Effect of boundary conditionapproximation on convergenceand accuracy of a finite volumediscretization of the transient

heat conduction equationMartin Joseph Guillot

Department of Mechanical Engineering, University of New Orleans,Louisiana, USA, andSteve C. McCool

Department of Engineering, Novacentrix, Inc., Austin, Texas, USA

AbstractPurpose – The purpose of this paper is to investigate the effect of numerical boundary conditionimplementation on local error and convergence in L2-norm of a finite volume discretization of thetransient heat conduction equation subject to several boundary conditions, and for cases withvolumetric heat generation, using both fully implicit and Crank-Nicolson time discretizations. The goalis to determine which combination of numerical boundary condition implementation and timediscretization produces the most accurate solutions with the least computational effort.Design/methodology/approach – The paper studies several benchmark cases including constanttemperature, 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 boundaryderivatives produces the most accurate solutions on the coarsest meshes with the least computationtimes. The Crank-Nicolson method allows up to 16X larger time step for similar accuracy, with nearlynegligible additional computational effort compared with the implicit method.Practical implications – The findings can be used by researchers writing similar codes forquantitative guidance concerning the effect of various numerical boundary condition approximationsfor 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 thefinite volume discretization for a wide range of benchmark cases and common time discretization methods.Keywords Heat conduction, Finite volume, L2 convergence, Photonic curingPaper type Research paper

IntroductionThe heat conduction equation has many applications in engineering and physics.While analytical solutions exist for a number of cases, most problems of engineeringinterest do not have analytical solutions, and so numerical methods are appliedto obtain approximate solutions. Commonly used methods include the finite difference,finite element, and finite volume methods (Pletcher et al., 2013), with each methodhaving advantages and disadvantages. The finite difference method is easiest toimplement, but is limited to structured or block-structured domains, making it difficultto implement on complex geometries. The finite volume and finite element methods aremore difficult to code, but offer greater flexibility to model arbitrary geometry, and inimplementing boundary conditions.

International Journal of NumericalMethods for Heat & Fluid FlowVol. 25 No. 4, 2015pp. 950-972©EmeraldGroup Publishing Limited0961-5539DOI 10.1108/HFF-02-2014-0033

Received 10 February 2014Revised 12 August 2014Accepted 13 August 2014

950

HFF25,4

The current authors previously developed a thermal simulation tool using the finitevolume method with a fully implicit time discretization of the transient heat conductionequation 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 lightfrom xenon flashlamps and forced convective cooling to quickly (on the order ofmilliseconds) thermally process thin films, such as the sintering of metal inks. Typically,these films are processed on low temperature substrates such as plastic and paper.First described by Schroder et al. (2006) and Schroder (2011), photonic curing israpidly finding many applications in the manufacture of flexible electronics and inother manufacturing processes. Typical printed electronics applications includeRFID tags, photovoltaics, large area displays, “electronic” paper and disposablesensors (Zardetto et al., 2011). Printed electronic circuits are manufactured usinga range of materials and processes, but one common method of forming theconductors is to use a printer to deposit silver or copper nanoparticles suspended insolvents and binders onto inexpensive flexible substrates such as polyethyleneterephthalate (PET), polyethylene naphthalate (PEN), or even paper. The metal inkcircuit patterns are then sintered to make them highly conductive.

Inexpensive substrates such as PET have a maximum working temperature of onlyabout 1501C. Because of this limitation, traditional sintering methods such as an ovenor IR heater that bring the entire film stack (ink pattern and substrate) into thermalequilibrium with the surroundings are constrained by the maximum workingtemperature of the substrate. This limitation forces a long processing time – oftenminutes, which means that either the oven is physically large or the throughput islimited by the oven. Either technique is economically undesirable.

The pulsed light produced during photonic curing produces a time-varying heatflux that can act as a surface heat flux or a volumetric heat source, depending on theabsorption properties of the material. Materials with high absorption coefficientsabsorb the energy near the surface, and then diffusion is the primary mechanism totransport the energy into the stack. Materials with low absorption coefficients absorbthe energy volumetrically throughout the stack, and diffusion then redistributes theenergy. Forced convective cooling is used in between pulses to cool the film andreduce heat flow into the substrate. The combination of pulsed radiant heatingand convective cooling makes it possible to heat a thin film on a low temperaturesubstrate to a temperature far beyond the maximum working temperature of thesubstrate without damaging the substrate. The process requires that the time at hightemperature be very short – of order a few ms. The peak pulse power, pulse length,number of pulses, rate of convective cooling, and pulse repeat frequency can beindependently varied to achieve desired thermal profiles throughout the film andsubstrate. Additionally, the phonic curing hardware allows the temperature at thebottom of the substrate to be clamped at a specific temperature using a vacuumchuck and bringing the bottom surface of the substrate into contact with a constanttemperature mass.

Because of the large parameter space, as well as material stack combinationsaffecting the temperature profile, it is critical to have a fast running simulation tool thatcan provide accurate simulations of photonic curing processes, where the user can varythe many parameters that affect the results, and quickly view the resulting temperatureprofiles. This allows users to quickly zero in on the parameters that produce the desiredtemperature profile in a film stack for a given process. The simulation tool previouslydeveloped couples the finite volume discretization to a graphical user interface written

951

Effect ofboundarycondition

approximation

in LabView®, and a database of thermally dependent material properties to which theuser can add user defined materials.

