Study of the diffusion operator by the SPH method

Transcription

1 IOSR Jornal of Mechanical and Civil Engineering (IOSR-JMCE) e-issn: ,p-ISSN: X, Volme, Isse 5 Ver. I (Sep- Oct. 204), PP 96-0 Stdy of the diffsion operator by the SPH method Abdelabbar.Nait El Kaid, Pr: Ghita.Mangob 2 ( LMMI Laboratory, University Hassan st, FST Settat, 26000, Settat Morocco 2 LMMI Laboratory, University Hassan st, FST Settat, 26000, Settat Morocco Abstract: The Smoothed Particle Hydrodynamics (SPH) method is a prely Lagrangian method which was developed initially to simlate the dynamic of astrophysical flids (Lcy 977, Gingold and Monaghan 977, Benz 990). The basis of the SPH method drift fndamentally of the Lagrangian description of flids and that, considering that the movement of a control volme of a flid can be represented by a particle carrying all physical information (mass, solicitation, energy,... ). In this work, we stdy the nmerical behavior of the diffsion operator in the SPH method) with two different formlations. The first formlation is called the gradient formlation given by G_Li and the second one is called Laplace. An Application to the problem of heat condction in a sqare plate is considered. The differential operators are discretized by the SPH method while for the time derivatives,we se an explicit schema based on velocity Verlet algorithm. Keywords: Condction, diffsion, particle method, SPH method I. Principal And Mathematical Formlation Of The SPH Method Principal of the SPH method The SPH method is based on three essential steps: The representation of a fnction by its integral with the definition of the interpolation kernel («Kernel approximation»). The particle approximation with the passage of the integral form to a discrete sm of the contribtions of particles in the inflence domain. The application of the formalism to the conservation eqations for the desired application.. Representation of a fnction by its integral Let a fnction f defined and continos on a domain. The concept of the representation of f by its integral sed in SPH is based on the following eqation: f x = f x δ x x dx () Where f is a fnction of position vector x and δ(x x )the Dirac fnction defined as follows: δ x x x=x = { 0 x x (2) By replacing the Dirac fnction δ by the interpolation kernel W(x x, h), we obtain the integral interpolating of the fnction f(x) over noted < f(x) >. It is then written in the form: < f(x) >= f x W x x, h dx (3) Where W mst verify over the condition of nity, convergence to the Dirac fnction when the smoothing length h tends to 0 and the sbstrate mst be compact. These three conditions are given respectively by the following eqations 4, 5 and 6: W x x, h. dx = (4) lim h 0 W(x x, h) = δ(x x ) (5) W x x, h = 0 if x x > kh (6) Where κis a constant of the interpolation fnction. If the interpolation kernel is differentiable, we can constrct the differential interpolating of the fnction f noted<. f(x) >, let : <. f x >= f x. W x x, h dx = f x. W x x, h dx (7) We immediately notice than the gradient operator is transmitted to the interpolation fnction and the estimation of the derivative of a fnction can then be determined from the fnction vales and derivative of the kernel. II. Choice of the kernel The chosen kernel to approximate the Dirac fnction can be for example a Gassian fnction or a spline. The respect of conditions (4) and (5) is helping to bild sch fnctions. In practice, the spport of the kernel is not the comptational domain bt a compact spport D (ie beyond a size eqivalent to n times the length of the smoothing,the kernel vale is zero). The most crrently sed kernels have a spport with aray 96 Page

