COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Numerical Study of Two-dimensional Transient Heat Conduction using
Finite Element Method
Lee Yuk Choi 1*, Wong Sau Keong 1, Ooi Hooi Woon 1, Sia Chee Kiong 2
1
2
Faculty of Mechanical Engineering, Kolej Universiti Teknikal Kebangsaan Malaysia (KUTKM), Locked Bag 1200,
Ayer Keroh, Melaka, Malaysia
Department of Material and Design Engineering. Kolej Universiti Teknologi Tun Hussein Onn (KUiTTHO),
Locked Bag 101, Batu Pahat, Johor, Malaysia
Email: yukchoi@ kutkm.edu.my
Abstract A mathematical formulation applied to a numerically study of two dimensional transient heat conduction
in a building slab to determine the temperature distribution as a function of time. The finite element method is a
method of choice and it is a numerical procedure for analyzing structures and continua. Usually the problem
addressed is too complicated to be solved satisfactorily by classical analytical methods. The Galerkin method is the
method of choice in the formulation, which involves transient heat conduction by using the bilinear quadrilateral
element discretization; the integration of weighted residual of the differential equation and also the boundary
condition was perform using finite element method. A computer program is developed using Matlab to run the
analysis. The finite element analysis software MSC Nastran has also been used to analyze the temperature
distribution and for the comparison purposes. It was found that temperature declined in the early stage for delta time,
Δt =1 second which is more efficient. For comparison with MSC Nastran, different delta time for 32 and 200
elements is taking into consideration. The result of the analysis shows that the error is less than 1% by using Matlab
and MSC Nastran, which was deemed satisfactory. There are 3 nodes selected for comparison, which is Node 5-ML,
Node 23-ML and Node 41-ML for Matlab simulation tools and Node 5-MSC, Node 23-MSC and Node 41-MSC is
from MSC Nastran simulation tools. It can be said that with 32 elements, Δt =0.4 s showed not much difference as
most of the point situated at the same exponential curve line for both MSC Nastran and Matlab simulation tools. It
achieves saturation at 213oC for Node 5-ML and Node 5-MSC, 175oC for Node 23-ML and Node 23-MSC and 67oC
for Node 41-ML and Node 41-MSC.
Key words: heat conduction, transient, finite element method, bilinear element, temperature
INTRODUCTION
Many engineering applications rely on the dynamics of heat intensity and flow within objects. Two-dimensional heat
transfer problem find significant importance in building because thermal bridge in walls, windows and other
component can have significant effects on energy performance and occupant comfort. The insulating value of a
material is not sufficient to determine the energy performance of a wall or other component in when the material is
used because the entire area of the wall is not completely filled with insulating material. With theory of twodimensional heat transfer and utilizing the finite element method, it becomes easy to analyze the heat flow.
Therefore, knowledge of the temperature and its transient distribution is vital in design and implementation. [1]
Time dependent problem of engineering involve both in space and time. The time variations start from some initial
conditions and then propagate through time. It can efficiently solve with the finite element method to consider the
temperature for each nodal point. In this study, finite element method is used to determine the two–dimensional
transient heat conduction in a slab. It is used to solve the two dimensional transient heat conduction problems.
Galerkin’s method uses a set of governing equations in the development of an integral form. Starting with a
governing equation weighted residual integrals are evaluated at each element to form a system of linear algebra
equation. [2]
⎯ 632 ⎯
Matlab is a high performance language for technical computing. It can be programmed in an easy to use environment
where problem and solution are expressed in familiar mathematical notation. One attractive aspect of Matlab is that
it is relatively easy to learn. The finite element method is a well – defined method for which Matlab can be very
useful as a solution tool. Matrix and vector manipulation are essential parts in this method. To assess the accuracy of
the method, further comparison of the result with MSC Nastran is also presented.
FINITE ELEMENT FORMULATION
This section presents the finite element formulation for partial differential equation. Laplace’s and Poisson’s
equation are used in a finite element for two- dimensional and axisymmetric problem developed and transient (initial
value) problems are considered.
