contain a breakpoint [14]. This strategy turns implementation

more complex and integrations more expensive. This work presents a practical way to avoid several

integrations per element while retaining the accuracy of the numerical formulation at the surface,

leading to a simpler and cheaper NEFEM. The basic idea is to employ NURBS curves of very high

order when describing the surface so as to make numerical integrations essentially insensitive to

continuity reductions at breakpoints.

In support of the proposed approach, theoretical estimates for the convergence rate of quadratures

are provided along with numerical experiments, explaining why the use of higher-order NURBS

yields faster decay of integration errors. For the simulations, the discontinuous Galerkin (DG)

formulation [15, 16] is adopted for the spatial discretization of the numerical solution. These

simulations are carried through a parallel solver named Veritas2D [17], where examples of the

NEFEM technique with NURBS of higher order are discussed minding particularly applications in

compressible aerodynamics. Code Veritas2D was developed to simulate the Euler equations of gas

dynamics in unstructured meshes of triangles. It offers several options for the numerical Riemann

flux, see [18], while viscous terms are accounted by the BR2 scheme [19, 20]. The solver has also

superior shock resolution capabilities, combining sub-cell shock capturing schemes [21, 22, 23]

with a novel mesh-refinement strategy [24].

This paper is organized as follows. Section 2 presents the DG formulation for generalized

advection-diffusion problems (being the Euler equations a particular case) while focusing on

unstructured meshes of triangles. In Section 3, the basic concepts on NURBS curves are presented.

Mapping relations between reference domains and actual mesh elements are considered in Section

4. Then, boundary and interior integrations required by the DG method are discussed in Section 5

Copyright c© 2015 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids (2015)Prepared using fldauth.cls DOI: 10.1002/fld

A HIGHER-ORDER SURFACE TREATMENT APPROACH FOR DG METHODS 3

for both straight-sided and curved elements. Finally, in Section 6, the use of higher-order NURBS is

advocated through theoretical analysis and numerical experiments. The proposed approach is then

exemplified in Section 7 by means of practical aerodynamics simulations. At last, our conclusions

summarize the study in Section 8.

2. THE DG FORMULATION

Here the discontinuous Galerkin discretization is discussed for non-linear advection-diffusion

problems in general. For simplicity’s sake, a two-dimensional setting is assumed. Regarding

notation, we employ bold font for column arrays of physical quantities and arrows on top of vectors

with Cartesian (x and y) components.

The problems of interest are governed by systems of equations of the form

∂Q

∂t+ ~∇ · ~Fi(Q) = ~∇ · ~Fv(Q, ~∇Q) , (1)

where ~Fi = Fx,Fyi and ~Fv = Fx,Fyv are respectively called the inviscid and viscous flux

vectors. A decomposition of Eq. (1) into two first-order equations (regarding the spatial derivative

of Q) is usually carried within the DG framework, namely,

∂Q

∂t+ ~∇ · ~Fi(Q) = ~∇ · ~Fv(Q, ~G) , (2)

~G = ~∇Q , (3)

in which the gradient variable ~G was introduced. The so-called hp discretization is then carried.

In what follows, unstructured meshes of triangles are considered. By h discretization, one refers to

dividing the solution domain Ω into non-overlapping triangular elements Ωe such that Ω =⋃

e Ωe.

The p discretization consists of approximating the numerical solution Q within each element by a

weighted sum of local basis functions φn, normally polynomials of degree up to a prescribed integer.

Hence, for each element,

Q(x, t) =

N∑

n=1

cn(t)φn(ξ1, ξ2) , (4)

where the basis functions φn(ξ1, ξ2) are defined in a reference standard triangle st. Reference

elements are discussed in Section 4, along with mapping relations between such elements and actual

mesh elements. The number of basis functions N employed per element must be sufficient to span

a polynomial space of degree P , so that N ∝ PD, being D the dimension (in physical space) of the

considered problem.

Orthogonal polynomial basis functions are often employed [25], so that

∫

st

φnφm dξ1dξ2 ∝ δnm , (5)

in which δnm is the Kronecker delta. However, this orthogonality holds only for straight-sided

triangles. As a result, curved elements will have non-diagonal mass matrices, as discussed further

on.

The next step is to require the projection of Eq. (2) to vanish within the local approximation space,

i.e. for m = 1, 2, . . . , N, one should have

∫

Ωe

φm∂Q

∂tdΩe =

∫

Ωe

φm~∇ · (~Fv − ~Fi) dΩe , (6)

which, by using vector calculus identities, can be rewritten as

∫

Ωe

φm∂Q

∂tdΩe =

∮

∂Ωe

φm (~Fv − ~Fi) · ~ndℓe −

∫

Ωe

(~Fv − ~Fi) · ~∇φm dΩe , (7)



where Gauss’ theorem was applied to provide the boundary integral, being ~n the unit vector normal

to the boundary ∂Ωe and pointing outside Ωe. Analogously, Eq. (3) can be manipulated to yield∫

Ωe

φm~G dΩe =

∮

∂Ωe

φmQ~ndℓe −

∫

Ωe

Q ~∇φm dΩe . (8)

To close the formulation, the boundary integrals in Eqs. (7) and (8) must be adapted to take

into account information from neighbouring elements (or boundary conditions). This is done

by introducing the so-called inviscid Fi and viscous Fv numerical fluxes, which respectively

approximate the products ~Fi · ~n and ~Fv · ~n , as well as the numerical average Q, used when

evaluating Q for the boundary integral in Eq. (8). The numerical fluxes are functions of properties

from both elements sharing each considered interface. In the DG method, one can use for Fi

practically any Riemann solver employed for high-resolution methods [18], while for Fv one need

more specific (DG-tailored) techniques, see [26].

