2D capacitance extraction with direct boundary methods

Engineering Analysis with Boundary Elements 58 (2015) 195–201

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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound

2D capacitance extraction with direct boundary methods M. Borkowski n Faculty of Electrical and Computer Engineering, Rzeszow University of Technology, Rzeszów, Poland

art ic l e i nf o

a b s t r a c t

Article history: Received 10 October 2014 Received in revised form 12 March 2015 Accepted 24 April 2015 Available online 28 May 2015

The paper presents the algorithm of hierarchical capacitance extraction based on direct boundary methods. Three selected methods, i.e. Boundary Element Method, direct Trefftz method (based on THcomplete functions) and regular direct Boundary Element Method (direct Trefftz–Kupradze method), are compared for their effectiveness. The algorithm employs binary tree decomposition of the problem domain. Coupling capacitance matrix is calculated in hierarchical process with simultaneous dynamical updating library with basic element matrices. Numerical examples presented in the paper concern 2D planar transmission line structures composed of isotropic dielectric layers. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Direct boundary methods 2D capacitance extraction BEM Direct Trefftz method Direct regular BEM

1. Introduction Parasitic extraction is a problem that has been studied for over forty years [13,26]. When microelectronic device operates at high frequencies non-zero parasitic element values cause a number of problems such as interconnect delays, cross-talks, signal integrity or power consumption. With advancing minimization of process technology extraction of parasitic parameters is becoming more and more important. The biggest technical problem of parasitic extraction concerns parasitic capacitances. It arises due to non-homogeneous dielectric media surrounding conducting parts of the device. Since analytical ways of solving the problem are limited to very simple geometries, nowadays mainly numerical methods are used. State of the art and a brief review can be found in [19,31]. Applications of boundary integral method employ popular Boundary Element Method (BEM) both in direct (dBEM) and indirect (iBEM) formulations. The first paper, to the author's knowledge, that applied dBEM calculations for parasitic capacitance extraction was [10]. The following years brought application of the fast multipole method [1], hierarchical algorithms [7,11,22,27] and the so-called QuasiMultiple Medium method [30,32]. Unlike BEM that employs the singular fundamental solutions, Trefftz methods [4,15,20] utilize regular functions satisfying the governing equations. The advantages of this approach are simpler

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http://dx.doi.org/10.1016/j.enganabound.2015.04.017 0955-7997/& 2015 Elsevier Ltd. All rights reserved.

implementation, no difficulties with singularities and consequently smaller quadrature error. Unfortunately, instability and poor conditioning arise due to the fact that boundary integral equations in Trefftz methods are Fredholm equations of the first kind. Despite that Trefftz methods were successfully applied in wide range of problems and the possible applications can be found in [20,21]. Generally speaking, regularity of the functions satisfying governing equations can be obtained in two different ways. The first one is the application of regular, the so-called TH-functions [14] and results in Trefftz–Herrera method (THM). The second one consists in placing source points outside the domain of interest. This approach is known under the name Trefftz–Kupradze method (TKM), regular BEM, method of fundamental solutions and many others [9]. Both these methods can also be classified into direct and indirect formulations. Direct counterpart of THM is known as direct Trefftz method (dTM) [17,21], while direct formulation of TKM is often called regular direct BEM (rdBEM) [29]. Another method based on Trefftz idea is referred in the literature as Trefftz Finite Element Method (TFEM), or Hybrid Trefftz Method [18,25]. Generally it can be regarded as the combination of discretization of the domain of interest as in Finite Element Method (FEM) with Trefftz type approximation of the solution inside each element. Although this approximation is usually conducted with the use of TH-complete functions, in recently published works fundamental solutions are also employed [6]. In this way the governing equation is satisfied inside each element but not on the element boundary. Interelement continuity has to be achieved with additional approximation

