Lyapunov exponents and adaptive mesh refinement for

high-speed flows using a discontinuous Galerkin

algorithm

R.C.Mouraa,1, A.F.C.Silvaa,2,∗, E.D.V.Bigarellab,3, A.L.Fazendac,4,M.A.Ortegaa,5

aITA, Technological Institute of Aeronautics, Sao Jose dos Campos, SP, BrazilbEMBRAER, Commercial Aviation, Sao Jose dos Campos, SP, Brazil

cUNIFESP, Federal University of the State of Sao Paulo , Sao Jose dos Campos, SP,Brazil

Abstract

This paper addresses two very important improvements to a discontinu-ous Galerkin platform: (i) Accurate shock demarcation using FTLE — FiniteTime Lyapunov Exponent; (ii) Point-implicitness of the artificial viscosityfield calculation algorithm. When dealing with high-speed flows, within thetransonic, supersonic or hypersonic regimes, the presence of shock-wave phe-nomena dictates the main features of the flow. Shock waves are very thinstructures across which the flow properties vary in a practically discontinu-ous fashion. Therefore, within the realm of computational fluid dynamics,the employment of adaptive mesh refinement is crucial for the proper cap-turing of shock waves and related flow transitions. This work presents anelegant approach for the precise demarcation of shocks, having in mind thesubsequent application to local grid refinement. The proposed technique is

∗Corresponding authorEmail addresses: rodrigomoura.t10@gmail.com (R.C.Moura),

andre.fernando.t10@gmail.com (A.F.C.Silva), enda.bigarella@gmail.com(E.D.V.Bigarella), alvaro.fazenda@unifesp.br (A.L.Fazenda),marcos.ts.ortega@gmail.com (M.A.Ortega)

1Assistant professor.2Assistant professor.3Senior aerodynamicist.4Associate professor.5Professor.

Preprint submitted to Journal of Computational Physics February 11, 2015

based on the use of an operator originally developed within the context ofdynamical systems, namely, the Finite Time Lyapunov Exponent (FTLE).When applied to the dilatation field (divergence of velocity) of the flow, suchoperation is capable of highlighting shock waves in a remarkable way. The ar-tificial dissipation in our original discontinuous Galerkin code is representedby a distributed field, which results from the solution of a partial differen-tial equation. The modal artificial dissipation model brings an even higherdegree of stiffness to the scheme than the typical stiffness related to the vis-cous terms. To overcome this inconvenience we have treated the dissipationequation implicitly, what characterizes the overall scheme as a devoted-point-implicit numerical tool (devoted in the sense that the point-implicit strategyis applied only to the dissipation equation). In order to demonstrate thecapabilities of these new ideas, different test cases chosen especially from theaerospace context are tackled and discussed.

Keywords:Modal-based discontinuous Galerkin, Finite time Lyapunov exponents,Point-implicit algorithm

1. Introduction

The physical phenomena that take place in high-speed flows is ofparamount importance specially in the aerospace context. For such flows,the presence of structures known as shock waves, through which the fluidproperties experience an abrupt change, dictates the main features of theflow field. In typical aeronautical conditions, the thickness of a shock wave isso small (about 10-7m) that it may be regarded as a discontinuity. This fea-ture represents an extremely difficult challenge for numerical schemes whichare designed to simulate such flows.

When a numerical scheme is used to solve a set of partial differentialequations (PDE) modeling the physical phenomena taking place, a discretedomain is chosen where an algebraic approximation to the PDEs is actu-ally solved. Regardless the numerical formulation adopted, the local errorand hence the accuracy of the solution are functions of the local mesh size.In the vicinity of shocks, numerical schemes normally introduce unphysicalwiggles, commonly related to the Gibbs’ phenomenon. The methodologiesdeveloped to overcome this drawback usually reduce the local accuracy tofirst order in these regions, smearing the shock waves and increasing the as-

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sociated numerical thickness. In such cases, only a local mesh refinementwould diminish the numerical dissipation and provide a better shock resolu-tion without the excessive computational cost that would follow from globalmesh refinement. Thus, the use of local mesh refinement has become a veryuseful tool when dealing with high-speed flows and has drawn the interestof the scientific community [1]. In the past thirty years a large numberof works addressed several algorithms and convergence-efficiency methods,what established the relevance of a tool with such characteristics [2, 3, 3].Among those efforts, many used a multigrid-like approach consisting of sev-eral layers of progressively refined meshes that are overlaid only in the regionswhere the refinement is needed. Here, the main advantage is that there is nochange in the main mesh, which is very interesting, specially when dealingwith complex geometries. In this work, however, it was adopted a differentline of thought similar to the work of [4]. The initial mesh is subjected to aniterative process, where successive refinements are produced, each of whichis based on preliminary solutions obtained in the coarser grids.

One of the most fundamental issues to be addressed when performingan adaptive mesh refinement is to specify the regions which need to be sub-jected to the procedure. In practice this corresponds to delimit regions oflarge gradients. In this paper, an operator originally developed within thecontext of dynamical systems, namely, the Finite Time Lyapunov Exponent(FTLE)[5], is used to perform the task.

The study to be reported herein has profited from a previously developed,two-dimensional discontinuous Galerkin code [6]. In terms of artificial dissi-pation, two methods were tested and compared: the Persson and Peraire [7]and Barter and Darmofal [8]. The last approach is preferred simply becausethe product of the calculation is a smooth artificial viscosity field. On theother hand, one has to “pay” for this, because to obtain this distributionone has to solve an extra PDE. It is well known that diffusion terms aresynonymous of numerical stiffness. To face this new hurdle we decided foran implicit approach to the artificial viscosity PDE.

This paper is organized as follows. A short presentation of the discon-tinuous Galerkin methodology that is employed in this work, together witha brief explanation on the point-implicit approach is given in Section 2, fo-cusing only on the main aspects of the technique. A brief introduction tothe FTLE operator is presented in Section 3. The main aspects concerningthe refinement procedure are addressed in Section 4. The numerical formu-lation of the solver used for the final simulations, along with the test cases

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of interest, is described in Section 5. Final comments and future researchpossibilities are discussed in Section 6.

