A fast boundary cloud method for 3D exterior electrostatic analysis

Transcription

1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 24; 59: (DOI:.2/nme.949) A fast boundary cloud method for 3D exterior electrostatic analysis Vaishali Shrivastava and N. R. Aluru, Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 68, U.S.A. SUMMARY An accelerated boundary cloud method (BCM) for boundary-only analysis of 3D electrostatic problems is presented here. BCM uses scattered points unlike the classical boundary element method (BEM) which uses boundary elements to discretize the surface of the conductors. BCM combines the weighted least-squares approach for the construction of approximation functions with a boundary integral formulation for the governing equations. A linear base interpolating polynomial that can vary from cloud to cloud is employed. The boundary integrals are computed by using a cell structure and different schemes have been used to evaluate the weakly singular and non-singular integrals. A singular value decomposition (SVD) based acceleration technique is employed to solve the dense linear system of equations arising in BCM. The performance of BCM is compared with BEM for several 3D examples. Copyright 24 John Wiley & Sons, Ltd. KEY WORDS: meshless methods; boundary cloud method (BCM); fast algorithms; SVD; BEM; exterior electrostatics. INTRODUCTION Computational analysis of micro-electro-mechanical systems (MEMS) requires accurate analysis of exterior electrostatic problems [ 4]. Boundary integral formulations [5] are powerful computational techniques for rapid analysis of exterior problems, especially when combined with fast algorithms based on multipole expansions [6], Fast Fourier transform (FFT) [7] or singular value decomposition (SVD) [8 ]. Boundary integral formulations are best suited for Correspondence to: N. R. Aluru, Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 68, U.S.A. aluru@uiuc.edu Contract/grant sponsor: National Science Foundation; contract/grant number: NSF Contract/grant sponsor: National Science Foundation; contract/grant number: NSF-7623 Contract/grant sponsor: Natural Science Foundation; contract/grant number: NSF-266 Contract/grant sponsor: Natural Science Foundation; contract/grant number: NSF Received 8 December 22 Revised 5 May 23 Copyright 24 John Wiley & Sons, Ltd. Accepted 6 July 23

2 22 V. SHRIVASTAVA AND N. R. ALURU exterior electrostatic analysis as they require meshing of the conductor surface only. Although the meshing of the surfaces can be less involved compared to the meshing of the entire volumes, it can still be an arduous task. An attractive alternative is to combine meshless techniques with boundary integral formulations to circumvent the problems associated with meshing. Boundary integral formulations generate dense matrices to compute the unknown charge densities on the surface of the conductor. The use of classical direct methods, such as Gaussian elimination, requires O(N 3 ) operations, where N is the number of nodes or panels on the surface of the conductor. Iterative methods [ 3] require O(N 2 ) operations, in each iteration, to calculate a dense matrix vector product. The concept of SVD has been introduced in References [8, 9] to compute the dense matrix vector product. This approach uses the fact that large parts of the integral operator matrix have numerically low rank and SVD can be used to compress these parts. The complexity of this approach is typically O(N log N) [8 ]. Boundary integral formulations when combined with scattered point or meshless techniques are attractive computational methods for exterior problems as they alleviate the task of mesh generation. Recently, several meshfree methods have been proposed in the literature (see e.g. Reference [4] for an overview) that combine moving least-squares approach with Galerkin as well as collocation techniques for interior problems. A boundary node method [5, 6] is a combined boundary integral/meshless approach for boundary-only analysis of partial differential equations. In a boundary node method, given a scattered set of points, meshless interpolation functions are constructed by using a moving least-squares approach [7]. Use of Cartesian co-ordinates in a moving least-squares approach can lead to singular coefficient matrices if the points on the boundary lie on the same plane. The boundary node method circumvents this problem by employing curvilinear co-ordinates in the moving least-squares approach to construct interpolation functions. The definition of these co-ordinates is non-trivial for complex geometries. Recently, a boundary cloud method [8] has been introduced which also uses a combined boundary-integral/scattered point approach for boundary only analysis of partial differential equations. The boundary cloud method employs a generalized least-squares approach to construct interpolation functions. While the method proposed in Reference [8] allows the use of Cartesian co-ordinates, it forbids putting points along the edges and corners where the normal to the boundary is not well defined. The boundary cloud method has been extended in Reference [9] so that points can be placed at the corners for 2D problems. The basic idea is to use a varying polynomial basis to eliminate the singularity encountered in the moving least-squares approach when all the points lie along a straight line. In this paper, the boundary cloud method is extended for 3D problems. A cloud-by-cloud polynomial basis is used so that Cartesian co-ordinates can be employed and points can be put along the edges and corners for 3D problems. The boundary cloud method is also combined with the SVD-based acceleration technique for rapid solution of the 3D exterior electrostatic problem. The rest of the paper is organized as follows: Section 2 summarizes the governing equation and the boundary integral formulation for the 3D electrostatic problem, introduces the varying basis approach to compute the interpolation functions, describes the discretization and integration of the boundary integrals and provides an order of magnitude estimate for the CPU time required by BEM and BCM. The SVD-based acceleration technique is summarized in Section 3. In Section 4, we present some 3D examples in which we compare the classical BEM and BCM in terms of the accuracy of the solutions obtained and the CPU time required by both the methods. Finally, Conclusions are presented in Section 5. Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59:29 246