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of the physical quantities on the element boundary. It is enforced with the help of modified variational principle with the so-called frame field defined independently for each element on its boundary. Literature review undertaken by author reveals that though some commercial software that deals with capacitance extraction (ANSYS, Simbeor) employs Trefftz idea, it is only applied in the latter mentioned formulation [12,28]. Taking this fact into consideration, the aim of this paper is to show this possibility. Due to that need algorithm suitable for any direct boundary method is presented. dBEM, dTM and rdBEM are used as PDE solution engines. The efficiency of these methods was verified in our previous work for simple Laplace problems [4]. In this paper, direct boundary methods are applied to capacitance extraction and in this way Trefftz methods can be compared with dBEM, which was already used for that purpose. The paper is organized as follows: Section 2 contains general formulation of the problem; next, the theory of direct boundary methods derived from inverse variational formulation is given in Section 3; Section 4 presents the details of the extraction algorithm; the two last sections contain some numerical examples (compared with results given by Linpar software [8]) and conclusions.

2. Formulation of the problem Let one consider system of N n 4 2 conductors. The charge Qm induced on m-th conductor is given by Qm ¼

Nn X

C mn un ;

m ¼ 1; 2; …; N n ;

u¼u q¼

ð2Þ

where C ¼ ½C mn Nn Nn , u ¼ ½u1 ; …; uNn T , Q ¼ ½Q 1 ; …; Q Nn T . C matrix is square and symmetric and is called the generalized capacitance matrix (GCM) [16]. For simplicity, in this paper planar transmission lines are considered (Fig. 1). Under some assumptions, calculation of coupling capacitances of such systems can be regarded as 2D Laplace problem. Due to the fact that each dielectric layer can have different relative permittivity, the compatibility conditions on each dielectric interface need to be satisfied. Thus, each k-th dielectric layer can be described by  2  ∂ u ∂2 u εk ∇2 u ¼ εk þ 2 ¼ 0 in Ω ð3Þ 2 ∂x ∂y

∂u ¼q¼0 ∂n

on Γ q

εa qa þ εb qb ¼ 0;

ua ¼ ub on Γ i ;

ð5Þ ð6Þ

3. Direct boundary methods Classical Laplace problem can be transformed into variational formulations. One of them i.e. inverse variational formulation Z Z Z ∂w dΓ þ qw dΓ ¼ 0 u∇2 w dΩ  u ð7Þ Ω Γ ∂n Γ where w , weighting function, is a starting point for direct boundary methods. Partially unknown functions u and q have physical interpretation. They are potential and flux on the boundary and they can be approximated at the boundary using any suitable interpolation polynomials (e.g. Lagrange) u¼

Nν X

ν¼1

n¼1

Q ¼ Cu

ð4Þ

where εk is the relative permittivity of the k-th layer, ∇2 the Laplacian operator, u and q the potential and its derivative in the normal direction, respectively, n the unit outward normal vector to Γ . Γ u the boundary with Dirichlet condition prescribed (surfaces of conductors), Γ q the boundary with homogeneous Neumann condition prescribed (external surfaces of dielectrics) – “magnetic wall” [2,24], Γ i the interface between two dielectric layers.

ð1Þ

where un is the voltage between n-th conductor and the ground of the system. Element C mn (m a n) is the coupling capacitance between conductors m and n, C mm is the self-capacitance (total capacitance) of the m-th conductor. Eq. (1) can be rewritten for Nn conductors in matrix form:

on Γ u

q¼

Nν X

ν¼1

uν N ν ðξ Þ

ð8Þ

qν N ν ðξÞ

ð9Þ

where variable ξ A ½  1; 1, N ν is the ν-th interpolation polynomial, Nν the number of function approximation nodes on Γ . Proper choice of weighting function w transforms Eq. (7) into boundary integral equation, which next has to be fulfilled in the set of Nμ collocation points. Taking into account Eqs. (8) and (9), Eq. (7) can be rewritten as Nν X