For example, when employing the BR2 scheme for the viscous numerical flux, one has

Q =1

2(Q+ +Q−) and Fv =

~n

2·(

~Fv(Q+, ~∇Q+ + η ~δ

+

s ) + ~Fv(Q−, ~∇Q− + η ~δ

−

s ))

, (9)

where ~∇Q is evaluated directly from Eq. (4) for each element sharing the considered interface.

Also, η is a penalty parameter which must be greater than the number of edges per element (fixed

as η = 4 in the code Veritas2D). At last, ~δs are side-related functions defined for each side s of Ωe

by the formula∫

Ωe

φm~δs dΩe =

1

2

∫

∂Ωse

φm(Q+ −Q−)~ndℓse , m = 1, 2, . . . , N . (10)

Now, defining ~F = ~Fv − ~Fi and F = Fv − Fi, one can write Eqs. (7) and (8) as∫

Ωe

φm∂Q

∂tdΩe =

∮

∂Ωe

φm F(Q±, ~∇Q± + η ~δ±

s ) dℓe −

∫

Ωe

~F(Q, ~G) · ~∇φm dΩe , (11)

∫

Ωe

φm~G dΩe =

∮

∂Ωe

φmQ(Q±)~n dℓe −

∫

Ωe

Q ~∇φm dΩe , (12)

while for both equations above, (±) is simply used to denote that information from outside (+) and

inside (−) element Ωe must be taken into account. When dealing with the first integral in Eq. (12),

two vector components must be considered as the gradient variable is represented by the expansion

~G =

N∑

n=1

γx,γyn φn . (13)

Finally, Eqs. (11) and (12) can be written in vector form, such as

Me ∂

∂t

c0...

cN

=

∮

∂Ωe

F(Q±, ~∇Q± + η ~δ±

s )

φ0

...

φN

dℓe −

∫

Ωe

~F(Q, ~G) · ~∇

φ0

...

φN

dΩe , (14)

where Mem,n is the so-called mass matrix of element Ωe. For straight-sided triangles, the Jacobian

Je of the mapping between Ωe and st is a constant (half of the triangle’s area, as shown in Section

4), and therefore, through Eq. (5),

Mem,n =

∫

Ωe

φmφn dxdy = Je

∫

st

φmφn dξ1dξ2 ∝ δmn , (15)

which makes the mass matrix diagonal. In this case, Me is trivially invertible and the coefficients

cn in Eq. (14) can be readily integrated in time. This does not hold for curved elements, where full

mass matrices are common and the need for actual matrix inversion exists.



3. FUNDAMENTALS ON NURBS CURVES

In this section, NURBS are defined and the relevant basic concepts are introduced. A short survey

about NURBS curves, including applications, can be found in [5]. For an extensive and detailed

treatment on the matter, the reader is referred to [13].

A NURBS curve of degree q is a piecewise rational function defined in parametric form by

~C(λ) =

(

ncp∑

i=0

wiBi,q(λ)~Pi

)

/

(

ncp∑

i=0

wiBi,q(λ)

)

, 0 ≤ λ ≤ 1 , (16)

where ~Pi (i = 0, . . . , ncp) are the control points, wi are their respective weights and Bi,q(λ) are

B-splines (from ‘basis-splines’) of degree q, defined recursively by the relations

Bi,0(λ) = 1 if λ ∈ [λi, λi+1) , (17)

Bi,0(λ) = 0 if λ /∈ [λi, λi+1) , (18)

Bi,k(λ) =λ− λi

λi+k − λiBi,k−1(λ) +

λi+k+1 − λ

λi+k+1 − λi+1Bi+1,k−1(λ) , (19)

for k = 1, . . . , q, where λi (i = 0, . . . , nk) are the knots or breakpoints, which are ranked in

ascending order such that 0 ≤ λi ≤ λi+1 ≤ 1. All together, the values of λi define the knot vector

Λ, namely

Λ = 0, ..., 0, λq+1, ..., λnk−q−1, 1, ..., 1 , (20)

which univocally describes the basis functions Bi,q(λ).The number of breakpoints (nk + 1) and the number of control points (ncp + 1) are related to the

NURBS degree q by the formula

nk = ncp + q + 1 . (21)

One should note that the initial and final breakpoints have (in the knot vector) a multiplicity of

q + 1. For the remaining breakpoints, the multiplicity in the knot vector determines the decrease

in the number of continuous derivatives of the NURBS function at the considered breakpoint. For

example, if all “interior” breakpoints have unit multiplicity, a NURBS of degree q shall have a

continuity of order q − 1 at its breakpoints. Anywhere else, NURBS are infinitely smooth once the

Bi,q(λ) functions are polynomials.

It is also worth mentioning that the B-splines used as basis functions for the NURBS construction

are normalized in the sense that

ncp∑

i=0

Bi,q(λ) = 1 , ∀ λ ∈ [0, 1] . (22)

Therefore, if wi = 1 for all i, see Eq. (16), the NURBS becomes a piecewise polynomial curve

(a spline). In general, NURBS are rational functions and this is the reason why NURBS can be

exact when representing conics, such as ellipses, hyperbolas, etc. Also, the first two letters in the

acronym NURBS (Non-Uniform Rational B-Spline) refer to the fact that breakpoints need not to be

equispaced in the knot vector, which allows for more flexibility when adjusting the parametrization

characteristics.

