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Finite Elements in Analysis and Design 41 (2005) 703 728

www.elsevier.com/locate/finel

Least-squares variational principles and the finite element method:theory, formulations, and models for solid and fluid mechanics

J.P. Pontaza

Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843-3123, USA

Received 31 August 2004; accepted 7 September 2004

Abstract

We consider the application of least-squares variational principles and the finite element method to the numerical

solution of boundary value problems arising in the fields of solid and fluid mechanics. For many of these problems

least-squares principles offer many theoretical and computational advantages in the implementation of the corre-

sponding finite element model that are not present in the traditional weak form Galerkin finite element model. For

instance, the use of least-squares principles leads to a variational unconstrained minimization problem where com-

patibility conditions between approximation spaces never arise. Furthermore, the resulting linear algebraic problemwill have a symmetric positive definite coefficient matrix, allowing the use of robust and fast iterative methods for

its solution. We find that the use of high p-levels is beneficial in least-squares based finite element models and

present guidelines to follow when a low p-level numerical solution is sought. Numerical examples in the context

of incompressible and compressible viscous fluid flows, plate bending, and shear-deformable shells are presented

to demonstrate the merits of the formulations. 2005 Elsevier B.V. All rights reserved.

Keywords: Least-squares finite element formulations; Incompressible flow; Compressible flow; Plates; Shells

1. Introduction

Finite element formulations based on the weak form Galerkin procedure are nowadays the preferredapproach to develop finite element models for problems arising in solid mechanics (e.g., linear and

Tel./fax: +1 979 693 9851.E-mail address: [email protected] (J.P. Pontaza).

0168-874X/$ - see front matter 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.finel.2004.09.002
http://www.elsevier.com/locate/finelmailto:[email protected]:[email protected]://www.elsevier.com/locate/finel

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704 J.P. Pontaza / Finite Elements in Analysis and Design 41 (2005) 703 728

nonlinear dynamic analysis of structures). The success of these formulations is attributed, in its majority,to the fact that a large class of problems in solid mechanics nicely fit into the mathematical frameworkof unconstrained minimization, i.e. global minimization of quadratic (energy) functionals. In this case,

given a conforming discretization, the finite element solution is the minimizer of an energy functional onthe trial space and represents the best possible approximation in the energy norm [1].

Problems that do not fit into the mathematical framework of unconstrained minimization lack many of

the attractive properties of the (unconstrained) variational setting. For example, problems of the saddle-point type, arising in mixed formulations. For such problems, implementation of weak form Galerkin

finite element models without accounting for restrictive conditions leads to a non-optimal finite elementapproximation and un-reliable numerical results. A typical example is given by the primitive variableformulation of the Stokes problem for which the velocity and pressure approximation spaces cannot be

chosen independently and must satisfy an infsup condition [2].In the context of the Stokes and/or the NavierStokes equations, various finite element models attempt-

ing to fully or partially recover some of the properties of the (unconstrained) variational setting have beenproposed, and among them the Galerkin-Least-Squares and stabilized Galerkin formulations have beenextensively researched (see, e.g., [3,4]). These approaches have failed to achieve widespread use and

acceptance due to their explicit dependence on various mesh-dependent calibration parameters that needto be fine-tuned from application to application.

Finite element models based on least-squares variational principles are attractive alternatives to the

weak form Galerkin models and their stabilized versions, and are the focus of this work. Finite elementformulations based on least-squares principles give rise to unconstrained minimization problems througha variational framework of residual minimization. The idea is to define the least-squares functional

as the sum of the squares of the governing equations residuals measured in suitable norms of Hilbertspaces. Provided the governing equations (augmented with suitable boundary conditions) have a unique

solution, the least-squares functional will have a unique minimizer. Thus, by construction, the least-squaresfunctional is always positive and convex, ensuring coerciveness, symmetry, and positive definitivenessof the bilinear form in the corresponding variational problem. Moreover, if the induced energy norm

is equivalent to a norm of a suitable Hilbert space, optimal properties of the resulting least-squaresformulation can be established.

Direct application of least-squares principles to develop finite element models for 2mth-order dif-

ferential operators require that the finite element space be spanned by functions that belong to theHilbert space H2m, in contrast to weak form Galerkin models which require only Hm regularity. The

lower smoothness requirements associated with the weak form Galerkin models is due to the weakeneddifferentiability of the operators, induced by the integration by parts step, which is absent in least-

squares formulations. The necessary degree of global differentiability that needs to be retained in theleast-squares numerical solution is not only dictated by the differentiability requirements of the gov-erning equation(s) under consideration but also depends on the norms used to measure the residuals inthe least-squares functional. The higher smoothness requirement is considered the major drawback of

least-squares based formulations and is the reason why the methodology has not yet gained widespreadacceptance.

To reduce the higher regularity requirements and allow the use of practical C0 element expansions inthe least-squares finite element model, the governing equation(s) are first transformed into an equivalentfirst-order system and the least-squares functional defined by measuring their residuals in terms of L2norms only. This approach renders the formulation practical, in the sense that existing computational
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J.P. Pontaza / Finite Elements in Analysis and Design 41 (2005) 703 728 705

frameworks based on C0 expansions are easily adapted to the least-squares formulations. Transformationof the governing equations to an equivalent first-order system may necessarily imply that additionalindependent variables need be introduced, implying an increase in cost. However, the auxiliary variables

may be argued to be beneficial as they may represent physically meaningful variables, e.g. fluxes orstresses, and will be directly approximated in the model.

Benefits from working in an unconstrained variational setting include the fact that stability requirements

such as infsup conditions will never arise and that the resulting algebraic problem will have a symmetricpositive definite (SPD) coefficient matrix. In order to fully emulate the variational setting, one must

define a L2 least-squares functional that induces an energy norm that is equivalent to the H1 norm. If

this is achieved, the least-squares finite element solution can be shown to be an optimal approximationin the H1 norm [5]. Formulations that are not H1 norm equivalent are referred to as non-equivalentformulations. In practice, we find that H1 coercivity of the least-squares functional may sometimes onlybe achieved by requiring the least-squares functional to be defined in terms of computationally impractical

norms, e.g. H1

norms or H1

norms, the latter requiring the use C1

element expansions for an exactrepresentation.