2. The discontinuous Galerkin framework

Let Ω ⊂ Rd, for d ≥ 1, where R is the real line, be an open, bounded setwith sufficiently smooth boundary. Let Hm(Ω) be such that

Hm(Ω) = ψ ∈ L2(Ω): for each nonnegative multi-index α with | α |≤ m,

the distributional derivative Dαψ belongs to L2(Ω).

Hm is a Sobolev vector space of functions ψ(x, y) ∈ L2(Ω). In this work, thetwo-dimensional compressible flow of an inviscid gas will be treated. Suchkind of flow can be mathematically modeled by the Euler equations. Thedomain of interest is denoted by Γ ⊂ Hm and, given adequate initial andboundary conditions, one has in divergence form [9],

∂U∂t

+∂Fx

∂x+∂Fy

∂y= 0 in Γ , (1)

where (x, y) ∈ Γ, t ∈ (0, T ], U = U(x, y, t), F = f(U). The above equationsrepresent, physically, the conservation of mass, the two components of themomentum equation and the energy equation. The conserved variables vectorU and the Cartesian components of the inviscid flux vector ~F are respectivelywritten as

U = ρ, ρu, ρv, eT , (2)

Fx =ρu, ρu2 + p, ρuv, (e+ p)u

T, (3)

Fy =ρv, ρvu, ρv2 + p, (e+ p)v

T, (4)

where ·T indicates the transpose of a vector, ρ stands for density, u and vare the Cartesian components of the velocity vector, e = ρ[ei+(u2 +v2)/2] isthe total energy per unit volume, and ei is the specific internal energy. Notethat this system has five unknown variables but only four equations. Thelast equation needed to close the model is provided by the equation of statefor a perfect gas (this approximation is quite reasonable for the atmosphericair in a wide range of pressures and temperatures), namely, p = ρ(γ − 1)ei,

4

where γ is the fluid ratio of specific heats, which assumes the value γ = 7/5for the air.

As for the standard finite element formulation, DG works with the weakform of the PDEs. The two-dimensional Euler equations (Eq. (1)), can berewritten in divergence form as

∂U∂t

+ ~O · ~F = 0 , (5)

where ~O · ~F is the divergent operator actuating over the two-dimensionalinviscid flux tensor ~F , whose rows correspond to the original flux vectors(Eq. (4)), that is,

~F =

[FxTFyT

]. (6)

This equation is then multiplied by the test function φn (in the classicalGalerkin method, the test functions are the same as the basis functions) andthe term with the divergence is split in two by the use the formula for thedivergence of a product between a scalar field and a vector field to obtain

φn∂U∂t

+ ~O · (φn ~F)− ~F · ~Oφn = 0 . (7)

In each element, the solution vector U is locally approximated by a poly-nomial expansion with NP = (P+1)(P+2)

2modes, where P stands for the order

of the polynomial expansion, as

U =

NP∑m=1

Cm(t)φm(ξ1, ξ2) , (8)

whose temporal derivative is given by

∂U∂t

=∂

∂t

N∑m=1

Cmφm =N∑m=1

φmdCm

dt= φ1, ..., φN

d

dt

C1···

CN

. (9)

5

By integrating Eq. (7) over each individual element Ωe, one obtains∫∫Ωe

φn∂U∂tdΩe =

d

dt

C1···

CN

∫∫Ωe

φn φ1, ..., φN dΩe (10)

= Men

d

dt

C1···

CN

(11)

=

∫∫Ωe

~F · ~Oφn dΩe −∫∫

Ωe

~O · (φn ~F) dΩe (12)

=

∫∫Ωe

~F · ~Oφn dΩe −∮∂Ωe

φn ~F · ~n d∂Ωe , (13)

where Men stands for the n-th line of the so called mass matrix. This equation

is known as the weak form of the original PDE system Eq. (1).In this work, artificial viscosity terms are introduced to the original Euler

equations in order to improve shock capture capabilities and prevent theaforementioned related issues. When using such techniques, the modifiedEuler equations can be typically written as

∂U∂t

+ ~O ·(~F(U)− ~Fv(U , ~∇U)

)= 0 , (14)

so that ~Fv = (Fxv ,Fyv ) stands for a viscous flux vector that depends explicitly

not only on the state vector U , but also on the state vector gradient ~∇U . Asoriginally proposed in [10], one should then split the 2nd-order equation intoa system of two 1st-degree equations, namely,

∂U∂t

+ ~O ·(~F(U)− ~Fv(U , ~G)

)= 0 , (15)

~G − ~OU = ~0 , (16)

where Eq. 16 receives specific treatment, being totally solved within eachtime level. After some manipulation, the auxiliary equation comes to

∫Ωe

φn ~G dΩe =

∮∂Ωe

φnU(U−,U+)~n d`e −∫

Ωe

U~OφndΩe . (17)

In this work, the second scheme of Bassi and Rebay [11], also known

as BR2, is used to calculate the numerical averages for ~Fv and U at theboundaries and the viscous flux is calculated as

6

~Fv = ε(U)~∇U . (18)

The key issue now is how to determine the numerical dissipation magni-tude (ε) throughout the domain. For the test cases presented in this papertwo approaches were adopted.