ν¼1

Z uν

Γ

Nν

Z Nν X ∂w μ dΓ þ qν N ν w μ dΓ ¼ 0 ∂n Γ ν¼1

ð10Þ

where w μ is the μ-th weighting function, μ ¼ 1; 2; …; N μ . Some of the values uν , qν are known due to boundary conditions. Usually they are moved to the RHS and the unknowns are found by solving the system of linear equations. 3.1. Direct Boundary Element Method (dBEM) Standard BEM employs the singular fundamental solution Gn as the weighting function w . For 2D Laplace problem, Gn is given by Gn ðx; xμ Þ ¼ 

1 lnðjx  xμ jÞ 2π

ð11Þ

and it satisfies non-homogeneous equation ∇2 Gn ðx; xμ Þ ¼  δðx  xμ Þ

ð12Þ

where x is the source point, xμ the collocation point, δ the Dirac delta function. The weighted integral over Ω in Eq. (7) can be simplified to general formula Z u∇2 Gn ðx; xμ Þ dΩðxÞ ¼  cðxμ Þuðxμ Þ ¼  cμ uμ ð13Þ Fig. 1. Non-homogeneous multi-layer 2D planar transmission line as an example of system of Nn conductors.

Ω

where coefficient cμ depends on the position of xμ . For 2D

M. Borkowski / Engineering Analysis with Boundary Elements 58 (2015) 195–201

problems 8 1 3 xμ A Ω > > > < β 3 xμ A Γ cμ ¼ 2π > > > : 03x 2 μ=Ω ¼ Ω [ Γ

ð14Þ

ð15Þ which is a discrete boundary integral equation. 3.2. Regular direct Boundary Element Method (rdBEM) From Eq. (14) one can see that cμ equals zero for points lying outside the domain Ω. Applying Eqs. (11), (13) and (14) to Eq. (7), the following boundary integral equation can be obtained as Z Z Nν Nν X X ∂Gn ðx; x^ μ Þ uν N ν qν N ν Gn ðx; x^ μ Þ dΓ ¼ 0 ð16Þ  dΓ þ ∂n Γ Γ ν¼1 ν¼1 rdBEM [23] can be viewed as a direct counterpart of iTKM, and similar to it, needs to have determined the way of distribution of the collocation points and their distances from the real boundary. Here, collocation points are translated in the normal direction from the boundary nodes by distance λ. 3.3. Direct Trefftz method (dTHM) Another way of making Eq. (7) regular boundary integral equation is to employ non-singular T-complete functions as weights w [17,21]. T-functions for 2D interior Laplace problem take the form un A

f1

when j ¼ 0;

ρj cos jθ; ρj sin jθ; j ¼ 1; 2; 3; …g

ð17Þ

where ρ ¼ ρðx; x0 Þ and θ ¼ θðx; x0 Þ are the plane polar coordinates, and x0 is the Trefftz origin. The normal derivative of T-function qn ¼ ∂un =∂n. Thus, for the sake of notation uniformity with dBEM, non-singular solutions can be expressed in terms of x and x0 , i.e. un ¼ un ðx; Þ, qn ¼ qn ðx; x0 Þ. Since each unμ T-function fulfils Eq. (3), boundary method procedure transforms Eq. (7) into Z Z Nν Nν X X uν N ν qnμ ðx; x0 Þ dΓ þ qν N ν unμ ðx; x0 Þ dΓ ¼ 0 ð18Þ  ν¼1

Γ

ν¼1

Γ

which is the final equation for direct Trefftz method. 3.4. Matrix form of direct boundary methods and boundary capacitance matrix Eqs. (15), (18), (16) constitute linear systems of equations which may be written in matrix form as Hu ¼ Gq

system respectively on the RHS and LHS, and solving for unknown values " # " # q1 u1 ½H1 H2  ¼ ½G 1 G 2  ð20Þ q2 u2 "

where domain Ω is locally seen from point xμ at an angle β. Substituting Eq. (13) into Eq. (7), via Eq. (10) one can get Z Z Nν Nν X X ∂Gn ðx; xμ Þ dΓ þ cμ uμ  uν N ν qν N ν Gn ðx; xμ Þ dΓ ¼ 0 ∂n Γ Γ ν¼1 ν¼1

