Solving Poisson equations by boundary knot method

Transcription

1 International Worksho on MeshFree Methos 23 1 Solving Poisson equations by bounary knot etho W. Chen 1 Abstract: The bounary knot etho (BKM) is a recent eshfree bounary-tye raial basis function (RBF) collocation technique. Coare with the etho of funaental solution, the BKM uses the nonsingular general solution instea of the singular funaental solution to evaluate the hoogeneous solution, while as such the ual recirocity etho (DRM) is still eloye to aroxiate the articular solution. However, it is note that the nonsingular general solution of Lalace equation is a constant, the BKM can not thus irectly alie to it. This aer is an extension of reference [1], where a sile BKM schee was resente for solving Lalace equations. The schee oens the oor to use the BKM for general linear an nonlinear robles. 1 Introuction Aong tyical eshfree bounary-tye nuerical schees are the local bounary integral equation (MLBIE) etho [2], the bounary noe etho (BNM) [3], bounary oint interolation etho [4], an the etho of funaental solutions (MFS) [5]. The essence of the eshless MLBIE an BNM is basically a cobination of the oving least square (MLS) technique with a variant bounary eleent schee, whereas the MFS is a bounary-tye raial basis function (RBF) collocation schee. Both the MLBIE an the BNM involve singular integration an hence are atheatically ore colicate in coaring with the coonly use finite eleent etho (FEM). In aition, their low-orer aroxiations also owngrae coutational efficiency. In fact, since the BNM still requires eshes in its nuerical integration, it is not a truly eshless schee. The MLBIE oes not require a esh at all but is very exensive an not easily use. On the other han, the MFS ossesses integration-free, sectral convergence, easy-to-use, an inherently eshfree erits. In recent years, the MFS, also known as regular bounary eleents, revives, artly thanks to its cobining with the ual rerocity etho (DRM) for inhoogeneous robles [5]. Because of the use of singular funaental solution, the MFS nevertheless requires a controversial fictitious bounary outsie hysical oains, which largely iees its ractical use for colex geoetry robles. As an alternative RBF aroach, Chen an Tanaka [6] recently eveloe the bounary knot etho (BKM), where the erlexing artificial bounary is eliinate via the nonsingular general solution. Just like the MFS an ual recirocity BEM (DR-BEM) [7], the BKM also alies the DRM 1 Siula Research Laboratory, P. O. Box. 134, 1325 Lysaker, Norway, (wenc@siula.no).

2 2 W Chen to evaluate the articular solution. The etho is syetric, sectral convergence, integration-free, eshfree, easy to learn an ileent, an has successfully been alie to the Helholtz, iffusion, an convection-iffusion robles. Unfortunately, the nonsingular RBF general solution of Lalace equation, however, is a constant rather than a RBF, which akes the BKM not suitable for the robles of this kin. Chen [1] eveloe a sile BKM schee to solve this trouble issue. This aer further extens this strategy to the BKM solution of Possion equation robles. 2 BKM schee for Possion equation Here we introuce the BKM with a Possion equation roble u = f ( x) ( x) R( x) u( x) = N( x) 2, x Ω, (1) u =, x Su, (2a), x ST, (2b) where x eans ulti-iensional ineenent variable, an n is the unit outwar noral. The governing equation (1) can be restate as or ( x) δu 2 2 u + δ u = f + ( x) δu 2 2 u δ u = f, (3b) where δ is an artificial araeter. Eqs. (3a,b) are resectively Helholtz an oifie Helholtz equations. This strategy can be unerstoo that the use of nonsingular general solutions of Helholtz-like equation with sall characteristic araeter δ aroxiates the constant general solution of Lalace equation. The solution of the Possion roble can be slit as the hoogeneous an articular solutions (3a) u = u h + u, (4) The latter satisfies the governing equation but not bounary conitions. To evaluate the articular solution, the inhoogeneous ter is aroxiate by f N L ( x) β ϕ( r ) + = 1, (5) where β are the unknown coefficients. N an L are resectively the nubers of knots on the oain an bounary. The use of interior oints is usually necessary to guarantee the accuracy an convergence of the BKM solution of inhoogeneous robles. r = x x reresents the Eucliean istance nor, an ϕ is the raial basis function. By forcing aroxiation reresentation (5) to exactly satisfy governing equations at all noes, we can uniquely eterine β = A 1 ϕ { f ( x i )}, (6) where A ϕ is the nonsingular RBF interolation atrix. Then we have u = N + L = 1 ( x x ) β φ, (7)

