HFF 25,4 Effect of boundary condition approximation on convergence and accuracy of a finite volume 950 discretization of the transient Received 10 February 2014 Revised 12 August 2014 heat conduction equation Accepted 13 August 2014 Martin Joseph Guillot Department of Mechanical Engineering, University of New Orleans, Louisiana, USA, and Steve C. McCool Department of Engineering, Novacentrix, Inc., Austin, Texas, USA
Abstract Purpose – The purpose of this paper is to investigate the effect of numerical boundary condition implementation on local error and convergence in L2-norm of a finite volume discretization of the transient heat conduction equation subject to several boundary conditions, and for cases with volumetric heat generation, using both fully implicit and Crank-Nicolson time discretizations. The goal is to determine which combination of numerical boundary condition implementation and time discretization produces the most accurate solutions with the least computational effort. Design/methodology/approach – The paper studies several benchmark cases including constant temperature, convective heating, constant heat flux, time-varying heat flux, and volumetric heating, and compares the convergence rates and local to analytical or semi-analytical solutions. Findings – The Crank-Nicolson method coupled with second-order expression for the boundary derivatives produces the most accurate solutions on the coarsest meshes with the least computation times. The Crank-Nicolson method allows up to 16X larger time step for similar accuracy, with nearly negligible additional computational effort compared with the implicit method. Practical implications – The findings can be used by researchers writing similar codes for quantitative guidance concerning the effect of various numerical boundary condition approximations for a large class of boundary condition types for two common time discretization methods. Originality/value – The paper provides a comprehensive study of accuracy and convergence of the finite volume discretization for a wide range of benchmark cases and common time discretization methods. Keywords Heat conduction, Finite volume, L2 convergence, Photonic curing Paper type Research paper
Introduction The heat conduction equation has many applications in engineering and physics. While analytical solutions exist for a number of cases, most problems of engineering interest do not have analytical solutions, and so numerical methods are applied to obtain approximate solutions. Commonly used methods include the finite difference, finite element, and finite volume methods (Pletcher et al., 2013), with each method having advantages and disadvantages. The finite difference method is easiest to International Journal of Numerical Methods for Heat & Fluid Flow implement, but is limited to structured or block-structured domains, making it difficult Vol. 25 No. 4, 2015 to implement on complex geometries. The finite volume and finite element methods are pp. 950-972 © Emerald Group Publishing Limited more difficult to code, but offer greater flexibility to model arbitrary geometry, and in 0961-5539 DOI 10.1108/HFF-02-2014-0033 implementing boundary conditions. The current authors previously developed a thermal simulation tool using the finite Effect of volume method with a fully implicit time discretization of the transient heat conduction boundary equation to simulate a newly emerging manufacturing process termed “photonic curing” (Guillot et al., 2012). Photonic curing is a process that uses short duration pulses of light condition from xenon flashlamps and forced convective cooling to quickly (on the order of approximation milliseconds) thermally process thin films, such as the sintering of metal inks. Typically, these films are processed on low temperaturesubstratessuchasplasticandpaper. 951 First described by Schroder et al. (2006) and Schroder (2011), photonic curing is rapidly finding many applications in the manufacture of flexible electronics and in other manufacturing processes. Typical printed electronics applications include RFID tags, photovoltaics, large area displays, “electronic” paper and disposable sensors (Zardetto et al., 2011). Printed electronic circuits are manufactured using a range of materials and processes, butonecommonmethodofformingthe conductors is to use a printer to deposit silver or copper nanoparticles suspended in solvents and binders onto inexpensive flexible substrates such as polyethylene terephthalate (PET), polyethylene naphthalate (PEN), or even paper. The metal ink circuit patterns are then sintered to make them highly conductive. Inexpensive substrates such as PET have a maximum working temperature of only about 1501C. Because of this limitation, traditional sintering methods such as an oven or IR heater that bring the entire film stack (ink pattern and substrate) into thermal equilibrium with the surroundings are constrained by the maximum working temperature of the substrate. This limitation forces a long processing time – often minutes, which means that either the oven is physically large or the throughput is limited by the oven. Either technique is economically undesirable. The pulsed light produced during photonic curing produces a time-varying heat flux that can act as a surface heat flux or a volumetric heat source, depending on the absorption properties of the material. Materials with high absorption coefficients absorb the energy near the surface, and then diffusion is the primary mechanism to transport the energy into the stack. Materials with low absorption coefficients absorb the energy volumetrically throughout the stack, and diffusion then redistributes the energy. Forced convective cooling is used in between pulses to cool the film and reduce heat flow into the substrate. The combination of pulsed radiant heating and convective cooling makes it possible to heat a thin film on a low temperature substrate to a temperature far beyond the maximum working temperature of the substrate without damaging the substrate. The process requires that the time at high temperaturebeveryshort– of order a few ms. The peak pulse power, pulse length, number of pulses, rate of convective cooling, and pulse repeat frequency can be independently varied to achieve desired thermal profiles throughout the film and substrate. Additionally, the phonic curing hardware allows the temperature at the bottom of the substrate to be clamped at a specific temperature using a vacuum chuck and bringing the bottom surface of the substrate into contact with a constant temperature mass. Because of the large parameter space, as well as material stack combinations affecting the temperature profile, it is critical to have a fast running simulation tool that can provide accurate simulations of photonic curing processes, where the user can vary the many parameters that affect the results, and quickly view the resulting temperature profiles. This allows users to quickly zero in on the parameters that produce the desired temperature profile in a film stack for a given process. The simulation tool previously developed couples the finite volume discretization to a graphical user interface written HFF in LabView®, and a database of thermally dependent material properties to which the 25,4 user can add user defined materials. The geometric, material, and pulse parameters are all specified using the interface, and are the same for the simulation as for the hardware settings of the equipment used in the actual photonic curing process. The output is a graphical time-temperature history in all locations of the stack viewable within the same 952 interface. The goal of the tool is to have the simulation be as fast as possible without sacrificing accuracy, so that the user can