2.1. The CTE-based artificial viscosity

This approach was first proposed by Person and Peraire [12] and corre-sponds to a element-wise constant artificial viscosity field. For the evaluationof the dissipation coefficient ε inside each element Ωe, an oscillation sensorσ must be calculated. Using u and u to respectively represent the complete(within the polynomial space of order P ) and truncated (containing only themodes associated with the space of order P −1) expansions of some propertyderived from the expansion of U (see Eq. (8)), namely,

u =

NP∑m=1

cmφm(ξ1, ξ2) , u =

NP−1∑m=1

cmφm(ξ1, ξ2) , (19)

the sensor σ can be obtained as the squared ratio between the L2 norms(within Ωe) of q − q and q, given by

σ =‖u− u‖2

L2

‖u‖2L2

=

(N∑

m=N ′

c2m‖φm‖

2L2

)/

(N∑m=1

c2m‖φm‖

2L2

), (20)

where N = NP and N ′ = NP−1 + 1.After calculating the oscillation sensor σ, the dissipation ε is introduced

gradually by means of a smooth switching,

ε =

0 if s < so − κ ,12εo

[1 + sin π(s−so)

2κ

]if so − κ ≤ s ≤ so + κ ,

εo if s > so + κ ,

(21)

where s = log10 σ, so = −(A + B log10 P ) and εo = Che/P . In the presentwork, typical values of the model parameters in the bidimensional contextwere A ∼= 4, B ∼= 4, C ∼= 0.1 and κ ∼= 0.5, but they are strongly dependenton the particular case being simulated and must be adjusted carefully to

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provide optimal results (sharp and yet non-oscillatory shock profiles). Foreach (triangular) element, the largest edge length was used to represent theelement size, he, which is a somewhat conservative choice. Also, regardingnumerical implementation, it is advisable to add a small constant (∼= 10−10)to the value of σ when calculating s = log10 σ in order to avoid the divergenceof the logarithmic function in the limit of a zero argument.

Finally, regarding the implementation of boundary conditions, we haveproceeded as follows. For periodic boundaries, the procedure is natural: onceelements linked through the periodicity condition should see each other asordinary neighbors, the choice of U± and ~G± is made just as if the boundarywere an internal interface. For farfield, free-slip wall and inflow/outflowboundaries, the conditions θ+ = 0 (equivalent to ε+ = 0) and ~OU+ = ~OU−were adopted.

2.2. The PDE-based artificial viscosity

In addition to the methodology presented in the last section, the tech-nique due to Barter and Darmofal [8, 13, 14] was also implemented, where amodal dissipation function is used. This approach prevents the strong discon-tinuities that may appear between elements when the CTE-based approachis adopted. In this model, the dissipation coefficient ε is given by

ε =

0 if θ < θL ,12θH

[1 + sin π

(θ−θLθH−θL

− 12

)]if θL ≤ θ ≤ θH ,

θH if θ > θH ,

(22)

where θH = λmaxh/P and θL = θH/100. The parameter λmax correspondsto the maximum wave speed, namely, λmax = max |u|+ c, |v|+ c for theEuler’s equations, and the scalar field h = h(x, y) is used to represent thelocal element size, defined as h = (hx + hy)/2, in which hx(x, y) and hy(x, y)are two-dimensional linear functions whose values at each mesh vertex (xk, yk)are defined by the expressions

hx(xk, yk) =1

nk

nk∑n=1

`xn , hy(xk, yk) =1

nk

nk∑n=1

`yn , (23)

where the index n runs through all nk elements sharing the vertex (xk, yk),and the lengths `xn and `yn are the dimensions of the box containing the tri-angular element n formed by the vertices (xn1 , yn1 ), (xn2 , y

n2 ), (xn3 , y

n3 ), being

8

given by

`xn = max |xn1 − xn2 |, |xn2 − xn3 |, |xn3 − xn1 | , (24)

`yn = max |yn1 − yn2 |, |yn2 − yn3 |, |yn3 − yn1 | . (25)

The dissipation coefficient in Eq. (22) is triggered by a smooth oscillationsensor, θ = θ(x, y, t), which is driven by a specific partial differential equationof diffusive character (see more details in [13]), namely,

∂θ

∂t= ~O ·

[5

τ

(∂θ

∂xh2x x+

∂θ

∂yh2y y

)]+

1

τ

(h

PλmaxS − θ

), (26)

where τ would be a time-scaling parameter given by

τ =min hx, hy

3Pλmax, (27)

and the variable S consists of an element-wise oscillation sensor, obtained by

S =

0 if s < so − κ ,12

[1 + sin π(s−so)

2κ

]if so − κ ≤ s ≤ so + κ ,

1 if s > so + κ ,

(28)

in which s = log10 σ, so = −(4 + 4.25 log10 P ) and κ = 0.5. Here, σ cor-responds to the same oscillation sensor used for the CTE-based artificialviscosity (see Eq. (20)), actuating over the density expansion of the elementbeing considered. Again, it is advisable to add a small constant (∼= 10−10) tothe value of σ when calculating s = log10 σ in order to avoid the divergenceof the logarithmic function in the limit of a zero argument.

When using the PDE-based dissipation model, Eq. (26) can be appendedto the physical system of equations (Eq. (1) to (4)), resulting in the followingextended (or augmented) system:

∂U∂t

+ ~O ·(~F(U)− ~Fv(U , ~G)

)= B(U) , (29)

~G − ~OU = ~0 , (30)

9

where

U = ρ, ρu, ρv, e, θT , (31)

Fx =ρu, ρu2 + p, ρuv, (e+ p)u, 0

T, (32)

Fy =ρv, ρvu, ρv2 + p, (e+ p)v, 0

T, (33)

B =

0, 0, 0, 0, (hλmaxS/P − θ)/τT

, (34)

Fxv = Axv(U) Gx , (35)

Fyv = Ayv(U) Gy . (36)

The matrices Axv and Ay

v are designed to act over ρH = e+p, the enthalpyper unit volume, instead of simply over the energy per unit volume, e. Thisis done in order to avoid small wiggles (in the vicinity of shocks) that wouldhappen otherwise [13]. Since it can be shown that

∂ρH∂x

=1

2(γ − 1)(u2 + v2)

∂ρ

∂x+ (1− γ)u

∂(ρu)

∂x+ (37)

+ (1− γ)v∂(ρv)

∂x+ γ

∂e

∂x,

∂ρH∂y

=1

2(γ − 1)(u2 + v2)

∂ρ

∂y+ (1− γ)u

∂(ρu)

∂y+ (38)

+ (1− γ)v∂(ρv)

∂y+ γ

∂e

∂y,

the viscous matrices Axv(U) and Ay

v(U) are given by

Axv =

hxh

ε 0 0 0 00 ε 0 0 00 0 ε 0 0

12(γ − 1)(u2 + v2)ε (1− γ)uε (1− γ)vε γε 0

0 0 0 0 5hxh/τ

, (39)

