Local Discontinuous Galerkin Formulations for Heat Conduction Problems Involving High
Gradients and Imperfect Contact Surfaces
R. Kanapady∗, A. Jain†, K.K. Tamma‡ and S. Siddharth§
Abstract
A Local Discontinuous Galerkin (LDG) method is described here which provides a unified mathematicalsetting and framework for solving various kinds of heat conduction problems to include thermal contactconductance/resistance, sharp/high gradient problems and the like. For these applications, the LDG methoddoes not require much modifications to the basic formulation or the need to employ ad hoc approaches aswith the Continuous Galerkin (CG) finite element methods. In this paper, we describe the LDG formulationfor elliptic heat conduction problems which is then extended to parabolic problems. The advantages of theLDG method over the CG method are shown using two classes of problems—problems involving sharp/highgradients, and imperfect contact between surfaces. So far, interface/gap elements have been primarily usedto model the imperfect contact between two surfaces to solve thermal contact resistance problems. The LDGmethod eliminates the use of interface/gap elements and provides a high degree of accuracy. It is furthershown in the problems involving sharp/high gradients, that the LDG method is less expensive (requires lessnumber of degrees of freedom) as compared to the CG method to capture the peak value of the gradient.Several illustrative 1-D/2-D applications highlight the effectiveness of the present the LDG formulation.
1. Introduction
The Local Discontinuous Galerkin (LDG) method pre-sented here provides a unified framework for solvingvarious kinds of heat conduction problems like thermalcontact resistance and sharp/high gradient problemswithout much modifications to the basic formulation.This is a significant advantage over the CG methodin these situations. As in contact resistance prob-lem [1–3], the CG method requires modifications suchas the use of gap/interface elements to capture thetemperature jump at the interface. In the case ofsharp or high gradient problems, the CG method re-quires to be coupled with some special methods suchas curvilinear spectral overlay method [4] to capturethe peak value of the gradient field or requires a verysmall element size distribution in the region of thesharp gradients.
In this paper, we focus primarily on the formulationsand 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 [4]. 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.
Thermal contact resistance plays a very important rolein applications such as electronics packaging, layeredstructures, nuclear reactors, 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 [5]. 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 useof 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 the heat conduction elliptic equa-tions with extensions to parabolic equations for time
∗Research Associate: Department of Mechanical Engineering, University of Minnesota, 111 Church St. SE, Minneapolis, MN55455, Ph: (612) 626-8101, Fax: (612) 626-1596, [email protected]
†Graduate Research Assistant:[email protected]‡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,[email protected]§Graduate Research Assistant:[email protected]
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43rd AIAA Aerospace Sciences Meeting and Exhibit10 - 13 January 2005, Reno, Nevada
dependent problems is described. In section 3, theLDG method is applied to high or sharp gradient prob-lems. In section 4, the LDG formulation for thermalcontact resistance problem is proposed and several nu-merical examples are solved followed by concluding re-marks in section 5.
2. Local Discontinuous Galerkin
Method
The advantage of applying the LDG method to ellip-tic problems relies on the ease with which it handleshanging nodes, elements of general shapes, and localtrial functions of different types. These properties alsomake the LDG method ideally suited for hp-adaptivityand allow it to be easily coupled with other meth-ods [6]. The LDG method for second order ellipticproblems have been derived in [6,7]. However, for com-pleteness, the LDG formulation for heat conduction el-liptic equations with extensions to parabolic equationsfor time dependent problems is described next.