3 FAST BOUNDARY CLOUD METHOD 22 Ω R inf Figure. A system of n ideal conductors embedded in a uniform lossless dielectric medium. The reference potential plane is a sphere of radius R inf = R. 2. BOUNDARY CLOUD METHOD FOR EXTERIOR ELECTROSTATICS 2.. Governing equation and boundary integral equation Consider a system of n ideal conductors embedded in a uniform lossless dielectric medium (Figure ). A reference potential plane (a sphere in 3D with radius R ref ) is defined as a surface enclosing the conductors. The electrostatic potential is prescribed (constant) on the surface of the conductors and the objective is to compute the surface charge density. The governing equation for the problem is 2 = in Ω () i = g i on dω i (2) where i is the electrostatic potential of the ith conductor, dω i is the domain of the ith conductor, Ω is the infinite domain exterior to the conductors and g i is the prescribed electrostatic potential on conductor i. The boundary integral equation for the electrostatic problem is given by (see e.g. References [5, 2 22] for details) (p) = dω ε G(p, q)σ(q) dγ q + C (3) σ(q) dγ q = C T (4) dω Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59:29 246

4 222 V. SHRIVASTAVA AND N. R. ALURU Non singular cloud Singular cloud Star point "t" Node Figure 2. Definition of singular and non-singular clouds. where dω = n i= dω i, ε is the dielectric constant of the medium, p is the source point and q is the field point which moves along the boundary of the conductors. In three dimensions, G(p, q) = /4πr(p,q), where r(p,q) is the distance between the source point p and the field point q. C = G(R )C T, where C T is the total charge of the system and G(R ) as R. It is commonly assumed that =, i.e. C = and this leads to a wellposed problem in which the charge density can be obtained directly from Equation (3). The unknown charge density σ = / n is computed by using the boundary cloud method. For a two conductor system, with a prescribed potential of V on conductor and V on conductor 2, once the charge distribution on the conductors has been computed using the BCM, the capacitance of conductor can be computed using [2] Cap = σ(q) dγ q (5) dω where dω is the surface of conductor and σ is the charge distribution on the surface of conductor Construction of approximation functions In the boundary cloud method, the surface of the domain is discretized into scattered points. The points can be sprinkled randomly covering the boundary of the domain. Approximation functions are constructed by centering a weighting function at each point or node. A varying basis is used to approximate the unknown quantity σ. Given a point t (point t is referred to as the star point see Figure 2), the charge density in the vicinity of point t is approximated by σ(x,y,z)= p T (x,y,z)a t (6) where p is the m varying base interpolating polynomial vector and a t is an m unknown coefficient vector for point t. a t has been assumed constant for a given point t. The varying basis vector p is constructed by classifying the clouds into two types: singular and nonsingular (see Figure 2). When all the points inside a cloud lie on the same surface, the cloud is defined as singular, else, it is defined as non-singular. Using a linear polynomial basis, the Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59:29 246

5 FAST BOUNDARY CLOUD METHOD 223 base interpolating polynomial is given by { [ x y z] : m = 4 basis for non-singular cloud p T (x,y,z)= [ x y] or [ y z] or [ x z] : m = 3 basis for singular cloud (7) For a point t, the unknown coefficient vector a t is computed by minimizing J t = NP w i (x t,y t,z t ) [ p T ] 2 (x i,y i,z i )a t ˆσ i (8) i= where NP is the number of nodes and w i (x t,y t,z t ) is the weighting function centred at (x t,y t,z t ) and evaluated at node i whose co-ordinates are (x i,y i,z i ). ˆσ i is the nodal parameter for node i. The weighting function is typically a cubic spline or a Gaussian function. A cubic spline weighting function is convenient to use for rectangular geometries and a Gaussian weighting is convenient for spherical geometries. The cubic spline weighting function is given by w i (x t ) = ( ) xt x i ŵ (9) d x d x : x< 2 (x) 3 /6 : 2 x 2/3 x 2 ( + x/2) : x ŵ(x) = 2/3 x 2 ( x/2) : x (x) 3 /6 : x 2 : x>2 where x = (x t x i )/d x and d x denotes the support size of the weighting function in the x-direction. A multi-dimensional weighting function can be constructed as a product of onedimensional weighting functions. In three dimensions, the weighting function is given by w i (x t,y t,z t ) = ( ) ( ) ( ) xt x i yt y i zt z i ŵ ŵ ŵ () d x d x d y d y d z d z The Gaussian weighting function is given by () w i (x t ) =ŵ(x t x i ) (2) exp (x/c x) 2 exp (d x/c x ) 2 : x <d ŵ(x) = exp x) 2 x : x d x (3) where x = (x t x i ), d x denotes the support size of the weighting function in the x-direction and c x is the dilation parameter which has been chosen to be d x /3 in this paper (see Reference [23] Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59:29 246

6 224 V. SHRIVASTAVA AND N. R. ALURU for a discussion). A multi-dimensional weighting function can be constructed as a product of one-dimensional weighting functions. The stationarity of J t leads to P T WPa t = P T Wˆσ (4) Equation (4) can be rewritten as C t a t = A t ˆσ (5) where C t is an m m matrix and A t is an m NP matrix defined as follows: C t = P T WP (6) A t = P T W (7) where P is an NP m matrix, ˆσ is an NP vector and W is a square NP NP diagonal matrix and their definitions are given by p T (x,y,z ) ˆσ p T (x 2,y 2,z 2 ) ˆσ 2 P =, ˆσ =.. p T (x NP,y NP,z NP ) ˆσ NP w (x t,y t,z t ) w 2 (x t,y t,z t ) W =... w NP (x t,y t,z t ) The charge density σ, can be expressed as σ(x,y,z)= N(x,y,z)ˆσ (9) where N(x,y,z) isa NP approximation vector defined by N(x,y,z)= p T (x,y,z)c t A t (2) Equation (9) can be rewritten in a matrix form as σ =[N]ˆσ (2) where [N] is an NP NP approximation matrix. At any point (x,y,z), the unknown is evaluated by σ(x,y,z)= NP N I (x,y,z)ˆσ I (22) I= where N I is defined as the approximation function for node I and is given by N I (x,y,z)= p T (x,y,z)c t p(x I,y I,z I )w I (x t,y t,z t ) (23) Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59: (8)