1. Governing equation The finite element formulation of Laplace’s equation is [3]:
∇ 2T = 0
(1)
While Poisson’s equation is
∇ 2T = g
(2)
Since the Poisson’s equation is more general than Laplace’s equation, therefore Poisson’s equation is considered in
the following formulation. Poisson’s equation in terms of the Cartesian coordinate system can be written as:
∂ 2T ∂ 2T
+
= g ( x, y )
∂x 2 ∂y 2
(3)
Integration of weighted residual of the differential equation and boundary condition yields:
I = ∫ N(
xy
∂ 2T ∂ 2T
∂T
+ 2 − g ( x, y ))dydx − ∫ N
dΓ
2
Γ
e
∂x
∂y
∂t
(4)
where Γe are boundaries for essential conditions.
In order to develop the weak formulation of Eq. (4), the equation by parts is applied to reduce the order of
differentiation within the integral. The subsequent of the integration by parts, consider a two dimensional domain
first of all, evaluate the first term Eq. (4) [7], yield:
∫
xy
N
∂ 2T
dydx
∂x 2
(5)
The domain integral can be expressed as:
∫
y2
y1
⎛ x2 ∂ 2T ⎞
⎜ ∫x1 N 2 dx ⎟ dy
∂x
⎝
⎠
(6)
where y1 and y2 are the minimum and maximum value of the domain in y-axis and the x1 and x2 are the minimum
and maximum values of the domain in x-axis.
Rewriting the expression using the domain and boundary integration will result in:
∂N ∂T
∂T
∂T
dxdy + ∫ N
nx d Γ − ∫ N
nx d Γ
∂x ∂x
∂x
∂x
xy
Γ1
Γ2
−∫
(7)
in which nx is the x- component of the unit normal vector which is assumed to be positive in the outward direction.
Combine the two boundary integrals as
∂N ∂T
∂T
dxdy + v∫ N
nx d Γ
∂x ∂x
∂x
xy
Γ
−∫
(8)
where the boundary integral in the counter-clock wise direction. Similarly, the second term in Eq. (4) can be written
as
∂N ∂T
∂T
dxdy + v∫ N
ny d Γ
∂
y
∂
y
∂y
Γ
xy
−∫
(9)
⎯ 633 ⎯
Finally, the equation (4) can be written as:
⎛ ∂N ∂T ∂N ∂T ⎞
∂T
+
I = −∫ ⎜
dΓ
⎟ dxdy − ∫xy Ng ( x, y ) dxdy + ∫Γ N
∂
∂
∂
∂
∂n
x
x
y
y
⎠
xy ⎝
The symbol
v∫
(10)
denotes the line integral around a closed boundary and replaced by
∫
for simplicity and is known
as Green’s theorem become it is two-dimensional problems.
The first volume integral becomes a matrix term while both the second volume integral and the line integral become
vector terms. In the context of heat conduction, the second volume integral is related to heat source or sink within
the domain and line integral denotes the heat flux through the natural boundary.
2. Bilinear rectangular element The shape function for this element can be derived from the following
interpolation function:
T = a1 + a2 x + a3 y + a4 xy
(11)
the function is linear in both x and y. Applying the same procedure will give the following shape function:
N1 =
1
(b − x)(c − y ) ,
4bc
N3 =
1
(b + x)(c + y )
4bc
(12, 13)
N2 =
1
(b + x)(c − y ) ,
4bc
N4 =
1
(b − x)(c + y )
4bc
(14, 15)
where 2b and 2c are the length and height of the element.
y
4
c
-b
3
b
-c
1
x
2
Figure 1: Bilinear Element
The shape functions in Eqs. (12) - (15) can be obtained by calculating the product of two sets of one-dimensional
shape function. The linear shape functions in the x-direction, as shown in Fig. 1, consider the x-direction with nodes
located at x = −b and x = b be
ξ1 ( x) =
1
(b − x ) ,
2b
ξ 2 ( x) =
1
(b + x)
2b
(16, 17)
Similarly, the linear shape functions in y- direction are
η1 ( x) =
1
(c − y ) ,
2c
η1 ( x) =
1
(c + y )
2c
(18, 19)
The shape functions obtained by the products as shown are called the Lagrange shape functions.
