AIAA-2006-0590

Local Discontinuous Galerkin Method for Parabolic Problems Involving Imperfect Contact

Surfaces

A. Jain∗ R. Kanapady† and K.K. Tamma‡

Abstract

The Local Discontinuous Galerkin (LDG) method that is presented here provides a unified frameworkand an elegant approach for solving various kinds of heat conduction problems like thermal contact resistanceand sharp/high gradient problems without much modifications to the basic formulation. In this paper, wedescribe the LDG formulation for parabolic heat conduction problems. The advantages of LDG method overthe Continuous Galerkin (CG) finite element method are shown using two classes of problems—problemsinvolving sharp/high gradients, and imperfect contact between surfaces. So far, interface/gap elements havebeen primarily used to model the imperfect contact between two surfaces to solve thermal contact resistanceproblems. The LDG method eliminates the use of interface/gap elements and provides a high degree ofaccuracy. It is further shown in the problems involving sharp/high gradient, that the LDG method is lessexpensive (requires less number of degrees of freedom) as compared to the CG method to capture the peakvalue of the gradient. Several illustrative 1-D/2-D applications highlight the effectiveness of the presentLDG formulation.

1. Introduction

The discontinuous Galerkin (DG) methods are locallyconservative, higher-order accurate, and stable meth-ods which have the ability to solve problems involvingirregular meshes with hanging nodes, complex geome-tries and approximations that have polynomials ofdifferent degrees in different elements. These proper-ties render them capable of capturing highly complexsolutions presenting discontinuities with high resolu-tion. These methods have been successfully applied toa variety of problems in fields of computational fluiddynamics, heat conduction and elasticity [1].

The Local Discontinuous Galerkin (LDG) methoddescribed in this paper, provides a unified frameworkfor solving various kinds of time dependent heat con-duction problems involving thermal contact resistanceand sharp/high gradients without much modificationsto the basic formulation. This is a significant ad-vantage over the CG method in these situations. Asin contact resistance problem [2–4], the CG methodrequires modifications such as use of gap/interface el-ements to capture the temperature jump at the inter-face. In the case of sharp or high gradient problems,the CG method requires to be coupled with somespecial methods such as curvilinear spectral overlaymethod [5] to capture the peak value of the gradientfield or requires a very small element size distributionin the region of sharp gradients.

In this paper, we focus primarily on the time depen-dent applications of the LDG method in the problemsinvolving high localized gradients and thermal con-tact resistance problems. The CG method employinglinear elements is computationally expensive and isincapable of capturing the shape and peak value ofgradient field with high resolution [5]. However, thesedisadvantages can be overcome by the LDG method,since it uses completely discontinuous approximations.The results from the numerical examples show thatthe LDG method is very powerful and computation-ally inexpensive as compared to the CG method incapturing the shape and peak value of the gradientfield. The convergence study shows that the implicittime integration schemes have the same order of accu-racy with the LDG method as with the CG method.

Thermal contact resistance plays a very important rolein applications such as electronics packaging, layeredstructures, nuclear reactor, space craft structures andheat exchangers which involve imperfect contact be-tween two surfaces. Several analytical, experimentaland numerical models are available which predict thethermal contact resistance between two surfaces veryaccurately [6]. In this paper, we propose a new for-mulation for solving thermal contact resistance prob-lems using the LDG method to capture the highlylocalized temperature jump in the contact zone fora given value of the thermal contact resistance. TheCG method employs interface/gap elements to cap-ture the temperature jump at the interface. The use

∗Graduate Research Assistant:jainamit@me.umn.edu†Research Associate: Department of Mechanical Engineering, University of Minnesota, 111 Church St. SE, Minneapolis, MN

55455, Ph: (612) 626-8101, Fax: (612) 626-1596, ramdev@me.umn.edu‡To receive correspondence, Professor and Technical Director, Department of Mechanical Engineering, University of Minnesota,

111 Church St. SE, Minneapolis, MN 55455, Ph: (612) 625-1821, Fax: (612) 624-1398,ktamma@tc.umn.edu

1American Institute of Aeronautics and Astronautics

44th AIAA Aerospace Sciences Meeting and Exhibit9 - 12 January 2006, Reno, Nevada

AIAA 2006-590

Copyright © 2006 by Dr. K.K Tamma. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

of these interface/gap elements introduces additionaldegrees of freedom at the interface. However, theLDG method for thermal contact resistance problemsdeveloped in this paper overcomes these drawbacks.The proposed method eliminates the use of any inter-face/gap elements and hence, the degrees of freedomfor the problem remain same.