Ayv =

hyh

ε 0 0 0 00 ε 0 0 00 0 ε 0 0

12(γ − 1)(u2 + v2)ε (1− γ)uε (1− γ)vε γε 0

0 0 0 0 5hyh/τ

. (40)

By analogy with Eqs. (10) to (12), the discretization of Eq. (29) wouldresult in

10

Me ddt

C1···CN

=∫

Ωe

(~F(U)− ~Fv(U , ~G)

)· ~O

φ1···φN

dΩe +

∫Ωe

φ1···φN

B(U) dΩe

−∑

s

∫∂Ωs

e

φ1···φN

(F(U−,U+, ~n)− Fv(U−, ~G−,U+, ~G+, ~n)

)d`

(s)e , (41)

where the calculation of F should be performed by means of a numerical fluxas the Lax-Friedrich’s, except for the last component, the one related to θ,in which one should do F θ = 0 for the sake of consistency. Additionally, theevaluation of Fv can be performed through the BR2 scheme, which yieldsthe expression

Fv(U−, ~G−,U+, ~G+, ~n) =1

2

(~Fv(U−, ~G−) + ~Fv(U+, ~G+)

)· ~n , (42)

where the values ~G− and ~G+ are evaluated by the following auxiliary equa-tions

~G± = ~OU± + η ~D±s , (43)∫Ωe

φn ~Ds dΩe =1

2

∫∂Ωs

e

φn(U+ − U−)~n d`(s)e , (44)

and the internal vector function ~G (see the first integral in Eq. (41) is givenby

~G = ~OU + ~D , (45)

~D =∑s

~Ds . (46)

Finally, regarding the implementation of boundary conditions, one canproceed as discussed for the element-wise constant artificial viscosity model.

2.3. The point-implicit smooth artificial viscosity approach

Preliminary tests have shown that the PDE-based artificial dissipationmodel has a direct impact on the CFL number for explicit time marching.Since the time scale does not have to be the same for all equations, an implicitscheme was adopted to advance the artificial viscosity equation (Eq. 26) in

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time. All the remaining equations are advanced by means of an explicit timemarching.

Equation (26) can be represented by the following model

∂θ

∂t= ~∇· ~Fv(θ,~g) + S(θ) , (47a)

~g = ~∇θ . (47b)

If the Galerkin procedure of Section 2 is applied together with the BR2scheme, the following results

Med

dt

c1···cN

ne

= Re(θn) (48)

where Me is the mass matrix, ci are the polynomial expansion coefficientsof the variable θ, Re stands for the entire expression that appears on theright hand side of the Eq. 47, and n refers to the present time step. On theother hand, within the implicit time-stepping context, the right hand side isevaluated at a future time step (n+ 1) and a first-order Taylor expansion isused. Note that only the immediate neighborhood appears explicitly here.Then, circumstances are such that Eq. (48) can be rewritten as

Me

∆t(θn+1e − θne ) =

Me∆θe∆t

= Re(θn+1) ≈ Re(θ

n) +

[∂Re

∂θ

]n∆θ (49)

= Re(θn) +

[∂Re

∂θe

]n∆θe +

Neighe∑k=1

[∂Re

∂θek

]n∆θek .

where Neighe stands for the number of neighbors of the element e.The grouping of the terms related to the central element, yields the fol-

lowing expression

(Me

∆t−[∂Re

∂θe

]n)∆θe = Re(θ

n) +

Neighe∑k=1

[∂Re

∂θek

]n∆θek . (50)

The major difficulty now is related to the summation in the right hand side,since it couples the target elements and its neighbors. Following [15], the

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evaluation of Eq. (50) is done iteratively and the influence of the neighboringelements is partially disregarded. So, the previous equation can be rearrangedas

(Me

∆t−[∂Re

∂θe

]∗)∆θe = Re(θ

∗) , (51)

where the superscript ()∗ means that the term is being evaluated using thelatest available solution in the adjacent element. At first sight, this proceduremay be regarded as a point Gauss-Seidel method at each mesh element.Therefore, it seems that one would need to perform several mesh sweeps inorder to accurately evaluate the term ∆θe or even seek out the optimumsequence of elements to perform this sweep. However, preliminary tests onthe PDE-based viscosity model have shown that just one sweep is enough toprovide good results, and the elements order has little influence in the finalsolution (the testing involved a random sweep and an “upwind” sweep).

Note that the proposed procedure introduces new approximations thatcertainly will impact the stability envelope of the implicit method. However,high CFL numbers are still possible. In fact, for practical simulations, theCFL number is severely restricted by the explicit approach and, as shown, theimplicit scheme has behaved robustly and quite well. Moreover, the diffusiontime scale related to the artificial viscosity can be made much smaller thanthe convective time scale of the Euler’s equations. So, the CFL numberemployed in the implicit scheme is often set to be 100 times larger than theEuler’s equations CFL number.

3. The finite time Lyapunov exponent

Recently, the development of the dynamical systems theory (speciallyin the field of non-linear dynamics and chaos) and its application in fluiddynamics has provided interesting insights into the physics of a variety offlows using the so called Lagrangian approach to the problem [16, 17, 18].As opposed to the Eulerian scheme, the attention is here focused on themovement of each individual particle of fluid.

It is known that even unsteady flows may admit material lines which havean attractive or repulsive character with regard to the fluid particles [19].Specially, following [5], the material lines derived from individual particletrajectories are called Lagrangian Coherent Structures (LCS). A stable LCS

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is defined as the locus of points whose trajectories converge to a particularregion when t→∞ (forward integration). Analogously, an unstable LCS isthe locus of points whose trajectories converge to a particular region whent→ −∞ (backward integration).

Shaden et al. [5] have demonstrated that the LCSs can be calculated usingthe FTLE operator. Specifically, one can show that the LCSs correspondto the ridges of the scalar field calculated by the FTLE. These ridges canbe understood as attracting (or repelling) structures whose dimensionalitycorresponds to the considered flow dimension subtracted by one. For the two-dimensional flows of interest in the present research, the calculated LCSs arecurves representing shock waves.