7 FAST BOUNDARY CLOUD METHOD 225 Nodes P Cells Singular integration (regularization scheme) Figure 3. Discretization of the boundary for integration Discretization, integration and matrix form In the boundary cloud method, the boundary of the domain is discretized into quadrilateral cells for integration purpose (see Figure 3). Each cell contains a certain number of nodes. The number of nodes can vary from cell to cell. The cell can be of any shape and size and we use a quadrilateral cell only for illustration purpose. The only restriction is that the union of all the cells should be equal to the boundary of the domain. Assuming the boundary is discretized into NC cells, the boundary integral equation for the electrostatic problem can be written as (P ) = NC k= dω k 4πεr(P,Q k ) σ(q k) ds Q (24) where dω k is the surface of the kth cell, Q k is the field point in the kth cell and σ(q k ) is the charge density. Depending on the location of the source point P and the field point Q, various techniques are used to evaluate the integrand in Equation (24). When P and Q are in the same cell, analytical integration or regularization techniques can be used. When P and Q are in different cells, the integration can be performed by using a regular Gauss quadrature Cell does not contain the source point. When the source point P does not lie in a cell, regular Gauss quadrature is used for numerical integration (either 3 3 or 4 4 Gauss quadrature is typically adequate). For a given cell k, the boundary integral equation (24) can be written as I k = NG k j= η j 4πεr(P,Q j ) σ(q j ) (25) where NG k is the total number of Gauss points in cell k and η j is the weight of Gauss point j. Substituting the approximation for σ given in Equation (22) into Equation (25), we obtain I k = NP NG k i= j= η j 4πεr(P,Q j ) [N i(q j )ˆσ i ] (26) Cell contains the source point. When a cell contains the source point, the order of singularity associated with the integrand in Equation (24) is O(/r), which is a weak singularity for two-dimensional surface integrals. Regular numerical integration of the product of Green s Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59:29 246

8 226 V. SHRIVASTAVA AND N. R. ALURU function with the linear polynomial basis can be inaccurate. The regularization method suggested by Chati and Mukherjee [5] has been used to compute the weakly singular integrals. The integration over the kth cell (see Equation (24)) is given by I2 k = NP i= [ = NP 4πε i= dω k 4πεr(P,Q k ) pt (Q k ) ds Qk ] [ C P p(x i,y i,z i )w i (x P,y P,z P )ˆσ i ] [ ][ ] I2k I22 k I23 k I24 k C P p(x i,y i,z i )w i (x P,y P,z P )ˆσ i (27) where I2k = dx dy dω k r(p,q k ) (28) I2k 2 = x dx dy dω k r(p,q k ) (29) I2k 3 = y dx dy dω k r(p,q k ) (3) I2k 4 = z dx dy r(p,q k ) (3) dω k and the cloud is centred at the source point P and C P has been evaluated at point P. Consider evaluating Equations (28) (3) over a quadrilateral cell as shown in Figure 4(a). The cell shown contains the source point P as one of its vertices. The cell is divided into two triangular regions, such that, the source point P is one of the vertices of each of the triangles. Each of these triangles is mapped onto the parent space r s (Figure 4(b)) using well-known shape functions for T3 triangles. In the parent space, the integrals in Equations (28) (3) can be rewritten as I2 k = n T I2 2 k = n T I2 3 k = n T I2 4 k = n T s= r= s i = s= r= s= r= s i = s= r= s= r= s i = s= r= s= r= s i = s= r= r(p,q) J ( i) dr ds (32) x r(p,q) J ( i) dr ds (33) y r(p,q) J ( i) dr ds (34) z r(p,q) J ( i) dr ds (35) If the source point P does not lie at one of the vertices of the cell, then the cell is divided into 4 triangles, such that, point P is one of the three vertices of each of the triangles. x = 3 i= N i ˆx i where N = r s, N 2 = r and N 3 = s. y and z are also mapped in the same way as x. Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59:29 246

9 FAST BOUNDARY CLOUD METHOD 227 η 2 P z T2 T - - η (d) x (,) y s (a) ρ = ρ P (b) (,) r (c) θ = π /2 θ Figure 4. Mapping for weakly singular integrals. where n T is the number of triangles into which the cell is divided, such that, P is one of the vertices of each of the triangles (n T = 2 in Figure 4(a)) and J ( i) is the Jacobian of the transformation of points in triangle i from x y z co-ordinates to r s co-ordinates and it is given by x x y z r s r s z y r s y y z x J = = r s r s x z (36) r s z z x y r s r s y x r s J = J (37) The flat triangles are then mapped from the r s co-ordinate system onto a rectangle in the ρ θ space (Figure 4(c)), that is, r = ρ cos 2 θ, s = ρ sin 2 θ (38) Using the transformation in Equation (38), the integrals take the following form: I2k = n T θ=π/2 ρ= r(p,q) J ( i) ρ sin(2θ) dρ dθ (39) I2 2 k = n T i = θ= ρ= θ=π/2 ρ= i = θ= ρ= x r(p,q) J ( i) ρ sin(2θ) dρ dθ (4) Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59:29 246