Ideally, a least-squares finite element model with C0 practicality and full (mathematical) optimality

is to be developedunfortunately, this can seldom be achieved and we are sometimes forced to choosepracticality over optimality. Nevertheless, if H1 coercivity of the least-squares functional cannot beestablished it does not imply that the resulting method is not optimal. It simply means that its optimality

cannot be determined a priori using standard elliptic theory. In particular, we find that use of high p-levels is desirable for least-squares based finite element models and that such practice provides a desirablebalance between practicality and optimality.

The objective of this work is to present least-squares based finite element models for problems ofengineering relevance in the fields of fluid and solid mechanics, exemplified by incompressible and

compressible viscous fluid flow, the bending of thin plates, and shear-deformable shell structures. Indoing so, we show that the use of high p-levels mitigates the ills associated with non-equivalence. Inaddition, we consider collocation least-squares procedures, which can be used to obtain least-squares

numerical solutions at low p-levels.An overview of the paper is as follows. In Section 2, we present an abstract least-squares formulation for

an initial boundary value problem. We briefly review the mathematical theory of least-squares based for-

mulations, specifically the notion of norm-equivalence of least-squares functionals and its consequencesin the form of optimal a priori error estimates. The formulations and theory set forth in Section 2 are

presented in a general setting and readily specialize to treat boundary value problems in fluid and solidmechanics. Sections 3 and 4 are devoted to numerical examples for incompressible and compressible

fluid flow, and plates and shells. In Section 5 we present a summary and concluding remarks.

2. An abstract least-squares formulation

In this section we present the steps involved in developing and arriving at a least-squares based finiteelement model. We wish to present the procedure in a general setting, and to this end present the procedure

in the context of an abstract initial boundary value problem.First, we introduce notation that will be used throughout this section and in the remainder of this work.

Given the abstract initial boundary value problem, our first task is to form the least-squares functional;

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which may be defined to yield a spacetime coupled or decoupled formulation. The variational problemstatement is obtained by defining the least-squares minimization principle in infinite-dimensional spacesand the corresponding finite element model obtained by restricting the spaces to finite dimensional

subspaces. These closed spaces are spanned at the element level by piecewise nodal/modal polynomialexpansions.

Of paramount importance is to establish whether or not the resulting least-squares based formulation is

within the ideal mathematical setting, i.e., whether or not the least-squares functional defines an equivalentnorm in a suitable Hilbert space. Although norm-equivalence is always desirable, it sometimes needs to

be sacrificed to yield practical finite element models. We discuss the properties that are lost when thiscompromise between optimality and practicality is made, and show through a numerical example, howthe use of high p-levels mitigates the ills of non-equivalence.

In conclusion, we present a point collocation least-squares formulation, where the least-squares func-tional is defined as the sum of the squares of the equations residuals evaluated at a finite number of

(collocation) points. Under certain conditions, this formulation may be used to obtain accurate least-squares solutions at low p-levels.

2.1. Notation

Let be the closure of an open bounded region in Rd, where d= 2 or 3 represents the number ofspace dimensions, and x=(x1, . . . , xd)=(x,y,z) be a point in =j, where j= is the boundaryof.

For s0, we use the standard notation and definition for the Sobolev spaces Hs () and Hs () with

corresponding inner products denoted by (

,

)s, and (

,

)s, and norms by

s, and

s,, respectively.

Whenever there is no chance of ambiguity, the measures and will be omitted from inner productand norm designations. We denote the L2() and L2() inner products by (, ) and (, ), respectively.By Hs () we denote the product space [Hs ()]d. We denote by H10 () the space consisting of H1()functions that vanish on the boundary and by L2() the space of all square integrable functions withzero mean with respect to .

2.2. The abstract problem

Consider the following abstract initial boundary value problem:

Lt(u) +Lx(u) = f in (0, ], (1)

G(u) = h on (0, ], (2)

where > 0 is a real number denoting time,Lt andLx are linear first-order partial differential operatorsin time and space respectively, acting on the vector u of unknowns, f is a known vector valued forcing

function, G is a trace operator acting on u, and h represents a known vector-valued function on theboundary. We assume initial conditions are given such that the problem is well posed and a unique

solution exists.

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2.3. L2 least-squares formulation

The L2 least-squares functional associated with the abstract initial boundary value problem is con-

structed by summing up the squares of the equations residuals in the L2 norm and is given by

J(u; f, h) = 12 (Lt(u) +Lx(u) f20,(0,] + G(u) h20,(0,]). (3)It is easy to see that the minimizer of (3) solves (1)(2) and viceversa.

Note that in presenting the abstract initial boundary value problem and defining its associated least-squares functional we made two restrictions: (1) the temporal and spatial partial differential operators

are of first-order and (2) the least-squares functional is defined exclusively in terms of L2 norms. Theserestrictions are necessary in order to ensure a pre-determined level of practicality in the resulting least-

squares finite element model: specifically, the permission to use finite element spaces with merely C0

regularity across inter-element boundaries. This is done with the understanding that the resulting finite

element formulation may depart from the ideal mathematical setting and hence may not yield optimal apriori error estimates. Nevertheless, as we shall demonstrate with the aid of numerical examples, suchdepartures from the ideal mathematical setting will not result in disaster (as a violation of an infsupcondition would, in a weak form Galerkin formulation). In fact, least-squares formulations that depart

from the ideal mathematical setting show remarkable robustness and are ableto recover optimal asymptoticconvergence behavior provided the finite element spaces are spanned by sufficiently high p-levels.

If the partial differential equations (PDEs) under consideration are not of first-order, the C0 practicalityof the least-squares finite element model comes at an extra cost, implied in restriction (1); which requiresthat the partial differential operators be of first-order. This can always be achieved by introducing auxiliary

variables until a first-order system is attained. The added cost might be viewed as beneficial, in the sensethat the auxiliary variables may have physical relevance to the problem under consideration, e.g., fluxesor stresses.