see the effects of parameter changes on the thermal profile in near “real-time” fashion. This allows the user to much more easily shape the thermal profile in a film stack to achieve the desired process results and to determine the settings to use for the photonic curing equipment. To accomplish this, it is important to know how the various approximations to boundary conditions and time stepping methods affect solution convergence rate and accuracy, and simulation run time for each of the boundary condition types encountered in photonic curing. To achieve the goal of developing an accurate fast running simulation tool, a systematic study of the various boundary condition approximations and time stepping methods of a finite volume discretization of the transient 1-D heat conduction equation was undertaken in order to determine which combination of time stepping method and boundary condition approximation produced the most accurate results with the least computational effort. The most common time discretization methods applied to the heat conduction equation are fully implicit and Crank-Nicolson. While both methods are unconditionally stable, the fully implicit method is only formally first-order time accurate. The Crank-Nicolson method is formally second-order time accurate, but can suffer from unphysical oscillations when there is a discontinuity in the initial conditions or between the initial conditions and the boundary conditions, due to the amplification factor of the highest frequency component of the error being near unity (Britz et al., 2003). There is a large volume of literature describing the various methods applied to transient heat conduction equation (e.g. Blackwell and Hogan, 1993; Al-Odat et al., 2013; Charoensuk and Vessakosol, 2010; Tian et al., 2005; Zarghami, 2014; Hor-Yen et al., 2013), and it is well known that the finite volume method using central differences for the derivative is spatially second-order accurate. However, there is surprisingly little in the literature that systematically discusses the various boundary condition approximations and volumetric heating coupled with time stepping methods, and quantifies their effect on convergence rate and solution accuracy. Makenzie and Morton (1992) discuss solution error and convergence properties of a finite volume discretization of the steady convection-diffusion equation for two cases involving Dirichlet boundary conditions. Berg and Nordstrom (2013) discuss convergence properties for an incompletely parabolic system using the finite difference method subject to Dirichlet and heat flux boundary conditions. Cho et al. (2002) present convergence in the L2-norm for the transient heat conduction equation subject to convection boundary conditions using the finite element method with Crank-Nicolson time discretization. They show the dependence of time step and mesh size for maintaining theoretical convergence under mesh refinement. Eymard et al. (2000) demonstrate second-order convergence in the L∞-norm of the transient heat conduction equation, including source term, but limited to Dirichlet boundary conditions. Hesthaven and Warbution (2008), apply an explicit discontinuous Galerkin finite element discretization to the 1-D transient heat conduction equation. They demonstrate sub-optimal convergence when polynomials of odd order are used, Effect of e.g. linear polynomials display first-order convergence. The analysis is limited to boundary homogenous Dirichlet boundary conditions. Jaluria (2010) used experimental data combined with numerical simulation to assess the accuracy of thermal process condition simulation methods. approximation The goals of this effort were to investigate the fully implicit and Crank-Nicolson time stepping methods with the full range of boundary conditions needed to simulate 953 photonic curing, and the various approximations to the boundary conditions, in order to compare the convergence rates in the L2-norm for highly resolved discretizations, and to quantify the local error and computation run times for two cases representative of photonic curing for coarse discretizations. The objective was to determine mesh and time step requirements to obtain specified accuracy (e.g. 2 percent) for the representative cases. In principle, the Crank-Nicolson method, because it is second-order accurate in time, allows larger time steps than the implicit method while retaining similar accuracy. However, the increased accuracy comes at increased computational effort due to the greater complexity of the tridiagonal coefficients in the discrete equation. So the question is whether the additional accuracy of the Crank-Nicolson method allows a large enough time step to be taken when compared with the implicit method to overcome the additional computational expense per time step, so that total simulation time is reduced. The first part of the study quantifies the convergence rates in the L2-norm for highly refined meshes and times steps, applying the various boundary conditions and time stepping methods for several cases where analytical solutions are available. The second part of the study quantifies local solution error and CPU time for two cases representative of photonic curing using coarse mesh spacing and time steps.
Heat conduction equation The 1-D transient heat conduction equation with volumetric source heating in a domain 0 ⩽ x ⩽ L is written as: @T @ @T rc ¼ k þq000 (1) P @t @x @x 3 where ρ(kg/m ), cP(J/kg-K), and k(W/m-K) are the density, specific heat at constant pressure, and thermal conductivity, respectively, of each material, and q000(W/m3)is a volumetric source term. In general, the properties are temperature dependent. 2 The thermal diffusivity is defined as α ¼ k/ρcp (m /s).
Boundary and initial conditions Photonic curing simulation involves several boundary condition types, including Dirichlet, Robin, and Neumann, which correspond to prescribed temperature, convection, and heat flux boundary conditions, respectively. They are written as:
TB ¼ TBðÞt Dirichlet 7 dTB ¼ ðÞ k dx h1 TB T1 Robin (2) 00 ðÞ¼ dTB qB t k dx Neumann HFF where h∞ is the convective heat transfer coefficient, TB is the temperature on the 25,4 boundary and T∞ is the ambient temperature. For the Neumann boundary condition, heat flux is positive in the positive x direction and vice versa. The sign of the (±)in the Robin boundary condition is (+) on the left boundary (top surface) and (−)onthe right boundary (bottom surface). The initial temperature distribution T(x,0) ¼ Tinit(x) is known. 954 Finite volume discretization Equation (1) is spatially discretized using the finite volume method. The domain is divided into N finite volumes (cells) and Equation (1) is integrated over each volume, with average cell temperatures defined at the cell centers. The domain discretization and nomenclature are illustrated in Figure 1. The cell centers are located at xi, and the cell boundaries are located at xi ± 1/2.Thegridspacingis defined by: D ¼ ; D ¼ xi xi þ 1=2 xi 1=2 xi 7 1=2 xðÞi þ 1=2 7 1=2 xðÞi 1=2 7 1=2 (3) Integrating Equation (1) over a typical finite volume produces the semi-discrete equation, written as: "# n þ 1 n þ 1 r D dTi ¼ y dT dT þ 000 n þ 1D cp x i k k qi xi dt dx i þ 1=2 dx i 1=2 "# n n þðÞ y dT dT þ 000 nD 1 k k qi xi (4) dx i þ 1=2 dx i 1=2 The current time level is denoted by n and the new time level by n+1. A value of θ ¼ 1 corresponds to the fully implicit method, and θ ¼ 1/2 corresponds to Crank-Nicolson. Discretizing the time derivative, and using central differences for the spatial derivatives in Equation (4) yields the fully discrete set of equations for the interior cells, 2 ⩽ i ⩽ N−1, written in general form as: n þ 1 þ n þ 1 þ n þ 1 ¼ n þ n þ n þ AiTi 1 BiTi CiTi þ 1 DiTi 1 EiTi FiTi þ 1 Gi (5) Defining:
k 7 = Dt b ¼ i 1 2 i 7 1=2 ðÞr D D (6) cP i xi xi 7 1=2
xi –1/2 xi +1/2
xi –1 xi xi +1 Figure 1. Δ Finite volume xi discretization Δ Δ of 1-D domain xi –1/2 xi +1/2 then: Effect of ¼ yb ; boundary Ai i 1=2 condition ¼ þyb þyb ; approximation Bi 1 i 1=2 i þ 1=2 ¼ yb Ci i þ 1=2 955 ¼ ðÞ y b ; Di 1 i 1=2 ¼ ðÞ y b þb ; Ei 1 1 i 1=2 i þ 1=2 ¼ ðÞ y b Fi 1 i þ 1=2 q000Dt G ¼ i (7) i r cp i
Numerical implementation of boundary conditions The boundary cells must incorporate the given boundary conditions. In every case, the spatial derivative appearing in Equation (4) on the boundary is approximated with an expression that incorporates the boundary condition information. In this study we will impose heat flux boundary conditions on the right boundary. In some cases we impose an adiabatic boundary condition on the right boundary, and in others we use the analytical solutions to apply an appropriate heat flux boundary condition in order to simulate finite layers. We study the effect of boundary condition implementation by using various boundary condition implementations on the left boundary. The left boundary mesh is shown in Figure 2. The Dirichlet boundary condition can be imposed by writing a one sided difference for the derivative appearing on the left boundary and applying the known temperature on the boundary, or by imposing a convective (Robin) boundary condition, setting the ambient temperature to the desired surface temperature, and setting the convection coefficient to an arbitrarily high number. This is known as a penalty method, or weak implementation, and has the advantage that the numerical boundary condition does not require separate coding for Dirichlet and convective boundary conditions (Becker et al., 1981). The Neumann boundary condition is applied by replacing −kdT/dx on the 00 boundary with the known value of the heat flux, qB, and, thus, requires no spatial approximation. One sided difference expressions for the boundary derivative can be first- or second-order spatially accurate. In principle, since the overall method is spatially second-order accurate, it is best to use a second-order accurate expression on the boundary as well. However, it is not uncommon to see the derivative approximated
TB T1 T2
x x1 x x2 1–1/2 1+1/2 Figure 2. Δ Δ x1 x2 Left boundary mesh HFF with a first-order expression, (Versteeg and Malalasekera, 2007). We use both, and 25,4 show that the first-order accurate expression can display second-order spatial convergence under certain restrictions. The first-order accurate expression is simply: dT T T ¼ 1 B þO½ Dx (8) D = dx 1 1=2 x1 2 956 The second-order accurate expression involves TB, T1, and T2, and can be derived by fitting a second-order polynomial through the three points, and then taking the derivative of the polynomial at the boundary. The expression is written as: dT 8=3T þ9=3T 1=3T ¼ B 1 2 þ D 2 D O x (9) dx 1 1=2 x1 It is noted that the first-order accurate expression is not restricted to constant spacing meshes, whereas the second-order accurate expression assumes that Dx2 ¼ Dx1. Equation (9) can easily be extended to non-constant spacing, but then the coefficients become expressions involving Dx1 and Dx2. Although the finite volume method developed for the study does allow variable spacing, we restrict the results in this effort to constant mesh spacing since we are primarily interested in investigating the boundary condition approximations and time stepping methods. For the Dirichlet boundary condition, the surface temperature, TB is known. However, for the convection boundary condition, the surface temperature is unknown, and so an expression must be developed for the surface temperature. Two approaches are investigated. The first is to approximate the derivative on the boundary with a first- or second-order expression, as given by Equations (8) and (9), respectively, substitute that into the boundary condition given by the second expression in Equation (2), and then solve for the boundary temperature, TB. This value of TB is then used in the boundary condition. The second approach is to extrapolate the temperature values of the first two cells to the boundary. In each case, the surface temperature can be written in the form:
TB ¼ a1T1 þb1T2 þd1T1 (10)
where the coefficients for a1, b1 and d1 for each approximation are given in Table I. Then the Robin boundary condition can be written as: dT k ¼ h1ðÞa1T1 þb1T2 þðÞd1 1 T1 (11) dx 1 1=2
Coefficient Boundary condition hc1 c2 c3 c4 a1 b1 d1 Dirichlet first order 0 2 0 2 0 0 0 0 Table I. Dirichlet second order 0 3 1/3 8/3 0 0 0 0 Coefficients for left Neumann 0 0 0 0 1 0 0 0 D þ D Dx1 2 x1 x2 0 Robin Extrapolation h∞ 00 00 Dx1 þ Dx2 boundary cell for Dx1 þ Dx2 k h1Dx1=2 given boundary First order h∞ 00 00 0 k þ h1Dx1=2 k þ h1Dx1=2 condition D 8=3k 1=3k h1 x1 approximation Second order h∞ 00 00 = þ 1D 8=3k þ h1Dx 8=3k þ h1Dx1 8 3k h x1 1 This is substituted into Equation (4) to develop the discrete equation for the Effect of surface cell. The tridiagonal coefficients for the boundary cell are written in a general boundary form as: condition A1 ¼ 0 approximation ! D ¼ þy b þb þ ha1 t B1 1 c1 1 1=2 1 þ 1=2 957 rcp Dx1 1 ! hb Dt C ¼ y c b þb 1 1 2 1 1=2 1 þ 1=2 r D cp 1 x1 D ¼ 0 1 ! D ¼ ðÞ y b þb þ ha1 t E1 1 1 c1 1 1=2 1 þ 1=2 rcp Dx1 1 ! D ¼ ðÞ y b þb hb1 t F1 1 c2 1 1=2 1 þ 1=2 rcp Dx1 1 000n þ 1 00 q Dt c q þðÞ1 d hT1 Dt G ¼ c b T þ 1 þ 4 B 1 (12) 1 3 1 1=2 B ðÞrc r D P 1 cp 1 x1 The coefficients for the various boundary conditions are also given in Table I.
Test cases The effects of solution method and boundary condition implementation on convergence rate and solution accuracy are investigated using five cases: constant temperature; convection; constant heat flux; time-varying heat flux; and volumetric heating. All cases are a single layer of copper initially at uniform temperature, Tinit ¼ 251C. The properties of copper are assumed to be constant, with thermal conductivity, 3 k ¼ 401 W/m-K, density, ρ ¼ 8,960 kg/m , and specific heat, cp ¼ 382.5 J/kg-K. Various thicknesses and simulation times were chosen for the different cases, and are representative of various photonic curing processes. The analytical solution for Case 1 is found in (Carslaw and Jaeger, 1959), and the analytical solutions to Cases 2-4 are found in (Beck, 1992). Case 5 is a spatially exponentially varying volumetric source with convection boundary conditions. The exponential source is in the form of Beer’s law, which is written as: Zx I x ¼ I sðÞ1 m e (13) 2 where Ix is the intensity (W/m ) at location x, Is is the intensity at the surface, η is the absorptivity (m 1), and μ is the reflectance. In this work we assume μ ¼ 0. Differentiating Equation (13) gives the volumetric heating, q000, in Equation (1) as: 000 Zx q ¼ I sZðÞ1 m e (14) Sahin (1992) developed a solution to this case using separation of variables to separate the solution into steady and unsteady components. The volumetric source term is included in the steady solution. The unsteady solution reduces to a convection problem with an exponential initial temperature distribution. HFF Cases 1 and 2 are used to determine the effect of various boundary condition 25,4 approximations on the convergence in the L2-norm. Cases 3 and 4 apply Neumann boundary conditions, which require no approximations, and so only compares time stepping methods. Case 4 provides a time-varying surface heat flux, whereas Case 5 represents the situation where materials absorb the energy deposited by the pulsed lamps though volumetric heating as opposed to surface heating. 958 For these cases, we look at the local error using coarse meshes and time steps to determine minimum required discretization parameters that will still achieve desired accuracy.
