Solving Poisson s equations by the Discrete Least Square meshless method

Transcription

1 Boundary Elements and Oter Mes eduction Metods XXV 3 Solving Poisson s equations by te Discrete Least Square mesless metod H. Arzani & M. H. Afsar Said aaee University, Lavizan, eran, ran Department of Civil Engineering, ran University of Science & ecnology Abstract A mesless metod is proposed in tis paper for te solution of two-dimensional elliptic problems. e proposed metod does not require any mes so it is truly a mesless metod. e approac termed generically te Discrete Least Square mesless metod is applied to discrete te governing differential equations in inner and boundary nodes. A functional is defined as te sum of te squared residual of te governing differential equation and te boundary conditions at te nodal points. Moving least-square (MLS) interpolation is used to construct te sape function s values, wic ave ig continuity in te problem domain. o evaluate te accuracy of te metod as an alternative mesless metod te development and teory of tis new approac is presented in te context of te solution of D elliptic equations. Numerical results sow tat te metod possesses ig accuracy wit low computational effort. Keywords: mesless metod, Discrete Least Square, elliptic problems. ntroduction e idea of using finite difference simplicity and finite element capability of andling complex geometries are te subect of many researces. is is mainly because te mes generation part of te solution as sown to be a very time consuming callenge especially in finite element applications. e idea of developing metods requiring no mes as led to te emerging of a new class of te so-called mesless metods. Many of te mesless metods developed so far require background mes to carry out numerical integration. e integration cells, owever, need not be compatible wit nodes and tus tey can be generated more easily tan te FEM meses. e existing mesless metods can W ransactions on Modelling and Simulation, Vol 4, 006 W Press SSN X (on-line) doi:0.495/be06003

2 4 Boundary Elements and Oter Mes eduction Metods XXV be generally divided in two main categories depending on te way te discrete equations are formed. First are te metods based on te weak form of te given differential equation. All tese metods use one form of te weigted residual suc as Galerkin or Petrov-Galerkin for discretization of te governing differential equations. n tis category, one can find Smoot Particle Hydrodynamics (SPH) by Monagan [], wic is te oldest of te mesless metods, eproducing Kernel Particle metod (KPM) []. ese metods use finite integral for function approximation; Partition of Unity (PU) metod [3]; p cloud metod [4]; Diffuse Element Metod (DEM) by Nayroles et al. [5]; Element Free Galerkin (EFG) by Belytscko et al. [6]. Atluri and Zu [7] and Zu et al. [8] suggested te local Petrov-Galerkin and local boundary integral equation (LBE) approac in wic integration is performed locally on eac subdomain. A common feature of all tese metods is te need for numerical integration requiring a mes of quadrature points in te domain. Construction of appropriate integration cells, owever, is a difficult ob and can make mesfree metods less effective. For tese reasons, Beissel and Belytscko [9] suggested a nodal integration procedure instead of using Gaussian quadrature in establising te coefficients of te system of equations. J.X. Zou et al [0] proposed a nodal integration procedure based on Voronoi diagram for general Galerkin mesless metods. Some of te mesless metods use finite series for function approximation wic include Polynomial Point nterpolation Metod (PPM), adial Point nterpolation Metod or adial Basis Function (BF) by Cen et al. [] and well-known Moving Least Square Metod (MLS) described by Lancaster and Salkauskas [] and used by Nayroles et al. [5]. Second are te metods starting directly from te governing equation suc as finite point metod by Onate et al. [3]. ese metods are often arrived at using a point collocation weigted residual formulation of te problem. e collocation metod, owever, can suffer from te stability problem as tat encountered in te nodal integration. n addition, it requires iger-order derivatives of sape functions and results in non-symmetric stiffness matrices. n tis paper we use a fully least square approac in bot of te governing differential equation discretization and function approximation wic are te main components of every mesless metod. e outline of tis paper is organized as follows. e moving least square approximation for establising sape functions is briefly described in section. e Discrete Least Square metod for discretizing te governing differential equation is presented in section 3. wo elliptic problems solved and te results are presented in section 4. And we close wit some concluding remarks in section 5. Moving Least Square (MLS) metod e metod of Moving Least Squares (MLS) as been widely used for function approximation by mesless community. e advantages of MLS are tree folds: first, tere is no need for explicit meses in te construction of MLS sape functions. Second, ig order continuity of sape functions so constructed W ransactions on Modelling and Simulation, Vol 4, SSN X (on-line) 006 W Press