3 International Worksho on MeshFree Methos 23 3 where the RBF φ is relate to the RBF ϕ through governing equations. In this stuy, we chose the first an secon orer general solutions as the RBFs φ an ϕ, which can be calculate with eqs. (3a,b) resectively by [8] n u () r = Q ( γ r) J n 2 1+ ( γr), (8a) an n u () r = Q ( τ r) I n 2 1+ ( τr), n 2, (8b) where n is the iension of the roble; Q =Q -1 /(2**γ 2 ), Q =1; enotes the orer of general solution; J an I reresent the Bessel an oifie Bessel function of the first kin. On the other han, the hoogeneous solution u h has to satisfy both governing equation an bounary conitions. By eans of the nonsingular general solution, the unsyetric an syetric BKM exressions are given resectively by u h u ( x) = a u ( r ) h L ( x) = u ( r ) k= 1 L + LN s s as 1 L + 1 α, (9a) k k ( r ) s, (9b) n where k is the inex of source oints on bounary, α k are the esire coefficients; n is the unit outwar noral as in bounary conition (2b), an L an L N are resectively the nubers of knots on the Dirichlet an Neuann bounary surfaces. The inus sign associate with the secon ter is ue to the fact that the Neuann conition of the first orer erivative is not self-aoint. In ters of reresentation (9b), the collocation analogue equations (3a) (or (3b)) an (2a,b) are written as Ls + LN ( ) ( ris ) asu ris as = R( xi ) u ( xi ), (1) L ( r ) + L 2 N s u ( rs ) as 2 L as 1 L + 1 = N ( r ) ( x ) ( x ), (11) + LN ls asu ( rls ) as = ul u ( xl ). (12) 1 L + 1 Note that i, s an are recirocal inices of Dirichlet (S u ) an Neuann bounary (S Γ ) noes. l inicates resonse knots insie oain Ω. Then we can eloy the obtaine exansion coefficients α an inner knot solutions u l to calculate the BKM solution at any other knots. 3 Nuerical results an iscussions The teste 2D an 3D Possion equation exales have accurate solutions 3 3 u = x y 2xy + x + 1, (13a) u = x yz xy z + xyz + x + 1. (13b) Figs. 1 an 2 show the teste 2D an 3D irregular geoetries, where the 3D ellisoi cavity locates at the center of the cube with the characteristic lengths 3/8, 1/8 an 1/8. Excet Neuann bounary conitions on x= surface of the 3D case, the otherwise bounary are all Dirichlet tye.

4 4 W Chen Fig. 1. A 2D irregular geoetry. Fig. 2. A cube with an ellisoi cavity We note that the unsyetric (eq. (9a)) an syetric (eq. (9b)) BKM schees rouce insignificant ifferences of accuracy. Therefore, Tables 1-3 only islays the unsyetric BKM L 2 nors of relative errors, which were calculate at 492 sale noes for 2D an 1 sale noes for 3D. Note that the abbrevations Hel an MHel in Tables 1 an 2 ean that the general solutions of Helholtz an oifie Helholtz equations (see eqs. (3)) are resectively use. Here the absolute error is taken as the relative error if the absolute value of the solution is less than.1. It is foun that a few inner noes are usually necessary to significantly irove the solution accuracy an stability. The first an secon nubers in the bracket of Tables reresents resectively the nubers of bounary an inner noes. It is observe fro Tables 1-3 that the accuracy of our nuercial exerieents is insensitive to artificial araeter δ an quite high. The convergence is also stable. Table 1. L 2 relative errors of 2D Possion equation uner a oain shown in Fig. 1. Hel (9+3) Mhel (9+3) Hel (13+3) Mhel (13+3) δ= e e e-4 3.5e-4 δ=.2 5.1e-3 5.1e e e-6