Case 1 – constant temperature (Dirichlet) The first case applies a constant temperature boundary condition on a finite layer. Defining non-dimensional variables, θC ¼ (T−Ti)/(T0−Ti), the Fourier number 2 Fo ¼ (α/L )t, ξ ¼ x/L, and λn ¼ (n+1/2)π, the analytical solution is written as:
X1 ðÞ l y ðÞ¼; exp nFo ðÞl x T x t 1 2 l sin n (15) n¼0 n
The temperature on the left boundary is 1,0001C for tW0. Adiabatic conditions are imposed on the right boundary. The numerical computations were performed on a50μm finite layer and the total simulation time was 5.0 μsec. The solution using 15,625 time steps on an 800-cell mesh is shown in Figure 3. Thefigure shows the results using the implicit method and first-order accurate boundary condition, but the results using the Crank-Nicolson method and other boundary conditions are virtually identical as viewed at this scale. It is noted that the Crank-Nicolson method produces spurious oscillations in the solution if steps are not taken to damp them. The oscillations are damped using the implicit method for one time step, and then switching to the Crank-Nicolson method. Britz et al. (2003), note that this technique damps the oscillations while retaining second-order time accuracy. It was found in this study that this method works very well for this case. Without using the implicit method for the first time step, large spurious temporal oscillations in the solution occurred near the left
1,200
Case 1: Constant Surface Temperature, Ts = 1,000°C Solution at 0.2, 1, 3, 5μsec 1,000 t = 0.2μsec also solution to Case 2b
800 Numerical Analytical 600
400 0.2 1.0 3.0 5.0
200 Temperature, T, (°C) T, Temperature, 0
Figure 3. –200 Case 1 solution: –5 0 5 10 15 20 25 30 35 40 45 50 55 constant surface Depth, x, (microns) temperature Note: Numerical and analytical solutions boundary. However, when damping was applied, the oscillations were completely Effect of damped, and the Crank-Nicolson solution method performed very well, as we show boundary when discussing the convergence in the L2-norms. condition approximation Case 2 – convection (Robin) Case 2 is a semi-infinite solid with a convection boundary condition applied to the 6 2 left end. Case 2a applies a finite convection coefficient, h∞ ¼ 10 × 10 W/m and 959 ambient temperature, T∞ ¼ 1,000 1C. These conditions produce a time dependent temperature on the left boundary. Case 2b approximates a constant temperature boundary condition weakly by specifying a convection boundary condition with heat 20 transfer coefficient, h∞ ¼ 10 , and setting the ambient temperature to 1,000 1C. The total simulation time is 1 μsec. In both cases, adiabatic conditions are imposed on the right boundary. The solution to Case 2a is shown in Figure 4. The solution to Case 2b is the same as Case 1, but limited to the first 1 μsec. The numerical simulations are run on a finite layer of 50 μm, but the simulation time is such that thermal effects do not reach the right boundary, and so the results can be compared to the semi-infinite solution. Defining the Biot number as Bi ¼ hL/k, the analytical solution is given by: x x y ðÞ¼x; t erfc exp BixþFo Bi2 erfc þFo1=2 Bi (16) c 2Fo1=2 2Fo1=2 where θc ¼ (T−Ti)/(T∞−Ti).
Case 3 – constant heat flux (Neumann) 00 ¼ ; = 2 Case 3 is a finite layer solid with constant surface heat flux of qs 2 000 kW cm run for a simulation time of 3.0 μsec. The numerical simulations are run on a finite layer of 15 microns. The analytical solution for a semi-infinite solid is given as: pffiffiffiffi ! ! 2 at 1 x2 x x TxðÞ ; t T ¼ q00 exp erfc (17) i k s p1=2 4Fo 2Fo1=2 2Fo1=2
For this case, in order to simulate a finite layer, we apply the heat flux as given by the analytical solution to the numerical simulation on the right boundary. The temperature
250 Case 2a: Convection BC, h∝=10E6 Solution at 0.2, 0.4, 0.6, 0.8, 1.0μsec
200 Numerical Analytical
150
100
Temperature, T, (°C) T, Temperature, Figure 4. Case 2a solution: 50 convection boundary condition, 0 5 10 15 20 25 30 35 40 45 50 6 2 h∞ ¼ 10 × 10 W/m -K Depth, x, (microns) HFF distribution on an 800-cell mesh using 15,625 time steps is shown in Figure 5 at 25,4 0.6 μsec intervals.
Case 4 – time-varying heat flux – (Neumann) Case 4 is a three pulse case that is representative of photonic curing, i.e., repeated short pulses, except that we do not consider convection in between pulses. The pulse length 960 for this simulation was 500 μsec with a pulse repetition frequency of 500 Hz. The simulation was performed on a 5,000 micron copper layer with a pulse intensity of 00 ¼ = 2 μ qs 210 kW cm and total simulation time of 9,000 sec. The case consists of periods of pulsed heating followed by periods of adiabatic diffusion. Although the adiabatic periods are not technically “cooling,” but rather a redistribution of temperature due to diffusion, we refer to these periods as cooling because the surface temperature cools during those times. While there is no purely analytical solution for this case, semi-analytical solutions for semi-infinite solids can be found using Green’s function solutions. Linearity of the heat conduction equation allows the solution to be computed as a superposition of two solutions. During heating cycles, the problem can be separated into two problems. The first is a constant heat flux boundary condition with uniform initial conditions, whose solution is denoted by Tq(x,t), and the second is an adiabatic evolution of temperature from a non-uniform initial temperature distribution, whose solution is denoted by Tc(x,t). During cooling cycles the problem is strictly an evolution of temperature from an initial non-uniform temperature distribution. During the heating cycles the solution to the heat flux problem, Tq(x,t), is purely analytical and is given by Equation (17), and during the cooling cycles Tq(x,t) ¼ 0. The solution to the evolution of the temperature, Tc(x,t), from non-uniform initial conditions, ToðÞx; to , beginning at time to is given by the Green’s function solution, (Carslaw and Jaeger, 1959), and is written as: Z "#! ! 1 L ðÞx x0 2 ðÞxþx0 2 T ðx; tÞ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T ðÞx; t exp þexp dx0 (18) c o o aðÞ aðÞ 4apðÞt to 0 4 t to 4 t to
Equation (18) is integrated numerically using eight-point Gaussian quadrature. Tc(x,t) is found by integrating forward using initial conditions at the beginning of each new heating cycle. During the first heating cycle Tc(x,t) ≡ 0 because the problem begins from
1,400 Case 3: Constant Heat Flux: q=2,000kW/cm2 μ 1,200 Solution every 0.6 sec Total simulation time: 3.0μsec
1,000 Analytical 800 Numerical
600
400
Figure 5. (°C) T, Temperature, Case 3 solution: 200 constant surface heat flux, 0 q ¼ 2,000 kW/cm2 –5 0 5 10 15 20 Depth, x, (microns) uniform initial conditions, and so during the first heating cycle, the “semi” analytical Effect of solution is purely analytical and is equivalent to Case 3. The solution to Case 4 is shown boundary in Figures 6 and 7. Figure 6 shows the surface time history and Figure 7 shows the temperature distribution within the layer at the midpoints of the heating pulse (P) and condition cooling cycles (C). approximation
Case 5 – exponentially varying volumetric source with convection 961 Sahin (1992) writes the solution to this problem as: x pffiffiffi x yxðÞ¼; t erfc pffiffiffi exp Bi2tþBix erfc Bi tþ pffiffiffi 2 t 2 t Bi Biþ1 x þ erfc pffiffiffi expðÞ x c Bi 2 t 1 Biþ1 pffiffiffi x expðÞtþx erfc tþ pffiffiffi 2 Bi 1 2 t 1 pffiffiffi x þ expðÞt x erfc t pffiffiffi 2 2 t 1 pffiffiffi x þ exp Bi2tþBix erfc Bi tþ pffiffiffi (19) BiðÞ Bi 1 2 t 2 where θ is defined as for Case 2, ξ ¼ ηx, τ ¼ αη t, Bi ¼ h/ηk, and c ¼ h(T∞−To)/Is(1−μ). The simulation conditions were identical to Case 3, except that the incident flux was absorbed volumetrically rather than as a surface flux. The temperature distribution is shown in Figure 8. For comparison, the solution to Case 3 is shown as dashed lines. The temperature distributions are similar. This is not surprising, since the same amount of incident energy is absorbed in both cases. However, the volumetric and surface heating cases deviate significantly in approximately the first 5 microns. The volumetric case temperature distribution tends to level off, but the surface heating case it continues to increase as the surface is approached.