2.3.1. Spacetime coupled formulation

In addition, note that prior to defining functional (3) we did not replace the temporal operator with a

discrete equivalent. This results in a fully spacetime coupled formulation, implied in the definition offunctional (3) where the L2 norm is defined in spacetime, i.e., 0,(0,] denotes the L2 norm of theenclosed quantity in spacetime:

u20,(0,] =

0

|u|2 d dt.

This implies, for example, that a two-dimensional time-dependent problem will be treated as a three-dimensional problem in spacetime domain. When dealing with the stationary form of the equations theintegral over time domain is simply dropped.

In the spacetime coupled approach, the effects of space and time are allowed to remain coupled. There

is no approximation of the initial boundary value problem. Instead, a basis is introduced in time domainto represent the time evolution of the independent variables.

Invariably, we as analysts would like to simulate and study the time evolution of an initial boundary

value problem for large values of time. Taking into consideration modelling issues, we realize that thiswould require a spacetime mesh with a large number of elements in time domain. The size of the

resulting set of assembled algebraic equations could be large and prohibitively expensive in terms of

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available computer memory and non-optimal in terms of CPU solve time. To alleviate the drawbacks, weadopt a time-stepping procedure in which the solution is obtained for spacetime strips in a sequentialmanner. The initial conditions for the current spacetime strip are obtained from the latest space plane

from the previous spacetime strip. Hence, for each spacetime strip we solve a true initial boundaryvalue problem, by minimizing the following functional in spacetime domain:

J(u; f, h) = 12 (Lt(u) +Lx(u) f20,[ts ,ts+1] + G(u) h20,[ts ,ts+1]), (4)where the interval [ts , ts+1] can be taken arbitrarily large, i.e., there are no restrictions on the size of theinterval.

Since the initial boundary value problem in each spacetime strip is represented and solved withpredetermined accuracy of order p, the question of time-stepping stability does not arise [6,7]. The only

issue that remains is temporal accuracy; which we can control by hp refinements in time. Furthermore,the L2 least squares functional can provide an error measure for adaptive h, p, or hp refinements in

spacetime.

2.3.2. Spacetime decoupled formulation

In a spacetime decoupled formulation, discretization in space and time are done independently. Tra-ditionally, the temporal operators are represented by truncated Taylor series expansions in time domain.Such formulations result in an inherent approximation of the initial boundary value problem and thus the

investigation of stability is essential. Representation of the temporal operator by high-order approxima-tions, such as multi-step schemes, are only conditionally stable; imposing severe limitations on the size

of the allowable time increment.In a spacetime decoupled formulation the temporal operator in Eq. (1) is replaced by a discrete

equivalent:

Lt(u) Lt(us+1, usq ),where the dependence of the discrete operator on the time incrementtis explicit as well as its dependenceon histories of previous time steps. For sufficiently small t, the modified problem is equivalent to theoriginal problem. To march the problem in time using a least-squares spatial finite element model, we

must minimize the following space functional at each time step:

Jt(u; f, h) = 12 (Lt(us+1, usq ) +Lx(us+1) fs+120, + G(us+1) hs+120,). (5)

2.4. The variational problem

Having defined the least-squares functional (3), the abstract least-squares minimization principle canbe stated as:

find u X such that J(u; f, h)J(v; f, h) v X, (6)where X is a suitable vector space, e.g. X = H1( (0, ]), and we assume that the functions f, h aresufficiently regular, e.g. f L2( (0, ]) and h L2( (0, ]).

The EulerLagrange equation for this minimization problem is given by the following variationalproblem [1]:

find u X such that B(u, v) =F(v) v X, (7)
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where B is a symmetric form given by

B(u, v)

=(L(u),L(v))0,

(0,

] +(G(u),G(v))0,

(0,

]and F is a functional given by

F(v) = (f,L(v))0,(0,] + (h,G(v))0,(0,],where L=Lx +Lt.

The inclusion of the boundary residual in (3) allows the use of spaces X that are not constrained tosatisfy the boundary condition (2). In such a case, the boundary condition (2) is enforced in a weak sensethrough the least-squares functional. This is a tremendous advantage of least-squares based formulations,

as it allows boundary conditions that are computationally difficult to impose to be efficiently included inthe least-squares functional. An example where this property becomes extremely useful is for viscous or

inviscid compressible flow (see Section 3), where characteristic-based boundary conditions need to beprescribed at outflow/inflow boundaries. Of course, if the boundary condition (2) can be easily imposedand included in the space X, we omit the residual associated with the boundary term in (3).

The abstract expressions given above for the symmetric form B and functional F are only validwhen the partial differential operators are linear. For the case when the partial differential operators arenonlinear, the following more general expressions apply:

B(u, v) = (L(u), L(u))0,(0,] + (G(u), G(u))0,(0,]and

F(v) = (f, L(u))0,(0,] + (h, G(u))0,(0,],where it is understood that u = v. In general, when the partial differential operators are nonlinear, theresulting form will be non-symmetric. Symmetry of the form is restored only when the EulerLagrangeequation is suitably linearized, e.g., by Newtons method.

2.5. The finite element model

The finite element model is obtained by either restricting (7) to the finite-dimensional subspace Xhp ofthe infinite-dimensional space X, or equivalently by minimizing (3) with respect to the chosen approxi-

mating spaces. This process leads to the discrete variational problem given by

find uhp Xhp such that B(uhp, vhp) =F(vhp) vhp Xhp. (8)We proceed to define a discrete problem by choosing appropriate finite element spaces for each of thecomponents of the vector-valued function u. There are no restrictive compatibility conditions on the

discrete spaces, so we choose the same finite element space for each of the primary variables.Consider the stationary, two-dimensional case u = (u, v) and letPh = {Q} be a family of quadrilateral

finite elements e that make up the connected model h

. We map e to a bi-unit square e = [1, 1] [1, 1], where = (1, 2) = (, ) is a point in e. Over a typical element e, we approximate u

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by the expression

u(, ) uhp

(, ) =n

j=1

jj(, ) in e, (9)

where j are nodal/modal C0 basis functions andj their associated coefficients. We proceed to generate

a system of linear algebraic equations at the element level using (8), where the integrals are evaluatedusing Gauss quadrature rules.

