Parallel domain decomposition meshless modeling of dilute convection-diffusion of species

Transcription

1 Boundary Elements XXVII 79 Parallel domain decomposition meshless modeling of dilute convection-diffusion of species E. Divo 1,. Kassab 2 & Z. El Zahab 2 1 Department of Engineering Technology, University of Central Florida, Orlando, FLLLLLLLlorida L , U.S.. 2 Department of Mechanical, Materials, and erospace Engineering, University of Central Florida, Orlando, Florida , U.S.. bstract meshless formulation and computational modeling of non-reacting dilute species transport is presented in this paper. Two species are considered for the problems in the example section: methane (CH 4 ) and air. parallel domain decomposition meshless method based on Multiquadric RBF collocation is developed herein. This approach reduces conditioning numbers of the resulting coefficient matrices while accelerating the solution process and reducing the memory requirements. Numerical examples are presented to validate the approach by comparing the meshless solutions to CFD-FVM solutions provided by the commercial package, Fluent Introduction The underlying algorithms of classical numerical methods such as finite element methods (FEM) and finite volume methods (FVM) have matured and afford the analyst the ability to routinely and accurately analyze a wide range of complex phenomena. However, the most commonly used methods require domain meshing due to the fact that rely on locally-supported polynomial expansions for the unknown field variables. Mesh generation is far from automated and in particular in computational fluid dynamics (CFD) applications require a large amount of effort and human interaction to generate meshes of sufficient quality to produce acceptable results.

2 80 Boundary Elements XXVII In recent years, meshless or mesh-free methods, reminiscent of spectral tech- niques [8] in their algorithmic structure, have been under intensive development in an attempt to overcome the difficulty of meshing the solution domain. The term mesh-free method refers to any numerical technique that relies on a set of unstructured nodes, that is, without pre-defined connectivity or relationship among the nodes, to solve a particular field problem. The key advantage of mesh-free methods as compared to traditional mesh-based methods is the expected large savings in labor dedicated to mesh generation. Unlike spectral methods that rely on global orthogonal expansion using typically, Legendre or Chebyshev polynomials, the meshless method of interest in this paper belongs to a class of that techniques that adopts radial basis function (RBFs) [9] expansions for the dependent variable(s). Domain decomposition [5, 10] has been shown to effectively reduce storage and RM access requirements as well as significantly reduce matrix conditioning in applications to scalar problems. In this paper, we develop a parallel domain decomposition collocation meshless method for dilute convection-diffusion of species. The species densities is approximated using the inverse Multiquadrics expansion RBF. Two cases are considered: (1) a parallel plates channel problem with and without domain decomposition and (2) a T-shape channel decomposed into five sub-domains. 2 Meshless formulation of dilute convection-diffusion of species Transient incompressible fluid flow is governed by the coupled set of non-linear partial differential equations, and these are the continuity, Navier-Stokes, and energy transport equations which are collectively often referred to as the Navier- Stokes equations: V =0 ρ V t + ρ( V ) V = p + µ 2 V + ρ f ρc T t + ρc( V )T = k 2 T +Φ where V is the flow velocity vector, ρ is the bulk density of the flow, p is the field pressure, µ is the absolute fluid viscosity, f is a specific body force, c is the fluid specific heat, T is the field temperature, k is the fluid thermal conductivity,and Φ is the viscous heat dissipation. ll field variables are functions of space (x) and time (t), in a fixed domain Ω surrounded by a closed boundary Γ. If a new species () is introduced into the bulk flow in dilute form (small concentration), then the above equations can be solved independently under the assumption that the overall flow field remains approximately incompressible. The transport of such species is governed by the following differential equation, coupled one-way to the Navier-Stokes equations (Eqn. (1)): ρ +( V t )ρ = D B 2 ρ (2) (1)