10 228 V. SHRIVASTAVA AND N. R. ALURU I2 3 k = n T i = I2 4 k = n T i = θ=π/2 ρ= θ= ρ= θ=π/2 ρ= θ= ρ= y r(p,q) J ( i) ρ sin(2θ) dρ dθ (4) z r(p,q) J ( i) ρ sin(2θ) dρ dθ (42) As ρ is the measure of the distance between the source point and the field point, the integrals in Equations (39) (42) are now regularized. In other words, ρ in the numerator cancels the O(/r) singularity. The integrals in Equations (39) (42) can be evaluated by regular Gauss quadrature. The final mapping involves the use of quadratic (Q4) shape functions to map the rectangle from the ρ θ space onto the standard square in η η 2 space (Figure 4(d)), using ρ = η +, θ = π η and the Jacobian of the final transformation (43) J 2 = π/8 (44) The final form of the integrals is I2 k = n T i = I2 2 k = n T i = I2 3 k = n T i = I2 4 k = n T i = η2 = η = η 2 = η = η2 = η = η 2 = η = η2 = η = η 2 = η = η2 = η = η 2 = η = r(p,q) J ( i) J 2 ρ sin(2θ) dη dη 2 (45) x r(p,q) J ( i) J 2 ρ sin(2θ) dη dη 2 (46) y r(p,q) J ( i) J 2 ρ sin(2θ) dη dη 2 (47) z r(p,q) J ( i) J 2 ρ sin(2θ) dη dη 2 (48) For a quadrature point in the η η 2 space, ρ and θ are computed by using Equation (43), and subsequently r and s are computed by using Equation (38). Finally, the regular Gauss integration can be used to evaluate the integrals in Equations (45) (48). The integrals are then substituted back into Equation (24) after evaluation. In summary, for a source point P, the discretized form of Equation (24) can be written as = NC k= Equation (49) can be rewritten in the matrix form as I k + I2 NC (49) Gσ = (5) Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59:29 246

11 FAST BOUNDARY CLOUD METHOD 229 where G ij = NC k= dω k 4πεr(P i,q k ) NP j= N j (Q k ) dγ qk (5) and σ are the NP right-hand side and unknown vector, respectively. The next step is to solve the matrix problem in Equation (5). Direct solution via Gauss elimination requires O(NP 2 ) storage and O(NP 3 ) time, where NP is the number of nodes. Hence, it is impractical for large problems. The system of equations can be solved iteratively by using GMRES [2]. The solution process in the iterative scheme requires multiplication of matrix G with a sequence of recursively generated vectors which costs O(NP 2 ) operations. The CPU time can be reduced by applying the SVD-based acceleration technique, which is described in detail in References [8 ], and the salient points of the technique are summarized in Section Computation cost in BEM and BCM The boundary integral equation for the electrostatic problem can be written as (P ) = NC or NE k= dω k 4πεr(P,Q k ) σ(q k) ds Q (52) where NC is the number of cells in BCM, NE is the number of elements in BEM, dω k is the surface of the kth cell in BCM or kth element in BEM and Q k is the field point in the kth cell. The matrix form is given by Equation (5). The G matrix in Equation (5) can be computed by using the BCM as shown in Algorithm and by using the BEM as shown in Algorithm 2. Algorithm Constructing the coefficient matrix G in BCM. for i = to NP (number of nodal points) do Singular Integration (i, cell containing i) for j = tonc (number of cells) do for k = tong j (number of Gauss points in cell j) do Compute G(i, cloud(k)) Regular Integration(k) / cloud(k) denotes the points inside the cloud of Gauss point k / end for end for end for Algorithm 2 Constructing the coefficient matrix G in BEM. for i = to NP (number of nodal points) do Singular Integration (i, element containing i) for j = tone (number of elements) do for k = tong j (number of Gauss points in element j) do Compute G(i,connectivity(j)) Regular Integration(k) / connectivity(j) denotes the points of element j / end for end for end for Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59:29 246

12 23 V. SHRIVASTAVA AND N. R. ALURU In Algorithms and 2, NP denotes the number of nodes, NG j is the number of Gauss integration points in the jth element/cell (we assume NG BEM j = NG BCM j ), NE is the number of elements in BEM (we assume NE NP) and NC is the number of cells in BCM (NC < NP). In BEM, the number of nodal points whose shape functions are involved at an integration point, N BEM = 4, for bilinear elements and in BCM, the number of nodal points whose shape functions are involved at an integration point, N BCM = number of nodal points in the cloud of a node = n p > 4. The CPU time t for computing a single term in the G matrix is given by t = O[N BCM (or N BEM ) NG j NC(or NE)] (53) Therefore, the relationship between the CPU time required for the BEM and the BCM can be written as [ ] [ ] t BCM NBCM NG j NC np NC = O = O (54) t BEM N BEM NG j NE 4 NE Typically, for NC = NE and cloud size, n p = 9, BCM is about 2.25 times slower than BEM. But, BCM needs fewer number of cells, NC, compared to the number of elements, NE, in BEM to give results of similar accuracy. We show in Section 4 that the CPU time required by BCM is less than that used by BEM for a similar solution. 3. ACCELERATION TECHNIQUE 3.. Basic idea The idea behind the acceleration technique is to exploit the fact that the assembled matrix G via BCM has large sections which are rank deficient. The reason behind this is that, typical Green s functions vary smoothly. Each entry in the assembled matrix denotes interaction between two points a source point and a field point. If the distance between them is relatively large, the influence of a source point on a far-away field point is almost the same as that of its neighbours (owing to smoothly varying Green s function). Thus, the numerical rank of such portions of the matrix is small. SVD [, 3] can be employed to exploit the rank-deficient nature of the coefficient matrices generated by the boundary cloud method. The SVD of a matrix A is given by where U, S, and V T are n n matrices, A = USV T (55) UU T = V T V = I (56) and S is a diagonal matrix of singular values with s s 2 s n. The numerical rank of a matrix to precision c is given by integer r such that s r /s <c. The matrix A can be approximated by Ā = Ū S V T (57) where Ū is an n r matrix, S is an r r matrix and V is an r n matrix, such that A Ā <c (58) Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59:29 246

