COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Promotion of Frontier Science Research by Aid of Automatic Program
Generation Technology
Bangxian Wu 1,2*, Huashan Qian 2, Shui Wan 3
1
2
3
Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing, 100080 China
Beijing Fegen Software Company, Beijing, 100098 China
Southeast University, Nanjing, 210096 China
Email: [email protected]
Abstract Based on a new DIY concept for software development, an automatic program-generating technology
attached on a software system called as Finite Element Program Generator (FEPG) provides a platform of developing
programs, through which a scientific researcher can submit his special physico-mathematical problem to the system in
a more direct and convenient way for solution. Two examples, including the solution of a thermoelastic problem with
non-Fourier heat conduction and the numerical simulation of the flow and heat transfer in heat pipes are shown in this
paper to illustrate the usage and superiority of this technology.
Key words: automatic program generation, non-Fourier heat conduction, heat pipe, thermoelastics, stabilized finite
element method
INTRODUCTION (WHY FEPG?)
With the development of science and technology at an unprecedented high speed in 21st century, many
cross-disciplines or new branches of sciences emerge. In energy and material science areas, for instance, heat transfer
under extreme conditions, non-Fourier thermoelastics problems, viscoelastic flow problems, etc. stimulates the
development of new theories. On the other hand, technical innovation is often accompanied by various new interactive
multi-physics problems to come up, which asks for more effective solution method. A general-purpose commercial
software of computational fluid dynamics (CFD), which was developed based on the classical fluid mechanics model,
fails to meet the above need. For a special physical and mathematical model (including new controlling PDE and/or
new constitutive equations), or a multi-physics problem with cross disciplines, it is a tough task for the researcher to
write program by himself. Recently, people has been dreaming of a way that can free scientists from this kind of burden
and make them be able to concentrate on the task of their specific science problem and its solution strategy [1]. The
automatic program generation technology attached on a software system called as FEPG provides a platform of
developing software, on which a scientific researcher can describe his problem in a more direct and convenient way
and then solve it with high efficiency [2].
DIY CONCEPTS IN SOFTWARE DEVELOPMENT
Just as computer hardware can be assembled with DIY (do-it-yourself) concept, it is also realizable for software. DIY
concepts for software mainly refer to that: software designed as composed of components assembled; the calculating
program is automatically generated by so-called program generator based on a couple of files written in a special
modeling language; etc. They are integrated in a system called as FEPG (Finite Element Program Generator) for
numerical computation. Its execution manner is completely different with usual. In the program generator mode, a
couple of files of describing the partial differential equations to be solved for each physical field as well as that of
describing the solving algorithm (which can be either chosen from the system library or designed by the researcher)
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Figure 1: Different software execution modes
and the solving procedures should be prepared, in stead of importing parameters to solve a problem in the normal
software mode (see Fig. 1). The system would be able to automatically generates the calculating program based on
these files and then solve the problem. Two examples are chosen in this paper to illustrate the usage and superiority of
automatic program generation technique. Both are solved by using finite element method.
THERMOELASTIC PROBLEM WITH NON-FOURIER HEAT CONDUCTION
For super rapid heating of material or micro-scale heat transfer, the finite speed of thermal propagation and the
relaxation behavior must be taken into account. The classical heat conduction theory does no longer work. Instead,
wave heat conduction theory has been widely accepted. Since 90’s of last century, different non-classical
thermo-elastic theories appear and develop rapidly. Numerical simulation becomes one of the important means for the
verification, correction and development of new theories.
In the so-called dual-phase-lag model for non-Fourier heat conduction, the correlation of the heat flux with the
temperature gradient is expressed by Eq. (1), in which two time constants (the thermal relaxation time τ 0 of heat flux
and the thermal retardation time τ 2 of temperature gradient) are included [3]:
q i + τ 0 q& i = − k (T ,i + τ 2 T&,i )
(1)
where q and T , q& and T& are heat flux and temperature and their time derivatives, respectively.
Consider a thin aluminum film irradiated by laser on one of its surfaces (see Fig. 2), the deformation of the film and the
distribution of the temperature and the stress/strain inside the film are to be investigated.
Figure 2: Thin film irradiated by
pulsed laser
Figure 3: Temperature profile and deformation
of film at 36th time step
The change of the energy density of the laser irradiated on unit surface area with time can be expressed by:
q(τ ) =
Q0τ −τ / τ p
e
, x=0
Aτ p2
(2)
where τ p is the characteristic time of the laser. τ 0 ， τ 2 and τ p are on order of μ S .