3 Boundary Elements XXVII 81 Where ρ is the partial density of species, andd B is the binary diffusivity of species into the bulk flow. Equation (2) can be solved independently if the bulk flow velocity field, V, is predetermined. In this paper we use a pressurecorrection based meshless method as described in [12]. Expressing the species density transport Eqn. (2) in two-dimensional Cartesian form, see [13], yields: ρ t with generalized boundary conditions: + u ρ x + v ρ y = D B 2 ρ (3) β ρ n + γρ = σ (4) where the space-time dependency (x, y, t) has been omitted for simplicity. Here u and v are the horizontal and vertical velocity components of the bulk flow respectively. The parameters β, γ,andσ are specified and determine the type of boundary condition. The meshless approximation begins by globally expanding the species density over a number NB + NI of independent points in the domain Ω(x, y) and on the boundary Γ(x,y)as: ρ (x, y, t) = NB+NI j=1 α j (t)χ j (x, y) (5) where NB: boundary expansion points, NI: internal expansion points, α j (t):transient expansion coefficient, and χ j (x, y): global expansion functions. mong the family of global expansion functions the radial-basis functions (RBF) have shown to effectively model scattered data, see [6, 9]. The Multiquadric RBF, see [5, 7], stands out in comparison to other RBF s for its ability to accurately interpolate field variables over generalized domains and point distributions. The Multiquadric RBF s are defined as: χ j (x, y) = [ r 2 j (x, y)+c 2] n 3 2 (6) where n: exponent parameter (positive integer), c: shape parameter, and r j : Euclidean distance from (x, y) to (x j,y j ). In Eqn. (5), the profile of the expansion flattens as the values of n and c increase which will give a smooth and accurate global representation of the field variable. Nevertheless, the drawback of having a flat representation of the field variable lies in the linear dependence that arises among the system of linear equations resulting in a high matrix conditioning number [7]. Thus, it is necessary to choose n and c values to yield an admissible matrix conditioning number. For the current formulation, n is taken as 1 yielding the socalled inverse Multiquadric. The species density transport Eqn. (3) can be numerically evaluated between time steps p and p +1by replacing the time derivative with a first order backward differencing approximation as:

4 82 Boundary Elements XXVII ρ p+1 ρp + u t p+1 ρp+1 ρp+1 + vp+1 = D B 2 ρ p+1 (7) x y or, placing the current time step values, p +1, on the left-hand side yields: D B 2 ρ p+1 ρp+1 up+1 ρp+1 vp+1 x y ρp+1 t = ρp t t this point, the global expansion of the species density in Eqn. (5) can be introduced. Notice that the spatial differential operators and the summations are interchanged and the differential operators are applied directly to the expansion functions χ j (x, y) as: (8) NB+NI j=1 α p+1 j (D B 2 χ j u p+1 χ j x χ vp+1 j y χ j t )= ρp t (9) where ρ p is known from the previous time step starting at ρ0. The same procedure can be applied to the boundary condition relation in Eqn. (4) as: NB+NI j=1 α p+1 j (β χ j n + γχ j)=σ (10) The coefficients α p+1 j corresponding to the time step p +1can be found by collocating the boundary condition expansion in Eqn. (10) at points i =1..NB and the governing equation expansion in Eqn. (9) at points i = NB+1..NB+NI. This process yields the following set of linear simultaneous equations: [ ] ψj { } { } (x i,y i ) α p+1 σ(xi,y η j (x i,y i ) j = i ) 1 t ρp (x i,y i ) where: NB+NI NB+NI ψ j (x i,y i )=β χ j(x i,y i ) n η j (x i,y i )=D B 2 χ j (x i,y i ) u p+1 (x i,y i ) χ j(x i,y i ) x NB+NI 1 + γχ j (x i,y i ) v p+1 (x i,y i ) χ j(x i,y i ) y NB+NI 1 (11) χ j(x i,y i ) t (12) the left-hand side coefficient matrix is fully-populated and normally found to have very large conditioning numbers. Both of these features are major draw-backs for the implementation of this technique to large-scale multi-dimensional problems. Evidently, pre-conditioning this matrix is a mandatory step in the solution of realistic systems, see [5, 10] for more detail. This issue will be addressed next and the salient features of a typical coefficient matrix resulting from mesh-free expansions will be illustrated.