13 FAST BOUNDARY CLOUD METHOD 23 The multiplication of a vector with matrix A is done by multiplying the vector in turn by V T, S and Ū. The number of operations is O((2n + )r) and for r>n this process is more efficient compared to the O(n 2 ) operations that are normally required to compute the dense matrix vector product. The concept of SVD has been used to explain the theory behind this method. But SVD of a matrix is computationally expensive. Therefore, Gram Schmidt [, 3] process is used to compute the numerical rank r to precision c of a matrix A of size n n. The matrix A is decomposed as follows: A = Ū V T (59) where Ū is an n r orthogonal matrix which spans the column space of A and r is the numerical rank of the matrix A to precision c r. The process of multiplying a vector with matrix A takes O(2nr) operations Procedure The procedure to rapidly compute the matrix vector product can be broadly divided into two steps:. Preprocessing This step generates the compressed form of the matrix. The complexity of the preprocessing step is dependent on computing the UV decomposition of the submatrices. 2. Solve the linear system using GMRES The compressed representation is then used to compute the matrix vector products in an iterative solver (like GMRES). These two steps are described in great detail in References [8 ]. The preprocessing step typically costs about 4 or 5 dense matrix vector products and it can be shown that the complexity of the acceleration technique is O(N log N) for /r type Green s function. 4. RESULTS In this section, we present results on electrostatic analysis of cubic, spherical and cylindrical conductors and multiple conductor problems, i.e. a carbon nanotube over a ground plane and a bus crossing example. Equation (5) has been used to evaluate the capacitance of the conductor. The analytical expressions for the capacitance of a sphere and cylinder are given in Reference [24]. The dense linear system of equations generated by the BCM have been solved using the accelerated iterative solver and its computational complexity is addressed. The numerical precision for computing the rank is 4 (referred to as c in Section 3). The convergence rate and accuracy obtained from the varying linear polynomial basis BCM and linear BEM have been compared. The error is defined by u e u c, where u e is the exact or the analytical solution for the problem and u c is the computed solution. If an exact solution is not available, then u e denotes a reference solution. All computations have been done on an alpha 4 MHz computer. 4.. Cubic capacitor Figure 5 shows the configuration of the cubic conductor ( units) problem. The approximation functions in BCM are computed by using a cubic spline weighting function. The Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59:29 246

14 232 V. SHRIVASTAVA AND N. R. ALURU.8.6 z y x.4.2 Figure 5. A cubic conductor of dimensions units. The potential applied on the surface of the conductor is unit. The theoretical value of the capacitance of the conductor is 8.3ε units. 2 Surface Charge Density.5 z y.5 x 2 Figure 6. Surface charge density plot ( e ) for a uniform distribution of 488 points on the surface of the cube. surface of the cube has been discretized by using quadrilateral cells/elements in BCM/BEM. Each cell/element has 4 Gauss points for numerical integration. Figure 6 shows the plot of surface charge density obtained by BCM for a uniform distribution of 488 points on the surface of the cube. Figure 7 shows the convergence of BCM and BEM for evaluating the capacitance of the conductor. We take the solution obtained from FastCap [25], which is 8.3ε fora cubic capacitor, as the reference solution. The convergence rate of BCM is.36 and that of BEM is.24. Figure 8 shows the comparison of the CPU Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59:29 246

15 FAST BOUNDARY CLOUD METHOD BCM linear basis BEM linear element.8 log(error in capacitance) log(h) Figure 7. Convergence plot of BCM and BEM for the cube example. The convergence rate of BCM is γ =.36 and the convergence rate of BEM is γ = BCM Shape function in BCM BEM Computation time (secs) Number of nodes Figure 8. Comparison of CPU time used by BCM and BEM for the cube example. Also, shown is the CPU time of approximation function calculation in BCM. time required by BCM and BEM. It also shows the time required for the generation of shape functions in BCM, which is quite negligible as compared to the total CPU time. Using Equation (54), we estimate the ratio of the CPU time required by BCM and BEM as t BCM t BEM = O ( np NC 4 NE ) 2.25, n p = 9 (cloud size) and NC = NE (6) Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59:29 246