3.1. Calculation methodology

Basically, the FTLE is a measure of the local repulsion (or attraction)of neighbor particles traveling on a given flow field. In practice, it can bedefined as the time-averaged maximum exponential rate of repulsion betweenparticles initially very close. Such particles must be convected by a prescribedvector field during a suitable time interval.

Let ~τ t0+Tt0 (x, y) be the position vector at time t0 + T of a particle that,

in time t0, occupies the position (x, y) and let ~δi = (δx, δy) be an infinites-

imal vector pointing toward an arbitrary direction. The initial difference ~δibetween the positions of two close particles will be amplified to the final dis-tance ~δf = ~τ t0+T

t0 (x+ δx, y + δy)− ~τ t0+Tt0 (x, y) by the “traveling” function ~τ

after a time interval T . Note that positive values of T correspond to a for-ward integration of the particles trajectories in time and negative values ofT correspond to a backward integration in time. The FTLE value calculatedat (x, y), denoted by σTt0(x, y), is given by

σTt0(x, y) = max~δi

1

Tln

(|~δf ||~δi|

), (52)

where ~δi is allowed to vary within a close perimeter of neighbor particles inorder to provide the maximum local repulsion rate.

Note that, once the parameters t0 and T are chosen, σTt0(x, y) is a func-tion of the position ~p = (x, y) alone and therefore represents a scalar field.Numerically, an efficient way to calculate this quantity over a entire domain

14

Figure 1: Schematic illustration for the calculation of the FTLE showing a target nodeand its satellite nodes.

was proposed by Padberg et al. [20]. The key idea is to generate an equally-spaced Cartesian mesh whose nodes represent the initial position of particles.Then, one can evaluate, for every grid node, the value of ~τ t0+T

t0 (x, y) by inte-grating the trajectories of each node-related particle. For sufficiently refinedgrids, the FTLE can be approximated by

σTt0(~p) =1

|T |ln

(maxj

∣∣~τ t0+Tt0 (~p)− ~τ t0+T

t0 (~nj(~p))∣∣

|~p− ~nj(~p)|

), (53)

where ~nj(~p) represents each of the eight mesh positions that are closest to~p, namely, the satellite particles presented in Fig. 1. Note that the term~τ t0+Tt0 (~nj(~p)) represents the position of each of the satellite particles at the

end of the time interval T . This expression is preferable when compared toother ones available in the literature because it avoids the numerical calcula-tion of derivatives. The finer the mesh used to calculate the FTLE, the moreaccurate will be the resulting scalar field.

Note that according to the LCS definition previously presented, a highlocal value of the FTLE scalar field for either a positive T (progressive in-tegration) or negative T (regressive integration) would correspond to LCSs.

15

Actually, it can be shown [5] that one distinctive feature of the resultingscalar field is that it presents a constant value at the LCSs and a null valueelsewhere. This property will play a fundamental role in the present worksince the shock regions are intended to be equally demarcated regardless ofthe shock strength.

3.2. The velocity field analogy

As previously stated in the introduction, a major issue regarding therefinement process is to determine the regions where the abrupt variationof the fluid properties requires a finer mesh in order to achieve a betterlocal accuracy. Several shock sensors have already been proposed in theliterature for a wide range of numerical methods. In the present work thedilatation field (velocity divergence) was adopted following the suggestionsof [21] and [22], since such variable reaches strong negative values at shockwaves. Actually, shock waves correspond to ridges of negative magnitude inthe dilatation field.

It is now useful to imagine a velocity field given by the dilatation gradi-ent vector field. The shock wave will act as a repulsive structure to particlesfollowing this velocity field since the gradient vector will point to oppositedirections on each side of the shock wave. Thus, the FTLE based on progres-sive integration will provide a maximum value along the entire shock wave.Obviously, the finer the mesh used to evaluate the FTLE, the better will bethe resolution of the shock location but also the more expensive will be thecalculation. Figure 2 shows the dilatation field close to the shock wave aheada circular cylinder subjected to a supersonic flow. The dilatation gradientfield is represented by the black arrows. Note that particles following such(virtual) velocity field would be repelled away from each other depending onwhich side of the shock wave they are placed initially.

4. The refinement procedure

Having in mind complex geometries, the use of unstructured meshes isa logical choice. The meshes we have used were created using the softwareDistmesh [12], a Matlab c© code developed by Persson [23]. This subroutineis able to generate unstructured meshes for geometries specified via implicitfunctions, iteratively improving an initial mesh by enforcing an equilibriumof forces at the element edges. It uses three basic inputs to generate anytriangular mesh:

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Figure 2: Dilatation field of the shock wave ahead a circular cylinder subjected to a Mach-2.5 uniform flow. The vectors represent the velocity field that (virtual) particles wouldfollow according with the proposed analogy.

• The minimum mesh length h0;

• A scalar field fd(x, y) that corresponds to the distance of point (x, y) tothe closest boundary, yielding a negative or a positive value for pointsinside or outside the domain, respectively;

• The element size function fh(x, y) that corresponds to the relative localmesh length scaled by h0.

The function fd(x, y) is solely settled by the domain boundary geometryand, therefore, only needs to be determined once. The minimum mesh lengthh0 is a global measure of the mesh coarseness and the principal parameter tobe changed in a mesh convergence analysis. Thus, the main objective hereis to determine the element size function fh(x, y) so that the vicinities of theshock waves are refined.

The mesh refinement procedure is based on an iterative process that fol-lows basically four steps as illustrated by Fig. 3. Beginning with an initialmesh which does not present any specific refinement besides that needed toprovide a better description of the boundary conditions, the first step is touse a numerical solver to calculate the solution which represents a prelim-inary approximation to the desired flow pattern. In the second step, the

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Figure 3: Schematic illustration showing the major steps of the refinement procedure.

flow properties, represented by the averages at the elements, and stored atthe triangles barycenters, are interpolated unto an auxiliary equally-spacedCartesian mesh, in which all further calculations will take place — the finerthe mesh, the more accurate the following calculations will be.