This paper is organized as follows. In section 2, theLDG formulation for parabolic heat conduction equa-tions. In section 3, the LDG method is applied to highor sharp gradient problems. In section 4, the LDGformulation for thermal contact resistance problem isproposed and several numerical examples are solvedfollowed by concluding remarks in section 5.

2. The Local Discontinuous

Galerkin Method

The advantage of applying the LDG method to ellipticproblems relies on the ease with which it handles hang-ing nodes, elements of general shapes, and local trialfunctions of different types. These properties makethe LDG method ideally suited for hp-adaptivity andallow it to be easily coupled with other methods [7].The LDG method for second order elliptic problemshave been derived in [7,8]. However, for completeness,the LDG formulation for parabolic heat conductiontime dependent problems follows next.

To illustrate the definition of the LDG method, con-sider the heat conduction equations for a body Ω en-closed by a closed surface ∂Ω

ρc∂T

∂t+ ∇ · q = b in Ω

the boundary and initial conditions

T = T on ∂ΩT

q · n = q on ∂Ωq

T (Ω, 0) = T0

(1)

where n is the outward unit normal to the boundary, bis the heat source, ρ and c are the density and specificheat respectively, T and q are the prescribed temper-ature and flux respectively.The constitution equation can be written as

q = −K∇T (2)

where K is the thermal conductivity tensor.Let τ be a triangulation of Ω into elements Ωe.It may have hanging nodes and elements of variousshapes. The LDG method determines an approxima-tion (Th,qh) to (T,q) belonging to the local finite

element space V(Ωe) × ((W)(Ωe))2, typically consist-

ing of polynomials, for each Ωe ∈ τ . This approx-imate solution is obtained by imposing that, for all(v,w)∈ V(Ωe) × ((W)(Ωe))

2,

∫

Ωe

v(ρc∂Th

∂t+ ∇.qh − b) dΩe = 0 (3)

∫

Ωe

w(qh

K+ ∇Th)) dΩe = 0 (4)

where v, w are the weighting functions. Integrating byparts the terms associated with the divergence in Eq.(6.3) and gradient in Eq. (6.4), the weak form is givenby

∫

Ωe

vρc∂Th

∂tdΩe −

∫

Ωe

qh.∇v dΩe+

∫

∂Ωe

vqh · n ds −

∫

Ωe

vb dΩe = 0

(5)

and∫

Ωe

K−1qh.w dΩe −

∫

Ωe

Th∇ · w dΩe+

∫

∂Ωe

Thw.n ds = 0

(6)

We replace Th and qh in the boundary terms by theso called ‘numerical fluxes’ which are discrete approx-imations to the traces of T and q on the boundary ofthe elements. Summing up for all the elements in thedomain we have,

n∑

(

∫

Ωe

vρc∂Th

∂tdΩe −

∫

Ωe

qh.∇v dΩe+

∫

∂Ωe

vqh · n ds −

∫

Ωe

vb dΩe) = 0

(7)

and

n∑

(

∫

Ωe

K−1qh.w dΩe −

∫

Ωe

Th∇ · w dΩe+

∫

∂Ωe

Thw.n ds) = 0

(8)

where n is the number of elements and Th and qh

are the so-called ‘numerical fluxes’. The proper choiceof the numerical fluxes, Th and qh, is the most deli-cate and important aspect of the LDG method whichrenders the underlying accuracy and stability of themethod. In order to define these numerical fluxes, wehave to introduce some notation. Let Ω+

e and Ω−e be

two adjacent elements of τ ; let x be an arbitrary pointof the set r = ∂Ω+

e ∩ ∂Ω−e (which is not necessarily

an entire edge of an element in τ) and let n+ andn− be the corresponding outward unit normals at thatpoint [9]

2American Institute of Aeronautics and Astronautics

If the boundary ∂Ωe of any element e /∈ ∂ΩT and ∂Ωq,

the numerical fluxes Th and qh are defined by

Th = 〈Th〉 + β · [[Th]] − γ[[qh]] (9)

qh = 〈qh〉 − α[[Th]] − β[[qh]] (10)

and, if ∂Ωe ∈ ∂ΩT or ∂Ωq, then

qh =

q+h − α[[Th]] on ∂ΩT

q on ∂Ωq.(11)

Th =

T on ∂ΩT

T−h − γ[[qh]] on ∂Ωq.