The global system of equations is assembled from the element contributions to ensure the uniquenessof the degrees of freedom at element interfaces. The assembled system of equations can be written as [K11] [K12]

[K12]T [K22] {1}

{2}

= {F1}

{F2}

, (10)

where {1}, {2} are the nodal/modal unknown coefficients associated with u and v. For details onstandard finite element methods, such as mapping ee, numerical integration in e, and the assemblyprocess, see [8].

2.6. Norm-equivalence and its implications

Our ultimate goal is to use (7) to compute approximate solutions to (1)(2). Clearly, the least-squaresfunctional is consistent in the sense that for sufficiently smooth data f, h and smooth solutions u of (1)(2),

J(u; f, h) = 0. Furthermore, by construction, the least-squares functional is convex and positive. Whichallows us to define an energy norm:

E =J(; 0, 0)1/2 : X R (11)and an associated energy inner product:

((, ))E : X X R. (12)Then, ifXhp X,

(1) the variational problem (8) has a unique solution given by uhp Xhp, and(2) uh is the orthogonal projection ofu with respect to the energy inner product (12), and thus represents

the best possible solution in the energy norm (11).

In addition, if{i}ni=1 spans Xhp; the variational problem (8) is a linear system of algebraic equationswhose coefficient matrix K has entries given by

Kij = ((j,i ))E. (13)Thus, the coefficient matrix K is a Gramm matrix with respect to the energy inner product (12) and thusis symmetric and positive definite. As a result, the system KU = F has a unique solution.

Note that so far we have said nothing about norm-equivalence and already we have established that theleast-squares finite element model will yield a convex, unconstrained minimization problem with a unique

minimizer that coincides with the best possible approximate solution to (1)(2) in a well-defined norm.

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In addition we have also established that the resulting discrete algebraic problem will have a symmetricpositive definite coefficient matrix. This explains the robustness of least-squares based formulations, evenwhen they depart from the ideal mathematical setting (which has not yet been established).

Establishing norm-equivalence (and hence, and ideal mathematical setting) will enable us to predictthe asymptotic behavior of how uhp approaches u. For simplicity, consider the stationary version of (1)with homogeneous boundary conditions and let X = H10(). If the coercivity relation

C2u1,L(u)0,C1u1, (14)holds for all smooth solutions u of (1)(2), then the L2 least-squares functional defines an equivalentnorm in H1 in the sense

12 C

22u21,J(u) 12 C21u21, (15)

or equivalently (and for the more general case),

12

C22u2Xu2E 12 C21u2X (16)and optimal h-convergence rates should be attained as follows [5]:

u uhpr,Chp+1r , r = 0, 1 (17)for smooth solutions u, with C independent ofh. If norm-equivalence cannot be established the constant

C may be dependent on the mesh parameter h and/or not much can be said about the optimality of theconvergence rates of the resulting formulation. This, however, does not imply that the resulting method

is non-optimal. It simply means that its optimality cannot be determined a priori using standard elliptictheory.

We illustrate the behavior of a non-equivalent formulation with the numerical solution of the stationary

incompressible NavierStokes equations in its velocitypressurevorticity based first-order form, whichfails to define an H1 coercive formulation for the pure velocity boundary condition [9]. For the purposes

of this demonstration, we solve the well-known two-dimensional lid-driven cavity problem in a seriesof meshes and for different p-levels such that the total number of degrees of freedom remains constant.For each case we plot the u-velocity profile along the geometric vertical mid-line of the cavity and the

v-velocity profile along the geometric horizontal mid-line of the cavity. We take the target solution tobe that reported and tabulated by Ghia et al. [10], frequently used and widely accepted as a validationbenchmark.

Figs. 1 and 2 show the u- and v-velocity profiles along the geometric vertical and horizontal mid-

lines of the cavity. For the 60 60 finite element mesh with a p-level of 1 (i.e., bi-linear elements), thepredicted velocity profiles are surprisingly of extremely poor quality. Initially one might be disappointedat the performance of the least-squares based formulation, as the 60 60 bi-linear finite element meshwill give considerably better results with a weak form Galerkin formulation. However, knowing that

the least-squares functional we used to develop the finite element model does not define an equivalentnorm in Xhp X, we conjecture that the constant C in Eq. (17) depends on the mesh parameter h andthus expect a poor numerical solution. To keep the cost of the computation comparable and the total

number of degrees of freedom constant we consider a 30 30 finite element mesh with a p-level of 2(i.e., bi-quadratic elements). The predicted velocity profiles are significantly improved, however not yet

completely satisfactory. Still we are led to believe that the constant C depends on the mesh parameter h

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y 6060, p = 1

3030, p = 2

1010, p = 6

Ghia et al.

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

0.50 0.25

u - velocity

0.00 0.25 0.50 0.75 1.00

Fig. 1. u-velocity profiles along the vertical mid-line of the cavity at flow conditions Re = 103.

x

1.00

3030, p = 2

1010, p = 6

Ghia et al.

0.750.50

v-velocity

0.6

0.5

0.4

0.3

0.2

0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.60.00 0.25

6060, p = 1

Fig. 2. v-velocity profiles along the horizontal mid-line of the cavity at flow conditions Re = 103.

but with a weakened dependence at this p-level. Finally, we consider a 10 10 finite element mesh witha p-level of 6, where the total number of degrees of freedom is the same as for the previous two cases.The predicted velocity profiles are in excellent agreement with the benchmark solution and we are led

to believe that at this p-level the dependence of the constant C on the mesh parameter h is negligible ornonexistent. Typically a p-level of 4 is sufficient to assure negligible or nonexistent dependence on themesh parameter h.

The above illustrative example shows that if a non-equivalent least-squares functional is used to developthe finite element model, high-order expansions are desirable. If low-order expansions are to be used (i.e.,

p-levels of 1 or 2), it is best to use non-standard least-squares procedures such as collocation.