The geometric, material, and pulse parameters are all specified using theinterface, and are the same for the simulation as for the hardware settings ofthe equipment used in the actual photonic curing process. The output is a graphicaltime-temperature history in all locations of the stack viewable within the sameinterface. The goal of the tool is to have the simulation be as fast as possible withoutsacrificing accuracy, so that the user can see the effects of parameter changeson the thermal profile in near “real-time” fashion. This allows the user to much moreeasily shape the thermal profile in a film stack to achieve the desired process resultsand to determine the settings to use for the photonic curing equipment.To accomplish this, it is important to know how the various approximations toboundary conditions and time stepping methods affect solution convergence rateand accuracy, and simulation run time for each of the boundary condition typesencountered 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 timestepping methods of a finite volume discretization of the transient 1-D heat conductionequation was undertaken in order to determine which combination of time steppingmethod and boundary condition approximation produced the most accurate resultswith the least computational effort. The most common time discretization methodsapplied to the heat conduction equation are fully implicit and Crank-Nicolson. While bothmethods are unconditionally stable, the fully implicit method is only formallyfirst-order time accurate. The Crank-Nicolson method is formally second-order timeaccurate, but can suffer from unphysical oscillations when there is a discontinuity in theinitial conditions or between the initial conditions and the boundary conditions, due to theamplification 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 totransient 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 forthe derivative is spatially second-order accurate. However, there is surprisingly littlein the literature that systematically discusses the various boundary conditionapproximations and volumetric heating coupled with time stepping methods,and quantifies their effect on convergence rate and solution accuracy. Makenzie andMorton (1992) discuss solution error and convergence properties of a finite volumediscretization of the steady convection-diffusion equation for two cases involvingDirichlet boundary conditions. Berg and Nordstrom (2013) discuss convergenceproperties for an incompletely parabolic system using the finite difference methodsubject to Dirichlet and heat flux boundary conditions. Cho et al. (2002) presentconvergence in the L2-norm for the transient heat conduction equation subjectto convection boundary conditions using the finite element method withCrank-Nicolson time discretization. They show the dependence of time step andmesh size for maintaining theoretical convergence under mesh refinement. Eymardet al. (2000) demonstrate second-order convergence in the L∞-norm of the transientheat conduction equation, including source term, but limited to Dirichlet boundaryconditions. Hesthaven and Warbution (2008), apply an explicit discontinuousGalerkin finite element discretization to the 1-D transient heat conduction equation.

952

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They demonstrate sub-optimal convergence when polynomials of odd order are used,e.g. linear polynomials display first-order convergence. The analysis is limited tohomogenous Dirichlet boundary conditions. Jaluria (2010) used experimental datacombined with numerical simulation to assess the accuracy of thermal processsimulation methods.

The goals of this effort were to investigate the fully implicit and Crank-Nicolsontime stepping methods with the full range of boundary conditions needed to simulatephotonic curing, and the various approximations to the boundary conditions, in orderto 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 representativeof photonic curing for coarse discretizations. The objective was to determine meshand time step requirements to obtain specified accuracy (e.g. 2 percent) for therepresentative cases.

In principle, the Crank-Nicolson method, because it is second-order accurate intime, allows larger time steps than the implicit method while retaining similaraccuracy. However, the increased accuracy comes at increased computational effortdue 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 allowsa large enough time step to be taken when compared with the implicit method toovercome the additional computational expense per time step, so that total simulationtime is reduced.

The first part of the study quantifies the convergence rates in the L2-norm for highlyrefined meshes and times steps, applying the various boundary conditions andtime 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 casesrepresentative of photonic curing using coarse mesh spacing and time steps.

Heat conduction equationThe 1-D transient heat conduction equation with volumetric source heating in a domain0⩽ x⩽L is written as:

rcP@T@t

¼ @

@xk@T@x

� �þq000 (1)

where ρ(kg/m3), cP(J/kg-K), and k(W/m-K) are the density, specific heat at constantpressure, and thermal conductivity, respectively, of each material, and q000(W/m3) isa volumetric source term. In general, the properties are temperature dependent.The thermal diffusivity is defined as α¼ k/ρcp (m

2/s).

Boundary and initial conditionsPhotonic curing simulation involves several boundary condition types, includingDirichlet, Robin, and Neumann, which correspond to prescribed temperature,convection, and heat flux boundary conditions, respectively. They are written as:

TB ¼ TB tð Þ Dirichlet

7k dTBdx ¼ h1 TB�T1ð Þ Robin

q00B tð Þ ¼ �k dTBdx Neumann

(2)

953

Effect ofboundarycondition

approximation

where h∞ is the convective heat transfer coefficient, TB is the temperature on theboundary 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 (±) inthe Robin boundary condition is (+) on the left boundary (top surface) and (−) on theright boundary (bottom surface). The initial temperature distribution T(x,0)¼Tinit(x)is known.

Finite volume discretizationEquation (1) is spatially discretized using the finite volume method. The domainis divided into N finite volumes (cells) and Equation (1) is integrated over eachvolume, with average cell temperatures defined at the cell centers. The domaindiscretization and nomenclature are illustrated in Figure 1. The cell centersare located at xi, and the cell boundaries are located at xi± 1/2. The grid spacing isdefined by:

Dxi ¼ xiþ 1=2�xi�1=2; Dxi7 1=2 ¼ x iþ 1=2ð Þ7 1=2�x i�1=2ð Þ7 1=2 (3)

Integrating Equation (1) over a typical finite volume produces the semi-discreteequation, written as:

rcpDx� �

i

dTi

dt¼ y k

dTdx

� �nþ1

iþ1=2� k

dTdx

� �nþ1

i�1=2þ q000i� �nþ1

Dxi

" #

þ 1�yð Þ kdTdx

� �n

iþ 1=2� k

dTdx

� �n

i�1=2þ q000i� �n

Dxi

" #(4)

The current time level is denoted by n and the new time level by n+1. A value of θ¼ 1corresponds to the fully implicit method, and θ¼ 1/2 corresponds to Crank-Nicolson.Discretizing the time derivative, and using central differences for the spatialderivatives in Equation (4) yields the fully discrete set of equations for the interior cells,2⩽ i⩽N−1, written in general form as:

AiTnþ 1i�1 þBiT

nþ 1i þCiT

nþ 1iþ 1 ¼ DiT

ni�1þEiT

ni þFiT

niþ 1þGi (5)

Defining:

bi7 1=2 ¼ki7 1=2Dt

rcPð ÞiDxiDxi7 1=2(6)

xi –1/2

Δxi –1/2 Δxi +1/2

Δxi

xi –1 xi xi +1

xi +1/2

Figure 1.Finite volumediscretizationof 1-D domain

954

HFF25,4

then:

Ai ¼ �ybi�1=2;

Bi ¼ 1þybi�1=2þybiþ1=2

� �;

Ci ¼ �ybiþ 1=2

Di ¼ 1�yð Þbi�1=2;

Ei ¼ 1� 1�yð Þ bi�1=2þbiþ 1=2

� �� �;

Fi ¼ 1�yð Þbiþ 1=2

Gi ¼q000i Dtrcp� �

i

(7)

Numerical implementation of boundary conditionsThe boundary cells must incorporate the given boundary conditions. In every case, thespatial derivative appearing in Equation (4) on the boundary is approximated with anexpression that incorporates the boundary condition information. In this study we willimpose heat flux boundary conditions on the right boundary. In some cases we imposean adiabatic boundary condition on the right boundary, and in others we use theanalytical solutions to apply an appropriate heat flux boundary condition in order tosimulate finite layers. We study the effect of boundary condition implementationby using various boundary condition implementations on the left boundary. The leftboundary mesh is shown in Figure 2.

