A Generalized Minimal Residual Acceleration of the Charge Iteration Procedure

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A Generalized Minimal Residual Acceleration of the Charge Iteration Procedure

G. Aiello, S. Alfonzetti, G. Borzì

To cite this version:

G. Aiello, S. Alfonzetti, G. Borzì. A Generalized Minimal Residual Acceleration of the Charge Iteration Procedure. Journal de Physique III, EDP Sciences, 1997, 7 (10), pp.1955-1966. �10.1051/jp3:1997234�.

�jpa-00249693�

A Generalized Minimal Residual Acceleration of the Charge

Iteration Procedure (*)

G. Aiello, S. Alfonzetti (**) ^{and G. Borzi}

Dipartimento Elettrico, Elettronico

Sistemistico, Universith di Catania, Viale A. Doria 6, 95125 Catania, Italia

(Received 20 March 1997, revised 19 June 1997, accepted 7 July 1997)

PACS 02.70.Dh Finite-element and Galerkin methods

PACS.41.20.Cv Electrostatics; Poisson and Laplace equations, boundary-value problems

Abstract. Charge iteration is

procedure for the computation of unbounded electrostatic fields induced by voltaged conductors It is based

the iterative improvement of

Dirichlet

condition

fictitious boundary enclosing all the conductors. In this paper the improvement

is shown to be equivalent to the

of Richardson's method

reduced system. A conspic-

uous

reduction in computational and memory requirements is achieved by

of GMRES

acceleration

this reduced system

Rdsumd. L'it6ration de charge est

proc4dure _pour le calcul de champs dlectrostatiques produits _par des conducteurs

tension Elle est fondde

l'amdhoration itdrative de la

condition de Dirichlet

hmite fictive qui renferme _tous les conducteurs Darts

papier

il est ddmontrd que l'amdlioration _est dquivalente h l'usage de la m#thode de Richardson _pour

un

systAme r4duit. Une grande r4duction de la demande de temps ^{de calcul} et de m6moire est obtenue

moyen de l'acc61dration GMRES

systAme rdduit

1. Introduction

A wide variety ^of techniques have been developed to give ^the Finite Element Method (FEM)

the capability of dealing with electromagnetic problems in unbounded domains. For the _case of static and quasi-static problems the major techniqiies _are ballooning ill, asymptotic boundary

conditions [2], ^{infinite elements} [3j, hybrid FEM/BEM _[4j and co-ordinate transformations [5,6].

In previous works the authors devised

FEM procedure, named charge iteration, to deal with unbounded electrical field problems created by voltaged conductors ii, 8j. The _proce- dure makes _use of

fictitious boundary enclosing all the conductors and the possibly _non- homogeneous dielectrics. On this boundary _a Dirichlet condition is initially guessed by and the resulting bounded problem is solved by the FEM method. The field calculated is used to

improve the Dirichlet condition _on the fictitious boundary and the procedure is iterated. If the fictitious boundary is placed at _an appropriate distance, convergence takes place whatever

(*) This _paper

presented at NUMELEC'97

(** ^{Author for} correspondence (e-mail. alfotldees.unict.it)

@ Les Editions de Physique 1997

the first guess for the Dirichlet condition _on it iii. A remarkable improvement ^{is obtained} by introducing overrelaxing in the iterative procedure _[9]. It dramatically reduces the number of iterations and minimizes the distance between the fictitious boundary and the conductors, reducing the size of the FEM problem. The critical point of overrelaxed charge iteration lies in the choice of the optimal relaxation parameter. Although _some heuristic rules have been

given it is not completely satisfactory.

A generalization of the procedure _was introduced by the authors in [10], employing _{a more} general non-homogeneous Robin boundary condition _on the fictitious boundary, resembling _an

asymptotic _one. By this approach the number of iterations _are reduced, but the increased _num- ber of unknowns in the FEM system counter-balances this reduction: the minimum computing

time required for the solution of the system ^{is almost the} same as that needed by overrelaxed

charge iteration with _an optimal relaxation parameter. Generalized charge iteration is less sensitive to the choice of the coefficient of the Robin condition, but _on the other hand it _seems to be less accurate and requires _more computing time to set up the system.

In this _paper the basic charge iteration procedure is modified with the introduction of _a

more robust approach based _on the Generalized Minimal Residual (GMRES) algorithm _[11].

The structure of the paper is _as follows. In Section 2 the charge iteration procedure is briefly

recalled. In Section 3 _a brief _survey of conjugate gradient-like solvers is provided. In Section 4 the utilization of GMRES to accelerate charge iteration is illustrated. In Section 5 _some

numerical examples _are provided. The authors' conclusions follow in Section 6.

2. The Charge Iteration Procedure

Let _us consider _a system of NC conductors with given potentials Vk, k

=

1,

,

NC, embedded in _an unbounded dielectric medium with possibly localized non-homogeneities. ^The electric

potential _u satisfies the Laplace equation:

V 6VUlr)

° li)

where _c is the electrical permittivity and _r is the generic point vector, ^with the appropriate Dirichlet boundary conditions _{on the conductor} surfaces Bk, k

=

1,..., Nci

u(r)

Vk r E Bk (2)

whereas it vanishes at infinity.