3. Computation of K matrix To compute the element matrix for Poisson’s equation using the bilinear shape
functions.
⎛ ⎧ ∂ N1 ⎫
⎞
⎧ ∂ N1 ⎫
⎜⎪
⎟
⎪
⎪ ∂y ⎪
∂
x
⎜⎪
⎟
⎪
⎪
⎪
⎜ ⎪ ∂ N2 ⎪
⎟
⎪ ∂ N2 ⎪
⎜⎪
⎟
⎪
⎪ ∂y ⎪
∂
x
⎪
⎪ ⎡ ∂ N1 ∂ N 2 ∂ N 3 ∂ N 4 ⎤ ⎪
⎪ ⎡ ∂ N1 ∂ N 2 ∂ N 3 ∂ N 4 ⎤ ⎟
⎡⎣ K e ⎤⎦ = ∫ ⎜ ⎨
⎬⎢
⎥ + ⎨∂ N ⎬ ⎢
⎥ dx dy
xy ⎜ ∂ N
∂
∂
∂
∂
/
x
x
x
x
⎦ ⎪ 3 ⎪ ⎣ ∂ y ∂ y ∂ y ∂ y ⎦⎟
3 ⎪⎣
⎪
⎜
⎟
⎪ ∂y ⎪
⎜⎪ ∂ x ⎪
⎟
⎪
⎪
⎜ ⎪∂ N ⎪
⎟
∂
N
⎪ 4⎪
⎜⎜ ⎪ 4 ⎪
⎟⎟
⎪⎩ ∂ x ⎪⎭
⎪⎩ ∂ y ⎪⎭
⎝
⎠
⎯ 634 ⎯
(20)
Where Ni are bilinear shape functions.
Performing the integrations for all terms result in the following element matrix for bilinear rectangular element.
⎡ k11k12 k13 k14 ⎤
⎢
⎥
k12 k22 k23 k24 ⎥
e
⎢
⎡⎣ K ⎤⎦ =
⎢ k13 k23 k33 k34 ⎥
⎢
⎥
⎢⎣ k14 k24 k34 k44 ⎥⎦
(21)
Since the matrix is symmetry, therefore the stiffness for other element can be written as:
k22 = k11
(22)
k23 = k14 = k41
(23)
k24 = k13 = k4
(24)
k33 = k11
(25)
k34 = k12 = k43 = k21
(26)
k44 = k11
(27)
4. Boundary Integral
⎛ ∂w ∂T ∂w ∂T ⎞
∂T
+
I =∫ ⎜
dΓ
⎟ d Ω + ∫Ω wg ( x, y ) d Ω − ∫Γn w
Ω ∂x ∂x
∂y ∂y ⎠
∂n
⎝
(28)
The boundary integral in Eq. (28) is
∫
Γn
w
∂T
∂T
dΓ = ∑ ∫ w dΓ
Γ
e
∂n
∂n
(29)
where subscript n denotes natural boundary, and superscript e denotes element boundary.
The boundary integral along the element boundary becomes:
⎧ xj −x ⎫
q hij ⎧1⎫
x j ⎪ x j − xi ⎪
∂T
∫Γe w ∂n d Γ = q ∫xi ⎨ x − xi ⎬dx = 2 ⎨⎩1⎬⎭
⎪⎩ x j − xi ⎪⎭
(30)
Where
hij = x j − xi
(31)
denotes the length of element boundary.
One common boundary condition in heat transfer is heat convection at the boundary. This boundary condition is
expressed as:
∂T
= − a(T − T∞ )
∂n
(32)
where a denotes heat convection coefficient, T∞ denotes ambient temperature.