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2.7. A least-squares collocation formulation

In this subsection we present a least-squares finite element point collocation formulation, which does

not directly fit into the mathematical framework presented earlier. To define the least-squares point

collocation formulation, we chose a finite set of points {xm}Cm=1 in h and {xn}Cbn=1 on h, and define theleast-squares functional as the sum of the weighted squares of the residuals evaluated at the points xmand xn:

J() =I

i=1

Cm=1

im (RiL(, xm))

2 +J

j=1

Cbn=1

j n(RjG(, xn))

2. (18)

In Eq. (18), RiL denotes the residual associated with the ith partial differential operator and RjG the

residual associated with the jth boundary operator,

RiL =Lix(uhp) fi i = 1, 2, . . . , I ,R

jG

= Gj(uhp) hj j = 1, 2, . . . , J .In addition, the weights im and j n may be different for each governing equation and/or collocationpoint.

Upon minimization of (18) with respect to the nodal/modal coefficients in we arrive at the system ofequations, which can be written as [K11] [K12]

[K12]T [K22] {1}

{2}

= {F1}

{F2}

, (19)

where {1}, {2} are the modal/nodal unknown coefficients associated with u and v. A reliable least-squares collocation solution is achieved when the number of collocation residual equations (C I ) +(Cb J ) is equal to the number of nodal/modal degrees of freedom Ndof Nbc, where Nbc is the numberof prescribed Dirichlet boundary conditions on . Denoting the ratio between the number of collocationresidual equations and the number of nodal degrees of freedom by , i.e.,

= (C I ) + (Cb J )Ndof Nbc , (20)

ideally we would like asclosetounityaspossible.Forhigh p-levels, 1.0fromaboveasthe p-level isincreased. At low p-levels, may be forced to unit value by judicious choice of the number of collocation

points. Additional discussion, including illustrative examples, may be found in [11]. Numerous othertypes of least-squares formulations my be found in the review of Eason [12].

3. Viscous incompressible and compressible fluid flows

The majority of work for incompressible flow has been done in the context of collocation least-squares

models at low p-levels. This is the preferred approach in the work of Jiang [13,14], Jiang and Chang[15], Jiang and Povinelli [16] and Tang et al. [17], although they refer to their collocation solutions as

reduced integration solutions; because their collocation points coincide with the GaussLegendre reduced
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integration points. The use of high p-levels in least-squares formulations has been demonstrated by Prootand Gerritsma [18] in the context of Stokes flow, and by Pontaza and Reddy [19,7,11] for the steady andunsteady NavierStokes equations. Here, we demonstrate exponentially fast decay of error measures,

resulting from p-refinement, with a (incompressible flow) verification benchmark.Least-squares formulations for compressible flows have been limited to inviscid flows (i.e., the com-

pressible Euler equations) [2022]. Here, we present numerical results for viscous compressible flows

in the subsonic and transonic regimes. Such formulations are aimed at flow problems characterized byincompressible flow in parts of the domain with imbedded regions where compressibility effects are

significant and cannot be neglected. Numerical results are presented for flow past a NACA0012 airfoil atMach 0.85, Re = 500 and Re = 2000.

3.1. Governing equations

We consider the solution of the NavierStokes equations governing viscous compressible flow of anideal, Newtonian gas. The governing equations for incompressible flow are but a special case, so weconsider the more general form of the equations. The governing equations in dimensionless form can be

written as:Find the density (x, t), velocity u(x, t ), pressure p(x, t), and temperature T (x, t ) such that

j

jt+ (u ) + ( u) = 0 in (0, ], (21)

ju

jt+ (u )u + p 1

Re [(u) + (u)T] 1

Re( u) = f in (0, ], (22)

jT

jt+ (u )T 1

P e (kT ) ( 1)M2 Dp

Dt= ( 1)M

2

Re in (0, ], (23)

1 + M2p = T in (0, ], (24)where M is the Mach number, Pr is the Prandtl number, Re is the Reynolds number, Pe = RePr is thePeclet number, is the ratio of specific heats, is the dynamic viscosity, is the bulk viscosity, k is thecoefficient of thermal conductivity, and is the viscous dissipation function,

= ( u)2 + 2D : D,with D = 12 [(u) + (u)T] denoting the deformation tensor. We assume initial and boundary conditionsare given such that the problem is well posed.

The equations have been normalized using the reference values (ref, uref, pref, Tref) as ref = 0,uref = u0, pref = 0RT0, Tref = T0, and a reference length L, where (0, u0, T0) is a given characteristicstate and R is the universal gas constant. This gives a Reynolds number as Re = 0u0L/0, a Prandtlnumber as P r = cp0/k0, and a Mach number as M = u0/

RT0. The dimensionless pressure p has

been defined as

p = p pref0u

20

,
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where p denotes the dimensional (total) pressure. The dynamic viscosity and coefficient of thermalconductivity are modelled as temperature dependent using Sutherlands law,

= (T )0

=

T

T0

3/2 T0 + S

T + S, k = k(T)

k0=

T

T0

3/2 T0 + Sk

T + Sk,

where S = 110 K and Sk = 194 K for atmospheric air.When M = 0 and isothermal conditions are considered (jT = 0) the incompressible NavierStokes

equations are recovered:

u = 0 in (0, ], (25)ju

jt+ (u )u + p 1

Re [(u) + (u)T] = f in (0, ]. (26)

IfM=0 and the flow experiences non-isothermal conditions (jT = 0), the low-speed compressible flowequations are recovered with the proper equation of state.

Recall from Section 2, that to ensure the C0 practicality of the least-squares finite element modelthe governing equations must be recast as an equivalent first-order system and the least-squares func-tional defined in terms of L2 norms only. The NavierStokes equations, in particular the conservation

of momentum and energy equations, contain second-order derivatives implying u H2, T H2 asa minimum requirement. A least-squares finite element model can indeed be developed by using theNavierStokes equations in their strong form, however such formulation would require C1 regularity of

the finite element spaces across inter-element boundaries as a minimum requirement (see [11] for C1

formulations and numerical examples).