5 Boundary Elements XXVII 83 Boundary Point Internal Point Figure 1: Problem domain and typical boundary and internal point collocation. 3 Parallel domain decomposition implementation In order to pre-condition a set of algebraic equations, one may pre-multiply the system of equations by a matrix that closely approximates the inverse of the coefficient matrix and that is also cheaply evaluated from the computational standpoint. Incomplete LU pre-conditioning is such an approach. n alternative to mitigate the effect of large conditioning numbers is to reduce the number of degrees of freedom in the algebraic system. This can be accomplished by decomposing the problem domain Ω into multiple sub-domains Ω k separated by artificially created interfaces Γ I, see [10]. The problem is started from initial conditions which are used as boundary conditions at the artificially created interfaces at time level p =0.Once each sub-domain is a well-posed boundary value problem, the standard collocation approach described in the previous section may be followed to render independent expansions at each sub-domain. simple iteration implementation can follow to force continuity conditions of the field variable and normal derivatives at the interfaces. If a single-region point collocation approach as the one shown in Fig. 1 is followed, a system of simultaneous linear equations of size N,whereN = NB+NI, will be formed as: Ω [] N N {α} N 1 = {b} N 1 (13) where {α} represents the vector of coefficients for the global expansion. The number of floating point operations required to arrive at the algebraic system above is proportional to N 2 as well as direct memory allocation also proportional to N 2. The solution to the algebraic system for the coefficients {α} can be performed using a direct solution method such as LU decomposition requiring floating point operations proportional to N 3 or an indirect method such as Bi-conjugate Gradient or Generalized Minimum Residual (GMRes) which, in general, require floating point operations proportional to N 2 to achieve convergence. On the other hand, if a the multi-region iteration approach were to be adopted, the domain is divided into K sub-domains and each one is independently filled with collocation points. See Fig. 2 for a typical domain decomposition with K =4.

6 84 Boundary Elements XXVII 3(1) 3(2) 3(3) 3(4) 4(1) I 2(1) + I 4(2) II 2(2) + II 4(3) III 2(3) + III 4(4) 2(4) (1) 1(2) 1(3) 1(4) Figure 2: Typical domain decomposition and collocation. The information between neighboring sub-domains separated by an interface will be passed through using their respective radial-basis expansions. Therefore, the collocation points at the artificially created interfaces do not need to coincide. The boundary value problem will now be solved independently over each subdomain starting by imposing the initial condition at the artificially created interface. For instance, the boundary value problem of sub-domain Ω 1 is transformed into the algebraic analogue of corresponding influence coefficient matrices and nodal boundary values as: Ω 1 [] n n {α} n 1 = {b} n 1 (14) The composition of this algebraic system requires floating point operations proportional to the square of the number of boundary and internal collocation points (n 2 ) in the sub-domain Ω 1. Memory allocation is also proportional to the same number (n 2 ). This new proportionality number n is roughly equivalent to: n N K (15) as long as the collocation density along the boundaries is equivalent to the collocation density in the interior and the artificially created interfaces are selected to perpendicularly sub-section a longitudinal direction. Direct memory allocation requirement for later algebraic manipulation is now reduced to a proportion of n 2 as the system matrices can easily be stored on disk for later use after the boundary value problems on remaining sub-domains have been effectively solved. For the example shown here where the number of subdomains is K = 4 the new proportionality value n is approximately equal to n N/4. This simple multi-region example reduces the memory requirements to about n 2 /N 2 =(1/16) = 6.25% of the single-region approach. The solution of all of the four resulting algebraic systems for all four sub-domains require a number of floating point operations proportional to K (n 3 /N 3 )=4 (1/64) = 6.25% of the single-region approach if a direct algebraic

