AIAA 93-2770 Flux Based Finite Volume Representations For General Thermal Problems R.V. Mohan and K.K. Tamma University of Minnesota Minneapolis, MN

AIAA 28th Thermophysics Conference July 6-9, 1993 / Orlando, FL

For permlulon to copy or republlsh, contact the Amerlcsn Instltute of Aeronautlcs and Astronautics 370 L'Enfant Promenade, S.W., Washlngton, D.C. 20024

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Flux-Based Finite Volume Representations for General Thermal Problems

Ram V. Mohan Graduate Research Assistant

Kumar K. Tamma Associate Professor

Department of Mechanical Engineering Institute of Technology 111 Church Street S.E. University of Minnesota Minneapolis, MN 55455

Y

Abst rac t

The present paper describes new Flux Based Finite Volume element representations for general thermal prob- lems in conjunction with a generalized trapezoidal 7~ - fam- ily of algorithms which are formulated following the spirit of what we term as the Lax-Wendroff based Finite Volume formulations. In comparison to the traditional control vol- ume developments and representations adopted in the nu- merical solution of thermal problems, the new flux based representations introduced in this paper offer an improved physical interpretation of the problem along with computa- tionally convenient and attractive features. The space and time discretization emanate from a conservation form of the governing equation for thermal problems, and in conjunc- tion with the flux - based element representations give rise to a physically improved and locally consemtive numerical formulations. The present representations developed in this paper seek to involve improved locally conservative proper- ties, improved physical representations and computational features. Developments are presented here based on a two dimensional, bilinear finite volume element and can be, ex- tended for other cases. Time discretization based on a 7~ - family of algorithms in the spirit of a Lax-Wendroff based Finite Volume formulations are employed. Numerical ex- amples involving linear/nonlinear steady and transient sit- uations are shown to demonstrate the applicability of the present representations for thermal analysis situations.

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In t roduct ion

The accurate prediction of steady/transient tbermal response in structures and materials (isotropic and com- posite) is of considerable practical importance for numer- ous engineering applications. Nonuniform and/or nonlinear heating may have a significant effect on the performance characteristics and is of utmost concern in the development of advanced structural materials, component behavior, and

in the design/analysis of engineering structures. Computa- tional techniques have matured to significaot levels for ther- mal problems. Finite Element Methods, Finite Difference Methods and Finite Volume Methods have been the more prominent approaches to-date for general thermal analysis situations.

The Finite Element Method(FEM) provides an ef- fective approach for complex geometries, and a systematic and general way of handling the houndarywnditions. The finite element approach follows a philosophically different approach than the finite volume methods and the finite dif- ference method. In elasticity problems, for eg., there exists a variational extremum principle, such that the minimiza- tion of potential energy or the application of the principle of virtual work leads, naturally, to the formulation of the discrete model. In tbermal problems such a natural for- mulation with a clear physical interpretation does not exist. Galerkin finite element formulations for the heat conduction problems based on the differential equations lead to finite element equations whicb may he locally non conservative in nature. Someof the related papers appear in references[l-51 and references thereof.

The Finite Difference Method(FDM) is another com- monly used numerical technique for thermal problems in which the approximate solution is obtained by directly dis- cretizing the governing differential equation. This form of the finite difference approximations bas been historically re- stricted to rectangular domains and meshes, and later have been extended to include applicability to general, ortbogo- nal curvilinear coordinate systems 16, 7).

Another approach that has received attention for thermal analysis is the Control Volume approach. In this approach, an energy balance is applied to discrete control volumes. and, only where surface fluxes require approdma- tion discretization is used. The property of the control vol- ume approach is that the resulting finite differenceequations are conservative, and the discrete equations maintain an ac- curate account of energy flows in the domain. i& 91. In the

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applicationdthe finite difference and control volumemeth- ods, the.coordinate system must be defined over the entire solution domainprior to effecting the discrete method. The finite alenmnianethod, however removes the above disad- vantage by AEzing a coordinate system which is local to each individual element. Another approach is the use of fi- nite element philosophy (such as isoparametricformulations based on a finite volume) directly to the conservative form of the heat .conduction equation, and obtaining the discrete form based on the energy balance. This approach, called the Finite Vnlnme Method(FVM), has been cited to have the benef i tewhhehi te element method in its applicability to a general curvilinear domain, while preserving the con- servation of energy [10-13].