(12)

where the average and jump operators can be definedas

〈Th〉 = (T+h + T−

h )/2

[[Th]] = T+h n+ + T−

h n−

〈qh〉 = (q+h + q−

h )/2

[[qh]] = q+h · n+ + q−

h · n−

(13)

where α is of the order Ø(h−1) and β is such that|β · n| = 1

2 .

For the stability and accuracy of the method, α mustbe > 0, β can be arbitrary and γ can be 0. The ap-proximations and the weighting fields are taken as

T =l

∑

NTi Ti

q =

l∑

N qi qi

v = NT

w = Nq

(14)

where l is the number of basis function. AssumingNT = Nq = N, the following semi-discrete equationsare obtained by substituting eqns. (6.9)–(6.14) in eqns.(6.7)–(6.8),

[

MTT MTq

MqT Mqq

] (

T

q

)

+

[

KTT KTq

KqT Kqq

] (

T

q

)

=

(

bT

bq

) (15)

which can be written as

Md + Kd = B (16)

where,

MTq = MqT = Mqq ≡ 0

MTT = ρc

n∑

∫

Ωe

NT N dΩe

KTT = f(α)

KTq =

n∑

∫

Ωe

BT N dΩe + f(β)

KqT =n

∑

∫

Ωe

BT N dΩe + f(β)

Kqq = K−1n

∑

∫

Ωe

NT N dΩe + f(γ)

bT =

∫

∂Ωe

NT b + f(α)

bq = f(T )

(17)

where f(β), f(α) and f(γ) includes boundary integralscontaining numerical flux terms. Since Mqq = 0, wecannot take the inverse of M, especially for explicittime integration methods which are described in thenext section. Furthermore, for ease of solution of q, weassume γ = 0 in Eq. (6.9) for the LDG method. As-suming γ as 0, allows us to actually eliminate the vari-able qh from the equations in an element-by-elementfashion. It is due this local solvability feature, thatthe method is called LDG method. This assumptionmakes the matrix Kqq a block diagonal matrix, allow-ing to solve Eq. (6.12) for q and substitute in the Eq.(6.11) to get,

MTT T + (KTT −KTqKqq−1KqT )T = bT −Kqq

−1bq

(18)which can be written as

MT + KT = B (19)

which is a first order semi- discrete ordinary differen-tial equation that can be solved by traditional timeintegration techniques that are employed for the CGmethod for parabolic equations which are describednext.

3. Time discretization

(

M + θ∆tK)

Tn+1 =[

M− (1 − θ) ∆tK]

Tn + ∆tB(20)

Tn = Tn+1 (21)

3American Institute of Aeronautics and Astronautics

4. High gradient problem

Linear Lagrangian elements are commonly used in fi-nite element analysis. These elements are however,limited in their capability when it comes to captur-ing high localized gradient fields with traditional finiteelement of continuous formulation [5]. It is shown fur-ther by numerical examples, that a very refined meshis required to capture the peak of the gradient field.However, since the LDG method uses completely dis-continuous approximations [10], it is able to capturethe peak of the gradient field with much less numberof degrees of freedom. The results from the numericalexamples solved in the paper (see appendix) show thatthe LDG method is computationally efficient and hasa higher order of convergence as compared to the CGmethod.

5. Thermal contact resistance

problems

Thermal contact resistance is an important considera-tion in applications such as electronics packaging, lay-ered structures, nuclear reactor, space craft structuresand heat exchangers. Several analytical, experimentaland numerical models have been developed in the pastfor predicting the thermal contact resistance betweentwo surfaces. The goal of this paper is not to develop anew model to predict thermal contact resistance, butto apply the LDG method to capture the highly local-ized temperature jump in the contact zone for a givenvalue of the thermal contact resistance.Imperfect contact between two contacting surfaces re-sults in thermal contact resistance. Also, the contactbetween two finished surfaces is not perfect. At themicroscopic level, all the surfaces can be consideredas rough. The real contact between two surfaces oc-curs at certain spots known as ‘asperities’. The realcontact area is always less than the apparent contactarea. The contact zone is characterized by the asper-ities and cavities. Thus, the heat exchange betweenthe two surfaces takes place by conduction throughthe spots, conduction through the gas contained in thecavities and radiation between cavity surfaces. Here,we neglect the gas contribution and radiation effectsto demonstrate the basic features of the formulation.The thermal contact resistance, R, can be determinedusing the relation suggested in [11], which is