In order to reduce the unbounded problem to a bounded _one, the unbounded domain is

truncated with _a fictitious boundary BF enclosing all the conductors and the dielectric _non-

homogeneities (see Fig. 1). The problem _now is the imposition of _{a proper} boundary condition

on BF. This _can be achieved by _means of the second Green's formula:

U(~F)

/~ ^G(rF> r)~j)~ ~(~)~~jl'~~ld~ ^{~F E ~F, r E ~M (3)}

M

where EM is _a surface between the system and BF, _n is the normal to this surface pointing

outward and G is the free-space Green's function. Note that since EM and BF do not intersect the integrand in (3) is not singular. Applying the FE discretization to ii and (3) the following

algebraic equations _are obtained iii

AV

=

Bo AFVF (4)

VF

VFO + HV (5)

Fig. 1. Conductor system and fictitious boundary.

where A is the global Dirichlet matrix (symmetric and positive definite), V and VF _are the vectors of the unknown potential values at the nodes inside the domain and

BF, respectively, Bo is the known-term _vector due to the conductor potentials, whereas -AFVF is due to the

boundary conditions _on BF. In (5) VF is given by two terms: the first _one, VFD, is due to the conductor potentials, whereas the second _one, HV, is due to the unknown nodal potential

values of the elements whose sides lie _on EM

The system formed by (4) and IS) is suitable for iterative solution. Starting from _a first guess Vf~ for VF, equation (4) is solved for V(°); then equation IS) is used to improve the guess for VF. At the generic step _p:

AV(P)

=

Bo AFV)~~ (6)

V)~+~~

VFO + HV(P). ₍₇₎

In order to study this iterative procedure, _we formally solve (6) for V(P) and substitute in (7)

to get:

v(P+1) _{~ ~} p~(p) _j~)

where:

vo

v~o ₊ HA-IBO j9)

P

=

-HA~~AF. (lo)

Then, if the spectral radius of matrix P is lower than one, the procedure _converges to the solution of the unbounded problem whatever the first _guess Vf~. By experimental _[7] and theoretical _[8] studies it has been shown that this condition is satisfied if the _mean distance between the conductors and the fictitious boundary is greater than the _mean radius of the conductor system.

If overrelaxation is employed, equation (8) changes ^into:

v)~~~~

>vo ₊ Ill >)I + >Pjv)~~. ill

With _a proper choice for the relaxation parameter I, the procedure _{converges even} when the

basic procedure does not [9]. Unfortunately only heuristic rules for choosing J exist, and the

procedure _seems to be _very sensitive to the value selected.

There is another way of looking at the iterative algorithm used to solve the system (4)-(5).

In fact equations (8) and (11) _can be interpreted _as the iterative relations employed to solve the reduced algebraic systems:

(I P)VF

=

Vo (12)

and

Iii P)VF

JVO (13)

respectively, by _means of Richardson's method [12]. ^{Since it is well known that this method is} a

very weak (possibly non-converging) solver, _{one can} think of finding _{a more} robust substitute.

In the next Section a brief _survey of conjugate gradient (CG)-like solvers is given, so as to better illustrate the _reasons why GMRES has been selected.

3. A Brief Survey of CG-Like Algorithms

We start by noting that the non-symmetric matrix I P is not directly available. However,

it is still possible to perform matrix-vector multiplications with such _a matrix, their main computational cost being given by the solution of the linear system (4). The substitute must be _a solver which _uses the coefficient _matrix only to perform matrix-vector multiplications and

so iterative solvers such _{as some} of the classical stationary iterative methods (Gauss-Jacobi, Gauss-Seidel, SOR and SSOR) _{[12] are not} applicable.

On the contrary, non-stationary ^iterative methods, such _as conjugate gradient-type solvers,

are good candidates owing to the fact that they require only matrix-vector multiplications, if used without preconditioning. It is well known that the classical conjugate gradient algorithm

is not applicable to systems having non-symmetric coefficient matrices, like system (12), but

a

wide variety ^of extensions of the original algorithm have been proposed ⁱⁿ literature to deal with non-symmetric matrices. The most well-known _are Biconjugate ^Gradient (BiCG)

[13], Quasi Minimal Residual (QMR) _[14], GMRES [11], Conjugate ^Gradient Squared (CGS)

[15j ^and Biconjugate Gradient Stabilized (BicGstab) [16j. All these solvers _are polynomial

accelerations of the basic Richardson method. They find _an approximate solution of the generic linear system:

Ax

=

b (14)

with _an initial guess xo in the Krylov subspace generated by the starting ^residual ro

b Axo Km(A, ro)

=

A (ro, Aro;.., A~~~ro) (15)

where A (vi, .,vm) denotes the subspace spanned by _{vi, ,vm} and _m depends _on the

number of iterations performed.