Governing Equations
Heat conduction equations for a body Ω enclosed by aclosed surface ∂Ω which is initially at temperature T0,can be written as
ρc∂T
∂t+ ∇ · q = b in Ω
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. Considering theweighted residual equation on each element Ωe, wehave
∫
Ωe
v(ρc∂T∂t
+ ∇.q − b) dΩe = 0 (3)∫
Ωe
w(K−1q + ∇T )) dΩe = 0 (4)
where v, w are the weighting functions. Integrating byparts the terms associated with the divergence in Eq.3and gradient in Eq.4, the weak form is given by
∫
Ωe
vρc∂T
∂tdΩe −
∫
Ωe
q.∇v dΩe+
∫
∂Ωe
vq · n ds −
∫
Ωe
vb dΩe = 0
(5)
and∫
Ωe
K−1q.w dΩe −
∫
Ωe
T∇ ·w dΩe+
∫
∂Ωe
Tw.n ds = 0
(6)
We replace T and q in the boundary terms by the socalled ‘numerical fluxes’ which are discrete approxima-tions to the traces of T and q on the boundary of theelements. Summing up for all the elements in the do-main we have,
n∑
(
∫
Ωe
vρc∂T
∂tdΩe −
∫
Ωe
q.∇v dΩe+
∫
∂Ωe
vq · n ds −
∫
Ωe
vb dΩe = 0)
(7)
and
n∑
(
∫
Ωe
K−1q.w dΩe −
∫
Ωe
T∇ ·w dΩe+
∫
∂Ωe
Tw.n ds = 0)
(8)
where n is the number of elements and T and q arethe so called ‘numerical fluxes’. If the boundary ∂Ωe
of any element e /∈ ∂ΩT and ∂Ωq, the numerical fluxes
T and q are defined by
T = 〈T 〉 + β · [[T ]] + γ[[q]] (9)
q = 〈q〉 − α[[T ]] − β[[q]] (10)
and if ∂Ωe ∈ ∂ΩT or ∂Ωq , then
q =
q+ − α[[T ]] on ∂ΩT
q on ∂Ωq.(11)
T =
T on ∂ΩT
T− + γ[[q]] on ∂Ωq.(12)
where the average and jump operators can be definedas
〈T 〉 = (T+ + T−)/2
[[T ]] = T+n+ + T−n−
〈q〉 = (q+ + q−)/2
[[q]] = q+ · n+ + q− · n−
(13)
where α is of the order Ø(h−1) and β is such that|β · n| = 1
2 .
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α, β, γ determine the stability of the method. For thestability and accuracy of the method, α must be > 0,β can be arbitrary and γ can be zero. The approxi-mations 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.Assuming NT = Nq = N, the following semi-discreteequations are obtained by substituting eqns. (9)–(14)in eqns. (7)–(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 =n
∑
∫
Ωe
NT N dΩe + f(γ)
bT =
∫
∂Ωe
NT b + f(α)
bq = f(T )
(17)
where f(α), f(β) and f(γ) includes boundary integralscontaining numerical flux terms. Since Mqq = 0, wecannot take the inverse of M , especially for explicittime integration methods. Furthermore, for ease of so-lution of q, we assume γ = 0 in Eq. 9 for the LDGmethod. This assumption makes the matrix Kqq ablock diagonal matrix, allowing to solve Eq. 15 for q
and substitute in the Eq. 16 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.
3. 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 the cap-turing of high localized gradient fields with the tradi-tional finite element of continuous formulation [4]. Itis shown further by numerical examples, that a veryrefined mesh is required to capture the peak of thegradient field.However, since the LDG method uses completely dis-continuous approximations [8], it is able to capturethe peak of the gradient field with much less numberof degrees of freedom. The results from the numeri-cal examples solved in the paper show that the LDGmethod is computationally efficient as compared to theCG method.
Numerical Examples
Fin problem: Let us consider the steady state heatconduction problem in a fin (6 inch in length) withheat source [4]:
b(x) = 2s2sech2[s(x − 3)] tanh[s(x − 3)] (20)
with the boundary conditions
T = − tanh(3s) at x = 0
T = tanh(3s) at x = 6(21)
The exact solution for this problem is
T = tanh[s(x − 3)] (22)
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 ana-lytical solution in Figure 1(a). Results from both themethods exactly match the analytical solution. How-ever, in the case of the flux distribution, (Figure 1(b)),while the CG method shows poor agreement with theanalytical solution, the LDG method shows excellent
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0 1 2 3 4 5 6−1.5
−1
−0.5
0
0.5
1
1.5
x
Tem
pera
ture
Traditional FEMLDGExact
0 1 2 3 4 5 6−40
−35
−30
−25
−20
−15
−10
−5
0
5
x
Flu
x
ExactLDGTraditional FEM
Figure 1: (a) Comparison of exact, CG and LDG method 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 method 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).