16 234 V. SHRIVASTAVA AND N. R. ALURU Table I. Comparison of BCM and BEM solutions for the cube example using similar CPU time. BCM BEM BCM BEM Number of nodes Number of cells/elements Error in capacitance (%) CPU time (s) Table II. Comparison of CPU time used by BCM and BEM for the same solution for the cube example. BCM BEM Number of nodes Number of cells/elements Error in capacitance (%).6.6 CPU time (s) From Figure 8, we observe that the actual CPU time used by BCM compared to that used by BEM for various discretizations is well within Table I compares the error in the computed value of the capacitance of a cubic conductor using BCM and BEM for similar CPU time. We observe that BCM gives a smaller error compared to BEM. Typically, the regular Gauss quadrature becomes inaccurate in evaluating the singular or near-singular integrals. The regularization scheme is more accurate in the region near the source point. In BCM fewer number of cells are used compared to the number of elements used in BEM. Therefore, each cell in BCM contains more number of nodal points as compared to the number of nodal points contained in an element in BEM. As a result, the accurately integrated region (or the region in which the regularization scheme is used) is larger in BCM compared to that used in BEM, which probably contributes to the smaller error with BCM. Table II shows the comparison of the CPU time used by BCM and BEM for computing approximately the same solution. From the results shown in Tables I and II, we can infer that BCM gives more accurate results compared to BEM for similar CPU time and BCM is faster than BEM for the same error in the solution. Figure 9 shows the scaling of the CPU time with the number of nodes. We observe the complexity to be of O(N log N) for computing the charge densities using the accelerated iterative solver, where N is the number of nodes on the surface of the conductor Spherical capacitor Figure shows a spherical conductor of radius.9 units. The surface of the sphere has been discretized using quadrilateral cells/elements. Gaussian weighting function and a 9 point cloud has been used to compute the shape functions in BCM. The analytical solution for the surface charge density on the surface of a sphere is constant, given by ε /R, where ε is permittivity of the medium, is the applied potential and R is the radius of the sphere. For R =.9, = and ε =, the surface charge density is. units. Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59:29 246

17 FAST BOUNDARY CLOUD METHOD FastBCM O(NlogN).8 Solve time (secs) Number of nodes Figure 9. Scaling of the CPU time with the number of nodes for the cubic conductor problem. The solid line shows the CPU time used for computing the matrix vector product in GMRES with the compressed form and it scales same as the dash dot line, i.e. O(N log N). Figure. Discretized sphere of radius.9 units. The potential applied on the surface of the conductor is unit. The theoretical value of capacitance of the conductor is.3ε. Figure shows the charge density along the circumference of the sphere and the constant value is. units. After obtaining the charge density, the capacitance is computed by using Equation (5). The analytical solution is 4πεR [24], which reduces to.3ε for R =.9. Figure 2 shows the convergence of BCM and BEM for evaluating the capacitance of the conductor. The convergence Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59:29 246

18 236 V. SHRIVASTAVA AND N. R. ALURU Figure. Charge density along the circumference of the sphere BCM linear basis BEM linear elements 3.2 log(error in capacitance) log(h) Figure 2. Convergence plot of BCM and BEM for the sphere example. The convergence rate of BCM is γ =.7 and the convergence rate of BEM is γ =.98. rate of BCM is.7 and that of BEM is.98. Figure 3 shows the comparison of the CPU time required by BCM and BEM and the time required for the generation of shape functions in BCM. Again we observe that the ratio of the CPU time used by BCM and BEM for various discretizations is well within Figure 4 shows the scaling of the CPU time with the number of nodes. We observe that it scales as O(N log N), where N is the number of nodes on the surface of the conductor. Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59:29 246

19 FAST BOUNDARY CLOUD METHOD BCM Shape function in BCM BEM 7 Computation time (secs) Number of nodes Figure 3. Comparison of CPU time used by BCM and BEM for the sphere example..2 Fast BCM O(NlogN). Solve time (secs) Number of nodes Figure 4. Scaling of the CPU time with the number of nodes for the spherical conductor problem. The solid line shows the CPU time used for computing the matrix vector product in GMRES with the compressed form and it scales same as the dash dot line, i.e. O(N log N) Cylindrical capacitor The charge density on the surface of a cylindrical conductor of radius unit and length 5 units has been evaluated. The potential on the surface of the conductor is unit. The theoretical value of the capacitance is given by [ (b/a).76 ]εa, where 2b is the length and a is Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59:29 246

20 238 V. SHRIVASTAVA AND N. R. ALURU Figure 5. A cylindrical conductor of radius unit and length 5 units. The potential applied on the surface of the conductor is unit. The theoretical value of capacitance of the conductor is 2.9ε..2.4 BCM linear basis BEM linear elements log(error in capacitance) log(h) Figure 6. Convergence plot of BCM and BEM for the cylindrical conductor. The convergence rate of BCM is γ =.74 and the convergence rate of BEM is γ =.57. the radius [24]. The surface of the tube has been discretized using quadrilateral cells/elements in BCM/BEM (see Figure 5). Each cell/element has 9 Gauss points for numerical integration. Gaussian weighting function with a 9 point cloud has been used to generate the shape functions. Figure 6 shows the convergence plot of BCM and BEM for evaluating the capacitance of the conductor. The convergence rate of BCM is.74, which is slightly higher than that of Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59:29 246