The velocity divergence field is then calculated and the FTLE operatorcan be used to determine the shock waves location as the set of nodes (inthe auxiliary mesh) whose calculated FTLE values exceed a suitable cutoffvalue. A new element size function f ih(x, y) is obtained based on the radialdistance of any point (x, y) to the position of the marked shock-related nodes.A new mesh is then generated using the Distmesh and the entire procedureis repeated until the shock waves are sufficiently resolved in the numericalsolution.

It is worth mentioning that the initial mesh needs to have a minimumdegree of refinement in order to provide good results for the first shock de-marcations. Since the shock location may slightly change between successiveiterations due to the coarseness of the intermediary meshes, it is advisableto perform a few steps of progressive refinement instead of a single one.

5. Numerical results

In this Section, results are presented to validate the proposed enhance-ments to our DG platform. At first, improvements due to the implementationof the point-implicit scheme are discussed. Then, the results for the FTLE-

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enhanced adaptive refinement framework are presented and discussed. Wecall the attention of the reader to a very important point: the NACA 0012and the forward-facing step cases were tackled with the help of the CTE-based artificial viscosity (Section 2.1), whilst the supersonic flows past thewedge and the cylinder were simulated with the PDE-based artificial viscosityscheme (Section 2.2).

5.1. The devoted point-implicit artificial viscosity performance

The artificial viscosity model due to Barter and Darmofal [8], when im-plemented together with an explicit temporal marching, leads to a significanttightening of the stability envelope of the overall simulation. This limitationis due to the extra equation that models the diffusion of the parameter θ,whose stability envelope ends up constraining the entire system. The delaycaused by using such small CFL numbers renders this approach practicallyunfeasible.

The improvements obtained by implementing the point-implicit scheme tothe last equation can be measured by comparing the CFL numbers. Whereasthe CFL numbers for the explicit schemes were not much larger than 10−5,the implicit formulation allowed numbers up to 10−2. In order to assess theconsistency of the solutions, two important test cases shall be tackled in thenext Section.

As already mentioned in Section 2.3, the point implicit algorithm is verysimilar to the Gauss-Seidel method. Having the latter in mind, one wouldargue about the need of performing several mesh sweeps to accurately solvethe problem. Moreover, focus is set upon the sequence of mesh elements tobe considered in the updating phase. The idea is to investigate if differentsequences would lead to different values for each step.

In order to address these issues, an one-dimensional test is proposed. Fora fixed Mach number distribution, such as M(x) = 2 − arctan(100x), thesteady-state solution of the oscillation sensor, θ, is calculated by evolvingEq. (26) using the proposed implicit time marching scheme. Figures 4 and 5present the final solution for θ and the residual history for a single-step pointimplicit scheme with different swap ordering choices. As the reader can grasp,the swap ordering has little effect on the convergence rate. For the sake ofsimplicity and computational cost savings, hereafter the single-step approachwas adopted since the obtained results were quite satisfactory.

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(a) Ordered swap.

(b) Random swap.

Figure 4: Steady-state solution and residual history for the oscillation indicator θ using ap = 1 representation.

5.2. Adaptive refinement using the FTLE

In order to verify the capabilities of the proposed refinement methodology,an Euler equations numerical solver was implemented to simulate the physicsof inviscid compressible air flows. The code called Veritas2D is based on theDiscontinuous Galerkin (DG) scheme for unstructured meshes. The treat-

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(a) Ordered swap.

(b) Random swap.

Figure 5: Steady-state solution and residual history for the oscillation indicator θ using ap = 3 representation.

ment of shocks is accomplished by means of artificial viscosity approaches orvia an ENO-based limiting technique. The reader is referred to the works ofMoura [6] and da Silva [24] for a detailed description of the DG formulationand of the shock-capturing approaches available in the code. In addition, thehigh-order NURBS treatment of curved boundaries is employed (for specifics

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and details of implementation see the work of Silveira et al. [25]). (NURBSstands for “Non-Uniform Rational B-Splines”.)

Three test cases were chosen to demonstrate the refinement procedureproposed in Section 3, namely, a transonic airfoil, the supersonic flow pasta circular cylinder and the forward facing step in a supersonic channel. Allresults were obtained with a polynomial order of 2 (which yields third-orderspatial accuracy at smooth regions), and with the inviscid Lax-Friedrichs [26]and the viscous BR2 [11, 27] numerical fluxes. It is important to say thatthe refinement methodology here proposed can be used as well with classicallow-order formulations.

5.2.1. The transonic flow over a NACA-0012 airfoil (cutoff analysis)

The first test case is the transonic flow past an standard NACA-0012airfoil with sharp trailing edge. The chosen freestream Mach number is 0.80and the angle of attack is set to 1.50 degrees. The flow is solved withina circular domain centered at the profile with a far-field outer diameter of20 chords. Figure 6(a) shows a global view of the initial mesh used for therefinement procedure. Each individual mesh thereafter will only differ on thenear-shock refinement pattern.

Probably the first thing someone might argue is that it is easier to use thedilatation field directly to build the element size function. In order to addressthis issue, a block-like refinement was performed in the regions enclosing theshock wave in the upper surface and the weaker one in the lower surface. Theelement sizing for both regions is uniform and the value of h0 is the same.Note that the stronger the shock wave, the bigger the dilatation absolutevalue at that point. Thus, to use the dilatation field directly would lead toa stronger refinement in the upper surface.

In order to compare the efficiency of both the dilatation field and theFTLE to determine the location of the shock waves, the scalar fields werenormalized by its maximum values. A point is said to belong to a shock waveif the normalized scalar field value at that point is larger than an adjustablecutoff value. The right side of Fig. 7 highlights the grid nodes that are markedby the FTLE as belonging to the shock wave for a given cutoff value; andthe left side shows the grid nodes that were marked by the dilatation fieldfor the cutoff value that provide the same number of points marked by theFTLE. Note that for the same number of grid nodes, the FTLE is capableof marking a longer extension of the upper shock wave or even capture thelower surface shock wave when the dilatation field alone is not capable of.