Wishing to retain the C0 practicality of the finite element model we realize that auxiliary variablesneed to be introduced to recast the NavierStokes as an equivalent first-order system. Here we use the vor-ticity based first-order system for the incompressible NavierStokes and the velocity/temperature gradient

based first-order system for the compressible NavierStokes equations. The least-squares formulation andfinite element model follow from the outline given in Section 2.

3.2. Kovasznay flow

The benchmark problem to be used for the purposes of verification is an analytic solution to the two-dimensional, stationary incompressible NavierStokes due to Kovasznay [23]. The spatial domain in

which Kovasznays solution is defined is taken here as the bi-unit square

= [0.5, 1.5

] [0.5, 1.5

].

The solution is given by

u(x,y) = 1 ex cos(2y),

v(x,y) = 2

ex sin(2y),

p(x,y) = p0 12 e2x , (27)where = Re/2 (Re2/4 + 42)1/2, p0 is a reference pressure (an arbitrary constant), and we chooseRe = 40.

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2 3 4 5 6 7 8 9 101018

LS functional

|| u - up

||

|| v - vp

||

|| p - pp

||

|| - p

||

expansion order, p

L2norm

1016

1014

1012

1010

108

10610

4

102

100

Fig. 3. Decay of the least-squares functional and convergence of the velocity, pressure, and vorticity fields to the Kovasznaysolution in the L2 norm. p-convergence.

Here, we perform a p-refinement study. For such a study, we choose and fix a spatial discretization

h

, and systematically increase the p-level of the element approximation functions. Fig. 3 shows the

discretization of the domain, h

, for the p-refinement study. The discretization consists of 8 quadrilateralfinite elements e. Having chosen appropriate error measures, these measures should decay exponentiallyfast as the p-level is increased. In a logarithmic-linear scale the expected rate of convergence would appear

as a straight line.The exact solution, given by Eq. (27), is used to prescribe Dirichlet velocity boundary conditions

on and pressure is specified at a point. No boundary conditions for the vorticity are necessary. The

linearization is done using Newtons method and the resulting linear algebraic system of equations witha symmetric positive definite coefficient matrix solved using Cholesky factorization at each Newton step.

Nonlinear convergence was declared when the relative norm of the residual in velocities, uhp/uhp,was less than 104, which typically required three to five Newton iterations.

In Fig. 3 we plot the L2 least-squares functional and L2 error of the velocity, pressure, and vorticity

fields as a function of the expansion order in a logarithmic-linear scale. Exponentially fast decay (spectralconvergence) of the L2 least-squares functional and L2 error is observed. Decay of the least-squaresfunctional implies that the L2 norm of the residuals of the governing equations are going to zero, i.e.,

conservation of mass and momentum are being satisfied. Decay of the L2 norms of the difference between

the numerical solution and the analytic solution indicates that the numerical solution approaches the exactsolution.

Of importance is to test whether the formulation is able to achieve the optimal asymptotic convergencerate in geometrically distorted meshes. To this end, we perform a p-refinement study for the geometrically

distorted mesh shown in Fig. 4.In Fig. 4 we plot the L2 least-squares functional and L2 error of the velocity, pressure, and vorticity

fields as a function of the expansion order in a logarithmic-linear scale for the analysis in the geometrically

distorted mesh. Exponentially fast decay (spectral convergence) of the L2 least-squares functional and

L2 error is observed. As expected, for the distorted mesh, slightly higher p-levels are needed to obtain

the same level of accuracy when compared to the geometrically undistorted mesh results.

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2 3 4 5 6 7 8 9 10 11 1210

18

1016

1014

1012

1010

108

106

10

4

102

100

LS functional

|| u - up

||

|| v - vp

||

|| p - pp

||

|| - p

||

expansion order, p

L2norm

Fig. 4. Distorted mesh study: decay of the least-squares functional and convergence of the velocity, pressure, and vorticity fieldsto the Kovasznay solution in the L2 norm. p-convergence for the geometrically distorted mesh.

3.3. Compressible flow past a NACA0012 airfoil

We present results for two of the standard benchmarks problems of the GAMM workshop [24]. The flowconditions considered here correspond to a free-stream Mach number of 0.85 and free-stream Reynolds

numbers of 500 and 2000. The airfoil is of unit chord length and is placed in the finite region =[4.0, 8.0] [5.0, 5.0]. The leading edge of the airfoil lies at (x,y) = (0, 0), so that the inflowboundary is located 4 chord-lengths in front of the leading edge and the outflow boundary 7 chord-

lengths downstream of the trailing edge. The top and bottom boundaries are located each 5 chord-lengthsabove and below the airfoil.

The issue of open boundary conditions of inflow and outflow type for the compressible NavierStokesequations is of paramount importance and non-trivial. It has been the subject of extensive research andnumerical experimentation (see, for example, [2529]). Here, we apply inflow and outflow boundary

operators which ensure maximum energy dissipation and result in a strongly well-posed problem. Inthe limit of vanishing viscosity, the boundary operators recover the Euler characteristics. The complex

boundary conditions in terms of characteristic variables are enforced in the finite element model throughthe least-squares functional. Details and derivation of the well-posed boundary operators may be found in[28]. No-slip boundary conditions are specified on the airfoil surface: u = v = 0, together with prescribedtemperature corresponding to the free-stream total temperature.

We are interested in a point collocation solution (see Section 2.7), obtained by requiring the least-

squares functional to vanish at specific points in the computational domain. The connected model, h

, ischaracterized by the total number of elements Nel, number of elements around the airfoil Nel,s, a far-field

element size , and the size of adjacent elements to the airfoil . We present results for a bi-linear finiteelement mesh with Nel = 11, 250, Nel,s = 120, = 0.001, = 0.30. Fig. 5 shows the mesh, where aclose-up view of the element distribution around the trailing edge is also shown.