7 Boundary Elements XXVII 85 solution method is employed, or a number of floating point operations proportional to K (n 2 /N 2 )=4 (1/16) = 25% of the single-region approach if an indirect algebraic solution method is employed. For both, floating point operation count and direct memory requirement the reduction is dramatic. However, as the first set of solutions for the sub-domains were obtained using guessed boundary conditions along the interfaces, the global solution needs to follow an iteration process and convergence criteria. The more significant reduction is revealed in the RM requirements as only the memory needs for one of the sub-domains must be allocated at a time. The rest of the system matrices for the remaining sub-domains are temporarily stored on disk until access and manipulation is required. With respect to the algebraic solution of the system of equations for each subdomain of the domain decomposition, if a direct approach such as the LU decomposition method is employed for all sub-domains, the LU factors of the coefficient matrices for all sub-domains can be computed only once at the first iteration step and stored on disc memory for later use during the iteration process, unless the sub-domains are kept small enough to store the LU factors in RM on each processor when the algorithm is implemented in parallel. Thereon, only a forward and a backward substitution will be required at each iteration to arrive at a solution for the system in hand. This feature allows a significant reduction in the operational count through the iteration process until convergence is achieved, as only a number of floating point operations proportional to n as opposed to n 3 is required at each iteration step. To this computation time is added the access to disk (if the option of storing the factors in RM is not feasible) at each iteration step which is usually larger than the access to RM. 4 Time progression iteration process Once each boundary value problem is independently solved in each sub-domain, a time progression iteration process must follow to continuously ensure continuity of the field variable and its normal derivative across the interfaces. This is accomplished by setting up a process in which each interface is treated differently on both sides: on one side a first kind boundary condition is applied while at the other side a second kind boundary condition is applied. The resulting boundary values are then averaged with their corresponding values across the interfaces. This process is illustrated below in Fig. 3 for a single point on the interface between sub-domain Ω 1 and Ω 2. In Fig. 3 the imposed boundary conditions are ρ p on the left-hand side of the interface and ρ p / n on the right-hand side. fter the solution of the boundary value problem on each sub-domain the resulting (provisional) values are ρ p+1 / n on the left-hand side of the interface and ρp+1 on the right-hand side. These resulting values are averaged to generate the boundary conditions for the

8 86 Boundary Elements XXVII I 2(1) I 4(2) I 2(1) I 4(2) I 2(1) I 4(2) p ˆ ˆ n p p n 1 p 1 ˆ 1 p p ˆ n 1 Figure 3: Iteration process averaging across an interface. next time step iteration as: ρ p+1 ρ p+1 n = ρp + ρp+1 2 = ρp n 1 ( ρp 2 n + ρp+1 n ) (16) ensuring continuity of the field variable and its normal derivative at each time step iteration. Notice that in order to achieve truly time-accurate analysis, sublevel iterations must be performed at each time step. The iteration process may be stopped once steady-state is achieved or at any time level. Steady-state may be verified by defining a residual as: r p+1 = 1 NB + NI NB+NI i=1 (D B 2 ρ p+1 i u p+1 i ρ p+1 i x vp+1 i ρ p+1 i y )2 (17) and comparing it to that one of the previous time step, p. Once the residual difference reaches a preset convergence criterion or an oscillative behavior, the iteration process may be stopped and the resulting field values approximate the steady-state solution. 5 Numerical results Two examples are now presented to validate the formulation by comparing the meshless results to Finite Volume Method (FVM) solutions provided by the commercial CFD package Fluent 6.1. The first example consists of a 10 2cm channel through which air enters at 0.25m/s. leak of Methane enters through a section of 0.2cm at the center of the air inlet. The FVM mesh, 1-region, and 5-region meshless point distribution is presented in Fig. 4 along with the corresponding velocity magnitudes. The residual evolution for the 1-region meshless solution is shown in Fig. 5 along with the resulting Methane partial density distribution. The second example consists of a T-valve with 3mm inlets and outlets. ir enters at 0.25m/s through the two inlets while a leak of Methane enters through a section of 0.34mm at the center of the left wall. The FVM mesh and 5-region meshless point distribution is presented in Fig. along with the corresponding

