Towards a mesh-free computation of transport phenomena · Towards a mesh-free computation of transport phenomena Bo2idar ~arler Laboratory for Flciid Dynat~zics and Thermodynnmics,

Towards a mesh-free computation of transport phenomena

Bo2idar ~ a r l e r Laboratory for Flciid Dynat~zics and Thermodynnmics, Faculty of Mechanical Engineering, Crniversity of Ljubljana, SI-l000 Ljubljana, Slovenia

Abstract

This paper formulates a computatiorlal procedure based on the Trefftz method for the solution of nonlinear transport phenomena. This new unified approach is particularly important when solving coupled, nonlinear, inhomogenous, anisotropic, multiphase, and multifield heat and mass transfer problems. Physical system represents the general transport equation, standing for a broad spectra of mass, energy, momentum: and species transfer problems. This equat,ion is cast into non-linear Poisson form and expanded with respect to t,he transport variable. Fully implicit time-discretizat,ion is used. The particular solution method is applied as a general solution framework. The solution of the inhomogenous part is based on the radial basis function global approximation, and the solution of the homogenoiis part is based on the Trefftz method Laplace equation fundamental solution global approximation. The discrete approximative met,hod results in a global point-collocation based mesh and hence eliminates the need for polyg~nizat~ion of the computational boundary and domain.

1 Introduction

1.1 Impulses

The development of efficient algorithrns for the numerical solution of par- tial differential equations (PDEs) is of major interest in applied sciences and engineering. Thc most popular discrete approximative neth hods for PDEs are nowadays the finite difference (FDM), the finite volume (FVM), the finite element (FEM), thc spectral (SM). and the boundary element (BEN) methods. Despite the powerful featlures of these methods, there are often substantial difficult,ies in appliying t>hem to realistic, geonletrically complex

Transactions on Modelling and Simulation vol 28, © 2001 WIT Press, www.witpress.com, ISSN 1743-355X

three-dimensional transient problems [l]. A common drawback of the men- tioned methods is the need to create a polygonization. either in the domain and/or on its boundary. This type of meshing is often the most time con- suming part of the solution process and is far from being fully automated.

1.2 Polygonization in multiphase transport phenomena problems

The numerical solution of coupled heat and mass transfer problems is be- coming increasingly important as a result of the computational modelling needs in diverse modern technologies. X broad class of such heat, mass, momentum, and solute transfer problems involves two or more phases, se- pareted by free (steady state) or moving (transient,) interphase boundaries. Due to the existence of complex shaped interphase boundaries, most of the numerical sirnulations of engineering gas-liquid and liquid-solid two-phase flows conducted so far have been based on averaged field equations with con- stitutive interphase relations solved on a fixed mesh. However, the diversity of the possible involved length scales. inhomogeneities, and anisotropics, usually requires the adaptation of the mesh with respect to high field gradi- ents and subsequent re-meshing. Recent rapid progress in computer perfor- mance however gives us a large prospect of rea,lizirig more detailed numerical simulation of multiphase systems. Physically sound information can unfor- tunatelly be perceived only from the numerical approaches which explicitly take into account the moving boundaries. The principal bot,t,leneck in these type of numerical methods is the time consurning re-meshing of the evolv- ing interphase boundaries and phase domains which limits such methods to problems with quite trivial phase patterns. The polygonization problem is thus even more pronounced in such type of front-tracking approach.

1.3 Fundamental solution based methods

One method for alleviating the described difficulties is tjo use the BEM [2], which requires only boundary polygonization in special cases like the potential flow. Realistic viscous laminar or turbulent situations may not be solved by these methods without domain polygonization either and lead to the boundary-domain integral methods (BDIM). In recent, years there has been corisiderable int,crcst to develop "mesh-frce" methods which do not involve extensive polygonization. hIost common among such nlethods is the d u d reciprocity boundary element method (DR.BEM) [3]. In this method, the weighted residual formulation of the governing PDE is reduced to a set, of boundary integrals only by using global interpolation of the domain integrals. One of the drawbacks of boundary integral methods is t,hat formula.t,ion involves the evaluation of hypersingular, strongly singular, singular or near-singular integrals. The accurate evaluation of t,hese integrals is usually conlputationally difficult and expensive. To circumvent the problem of having to evaluate boundary integrals, a class of boundary


collocation met,hods classified under the generic name "Trefftz method" can be used. Here the solutior~ is represented using layer potent,ials on non- physical surfaces, thereby circumventing the need to evaluate any integrals. By adopting such a method, one also avoids the problem of complicated surface polygonization, required in bot,h the traditional BDIM as well in t,he DRBEM [4]. This paper represents a logical upgrading of our DRBEM solution [S] of the general transport equation in sense of solving it on a complete "mesh-free" basis. The developments of Golberg [6] for linear and Balakrishnan and Ramachandran [7] for non-linear Poisson equat,ion have been used as a starting point for this study. Present paper explicitly takes into account all engineering boundary condition types and specifics of the Poisson equat,ion, originating from the general transport equation.