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In all these above approache, the semidiscretization process for the spatial discretization in a steady or transient analysis is done by approximating the temperatures within each element or control domain and subsequently solving for the unknown temperature field. Such approximations are quite standard today and form the basis of many com- mercial codes. In addition to these, discretizations reqnir- ing mixed variational principles in which different variables are employed as unknowns(mixed methods) have also been suggested [14]. However, these mixed approximations tend to be in general costlier for thermal problems for solving the dependent variables ( for example, both temperature and heat flux are solved for, and, typical discretized ele- ment representations inherit similar characteristics as the traditional Zormulations frequently employed). More recent research e h t s propose alternate formulations based on flux based element representations via the finite element method for thermal analysis [15, 161. Employing such a tecbnol- ogy and formulatiom for the finite volume method, helps to capitalize on both the local conservative properties of the finite volume method and the advantages of the flux based representations leading to an improved physical iuterpreta- tion and numericalattributes for thesolution of the thermal problems.

The present paper focuses attention on providing such an improved physical interpretation and numerical at- tributes in the context of the finite volume method. The dis- tinguished advantages and features of the present methodol- ogy and formulations include: (1) Improved physical inter- pretation, (2) Permit independence of element integral eval- uations from thermophysical nonlinearities,(3) Permit nat- ural introduction and implementation features for general linear/nonlinear boundary effects, and (4) Offer good com- putationally convenient and attractive features for general steady/transient linearfnoniinear thermalanalysis. The tra- ditional locally conservative finite volume discretized repre- sentations approximate only the temperature field and sub sequently involve the solution of the unknown temperature field. The present flux based finite volume representations, although they employ approximations of both temperatures and heat fluxes in the semidiscretization process, involve

'v the solution of only a single field variable; however, they in- herit several.computationally convenient and attractive fea- tures with improved physical interpretations and physical attributes as described earlier. Numerical examples are de-

scribed to validate applicability of the present formulations for general thermal problems.

Geometry and E lemen t Definition

The typical solution domain is shown in Fig. 1 which is subdivided into linear quadrilateral finite elements. A typical element is shown in Fig. 2, with the local coordi- nate system ( ( , q ) . For the individual element, the local node numbers range from 1 through 4 as shown and the temperature field T, the heat fluxes qr,qv and the global CD- ordinates x,y are interpolated using bilinear isoparametric formulations of the form

and

y = ~ N ; Y ; ( 5 ) k l

where the shape functions N; are defined by

Radiarioo Hcat T - k

Smfacz Hcat f low

Fig, 1: Typical solution domain

Fig. 2: A typical finite volume element

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U

The x and y derivatives of the temperature can be deter- mined as

where the x and y derivatives of the shape functions appear- ing in the Equation (7) are determined by

Considering a general line segment shown in Fig. 3, the normal vector while traversing along from point 1 to point 2 is defined by

t,

1

Fig. 3: Typicalsurface line segment

With the above definitions, the associated numerical formulations for the flux based finite volume representations in conjunction with a generalized 77 - family of trapezoidal formulations are described next for applicability to general thermal analysis.

Development Of Flux-Based F in i t e Volume Repre- sentations

In this section the Flux Based Finite Volume for- mulations in conjunction with a generalized trapezoidal 7~ - Family of representation is described. Following Tamma and Xamburu 115, 161, we demonstrate next the derivation of the time discretized formulations followed by the space

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discretization in the spirit of a Lax4%admff..based Finite Volume Flux Based Formulation.

T i m e Discretization

The classical heat conduction problem is first cnnve- niently represented in the following conservation form as

where

where T is the temperature field. In general, typical thermc- physical properties such as K, p, c, etc., may be temperature dependent. Typical boundary conditions which may exist include prescribed temperature fields acting on a segment rI of the boundary and heat flux conditions acting on r2, due to surface heating, convection and radiation. Here also the tbermophysical properties may be temperature depen- dent. These characteristics together with radiation makes the thermal problem nonlinear. The initial conditions in the body are assumed known.