21 FAST BOUNDARY CLOUD METHOD Shape function in BCM BCM BEM 8 Computation time (secs) Number of nodes Figure 7. Comparison of CPU time used by BCM and BEM for the cylinder example Fast BCM O(NlogN) Solve time (secs) Number of Nodes Figure 8. Scaling of the CPU time with the number of nodes for the cylindrical conductor problem. The solid line shows the CPU time used for computing the matrix vector product in GMRES with the compressed form and it scales same as the dash dot line, i.e. O(N log N). BEM, which is.57. Figure 7 shows the comparison of the CPU time required by BCM and BEM and the time required for the shape function generation in BCM. We again observe that t BCM /t BEM < Figure 8 shows the scaling of the CPU time with the number of nodes. We observe that it scales as O(N log N), where N is the number of nodes on the surface of the conductor. Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59:29 246

22 24 V. SHRIVASTAVA AND N. R. ALURU Table III. Comparison of BCM and BEM solutions and CPU time for the cylinder example using the same number of nodes. BCM BEM BCM BEM BCM BEM Number of nodes Number of cells/elements Error in capacitance (%) CPU time (s) Figure 9. Configuration of the two-conductor system. The radius of the nanotube is.67 nm and its length is 5 nm. The gap between the axis of the nanotube and the ground plane is 5 nm and the potential applied on the nanotube is 5 V. Table III compares the error in the computed value of the capacitance of a cylindrical conductor and the CPU time used by BCM and BEM for the same number of nodes. We observe that BCM gives a smaller error compared to BEM for the same number of nodes. Also, BCM takes less CPU time compared to BEM when the same number of nodes are used for discretization Two conductor problem cylindrical nanotube The configuration of the two-conductor problem is shown in Figure 9. The surface charge density of the cylindrical nanotube of radius.67 nm and length 5 nm at a distance of 5 nm from the ground plane has been evaluated using BCM. The applied potential on the conductor is 5 V. The surface of the tube has been discretized using quadrilateral cells. Each cell has 4 Gauss points for numerical integration. Gaussian weighting function with a 9 point cloud has been used to generate the approximation functions. Figure 2 shows the surface charge density on the tube. The computed solution is similar to that of an infinitely long tube with uniform surface charge distribution almost throughout the length tube, except at the ends. There is a charge concentration at the free and the closed ends. There is a charge depletion at the end close to the contact because of the screening of the nanotube surface by the contact. Figure 2 shows the charge density along different cross-sections of the 5 nm tube (z = 5 nm, z = nm, z = 37.5 nm and z = 25 nm). We Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59:29 246

23 FAST BOUNDARY CLOUD METHOD 24 Figure 2. Charge distribution on the surface of the 5 nm nanotube with an applied potential of 5 V y x 2 z 4 6 tube cross section 8 z = 5 z = z = 37.5 z = 25 Figure 2. Charge distribution along the different cross-sections of the 5 nm nanotube with an applied potential of 5 V. can see that a dipole is formed due to the accumulation of charges in the lower half of the tube facing the ground plane. Figure 22 shows the dependence of the capacitance on the distance to the ground plane, h, for a fixed length (i.e. 5 nm) of the device. We observe that the capacitance of the nanotube, Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59:29 246

24 242 V. SHRIVASTAVA AND N. R. ALURU C g, nf h, nm Figure 22. Dependence of the capacitance of the cantilever nanotube of radius.67 nm on the gap between the nanotube and the ground plane of the device, h. Here, C g denotes the capacitance evaluated by using Equation (5)..4.2 Quantum mechanical Statistical BCM.8 φ, volt Distance from the contact, nm Figure 23. Comparison of results from electrostatic analysis and the quantum mechanical computation [26]. computed by using Equation (5), decreases with h because with the increase in the distance from the nanotube to the ground plane the screened area of the nanotube surface grows, and this screening diminishes the capacitance. We have also compared the charge distribution obtained by solving the Laplace equation by using BCM with that obtained by quantum mechanical Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59:29 246

25 FAST BOUNDARY CLOUD METHOD z y x Figure 24. Configuration of the bus crossing problem z y x 4 6 Figure 25. Surface charge density plot ( e ) for a uniform distribution of 352 points on the surface of each conductor in the bus crossing problem. computation (see Reference [26] for details). Figure 23 shows the comparison. The difference in the results is within 6 7% Bus crossing problem The final example is a bus crossing geometry illustrated in Figure 24 with 4 conductors of dimension 5 units. The applied potential on the top two conductors is V and the bottom two conductors are grounded. In this example, we have discretized the surface of the conductors into large number of points (maximum number of nodes = 24584). Figure 25 shows Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59:29 246

26 244 V. SHRIVASTAVA AND N. R. ALURU.2.4 log (error in capacitance) log (h) Figure 26. Convergence plot for the bus crossing problem. The convergence rate of is γ = Fast BCM O(NlogN) 2 Sove time (secs) Number of nodes Figure 27. Scaling of the CPU time with the number of nodes for the bus crossing problem. The solid line shows the CPU time used in GMRES with the compressed form of the matrix and it scales same as the dashed line plot, i.e. O(N log N). the charge distribution on the surface of the 4 conductors. Figure 26 shows the convergence for evaluating the capacitance of the top conductor. In Figure 27, we have shown the scaling of the CPU time using the SVD-based acceleration scheme. We observe that it scales as O(N log N). Figure 28 shows a comparison of the storage space used by the compressed form of the matrix and the storage space used by the dense form of the matrix. As the discretization becomes finer and finer, i.e. the number of nodes on the surface of the conductor becomes large, we Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59:29 246