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(a)

(b)

Figure 6: The chosen mesh for the cutoff analysis: (a) A global overview of the meshused in the NACA-0012 transonic simulations. (b) A closer look showing a block-shapedrefined region in the mesh used for the cutoff analysis.

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Using the nodes previously marked by the FTLE, one can create therefined mesh presented in Fig. 8(a) and calculate the flow past the profileone more time. The resulting Mach number field is presented in Fig. 8(b).

5.2.2. Supersonic flow past a wedge

The second test is the supersonic flow past a wedge. The wedge angle is12 degrees and the incident flow, which is directed from left to right, has afreestream Mach number of 1.5. Thus, one should expect an oblique shockwave attached to the tip of the wedge with an angle of 64.36 degrees withrespect with the incident flow direction. This is a very interesting initialcase since we know exactly where the shock wave must be. Starting with auniformly refined mesh, three refinement cycles were performed. Figure (9)presents the final mesh and respective Mach number field, showing a verysharp shock transition.

5.2.3. The supersonic flow past a circular cylinder

The third test case is the supersonic flow past a circular cylinder. Theincident flow is directed downwards and has a freestream Mach number of2.5. Following the refinement steps presented in Section 3, three completecycles were performed. Figure 10 presents all meshes created throughoutthe process. An interesting feature of these meshes is the fact that therefinement degree is the same along the entire shock extension. This resultsfrom the FTLE property of delivering a constant value along the shock waveno matter how strong it is. Other methodologies tend to furnish a strongerrefinement near the stagnation point where the shock transition is stronger sothat the numerical results tend to display an increasing smearing of the shockwave away from the symmetry line. The successive solutions for both Machnumber and static pressure are presented in Fig. 11, where an expressiveimprovement in the shock resolution is achieved by efficiently employing therefinement only along the shock wave.

5.2.4. The steady supersonic flow through a channel with a forward facingstep

The fourth and last test case is a modification of the classical forward-facing step problem by Woodward and Colella [28]. The solution domain istwo units high in the inlet Section and six units long. The relative step heightwas modified from 20% to 17.5% so that the problem admits a steady-state

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(a)

(b)

Figure 7: The illustrations show the node positions which are marked for refinement usingsimply the dilatation field (left) and the corresponding FTLE based on the dilatation field(right) for the same number of marked points: (a) Using a cutoff of 40% of the maximumFTLE field magnitude. (b) Using a cutoff of 10% of the maximum FTLE field magnitude.

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(a) (b)

Figure 8: The final solution: (a) The final used mesh; (b) The final Mach number field.

(a) The final mesh. (b) The final Mach number field.

Figure 9: The supersonic wedge test case: M∞ = 1.5 and θ = 12.

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solution. For all cases the inlet Mach number is set to be 3. This particularcase was run using the CTE-based viscosity approach.

The initial mesh is essentially uniformly spaced with only a local refine-ment near the geometric discontinuity represented by the step, as presentedin Fig. 12(a). Figures 12(b) and 12(c) show, respectively, the velocity diver-gence and the Mach number fields simulated in this initial Mesh. After sevenrefinement cycles, the final mesh obtained is the one presented in Fig. 13(a).The corresponding velocity divergence and Mach number fields, showed inFigs. 13(b) and 13(c), attest the quality of the FTLE-based refinement pro-cedure. Again, one can see that the shock waves where equally refined alongtheir entire extension despite their different local intensity.

6. Conclusions

Firstly, a devoted-implicit artificial viscosity approach is presented in or-der to circumvent the restricting limitations that the smooth oscillation in-dicator model imposes to the overall explicit time marching. The numericalresults demonstrate a significant performance increase even for an extremelysimplified implementation that speeds up the simulation retaining the con-sistency.

This article also presented a new methodology to perform adaptive meshrefinement focused on shock capturing that is based on a dynamical systemsoperator named Finite Time Lyapunov Exponent (FTLE). After a brief in-troduction to the fundamental concepts regarding the FTLE, a velocity fieldanalogy was formulated that allowed the calculation of the shock waves lo-cation by applying the FTLE operator to the gradient of the dilatation field(divergence of the velocity). This procedure has shown to be more efficient inmarking the shock location when compared to the direct use of the dilatationfield, since a narrower number of points is used to represent the discontinuity.

Numerical tests were presented that not only demonstrated the method-ology efficiency but also the quality of the resulting simulations. Due to theoutstanding mathematical properties of the FTLE operator, the entire exten-sion along the shock waves is almost equally refined despite the local shockintensity what guarantees a uniformly sharp shock transition throughout itsentire extension.

Among future research possibilities, one can mention the extension tothree-dimensional geometries and unsteady flows. Moreover, a more accurateinterpolation procedure can be devised, “between” the actual mesh and the

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auxiliary mesh used to perform the shock location calculations, by using thehigh-order information from the polynomials within each element. At last,an advanced methodology can be developed to automatically perform theiteration steps which are currently carried out manually. Also, the implicitsmooth artificial viscosity can be used to improve even further the refinementresults here presented. The use of both improvements together remains as alogical next step.

7. Acknowledgments

The authors gratefully acknowledge the support for this research providedby FAPESP (Sao Paulo Research Foundation), under the Research Grant No.2012/16973-5.

References

[1] D. C. Arney, J. E. Flaherty, An adaptive local mesh refinement methodfor time-dependent partial differential equations, Appl. Numer. Math. 5(1989) 257–274.

[2] W. C. Rheinboldt, On a theory of mesh-refinement processes, SIAM J.Numer. Anal. 17 (6) (1980) 766–778.

[3] M. Berger, P. Colella, Local adaptive mesh refinement for shock hydro-dynamics, J. Comput. Phys. 82 (1989) 64–84.

[4] R. C. Ripley, F.-S. Lien, M. M. Yovanovich, Adaptive unstructured meshrefinement of supersonic channel flows, Int. J. Comput. Fluid Dyn. 18 (2)(2004) 189–198.