4. MAPPING RELATIONS

In this section, mapping relations between a rectilinear element ΩR and its curved counterpart

ΩC shall be discussed; in the process, standard reference elements mentioned in Section 2 will

be introduced, see Figs. 1 and 2. The NEFEM approach [7, 10] is here followed closely, so that

only one element edge is made curve, namely, the one opposite to the collapsed vertex. This vertex



originates when a square domain is mapped into a triangular domain, and is commonly seen in the

context of spectral/hp methods, see [25].

In order to map ΩR into ΩC (or vice-versa), two sequences of mappings are carried. For the first

one, see Fig. 1, ΩR is mapped into the standard square st while the standard triangle st is used

as an intermediary domain. For the second one, see Fig. 2, st is mapped into ΩC through the

domain λ, here named lambda–rectangle.

Regarding the first mappings sequence, in order to map a point X,Y ∈ ΩR into its

corresponding point ξ1, ξ2 ∈ st, one can use the linear relations

ξ1 =2(yC − yA)X + 2(xA − xC)Y + (yA − yC)(xB + xC) + (xC − xA)(yB + yC)

(xB − xA)(yC − yA)− (xC − xA)(yB − yA), (23)

ξ2 =2(yA − yB)X + 2(xB − xA)Y + (yB − yA)(xB + xC) + (xA − xB)(yB + yC)

(xB − xA)(yC − yA)− (xC − xA)(yB − yA), (24)

where the letters A, B and C refer to the vertices of ΩR or ΩC , as shown in Figs. 1 and 2, being C

the collapsed vertex. The relations above rely on a counter-clockwise nomination of the vertices. It

is worth mentioning that the Jacobian of the transformation above is given by

J =∂(X,Y )

∂(ξ1, ξ2)=

xAyB − xByA + xByC − xCyB + xCyA − xAyB4

, (25)

and equals half of the area of the triangle ΩR. In the sequence, to map a point ξ1, ξ2 ∈ st into

its corresponding point η1, η2 ∈ st, one must employ [25].

η1 = 21 + ξ11− ξ2

− 1 , and η2 = ξ2 . (26)

Figure 1. First mappings sequence — from the rectilinear element ΩR to the standard square st, passingby the standard triangle st.

Now, regarding the second mappings sequence, a point η1, η2 ∈ st is mapped into λ1, λ2 ∈λ through the linear relations

λ1 =1− η1

2λa +

1 + η12

λb , and λ2 =1 + η2

2. (27)

In the sequel, when mapping a point λ1, λ2 ∈ λ into its correspondent x, y ∈ ΩC , the NURBS

function ~C(λ) = Cx(λ), Cy(λ) representing the curved edge of ΩC is finally used. This last

transformation can be performed through a simple but ingenious formula suggested in [7], namely

xy

= (1− λ2)

Cx(λ1)Cy(λ1)

+ λ2

xC

yC

. (28)

In the convention here adopted, vertices A and B are respectively associated to the extrema λa and

λb of the curved edge parametrization, i.e. ~C(λa) = xA, yA and ~C(λb) = xB, yB, as shown in

Fig. 2. Using the above formula is advantageous because of its linearity in the variable λ2, see [7].



Figure 2. Second mappings sequence — from the standard square st to the curved element ΩC , passing bythe “lambda–rectangle” λ.

5. BOUNDARY AND INTERIOR INTEGRATIONS

This section addresses boundary and interior integrals mentioned in Section 2, such as the ones

figuring in Eqs. (11) and (12). Both straight and curved elements are considered.

Starting with boundary integrations, we introduce a generic boundary-related function B(~r, ~n),being ~r = x, y a position along the edges of Ωe and ~n the unit vector normal to the boundary

pointing outside the element. The relevant integrations can then be written as

∮

∂Ωe

B dℓe =∑

s

∫

∂Ωse

B(~rs, ~ns) dℓse (29)

where the summation is performed over all the edges, or sides (s), of Ωe.

In the context of spectral/hp element methods, a mapping to the standard interval Ωst = [−1, 1] is

frequently employed when dealing with one-dimensional integrations, so that Gauss-like quadrature

rules can be used, see [25]. Hence, for rectilinear edges, simple linear mappings can be used to yield

∫

∂Ωse

B(~rs, ~ns) dℓse =

ℓse2

∫

Ωst

B (~rs(ξ), ~ns(ξ)) dξ , (30)

in which ℓse is the length of the edge s of Ωe and the vector ~ns(ξ) is actually a constant in this case

of rectilinear edges.

Now turning to curved edges, the NURBS function ~C(λ) = Cx(λ), Cy(λ) describing the

considered edge for λa ≤ λ ≤ λb is employed so that

∫

∂Ωse

B(~rs, ~ns) dℓse =

∫ λb

λa

B (~rs(λ), ~ns(λ)) ‖ ~C′(λ)‖ dλ , (31)

where ‖ ~C′(λ)‖ =[

(C′x)

2 + (C′y)

2]1/2

, see [13] for efficient algorithms to evaluate the components

of ~C′(λ), and

~ns(λ) =

C′y(λ),−C′

x(λ)

‖ ~C′(λ)‖. (32)

Again, by using a simple linear mapping, it is possible to obtain λ = λ(ξ) so as to have

∫ λb

λa

B (~rs(λ), ~ns(λ)) ‖ ~C′(λ)‖ dλ =

λb − λa

2

∫

Ωst

B (~rs(ξ), ~ns(ξ)) ‖ ~C′(ξ)‖ dξ . (33)

Regarding interior integrals, we now consider a generic interior-related function I(~r), where here

~r refers to a position inside Ωe. In order to apply numerical quadratures, these two-dimensional



integrations must be carried within the standard square element st. The mapping sequences

discussed in Section 4 must then be used.