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According to the generalized terhmo-elastics theory with non-Fourier heat conduction [4], the problem is described by
the following controlling equations:
Displacement equation
ρ u&&i = (λ + μ )u j , ij + μ ui, jj − (3λ + 2μ )αT, i
(3)
Energy equation
k (T,ii + τ 2T&,ii ) = ρ c E (T& + τ 0T&&) + (3λ + 2 μ )αT0 (ε&kk + τ 0ε&&kk )
(4)
Constitutive stress/strain equation
σ ij = λε kk δ ij + 2με ij − (3λ + 2μ )α (T − T0 )δ ij
(5)
where ρ is density, ui is displacement tensor, ε ij and σ ij are strain and stress tensor, respectively. T is temperature,
λ , μ , α k and cE are Lame coefficients, linear thermal expansion coefficient, thermal conductivity, specific heat
capacity of the material, respectively.
Main features of the problem
(1) Non-Fourier heat conduction characterized by wave nature of the controlling equations, including the displacement
equation and the energy equation;
(2) Two-way coupling between temperature field and the displacement field.
The non-dimensional weak form of the above equations, which can be easily derived by aid of the method of
integration by parts and applying the Green-Gauss Theorem, are as follows:
r
r r
r
λ+μ
r
μ
&r& r
(U , w) +
(∇ ⋅ U , ∇ ⋅ w) +
(∇U , ∇w) = − (∇θ , w)
λ + 2μ
λ + 2μ
(6)
β 0 (θ&&, Q) + β 2 (∇θ&, ∇Q) + (θ&, Q) + (∇θ , ∇Q)
&& kk ), Q) − θ , X ⋅ Q dΓ −
= −δ ((Ε& kk + β 0 Ε
∫Γ
∫Γ β 2 θ&, X ⋅Q dΓ
(7)
(Σ, S ) = ( DE , S ) − (θ , S )
(8)
1
1
r
r
where （*，*）means internal product. w and Q are the weighting functions of vectorial displacement U and scalar
temperature θ , respectively. Γ1 is the irradiated surface. S is the weighting function of the stress tensor. D is the
coefficient matrix of the constitutive equation.
In using FEPG to solve this problem, a couple of files need to be prepared, mainly including PDE files for each
r
unknown variable field ( U , θ , Σ ), in which the mass matrix, damping matrix, stiff matrix and the right hand side of
the equations are concisely described, as well as a GCN file, in which the solving algorithm for each decoupled field,
and the flow chart of the solving procedure is described. For the solution of the displacement equation and the energy
equation, both containing terms with first and secondary time derivatives and space derivatives, implicit Newmark
algorithm with distributed stiff matrix is chosen from the system library. The stress value is found from the strain rate
by using Least Square method.
Figure 4:
Change of foil temperature with time
(front and back surfaces)
Figure 5:
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Swinging motion of foil with time
An example is given here for a suspended aluminum film irradiated by a laser with total heat quantity of 1.0 J. The film
is fixed at its upper end with its thickness of 50 μ m and its surface area of 5 × 5 mm2. The relaxation time of the heat
flux τ 0 and the characteristic time τ p of the pulsed laser are 1.0 μ S. The effects of relaxation parameters on the
thermo-mechanical behavior of the thin film are investigated. A couple of typical results are given in Figs. 3-5.
Fig. 3 shows the temperature profile in the film and the film deformation at 36th time step. In Fig. 4 it is shown that the
temperatures of the film on the front and the back surfaces approach to uniform. Based on the total energy
conservation, the final uniform temperature Tb of the film can be evaluated. Tb − T0 ≈ 82.4 K ，which coincides with the
data in Fig. 4 obtained by numerical simulation. Thus the correctness of the computation is indirectly verified. Fig. 5
shows the change of the non-dimensional deformation (lateral displacement) of the film with time.
FLUID FLOW AND HEAT TRANSFER IN HEAT PIPES
Heat pipes are two-phase heat transfer devices with extremely high effective thermal conductivity. Heat pipes offer an
effective alternative for removing heat without significant increases in operating temperature. With the working fluid
in a heat pipe, heat can be absorbed on the evaporator region and transported to the condenser region where the vapor
condenses releasing the heat to the cooling media. Heat pipe technology has found increasing applications in
enhancing the thermal performance of heat exchangers in microelectronics, energy and other industrial sectors, like
aerospace, solar technology and cooling of production tools, etc.
The difficulty of modeling flow and heat transfer in heat pipes lies in the coupled vapor/liquid flow and heat transfer
phenomena associated with phase change, liquid flow in wick structure and transient working conditions, etc.