The collocation points are chosen to lie at the center of the elements, so that the total number of interiorcollocation points, Ncoll, is equal to the total number of elements in the finite element mesh. In addition to

the interior collocation points, we chose collocation points at the inflow and outflow boundaries to enforce
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x

y

0.6

0.4

0.2

0.0

0.2

0.4

0.6 0.03

0.02

0.01

0.00

0.01

0.02

0.03

(a) (b)

0.5 0.0 0.5 1.0 1.5

Fig. 5. Computational domain and mesh for compressible flow past a NACA0012 airfoil. (a) Partial view of the connected model,

h

. (b) Close-up view of the element distribution near the trailing edge.

s/c

Cp

0.00 0.25 0.50 0.75 1.000.6

0.4

0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.21.4

collocation least-squares

Satofuka et al. (1987)collocation least-squaresSatofuka et al. (1987)

s/c

Cf

0.00 0.25 0.50 0.75 1.000.00

0.05

0.10

0.15

0.20

0.25

0.300.35

(a) (b)

Fig. 6. Predicted steady-state (a) pressure and (b) skin-friction coefficients for M

=0.85, Re

=500.

the open-type boundarycondition through the least-squares functional.These boundarycollocation points,

Ncoll,b, lie at the center of each element edge along the outflow boundary.

As explained in Section 2.7, a reliable least-squares collocation solution can only be achieved if a strictbalance between the number of collocation residual equations and number of nodal degrees of freedom issatisfied. Such balance is achieved when the effectivity (or reliability) index , given by Eq. (20), is close

to unity. For the problem under consideration, = 1.035, ensuring a reliable least-squares collocationsolution.

To treat the unsteady nature of the problem, we implement a spacetime decoupled formulation (see

Section 2.3.2). The temporal terms in the governing equations are represented by the generalized -

method family of approximations [31,32], which retain second-order accuracy in time. Even though bothsimulations pursued here posses a steady-state solution, a time accurate simulation is performed.

At each time step and for each Newton linearization step, the resulting system of equations with a SPDcoefficient matrix is solved using a matrix-free pre-conditioned conjugate gradient (PCG) algorithm with

a Jacobi preconditioner. PCG convergence was declared when the norm of the residual, KU F, wasless than 106. Nonlinear convergence was declared when the relative norm of the residual in velocitiesbetween two consecutive iterations was less than 104, which typically required two or three Newtoniterations per time step.

Fig. 6 shows the predicted steady-state pressure and skin-friction coefficients for the case M =0.85, Re = 500. Also shown in the figure, are the reported numerical results of Satofuka et al. [33],
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0.

78

0.75

1.00

1.0

0

1.03

1.03 0.96

0.96

0.69

(a)

(b)

Fig. 7. Steady-state (a) Mach and (b) pressure contours for M = 0.85, Re = 500.

Cp

0.00 0.25 0.50 0.75 1.00

0.6

0.4

0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

collocation least-squaresSatofuka et al. (1987)Mittal (1998)

s/cs/c

Cf

0.00 0.25 0.50 0.75 1.00

0.00

0.05

0.10

0.15

0.20

collocation least-squaresMittal (1998)

(a ) ( b )

Fig. 8. Predicted steady-state (a) pressure and (b) skin-friction coefficients for M = 0.85, Re = 2000.

using a finite-difference discretization scheme. We see excellent agreement between the predicted andbenchmark profiles. Fig. 7 shows steady-state Mach and pressure contours. In accordance with theGAMM benchmark results [24], we see a symmetric supersonic pocket near the mid-section of the

airfoil.Similarly, Fig. 8 shows the predicted steady-state pressure and skin-friction coefficients for the case

M = 0.85, Re = 2000. We compare our numerical results with those reported by Satofuka et al. [33]and Mittal [34], and find good agreement. From Fig. 9 we see that at this higher Reynolds number, thesupersonic pocket has moved towards the nose of the airfoil. Table 1 shows good agreement between the

computed drag coefficient and the values reported in the GAMM workshop [24].

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(a)

(b)

1.00

1.00

1.04

1.04

0.97

0.97

0.95

0.95

0.76

0.79

Fig. 9. Steady-state (a) Mach and (b) pressure contours for M = 0.85, Re = 2000.

Table 1

Drag coefficient for flow past a NACA0012 airfoil

Mach Re Parameter Computed Benchmark [24]

0.85 500 CD 0.2337 0.2261

0.2300

0.85 2000 CD 0.1188 0.1160

0.1164

4. Plates and shells

Finite element formulations for the analysis of plates and shell structures are traditionally derived

from the principle of virtual displacements or the principle of minimum total potential energy [1]. When

considering the limiting behavior of a shell as the thickness becomes small, for a given shell geometry andboundary conditions, the shell problem will in general fall into either a membrane dominated or bendingdominated statedepending on whether the membrane or bending energy component dominates the totalenergy. Displacement-based finite element models have no major difficulties in predicting the asymptotic

behavior of the shell structure in the membrane dominated case. However, computational difficulties arisein the case when the deformation is bending dominated [35]. A strong stiffening of the element matricesoccurs, resulting in spurious predictions for the membrane energy component. This phenomenon is known

as membrane-locking. In shear-deformable shell models, yet another form of locking occurs and presentsitself (again) in a strong stiffening of the element matrices, resulting in spurious predictions for the

shear energy component. This form of locking is also present in plate bending analysis when the side-to-

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thickness ratio of the plate is large (i.e., when modelling thin plates). This locking phenomenon is knownas shear-locking.

Least-squares finite element formulations for plates and shells have been shown to be robust with

regards to membrane- and shear-locking and to yield highly accurate results for displacements as well asstresses (or stress resultants) [36,37]. The formulations retain the generalized displacements and stressresultants as independent variables and, in view of the nature of the variational setting upon which the

finite element model is built, allows for equal-order interpolation. In the following we present numericalresults for the bending of a thin circular plate under uniform loading and for a barrel vault (cylindrical

shell) loaded by its own weight.