9 Boundary Elements XXVII 87 Figure 4: FVM, 1-region meshless, 5-region meshless, point distribution and velocity magnitude contours Mass X-Momentum Y-Momentum Energy Species Step Figure 5: Residual evolution of the 1-region meshless solution and FVM, 1-region meshless, 5-region meshless Methane partial density distribution ρ. velocity magnitudes. The FVM and 5-region meshless pressure distribution is presented in Fig. 6 along with the resulting Methane partial density distribution. The residual evolution for the 1-region meshless solution is shown in Fig Conclusion meshless formulation and computational modeling of non-reacting dilute species transport is presented in this paper. Two species are considered for the problems in the example section: Methane (CH4 ) and air. parallel domain decomposition meshless method based on Multiquadric RBF collocation is developed herein. This approach reduces conditioning numbers of the resulting coefficient matrices while accelerating the solution process and reducing the memory requirements. Numerical examples are presented to validate the approach by comparing the meshless solutions to CFD-FVM solutions provided by the commercial package, Fluent 6.1.

10 88 Boundary Elements XXVII Figure 6: FVM and 5-region meshless, point distribution, velocity magnitude contours, and Methane partial density distribution ρ. 102 Mass X-Momentum Y-Momentum Energy Species Step Figure 7: Residual evolution of the 5-region meshless solution. References [1] Belytscho, T., Lu, Y.Y., and Gu, L., Element-free Galerkin methods, Int. J. Num. Methods, Vol. 37, 1994, pp [2] tluri, S.N. and Zhu, T., new meshless local Petrov- Galerkin (MLPG) approach in computational mechanics, Computational Mechanics, Vol. 22, 1998, pp

11 Boundary Elements XXVII 89 [3] Melenk, J.M. and Babuska, I., The partition of unity finite element method: basic theory and application, Comp. Meth. ppl. Mechanics and Eng., Vol. 139, 1996, pp [4] Kansa, E.J., Multiquadrics- a scattered data approximation scheme with applications to computational fluid dynamics I, Comp. Math. ppl., Vol. 19, 1990, pp [5] tluri,s.n. and Zhu, T., new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Computational Mechanics, Vol. 22, 1998, pp [6] Franke, R., Scattered data interpolation: Test of some methods, Math Comput., Vol. 38, 1982, pp [7] Cheng,.H.-D., Golberg, M.., Kansa, E.J., Zammito, G., Exponential Convergence and H-c Multiquadric Collocation Method for Partial Differential Equations, Numerical Methods in P.D.E., Vol. 19, No. 5, 2003, pp [8] Gottlieb, D. and Orzag, S.., Numerical nalysis of Spectral Methods: theory and applications, Society for Industrial and pplied Mathematics, Bristol, England, [9] Powell, M.J.D., The Theory of Radial Basis Function pproximation, in dvances in Numerical nalysis, Vol. II, Light, W., ed., Oxford Science Publications, Oxford, pp [10] Divo, E., Kassab,.J., Mitteff, E., and Quintana, L. Parallel Domain Decomposition Technique for Meshless Methods pplications to Large- Scale Heat Transfer Problems, SME Paper: HT-FED [11] Harlow, F.H. and Welch, J.E., Physics of Fluids, Vol. 8, pp , [12] Divo, E. and Kassab,., Effective Domain Decomposition Meshless Formulation of Fully-Viscous Incompressible Fluid Flows, Proc. of BEM 27, WIT Press, Southampton, 2005, (in press.) [13] Turns, S., R., n Introduction to Combustion, 2nd Ed., McGraw-Hill, New York, 2000.