[5] S. Shadden, F. Lekien, J. Marsden, Definition and properties of La-grangian coherent structures from finite-time Lyapunov exponents intwo-dimensional aperiodic flows, Physica D: Nonlin. Phenom. 212 (3-4)(2005) 271–304.

[6] R. C. Moura, A high-order unstructured discontinuous Galerkin finiteelement method for aerodynamics, Div. Aero. Eng., Tech. Inst. Aero.[http://www.aer.ita.br∼flowsim], M.Sc. Aero. Mech. Eng. ), 2012.

28

[7] P.-O. Persson, J. Peraire, Sub-cell shock capturing for discontinuousGalerkin methods, in: Proc. 44th AIAA Aero. Sci. Meet. Exhib. no.AIAA paper 2006-112, Reno, NV, 2006.

[8] G. E. Barter, D. L. Darmofal, Shock capturing with higher-order, PDE-based artificial viscosity, in: Proc. 18th AIAA CFD Conf. no. AIAAPaper 2007-3823, Miami, Florida, 2007.

[9] J. D. Anderson, Computational fluid dynamics - The basics with appli-cations, McGraw-Hill, New York, 1995.

[10] F. BASSI, S. REBAY, A high-order accurate discontinuous finite el-ement method for the numerical solution of the compressible Navier-Stokes equations, Journal of Computational Physics (131) (1997) 267–279.

[11] F. Bassi, S. Rebay, G. Mariotti, S. Pedinotti, M. Savini, A high-orderaccurate discontinuous finite element method for inviscid and viscousturbomachinery flows, in: Proc. 2nd Eur. Conf. Turbomachinery, FluidDyn. Thermodyn. Technologisch Instituut, Antwerpen, Belgium, 1997,pp. 99–108.

[12] P.-O. Persson, G. Strang, A simple mesh generator in Matlab, SIAMRev. 46 (2) (2004) 329–345.

[13] G. E. Barter, Shock capturing with PDE-based artificial viscosity foran adaptive, higher-order discontinuous Galerkin finite element method,Department of Aeronautics and Astronautics, Massachusetts Institute ofTechnology, Cambridge, Thesis (PhD in Aerospace Engineering), 2008.

[14] G. E. Barter, D. L. Darmofal, Shock capturing with PDE-based artifi-

cial viscosity for DGFEM: Part I. Formulation, J. Comput. Phys.(229)(2010) 1810–1827.

[15] R. R. Thareja, J. R. Stewart, O. Hassan, K. Morgan, J. Peraire, Apoint implicit unstructured grid solver for the Euler and Navier-Stokesequations, Int. J. Numer. Meth. in Fluids 9 (1989) 405–425.

[16] E. Franco, D. Pekarek, J. Peng, J. O. Dabiri, Geometry of unsteady fluidtransport during fluid-structure interactions, J. Fluid Mech. 589 (2007)125–145.

29

[17] B. M. Cardwell, K. Mohseni, A Lagrangian view of vortex shedding andreattachment behavior in the wake of a 2D airfoil, in: Proc. 37th AIAAFluid Dyn. Conf. Exhib. Miami, FL, 2007.

[18] H. Salman, J. S. Hesthaven, T. Warburton, Predicting transport by La-gragian coherent structures with a high-order method, Theor. Comput.Fluid Dyn. 21 (2007) 39–58.

[19] G. Haller, Distinghished material surfaces and coherent structures inthree-dimensional fluid flow, Phys. Nonlin. Phenom. 149 (4) (2001) 248–277.

[20] K. Padberg, T. Hauff, F. Jenko, O. Junge, Lagrangian structures andtransport in turbulent magnetized plasmas, New J. Phys. 9 (11) (2007)400–415.

[21] S. K. Lele, Compact finite difference schemes with spectral-like resolu-tion, J. Comput. Phys. 103 (1992) 16–42.

[22] S. Premasuthan, C. Liang, A. Jameson, Computation of flows withshocks using spectral difference scheme with artificial viscosity, in: Proc.48th AIAA Aero. Sci. Meet. Exhib. no. AIAA Paper 2010-1449, Or-lando, Florida, 2010.

[23] P.-O. Persson, Mesh generation for implicit geometries, Ph.D. the-sis, Massachusetts Institute of Technology, Cambridge, USA (February2005).

[24] A. F. C. Silva, On the use of ENO-based limiters for discontinuousGalerkin methods for aerodynamics, Div. Aero. Eng., Tech. Inst. Aero.[http://www.aer.ita.br/∼flowsim], M.Sc. Aero. Mech. Eng. 2012.

[25] A. S. Silveira, R. C. Moura, A. F. C. Silva, M. A. Ortega, High-ordergeometric boundary conditions for a discontinuous galerkin scheme, In-ternational Journal for Numerical Methods in Fluids (submitted).

[26] W. J. Rider, R. B. Lowrie, The use of classical Lax-Friedrichs Riemannsolvers with discontinuous Galerkin methods, Int. J. Numer. Meth. Flu-ids 40 (2002) 479–486.

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[27] F. Brezzi, G. Manzini, L. D. Marini, P. Pietra, A. Russo, DiscontinuousGalerkin approximations for elliptic problems, Numer. Meth. PartialDiff. Equ. 16 (2000) 365–378.

[28] P. Woodward, P. Colella, The numerical simulation of two-dimensionalfluid flow with strong shocks, J. Comput. Phys. 54 (1984) 115–173.

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(a) Initial mesh.

(b) 1st iteration.

(c) 2nd iteration.

(d) Final mesh.

Figure 10: Set of meshes corresponding to successive iterations of the refinement procedure.

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(a) Initial mesh.

(b) 1st iteration.

(c) 2nd iteration.

(d) Final mesh.

Figure 11: Static pressure (left) and Mach number (right) fields for successive iterationsof the refinement procedure (corresponding to the meshes presented in Fig. 10).

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(a) Initial mesh.

(b) Dilatation field.

(c) Mach number field.

Figure 12: Initial mesh and solution for the forward facing step.

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(a) Final mesh.

(b) Dilatation Field.

(c) Mach number field.

Figure 13: Final mesh and solution for the forward facing step after 7 iterations.

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