ð19Þ R where each element of the matrices H μν ¼ Γ N ν ð∂w μ =∂nÞ dΓ , R Gμν ¼ Γ N ν w μ dΓ , and vectors u ¼ ½u1 u2 …uNν T , q ¼ ½q1 q2 …qNν T ; w μ depends on the used method. It should be also noted that for each element H μν , Gμν the respective integral on the boundary is limited only to the element on which the adequate ν-th interpolation polynomial is defined. As it was already written above, typically boundary method procedure comes down to collecting knowns and unknowns of the

197

½  G 1 H2 

q1 u2

#

" ¼ ½ H1 G2 

u1

#

q2

ð21Þ

briefly Ax ¼ b

ð22Þ

After obtaining unknown values uν and qν on the boundary, one can treat them as densities of layer potentials and with the use of Eq. (15) express u inside Ω as an influence of the single and double layer potentials: Z Z ∂Gn ðx; x0 Þ dΓ þ qðxÞGn ðx; x0 Þ dΓ uðx0 Þ ¼  uðxÞ ð23Þ ∂n Γ Γ When the boundary conditions are limited only to pure Dirichlet ones (i.e. Γ q ¼ ∅) unknown vector q is given by q ¼ Cu

ð24Þ

where C ¼ G  1H

ð25Þ

Obtained in such way matrix C can be transformed into stiffness matrix by multiplying C with matrix that relates boundary quantities. Thus BEM and FEM can be easily combined [5]. Eq. (24) expresses electric flux in normal direction (i.e. charge density in physical interpretation) in terms of potential in separate nodes on the boundary. In some particular, simple cases C can be equivalent to GCM. In general, let C be called a boundary capacitance matrix (BCM). Obtaining final GCM of the system requires additional transformations presented in next sections.

4. Capacitance extraction algorithm When dealing with real-world systems of conducting paths (Fig. 1) parasitic capacitance extraction is difficult due to surrounding nonhomogeneous dielectric media. One of the ways to get around this problem is to decompose whole domain into smaller, different material regions, and then coupling them with continuity conditions on interface. Taking into consideration geometry of the studied structures one can see that this way of decomposition can be not enough because it causes that domains with big aspect ratio may appear. It is particularly unfavourable when BEM is governing equation solver. In order to prevent that author decided on further decomposition of the layer (within each dielectric layer). 4.1. Hierarchical binary decomposition Hierarchical binary tree decomposition is chosen as a mesh generation method. The advantage of such kind of discretization is that it allows to increase mesh density where needed (around conductors) and keep small diversity of subdomain geometry types. For balanced and dense enough binary tree decomposition (Fig. 2) geometries of leaf-subdomains forming dielectric layer composed of limited set of elements (Fig. 3). They can be divided into two classes: purely dielectric domains (Fig. 3a) and domains containing conductors (Fig. 3b). This observation allows creating library of BCMs for those basic elements. What is more, it can be observed that mainly 5 types of leaf-subdomain elements (numbered 1, 4, 0, 2, 8 in Fig. 3a) form the

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M. Borkowski / Engineering Analysis with Boundary Elements 58 (2015) 195–201

under symbolic submatrix indices i.e. c denotes nodes on the conductors, d the nodes on external boundary with homogeneous Neumann condition, i the nodes on interface between subdomains. Each pair of indices denotes coupling between respective parts of the boundary, e.g. cd symbolizes coupling between nodes on external dielectric boundary and nodes on conductors. To obtain BCM of parent domain composed of two children subdomains Ωa and Ωb one has to enforce field continuity conditions (Eq. (6)). Merging and simple reorganization of BCMs of two subdomains give common system of equations: 2 32 3 2 3 C cc C cd C ci uc q 6 76 7 6 c 7 C di 6 C dc C dd 74 ud 5 ¼ 4 qd 5 ð27Þ 4 5 ðbÞ C ic C id εa C ðaÞ þ ε C u 0 i b ii ii

Fig. 2. Domain decomposition of a single dielectric layer (a) and a corresponding binary tree (b).