3 Boundary Elements and Oter Mes eduction Metods XXV 5 eliminates te necessity of using weak form of governing equations as required in finite element metod (FEM) using standard sape functions. n addition, iger order continuity, if required, is not introduced at te expense of increasing te unknown parameters as usually practiced in FEM. ird; te availability of smoot derivatives eliminates te need for costly procedure of gradient recovery, wic is usually required by standard FEM. n MLS, te function to be approximated is represented by: = m u (x) p i (x) a i (x) p (x) a(x) () i = Here (x) a x represents te unknown coefficients to be determined by te fitting algoritm. e polynomial bases of order m in one and two dimension are given by: p m ( x) = [, x, x,, x ] () m m m p is a set of linearly independent polynomial basis and () p (x) = p ( x, y) = [, x, y, x, xy, y,, x,, xy, y ] (3) n te MLS approximation, at eac point ( x ), a ( x) is cosen to minimize te sum of weigted squared residuals defined by: n [ ] J = = w( x-x u ) p (x ) a(x) (4) Were u is nodal value of te function to be approximated, n is te number of nodes and w ( x-x ) is te weigt function defined to ave compact support. e weigt functions are cosen to ave te following properties: ) w ( x-x ) > 0 On a subdomain ) w( x-x = 0 (5) Outside te subdomain 3) w ( x-x dω = Ω ) A normality property 4) w ( x-x ) A monotonically decreasing function 5) w( x-x ) δ ( s) as x-x = 0 wereδ (s), is te Dirac delta function. Many weigt functions are establised and used by different researcers. n tis paper, we use a cubic spline weigt function defined as: 3 4r + 4r for r w ( r) = 4r + 4r r for < r (6) for r > W ransactions on Modelling and Simulation, Vol 4, SSN X (on-line) 006 W Press

4 6 Boundary Elements and Oter Mes eduction Metods XXV n wic, r = s / smax, s = x x and s max is te radius of te support. Eqn (4) can be written in matrix form as J = (Pa u) W (Pa u) (7) Were u = ( u, u,..., un ) (8) p (x ) p (x ) p m (x ) p (x ) p (x ) p m (x ) P = (9) p (xn ) p (xn ) p m (xn ) and w( x x ) w( x x ) W(x) = 0 (0) 0 0 w( x xn ) e coefficients a are found by minimizing J wit respect to tese coefficients. Carrying out te differentiation: J = A(x)a(x) B(x)u = 0 () a Were A = P W(x)P () B = P W(x) (3) Solving te above equation for te unknown parameters. a(x) = A (x)b(x)u (4) e approximation of te unknown function can now be written as n u (x) = N (x)u (5) were te sape functions are defined as: N = p (x)a (x)b(x) (6) n tis case, u u ( x ), so te parameters u cannot be treated like nodal values of te unknown function. e sape functions are not strict interpolates since tey do not pass troug te data. e sape functions do not satisfy te Kronecker delta condition: = W ransactions on Modelling and Simulation, Vol 4, SSN X (on-line) 006 W Press