R =σ

1.25kmAa

[

p

c1

(

1.6177106σ

m

)−c2]

−0.951+0.0711c2

(22)where k is the mean thermal conductivity of the twomaterials , m is the mean absolute asperity slope, σ is

RMS surface roughness, p is the apparent mechanicalpressure, Aa is the apparent contact area, and c1 andc2 are experimental hardness parameters determinedwith micro-hardness tests.Let us consider a two layer slab with imperfect contactat the interface. Due to imperfect thermal contact atthe interface, there is a temperature jump at the in-terface. The interface conditions can be written as [2]

K(1) ∂T (1)

∂x= K(2) ∂T (1)

∂x(23)

which is due to the fact that the heat flux remains con-stant at the interface. Also the temperature jump atthe interface is proportional to the heat flux. Thus,

K(1) ∂T (1)

∂x=

1

R(T (2) − T (1)) (24)

where R is the ‘Thermal Contact Resistance’.

The CG Method

As said earlier, in the CG method, the interface condi-tions are modelled using interface/gap elements. Theimperfect thermal contact functional corresponding tothe above interface conditions is given in [2] as

ΠTI =

nI∑

k=1

∫ L(k)

0

1

2

L(k)

R(k)

(

∂T (k)

∂x

)2

dx

where nI is the number of imperfect thermal con-tacts, R(k) the thermal contact resistance, and L(k)

is the thickness of the kth imperfect thermal contact(L(k) → 0).

The LDG Method

However, the LDG method captures the temperaturejump at the interface without using any additionalinterface/gap elements. In the LDG method, theedges of the elements coinciding with the interface aretreated as boundary edges having a ‘Dirichlet bound-ary condition’, where the prescribed temperature is theunknown temperature at the interface. The interfacephysics is imposed by defining the numerical flux, q,at the interface i, as

qi−1 = qi =1

Ri

[[T ]] (25)

where Ri is the thermal contact resistance for the in-terface i.In the numerical examples to follow, the effectivenessof the LDG formulation in capturing the temperaturejump at the interface are illustrated.

4American Institute of Aeronautics and Astronautics

Numerical Examples

1-D Slab problem: The above formulation is usedto predict the transient heat conduction response ofa two-layer slab problem given in [2] for two differentvalues of thermal contact resistance. The two slabs areeach of length 0.1m and have the following materialproperties,

K = 46.3 WmK

ρc = 4.27× 106 Jm3K

R = 3.8 × 10−3&R = 3.8 × 10−2 m2KW

Initially the slab is at uniform temperature T = 0;thereafter the boundaries x = 0, L are kept at con-stant temperature T = Ti and T = 0, respectively.One linear element was used per slab. Explicit timeintegration scheme Euler forward (θ = 0), and implicitschemes including Crank-Nicolson (θ = 1

2 ), Galerkin (θ= 2

3 ) and Euler backward (θ = 1) were used to obtainthe transient heat conduction response. The resultswere obtained using ∆t = 120 s and are compared with

the analytical solution (Fig. 1(a) (R = 3.8×10−3 m2KW

)

& Fig. 2(a) (R = 3.8 × 10−2 m2KW

)) and are found tobe in excellent agreement. However, it should be men-tioned that there are no additional degrees of freedomat the interface in the LDG formulation as in case ofthe CG method. Figs. 1(b) and 2(b) show the temper-ature histories for different values of thermal contactresistances.As can be seen from the results, the jump at the inter-face increases with the increase in the thermal contactresistance.

Fig.(3) shows the convergence rates for the LDGmethod (1D). The convergence rate of L2 norm oferror in temperature for the LDG method for Crank-Nicolson scheme is 2.0 and is 1.0 for the Euler Galerkinscheme.

The next two examples show the application of theLDG method to two dimensional heat conductionproblems. Linear triangular elements are used.