Parasitic extraction does not require any knowledge of charge density at each interpolation node. Only total amount of charge induced on each conductor is needed. This fact can be utilized for improving the speed and memory requirements of the algorithm. Each time two BCMs are merged into one, common matrix of two adjacent subdomains, the rows and the columns of this matrix that correspond to the nodes lying on each specific conductor are condensed into one row and one column. Thus, at the expense of information about flux distribution it is possible to condense matrix from Eq. (27) by means of integrating flux q on each conductor. Nodes that lie on the interface can be removed by Schur's complement [33]. Let one assume that " # " # Ccc Ccd Cci   Z1 ¼ ; Z4 ¼ Cic Cid ; Z2 ¼ ; Z3 ¼ εa CiiðaÞ þ εa CðbÞ ii Cdc Cdd Cdi ð28Þ Then BCM for new parent-domain that merges two subdomains is obtained as C ¼ Z1  Z2 Z3 1 Z4

ð29Þ

Such BCM could also be stored in library and reused in subsequent calculations if there are more subdomains with the same geometry. Fig. 3. Leaf-domains of the binary tree decomposition of the problem domain.

mesh, which allows reducing library only to these elements. BCMs of the rest of domain types must be calculated every single time they are needed. Described method of domain discretization does not need to parametrize library elements, as it is in [7]. What is more, provided that the same degree of interpolation functions for all boundary elements is assumed, interpolation nodes lying on interfaces between adjacent subdomains coincide with each other. Thus, continuity conditions on domain interfaces can be imposed, so to speak, naturally, and there is no need to inter- or extrapolate function value on the boundary.

32

uðkÞ 76 cðkÞ 76 u 74 d 5 uðkÞ i

3 7 7¼ε 5

Obtained matrix contains information about conductors and the external boundary elements. Enforcing boundary condition (5) and applying Schur's complement once again give ð31Þ

and finally one can get

for individual leaf-subdomain Eq. (25) can be C ðkÞ ci C ðkÞ di C ðkÞ ii

The above-described approach allows for hierarchical synthesis of subdomains contained in each layer. Then, in analogical way each dielectric layer can be merged into a global matrix. Elimination of all internal nodes from the dielectric interfaces (and enforcing boundary condition (5)) simplifies Eq. (26): " #" #   Ccc Ccd uc qc ð30Þ ¼ Cdc Cdd ud 0

1 Cdc CG ¼ Ccc  Ccd Cdd

4.2. Hierarchical extraction Generally, expressed as 2 C ðkÞ C ðkÞ cd 6 cc 6 C ðkÞ C ðkÞ εk 6 dc dd 4 C ðkÞ C ðkÞ ic id

4.3. BCM to GCM transformation

2

qðkÞ c 6 ðkÞ 6 q k4 d qðkÞ i

3 7 7 5

ð26Þ

where nodes placed on each type of boundary are grouped together

CG uc ¼ Q c

ð32Þ

Last formula is equivalent to Eq. (2), and matrix CG is GCM of the system of conductors. It should be noted here that dBEM which employs fundamental solution lets one interpret elements of BCM as real capacitance values between individual boundary elements. This is not allowed when regular solutions are taken as weighting functions. In this case, only elements of matrix calculated for domain corresponding

M. Borkowski / Engineering Analysis with Boundary Elements 58 (2015) 195–201

to root of the binary tree have physical meaning. Elements of BCMs of sub-domains cannot be called capacitance in the literal sense.