5 Boundary Elements and Oter Mes eduction Metods XXV 7 if i = Ni ( x ) δi = (7) 0 if i were N i ( x ) is te sape function of node i evaluated at node and δi is te Kronecker delta. 3 Discrete Least Square (DLS) metod Consider te following differential equation L (u) = f n Ω (8) B (u) = g On Γ (9) Were L and B are te differential operator defined on te problem domain Ω and its boundary ( Γ ), respectively. e pilosopy of least square is to find an approximate solution tat results in minimum residual error wen substituted into equations (8) and (9). e first step is to assume te form of approximate solution ( u ), including a total of m parameters wic can be adusted to minimize te error. is is sometimes called a trial solution, and can be represented by: u(x) u (a, x) (0) Were a is te vector of unknown parameters and x represents te independent variables of te domain. e error is measured by te residuals tat result wen u is substituted into eqns (9) and (0). Were Ω and Ω (a, x) = L (u ) f for x in Ω () Γ (a, x) = B (u ) g for x on Γ () Γ are called interior and boundary residuals, respectively. Finally, a weigted sum of squared residuals is minimized over te domain ( Ω ), establising te best values of te parameters a. n te Discrete Least Square formulation, te squared residuals are evaluated and summed at te set of points x cosen to represent te problem domain Ω and its boundary ( Γ ). i α nb [ (a, x )] [ (a, x )] ne d (a) = + i = Ω i i = Γ i (3) were ne and nb are te number of points cosen on te domain Ω and te boundary Γ, respectively. e factor α in above equation is te relative weigt of te boundary residuals wit respect to te interior residuals. t is te same of weigt coefficient in general penalty metod for boundary condition imposing and is equal unit in tis paper. W ransactions on Modelling and Simulation, Vol 4, SSN X (on-line) 006 W Press

6 8 Boundary Elements and Oter Mes eduction Metods XXV Minimization of te eqn (3) leads to: ne (a, x ) (a, x ) Ω i nb [ ] Γ i = + [ ] = 0 = Ω (a, x ) = Γ (a, x ) a i α a i a i (4) i nn Substitution of u = N i ui = Nu in eqns (), () and (4) yields te final i= system of equations. KU=F System of algebraic equations sould be solved for te vector of unknown parameters U. Here nn = ne + nb denotes te total number of nodes used to represent te problem domain of it body. 4 Numerical nvestigation n tis section, two numerical examples in te area of elliptic problems are solved and results are presented to illustrate te performance of te proposed discrete least square mesless metod. We consider two dimensional steady state eat conduction or seepage equation in a omogeneous ortotropic body. φ φ D + D = S (5) y y Subect to appropriate Diriclete and Newmann boundary conditions φ = φ on Γu φ D = q on Γq n e residuals on interior and boundary nodes are defined by φ Ω = D S 0 =, Γu = φ φ φ Γ = D n q q n is te t component of te outward unit normal vector to te boundary Γ q. General differential operators in eqns (8), (9) are defined as. (6) (7) ( ) L( ) = D on Ω and =,, f = S (8) W ransactions on Modelling and Simulation, Vol 4, SSN X (on-line) 006 W Press

7 Boundary Elements and Oter Mes eduction Metods XXV 9 B(.) =.0, g = φ on Γu (.) (9) B(.) = D n, g = q on Γq Application of DLS metod leads to te following system of equations k ϕ = f (30) Were ϕ is te vector of unknown parameters[ φ, φ,..., φn. ne N l N nb k m lm = D D + [ BN ] [ ] i = i = l BN i l i i i ] (3) f l ne N nb = l D S i i + = i = i [ BN ] gi l i (3) 4. Poisson equation Consider te solution of te Poisson s equation. u(x, y) = sin πx cos πy Ω( x, y) : 0 x, Boundary conditions given as u = 0 x = 0 u = 0 x = u = 0 y = 0 u = 0 y = e exact solution of te governing equation is given by u = π sin πx cos πy { 0 y } Numerical solutions are obtained on two sets of nodal spacing. First wit nodes ( ) and second wit 676 nodes (6 6). Polynomial order is cosen 0 0 zero order p = [ x y ] = [ ] and subdomain of every node includes one nearest node on bot side and bot direction (np=.0, ns=3.0). Figures and sow numerical and exact solution of two sections (0., 0.5meter) of te problem domain for two nodal distributions. W ransactions on Modelling and Simulation, Vol 4, SSN X (on-line) 006 W Press