Figure 6: Sketch of a heat pipe model
The sketch of a typical heat pipe model is shown in Fig. 6. A cylindrical metal pipe with a wick layer attached to the
inner wall and two closed ends contains a quantity of liquid inside. The heat is added to one end of the pipe and
extracted from the other end of the pipe. In between there is an adiabatic section. Vapor flow in the central core and
water flow in the wick constitute a closed loop. The unified steady-state conservation equations for the vapor core and
the liquid in the wick structure includes [5]:
Continuity
∇ ⋅ ( ρ u) = 0
(9)
Momentum equation
1
μu
ρu ⋅ ∇u = −∇p + ∇ ⋅ ( μ ∇u ) −
K
ε
ε
1
(10)
2
where ε and K are the porosity and the permeability of the wick structure, respectively. For vapor core, ε =1 and a big
number can be taken for K .
Energy equation
ρ c u ⋅ ∇T = ∇ ⋅ (λeff ∇T )
(11)
Heat conduction equation inside the pipe wall is a special form of Eq. (11):
∇ ⋅ ( λs ∇ T ) = 0
(12)
The boundary conditions include no-slip velocity condition and adiabatic temperature condition on the wall and the
two ends of the pipe, as well as the wall heat flux, which is related to the heat added to (extracted from) the heat pipe:
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± Q = Aλs
∂T
∂r
(13)
where A is the local surface area of absorbing (or releasing) heat on the pipe wall.
The conditions on the interface boundary between the vapor core and the liquid in annular wick include:
Energy equilibrium, which relates the phase change with the energy transfer there [6]
λ
∂T
∂T
| v −λ eff
| l = q = m& h fg = ρvh fg
∂r
∂r
(14)
where h fg is the latent heat of the vapor, λ eff is the effective heat conductivity of the wick.
Mass continuity
ρvvv = ρl vl
(15)
Thermodynamic relations for saturated vapor, which is a state equation correlating pressure p with temperature T
P
1
1
R
=
−
ln int
Tint T0 h fg
P0
(16)
The condition on the wick / wall interface is the equality of the heat fluxes across the interface:
u = v = 0, λ eff
∂T
∂r
= λs
l
∂T
∂r
(17)
s
Main Features of the problem
(1) Two phase flow with phase change in which the absorption and the release of the latent heat of vaporization of the
working fluid are taken into account;
(2) Coupled multi-physics fields, including velocity fields of vapor and liquid, temperature field, heat flux field;
(3) Different regions with not only different material properties, but also different momentum equations for the flows
in vapor core and in porous medium;
(4) Conjugate heat transfer including the convection in the porous wick and the conduction inside the pipe wall.
(5) For the solution of the vapor flow field in a heat pipe, it is necessary to overcome the numerical difficulty resulted
from dominated convection and the incompressibility. A certain stabilization measure should be taken to prevent the
pressure oscillation. The Galerkin-Least Square (GLS) method is adopted here for solving incompressible
(a) Flow chart of heat pipe computation
(b) A segment of GCN file
Figure 7: Flow chart of heat pipe and its counterpart in GCN file
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Navier-Stokes equations. Three additional stabilization terms are added in the standard Galerkin formulation.
In FEPG system, GCN file describes the solving algorithm for each decoupled variable field as well as the solving
procedures for the problem to be solved. In Fig. 7, a segment of GCN file is listed on the right side, which corresponds
the flow chart on the left side.
The interface conditions plays the key role in the coupling between the momentum field ( u v p ) and the temperature
field T in the present numerical model of heat pipe. In the above flow chart, cycles 2 and 3 represent the iteration
process due to the non-linear convection term in the momentum equation. Cycle 1 completes the coupling between
temperature field and the velocity field. Newton-Raphson iteration or simple iteration can be chosen. The relaxation
factor can be automatically adjusted in FEPG system according to the relative change trend of the residue of the
equation at each step.
The weak form of the integrated continuity and momentum equation of vapor is derived from Eqs. (9) and (10):
ρ ( w, u ⋅ ∇u ) − ( ∇ ⋅ w, p ) + (q, ∇ ⋅ u ) + μ (∇w, ∇u )
+
∑ τ su (u ⋅∇w, Ru ) + ρ ∑ τ ps (∇q, Ru ) + ∑ ρτ ls (∇ ⋅ w, ∇ ⋅ u) = 0
1
e
e
(18)
e
∫
where (*,*) denotes internal product. That is, (a, b) ≡ a ⋅ b dΩ , where a, b can be two scalars, two vectors or two
tensors. w is the vectorial weighting function of u . q is the scalar weighting function of p . Ru is the residue of the
momentum equation:
Ru = ρu ⋅ ∇u − μ∇ 2 u + ∇p
(19)
There are three stabilized terms on the second line of Eq. (18). In GLS method for solving incompressible flow
problems, the stabilized terms play the role in overcoming the numerical instability resulted from advection
domination and circumventing LBB consistent condition when using equal order of interpolation for pressure p and
velocity v . The evaluation method for three stabilization parameters τ su , τ ps , τ ls proposed in paper [8] is adopted in
the present work with some modifications. The effect of flow direction on the selection of the element length is
accounted for.