For straight-sided elements, the transformations shown in Fig. 1 are followed directly to yield

∫

Ωe

I dΩe = Je

∫

st

I (~r(ξ1, ξ2)) dξ1dξ2 = Je

∫

st

I (~r(η1, η2))

(

1− η22

)

dη1dη2 , (34)

where Je is given in Eq. (25) and the factor (1− η2)/2 = ∂(ξ1, ξ2)/∂(η1, η2) is derived from the

relations figuring in Eq. (26).

In case of curved elements, one must follow the transformations shown in Fig. 2 but in inverse

order, obtaining

∫

Ωe

I dΩe =

∫

λ

I (~r(λ1, λ2))∂(x, y)

∂(λ1, λ2)dλ1dλ2 =

λb − λa

4

∫

st

I (~r(η1, η2))∂(x, y)

∂(λ1, λ2)dη1dη2 ,

(35)

in which (λb − λa)/4 = ∂(λ1, λ2)/∂(η1, η2) stems from the relations in Eq. (27), while the

remaining Jacobian factor can be obtained from Eq. (28), so that

∂(x, y)

∂(λ1, λ2)=

∂x

∂λ1

∂y

∂λ2−

∂x

∂λ2

∂y

∂λ1= (1 − λ2) cf (λ1) , (36)

where cf is what we call a curvature function, being given by the expression

cf (λ) = (yC − Cy(λ))C′

x(λ)− (xC − Cx(λ))C′

y(λ) . (37)

6. QUADRATURES AND HIGHER-ORDER NURBS

In this section we shall discuss how integrations are usually performed in a numerical setting

for straight-sided elements, and also how the NEFEM technique traditionally suggests that such

integrations should be carried for curved elements [14]. Then, it will be explained how and why

one can employ higher-order NURBS to reduce the complexity and overhead of these integrations.

More specifically, it will be shown that, when increasing the number of quadrature points, the error

in the integrations decays faster if NURBS curves of higher order are used. At the end of the

section, an additional advantage of employing higher-order NURBS is pointed out, namely, that

when interpolating a given geometry through a fixed number of points, NURBS of higher order

generally provide a better approximation of the real geometry.

6.1. Traditional quadrature approaches

The relevant integrals for straight-sided elements, see Eqs. (30) and (34), are of the form

∫

Ωst

f(η) dη , and

∫

st

f(η1, η2) dη1dη2 . (38)

It is a common practice to employ Gauss-like quadratures to evaluate these integrations in

a numerical setting. For instance, when applying the Gauss-Legendre rule, it is possible to

approximate any integral in the standard interval Ωst through a weighted sum, namely

∫

Ωst

f(η) dη ∼=

Q∑

i=1

Wif(ηi) , (39)

where the weights Wi are functions of the roots ηi (ordered such that −1 < η1 < · · · < ηQ < 1) of

the Qth degree Legendre polynomial PQ, see [25], being given by

Wi =2

1− η2i

[

P ′

Q(ηi)]−2

. (40)



Generally, the error of the approximation in Eq. (39) decreases with increasing Q. In particular,

if the integrand f(η) is a Pth degree polynomial, exact results are obtained when Q ≥ (P + 1)/2.

For two-dimensional integrations, the rule above just need to be applied separately in each

direction, yielding∫

st

f(η1, η2) dη1dη2 ∼=

Q∑

i,j=1

WiWj f(

η(i)1 , η

(j)2

)

. (41)

Within the context of computational aerodynamics, it is a common practice [11] to enforce exact

integration for polynomials of degree 2P + 1. For Gauss-Legendre quadrature, such requirement

means employing Q = P + 1 quadrature nodes (in each direction). The Veritas2D solver uses

however Q = P + 3 nodes in order to further alleviate aliasing errors that may stem from

underintegration of the inviscid flux vector, which, strictly speaking, involves rational functions

of the conserved quantities, see Eq. (49).

Now turning to curved elements, the relevant integrals, see Eqs. (33) and (35), are of the form

∫

Ωst

f(η)‖ ~C′(η)‖ dη , and

∫

st

f(η1, η2) cf (η1, η2) dη1dη2 , (42)

in which cf = cf (η1, η2) since λ1 = λ1(η1, η2), as discussed in Section 4. Within the NEFEM

approach, the traditional way to approximate these integrals is to subdivide the integration domains

taking into account the piecewise smooth nature of NURBS curves and then to perform one

quadrature per subdomain. The reader is referred to [14] for the specific details, but the basic idea

is to split a curved triangle into several subtriangles, all of them sharing the vertex C, the collapsed

one, and such that their curved edges cover the whole edge of the original element, but with no

breakpoints within them. The goal is to avoid breakpoints inside integration paths, once NURBS

curves generally have a reduced continuity at these particular points.

Provided that in general there is a mismatch between NURBS breakpoints and vertices of (actual)

elements touching the surface, managing such irregular splitting of integration domains is obviously

cumbersome in terms of implementation. A simpler approach is also mentioned in [14], in which

a curved edge is subdivided into equispaced integrations paths so that Gauss-Legendre composite

quadratures can be applied; it is argued that for most practical applications this approach is able

provide reasonable integration accuracy. However, the necessity of managing several subdomains

per curved element can still be considered an inconvenience, specially when three-dimensional

applications are minded.

This work presents a practical way to avoid the drawbacks of the aforementioned quadrature

approaches while retaining good integration accuracy for curved elements. The basic idea is to

employ NURBS of sufficient high order when describing the surface, so that numerical integrations

may be essentially insensitive to continuity reductions at breakpoints. It should be recalled, see

Section 3, that the continuity level at breakpoints is proportional to the NURBS degree. The

proposed approach carries therefore the potential to make NURBS-based boundary treatment

techniques simpler and less expensive.