8 30 Boundary Elements and Oter Mes eduction Metods XXV nodes,np=,ns=3 nodes,np=,ns= Numerical Analytical Numerical Analytical u(x,y) x,y= x,y u(x,y) x,y= x,y Figure : Section plot of Laplace solution wit nodes. 676nodes,np=,ns=3 676nodes,np=,ns= Numerical Analytical Numerical Analytical u(x,y) x,y= u(x,y) x,y= x,y x,y Figure : Section plot of Laplace solution wit 676 nodes. 4. Seepage problem We consider Seepage problem wit tis governing equation and Diriclete and Newmann boundary conditions. φ φ 0 x + = 0 Ω(x, y) : y 0 y Subect to φ = x, y = φ = 0.0 φ = 0.0 n x y = on oter boundaries, W ransactions on Modelling and Simulation, Vol 4, SSN X (on-line) 006 W Press

9 Boundary Elements and Oter Mes eduction Metods XXV 3 Problem domain discretizes 33(4 8) nodes ( x = y = 0. 05). A 0 0 polynomial of zero order p = [ x y ] = [] is used (np=). Same as previous example every subdomain includes two nearest neigbor nodes on bot side and bot direction (ns=3). Figure 3 sows a countor plot of φ results in Problem domain. As sown in figure te distribution of potential are smoot inparticular near te Newmann boundaries. Figure 3: Seepage problem solution (np=, ns=3). 5 Concluding remarks n tis paper, we present Discrete Least Squares (DLS) mesless metod for te solution of elliptic problems. A fully Least Squares metod is used in bot function approximation and te discretization of te governing differential equations. e mesless sape functions are derived using te Moving Least Squares (MLS) metod of function approximation. e discretized equations obtained via a discrete least squares metod in wic te sums of te squared residuals minimized wit respect to unknown nodal parameters. e proposed metod as te additional advantages of te producing symmetric, positive definite matrices even for non-self adoint operators. e metod is tested against two elliptic examples in two dimensional steady state forms. eferences [] Monagan, J.J., An introduction to SPH, Comput. Pys. Comm. 48 (988) [] Liu, W.K., Li, S., Adee, J. & Belytscko,., eproducing Kernel Particle metods, nt. Journal For Numerical Metods fluids, 0, (995) [3] Melenk, J.M. & Babuska,., e partition of unity finite element metod: basic teory and applications, Comput. Metods Appl. Mec. Engng.999, 39, W ransactions on Modelling and Simulation, Vol 4, SSN X (on-line) 006 W Press

10 3 Boundary Elements and Oter Mes eduction Metods XXV [4] Durate, C.A. & Oden, J.., HP clouds An -p mesless metod, Numer. Met. Partial Diff. Eqns. 996;, [5] Nayroles, B., ouzot, G. & Villon, P., Generalizing te finite element metod: diffuse approximation and diffuse elements, Comput. Mec. 0 (99), [6] Belytscko,., Liu, Y. & Gu, L., Element Free Galerkin metods, nt Journal For Numer Metods Engng, 37, (994) [7] Atluri, S.N, Zu,., A New mesless local Petrov-Galerkin (MLPG) approac in computational mecanics. Computational Mecanics 998;, 7-7. [8] Zu,., Zang, J. &. Atluri, S.N., A local boundary integral equation (LBE) metod in computational mecanics, and mesless discretization approac. Computational Mecanics 998;, [9] Beissel, S. & Belytscko,., Nodal integration of te Element Free Galerkin metod. Comput. Metods. Appl. Mec. Engng 39, (996) [0] Zou, J.X., Wen, J.B., Zang, H.Y. & Zang, L., A nodal integration and post- processing tecnique based on Voroni diagram for Galerkin mesless metods. Comput. Metods Appl. Mec. Engng. 003; 9, [] Cen, J.K. & Beraun, J.E., A generalized smooted particle ydrodynamics metod for nonlinear dynamic problems, Comput. Metods. Appl. Mec. Engng, 90 (000) [ ] Lancaster, P. & Salkauskas, K., Surfaces generated by moving least square metods Mat. Comput. 37 (98). [3] Onate, E., delson, S., Zienkiewicz, O.C., aylor,.l. & Sacco, C., A stabilized finite point metod for analysis of fluid mecanics problems, Comput. Metods. Appl. Mec. Engng. 39 (996), W ransactions on Modelling and Simulation, Vol 4, SSN X (on-line) 006 W Press