Fin and plate problem:This problem is related to heat removal from the

electronic devices. Fins are attached to the plate toincrease the heat dissipation. The ambient air is at200C and the lower side of the plate is subjected toa constant heat flux (1000 Wm−2s−1). The materialproperties and parameters of the surface geometry canbe found in [3].18 triangular elements (54 dofs) were used for thisproblem. The temperature distribution for the above

problem is plotted for three different values of mechan-ical pressure (Fig 4). Fig. 5 shows the temperaturejump in the contact zone as a function of apparentmechanical pressure. The temperature jump at the in-terface decreases with increase in apparent mechanicalpressure. This is because with increase in apparentmechanical pressure, the real contact area increases,reducing the thermal contact resistance. It should benoted here that the gas contribution is neglected in thecalculation of the thermal contact resistance. The val-ues of temperature jump at the interface for differentmechanical pressures agree well with those reportedin [3].

Steel block problem:

In this problem, a temperature gradient of 100oCis applied to two stainless steel blocks in contact. Thecomplete problem along with material properties andparameters of the surface geometry is given in [4].The problem is solved using 16 triangular elements(48 dofs) for a mechanical pressure of 0.9008 MPa, atemperature jump of 3.1oC is obtained which agreeswith that reported in [4]. The transient response wasobtained using Galerkin time integration scheme. Thetransient solution gave the same temperature jump (fora given mechanical pressure) as observed in [4](Fig. 6).

Fig.(7) shows the convergence rates for the LDGmethod (2D). The convergence rate in L2 norm of er-ror in temperature for the LDG method for Crank-Nicolson scheme is 2.0 and is 1.0 for the Euler back-ward scheme.

6. Conclusions

The LDG formulation for the elliptic heat conduc-tion problems is described which is then extended toparabolic problems. The LDG method is applied tothe problems involving sharp/high gradient and im-perfect contact between the surfaces by proper designof numerical fluxes at the interface. The results showthat the LDG method has significant advantages overthe FEM in the above problems. The LDG method iscomputationally more efficient than the CG method inproblems involving sharp/high gradients. The applica-tion of the LDG method to thermal contact resistanceproblems eliminates the use of interface/gap elements.Future work aims at combining the LDG method withthe CG method for thermal contact resistance prob-lems. This will involve using the LDG method nearthe interface and the CG method elsewhere.

5American Institute of Aeronautics and Astronautics

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Distance, x (m)

T/T

i

Analytical solutionEuler forward (θ = 0)Crank−Nicolson (θ = 1/2)Galerkin (θ = 2/3)Euler backward (θ = 1)

0 200 400 600 800 1000 12000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

time (s)

T/T

i

Solution using explicit scheme

Crank−Nicolson (θ = 1/2)

Galerkin (θ = 2/3)

x = 0.5L

Figure 1: (a) Transient analysis to steady state solution for the 1-D slab problem for R = 3.8× 10−3 m2KW

(b)

Temperature histories for two-layer slab problem for R = 3.8 × 10−3 m2KW

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Distance, x (m)

T/T

i

Analytical solutionEuler forward (θ = 0)Crank−Nicolson (θ = 1/2)Galerkin (θ = 2/3)Euler backward (θ = 1)

0 200 400 600 800 1000 12000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

time (s)

T/T

i

Solution using explicit scheme

Crank−Nicolson (θ = 1/2)

Galerkin (θ = 2/3)

x = 0.5L

Figure 2: (a) Transient analysis to steady state solution for the 1-D slab problem for R = 3.8× 10−2 m2KW

(b)

Temperature histories for two-layer slab problem for R = 3.8 × 10−2 m2KW

7. Acknowledgments

The authors are pleased to acknowledge Prof. B. Cock-burn for related technical discussions. Support in partby the AHPCRC, under contract DAAD-19-01-2-0014to the University of Minnesota, and computer grantsby the Minneapolis Supercomputer Institute (MSI) arealso acknowledged.

References

[1] Cockburn B. Discontinuous Galerkin Methods.Technical report, School of Mathematics, Univer-sity of Minnesota, 2003.

[2] Blandford G.E. and Tauchert T.R. Thermoelasticanalysis of layered structures with imperfect layercontact. Computers and Structures, 21(6):1283–1291, 1985.

[3] Wriggers P and Zavarise G. Thermomechani-cal contact—A rigorous but simple numerical ap-proach . Computers and Structures, 46(1):47–53,1993.

[4] Zavarise G, Wriggers P, Stein E, and SchreflerB.A. Real contact mechanisms and finite ele-ment formulation–A coupled thermomechanicalapproach. International journal for numerical

methods in engineering, 35:767–785, 1992.