Next, in the spirit of the Lax-Wendroff type formu- lations [6, 171, consider for illustrative purposes, the Taylon series expansion of a single scalar for U in time t = t", cor- rect to second order which is of interest here, (higher orders are also permissible) as

Atz - 2 Urn+' = U" + A t o m + -Un + O(At3) (12)

or equivalently, introducing a free parameter y~ (where 0 2 73- 2 1) results in

Urn+' = U" + AtP*

U"* = yru"+' + (1 - 7 T ) U "

(13)

(14) where

and At is the time step.

Since the governing beat conduction equation is valid at every time level, the conservation form of the heat con- duction equation at R + x level can be represented as

With this the temporal discretized form is obtained from

U"+' = U" + At[fI"* - E>* - F , 3 (16)

The space discretization process for the generdized 73 - family of thermal representations is described next by introducing a locally conservative finite volume procedure.

p* + E;* + F,;* = H"* ( 1 5 )

Fq. ( I ? ) as

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Space Discretization

In this a single, isolated finite volume element is first considered. The application of energy balance to this e l e ment will give rise to the element level matrices and flux vec- tors which are related to the nodal temperatures and nodal heat fluxes of the element. Once the appropriate capaci- tance and heat flux vectors are developed, assembly des in a similar sense of the finite element formulations can be employed to construct the global equation system from the element level equations.

'L/

The development of numerical equations is based on a single element shown in Fig. 4. The single control volume element is subdivided into four internal control volumes, each of which is associated with the corresponding nearest neighboring node of the element. In the linear quadrilateral element shown in Fig. 4, the control volume boundaries are chosen to be coincident with the element exterior boundaries and with the local coordinate surfaces defined by E = 0 and 7 = 0. This choice is consistent from element to element, in the whole formulation and this boundary selection makes the evaluation of the integrals defined in the formulation easier.

The energy balance is now applied to one such con- trol volume and is expressed as:

Net rate of conduction into control volume =

Rate of generation within control volume + Rate of w change of energy within the control volume

For the control volume associated with a typical node 3 in Fig. 4, this energy balance can be mathematically exoressed as

where the limits of the integral correspond to the control volume associated with the node 3. The subscripts e l and e2 in Eq. (17) refer to the energy flows into control volume 3 through surfaces which are on the exterior of the element and arises either from the physical domain boundary or from adjacent elements. In the case of adjacent elements, the heat flux cancels with each other, while in the caSe of the physical boundary, the boundary conditions determine the contribution of these terms.

The interior, terms which give rise to the conduction matrix coefficients are considered next. In general, the heat flow through a surface can be expressed as

In this case d 3 = dy; - d z j

1

Fig. 4 Single element with finite volume subdivision and heat flows

Introducing the above definitions, and in conjunction with generalized 7~ - family of algorithms, the following discretized equation system is obtained.

[C]"*{AT}"* = (Q.}"r"+(Qs}"f"+{Q=}"r" (21)

In the above equation, { Q a } corresponds to the conduction heat flux, {Qb} corresponds to the associated boundary flux acting on I'? of the element boundary, and (Q.} corresponds to the contribution due to the internal heat generation term. The computations are done at n + y ~ ( 0 5 7~ 5 1 ) based on the selection of the algorithm.

Based on the energy balances of each of the control volumes, the capacitance matrix entries are given by the following expressions

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(30) 1 = At FUFIZ + FzzEi I where Fll and FZZ involve the element integrals which are independent of nonlinear thermophysical properties and can also be evaluated in a closed form. X refers to the sum of all sub finite volumes. In a nonlinear transient analysis these integrals are computed only once and need not be repeat- edly updated. The quantities Fiz and F,, corresponding

t/ to the conduction fluxes qz and qv are evaluated at nodal points in terms of unknown temperature fields for a general anisotropic material following Fourier's law as follows.

These are evaluated at each node in a finite volume ele- ment. The nonlinear themnophysical properties are only .involved in the above form which can be included directly, without disturbing the element integrals. With the above formulations, the unknown quantity involved here are only the temperatures, unlikeother mixed formulations which in- volve the solution of both temperatures and heat flnx fields. In all of the above formulations the element integrals are evaluated along a line using one point Gaussian Quadrature like the traditional finite volume formulation.