The Dirichlet boundary condition can be imposed by writing a one sided differencefor the derivative appearing on the left boundary and applying the known temperatureon the boundary, or by imposing a convective (Robin) boundary condition, setting theambient temperature to the desired surface temperature, and setting the convectioncoefficient to an arbitrarily high number. This is known as a penalty method, or weakimplementation, and has the advantage that the numerical boundary condition doesnot require separate coding for Dirichlet and convective boundary conditions (Beckeret al., 1981). The Neumann boundary condition is applied by replacing −kdT/dx on theboundary with the known value of the heat flux, q00B, and, thus, requires no spatialapproximation.

One sided difference expressions for the boundary derivative can be first- orsecond-order spatially accurate. In principle, since the overall method is spatiallysecond-order accurate, it is best to use a second-order accurate expression on theboundary as well. However, it is not uncommon to see the derivative approximated

TB

T1 T2

x1–1/2 x1+1/2x1

Δx1 Δx2

x2Figure 2.

Left boundary mesh

955

Effect ofboundarycondition

approximation

with a first-order expression, (Versteeg and Malalasekera, 2007). We use both, andshow that the first-order accurate expression can display second-order spatialconvergence under certain restrictions. The first-order accurate expression is simply:

dTdx

����1�1=2

¼ T1�TB

Dx1=2þO Dx½ � (8)

The second-order accurate expression involves TB, T1, and T2, and can be derived byfitting a second-order polynomial through the three points, and then taking thederivative of the polynomial at the boundary. The expression is written as:

dTdx

����1�1=2

¼ �8=3TBþ9=3T1�1=3T2

Dx1þO Dx2

(9)

It is noted that the first-order accurate expression is not restricted to constant spacingmeshes, whereas the second-order accurate expression assumes that Dx2¼Dx1.Equation (9) can easily be extended to non-constant spacing, but then the coefficientsbecome expressions involving Dx1 and Dx2. Although the finite volume methoddeveloped for the study does allow variable spacing, we restrict the results in this effortto constant mesh spacing since we are primarily interested in investigating theboundary 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 anexpression must be developed for the surface temperature. Two approaches areinvestigated. The first is to approximate the derivative on the boundary with a first- orsecond-order expression, as given by Equations (8) and (9), respectively, substitute that intothe boundary condition given by the second expression in Equation (2), and then solve forthe 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 theboundary. In each case, the surface temperature can be written in the form:

TB ¼ a1T1þb1T2þd1T1 (10)

where the coefficients for a1, b1 and d1 for each approximation are given in Table I. Thenthe Robin boundary condition can be written as:

kdTdx

����1�1=2

¼ h1 a1T1þb1T2þ d1�1ð ÞT1ð Þ (11)

CoefficientBoundary condition h c1 c2 c3 c4 a1 b1 d1

Dirichlet first order 0 2 0 2 0 0 0 0Dirichlet second order 0 3 1/3 8/3 0 0 0 0Neumann 0 0 0 0 1 0 0 0Robin Extrapolation h∞ 0 0 0 0 2Dx1 þDx2

Dx1 þDx2

�Dx1Dx1 þDx2

0

First order h∞ 0 0 0 0 kkþh1Dx1=2

0 h1Dx1=2kþh1Dx1=2

Second order h∞ 0 0 0 0 8=3k8=3kþh1Dx1

�1=3k8=3kþh1Dx1

h1Dx18=3kþh1Dx1

Table I.Coefficients for leftboundary cell forgiven boundaryconditionapproximation

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This is substituted into Equation (4) to develop the discrete equation for thesurface cell. The tridiagonal coefficients for the boundary cell are written in a generalform as:

A1 ¼ 0

B1 ¼ 1þy c1b1�1=2þb1þ 1=2þha1Dtrcp� �

1Dx1

!

C1 ¼ �y c2b1�1=2þb1þ 1=2�hb1Dtrcp� �

1Dx1

!

D1 ¼ 0

E1 ¼ 1� 1�yð Þ c1b1�1=2þb1þ 1=2þha1Dtrcp� �

1Dx1

!

F1 ¼ 1�yð Þ c2b1�1=2þb1þ 1=2�hb1Dtrcp� �

1Dx1

!

G1 ¼ c3b1�1=2TBþq000nþ 11 DtrcPð Þ1

þ c4q00Bþ 1�d1ð ÞhT1� �

Dt

rcp� �

1Dx1(12)

The coefficients for the various boundary conditions are also given in Table I.

Test casesThe effects of solution method and boundary condition implementation on convergencerate 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, Tinit¼ 251C.The properties of copper are assumed to be constant, with thermal conductivity,k¼ 401W/m-K, density, ρ¼ 8,960 kg/m3, and specific heat, cp¼ 382.5 J/kg-K. Variousthicknesses and simulation times were chosen for the different cases, and arerepresentative of various photonic curing processes. The analytical solution for Case 1is found in (Carslaw and Jaeger, 1959), and the analytical solutions to Cases 2-4 arefound in (Beck, 1992). Case 5 is a spatially exponentially varying volumetric sourcewith convection boundary conditions. The exponential source is in the form of Beer’slaw, which is written as:

I x ¼ I s 1�mð Þe�Zx (13)

where Ix is the intensity (W/m2) at location x, Is is the intensity at the surface, η is theabsorptivity (m�1), and μ is the reflectance. In this work we assume μ¼ 0.Differentiating Equation (13) gives the volumetric heating, q000, in Equation (1) as:

q000 ¼ I sZ 1�mð Þe�Zx (14)

Sahin (1992) developed a solution to this case using separation of variables to separatethe solution into steady and unsteady components. The volumetric source term isincluded in the steady solution. The unsteady solution reduces to a convection problemwith an exponential initial temperature distribution.