That is, heat flux is proportional to the temperature difference of the body surface and the environment. Rewriting
this in a more general expression gives
∂T
= − aT + b
∂n
(33)
where a and b are known functions because T∞ is a known value. Substituting Eq. (37) into the element boundary
integral in Eq. (33) results in:
∫
Γ
e
w
∂T
d Γ = ∫ e w{− a ( x, y )T + b( x, y )} dΓ
Γ
∂n
(34)
As a result, the first term of Eq. (34) becomes, using linear shape functions to interpolate the element boundary give:
⎯ 635 ⎯
∫
Γ
w {−a ( x, y )T } d Γ = − ∫
e
sj
si
⎧ ss j −−ss ⎫
⎪ j i⎪
a ⎨ s−s ⎬
⎪⎩ s j − sii ⎪⎭
{
s j −s
s j − si
s − si
s j − si
}
⎡
⎤
⎧ Ti ⎫ hl 2 1⎥
dΓ ⎨ ⎬ = ⎢
⎩T j ⎭ 6 ⎢1 2 ⎥
⎣
⎦
(35)
where si , s j = coordinate values of the local axis located along the element boundary, ui , u j = nodal variables at the
element boundary.
This integral results in a 2 × 2 matrix for an element with two nodes on the boundary. This matrix should be added to
the system matrix. The remaining term in Eq. (34) can be dealt with in the same way.
The second term, heat rate vector:
∫ w{b( x, y)} d Γ = ∫
Γ
e
sj
si
⎧ ss j −−ss ⎫
⎪ j i⎪
b ⎨ s−s ⎬
⎪⎩ s j − sii ⎪⎭
{
s j −s
s j − si
s − si
s j − si
} d Γ = hT2 l ⎧⎨⎩11⎫⎬⎭
∞
(36)
5. Transient Analysis The governing equation for transient heat conduction is:
∂T 1 ⎛ ∂ 2T ∂ 2T ⎞
= ⎜
+
⎟ in Ω
∂t α ⎝ ∂x 2 ∂y 2 ⎠
(37)
ρC p
, for heat conduction problem with constant material
k
properties, k = the coefficient of heat conduction, ρ = density, and Cp = Specific heat.
where t = time, α is a known function, usually is equal to
The heat generation or heat sink is neglected in this study.
Applying the method of weighted residual in the same way as given is governing equation gives:
I =∫N
xy
1
∂T
dx dy +
α
∂t
∫
xy
⎛ ∂ N ∂T ∂ N ∂T ⎞
1
∂T
dΓ
+
⎜
⎟ dx dy − ∫Γ N
α
∂n
⎝ ∂x ∂x ∂y ∂y⎠
(38)
The method of weighted residual is applied to the spatial domain but not to the temporal domain regardless of
whether it is a steady state or transient problem. As a result, the difference between the transient and steady state
problems is the first term in Eq.(38). The other difference is that the variable T is a function of both space and time
for the transient problem.
The variable T = T(x, y, z) is interpolated within a finite element in a similar way as before using shape function [7].
n
T ( x, y, z ) = ∑ Ni ( x, y )Ti (t )
(39)
i =1
So, the bilinear rectangular element is:
⎡ 4 2 1 2⎤
⎢
⎥
A 2 4 2 1⎥
⎡⎣ M e ⎤⎦ = ⎢
36 ⎢1 2 4 2 ⎥
⎢
⎥
⎣⎢ 2 1 2 4 ⎦⎥
(40)
where A is the area of the rectangular element.
Therefore, the final matrix equation or global matrix for Eq(40) becomes:
{}
• t
[M ] T
+ [ K ]{T } = { R} ,
t
t
(41)
•
Where [M] = element capacitance matrix, [K] = global conductivity matrix and effect of conduction, { T }t =
temperature time dependent, { T }t = temperature vector, and { R } = global heat rate vector.
Because this equation should be true at any time, place superscript t in Eq. (41) to denote the time when the equation
is satisfied. Furthermore, matrices [M] and [K] are independent of time. Now, the parabolic differential equation has
transformed into a set of ordinary differential equations using the finite element method [7]. In order to solve the
equations, finite different method is used for the time derivative.