2 Governing equations

For the present purposes, a transport phenomena problem can be briefly dp- scribed in a general manner as the numerical solution of Eulerian transport, equation, defined on a fixed domain R with boundary F, of the kind

with p , a, t , v, and S standing for density, transport variable, time, velocity, source, and diffusion matrix

The dependent variable stands, for instance for t,he velocity component in each coordinate direction! or temperature, or the mass fraction of a chemical species. The function 3 denotes the relation between the transported and thc diffused variable such as for example relation between the enthalpy and the temperature,

The solution of the governing equat,ion for the dependent variable at final time t = to + At is sought, where to represents t,he initial time and At the positive time increment. The solution is constructed by the initial and boundary conditions that follow. The initial value of transport variable @(p , to) a t a point with position vector p and time to is defined through the known function Q.

q p , t ) = %,; P E n u r (3)

The boundary l- is devided int,o not necessarily connected parts rD, P'' and FR

r = PU P'u r R (4)

with Dirichlet, Ncumann and Robin type boundary conditions respectively. These boundary conditions are a t the boundary point p with normal nr


The involved parameters of t,he governing equation and boundary conditions are assumed to depend on the transport variable, space and time. The solution procedure thus inherently involves iterations. The governing equation is transformed as follows. The diffusion matrix

is split into constant isot,ropic part D I. wit,h I denoting identity matrix, and the remaining nonlinear anisotropic part D'

The transport equation is subsequently cast into Poisson form

v2Q;=,+... (8)

with

The inhomogenous t,errns are Taylor expanded as

Q % = : + + & ( Q - & ) , @ = @ + G , , ( 9 - 3 ) ) (10 )

with 'bar' denoting value a t previous iteration. The final form of the t ram- formed equation becomes

3. Solution procedure

3.1 Time discretization

The time discretization is made (for simplicity of presentation only) in a

fully implicit (backward Euler) manner v-herc subscript 0 represents the value at, the initial t,irne.


3.2 Method of particular solutions

The solution procedure is built within the framework of the method of particular solutions whcre the solution cP is represented in terms of the particular solution @ a n d homogenous solution Q0

The particular solution of the problem satisfies the Poisson equation

and not necessarily the boundary conditions. The homogenous solution satisfies the Laplace equation

wit,h the modified boundary conditions that originate from the original ones and the particular solution

The solution 1s established in Ncol collocation points p,; 1 = 1 .2 . . . . , lY,,,. of which ll;r ,,l belong to the boundary and f i ,,l to the domain

The boundary collocation points can in general be devided into Dirich- let, r ~ & , I Neumann: and IV&,~ Robin nodes, respectively

The present paper is limited to two dimensional Cartrsian system, e.g. p,. p, denote the Cartesian co-ordinates ( base vectors i,, i, ) of point p

3.3 Construction of particular solution

Let us first concentrate on t,he particular solution. The inhornogcnous tcrm is approximated by the N global approximation functions y,(p) and t,heir coefficicrlts <,,. This term can subsequent,ly be writt,en in the form


The coefficients (,, can be calculated through collocation equations in collocating points p,; J = 1.2, . . . , NcOl distributed over R and l?

and L\,, = A: - iVCO1 constraints

0 = 2Dn7 cl,; j = Ncol+l. Nco1+2, . . . . N (22)

The compact equivalent for Equations (21) and (22) is

with

t,hat gives

j = 1 , 2 , . . . , N C o 1 , m = 1 , 2 ; . . , N (2:)

and the Equation (20) can be written in the form

where subscript taltcs values X, y. By making similar global approximation function expmsions like in Equation (20) one can write


with the coefficients <C ,,, calculat,ed equivelently as in Equat,ion (26)

Cc rri a< k (31)

The global interpolation of the E-th component of O thus reads

G< (P) = @m (P) Q G< k (32)

and the global int~rpolat~ion of the E-th component of C . 6 is

The final divergence expressions in collocation points p, are

The particular solution Poisson problem can be represented in the form

V"' (p) = v,,, (p) [ H , + C . G, - ( H.aL +C G.<bJ ) ( a, - G, ) ] ( 3 5 )

It is solved by defining the harmonlc functions c, (p)

CLGm (P) = ~ 7 n (P) (36)

which allow the particular solution to be tlxplicitly extractcd

@*(p) = &,(p) 9;: [ B J + C . G] + (H.?,, - +C . G.'D3 ) ( a, - 6 ] ) ] (37) -

3.4 Construction of homogenous solution

The homog~nous solution satisfies the Laplace equation with the modified boundary conditions that are based on the boundary conditions of the original problem and the particular solution


The homogenous solution of the problern is structured within the Trefftz for- malism. It is represented by the Kr global approximation functions '41: (p) and their coefficients <E.