2 eqals to 2h or 3h. This will directly inflence the nmber of neighbors and therefore the accracy of the approximation. In the end we are proposing to se these kernels (where q= r /h): The cbic spline proposed by Monaghan [2], on a spport of size 2h: 2 3 q2 2 q3 if 0 q W r, h = C c 6 2 q 3 if q 2 (8) 0 otherwise Where the normalization constant C c allows verifying the condition 4 and is worth, 5 and 3 h 7πh 2 2πh3, respectively in, 2 and 3 dimensions. It is important to note that this kernel fnction depends only on the distance between a point i and its neighbor. The spport is then spherical and the vicinity of one ball is actally composed of all the neighboring balls to which r i < 2h. The cbic spline viscosity given by [3], on a spport of size h W r, h = C c 2 q q2 2 q3 if 0 < q (9) 0 if > q Where the normalization constant C c allows verifying the condition (4) and is worth respectively in, 2 and 3 dimensions. 5, 45 h 7πh 2 and 5 2πh 3, III. The bondary conditions The bondary conditions are a very important task for the SPH. To well scceed a simlation amonts largely to well model bondary conditions. There are different alternatives that we can adapt to the SPH method and this concerning modeling spatial bondary conditions, speed or inter-reaction. Based on the natre of the SPH, Obtaining the vale of a physical qantity at a given point involves the smoothing of the neighborhood of the point. Conseqently, a problem emerges for the points in the domain bondaries as well as locations that do not have sfficient space to smooth. In this work, we se phantom particles method; it is a qestion of going rond the area/field by series of particles which called phantom particles. This name comes from the fact that these particles are not sed for anything other than to get a neighborhood of sfficient smoothing. The nmber and the spatial distribtion of these sets of particles are determined so as to consider smoothing lengths of the sed kernel (h, for Müller, 2h for the case of a kernel of Spline type and 4h for the case of the Gassian). IV. Physical problem Let a solid material, occpying a D domain of the three-dimensional space x,x 2,x 3 and for the mechanical point of view is on rest. Its density ρ and its specific heat C depend a priori on the three coordinates x,x 2,x 3 (for a nonhomogeneos material) bt not on time t ρ = ρ(x, x 2, x 3 ), C = C(x, x 2, x 3 ) Spposethat this material receives,leads and absorbs some heat flow. So its temperatre = (x,x 2,x 3 : t) evolves over time and space. It is the same forthe vector heat flow density q. Q = q x, x 2, x 3 : t. Recall that by definition of q, D, the qantity of heat received by throgh S= per nit time is eqal to : q. n ds. q dx dx 2 dx 3 (0) S Where n is the norm to s external of, the stock of accmlated thermal energy throgh material by heating (internal energy of material) increases by : C dm ρc dx dx 2 dx 3 () So the variation (0) of the stock is eqal to the inflow () where ρc. q dx dx 2 dx 3 = 0 (2) From this integral, assming that all is continos, ie the integrant is continos, we can, based on the fndamental lemma, conclde that the integrant is itself zero, hence the following relationship: ρc =. q (3) By introdcing the "behavior'' law known as"forier" law 97 Page

3 q = k (4) Withk = k x, x 2, x 3 is the thermal condctivity So this relation expresses the fact that the heat flows are even more intense as the temperatre difference is greater, and the heat goes from hot spots to cold spots. The following formla is then dedced: ρc =. k (5) If the material is homogeneos, ρ, C and k are constants, hence. k = k. = k (6) Conseqently: With: λ = k ρc = cte = λδ (7) = λ 2 x 2 2 y 2 2 z 2 (8) This eqation is called heat eqation and λ thermal diffsivity V. Applications And Reslts. Application in D case Initially, we are limited to the case of one variable of space, which we will denote x : this is the case for example of a metal bar whose temperatre depends only on the abscissa x and time t. The eqation (8) is redced to: = λ 2 x 2 This eqation is a model for the (parabolic) eqation of the bondary layer. We pt:q T = [0,] 0,, The overall problem is then written: = λ 2 x 2 sr Q T CL: 0, t =, t = 0, t > 0 CI: x, 0 = v x, xϵ 0, where λ is the thermal diffsivity. (9). Discretization - Discretization of the differential operator by SPH Δ i = i ρ (t) 2 W i (20) x 2 = λ ρ (t) 2 W i (2) x 2 We note that with the SPH method, following the formalism leads to derive twice the kernel, ie Δ =.. Bt this formla is not symmetrical; conseqently it does not check Newton law. It will be replaced by : Δ i = ( (t) i (t)) 2 W i (22) x 2 ρ moyi with ρ moyi = 2(ρ i ρ ) Leading to: i = λ ρ i ρ ( (t) i (t)) 2 W i (23) x 2 ρ moyi Discretization of the temporal operator To treat time derivatives, we se an explicit scheme based on the velocityverlet algorithm [5] as follows: n i = n i 0.5 t D n n i D i (24) with D n i = d i(t) dt 98 Page