τ su = τ ps =α h / 2 | u |
(20)
τ ls = αh | u | / 2
(21)
where α = coth Pe − 1 / Pe accounts for the comprehensive effect of the local convection and diffusion.
Peclet number Pe , defined as the ratio of the local convection over the diffusion, is the indicator of convection
domination degree in the discretization.
Pe = ρ | u | h / 2μ
(22)
h is the measure of the element length.
PDE file for each variable field is written following the weak form equation. In Fig. 8 is listed the coefficient matrix
expression in the PDE file for vapor velocity field. It can be found that the file is easy to understand.
Figure 8: Coefficient matrix expressed in PDE file for vapor velocity field
Since the heat flux occurs on the pipe boundary, a FBC file is needed for specifying the boundary condition. Other files
include a GIO file for defining the main information on fields and elements, a data file (PRE file), and algorithm files
(SOLV file), chosen from the system library.
Numerical simulation is conducted as an example for a small cylindrical heat pipe with its length of 0.6 m and diameter
of 0.012 m. The working medium is water and the working temperature is 100ºC. Some typical results with heat
loading 200 W are shown in Figs. 9-12. The change of the vapor temperature and pressure along the vapor/liquid
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interface are shown in Figs. 9-10, separately. In Figs. 11-12 are shown the axial and radial velocity profiles of vapor at
different axial positions. Negative radial velocity implies the incoming of vapor from interface due to heat absorbing
from the outer surroundings. And a positive value of the vapor radial velocity corresponds to the outgoing of vapor in
the heat release section of the pipe. The numerical results are fairly reasonable.
Figure 9: Vapor temperature distribution on interface
Figure 11: Profile of axial vapor velocity
Figure 10: Vapor pressure distribution on interface
Figure 12: Profile of radial vapor velocity
SOMETHING BEYOND
For the thermoelastic problem with non-Fourier heat conduction, heat flux correlates with temperature gradient by Eq.
(1), in which relaxation time τ 0 of the heat flux and retardation time τ 2 of the temperature gradient are included.
Similarly in viscoelastic flow problems, such as the solution of polymer melt flow in a contracted channel, the stress is
correlated with the strain rate by Eq. (23) in a form very similar with Eq. (1):
τ + λ1 τ (1) = η 0 (γ (1) + λ 2 γ ( 2 ))
(23)
where τ and γ (1) , τ (1) and γ (2) are the stress and the strain rate, and their respective time derivatives, respectively.
λ1 and λ 2 are the relaxation time and retardation time, respectively, by which these two constitutive models in
different disciplines are distinguished with the classical theory of heat conduction and the classical theory of fluid
mechanics for pure viscous flow problems. From the similar equations, however, new controlling partial differential
equations with wave nature result for the former, and a new constitutive equation of hyperbolic nature with a
convective term results for the latter. Therefore, in accord with the need of technology development, modern science
brings us to different time scales with normal, at that scale the behavior and the process behind the observed
phenomena follows new physical mechanisms. For the effective solution of, e.g. viscoelastic flow problems, or for the
development of a new theory, the test of different formulations and different solution strategies and schemes is often
inevitable [9]. A lot of special unique features of FEPG make it a powerful tool for frontier sciences often associated
with new physical models including novel constitutive equations and/or covering interactive multiphysics phenomena.
CONCLUSION
Through the illustration of the above two examples, the general idea on the usage and the superiority of the automatic
program generation technique in frontier science research can be obtained. FEPG represents a new generation of
software in the sense that numerical work can be done more directly on an easy-to-understand interface, through which
a scientist is able to ‘tell’ the generator to mechanically translate mathematical descriptions into the corresponding
code [1]. As an open platform, its flexibility greatly extends users’ problem-solving ability in solving problems with
new physical models or new constitutive equations, in adopting new calculating algorithms and schemes, and in
solving complicated multiphysics problems.
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REFERENCES
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J. J.A, 2003; 46: 126-130.
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Jiaotong University Press, Xi’an, China, 2005 (CD-ROM).
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