6.2. Integration errors and NURBS degree

We start by stating a theorem regarding the approximation of continuous functions by polynomial

interpolations. This theorem may be somewhat surprising for some readers as polynomial

interpolation is indeed a subject haunted by a couple of misconceptions [27]. For example, it

guarantees convergence of (certain) polynomial approximations even when interpolating functions

with discontinuous derivatives. For the theorem’s proof, the reader is referred to [28].

Theorem: Let u(x) be a Lipschitz continuous function in Ωst = [−1, 1], i.e. |u(x)− u(y)| ≤κ|x− y| for some constant κ and all x, y ∈ Ωst. In addition, let un(x) be the nth degree Chebyshev

or Legendre interpolant of u(x). By these interpolants we mean the Lagrange interpolating

polynomial of u(x) through the set of n+ 1 Chebyshev of Legendre quadrature nodes. Under

such definitions, if u has ν derivatives, with the νth derivative being of bounded variation, then



‖u− un‖∞ ∝ n−ν as n → ∞, where ‖·‖∞ stands for the L∞ norm over Ωst. Moreover, if u is

analytic, exponential convergence is obtained, with ‖u− un‖∞ ∝ ρ−n, for some constant ρ > 1.

The reason why this theorem is important here is that, when performing integrations for curved

elements, as for instance the boundary integral in Eq. (42), one has actually the approximation

I =

∫

Ωst

f(η)‖ ~C′(η)‖ dη ∼= IQ =

Q∑

i=1

Wif(ηi)‖ ~C′(ηi)‖ . (43)

And since Gauss-Legendre quadratures provide exact results for Q ≥ (P + 1)/2 when the integrand

is a Pth degree polynomial, IQ can also be written as

IQ =

∫

Ωst

LQ−1(η) dη , (44)

where LQ−1(η) is the Legendre interpolant of f(η)‖ ~C′(η)‖ through the Q quadrature nodes. Hence,

the integration error is given by

I − IQ =

∫

Ωst

[

f(η)‖ ~C′(η)‖ − LQ−1(η)]

dη , (45)

which, by introducing ε(η) = f(η)‖ ~C′(η)‖ − LQ−1(η), can readily be bounded as

|I − IQ| ≤

∫

Ωst

|ε(η)| dη ≤

∫

Ωst

‖ε‖∞ dη ∝ (Q − 1)−ν , (46)

being ν the number of derivatives of f(η)‖ ~C′(η)‖, which is actually the number of derivatives of

‖ ~C′(η)‖ since f(η) can be assumed analytic.

Therefore, upon increasing Q, Eq. (46) states that quadrature errors decay faster when using

NURBS of higher degree (higher ν). In other words, for a fixed Q (say, given by the polynomial

order P of a simulation), a NURBS of higher order should yield smaller errors for the quadratures.

It is not difficult to show that similar results hold for interior integrations, and so the related proof is

here omitted for the sake of brevity. In what follows, numerical experiments are carried to support

the claims above both for boundary and interior quadratures.

6.3. Numerical experiments

For the numerical experiments we consider the hump-shaped curve given by

y =

exp[

x/(1− x2)]

if x < 1 ,

0 if x ≥ 1 ,(47)

where y = y/hb and x = x/sb, being hb and sb the height and the semi-width of the hump,

respectively. This function is commonly used in distribution theory and is known to be infinitely

differentiable for all x. Its shape is depicted in Fig. 3 for hb = sb = 1/2.

We then define two curved triangles placed upon the top-left side of the hump, also shown in Fig.

3, where the lower one is named Ωlower while the upper one is named Ωupper . In what follows, the

length of their curved edge and their area will be evaluated with increasing number of quadrature

points for different NURBS degrees. Since for all such quadratures the integrands are Lipschitz

continuous (recall the aforementioned theorem), convergence is guaranteed as Q → ∞. Therefore,

as Q increases, the error in the integrations can be measured by the variation

δ(Q) = |IQ − IQ−1| , (48)

in which IQ is used to indicate the numerical value of the integrations obtained with Q quadrature

points, either for length or area integrals.



Figure 3. Infinitely smooth hump alongside with the curved triangular elements Ωlower and Ωupper used forthe numerical experiments.

The NURBS curves were generated by interpolating points of the hump through the centripetal

method [5, 13], which can applied to interpolate an arbitrary set of points for any chosen NURBS

degree. It is important to mention that this algorithm is very simple and that its complexity is

practically degree-independent. Two hundred equispaced (in the x direction) hump points were

employed to generate NURBS curves of degrees q = 1, . . . , 5. Then the convergence of δ(Q) for

the boundary and interior integrals was tracked until up to Q = 200 quadrature nodes. The results

are shown in Figs. 4 and 5. As expected, the convergence is faster for NURBS of higher order. We

stress that several breakpoints are found inside the curved edges representing the top-left side of the

hump, but no subdivision is performed to account for continuity reductions at breakpoints.

Figure 4. Convergence of the curved edge’s length for Ωlower (top) and Ωupper (bottom); the plots on theright use log-linear axis and focus on smaller values of Q.



It is clear from the log-log plots in Figs. 4 and 5 that the asymptotic convergence slope is

steeper for higher-order NURBS, although this asymptotic behaviour may sometimes require a

large number of quadrature nodes to appear, as seems to be the case for Ωupper in particular.