Typical natural boundary conditions such as the flux contributions pin; acting on rz of the boundary (could be associated with an edge or surface) due to surface radia- tion, convection and surface heating are introduced via the boundary term { Q b } in a very natural and direct manner and in a way thereby avoiding the need of correct physical way, avoiding the need of'reevaluating element boundary integrals. These approximations are performed via

As an illustration, a typical load vector due to edge heating, edge convection and edge radiation are given by the line integrals, with the flux vectors taken in the physical for m for the domain r; and element load vectors are evaluated using the following form for an edge of length 1.

For the volumetric heat source 3, the corresponding load vectors are given by

The terms related to surface radiation, convection are evaluated in a similar way as in traditional finite volume formulations.

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In summary, the flux based finite volume discretized equations via the 7r - family of trapezoidal representations w are represented as

where represents the sum of all the sub finite volumes

For general steady state situations, the Eq. (39) readily reduce to

where the element integrals are described earlier (and the term At is not present, i.e. At = 1)

Based on these representations the following remarks can be made regarding the flnx b a d representations as applied to the finite volume method via the 7~ - family of time stepping algorithms:

Remark

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1. The discretized finite volume formulations involving flux based representations are shown in a different perspective (in comparison to traditional formulations customarily adopted) and offer several computation- ally attractive and convenient features and enable an improved physical interpretation for thermal modeling and analysis. This along with local conservation prop- erty of the finite volume method provides an effective modeling strategy for thermal analysis of structures.

2. The vector {Q,,] can be interpreted physically as that which is solely associated with the mode of conduction heat transfer. The vectors qz and q,, contained in {Qa] represent nodal heat flnx components in their respec- tive coordinate directions and can be evaluated using Fourier law as shown earlier. As a consequence, the in- tegrals in {Qa] are independent of temperature depen- dent thermal conductivities and thus provide a compu- tationally attractive feature since these integrals need to be computed only once during analysis. The varia- tions in thermal conductivities enter the formulations via the vectors qr and qv as shown earlier.

3. The vector { Q b ] can be interpreted physically as that which is solely associated with applied natural bound- ary conditions. As a consequence, typical boundary conditions can he introduced directly and naturally

via the term (q;ni) without disturbing the integral con- tained in { Q b ] . Thus, generd nodinear/linear bound- ary conditions such as heating, convection and radi- ation can be naturally and effectively introduced and the integrals associated with r2 also need to be com- puted only once in case of nonlinear transient analy- sis. Material thermophysical nonlinearities associated with the applied natural boundary conditio= and ef- fects due to radiation do not afect the associated in- tegrds contained in { Q b }

4. The quantity {Qc] physically represents the term as-

5. The parameter h ( 0 <_ 7~ 5 1) controls the accuracy and stability of the method, thereby permitting either explicit or implicit formulations via the time integra- tion procedure.

sociated with internal heating effects.

Test Problems

To validate the present developments the following test problems are considered. These test problems with var- ious material nonlinearitiee and radiation effects cover the general range of potential problems encountered in engineer- ing practice. In transient problems, the timeintegration has been performed using the generalized trapezoidal 7~ - family of algorithms. The test examples are described next.

Nonlinear Convecting and Radiating F in - Steady S t a t e Analysis

This problem validates the capadility of the pro- posed flux based finite volume representations for evaluating steady state (involving nonlinearities) thermal situations. The test problem considered is steady state analysis of a one - dimensional conducting-convecting-radiating fin as shown in Fig. 5 modeled using two dimensional plane elements. The temperature at one end is prescribed to be 1000" K and the other end is prescribed to be at 0" K. The fin con- vects and radiates to an ambient temperature of 0' K, with a convection coefficient of 0.001 W/m-K, and the emissivity constant of 1.0 x 10-gW/m - K'. The thermal conductiv- ity is assumed to vary linearly with temperature, leading to nonlinear material properties. A similar analogous model is considered in 118). This example demonstrates the appli- cability of the present approach for handling nonlineaxities involving materials and radiation. The temperature distri- bution along the length is shown in Fig. 6.