6American Institute of Aeronautics and Astronautics

−12 −11 −10 −9 −8 −7 −6−32

−30

−28

−26

−24

−22

−20

−18

ln (∆ t)

ln (

L 2 err

or in

tem

pera

ture

)

LDG : R = 2.66

−12 −11 −10 −9 −8 −7 −6−16

−15.5

−15

−14.5

−14

−13.5

−13

−12.5

−12

−11.5

−11

ln (∆t)

ln (

L 2 err

or in

tem

pera

ture

)

LDG : R = 1

Figure 3: (a) Convergence rate in L2 norm of error in temperature for the LDG method (1D) for Crank-Nicolsonscheme (b) Convergence rate of L2 norm of error in temperature for the LDG method (1D) for Galerkin scheme

[5] Belytschko T and Lu Y.Y. A curvilinear spectraloverlay method for high gradient problems. Com-

puter Methods in Applied Mechanics and Engin-

nering, 95:383–396, 1992.

[6] Bahrami M, Culham J R, Yovanovich M M, andSchneider G E. Review of Thermal Joint Re-sistance Models for Non-Conforming Rough Sur-faces in a Vaccum. Proceedings of HTC’03 ASMESummer Heat Transfer Conference, Las Vegas,Nevada, USA, July 21-23 2003.

[7] Castillo P, Cockburn B, Perugia I, and SchotzauD. An a priori error analysis of the local dis-continuous Galerkin method for elliptic problems.SIAM Journal on Numerical Analysis, 38:1676–1706, 2000.

[8] Cockburn B, Kanschat, Perugia I, and SchotzauD. Superconvergence of the local discontinuousGalerkin method for elliptic problems on Carte-sian grids. SIAM Journal on Numerical Analysis,39:264–285, 2001.

[9] Castillo P, Cockburn B, Perugia I, and SchotzauD. Local discontinuous Galerkin methods for el-liptic problems. Communications in Numerical

Methods in Engineering, 18:69–75, 2002.

[10] Cockburn B and Shu C.W. The local dis-continuous Galerkin time-dependent method forconvection-diffusion systems . SIAM Journal on

Numerical Analysis, 35:2440–2463, 1998.

[11] Song S and Yovanovich M.M. Explicit relativecontact pressure expression: dependence uponsurface roughness parameters and Vickers micro-hardness coefficients. AIAA paper, (87-0152),1987.

7American Institute of Aeronautics and Astronautics

Figure 4: (a) Problem description for plate and fin problem, (b) - (c) the temperature distributions for theplate and fin problem for three different values of apparent mechanical pressure, p = 100 Pa, p = 10000 Paand p = 10000000 Pa,

Appendix

Numerical Examples

Bar problem: Let us consider the steady state heatconduction problem in a bar (6 inch in length) withheat source [5]:

b(x) = 2s2sech2[s(x − 3)] tanh[s(x − 3)] (26)

with the boundary conditions

T = − tanh(3s) at x = 0

T = tanh(3s) at x = 6(27)

The exact solution for this problem is

T = tanh[s(x − 3)] (28)

where s = 40 and conductivity, K = 1. This problemhas a high gradient near x = 3. The above problemis solved by the CG and the LDG method using theLinear Lagrangian elements with a uniform mesh. Theresults are compared with the exact solution.The temperature distributions calculated using theCG and the LDG method are compared with the an-alytical solution in Fig.8(a). Results from both themethods exactly match the analytical solution. How-ever, in the case of the flux distribution, Fig. 8(b),while the CG method shows poor agreement with theanalytical solution, the LDG method shows excellentagreement with the analytical solution. It is furthershown that in order to capture the peak of the gra-dient field, the CG method requires a very refined

mesh (1001 degrees of freedom) (Fig. 8(c)) whereasthe LDG method captures the peak with much lessnumber of degrees of freedom (200).

Rectangular plate problem: Let us now considerheat conduction in a rectangular plate (0.5 × 6 in2)with heat source

b(x, y) = 2s2tanh[s(y − 3)] (29)

with the boundary conditions

T = − tanh(3s) at y = 0

T = tanh(3s) at y = 6

T,x = 0 at x = −0.25 and x = 0.25

(30)

where s = 40 and conductivity, K = 1. The aboveproblem is solved by the LDG method and also in AN-SYS using linear triangular elements. This problem isa good illustration of application of the LDG methodto two-dimensional high gradient problems. While theCG method fails to capture the peak value of the gra-dient field even with a very refined mesh, the LDGmethod captures the gradient with a relatively coarsermesh. It is found that the LDG method (360 degreesof freedom) is much less computationally expensive ascompared to the CG method (1204 degrees of freedom)(Fig. 9). Fig. 10 shows the convergence rates for theLDG method. The convergence rate of the continuousL2 norm of temperature for the LDG method is 2.026and is 1.27 for the gradient field.