4 teta t emperatre[t]) Stdy of the diffsion operator by the SPH method.2 Reslts temp=0 temps= temps=0.50 temps= x p articles [m] FigLaplace_method: DT = 0.00, NX = 50, NDT = Fig2finite difference_method: DT = 0.00, NX = 50, NDT = Application in 2D case The variables of space, which we denote x and y this is the case for example of a metal cavity in which the temperatre depends only on x,y and time t. The eqation (8) is redced to: = λ 2 x 2 2 y 2 This eqation serves as a model for (parabolic) eqation of the bondary layer. With initial and bondary conditions : x, y, t = 0 = for x, y 0, y, t = x, 0, t = 0 for t>0 (25) where λ is the thermal diffsivity. 2. Discretization of the differential operator by SPH Δ i = ρ ( 2 W i 2 W i ) (26) x 2 y 2 Then i = λ ρ ( 2 W i 2 W i ) (27) x 2 y 2 We note that with the SPH method, following the formalism leads to derive twice the kernel, ie Δ =.. Bt this formla is not symmetrical; conseqently it does not check Newton law. It will be replaced by: Δ i = ( i )( 2 W i 2 W i ) (28) x 2 y 2 ρ moyi 99 Page

5 With ρ moyi = 2(ρ i ρ ) ρ i ρ Which leads to: i = λ ( ρ i )( 2 W i 2 W i ) (29) moyi x 2 y 2 This approximation which we call Laplace method, we compare it to another that we call gradient method[4] : d i dt = λ ( i )( w i ) (30) 4 k i k ρ ρ i k i k 2.2 Discretization of the temporal operator To treat time derivatives, we se an explicit scheme based on the velocityverlet algorithm [5] as follows: i n = i n 0.5 t D i n D i n With D i n = d i dt VI. Nmerical Reslts The geometry of the problem is a sqare cavity of side L = 2. Condctivity coefficient is taken eqal to. The calclation program is tested for different vales of nx and ny, npr = nx ny with dt = and ndt = 000. We have fond that the Laplace method and the gradient method give the same reslt when the nmber of particles is small (00 real particles and 40 virtal particles (phantoms)). Bt when the nmber of particles increases, Laplace method gives reslts comparable to those fond by other athors while with the gradient method, we obtain practically the same reslt regardless of the nmber of particles engaged. In Figres 3 and 4, the reslts are presented with the two methods fornx = ny = 40. In figre 3, we give the reslts obtained by Li for the same nmber of particles. Fig3gradient_method Fig4Laplace_method 00 Page

6 Fig5Li_method: [] VII. Conclsion In this paper, we have proposed a model based on SPH method to simlate condction problem. The main advantage of this techniqe involves the approximation of fnctions on the edge arond the tool which is difficlt to achieve with other nmerical methods sch as finite element method. We are working now on extending the proposed model to simlation process of natral convection with consideration of a bill of more realistic behavior depending on both temperatre model, the flid velocity as well as in the case 2D and 3D. References [] G. R. Li, M. B. Li, Smoothed Particle Hydrodynamics - A Meshfree Particle Method, World Scientific Pblishing Co. Pte. Ltd. 5 Toh Tck Link, Singapore , 2003 [2] J.J.Monaghan, "An introdction to SPH", Compter Physics Commnications, volme 48, pages 89-96, 988 [3] M. Müller, D. Charypar, and M. Gross. Particle-Based Flid Simlation for Interactive pplications. Proceedings of 2003 ACM SIGGRAPH Symposim on Compter Animation, pp , 2003 [4] J.J. Monaghan, Smoothed Particle Hydrodynamics, School of Mathematical Sciences, Monash University, Vic 3800, Astralia, pages , 2006 [5] M.P. Allen and D.J. Tildesley, Compter simlation of liqids (Oxford University Press, Oxford,) p.8, Page