957

Effect ofboundarycondition

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Cases 1 and 2 are used to determine the effect of various boundary conditionapproximations on the convergence in the L2-norm. Cases 3 and 4 apply Neumannboundary conditions, which require no approximations, and so only comparestime stepping methods. Case 4 provides a time-varying surface heat flux, whereasCase 5 represents the situation where materials absorb the energy depositedby the pulsed lamps though volumetric heating as opposed to surface heating.For these cases, we look at the local error using coarse meshes and time stepsto determine minimum required discretization parameters that will still achievedesired accuracy.

Case 1 – constant temperature (Dirichlet)The first case applies a constant temperature boundary condition on a finitelayer. Defining non-dimensional variables, θC¼ (T−Ti)/(T0−Ti), the Fourier numberFo¼ (α/L2)t, ξ¼ x/L, and λn¼ (n+1/2)π, the analytical solution is written as:

yT x; tð Þ ¼ 1�2X1n¼0

exp �lnFoð Þln

sin lnxð Þ (15)

The temperature on the left boundary is 1,0001C for tW0. Adiabatic conditions areimposed on the right boundary. The numerical computations were performed ona 50 μm finite layer and the total simulation time was 5.0 μsec. The solution using15,625 time steps on an 800-cell mesh is shown in Figure 3. Thefigure shows the resultsusing the implicit method and first-order accurate boundary condition, but the resultsusing the Crank-Nicolson method and other boundary conditions are virtually identicalas viewed at this scale. It is noted that the Crank-Nicolson method produces spuriousoscillations in the solution if steps are not taken to damp them. The oscillationsare damped using the implicit method for one time step, and then switching tothe Crank-Nicolson method. Britz et al. (2003), note that this technique damps theoscillations while retaining second-order time accuracy. It was found in this study thatthis method works very well for this case. Without using the implicit method for thefirst time step, large spurious temporal oscillations in the solution occurred near the left

1,200

1,000

800

600

400

200

–200–5 0 5 10 15 20 25 30 35 40 45 50 55

0

Depth, x, (microns)

Tem

pera

ture

, T, (

°C)

0.2 1.0 3.0 5.0

NumericalAnalytical

Case 1: Constant Surface Temperature, Ts = 1,000°CSolution at 0.2, 1, 3, 5μsect = 0.2μsec also solution to Case 2b

Note: Numerical and analytical solutions

Figure 3.Case 1 solution:constant surfacetemperature

958

HFF25,4

boundary. However, when damping was applied, the oscillations were completelydamped, and the Crank-Nicolson solution method performed very well, as we showwhen discussing the convergence in the L2-norms.

Case 2 – convection (Robin)Case 2 is a semi-infinite solid with a convection boundary condition applied to theleft end. Case 2a applies a finite convection coefficient, h∞¼ 10× 106 W/m2 andambient temperature, T∞¼ 1,000 1C. These conditions produce a time dependenttemperature on the left boundary. Case 2b approximates a constant temperatureboundary condition weakly by specifying a convection boundary condition with heattransfer coefficient, h∞¼ 1020, and setting the ambient temperature to 1,000 1C.The total simulation time is 1 μsec. In both cases, adiabatic conditions are imposed onthe right boundary. The solution to Case 2a is shown in Figure 4. The solution toCase 2b is the same as Case 1, but limited to the first 1 μsec. The numericalsimulations are run on a finite layer of 50 μm, but the simulation time is such thatthermal effects do not reach the right boundary, and so the results can be comparedto the semi-infinite solution. Defining the Biot number as Bi¼ hL/k, the analyticalsolution is given by:

yc x; tð Þ ¼ erfcx

2Fo1=2

� ��exp BixþFo Bi2

� �erfc

x2Fo1=2

þFo1=2 Bi� �

(16)

where θc¼ (T−Ti)/(T∞−Ti).

Case 3 – constant heat flux (Neumann)Case 3 is a finite layer solid with constant surface heat flux of q00s ¼ 2; 000 kW=cm2 runfor a simulation time of 3.0 μsec. The numerical simulations are run on a finite layer of15 microns. The analytical solution for a semi-infinite solid is given as:

T x; tð Þ�Ti ¼2ffiffiffiffiat

p

kq00s

1p1=2

exp � x2

4Fo

!� x2Fo1=2

erfcx

2Fo1=2

� � !(17)

For this case, in order to simulate a finite layer, we apply the heat flux as given by theanalytical solution to the numerical simulation on the right boundary. The temperature

0 5 10 15 20 25 30 35 40 45 50

Depth, x, (microns)

NumericalAnalytical

Tem

pera

ture

, T, (

°C)

Case 2a: Convection BC, h∝=10E6Solution at 0.2, 0.4, 0.6, 0.8, 1.0μsec

250

200

150

100

50

Figure 4.Case 2a solution:

convection boundarycondition,

h∞¼ 10× 106W/m2-K

959

Effect ofboundarycondition

approximation

distribution on an 800-cell mesh using 15,625 time steps is shown in Figure 5 at0.6 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 shortpulses, except that we do not consider convection in between pulses. The pulse lengthfor 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 ofq00s ¼ 210 kW=cm2 and total simulation time of 9,000 μsec. The case consists of periodsof pulsed heating followed by periods of adiabatic diffusion. Although the adiabaticperiods are not technically “cooling,” but rather a redistribution of temperature due todiffusion, we refer to these periods as cooling because the surface temperature coolsduring 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 functionsolutions. Linearity of the heat conduction equation allows the solution to be computedas a superposition of two solutions. During heating cycles, the problem can beseparated into two problems. The first is a constant heat flux boundary condition withuniform initial conditions, whose solution is denoted by Tq(x,t), and the second is anadiabatic 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 anevolution of temperature from an initial non-uniform temperature distribution.During the heating cycles the solution to the heat flux problem, Tq(x,t), is purelyanalytical 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 initialconditions, To x; toð Þ, beginning at time to is given by the Green’s function solution,(Carslaw and Jaeger, 1959), and is written as:

Tcðx; tÞ ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4ap t�toð Þp Z L

0To x; toð Þ exp

� x�x0ð Þ24a t�toð Þ

!þexp

� xþx0ð Þ24a t�toð Þ

!" #dx0 (18)

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 newheating cycle. During the first heating cycle Tc(x,t)≡ 0 because the problem begins from

Depth, x, (microns)

NumericalAnalytical

Tem

pera

ture

, T, (

°C)

Case 3: Constant Heat Flux: q=2,000kW/cm2

Solution every 0.6μsecTotal simulation time: 3.0μsec

1,400

1,200

1,000

800

600

400

200

0–5 0 5 10 15 20

Figure 5.Case 3 solution:constant surfaceheat flux,q¼ 2,000 kW/cm2

960

HFF25,4

uniform initial conditions, and so during the first heating cycle, the “semi” analyticalsolution is purely analytical and is equivalent to Case 3. The solution to Case 4 is shownin Figures 6 and 7. Figure 6 shows the surface time history and Figure 7 shows thetemperature distribution within the layer at the midpoints of the heating pulse (P) andcooling cycles (C).

Case 5 – exponentially varying volumetric source with convectionSahin (1992) writes the solution to this problem as:

y x; tð Þ ¼ erfcx

2ffiffiffit

p� �

�exp Bi2tþBix� �

erfc Biffiffiffit

p þ x2ffiffiffit

p� �

þBic

Biþ1Bi

� �erfc

x2ffiffiffit

p� �

�exp �xð Þ�

�12

Biþ1Bi�1

� �exp tþxð Þerfc ffiffiffi

tp þ x

2ffiffiffit

p� �

þ12exp t�xð Þerfc ffiffiffi

tp � x

2ffiffiffit

p� �

þ 1Bi Bi�1ð Þ

� �exp Bi2tþBix� �

erfc Biffiffiffit

p þ x2ffiffiffit

p� �

(19)

where θ is defined as for Case 2, ξ¼ ηx, τ¼ αη2t, Bi¼ h/ηk, and c¼ h(T∞−To)/Is(1−μ).The simulation conditions were identical to Case 3, except that the incident flux wasabsorbed volumetrically rather than as a surface flux. The temperature distribution isshown 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 sameamount of incident energy is absorbed in both cases. However, the volumetric andsurface heating cases deviate significantly in approximately the first 5 microns.The volumetric case temperature distribution tends to level off, but the surface heatingcase it continues to increase as the surface is approached.

2,000

1,500

1,000

500

0 2 4 6 8

Time, t, (msec)

NumericalSemi-Analytical

Sur

face

Tem

pera

ture

, T, (

°C)

Time varying heat flux qpeak=210 kW/cm2"Pulse Length: 500μsecPulse Frequency: 500Hz5,000 micron copper layer.

Figure 6.Case 4 solution:

time-varying heatflux, surface

temperature history

961

Effect ofboundarycondition

approximation

Convergence in the L2-NormThe first part of the study focusses on quantifying the convergence rates in theL2-norm on refined spatial and temporal grids for each of the methods, boundaryconditions, and boundary condition approximations. Although both methods areformally second-order spatially accurate, boundary condition implementation doesaffect the convergence rate and accuracy of the solution. The L2-norm of the spatialerror at any time is defined as (Becker et al., 1981):

:e:L2 ¼Z L

0T xð Þ�Ta xð Þð Þ2dx

� �1=2

ffiXNi¼1

Ti�Ta;i� �2

Dxi

!1=2

(20)

where Ta is the analytical solution. The convergence rate in the L2-norm ona uniformly spaced mesh is given by:

:e:L2 ¼ CDxP (21)

where C is an arbitrary constant and P is the convergence rate of the solution.For second-order accurate methods P¼ 2.When plotted in log-log coordinates, the L2-norm

Tem

pera

ture

, T, (

°C)

Time varying heat flux qpeak=210kW/cm2"Pulse Length: 500μsecPulse Frequency: 500Hz5,000 micron copper layer.

1,600

1,400

1,200

1,000

800

600

400

200

0 0.05 0.1 0.15 0.2

Depth, x, (cm)

Semi-Analytical - Pulse

Semi-Analytical - CoolingNumerical - Pulse

Numerical - CoolingP1

P2P3

C1

C3

C2Figure 7.Case 4 solution:time-varying heatflux, in-depthtemperature profile

1,200

1,000

800

600

400

200

00 5 10 15 20

Depth, x, (microns)

Tem

pera

ture

, T, (

°C)

Case 5: Exponential Volumetric Heating: Is=2,000kW/cm2

�=4,000cm–1

Crank-Nicolson methodSolution every 0.6μsecTotal simulation time: 3.0μsec

AnalyticalNumericalSurface Heating

Figure 8.Case 5 solution:exponential sourceterm, surfaceintensity,Is¼ 2,000 kW/cm2

962

HFF25,4

should be a straight line with the slope being the convergence rate (as long as the erroris in the asymptotic range).

The L2 norms were computed for each of the cases. Simulations were run on sixmeshes ranging from 25 to 800 cells, and number of time steps ranging from 15,625 to4,000,000, both doubling each increment. Although this translates into different Dx andDt for each of the cases (we run for various lengths and simulation times), it is easy toshow from the definition of the L2-norm, that the effect on the L2-norm scales.

The L2 norms for Case 1 are presented in Figures 9-13. Figures 9 and 10 show thefirst-order boundary condition implementation for implicit and Crank-Nicolsonmethods, respectively. In Figure 10, all lines overlay, so for clarity, only the coarsestand finest time steps are shown. Figures 11 and 12 show the same for the second-orderimplementation. A trend that is apparent for each of the boundary conditionapproximations is that as the spatial mesh is refined, the fully implicit method requiresa much finer time step than the Crank-Nicolson method to maintain second-orderspatial convergence through the entire range of mesh refinements. At the fewer numberof time steps, the L2-norm for the implicit method flattens out as the mesh is refined,

–4

–5

–6

–7

–8

–9

–10

–11

–12

–13–17 –16.5 –16 –15.5 –15 –14.5 –14 –13.5 –13 –12.5

Ln(Δx)

nΔt=15,625nΔt=31,250nΔt=62,500nΔt=125,000nΔt=250,000nΔt=500,000nΔt=1,000,000nΔt=2,000,000nΔt=4,000,000