4.4. Global matrix solution Hierarchical process of obtaining BCMs allows condensing information about internal nodes and consequently reduces memory requirements. Disadvantage of such approach has increased time spent on multiple computations of formula (29). Another way to solve the problem is to merge all BCMs into one global matrix in order to perform integration of the flux on conductor elements and to apply Schur's complement only once in the process of calculations. This approach yields big, sparse matrices that have to be solved. This can be done by any appropriate method, particularly some iterative methods like GMRES. For the comparative purposes this procedure is also implemented and the results are presented in the next section.

199

5.1. Example 1 The first example presents convergence of the algorithm on a simple transmission line system i.e. three coupled stripline where the first path on left is grounded (Fig. 4). The reference capacitance matrix is equal to   107:5  53:7 C ex ¼ pF=m:  53:7 69:7 The results are depicted in Table 1. Each row of Table 1 contains total number of nodes on the conductors – Nw (total number of interpolation nodes – Nν ), capacitance matrix – CG , root mean square error (RMSE) between the reference and computed capacitance matrix – E, computation time in seconds – t. All methods present similar convergence, yet dBEM gives the smallest RMSE. What is more, the computation time in dBEM calculations is smaller than in regular boundary methods. 5.2. Example 2

5. Numerical examples Presented examples are limited to 2D planar structures. It is assumed that quasi-TEM wave propagates along infinite conductive paths and the thickness of each path is negligible. Infinite paths length assumption implies that parasitic capacitance values are determined per unit length. Although the whole problem domain is non-homogeneous, each dielectric layer is linear, homogeneous and isotropic. For the examples with given exact solution (simple parallel capacitor, microstripline) the number of experiments for different degrees of elements is conducted in [3] to choose a proper value of arbitrary parameters of each method i.e.:

 In dBEM interpolation nodes are shifted by nondimensional

Consider more complex system of conductors with ground plane placed at the bottom of the structure (Fig. 5). For the sake of compactness and clarity only separate interconnect capacitance values are analysed. Reference results are C 11 ¼ 228 pF=m, C 22 ¼ 341:8 pF=m, C 23 ¼  13:12 pF=m, C 15 ¼  28:82 pF=m. Fig. 6 presents numerical convergence for the selected capacitance matrix elements in terms of the total number of interpolation nodes of the model Nν. Reference solution is denoted by black densely dotted straight line, and dBEM, dTHM and rdBEM solutions are denoted respectively by solid red line, dashed green line, and dotted blue line. In this example Trefftz methods give slightly better results than dBEM. As can be easily checked, the relative percentage “errors” between obtained and the reference solutions in both examples

parameter 0:2  element_length from the singularities.

 The centre of each subdomain is taken as a centre of eigen 

expansion in dTHM. In rdBEM collocation nodes are shifted outside the boundary by non-dimensional parameter 1:5  subdomain_diagonal. The height of top/bottom dielectric layer (simulation of air surrounding transmission line) is equal to the width of the widest path of the top/bottom plane.

Since for more complex geometries it is very difficult or even impossible to obtain the analytical solutions, results obtained with Linpar software [8] are taken as a reference solutions. In below-mentioned examples the calculations are conducted with quadratic boundary elements. The relative permittivities and geometry details are presented in the figures. The heights of the dielectric layers, widths of the paths and distances between them are given in mm. Conductors are numbered by the consecutive integers from left to right and from bottom to top layer. The computations are performed on standard PC (Intel Core i5 2.3 GHz, 4 GB RAM) in Matlab 7.12.