2,000
Numerical Semi-Analytical 1,500
1,000
Time varying heat flux q" =210 kW/cm2 500 peak Pulse Length: 500μsec Figure 6. Pulse Frequency: 500Hz
Surface Temperature, T, (°C) T, Temperature, Surface Case 4 solution: 5,000 micron copper layer. time-varying heat flux, surface 02468 temperature history Time, t, (msec) HFF 1,600 25,4 " 2 Time varying heat flux qpeak=210kW/cm Pulse Length: 500μsec 1,400 Pulse Frequency: 500Hz 5,000 micron copper layer. 1,200 Semi-Analytical - Pulse Numerical - Pulse 1,000 P3 Semi-Analytical - Cooling P2 Numerical - Cooling 962 P1 800 C3 600 Figure 7. (°C) T, Temperature, C2 400 Case 4 solution: C1 time-varying heat 200 flux, in-depth temperature profile 0 0.05 0.1 0.15 0.2 Depth, x, (cm)
2 Case 5: Exponential Volumetric Heating: Is=2,000kW/cm 1,200 =4,000cm–1 Crank-Nicolson method Solution every 0.6μsec 1,000 Total simulation time: 3.0μsec
800 Analytical Numerical Surface Heating 600
Figure 8. 400
Case 5 solution: (°C) T, Temperature, exponential source 200 term, surface intensity, 0 2 0 5 10 15 20 Is ¼ 2,000 kW/cm Depth, x, (microns)
Convergence in the L2-Norm The first part of the study focusses on quantifying the convergence rates in the L2-norm on refined spatial and temporal grids for each of the methods, boundary conditions, and boundary condition approximations. Although both methods are formally second-order spatially accurate, boundary condition implementation does affect the convergence rate and accuracy of the solution. The L2-norm of the spatial error at any time is defined as (Becker et al., 1981): ! Z = 1=2 L 1 2 XN 2 2 :e: ¼ ðÞTxðÞ TaðÞx dx ffi Ti Ta;i Dxi (20) L2 0 i¼1
where Ta is the analytical solution. The convergence rate in the L2-norm on a uniformly spaced mesh is given by:
:e: ¼ CDxP (21) L2 where C is an arbitrary constant and P is the convergence rate of the solution. For second-order accurate methods P ¼ 2. When plotted in log-log coordinates, the L2-norm should be a straight line with the slope being the convergence rate (as long as the error Effect of is in the asymptotic range). boundary The L2 norms were computed for each of the cases. Simulations were run on six meshes ranging from 25 to 800 cells, and number of time steps ranging from 15,625 to condition 4,000,000, both doubling each increment. Although this translates into different Dx and approximation Dt for each of the cases (we run for various lengths and simulation times), it is easy to show from the definition of the L2-norm, that the effect on the L2-norm scales. 963 The L2 norms for Case 1 are presented in Figures 9-13. Figures 9 and 10 show the first-order boundary condition implementation for implicit and Crank-Nicolson methods, respectively. In Figure 10, all lines overlay, so for clarity, only the coarsest and finest time steps are shown. Figures 11 and 12 show the same for the second-order implementation. A trend that is apparent for each of the boundary condition approximations is that as the spatial mesh is refined, the fully implicit method requires a much finer time step than the Crank-Nicolson method to maintain second-order spatial convergence through the entire range of mesh refinements. At the fewer number of time steps, the L2-norm for the implicit method flattens out as the mesh is refined,
–4 L2-Norms Fully Implicit Method –5 Case 1: Constant Surface Temperature 1st Order Derivative –6 t = 1μsec
–7 )
L2 –8 nΔt=15,625 nΔt=31,250 –9 nΔt=62,500 n =125,000 Ln( || e Δt n =250,000 –10 Δt Figure 9. nΔt=500,000 n =1,000,000 Case 1: L2-norms, –11 Δt nΔt=2,000,000 constant temperature n =4,000,000 –12 Δt boundary condition, first-order accurate –13 –17 –16.5 –16 –15.5 –15 –14.5 –14 –13.5 –13 –12.5 derivative BC, fully Ln(Δx) implicit method
–5 L2-Norms Crank Nicolson Method –6 Case 1: Constant Surface Temperature 1st Order Derivative t = 1μsec –7
) –8 L2 nΔ =15,625 –9 t nΔt =4,000,000
Ln( || e Figure 10. –10 Case 1: L2-norms, –11 constant temperature boundary condition, –12 first-order accurate derivative BC, –13 –17 –16.5 –16 –15.5 –15 –14.5 –14 –13.5 –13 –12.5 Crank-Nicolson Ln(Δx) method HFF and requires 4,000,000 time steps to maintain second-order spatial convergence 25,4 throughout the entire range of mesh sizes. The Crank-Nicolson method maintains second-order spatial convergence even at the coarsest time step. Figure 13 compares the Crank-Nicolson and implicit methods at the finest (4,000,000) number of time steps using both the first-order and second-order boundary derivative approximations. The figure shows that at the finest time step, both fully implicit and Crank-Nicolson 964 methods produce lower error for the second-order approximation than for the first-order approximation, but the magnitude of the error is approximately the same for each time discretization. Cases 2a and 2b apply convection boundary conditions to a true convection problem and to a temperature prescribed boundary equivalent to Case 1, respectively. Case 2b applies the temperature boundary condition using the actual temperature boundary condition, and by weakly enforcing the boundary condition using a very high convection coefficient and setting the ambient temperature to the desired surface temperature. As discussed previously, the convection boundary condition is approximated by three methods, and so Case 2b is approximated by five different
–5 L2-Norms Fully Implicit Method –6 Case 1: Constant Surface Temperature 2nd Order Derivative –7 t = 1μsec
–8 )
L2 –9 nΔt=15,625 nΔt=31,250 || e n =62,500 –10 Δt Figure 11. nΔ =125,000
Ln( t nΔ =250,000 Case 1: L2-norms, –11 t nΔt=500,000 constant temperature n =1,000,000 –12 Δt boundary condition, nΔt=2,000,000 n =4,000,000 second-order –13 Δt accurate derivative –14 BC, fully implicit –17 –16.5 –16 –15.5 –15 –14.5 –14 –13.5 –13 –12.5 method Ln(Δx)
–5 L2-Norms Crank Nicolson Method –6 Case 1: Constant Surface Temperature 2nd Order Derivative –7 t = 1μsec
–8 )
L2 –9 nΔt=15,625
|| e n =4,000,000 –10 Δt
Figure 12. Ln( Case 1: L2-norms, –11 constant temperature –12 boundary condition, second-order –13 accurate derivative –14 BC, Crank-Nicolson –17 –16.5 –16 –15.5 –15 –14.5 –14 –13.5 –13 –12.5 method Ln(Δx) –5 L2-Norms at 4,000,000 time steps Effect of –6 Case 1: Constant Surface Temperature Ncells = 25-800 boundary –7 t =1μsec condition –8 approximation )
L2 –9 1st Order || e –10 2nd Order 965 Ln( –11 Figure 13. –12 Implicit Crank Nicolson Case 1: L2-norm –13 comparison at finest time step constant –14 –17 –16.5 –16 –15.5 –15 –14.5 –14 –13.5 –13 –12.5 temperature Ln(Δx) boundary condition