Ln(||

e|| L2

)

L2-Norms Fully Implicit MethodCase 1: Constant Surface Temperature1st Order Derivativet = 1μsec

Figure 9.Case 1: L2-norms,

constant temperatureboundary condition,first-order accuratederivative BC, fully

implicit method

–5

–6

–7

–8

–9

–10

–11

–12

–13–17 –16.5 –16 –15.5 –15 –14.5 –14 –13.5 –13 –12.5

Ln(Δx)

Ln(||

e|| L2

)

L2-Norms Crank Nicolson MethodCase 1: Constant Surface Temperature1st Order Derivativet = 1μsec

nΔt=15,625nΔt=4,000,000

Figure 10.Case 1: L2-norms,

constant temperatureboundary condition,first-order accurate

derivative BC,Crank-Nicolson

method

963

Effect ofboundarycondition

approximation

and requires 4,000,000 time steps to maintain second-order spatial convergencethroughout the entire range of mesh sizes. The Crank-Nicolson method maintainssecond-order spatial convergence even at the coarsest time step. Figure 13 comparesthe Crank-Nicolson and implicit methods at the finest (4,000,000) number of time stepsusing both the first-order and second-order boundary derivative approximations.The figure shows that at the finest time step, both fully implicit and Crank-Nicolsonmethods produce lower error for the second-order approximation than for thefirst-order approximation, but the magnitude of the error is approximately the same foreach time discretization.

Cases 2a and 2b apply convection boundary conditions to a true convection problemand to a temperature prescribed boundary equivalent to Case 1, respectively. Case 2bapplies the temperature boundary condition using the actual temperature boundarycondition, and by weakly enforcing the boundary condition using a very highconvection coefficient and setting the ambient temperature to the desired surfacetemperature. As discussed previously, the convection boundary condition isapproximated by three methods, and so Case 2b is approximated by five different

–5

–6

–7

–8

–9

–10

–11

–12

–13

–14–17 –16.5 –16 –15.5 –15 –14.5 –14 –13.5 –13 –12.5

Ln(Δx)

nΔt=15,625nΔt=31,250nΔt=62,500nΔt=125,000nΔt=250,000nΔt=500,000nΔt=1,000,000nΔt=2,000,000nΔt=4,000,000

Ln(||

e|| L2

)

L2-Norms Fully Implicit MethodCase 1: Constant Surface Temperature2nd Order Derivativet = 1μsec

Figure 11.Case 1: L2-norms,constant temperatureboundary condition,second-orderaccurate derivativeBC, fully implicitmethod

–5

–6

–7

–8

–9

–10

–11

–12

–13

–14–17 –16.5 –16 –15.5 –15 –14.5 –14 –13.5 –13 –12.5

Ln(Δx)

Ln(||

e|| L2

)

L2-Norms Crank Nicolson MethodCase 1: Constant Surface Temperature2nd Order Derivativet = 1μsec

nΔt=15,625

nΔt=4,000,000

Figure 12.Case 1: L2-norms,constant temperatureboundary condition,second-orderaccurate derivativeBC, Crank-Nicolsonmethod

964

HFF25,4

numerical boundary conditions total for each method. The same trends in the L2 normsseen in Case 1 are seen in these two cases. Rather than show all of the L2 norms, wefocus instead on comparing the various boundary condition and time discretizations atthe 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 inmind this is the finest time step), and the error depends on the boundary conditionapproximation, so we only show the L2 norms for the Crank-Nicolson method in thefigures. 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 methodperforms only slightly better than the first-order method. However, Figure 15 showsthat for a temperature boundary condition prescribed weakly, the second-order methodperforms significantly better than the first-order method. It is seen by comparing thetwo figures that the overall level of error is higher for the infinite convection coefficientthan for the finite convection coefficient. Figure 15 also shows that the first- andsecond-order approximations for the boundary derivative perform the same regardless

–5

–6

–7

–8

–9

–10

–11

–12

–13

–14–17 –16.5 –16 –15.5 –15 –14.5 –14 –13.5 –13 –12.5

Ln(Δx)

Ln(||

e|| L2

)

L2-Norms at 4,000,000 time stepsCase 1: Constant Surface TemperatureNcells = 25-800t =1μsec

1st Order

2nd Order

ImplicitCrank Nicolson

Figure 13.Case 1: L2-norm

comparison at finesttime step constant

temperatureboundary condition

–6

–7

–8

–9

–10

–11

–12

–13

–14

–15–17 –16.5 –16 –15.5 –15 –14.5 –14 –13.5 –13 –12.5

Ln(Δx)

Ln(||

e|| L2

)

L2-Norms at 4,000,000 time stepsCase 2a: Convection BCh∝ =10E6t =0.2μsec

1st order - CONV BC

2nd order - CONV BC

Extrapolation - CONVBC Figure 14.Case 2a: L2-norms

comparison,convection

boundary condition,h∞¼ 106W/m2-K

965

Effect ofboundarycondition

approximation

of whether the derivative is applied directly as a temperature boundary condition(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 17for the fully implicit and Crank-Nicolson methods, respectively. As with the previouscases, the Crank-Nicolson method performs better than the fully implicit methodthrough the entire range of time steps and mesh sizes. The implicit method displays thesame trends as the constant temperature case except that there is greater flattening ofthe L2 norms using fewer time steps as compared with Case 1. The Crank-Nicolsonmethod displays second-order convergence on all meshes and all time steps, whereasthe fully implicit method slowly converges to second-order spatial accuracy as the timestep is reduced. The overall magnitude of error on a given mesh is slightly lower for theCrank-Nicolson method.