5 International Worksho on MeshFree Methos 23 5 Table 2. L 2 relative errors of 2D Possion equation uner an ellitic oain. Hel (9+3) Mhel (9+3) Hel (13+3) MHel (13+3) δ= e e e-4 1.2e-3 δ= e e-5 4.6e-5 9.8e-5 Table 3. L 2 relative errors of 3D Possion equation uner a oain shown in Fig. 2 with the general solution of oifie Helholtz oerator (eq. (3b)). (66+8) Mhel (96+8) Hel (138+8) MHel (192+8) δ= e e e e-4 δ=.5 5.7e-3 3.e-3 2.9e-3 4.8e-5 4 Soe rearks For Helholtz-like robles, the BKM outerfors the DR-BEM an MFS significantly in ters of accuracy, syetricity, efficiency, stability, an atheatical silicity [9]. The resent stuy shows that the etho is also very efficient for Possion equation robles. The aor rawbacks of the BKM are severe ill-conitioning an costly full atrix for large syste robles, which is a subect resently uner investigation. Chen [1] also foun that the BKM has the unerlying links with the oain-tye RBF ethos such as the Kansa etho [11] an accoringly eveloe a oifie Kansa etho. In this stuy, we use the high-orer general solutions of the Helholtz an oifie Helholtz equations to evaluate the articular solution. It will be interesting to investigate the erforances of the nonsingular high-orer funaental solution of the Lalacian equation in the evaluation of the articular solution. This stuy oens the way to aly the BKM to the solution of all general linear an nonlinear robles, where the DR-BEM has been use successfully. The sall araeter δ is soewhat arbitrary esite the fact that our nuerical investigations foun it is not sensitive to the geoetry an noe ensity. Anyhow, the resent BKM schee has soe obvious avantages over the MFS in that the BKM only requires austing one araeter δ, while the MFS has to arrange all artifical noes outsie hysical oains. Acknowlegeents: The work reorte here is sonsore by Siula Research Laboratory. Prof. C.S. Chen kinly rovie the configuration ata of figure 1.

6 6 W Chen References [1] W. Chen, Bounary knot etho for Lalace an biharonic robles. Proc. of the 14th Noric Seinar on Coutational Mechanics, , Lun, Sween, Oct. 21. [2] S. N. Atluri an T. Zhu, New concets in eshless ethos, Int. J. Nuer. Methos Eng., 47, , 2. [3] Y.X. Mukheree an S. Mukheree, The bounary noe etho for otential robles, Int. J. Nuer. Methos Eng., 4, , [4] G.R. Liu, MeshFree Methos -- Moving beyon the finite eleent etho, CRC Press, 22. [5] M.A Golberg an C.S. Chen, The etho of funaental solutions for otential, Helholtz an iffusion robles. In Bounary Integral Methos - Nuerical an Matheatical Asects, (E. by M.A. Golberg), , Cout. Mech. Publ., [6] W. Chen an M. Tanaka, New Insights into Bounary-only an Doain-tye RBF Methos, Int. J. Nonlinear Sci. & Nuer. Siulation, 1(3), , 2. [7] P.W. Partrige, C.A. Brebbia an L.W. Wrobel, The Dual Recirocity Bounary Eleent Metho, Cout. Mech. Publ., [8] M. Itagaki, Higher orer three-iensional funaental solutions to the Helholtz an the oifie Helholtz equations, Engineering Analysis with Bounary eleents, 15, , [9] Y.C. Hon an W. Chen, Bounary knot etho for 2D an 3D Helholtz an convection-iffusion robles with colicate geoetry, Int. J. Nuer. Meth. Engng , 56(13), 23. [1] W. Chen, New RBF collocation schees an kernel RBFs with alications, Lecture Notes in Coutational Science an Engineering, Vol 26, 75-86, 22. [11] E.J. Kansa an Y.C. Hon, Circuventing the ill-conitioning roble with ultiquaric raial basis functions: alications to ellitic artial ifferential equations, Cout. Math. Als., 39, , 2.