numerical boundary conditions total for each method. The same trends in the L2 norms seen in Case 1 are seen in these two cases. Rather than show all of the L2 norms, we focus instead on comparing the various boundary condition and time discretizations at the finest time step. The L2 norms for Case 2a and 2b are presented in Figures 14 and 15, respectively. The Crank-Nicolson and implicit methods perform equally (keeping in mind this is the finest time step), and the error depends on the boundary condition approximation, so we only show the L2 norms for the Crank-Nicolson method in the figures. The extrapolation method performs the worst, followed by the first-order method, with the second-order method producing the lowest error on a given mesh. Figure 14 indicates that for a finite convection coefficient, the second-order method performs only slightly better than the first-order method. However, Figure 15 shows that for a temperature boundary condition prescribed weakly, the second-order method performs significantly better than the first-order method. It is seen by comparing the two figures that the overall level of error is higher for the infinite convection coefficient than for the finite convection coefficient. Figure 15 also shows that the first- and second-order approximations for the boundary derivative perform the same regardless
–6 L2-Norms at 4,000,000 time steps –7 Case 2a: Convection BC h∝ =10E6 –8 t =0.2μsec
–9 )
L2 –10 || e –11 1st order - CONV BC Ln( –12 2nd order - CONV BC Extrapolation - CONVBC Figure 14. –13 Case 2a: L2-norms comparison, –14 convection –15 boundary condition, –17 –16.5 –16 –15.5 –15 –14.5 –14 –13.5 –13 –12.5 6 2 h∞ ¼ 10 W/m -K Ln(Δx) HFF of whether the derivative is applied directly as a temperature boundary condition 25,4 (TEMP BC) or weakly as a convective boundary condition (CONV BC). The L2 norms for Case 3, constant surface heat flux, are shown in Figures 16 and 17 for the fully implicit and Crank-Nicolson methods, respectively. As with the previous cases, the Crank-Nicolson method performs better than the fully implicit method through the entire range of time steps and mesh sizes. The implicit method displays the 966 same trends as the constant temperature case except that there is greater flattening of the L2 norms using fewer time steps as compared with Case 1. The Crank-Nicolson method displays second-order convergence on all meshes and all time steps, whereas the fully implicit method slowly converges to second-order spatial accuracy as the time step is reduced. The overall magnitude of error on a given mesh is slightly lower for the Crank-Nicolson method. Case 4 is the three-pulse case that produces a time-varying surface heat flux. The L2 norms are shown in Figure 18 for both the fully implicit and Crank-Nicolson methods at the midpoint of the final heating cycle, and at 750 μsec into the final cooling cycle. For clarity, only the L2 norms at 4,000,000 time steps are shown. Several interesting
–2
–3 L2-Norms at 4,000,000 time steps –4 Case 2b: Constant surface temperature h∝ =10E20 –5 t =0.2μsec –6 )
L2 –7
|| e –8
Figure 15. Ln( –9 Case 2b: L2-norms 1st order - CONV BC, TEMP BC –10 2nd order - CONV BC, TEMP BC comparison, constant Extrapolation - CONV BC temperature –11 boundary condition –12 applied as convection –13 boundary condition, –17 –16.5 –16 –15.5 –15 –14.5 –14 –13.5 –13 –12.5 ¼ 20 2 h∞ 10 W/m -K Ln(Δx)
–7 L2-Norms Fully Implicit Method –8 Case 3: Constant Surface Heat Flux t = 0.6μsec –9
–10 )
L2 –11 nΔt=15,625 nΔ =31,250 –12 t nΔt=62,500 Ln( || e nΔt=125,000 –13 nΔt=250,000 nΔt=500,000 –14 n =1,000,000 Figure 16. Δt nΔt=2,000,000 Case 3: L2-norms, –15 nΔt=4,000,000 constant heat flux –16 boundary condition, –18.5 –18–17.5 –17 –16.5 –16 –15.5 –15 –14.5 –14 fully implicit method Ln(Δx) –8 L2-Norms Crank Nicolson Method Effect of –9 Case 3: Constant Heat Flux t = 0.6μsec boundary –10 condition
–11 approximation )
L2 –12
nΔt=15,625 –13 967 nΔt=4,000,000 Ln( || e –14 Figure 17. –15 Case 3: L2-norms, constant heat flux –16 boundary condition, –17 Crank-Nicolson –18.5 –18–17.5 –17 –16.5 –16 –15.5 –15 –14.5 –14 method Ln(Δx)
0
–1 nΔt=15,625, Crank-Nicolson nΔt=15,625, Fully Implicit –2 nΔt=4,000,000, Crank-Nicolson & Fully Implicit –3
–4 || ) t=4,250μsec L2 –5 mid-point of 3rd pulse
In( || e –6 t=5,250μsec Figure 18. –7 750μsec after end of last pulse Case 4: L2-norms, –8 time-varying heat flux boundary –9 condition, fully –10 implicit and Crank- –12.5 –12–11.5 –11–10.5 –10–9.5 –9–8.5 –8 Nicolson methods In(Δ x) trends are apparent. Both achieve second-order spatial convergence on all meshes. The overall integrated level of error is virtually the same for both methods on each mesh. The error during a cooling cycle is less than the error during a heating cycle for a given mesh and time step. This is not surprising, as during a heating cycle both the spatial and temporal temperature gradients are higher. This implies that to improve computational efficiency one could take a larger time step during cooling cycles, which are usually much longer in duration than heating cycles, and reduce simulation times substantially. The parameters of Case 5 are identical to Case 3, with the difference being that the incident heat flux is absorbed volumetrically through Beer’s law, rather than as a surface heat flux. We chose an absorption coefficient of 4,000 cm 1. Although this value is lower than the actual value for copper, we chose it to allow absorption through a greater depth of the layer. The higher value of copper would have caused the incident flux to absorb very near the surface, basically acting as a surface heat flux. With this value, 95 percent of the incident energy is absorbed in the first half of the layer. The L2 norms for this case are essentially identical to Case 3 (Figures 16 and 17) in both trend and magnitude. HFF Local error analysis 25,4 Having quantified the convergence properties for each of the methods and boundary condition implementations, we investigated local error next. In this part of the study we focussed on the time-varying heat flux and volumetric source heating, Cases 4 and 5, because these most closely resemble photonic curing. We quantify the maximum Dx and Dt that can be used and still achieve a given accuracy. For this study we chose 968 2 percent maximum error to be sufficient for our purpose. For Case 4 we investigated the local error at the midpoints of the heating and cooling cycles and noted the maximum error in the domain at each of the times. The length of the heating pulse determined the minimum number of time steps that could be used. We varied the number of time steps between 2 and 128 during the pulse, which for 9 ms total simulation time gave between 36 and 2,304 total number of time steps. The maximum local error at each of the selected times for 25-, 100-, and 800-cell meshes are presented in Tables II-IV, respectively. Several interesting trends are apparent. The error is highest during the first pulse and decreases with increasing time. This is not surprising since the spatial and temporal gradients are highest during the first pulse and decrease due to diffusion at later times. In all cases, the error for the fully implicit method is higher than the Crank-Nicolson method for an equal number of time steps. Concentrating on the midpoint of the first pulse (P1), because this is where the error is greatest, we see that on the 25-cell mesh, the fully implicit method starts out much higher than the Crank-Nicolson method for the fewest number of time steps, and gradually decreases as the number of time steps increases, whereas, the Crank-Nicolson method remains relatively constant. However, both are approaching a similar error as