Table 1 Convergence of the algorithm for different direct boundary methods. Nw (Nν)

CG ðpF=mÞ

dBEM 18 (225)



30 (415)



58 (691)



rdBEM 18 (225)

30 (415)

58 (691)

 

112:37  56:18

 56:18 66:07

108:73  55:36

 55:36 64:52

107:54

 53:77

 53:77

64:03

98:78

 49:36

 49:36

59:77

103:57

 51:78

 51:78

62:25

105:22  52:61

 52:61 63:14

90:00  46:27

 40:68 52:42

t (s)



3.50

0.411



2.90

0.697



2.83

0.628



7.30

0.641



4.42

0.373



3.56

0.568



14.39

0.962



5.24

0.763



4.12

1.010

dTHM 18(225)



30 (415)



58 (691) Fig. 4. Three coupled stripline from example 1.



E



102:21

 51:10

 51:10

61:41

104:30

 52:15

 52:15

62:41

200

M. Borkowski / Engineering Analysis with Boundary Elements 58 (2015) 195–201

Fig. 5. Transmission line structure from example 2.

It turns out that for the problems with small number of interpolation nodes the standard way of solving the system of linear equations is faster than hierarchical one. This situation changes when the problem geometry becomes more complex and the number of interpolation nodes increases. The greater the number of interpolation nodes, the faster the hierarchical algorithm is (comparing it to the standard one). For meshes with about 5000 nodes hierarchical algorithm is two times faster regardless of the employed boundary method. Similarly as in example 1 solving system of equations generated by dTHM is much slower than methods employing fundamental solution.

6. Concluding remarks The paper discusses the application of the direct boundary methods in capacitance extraction problem. To this end, the theory of direct boundary methods is given briefly and the general algorithm employing hierarchical binary decomposition of the problem domain is presented. Then, chosen methods i.e. direct BEM, direct Trefftz method and regular direct BEM are applied to compare their efficiency. The numerical results show that

 The accuracy of three methods is similar – none of the method is evidently better or worse than the others.

 Methods employing fundamental solution are approximately



two times faster than Trefftz method which uses non-singular solution. This should be attributed to the fact that matrices obtained in direct Trefftz method are usually poor conditioned, which results in longer solution process. For dense meshes the hierarchical algorithm has the advantage over the global matrix solution.

Since, there has been no application of direct Trefftz methods in capacitance extraction problem until now, thus the article shows that they can be successfully utilized. The work may be continued in several directions.

 Improvements in speed and efficiency of the algorithm should be made.

 Presented idea can be employed for more complicated geomeFig. 6. Convergence of the algorithm for selected capacitance matrix elements in example 2: reference solution ( ), dBEM ( ), dTM ( ), rdBEM ( )

tries in 2D and 3D.

 New ways of improving calculations with ill-conditioned Trefftz matrices should be explored.

are mostly lesser than 5%. They can be attributed to the differences in convergence of presented algorithm and the method of moments, which is implemented in Linpar software and the ways that both approaches deal with external boundary conditions. 5.3. Example 3 In the third example using the system presented in Fig. 5 comparison of time usage of hierarchical and direct solution procedure is made. Table 2 shows the results. Each row of the table corresponds to the calculations with the same total number of interpolation nodes Nν, which is given in the first column. For each method three consecutive columns contain RMSE of the capacitance matrix – E, and respectively time usage of direct (td) and hierarchical algorithms (th). For the same binary tree decompositions of the problem geometry calculations are conducted with both algorithms.

Acknowledgements Numerical experiments were conducted with the use of MATLAB software, purchased during realization of Project no. UDA-RPPK.01.03.00-18-003/10-00 “Construction, expansion and modernization of the scientific-research base at Rzeszów Table 2 Time comparison of hierarchical vs direct calculation of GCM. Nν

1263 2700 5211

dBEM

rdBEM

dTHM

E

td (s)

th (s)

E

td (s)

th (s)

E

td (s)

th (s)

11.29 9.44 8.22

0.724 1.537 4.401

0.994 1.573 1.773

11.75 9.40 7.98

0.603 1.377 3.906

0.960 1.267 1.955

11.27 10.94 9.02

1.4824 3.1487 8.1472

2.202 2.421 3.650

M. Borkowski / Engineering Analysis with Boundary Elements 58 (2015) 195–201

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