Case 4 is the three-pulse case that produces a time-varying surface heat flux. The L2norms are shown in Figure 18 for both the fully implicit and Crank-Nicolson methodsat 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

–6

–5

–4

–3

–2

–7

–8

–9

–10

–11

–12

–13–16.5 –16 –15.5 –15 –14.5 –14 –13.5 –13 –12.5

Ln(Δx)

Ln(||

e|| L2

)

1st order - CONV BC, TEMP BC2nd order - CONV BC, TEMP BC

Extrapolation - CONV BC

L2-Norms at 4,000,000 time stepsCase 2b: Constant surface temperature h∝ =10E20t =0.2μsec

–17

Figure 15.Case 2b: L2-normscomparison, constanttemperatureboundary conditionapplied as convectionboundary condition,h∞¼ 1020W/m2-K

–7

–8

–9

–10

–11

–12

–13

–14

–15

–16–18.5 –18 –17–17.5 –16.5 –16 –15.5 –15 –14.5 –14

Ln(Δx)

nΔt=15,625nΔt=31,250nΔt=62,500nΔt=125,000nΔt=250,000nΔt=500,000nΔt=1,000,000nΔt=2,000,000nΔt=4,000,000

Ln(||

e|| L2

)

L2-Norms Fully Implicit MethodCase 3: Constant Surface Heat Fluxt = 0.6μsec

Figure 16.Case 3: L2-norms,constant heat fluxboundary condition,fully implicit method

966

HFF25,4

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 eachmesh. The error during a cooling cycle is less than the error during a heating cycle fora given mesh and time step. This is not surprising, as during a heating cycle both thespatial and temporal temperature gradients are higher. This implies that to improvecomputational efficiency one could take a larger time step during cooling cycles,which are usually much longer in duration than heating cycles, and reduce simulationtimes substantially.

The parameters of Case 5 are identical to Case 3, with the difference being that theincident heat flux is absorbed volumetrically through Beer’s law, rather than asa surface heat flux. We chose an absorption coefficient of 4,000 cm�1. Although thisvalue is lower than the actual value for copper, we chose it to allow absorption througha greater depth of the layer. The higher value of copper would have caused the incidentflux to absorb very near the surface, basically acting as a surface heat flux. With thisvalue, 95 percent of the incident energy is absorbed in the first half of the layer. The L2norms for this case are essentially identical to Case 3 (Figures 16 and 17) in both trendand magnitude.

nΔt=15,625

nΔt=4,000,000

–8

–9

–10

–11

–12

–13

–14

–15

–16

–17–18.5 –18 –17–17.5 –16.5 –16 –15.5 –15 –14.5 –14

Ln(Δx)

Ln(||

e|| L2

)

L2-Norms Crank Nicolson MethodCase 3: Constant Heat Fluxt = 0.6μsec

Figure 17.Case 3: L2-norms,constant heat flux

boundary condition,Crank-Nicolson

method

0

–1

–2

–3

–4

–5

–6

–7

–8

–9

–10–12.5 –12 –11 –10 –9 –8–11.5 –10.5 –9.5 –8.5

In(Δ x)

In(||

e L2||

)

nΔt=15,625, Crank-NicolsonnΔt=15,625, Fully ImplicitnΔt=4,000,000, Crank-Nicolson & Fully Implicit

t=4,250μsecmid-point of 3rd pulse

t=5,250μsec750μsec after end of last pulse

Figure 18.Case 4: L2-norms,time-varying heat

flux boundarycondition, fully

implicit and Crank-Nicolson methods

967

Effect ofboundarycondition

approximation

Local error analysisHaving quantified the convergence properties for each of the methods and boundarycondition implementations, we investigated local error next. In this part of the study wefocussed 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 Dxand Dt that can be used and still achieve a given accuracy. For this study we chose2 percent maximum error to be sufficient for our purpose.

For Case 4 we investigated the local error at the midpoints of the heating and coolingcycles and noted the maximum error in the domain at each of the times. The length ofthe 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 mstotal 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 meshesare 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 isnot surprising since the spatial and temporal gradients are highest during the firstpulse and decrease due to diffusion at later times. In all cases, the error for the fullyimplicit method is higher than the Crank-Nicolson method for an equal number of timesteps. Concentrating on the midpoint of the first pulse (P1), because this is where theerror is greatest, we see that on the 25-cell mesh, the fully implicit method starts outmuch higher than the Crank-Nicolson method for the fewest number of time steps, andgradually decreases as the number of time steps increases, whereas, the Crank-Nicolsonmethod remains relatively constant. However, both are approaching a similar error as

tμsec

250P1

1,250C1

2,250P2

3,250C2

4,250P3

5,250C3

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 1.554 63.67 26.99 16.40 6.60 7.53 3.42 5.33 2.02 5.15 2.26 4.26 1.588 48.17 28.60 11.92 6.79 5.78 3.95 3.62 2.02 3.65 2.62 2.93 1.5916 39.19 29.01 9.47 6.83 4.82 4.08 2.77 2.02 3.21 2.70 2.25 1.5932 34.32 29.11 8.19 6.85 4.48 4.11 2.39 2.02 2.98 2.72 1.92 1.5964 31.77 29.14 7.52 6.85 4.30 4.12 2.21 2.02 2.85 2.73 1.76 1.59128 30.47 29.14 7.19 6.85 4.21 4.12 2.12 2.02 2.79 2.73 1.67 1.59

Table II.Case 4: time-varyingflux, maximumlocal percent error,25-cell mesh

tμsec

250P1

1,250C1

2,250P2

3,250C2

4,250P3

5,250C3

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 2.674 45.15 5.36 11.54 1.10 5.78 2.65 3.47 0.73 3.65 2.13 2.85 0.578 26.23 2.17 6.38 0.40 3.45 0.59 1.77 0.14 2.13 0.47 1.49 0.1016 15.03 1.95 3.53 0.44 2.01 0.25 0.94 0.13 1.24 0.15 0.80 0.1032 8.82 2.04 2.02 0.45 1.20 0.29 0.53 0.13 0.73 0.18 0.45 0.1064 5.52 2.06 1.24 0.45 0.76 0.31 0.33 0.13 0.47 0.19 0.28 0.10128 3.81 2.07 0.85 0.45 0.54 0.31 0.23 0.13 0.33 0.19 0.19 0.10

Table III.Case 4: time-varyingflux, maximumlocal percent error,100-cell mesh

968

HFF25,4

the time step is reduced. The same trend is apparent on the 100-cell mesh. Essentiallyon each of the 25- and 100-cell meshes, the error flattens out as the number of time stepsis increased because the spatial meshes are not refined enough to maintain thetheoretical time convergence rate, and so the error remains constant under furthertime step refinement. But the Crank-Nicolson method reaches that limit with farfewer time steps than the fully implicit method. For example, on the 100-cell mesh, theCrank-Nicolson method error has been reduced to 2.17 percent by eight time steps,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 oftime steps actually increases slightly, but ultimately decreases below the coarser meshas the number of time steps is increased. This is consistent with the behavior of the L2norms, which showed that finer meshes required finer time steps to reach thetheoretical convergence rate. Figure 19 shows the error at 250 μsec for both methods onthe 800-cell mesh as the number of time steps (during the pulse) is increased from 2 to128. The error for the Crank-Nicolson methods starts much lower and decreasesrapidly, so that by 16 time steps it is reduced to ~2 percent. The fully implicit methoddoes not fall below 2 percent until 128 time steps. The figure clearly shows, as is thetrend for the 25 and 100 cells meshes, that the error is approaching the same valueunder time step refinement, but the Crank-Nicolson method reaches the limit in farfewer time steps.