Our experience indicates that with some mesh refinement, the asymptotic rate of convergence can

be achieved with a significantly reduced number of nodes. What is however very interesting to

realize is that the main benefit of employing NURBS of higher order, namely, that using higher qyields smaller integration errors for a fixed Q, holds far before the asymptotic region. Moreover,

by inspecting the convergence rate for smaller values of Q in a log-linear plot (see the right-hand

side graphs in Figs. 4 and 5) there seems to be a region of exponential convergence which becomes

longer with increasing q.

Figure 5. Convergence of the curved triangle’s area for Ωlower (top) and Ωupper (bottom); the plots on theright use log-linear axis and focus on smaller values of Q.

Although not inconsistent with the theoretical estimates discussed in Section 6.2 (since strictly

speaking they cover only asymptotic behaviours), exponential convergence is only expected for

analytic integrands, which is not the case in the present approach, where breakpoints are allowed

to reside inside integration paths. Our interpretation is that, when the number of quadrature nodes

is not enough to capture the NURBS lack of continuity, the convergence follows that of an analytic

integrand. This would explain why the observed region of exponential convergence holds for higher

values of Q when the integrand smoothness is increased (by increasing q). A rigorous proof is

difficult to be pursued since the majority (if not all) of the theorems within approximation theory,

see [28], concern asymptotic convergence, i.e. Q → ∞. Our experience however seems to indicate

that this behaviour happens in general and is specially useful if one aims to exploit exponential

convergence for the accuracy of simulations through spectral/hp high-order methods. A test case

related to this claim is discussed in Section 7.

6.4. Representation capability

Before the next section, it is worth pointing out another advantage of employing NURBS of higher

order. It happens that, in some practical applications, the geometry of interest is only known through



the coordinates of a reduced number of points along its surface. For such cases, it is verified that

interpolating the known surface points with higher-order NURBS is beneficial because, for a fixed

number N of interpolation points, employing higher q usually provides a better approximation of

the real geometry.

Although, to the authors’ knowledge, there is no mathematical proof of the above claim in general,

our practice indicates its validity provided that the real geometry is smooth and that N/q is not

smaller than 2 or 3. It is understood that using “wise” interpolation algorithms (as the centripetal

method [5, 13], for instance) is also of crucial importance. This information is useful for the

practitioner that sometimes is not primarily interested in obtaining good convergence characteristics,

but that actually wants the results to be representative of the real geometry.

A last numerical experiment is provided to illustrate the reasoning behind the methodology

in question. The hump described in Eq. (47), again for hb = sb = 1/2, is now approximated by

NURBS curves with q = 1, . . . , 5 for an increasing number of interpolation points, which are again

taken to be equispaced (in the x direction) within Ωx = [−sb,+sb]. The distance between the

approximating NURBS and the real geometry, as measured in the L∞ norm over Ωx, is given in

Fig. 6 for each NURBS degree and for N up to 2048. Asymptotic slopes are also given in the

picture and indicate an algebraic convergence rate of order q + 1.

Figure 6. Approximation errors between interpolating NURBS curves (q = 1, . . . , 5) and the real humpgeometry for increasing number of interpolation points.

The results depicted in Fig. 6 basically indicate that, upon increasing N , the approximation errors

decay faster when using NURBS of higher degree for the interpolations. In other words, for a fixed

N (say, when limited information of the real geometry is available), a higher-order NURBS should

be able to reproduce the real geometry with superior accuracy. It is worth mentioning however that

when only a very small number of interpolation points is known, increasing the NURBS degree

can instead provide increasingly oscillatory curves which would be in fact worse approximations of

the actual geometry. This is precisely what happens for the lowest numbers of interpolation points

considered in Fig. 6, and this is why we suggest that N/q should not be smaller than 2 or 3.

7. TEST CASES AND PRACTICAL APPLICATIONS

In this last section, test cases are addressed to demonstrate the implementation of the (simplified)

NEFEM technique in the code Veritas2D, as well as to illustrate the benefits of using NURBS

curves of higher order both for canonical and practical compressible flows. First, the transonic

flow past a NACA 0012 profile is considered with and without curved elements, to emphasize the



particular necessity of performing an adequate boundary treatment in the context of high-speed

compressible flows. Then, in connection with Section 6.3, the internal flow through a channel

with a semi-circular bump on its lower surface is addressed to demonstrate that the exponential

convergence capability of spectral/hp methods can be expected to hold within the proposed approach

as long as NURBS curves of sufficiently high degree are employed. Finally, minding engineering

applications, the compressible flow over a two-element airfoil is simulated and the results compared

with experimental measurements in a scenario where limited information of the real geometry is

available. As mentioned in Section 6.4, sometimes this happens to be the case in practical situations.

All these flow problems are modelled by the Euler equations of gas dynamics, obtained by Eq.

(1) without the viscous flux term, which can be used however for shock-capturing techniques based

on artificial viscosity [21, 22]. For the Euler equations, vectors Q and ~Fi = Fx,Fy are given by

Q =

ρρuρve

, Fx =

ρuρu2 + pρuv

(e + p)u

, Fy =

ρvρvu

ρv2 + p(e + p)v

, (49)

in which ρ stands for density, u and v are respectively the velocity components in the x and ydirections, e = ρ[ei + (u2 + v2)/2] is the total energy per unit volume, and ei is the specific internal

energy. The static pressure p is obtained using the equation of state for a perfect gas, namely,

p = ρ(γ − 1)ei, where γ is the fluid ratio of specific heats, which assumes the value γ = 7/5 for

the air.