⎯ 636 ⎯
6. Transient Response The transient response computed by the recurrence formula will therefore approximate the
exact solution to the original matrix differential equations, but it will approach the exact solution arbitrarily closely
as Δt→ 0. The larger the time step, Δt, the transient response computed may become unstable [1]. The study of the
stability of numerical solution algorithms for ordinary differential equations is an important in numerical analysis
and backward difference technique is applied for time integration. However, the time step size is of course important
for accuracy.
7. Simulation by using MSC Nastran
1) Pre-processing First, the geometry is defined and discretised into a suitable finite element mesh using a variety of
meshing tools by defining the shapes of element. Generate the model material and model properties.
2) Solution In transient problems, we must define the loading history and define control parameters for the solution
by using the model function tools before creating a load or material properties. One must take into consideration is
the initial conditions where it use to define the temperature starting point for a transient analysis where these node
points must match the boundary condition temperature at time equal to zero.
3) Post- processing Before analyze, we need to define the initial time step size and total number of time steps by
done using the File Analyze operation to specifying the total solution time in the analysis control tools.
RESULTS AND DISCUSSIONS
The temperature distribution as a function of time inside a rectangular slab with a dimension of 0.2m x 0.1m was
considered in the analysis. The first configuration was using 32 elements while the second configuration was using
200 elements with different delta time, Δt. Material properties used in the formulation are tabulated in Table 1.
Table 1 Material properties
Material property value
Symbol
Material Building Slab
Heat conduction coefficient
(W/mC)
k
0.69
Convection Coefficient
(W/m2C)
hc
100
Density
(kg/m3)
ρ
1600
Specific Heat
(J/kgC)
c
0.84
Ambient Temperature
( oC )
Tamb
50
Initial Temperature
(oC)
Ti
350
Figure 2: Finite element mesh
⎯ 637 ⎯
1. Comparison between MSC Nastran (MSC) and Matlab (ML) for Temperature versus Time for 32 and 200
Elements at Δt =0.4s Figs. 2 and 3 showed a comparison between MSC Nastran and Matlab for temperature versus
time with 32 elements at Δt=0.4s and 5s. 3 nodes were selected for comparison, which was Node 5-ML, Node 23ML and Node 41-ML for Matlab simulation tools and Node 5-MSC, Node 23-MSC and Node 41-MSC was from
MSC Nastran simulation tools.
Figure 3: Comparison between MSC Nastran and Matlab for temperature versus time at 3 different nodes
(32 Elements) at Δt =0.4s
From Fig. 3, with 32 elements, Δt =0.4 s showed not much difference as most of the point located at the same
exponential curve line for both MSC Nastran and Matlab simulation tools. It achieved saturation at 213oC for Node
5-ML and Node 5-MSC, 175oC for Node 23-ML and Node 23-MSC and 67oC for Node 41-ML and Node 41-MSC.
There were no significant differences between the results either from MSC Nastran or Matlab.
Table 2: Quadrilateral Error Calculation for Node 5, 23 & 41 (32 elements), delta t =0.4 s at time = 40 s
Node
MSC Nastran (T)
Matlab (T)
Error %
Node 5
(0.1,0,0)
213.1911
213.57
0.17%
Node 23
(0.1,0.05,0)
175.8192
173.93
1.07%
Node 41
(0.1,0.1,0)
67.5198
67.44
0.11%
Table 2 showed the quadrilateral error calculations for Δt =0.4s with 32 elements and by taking the time 40 s into
discussion, where the maximum error been captured was 1.07% for Node 23 and the minimum error was 0.11% for
Node 41. This was mainly due to internal error or variation between this two simulation tools.