4.1. Governing equations

We consider circular cylindrical shells, where the shell mid-surfaceS is given by

S= {L < x1 < L , x22 + x23 = R2 | (x1, x2, x3) R3} R3, (28)where 2L and R are the length and radius of the shell. The shell mid-surface S, given by Eq. (28), can

be parametrized by the single chart = (1,2,3), : R2 S R3,1(

1, 2) = 1,2(

1, 2) = R sin(2/R),3(

1, 2) = R cos(2/R), (29)so that is the rectangle occupying the region

{(1, 2) | L < 1 < L, R< 2 < R} R2. (30)In Naghdis shear-deformable shell model [38], the membrane, bending, and shear strain measures

(, , ) are [37]

11 = u1,1, 212 = u1,2 + u2,1, 22 = u2,2 + u3R

, (31)

11 = 1,1, 212 = 1,2 + 2,1 +u2,1

R, 22 = 2,2 +

1

R

u2,2 + u3

R

, (32)

1 = u3,1 + 1, 2 = u3,2 + 2 u2

R (33)

and the equilibrium equations take the form

N11,1 + N12,2 + p1 = 0, (34)

N12,1 + N22,2 +M12,1

R+ M

22,2

R+ Q

2

R+ p2 = 0, (35)

Q1,1 + Q2,2 N22

R M

22

R2+ p3 = 0, (36)
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M11,1 + M12,2 Q1 = 0, (37)M12,2

+M22,2

Q2

=0, (38)

where u are the displacements of the shell mid-surface, u3 is the out-of-plane displacement, arerotations of the transverse material fibers originally normal to the shell mid-surface, and N, M,

Q are the membrane, bending, and shear thickness-averaged stress resultants. Here we employ the

convention that Greek indices range over 1 and 2 and that , denotes differentiation.The stress resultants are related to the strain measures through the following constitutive relations [38]:

N11 = tE(1 2) (11 + 22), N

22 = t E(1 2) (11 + 22), N

12 = tE(1 + )12, (39)

M11

=

t3E

12(1 2

)

(11

+22), M

22

=

t3E

12(1 2

)

(11

+22),

M12 = t3E

12(1 + )12, (40)

Q1 = tE2(1 + ) Ks1, Q

2 = tE2(1 + ) Ks2, (41)

where t is the thickness of the shell, E is theYoungs modulus, is the Poissons ratio, and Ks is the shear

correction factor for the isotropic material. If we letR we recover the (linear) shear-deformable platebending strain measures and governing equations, where membrane and bending effects are decoupled.

The equilibrium equations (34)(38) and constitutive relations (39)(41) are already of first-order and

are used to define the least squares functional. The least-squares formulation and finite element modelfollow from the outline given in Section 2.

4.2. Simply supported circular plate

Consider a simply supported, circular isotropic plate subjected to a uniformly distributed load ofintensity q0. The total domain of the plate is [t /2, t /2], where t is the thickness of the plate. Theundeformed mid-plane of the plate, , has dimensions [0, a] [0, 2]. Due to the symmetry only onequadrant of the plate need be modelled. The connected model of the quarter plate,

h, used for the analysis

is shown in Fig. 10.Here we are considering pure bending, and work with the following dimensionless definitions for the

generalized displacement and thickness-averaged stress resultants,

u3 = u3 Dq0a4

, = Dq0a3

, M = M

q0a2, Q = Q

q0a, (42)

where D = Et3/12(1 2) is the flexural rigidity of the plate. In the actual computations we simply takea = 1.0, q0 = 1.0, D = 1.0, and vary a/t.

It is known that for the traditional (i.e., based on the weak form Galerkin procedure) displacement-basedfinite element formulations, even high-order elements still do not display a good predictive capability,

particularly when the elements are geometrically distorted and used for stress predictions. Of particular

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Q1

Q1

Least-squares Galerkin

Fig. 10. Predicted shear force contours for a uniformly loaded simply supported circular plate with diameter-to-thickness ratio

100. The mesh is geometrically distorted with the p-level fixed at 4 for both least-squares and weak form Galerkin elements.

xz

20.0

15.0

10.0

5.0

0.0

5.0

10.0

15.0

20.0

y / a y / a

yz

0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.000.0

1.0

2.0

3.0

4.0

5.0

6.0analytic

Least-squaresGalerkin

analytic

Least-squaresGalerkin

( a ) (b )

Fig. 11. Transverse shear stresspredictions along a vertical radial line. Simply supported circular plate with diameter-to-thickness

ratio 100. p-level fixed at 4 for both least-squares and weak form Galerkin elements.

interest is the transverse shear stress prediction, as it is often the most difficult stress component to predictfor Galerkin shear-deformable plate elements. The circular plate problem is ideal to asses the accuracyof the prediction as the analytical solution is readily available in closed form [1] and the elements are

naturally distorted to accommodate the plate geometry.

In Fig. 10 we present contour plots of the predicted shear force, Q1, for a plate with = 0.30 anddiameter-to-thickness ratio of 100 (2a/ t = 100) at a p-level of 4. The Galerkin plate elements givespurious shear force predictions, displaying a localized locking behavior near the curved boundary. Onthe other hand, the least-squares plate elements give a smooth and highly accurate shear force prediction

throughout.In Figs. 11 and 12 we present the predicted transverse shear stress distributions. Fig. 11 shows the

distributions along a vertical radial line and Fig. 12 along the circular arc, with measured counterclock-

wise. We plot the stresses at the elements nodal mid- and end-points. For comparison, we also presentresults obtained using the displacement based Galerkin formulation using the same mesh and p-level.

Clearly, the stress predictions using the proposed least-squares formulation are superior.

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, degrees

z

0 10 20 30 40 50 60 70 80 9020.0

15.0

10.0

5.0

0.05.0

10.0

15.0

20.0

analyticLeast-squares

Galerkin

, degrees

rz

0 10 20 30 40 50 60 70 80 903.0

4.0

5.0

6.0

7.0

8.0

9.0

analyticLeast-squares

Galerki

Fig. 12. Transverse shear stress predictions along the plates circular arc. Simply supported circular plate with diame-

ter-to-thickness ratio 100. p-level fixed at 4 for both least-squares and weak form Galerkin elements.

4.3. Barrel vault

We consider a barrel vault loaded by its own weight. The barrel vault is a segment of a circular cylindricalshell whose mid-surface, after being parametrized by (29), is given by

=

(1, 2) | L < 1 < L, 29

R < 2

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