7.1. Transonic flow past a NACA 0012 profile

This test case was chosen not only to demonstrate the implementation of the NEFEM technique

(simplified according to the approach here advocated) within the Veritas2D solver, but also to stress

the necessity to employ adequate curved boundary conditions particularly when simulating high-

speed compressible flows with high-order schemes. The NACA 0012 airfoil is considered with a

freestream Mach number of 0.8 and an angle of attack of 1.25 degrees. The same base mesh is used

in two simulations, but the proper curved boundary conditions are only applied for one of them.

For the spatial discretization, a forth-order (P = 3) discontinuous Galerkin scheme is employed in

both cases. The “plain upwind” numerical flux proposed in [29] and the sub-cell shock capturing

technique due to Persson and Peraire [21] are used in the simulations. A novel adaptive mesh

refinement algorithm [24] applied at the shocks allowed for good convergence of the numerical

residuals (which dropped by about seven orders of magnitude). The results obtained are compared

in Figs. 7 and 8 in terms of Mach number pattern and pressure distribution, respectively.

Figure 7. Mach number pattern without (left) and with (right) proper curved boundary treatment.

What is very interesting to observe from Fig. 7 is the stripped pattern inside the supersonic regions

of the flow over the airfoil when no curved boundary treatment is provided. Upon closer examination



of the mesh, we concluded that the high-order discretization employed was accurate to the point of

resolving the actual polygonal surface given to the solver: a succession of ramps with different

slopes. As predicted by supersonic flow theory, the slope discontinuity between each two of these

ramps leads to the formation of expansion fans. Such feature is not observed when using traditional

low-order schemes, and by this we highlight the necessity of employing adequate curved boundary

conditions especially for high-speed compressible flow simulations via high-order methods.

Figure 8. Pressure distribution without (left) and with (right) proper curved boundary treatment.

It is well known that when dealing with unstructured high-order methods, an adequate treatment

of curved surfaces is required not only to guarantee that the expected high-order is maintained in the

vicinity of surfaces, but also to avoid steady-state convergence issues [2]. Indeed, further increasing

the polynomial degree P of the DG basis functions would hinder steady-state convergence for the

straight-sided mesh case. In addition, it is also known [30] that CP distributions, for instance,

are usually polluted by wiggles associated with the slope discontinuities at the surface between

adjacent boundary elements. This undesired feature is cured when adequate curved wall treatment

is provided, as shown in Fig. 8. It is worth mentioning that this is not a peculiarity of NURBS-

based approaches since alternative boundary treatment techniques (e.g. isoparametric FEM) can be

expected to provide similar wiggle-free results.

It is important to mention that geometries with discontinuous slopes can always be represented

by the union of different high-order NURBS curves. By using the points of reduced continuity as

starting or ending points of each NURBS patch, which are breakpoints by definition, integration

along curved edges will not have their accuracy affected. When there is only one surface point of

reduced continuity, say a kink, it might be convenient to use a single NURBS with its starting and

ending points coinciding with the kink. This is commonly the case for airfoils with sharp trailing

edges, such as the NACA 0012 profile here addressed. Surface points of reduced continuity can

however be locally detrimental to the accuracy of the DG solution itself in some situations, see [11].

7.2. Isentropic flow over a semi-circular bump

The present case can be considered a standard test example for spectral/hp methods. The basic

geometry is that of a uniform channel with a semi-circular bump on its lower surface. The numerical

domain along with its triangulation is shown in Fig. 9 and is essentially the same used by other

authors [29, 10] to access the spurious generation of entropy over the bump’s curved surface. This



test case is discussed in connection with Section 6.3, where it was pointed out that the exponential

convergence capability of spectral/hp methods can be expected not to be hindered as long as NURBS

curves of sufficiently high degree are employed according to the simplified NEFEM-based approach

advocated in this study.

Figure 9. Domain and triangulation employed for the isentropic flow over a semi-circular bump.

To force the left-right symmetry in the numerical solution, the same freestream boundary

conditions were applied on the inflow as well as on the outflow boundaries of the simulation

domain, where a Mach number of 0.3 was imposed as done in previous works [29, 10]. Despite

the notable increase in the Mach number (up to about 0.75) on the top of the bump, the flow is

still subsonic everywhere. The free-slip wall condition has been applied for the remaining upper

and lower surfaces of the domain. All the boundary conditions were imposed weakly through the

Lax-Friedrichs numerical flux formula [31], which was also the numerical flux employed for inter-

element communication inside the domain.

The NURBS curves used to represent the semi-circular bump were actually obtained from the

interpolation of 360 equispaced points distributed along the (full) circle of radius r = 1/2 centred

at (0, 0). Obviously, only a trimmed section of the full NURBS was used to represent the bump

itself. From these 360 interpolation points, NURBS curves of degree q = 1, . . . , 4 were generated

again through the centripetal method [5, 13]. In addition, a reference NURBS describing the (full)

circle exactly was generated with the list of control points Π and the knot vector Λ given below. The

bump corresponds to the section for which 1/4 ≤ λ ≤ 3/4, since the point (0,−r) was used both as

starting and ending point for the (full) NURBS.

Π = (0,−r); (−r,−r); (−r, 0); (−r, r); (0, r); (r, r); (r, 0); (r,−r); (0,−r) , (50)

Λ = 0, 0, 0, 1/4, 1/4, 1/2, 1/2, 3/4, 3/4, 1, 1, 1 . (51)

Such reference NURBS is used in the simulations to provide the same results that

would be obtained if the classical NEFEM approach was followed. This is the case

because the different breakpoints composing the knot vector Λ correspond to the points

(0,−r); (−r, 0); (0, r); (r, 0); (0,−r), respectively, so that no breakpoints are allowed inside

integration paths and no element subdivisions is necessary. The present test case is therefore

representative in terms of comparison between the classical NEFEM approach and the less complex

higher-order NURBS approach proposed in this study.