t 250 1,250 2,250 3,250 4,250 5,250 μsec P1 C1 P2 C2 P3 C3 nΔt IMP C-N IMP CN IMP CN IMP CN IMP CN IMP C-N 2 87.88 26.41 24.08 5.89 11.37 3.23 8.75 2.06 8.17 1.95 6.86 1.55 4 63.67 26.99 16.40 6.60 7.53 3.42 5.33 2.02 5.15 2.26 4.26 1.58 Table II. 8 48.17 28.60 11.92 6.79 5.78 3.95 3.62 2.02 3.65 2.62 2.93 1.59 Case 4: time-varying 16 39.19 29.01 9.47 6.83 4.82 4.08 2.77 2.02 3.21 2.70 2.25 1.59 flux, maximum 32 34.32 29.11 8.19 6.85 4.48 4.11 2.39 2.02 2.98 2.72 1.92 1.59 local percent error, 64 31.77 29.14 7.52 6.85 4.30 4.12 2.21 2.02 2.85 2.73 1.76 1.59 25-cell mesh 128 30.47 29.14 7.19 6.85 4.21 4.12 2.12 2.02 2.79 2.73 1.67 1.59
t 250 1,250 2,250 3,250 4,250 5,250 μsec P1 C1 P2 C2 P3 C3 nΔt IMP CN IMP CN IMP CN IMP CN IMP CN IMP CN 2 74.34 16.09 20.32 4.72 9.20 11.07 6.89 3.34 6.80 9.16 5.50 2.67 4 45.15 5.36 11.54 1.10 5.78 2.65 3.47 0.73 3.65 2.13 2.85 0.57 Table III. 8 26.23 2.17 6.38 0.40 3.45 0.59 1.77 0.14 2.13 0.47 1.49 0.10 Case 4: time-varying 16 15.03 1.95 3.53 0.44 2.01 0.25 0.94 0.13 1.24 0.15 0.80 0.10 flux, maximum 32 8.82 2.04 2.02 0.45 1.20 0.29 0.53 0.13 0.73 0.18 0.45 0.10 local percent error, 64 5.52 2.06 1.24 0.45 0.76 0.31 0.33 0.13 0.47 0.19 0.28 0.10 100-cell mesh 128 3.81 2.07 0.85 0.45 0.54 0.31 0.23 0.13 0.33 0.19 0.19 0.10 the time step is reduced. The same trend is apparent on the 100-cell mesh. Essentially Effect of on each of the 25- and 100-cell meshes, the error flattens out as the number of time steps boundary is increased because the spatial meshes are not refined enough to maintain the theoretical time convergence rate, and so the error remains constant under further condition time step refinement. But the Crank-Nicolson method reaches that limit with far approximation fewer time steps than the fully implicit method. For example, on the 100-cell mesh, the Crank-Nicolson method error has been reduced to 2.17 percent by eight time steps, 969 whereas, the fully implicit method only begins to approach that by 128 time steps (3.81 percent error). As the number of cells increases, the error for the fewest number of time steps actually increases slightly, but ultimately decreases below the coarser mesh as the number of time steps is increased. This is consistent with the behavior of the L2 norms, which showed that finer meshes required finer time steps to reach the theoretical convergence rate. Figure 19 shows the error at 250 μsec for both methods on the 800-cell mesh as the number of time steps (during the pulse) is increased from 2 to 128. The error for the Crank-Nicolson methods starts much lower and decreases rapidly, so that by 16 time steps it is reduced to ~2 percent. The fully implicit method does not fall below 2 percent until 128 time steps. The figure clearly shows, as is the trend for the 25 and 100 cells meshes, that the error is approaching the same value under time step refinement, but the Crank-Nicolson method reaches the limit in far fewer time steps.
t 250 1,250 2,250 3,250 4,250 5,250 μsec P1 C1 P2 C2 P3 C3 nΔt IMP CN IMP CN IMP CN IMP CN IMP CN IMP CN 2 74.07 23.54 20.02 6.40 9.07 16.58 6.76 4.89 6.84 13.77 5.41 4.10 4 43.88 9.74 11.20 2.82 5.58 6.83 3.35 2.17 3.69 5.66 2.76 1.82 8 24.62 4.48 6.00 1.37 3.22 3.14 1.66 1.03 1.98 2.61 1.39 0.85 Table IV. 16 13.24 1.70 3.12 0.54 1.75 1.21 0.82 0.38 1.07 1.01 0.70 0.31 Case 4: time-varying 32 6.92 0.43 1.60 0.12 0.92 0.32 0.41 0.08 0.56 0.27 0.35 0.06 flux, maximum 64 3.56 0.06 0.81 0.01 0.47 0.05 0.21 0.00 0.29 0.04 0.18 0.02 local percent error, 128 1.82 0.03 0.41 0.01 0.24 0.00 0.10 0.00 0.15 0.00 0.09 0.02 800-cell mesh
80
60 μ ncell = 800, t = 250 sec Implicit Method Crank-Nicolson Method 40 % Error
20 Figure 19. Case 4: percent 0 error under time step refinement, 0 50 100 150 800-cell mesh nΔt HFF For Case 5 we investigated the local error for the number of time steps ranging from 2 25,4 to 128 and for the number of cells ranging from 5 to 120. We present the results for 10, 20, and 40 cells in Table V at 1.5 and 3.0 μsec. On the 10 cell mesh, the level of error is unacceptably high at the fewest time steps for the implicit method. However, the Crank-Nicolson method begins to produce less than 4 percent error with only four time steps. Increasing the number of time steps to eight reduces the error to less than 970 1 percent. The implicit method does not fall below 1 percent error until 128 time steps; a factor of 16 difference. As with Case 4, the error tends to level out for the lower number of cells, but by 40 cells, the order accuracy of both methods is clear. The implicit method error is reducing linearly as the time step is reduced, whereas the Crank-Nicolson method is reducing quadratically.
Simulation times The computation times were recorded for each method using the 800-cell mesh and 32,000 time steps. The code was profiled using Intel Vtune Amplifier®, and the time spent in each subroutine was computed. The large number of time steps and cells was required to minimize the scatter in the computed CPU time. The only difference between the fully implicit method and the Crank-Nicolson method is in the subroutine that calculates the tridiagonal coefficients. Because simulation times varied somewhat from run to run, five runs were made for each method, and average times computed. The implicit method used 3.034 sec to compute the tridiagonal coefficients, whereas the Crank-Nicolson method used 3.410 sec. This represents approximately a 12.4 percent increase of the Crank-Nicolson method over the implicit method. The total simulation times (including time spent in other subroutines) was 11.874 sec and 12.241 sec for the fully implicit and Crank-Nicolson methods, respectively. This represents only a 3.1 percent increase in overall simulation time for the Crank-Nicolson method over the fully implicit method.