The global approximation functions have the property

i.e. they are fundamental solutions of the Laplace operator. This solution is for the two dimensional problems = z, g equal to

Let us define the boundary condition indicators in ordcr to be able to rcprc- sent t,he boundary collocation equations in a compact form. The Dirichlet, Neumann, and Robin type of boundary condition indicators Xv, X.Lr, and XR are

The coefficients G can be calculated by .Vr collocation equations in collo-


that gives

Note the unknown ai in Robin boundary nodes that have to be determined in the next step.

3.3 Complete solution

The complete solution to the problem can now be written in the following form

@(P) = 4:,!~)q:,,' X

The previous equat,ion can be written in the collocation nodes pk; k = 1 , 2 , . . . , NCo1, distributed over R and E'. This gives us the following system of f i col - 1% ,,l + NLl X ATa , l - ATr ,,l + ;,R,,, collocation equations for solution of the urlltnown a, in collocation points p, distributed over points that do not belong to the Dirichlet or Neumann part of the boundary. The solution of the following system of linear cquat,ions completes the definition of the unlcrlow~is in the domain and boiindary collocation points. With this t,he ecluat,iorl (51) can be itsed to represent the solution in arbitrary point P E O u r .


+&,,, Q,', [t), + C . G, -

4. Numerical implementation

The numerical irnplemcntation can be based on various radial basis functions. The following scalcd augmented thin plate splinrs can be used in two dimensions

y,71(p)=~:logr,; n = 1 . 2 . .

as a possible choice which does not involve any free parameter. Scaling constants and pi stand for the mean coordinates of the domain 0. The

adjacent harmonic functions $:,, for the thin plate splincs (53) (the selection is not unique!) are

The full system (52) can be solved by thc iterative techniques [B]. Ma.ny computational improvements to the described basic ideas arc possible. For example use of the compactly support,ed radial basis functions with multi- level approach as described in [g] (,hat modd allow for solutioll of very large problems.


5 . Conclusions

During the past three dxades the humanity has h e a ~ i l y invested tjheir re- sources in developing classical polygonizat,ion based discrete approxirrlative met,hods. The present paper for the first time shows a "mesh-free" formulation for the general transport equation. The proposed formulation has the following key features [ll]: (i) It requires neither domain nor surface discretization. It is polygon-free. (ii) The formulation is similar for 2D and 3D problems. (iii) It does not involve numerical integration. (iv) Ease of learn- ing. (v) Ease of coding. (vi) Competitiveness because of the man-power reduction involved for the meshing. The Laplace equation fundamental solution weighted and the augmented thin plate splines governed DRBEM has been in the last years applied to completely non-linear heat transfer and fluid flow problems. This include conductive-convective heat transport with non-linear material properties, non-linear boundary conditions, and phase change [12, 131, incompressible Navier-Stokes equations [14], and couplcd problems [l51 like natural convection [l61 in Darcy porous media and in Newtonian fluid with and without phase-change [17]. The method found related industrial application in DC casting of aluminium alloy billets 1181. The referenced spectra of problems are expected to be recalculated by t,he represented polygon-free approach.

Acknowledgment

The author would like to acknowledge many productive discussions with C.S. Chen and M.A. Golbcrg. Present paper represents a part of the projects (1) COST- P3: Sim.ulation of Physical Phenomena zn Technologzcal Applications that forms a part of the Slovenian national research programme Multiphase Systems supported b y the Slovenian Ministry for Science and Tech~lology, and ( 2 ) Computational Modelling for Estimation. of Radzonuclide Transport in Natural and Technologzcal Systems supported b y the Slovenian Nuclear Safety .Administration.

References

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[3] Partridge, P.W., Brebbia, C..A. & T;STrobel. L.C., The Dual Reciprocity Boumlary EIernent Method. Elsrvier, London, 19'32.

[4] Golberg, M A . & Chen, C S . , The method of fundamental solutions for potential, Helmholtz and diffusion problems, (ed.) Golberg, M A . , Boundary Integral Methods - Nnmerical and Mathematical Aspects, CMP, Southampton, pp. 103-176, 1998.


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[l21 Sarler, B. & Kuhn, G., Dual reciprocity boundary element method for convective-diffusive solid-liquid phase change problems-I. Formulation. En,g.Anal., 21, pp. 53-63, 1998.

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Towards a mesh-free computation of transport phenomena · Towards a mesh-free computation of transport phenomena Bo2idar ~arler Laboratory for Flciid Dynat~zics and Thermodynnmics,

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