27 FAST BOUNDARY CLOUD METHOD Storage used (bytes) Number of nodes Figure 28. Scaling of storage with the number of nodes for the bus crossing problem. The solid line shows the storage space occupied by the compressed matrix, while the dashed line shows the storage required with the dense form. see that a large memory requirement for the storage of the dense form limits the size of the problem that can be solved using the dense form. But it can be solved using the compressed form, as it requires less storage space. 5. CONCLUSIONS A fast boundary cloud method (BCM) for numerical solution of exterior thee-dimensional electrostatic problems has been presented in this paper. It combines scattered points and boundary integral equations. BCM uses a varying polynomial basis to construct the meshless approximation functions. Depending on whether the cloud is singular or non-singular, the base interpolating polynomial is defined to satisfy the linear completeness condition. The solution process has been accelerated by using the SVD-based technique to compress the matrix and to reduce the CPU time of the matrix vector product in an iterative solver (GMRES). Numerical results have been presented for several 3D electrostatic problems, comparing the classical boundary element method using bilinear shape functions and the boundary cloud method using linear polynomial basis. The boundary cloud method exihibits an advantage of providing more accurate solutions as compared to BEM. The convergence behaviour of both the methods is similar. For the same accuracy, BCM needs fewer nodes and cells compared to BEM and BCM is found to be faster than BEM. For the same number of nodes and cells/elements in BCM/BEM, BCM is found to be a factor of 2 slower but BCM gives a smaller error compared to BEM. We have also observed that the CPU time using the acceleration technique scales as O(N log N) for 3D electrostatic problems, where N is the number of nodes on the surface of the conductor. Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59:29 246

28 246 V. SHRIVASTAVA AND N. R. ALURU ACKNOWLEDGEMENTS This research was supported by the National Science Foundation under grants NSF , NSF- 7623, NSF-266 and NSF The authors thank Gang Li and Xiaozhong Jin for helpful discussions. REFERENCES. Aluru NR, White J. An efficient numerical technique for electromechanical simulation of complicated micro-electro-mechanical structures. Sensors and Actuators A 997; 58:. 2. Aluru NR, White J. A multilevel Newton method for mixed-energy domain simulation of MEMS. Journal of Microelectromechanical Systems 999; 8(3): Li G, Aluru NR. Linear, nonlinear and mixed-regime analysis of electrostatic MEMS. Sensors and Actuators A 2; 9: Senturia SD, Aluru NR, White J. Simulating the behavior of MEMS devices: computational methods and needs. IEEE Computational Science and Engineering 997; 4(): Kane JH. Boundary Element Analysis in Engineering Continuum Mechanics. Prentice-Hall: Englewood Cliffs, NJ, Greengard L, Rokhlin V. A fast algorithm for particle simulations. Journal of Computational Physics 987; 73(2): Phillips JR, White J. A precorrected-fft method for electrostatic analysis of complicated 3-D structures. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 997; 6(): Kapur S, Long DE. IES 3 : a fast integral equation solver for efficient 3-dimensional extraction. IEEE Computer Aided Design, 997, Digest of Technical Papers 997, IEEE/ACM International Conference 997; Kapur S, Zhao J. A fast method of moments solver for efficient parameter extraction of MCMs. Design Automation Conference, 997, Proceedings of the 34th Conference 997; Shrivastava V, Aluru NR. A fast boundary cloud method for exterior 2-D electrostatics. International Journal for Numerical Methods in Engineering 23; 56(2): Barett B, Berry M, Chan TF, Demmel J, Donato J, Dongarra J, Eijkhout V, Pozo R, Romine C, Vorst H. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM Publications: Philadelphia, PA, Saad Y, Schultz MH. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing 986; 7(3): Trefethen LN, Bau D. Numerical Linear Algebra. SIAM Publications 997; 4(5). 4. Belytschko T, Krongauz Y, Organ D, Flemming M, Krysl P. Meshless methods: an overview and recent developments. Computer Methods in Applied Mechanics and Engineering 996; 39: Chati MK, Mukherjee S. The boundary node method for three-dimensional problems in potential theory. International Journal for Numerical Methods in Engineering 2; 47: Mukherjee YX, Mukherjee S. The boundary node method for potential problems. International Journal for Numerical Methods in Engineering 997; 4(5): Lancaster P, Salkauskas K. Surface generated by moving least-squares methods. Mathematics of Computation 98; 37: Li G, Aluru NR. Boundary cloud method: a combined scattered point/boundary integral approach for boundary-only analysis. Computer Methods in Applied Mechanics and Engineering 22; 9: Li G, Aluru NR. A boundary cloud method with a cloud-by-cloud polynomial basis. Engineering Analysis with Boundary Elements 23; 27(): Shi F, Ramesh P, Mukherjee S. On the application of 2D potential theory to electrostatic simulation. Communications in Numerical Methods in Engineering 995; : Hall WS. The Boundary Element Method. Kluwer Academic Publishers: Dordrecht, Mukherjee S. Boundary Element Methods in Creep and Fracture. Applied Science Publishers: London, Belytschko T, Lu YY, Gu L. Element free Galerkin methods. International Journal for Numerical Methods in Engineering 994; 37: American Institute of Physics. Handbook (3rd edn). McGraw-Hill: New York, Nabors K, Kim S, White J, Senturia S. FastCap USER s GUIDE. MIT: Cambridge, MA, Rotkin SV, Bulashevich KA, Aluru NR. Atomistic models for nanotube device electrostatics. In Proceedings Electrochemical Society, Kamat PV, Guldi DM, Kadish KM (eds). ECS Inc: Salt Lake City, UT, Copyright 24 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 24; 59:29 246