Conclusions Based on the results presented, the following conclusions can be drawn. In all cases, both methods achieve second-order spatial accuracy for fine enough time steps. However, the Crank-Nicolson achieves second-order spatial accuracy with far fewer time steps than the fully implicit method. The second-order temporal accuracy of the Crank-Nicolson method manifests itself in requiring far fewer time steps to display second-order spatial accuracy. Although the conclusions are based on a 1-D analysis, the behavior should extend in a straight forward way to 2-D and 3-D structured
Nc ¼ 10 Nc ¼ 20 Nc ¼ 40 1.5 μsec 3.0 μsec 1.5 μsec 3.0 μsec 1.5 μsec 3.0 μsec nΔt IMP CN IMP CN IMP CN IMP CN IMP CN IMP CN 2 33.70 15.54 26.90 8.88 34.62 15.69 27.4 9.17 35.13 15.72 27.68 9.29 4 20.26 3.77 15.4 1.02 20.79 4.10 15.66 1.10 21.08 4.20 15.81 1.16 8 10.83 0.76 8.03 0.15 11.07 0.97 8.14 0.26 11.21 1.05 8.21 0.30 Table V. 16 5.52 0.21 4.05 0.10 5.58 0.11 4.07 0.04 5.64 0.16 4.09 0.04 Case 5: volumetric 32 2.83 0.16 2.06 0.09 2.82 0.05 2.04 0.03 2.83 0.02 2.05 0.01 heating, maximum 64 1.49 0.14 1.08 0.09 1.43 0.04 1.03 0.03 1.42 0.01 1.03 0.01 local percent error 128 0.82 0.14 0.58 0.09 0.73 0.04 0.53 0.03 0.72 0.01 0.52 0.01 meshes, as the numerical boundary discretizations are conducted in a 1-D sense on each Effect of boundary. It is not clear that this behavior would extend to 2-D and 3-D unstructured boundary meshes, as the numerical discretization procedures on the boundaries are not the same as for structured meshes. condition For the temperature prescribed boundary condition, the first-order accurate approximation approximation tended to flatten out somewhat less at the fewest number of time steps compared to the second-order accurate method using the fully implicit method. 971 However, the overall error on a given mesh was lower for the second-order accurate expression at the finest time step. The Crank-Nicolson method maintained second-order convergence throughout the range of meshes for all time steps for both the first- and second-order methods. At the finest time step, the overall level of error on a given mesh was equal for the implicit and Crank-Nicolson methods. For the convection boundary condition, the second-order boundary condition produced the smallest error on all meshes and for all time steps. The constant heat flux and volumetric boundary conditions produced trends in the L2 norms similar to the constant temperature boundary condition. The time-varying heat flux boundary condition showed that the error in the L2-norm was lower between pulses than during pulses, indicating that a larger time step could be taken between pulses. For coarse discretizations, the Crank-Nicolson method performs far better than the fully implicit method for the two cases studied. The Crank-Nicolson method allows up to a factor of 16 greater time step than the implicit method for equivalent error on the same mesh. Furthermore, the computational penalty per time step of the Crank-Nicolson method is nearly negligible, allowing substantially reduced run times. Finally, the Crank-Nicolson method combined with the second-order accurate expression for the boundary derivative applied as a convection boundary condition performed the best overall out of all of the numerical boundary condition approximations tested. This formulation also allowed temperature prescribed boundary conditions to be imposed weakly through an arbitrarily large convection coefficient.
References Al-Odat, M.Q., Ubertini, S. and Succi, S. (2013), “Numerical investigation of fast precooling of a cylindrical food product based on the hyperbolic heat conduction model”, International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 23 No. 6, pp. 960-978. Beck, J.V. (1992), Heat Conduction using Greens Functions, Hemisphere Publishing Co., New York, NY. Becker, E.B., Carey, G.F. and Oden, J.T. (1981), Finite Elements: An Introduction, Volume I, Prentice Hall, London. Berg, J. and Nordstrom, J. (2013), “On the impact of boundary conditions on dual consistent finite difference discretizations”, Journal of Computational Physics, Vol. 236, March, pp. 41-55. Blackwell, B.F. and Hogan, R.E. (1993), “Numerical solution of axisymmetric heat conduction problems using finite control volume technique”, Journal of Thermophysics and Heat Transfer, Vol. 7 No. 3, pp. 462-471. Britz, D., Osterby, O. and Strutwolf, J. (2003), “Damping of Crank-Nicolson error oscillations”, Computational Biology and Chemistry, Vol. 27 No. 3, pp. 253-263. Carslaw, H.S. and Jaeger, J.C. (1959), Conduction of Heat in Solids, 2nd ed., Clarendon Press, London. Charoensuk, J. and Vessakosol, P. (2010), “A high order control volume finite element procedure for transient heat conduction analysis of functionally graded materials”, Heat & Mass Transfer, Vol. 46 Nos 11-12, pp. 1261-1276. HFF Cho, J.R., Ha, D.Y. and Kim, T.J. (2002), “Convergence characteristics of the Crank-Nicolson-Galerkin 25,4 scheme for linear parabolic systems”, KSME International Journal, Vol. 16 No. 10, pp. 1264-1275. Eymard, R., Gallouet, T and Herbin, R. (2000), “The finite volume method”, in Ciarlet, P. and Lions, J.L. (Eds), Handbook for Numerical Analysis, 715-2022, Vol. VII, North Holland and Elseiver, Amsterdam, pp. 713-1020. Guillot, M.J., Schroder, K.A. and McCool, S.C. (2012), “Simulating the thermal response of thin films 972 during photonic curing” ASME 2012 International Mechanical Engineering Congress and Exposition, Vol. 7, Fluids and Heat Transfer, Parts A, B, C, and D, Paper No. IMECE2012-87674, Houston, Texas, available at: http://proceedings.asmedigitalcollection.asme.org/proceeding. aspx?articleID=1751325 (accessed November 9–15, 2012). Hesthaven, J.S. and Warbution, T. (2008), Nodal Discontinuous Galerkin Methods: Algorithms, Analysis and Applications, Springer, New York, NY. Hor-Yen, R., Chen, C.Y., Huang, C.T. and Chen, P.J. (2013), “Numerical study of anisotropic thermal conductivity fabrics with heating elements”, International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 23 No. 5, pp. 750-771. Jaluria, Y. (2010), “Role of experiments in the accurate numerical simulation of thermal processes”, International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 20 No. 5, pp. 467-491. Makenzie, J.A. and Morton, K.W. (1992), “Finite volume solutions of convection-diffusion test problems”, Mathematics of Computations, Vol. 60 No. 201, pp. 189-220. Pletcher, R.H., Tannehil, J.C. and Anderson, D.A. (2013), “Computational fluid mechanics and heat transfer”, 3rd ed., CRC Press, Taylor and Francis Group, Boca Raton, FL. Sahin, A.Z. (1992), “Transient heat conduction in semi-infinite solid with spatially decaying heat generation”, International Communication in Heat and Mass Transfer, Vol. 19 No. 3, pp. 349-358. Schroder, K.A. (2011), “Mechanisms of photonic curing”, Technical Proceedings of the 2011 NSTI Nanotechnology Conference and Trade Show, Vol. 2, pp. 220-223. Schroder, K.A., McCool, S.C. and Furlan, W.R. (2006), “Broadcast photonic curing of metallic nanoparticle films”, Technical Proceedings of the NSTI Nanotechnology Conference and Trade Show, Vol. 3, pp. 198-201. Tian, W., Wilson, K.S. and Chiu (2005), “A two-dimensional scheme for axisymmetric radiative heat transfer using the finite-volume method”, Numerical Heat Transfer, Part B, Vol. 47 No. 3, pp. 199-211. Versteeg, H. and Malalasekera, W. (2007), An Introduction to Computational Fluid Dynamics: The Finite Volume Method, 2nd ed., Prentice-Hall, Harlow. Zardetto, V., Brown, T.M., Reale, A. and De Carlo, A. (2011), “Substrates for flexible electronics: a practical investigation on the electrical, film flexibility, optical, temperature, and solvent resistance properties”, Journal of Polymer Science Part B: Polymer Physics, Vol. 49 No. 9, pp. 638-648. Zarghami, A. (2014), “Finite volume formulation of the thermal lattice Boltzmann method”, International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 24 No. 2, pp. 270-289.
Corresponding author Martin Joseph Guillot can be contacted at: [email protected]