Several simulations were performed with increasing polynomial degree P of the DG basis

functions for each one of the NURBS curves generated through interpolation, as well as for the

reference NURBS representing the semi-circular bump exactly. These cases were run until full

convergence to steady-state was obtained (residuals dropping to machine zero), when the entropy

generation along the curved wall was evaluated as an error both in the L∞ and L2 norms measured

along the bump’s surface. The results obtained are summarized in Fig. 10, where the similarity with

the plots figuring in the right-hand side of Figs. 4 and 5 is evident.

Such results support the discussion carried at the end of Section 6.3, where the observed region of

exponential convergence is enlarged when the level of smoothness of the NURBS curves employed

is increased (by increasing q). It is also interesting to note that sometimes, at least for simple

geometries like the one here considered, acceptable engineering accuracy (say, error levels of about



Figure 10. Entropy errors as measured in the L∞ (left) and L2 (right) norms along the curved surface forincreasing polynomial order P of the DG basis functions.

10−3) can be achieved even with NURBS of relatively low order, such as q = 2 or q = 3, which

are the usual degrees employed in most CAD softwares [5]. A more comprehensive set of tests is

planned to be addressed in a forthcoming study, where the specific role of mesh refinement should

be analysed in detail. Preliminary results indicate that accuracy becomes even better when finer

meshes are used alongside higher-order NURBS in general, which is of course not surprising.

7.3. Compressible flow over a two-element airfoil

In this last test case, we consider a typical scenario in practical applications where the geometry

of interest is known only by a limited number of points. Specifically, here we deal with the NLR

7301 airfoil with trailing edge flap, whose geometry and CP distribution can be found in [32]. We

consider the arrangement where the flap inclination is set to 20o (relative to the main element) and

the gap between the elements corresponds to 2.6% of the chord length, see again [32], case number

A-9, for details.

The surface of the main element is available through the coordinates of 125 points, while 81 points

are known along the flap’s surface. For some readers these may seem to be rather large numbers,

but they may not be sufficient to generate higher-order NURBS without oscillations, specially if

just a few points are provided in the vicinity of the parametrization limits, λ = 0 and λ = 1, where

wiggles usually start to manifest as the NURBS degree q is increased. Here, the parametrization

limits correspond to the trailing edge’s upper and lower points, for both main element and flap.

Our practice in such situations is to generate first a low-order NURBS (e.g. with q = 2 or q = 3)

by interpolating the known points, so as to obtain a larger number of points upon which one can

generate a higher-order NURBS with no oscillations. In our experience, this procedure was found

to be reliable in general and ill-conditioning issues were never observed. Here, both for the main

element and the flap, a second-order NURBS curve was generated first and 200 points were obtained

(equispaced in the λ variable) as new coordinates of the geometry. Such a number of points was then

sufficient to generate NURBS curves of degree q = 10, free from wiggles, for both elements. The



Figure 11. Higher-order NURBS geometries and local triangulation employed in the simulation; mainelement (top) and trailing edge flap (bottom).

final geometries are shown in Fig. 11, which also provides a local view of the triangulation used

for the numerical solution. This unstructured mesh was generated through the open-source software

DistMesh [33].

Figure 12. CP distribution comparison between numerical solution and experiments (circles for the mainelement and dots for the flap data).

A fourth-order accurate (P = 3) DG scheme with Roe’s numerical flux [34] was employed. With

such choices, convergence to steady-state has been obtained with the unscaled residuals dropping by



about five orders of magnitude. The simulation carried for this test case was designed to match the

experiments described in [32], where the Reynolds number was 2.51 million, the Mach number was

0.185 and the angle of attack was set to 6o. For such a high Reynolds number, the Euler equations

were expected to yield a good approximation of the flow field, and indeed the agreement found

for the pressure distribution was fairly good, see Fig. 12. Minor deviations can however still be

attributed to the lack of viscous effects in the formulation, especially behind the main element’s

trailing edge, where a viscous wake is expected to affect the CP distribution on the upper surface

of the flap. Still, this example demonstrates that the simplified higher-order NURBS approach

proposed in the paper is capable of providing results of very good quality even when a reduced

number of points is known from the geometry of interest.

8. CONCLUSION

The adequate treatment of curved surfaces is of great importance in the context of unstructured high-

order methods, otherwise one can expect not only loss of accuracy near curved surfaces, but also

steady-state convergence issues. NURBS-based boundary treatment techniques can be considered

superior when compared to other available approaches in terms of accuracy as well as regarding

compatibility with CAD softwares, a clear advantage concerning engineering applications. In this

study, the discontinuous Galerkin (DG) formulation is described as implemented in the code

Veritas2D, a parallel unstructured high-order solver developed to simulate the Euler equations

of gas dynamics. Basic concepts on NURBS curves are presented and the relevant integrations

required by the numerical discretization are discussed in detail for both straight-sided and curved

elements. Then, a new approach based on the use of higher-order NURBS is proposed, with the

benefits of reduced implementation complexity and computation overhead. In order to support the

proposed approach, numerical experiments are carried along with theoretical estimates. At last,

additional simulations are addressed as numerical examples to illustrate the advantages of using

higher-order NURBS in practical situations, minding particularly engineering applications in the

context of compressible aerodynamics.

ACKNOWLEDGEMENTS

The authors acknowledge the support for this study provided by FAPESP (Sao Paulo Research Foundation)through the Grant 2012/16973-5, and by CNPq (Brazilian Council of Research and Development) throughthe Grant 305147/2010-2.

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higher-order surface treatment for discontinuous galerkin

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