80

60

40

20

0

0 50 100

nΔt

ncell = 800, t = 250μsec

150

Implicit MethodCrank-Nicolson Method

% E

rror

Figure 19.Case 4: percent

error under timestep refinement,

800-cell mesh

tμsec

250P1

1,250C1

2,250P2

3,250C2

4,250P3

5,250C3

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 4.104 43.88 9.74 11.20 2.82 5.58 6.83 3.35 2.17 3.69 5.66 2.76 1.828 24.62 4.48 6.00 1.37 3.22 3.14 1.66 1.03 1.98 2.61 1.39 0.8516 13.24 1.70 3.12 0.54 1.75 1.21 0.82 0.38 1.07 1.01 0.70 0.3132 6.92 0.43 1.60 0.12 0.92 0.32 0.41 0.08 0.56 0.27 0.35 0.0664 3.56 0.06 0.81 0.01 0.47 0.05 0.21 0.00 0.29 0.04 0.18 0.02128 1.82 0.03 0.41 0.01 0.24 0.00 0.10 0.00 0.15 0.00 0.09 0.02

Table IV.Case 4: time-varying

flux, maximumlocal percent error,

800-cell mesh

969

Effect ofboundarycondition

approximation

For Case 5 we investigated the local error for the number of time steps ranging from 2to 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 isunacceptably high at the fewest time steps for the implicit method. However, theCrank-Nicolson method begins to produce less than 4 percent error with only four timesteps. Increasing the number of time steps to eight reduces the error to less than1 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 lowernumber 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 theCrank-Nicolson method is reducing quadratically.

Simulation timesThe computation times were recorded for each method using the 800-cell mesh and32,000 time steps. The code was profiled using Intel Vtune Amplifier®, and the timespent in each subroutine was computed. The large number of time steps and cells wasrequired to minimize the scatter in the computed CPU time. The only differencebetween the fully implicit method and the Crank-Nicolson method is in the subroutinethat calculates the tridiagonal coefficients. Because simulation times varied somewhatfrom 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 theCrank-Nicolson method used 3.410 sec. This represents approximately a 12.4 percentincrease of the Crank-Nicolson method over the implicit method. The total simulationtimes (including time spent in other subroutines) was 11.874 sec and 12.241 sec forthe fully implicit and Crank-Nicolson methods, respectively. This represents onlya 3.1 percent increase in overall simulation time for the Crank-Nicolson method over thefully implicit method.

ConclusionsBased 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 fewertime steps than the fully implicit method. The second-order temporal accuracy of theCrank-Nicolson method manifests itself in requiring far fewer time steps to displaysecond-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

Nc¼ 10 Nc¼ 20 Nc¼ 401.5 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 9.294 20.26 3.77 15.4 1.02 20.79 4.10 15.66 1.10 21.08 4.20 15.81 1.168 10.83 0.76 8.03 0.15 11.07 0.97 8.14 0.26 11.21 1.05 8.21 0.3016 5.52 0.21 4.05 0.10 5.58 0.11 4.07 0.04 5.64 0.16 4.09 0.0432 2.83 0.16 2.06 0.09 2.82 0.05 2.04 0.03 2.83 0.02 2.05 0.0164 1.49 0.14 1.08 0.09 1.43 0.04 1.03 0.03 1.42 0.01 1.03 0.01128 0.82 0.14 0.58 0.09 0.73 0.04 0.53 0.03 0.72 0.01 0.52 0.01

Table V.Case 5: volumetricheating, maximumlocal percent error

970

HFF25,4

meshes, as the numerical boundary discretizations are conducted in a 1-D sense on eachboundary. It is not clear that this behavior would extend to 2-D and 3-D unstructuredmeshes, as the numerical discretization procedures on the boundaries are not the sameas for structured meshes.

For the temperature prescribed boundary condition, the first-order accurateapproximation tended to flatten out somewhat less at the fewest number of time stepscompared to the second-order accurate method using the fully implicit method.However, the overall error on a given mesh was lower for the second-order accurateexpression at the finest time step. The Crank-Nicolson method maintained second-orderconvergence throughout the range of meshes for all time steps for both the first- andsecond-order methods. At the finest time step, the overall level of error on a given meshwas equal for the implicit and Crank-Nicolson methods. For the convection boundarycondition, the second-order boundary condition produced the smallest error on allmeshes and for all time steps. The constant heat flux and volumetric boundaryconditions produced trends in the L2 norms similar to the constant temperatureboundary condition. The time-varying heat flux boundary condition showed that theerror in the L2-norm was lower between pulses than during pulses, indicating thata larger time step could be taken between pulses. For coarse discretizations, theCrank-Nicolson method performs far better than the fully implicit method for the twocases studied. The Crank-Nicolson method allows up to a factor of 16 greater time stepthan the implicit method for equivalent error on the same mesh. Furthermore, thecomputational penalty per time step of the Crank-Nicolson method is nearly negligible,allowing substantially reduced run times. Finally, the Crank-Nicolson methodcombined with the second-order accurate expression for the boundary derivativeapplied as a convection boundary condition performed the best overall out of all of thenumerical boundary condition approximations tested. This formulation also allowedtemperature prescribed boundary conditions to be imposed weakly through anarbitrarily large convection coefficient.

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Corresponding authorMartin Joseph Guillot can be contacted at: mjguillo@uno.edu

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