Bimetallic P l a t e Subjec ted To Step Varying Tem- pera ture

This problem validates the capability of the proposed flux based finite volume representations for evaluating com- plex situations. This problem along with a ramp type tem- perature change was also analyzed by Cardona and Idelsohn [19] and Raiikar and Tamma (201 and a solution is available. The thermal model shown in Fig, 7, is composed of two different materials with considerable differences in their re

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spective thermal diffusivities. The model was subjected to a sudden change in temperature on one side as shown in Fig. 7. The finite element model consisted of 288 nodes, and a time step of 0.01 seconds was used with 7r = 0.0. Two cases of step and initial ramp temperature changes were consid- ered. Figure 8 and Fig. 9 shows the isotherm at time = 5 seconds and Fig. 10, Fig. 11 show the isotherm at time t = 15 seconds for both the cases. The results are in good agreement with Cardona and Idelsohn [IS], and Railkar and Tamma I201 which was done using transfinite element formu- ' lations. This example clearly demonstrates the applicability of the present formulations for wmplex situations.

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Two Dimensional Rec tangular P l a t e W i t h Convec- t ion and Radiation Along T h e Edges

This test example demonstrates the applicability of the present flnx based finite volume formulations with spe- cial attention to radiation. In this example a rectangular plate with nonlinear boundary conditions involving radia- tion and nonlinear material properties is considered. One quarter of the plate is modeled due to symmetry. The physi- cal plate and the finite element model are are shown in Fig, 12 and the material thermophysical properties are shown in Fig. 13. Comparison of results with the traditional fi- nite volume method demonstrates the effectiveness of the present formulations, as shown in Fig. 14.

Concluding Remarks U

The present paper described new flnx based finite volume formulations in conjunction with a generalized 7T - family of trapezoidal representations. The formulations involve approximation of both the temperature and con- duction flux fields. However, only the temperature field is solved for after invoking the Fourier law for the fluxes in the present form. The present formulations enable an im- proved physical interpretation and provide attractive com- putational features in conjunction with the 7T - family of trapezoidal representations for the Finite Volume Method (FVM). The developments been described based on a t w o dimensional bilinear finite volume element, and, the present development capitalizes on the features of both' finite 701- ume methods and flnx based representations. A generalized -/T - family of trapezoidal representations have also been described for the time discretization following the spirit of a Lax-Wendroff/Finite Volume based formulation. The ad- vantages and attractive features of the present representa- tions have been outlined. The architecture of the present representations and the various inherent characteristic fea- tures offer different perspectives and several computation- ally convenient features in comparison to traditional formu- lations for the solution of general thermal problems. The numerical examples clearly demonstrated the applicability to realistic thermal problems and the results are in excellent agreement with those via traditional foGulations.

Acknowledgments

The authors are very pleased to acknowledge sup port, in part, by NASA-Johnson Space Center / Lockheed Engineering and Space Sciences Co., Houston, Texas, and,by the Army High Performance Computing Research Center (AHPCRC), at the University of Minnesota on a contract from the Army Research Office. Additional support and computer grants were furnished by the Minnesota Super- computer Institute at the University of Minnesota, Min- neapolis, Minnesota.

References

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1121 G.E. Schneider and M. Zedan. Control volume based finiteelement formulation of the heat conduction equa- tion. Prog. Astronaut. Aeronaut., 86:305-327, 1983.

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, I..

K = KoA(l+7 T) KOA = lxllle-05 W-m/K 7 = 1.0

h = lxllle-03 W/m-K T h = 0.0 K TR = 0.0 K

Fig. 6 Temperature Distribution along the fin

u

I

-1

Region 1 Region 2

k = 0.346 k = 0.0173

c = 2.4127 c = 2.815 p = 1.0 p = 1.0

Fig. 7: Plate with bimaterial and numerical data

Fig. 8: .Isotherms in plate with bimaterials at time = 5 s w . - Case 1

Fig. 5: Nonlinear convective-radiative fin

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Fig. 9: Isotherms in plate with bimaterials at time = 5 secs. - Case 2

0 I 1 1 k I

Fig. 10: Isotherms in plate with bimaterials at time = 15 secs. - Case 1

ti

Fig. 11: Isotherms in plate with bimaterials at time = 15 secs. - Case 2

't

Plale geometry

Convection and radiation f ,' f / alongtheedges

f f f /J /J

Fdte elcmnt d e l of one quaner plate

Fig. convection and radiation

I?: Transient heat conduction in a plate with edge

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v

(a) Temp. Dependent K (b) Temp. Dependent h

(c) Temp.Dep. Emisrivity(c)

Fig. 13: Nonlinear temperature dependent thermai proper- ties

Fig. 14: Comparative transient temperature history of the center

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