0 1 2 3 4 5 6−40
−35
−30
−25
−20
−15
−10
−5
0
5
x
flux
10 elements
50 elements
100 elements
400 elements
600 elements
Figure 2: Flux field results for FEM for the high gradient problem. A very refined mesh is required to capturethe peak of the gradient field.
agreement with the analytical solution. It is furthershown that in order to capture the peak of the gradi-ent field, the CG method requires a very refined mesh(599 degrees of freedom) (Figure 2) whereas the LDGmethod captures the peak with much less number ofdegrees of freedom (200).
Rectangular plate problem: Let us next considerthe heat conduction in a rectangular plate (0.5 × 6in2) with heat source
b(x, y) = 2s2tanh[s(y − 3)] (23)
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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
Figure 3: (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).
Continuous Galerkin finite element methodLDG method
Refined mesh
Figure 4: The LDG method requires much less number of degrees of freedom than the CG method to capturethe peak of the gradient field.
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
(24)
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 LDG
method captures the gradient with a relatively coarsermesh. Figure 3(a) and Figure 3(b) show the resultsfor temperature and flux field respectively. It is foundthat the LDG method (360 degrees of freedom) is muchless computationally expensive as compared to the CGmethod (1204 degrees of freedom) (Figure 4). Thepeak value of the gradient field obtained from ANSYSusing a very refined mesh is also included in the figurefor comparison. Figure 5 shows the convergence ratesfor the LDG method. The convergence rate of the con-tinuous L2 norm of temperature for the LDG methodis 2.026 and is 1.27 for the gradient field.
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Figure 5: (a) Convergence rate of the temperature norm for the high-gradient problem (Rectangular plate) forthe LDG method.(b) Convergence rate of flux norm for the high-gradient problem (Rectangular plate) for theLDG method.
Figure 6: (a) Transient response of a two layer slab, R = .34 × 10−3 m2KW
∆t = 120 s and time = 18000 s.The LDG method captures the temperature jump due to imperfect thermal contact without the use of gap
elements. (b) Transient response of a two layer slab, R = 3.8 × 10−3 m2KW
∆t = 120 s and time = 18000 s.
(c)Transient response of a seven layer slab, R = .34× 10−3 m2KW
∆t = 120 s and time = 18000 s.
4. Thermal contact resistance
problems
Thermal contact resistance is an important considera-tion in applications such as electronics packaging, lay-ered structures, nuclear reactors, 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 contact
between 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 [9], which is
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Figure 7: (a) Problem description; plate and fin problem, (b) - (c) the temperature distribution for the plateand fin problem for three different values of contact pressure, p = 100 Pa, p = 10000 Pa and p = 10000000 Pa,
0 1 2 3 4 5 6 7 8 9 10
x 106
0
50
100
150
200
250
300
350
Contact pressure (Pa)
Tem
pera
ture
Figure 8: Effect of contact pressure on the temperature jump in the contact zone. As the contact pressureincreases, the contact area increase thereby decreasing the temperature jump in the contact zone.
R =σ
1.25kmAa
[
p
c1
(
1.6177106σ
m
)−c2]
−0.951+0.0711c2
(25)where k is the mean thermal conductivity of the twomaterials , m is the mean absolute asperity slope, σ isRMS 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 [1]
K(1) ∂T (1)
∂x= K(2) ∂T (1)
∂x(26)
and the heat flux remains constant across the inter-face. Also the temperature jump at the interface isproportional to the heat flux. Thus,
K(1) ∂T (1)
∂x=
1
R(T (2) − T (1)) (27)
7American Institute of Aeronautics and Astronautics
where R is the ‘Thermal Contact Resistance’.
The CG Method Approach
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 [1] 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 stating the additional relation-ship between the numerical fluxes at the interface i,as
qi−1 = qi =1
Ri
(Ti − Ti−1) (28)
where Ri is the thermal contact resistance for the inter-face i. In the numerical examples to follow, the effec-tiveness of the CG method and the LDG formulationemploying the above interface conditions in capturingthe temperature jump at the interface are illustrated.