From Fig. 4, it can be said that Δt =5 s, some differences can be observed at the convergence point for Node 5-ML &
Node 5-MSC and Node 23ML & Node 23-MSC. Node 5-MSC and Node 23-MSC reached the convergence point
faster for MSC Nastran in 23 seconds in comparison with Matlab, which was 36 seconds. Here we can conclude that
Matlab was using consistent time spacing to plot out the exponential curve line, different with MSC Nastran which
using non consistent time spacing to plot out the exponential curve line. It achieved steady state at 213oC for both
Node 5-ML and Node 5-MSC, 175oC for both Node 23-ML and Node 23-MSC and 67oC for Node both 41-ML and
Node 41-MSC.
⎯ 638 ⎯
Figure 4: Comparison between MSC Nastran and Matlab for temperature versus time 3 different nodes
(32 Elements) at Δt =5s
Table 3: Quadrilateral Error Calculation Nodes 5, 23 & 41 for 32 elements, delta t =5s at time = 40s
Node
MSC Nastran
(T)
Matlab
(T)
Error %
Node 5
(0.1,0,0)
213.1867
213.97
0.36%
Node 23
(0.1,0.05,0)
175.8173
174.22
0.91%
Node 41
(0.1,0.1,0)
67.51228
67.48
0.05%
Table 3 showed the quadrilateral error calculations for Δt =5s with 32 elements and time 40s where the maximum
error was 0.91% for Node 23 and the minimum error was 0.05% for Node 41. The error was even smaller than that
in Table 2 with Δt =0.4s. Whilst, Figs. 5 and 6 showed a comparison between MSC Nastran and Matlab for
temperature versus time with 200 elements at Δt =0.4s and 5s. Total of 3 nodes were taken for comparison, which
were Node 23-ML with Node 23-MSC, Node 116-ML with Node 116-MSC and Node 221-ML with Node 221-MSC
from each simulation tools.
Figure 5: Comparison between MSC Nastran and Matlab for temperature versus time for different nodes
with 200 Elements at Δt =0.4s
⎯ 639 ⎯
From Fig. 5, it can be said that by using 200 elements with Δt =0.4 s, not much significant differences were obtained
for Node 116-ML with Node 116-MSC and both achieve saturation at 175oC, reach the convergence point in
41seconds for both Node 116-ML and Node 116-MSC. Node 221-ML and Node 221-MSC are both achieve
saturation at 67oC, and reaches convergence point at 53 seconds where most of the data point located at the same
exponential curve line. For Node 32-ML and Node 32-MSC, both achieve steady state at 211oC and reaches
convergence point at 53 seconds. Fig. 5 also depicted the similar results as in Fig. 6.
Figure 6: Comparison between MSC Nastran and Matlab for temperature versus time for different nodes
with 200 Elements at Δt =5s
2. Comparison with the different delta time. Figs. 7 and 8 showed the temperature distribution with different delta
time with 32 elements and 200 elements at node 21 and 111, which were selected for comparison. From the
comparison, it can be said that with Δt = 0.4 s, it was the optimum time in this formulation compare with other delta
time.
Figure 7: Temperature distributions as function of time inside a rectangular element with 200 elements
at node 21 at different delta time.
Figure 8: Temperature distributions as function of time inside a rectangular element with 200 elements at
node 111 at different delta time.
⎯ 640 ⎯
By choosing delta time 0.4 s as the optimum value for convergence criterion, it was important to choose a proper
time step because it will reduce the CPU time. Furthermore, it also can give the flexibility of iterative computations.
CONCLUSION
In this paper, by comparing the difference and capabilities of MSC Nastran and Matlab, we can conclude that both
have the similar pattern for delta time 0.4 s for 32 and 200 elements. For delta time 5 s, MSC Nastran achieves it
steady state slightly faster in comparison with Matlab for 2 particular nodes. Both Matlab and Nastran have its own
capabilities in which Matlab is more on programming based and MSC Nastran is more on the application based
without involve in any of the complicated programming code. It can be said that by using fine mesh for rectangular
element, it will yield almost the same result. The result of the analysis showed that the error was less than 1% in
most cases by using both Matlab and MSC Nastran, which was deemed satisfactory.
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⎯ 641 ⎯