8American Institute of Aeronautics and Astronautics

102

103

104

105

106

107

0

50

100

150

200

250

300

350

Mechanical Pressure (Pa)

Tem

pera

ture

(o C)

Figure 5: Effect of apparent mechanical pressure on the temperature jump in the contact zone. As the appar-ent mechanical pressure increases, the contact area increase thereby decreasing the temperature jump in thecontact zone.

104

105

106

107

0

5

10

15

20

25

30

35

40

45

Mechanical pressure (Pa)

Tem

pera

ture

jum

p at

the

cont

act z

one

(o C)

Figure 6: Transient to steady state analysis for the 2-D steel block problem showing temperature jump versusmechanical pressure (θ = 2/3, ∆t = 10 secs, time = 5000 secs)

9American Institute of Aeronautics and Astronautics

−14 −13.5 −13 −12.5 −12 −11.5 −11 −10.5 −10 −9.5 −9−20

−19

−18

−17

−16

−15

−14

−13

−12

−11

−10

ln(∆t)

ln (

L 2 err

or in

tem

pera

ture

)

LDG : R = 2

−12 −11 −10 −9 −8 −7 −6 −5 −4−7

−6

−5

−4

−3

−2

−1

0

1

ln(∆t)

ln(L

2 err

or in

tem

pera

ture

)

LDG : R=1

Figure 7: (a) Convergence rate in L2 norm of error in temperature for the LDG method (2D) for Crank-Nicolsonscheme (b) Convergence rate in L2 norm of error in temperature for the LDG method (2D) for Galerkin scheme

0 1 2 3 4 5 6−1.5

−1

−0.5

0

0.5

1

1.5

x

tem

pera

ture

fiel

d

CG methodLDG methodExact solution

0 1 2 3 4 5 6−45

−40

−35

−30

−25

−20

−15

−10

−5

0

5

x

flux

field

CG methodLDG methodExact solution

0 1 2 3 4 5 6−40

−35

−30

−25

−20

−15

−10

−5

0

5

x

flux

field

11 dof

51 dof

251 dof

1001 dof

Figure 8: (a) Comparison of exact, FEM and LDG methods results for temperature field for the high gradientproblem. Both CG and the LDG method show excellent agreement with the exact solution. Both methodsemploy equal number of degrees of freedom (200). (b) Comparison of exact, CG and LDG methods results forflux field for the high gradient problem. The LDG method shows excellent agreement with the exact solutionwhile FEM fails to capture the peak of the gradient field. Both methods employ equal number of degrees offreedom (200) (c) Flux field results for the CG method for the 1-D high gradient bar problem showing sequenceof refinements to capture the peak of the gradient field

0 1 2 3 4 5 6−4000

−3000

−2000

−1000

0

1000

2000

3000

4000

y

Tem

pera

ture

0 1 2 3 4 5 6−5000

−4000

−3000

−2000

−1000

0

1000

2000

3000

4000

5000

y

Flu

x

0 500 1000 1500 2000 2500 3000 3500 4000 4500−4800

−4700

−4600

−4500

−4400

−4300

−4200

−4100

−4000

−3900

Degrees of freedom

Pea

k va

lue

of th

e flu

x fie

ld

Traditional finite element methodLDG method

Refined mesh

Figure 9: (a) The LDG method results for temperature field for the high gradient problem (Rectangular plate)(b) The LDG method results for flux field for the high gradient problem (Rectangular plate) (c) The LDGmethod requires much less number of degrees of freedom than the CG method to capture the peak of thegradient field

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−0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.10.5

1

1.5

2

2.5

X: −0.5229Y: 1.249

ln (h)

ln (

L 2 err

or in

tem

pera

ture

)

LDG: R = 2.026

−0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

ln (h)

ln (

L 2 err

or in

flux

) LDG: R = 1.27

Figure 10: (a) Spatial convergence rate of the steady-state temperature norm for the high-gradient problem(Rectangular plate) for the LDG method.(b) Spatial convergence rate of flux norm for the high-gradient problem(Rectangular plate) for the LDG method.

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