Numerical Examples
Slab problem: The above formulation is used topredict the transient heat conduction response of atwo-layer slab problem given in [1] for two differentvalues of thermal contact resistance. The two slabsare each of length 0.1m and have the following mate-rial properties,
K = 46.3 WmK
, ρc = 4.27 × 106 Jm3K
, R = .34 ×
10−3 m2KW
Initially the slab is at uniform temperature T = 0;thereafter the boundaries x = 0, L are kept at constanttemperature T = 10 and T = 0, respectively. Galerkin(θ = 2
3 ) and the Euler backward(θ = 1) implicit timeintegration methods are used to obtain the transientheat conduction response. The results are comparedwith the analytical solution (Figure 6) and are foundto be in excellent agreement. However, it should be
mentioned that there are no additional degrees of free-dom at the interface in the LDG formulation as in thecase of the CG method. Figure 6(c) shows the tem-perature distribution for a seven-layer slab. The sharptemperature jumps are clearly evident at the interfaceshaving same thermal contact resistance.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 toheat removal from electronic devices. Fins are attachedto the plate to increase the heat dissipation. The am-bient air is at 200C and the lower side of the plate issubjected to a constant heat flux (1000 Wm−2s−1).The material properties and parameters of the surfacegeometry can be found in [2].The temperature distribution for the above problemis plotted for three different values of contact pressure(Figure 7). Figure 8 shows the temperature jump inthe contact zone as a function of contact pressure. Thetemperature jump at the interface decreases with in-crease in mechanical pressure. This is because withincrease in mechanical pressure, the real contact areaincreases, reducing the thermal contact resistance. Itshould be noted here that the gas contribution is ne-glected in the calculation of the thermal contact resis-tance. The values of temperature jump at the interfacefor different contact pressures agree well with those re-ported in [2].
Steel block problem: In this problem, a tempera-ture gradient of 100oC is applied to two stainless steelblocks in contact. The complete problem along withmaterial properties and parameters of the surface ge-ometry is given in [3]. The problem is solved for a con-tact pressure of 0.9008 MPa, and a temperature jumpof 3.1oC is obtained which agrees with that reportedin [3] .
5. 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.
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Future work aims at combining the LDG method withthe CG method for thermal contact resistance prob-lems to further enhance the efficiency of the computa-tions. This will involve using the LDG method nearthe interface and the CG method elsewhere.
Acknowledgements
The authors are pleased to acknowledge Prof. B. Cock-burn for discussions related to elliptic problems. Thecomputational support by the Minneapolis Supercom-puting Institute (MSI) is also gratefully acknowledged.
References
[1] Blandford G.E. and Tauchert T.R. Thermoelasticanalysis of layered structures with imperfect layercontact. Computers and Structures, 21(6):1283–1291, 1985.
[2] Wriggers P and Zavarise G. Thermomechani-cal contact—A rigorous but simple numerical ap-proach . Computers and Structures, 46(1):47–53,1993.
[3] Zavarise G, Wriggers P, Stein E, and SchreflerB.A. Real contact mechanisms and finite el-ement formulation–A coupled thermomechanicalapproach. International journal for numerical
methods in engineering, 35:767–785, 1992.
[4] Belytschko T and Lu Y.Y. A curvilinear spectraloverlay method for high gradient problems. Com-
puter Methods in Applied Mechanics and Enginner-
ing, 95:383–396, 1992.
[5] Yovanovich M.M Bahrami M., Culham J. R. andSchneider G.E. Review of thermal joint resistancemodels for non-conforming rough surfaces in a vac-uum. Proceedings of HTC’03 ASME Summer Heat
Transfer Conference, 2003.
[6] Castillo P, Cockburn B, Perugia I, and Schotzau D.An a priori error analysis of the local discontinuousGalerkin method for elliptic problems. SIAM Jour-
nal on Numerical Analysis, 38:1676–1706, 2000.
[7] 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.
[8] Cockburn B and Shu C.W. The local discontinuousGalerkin time-dependent method for convection-diffusion systems . SIAM Journal on Numerical
Analysis, 35:2440–2463, 1998.
[9] Song S and Yovanovich M.M. Explicit relative con-tact pressure expression: dependence upon surfaceroughness parameters and Vickers microhardnesscoefficients. AIAA paper, (87-0152), 1987.
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