BiCG and QMR need _m steps in order to build (implicitly) Km(A,ro) and each step _re- quires two matrix-vector multiplications, _one of which with the transpose ^of the coefficient

matrix. GMRES builds (and stores) Km IA, ro) in _m steps, requiring only _one matrix-vector multiplication ^{for each} step. CGS and BicGstab _{require two} matrix-vector multiplications (not with the transpose) at each step, but after _m steps they have built K2m(A, ro), _so their speed, measured as the ratio between the dimension of the Krylov subspace and the number of matrix-vector multiplications performed, is the _{same as} that of GMRES and is twice that

of BiCG and QMR

Another dilserence between them is the _way they find the approximate ^{solution. BiCG finds}

an xm such that its residual is orthogonal to Km(AT, so) where _so is the so-called shadow residual, in general a randomly chosen vector QMR determines _xm by performing a pseudo-

minimization of the residual. GMRES finds _{an xm} which minimizes the residual at the cost

of explicitly forming and storing an orthonormal base for Km(A,ro). CGS and BicGstab

are hybrid methods which combine _an orthogonalization with _a pseudo-minimization of the residual. BiCG and QMR _are slower than the other algorithms and require a transpose ^matrix-

vector multiplication which, although possible, requires _more programming effort. Moreover

BiCG _can be _very unstable and _can show _{a very} irregular convergence On the other hand, CGS and BicGstab incorporate ^all the possible instabilities of BiCG; CGS _{converges more} irregularly than BiCG and BicGstab behaves poorly with real matrices having complex eigenvalues.

4. Accelerating Charge Iteration by GMRES

GMRES performs

true minimization of the residual and is thus the optimal method for

accelerating the charge iteration procedure _as it minimizes the number of matrix-vector mul-

tiplications (without taking into account the other operations required). Because the matrix-

vector multiplications in the charge iteration algebraic system _are much _more expensive ^than

in _a problem where the coefficient matrix is directly available, the GMRES algorithm has been chosen GMRES is well documented in literature, however _a brief outline is given in the

Appendix for the reader's convenience.

Implementation of the GMRES is quite simple. Most of the available codes, such _as the codes given in the Templates book ii?], _are written using ^the reverse communication scheme.

With this scheme the subprogram which executes the algorithm does _not perform the needed matrix-vector multiplication directly, but _{sets some} flags indicating the current status and the operation ^needed ii. _e. which vector must be multiplied), and then _returns to the caller. The caller executes the multiplication and calls back the subprogram, passing it the multiplied

vector.

In _our implementation the only difference lies

the computation of the residual when _a restart is performed. The residual _can be computed directly with the approximate solution, thus requiring _a matrix-vector multiplication, _as suggested in ii ii for large values of the restart-

ing parameter m. Otherwise the residual _can be computed using ^the orthonormal basis of the

Krylov subspace, _as explained in the appendix. We have used the latter option because, as

pointed out above, matrix-vector multiplications _are much _{more expensive} than in _a problem

where the coefficient matrix is directly available

The major drawbacks of GMRES _are the computing time and memory required to compute

and store the orthonormal base, which increase linearly with the number of iterations. So

restarting procedures _are often used. In _{our case} the computing time and memory required by GMRES for the orthonormal base _are only

small fraction of the total _ones, because it works

on

reduced algebraic system, the number of unknowns being the potential values _on the nodes of the fictitious boundary. Most of the computing time and _memory is spent on solving (4), _1.e.

performing matrix-vector multiplications. It is therefore convenient to _use long restarts which generally result in _a full GMRES due to the quick convergence characteristic of the charge

iteration procedure. The fact that the overrelaxed charge iteration converges with a suitable choice of

positive relaxation parameter indicates that the eigenvalues of the matrix I P have positive real parts. ^This fact, known _as the half-plane condition [15j, ^is a very welcome

property, because it _assures that GMRES _converges to the true solution _even with very short restarts (m

=

1).

5. Numerical Results

The aim of this section is to show by _means of _some examples the effectiveness of the algorithm

proposed and to compare it with other methods: overrelaxed charge iteration (OCI) with _an

y

x

< >< >< >

R D R

Fig. 2. Two-wire transmission line

Table I. Comparison between different methods.

Method iteration computing normalized

number time capacitance

GMRES-CI (m

lo) 7 1 5.04831

GMRES-CI (m

=

3) ^{lo 1.316 5.04811}

GMRES-CI (m

=

1) ^{23 2 868 5.04811}

OCI II

=

o.2) 19 1.579 5.04298

optimal relaxation parameter and co-ordinate transformations (CT). For the solution of the FEM algebraic system (4)

standard conjugate gradient with diagonal scaling _[17] has been used in all _cases.

The first example _{concerns a} two-wire transmission line constituted by two infinite par- allel conducting circular cylinders of radius R whose centers _are separated by _a distance of D

=

2.4R, voltaged with opposite unitary potentials (see Fig. 2). For this system an ana-

lytical solution is available and the _exact capacitance _per unit length is C

=

5.04785 _co (18],

so that the accuracy of the method _can be tested. Owing to the symmetry of the system, the analysis _can be restricted to the first quadrant only, by imposing homogeneous Neumann and Dirichlet boundary conditions _on the _z- and y-axis, respectively. The fictitious boundary

was selected _as constituted by two circumferences of radius 1.14R centered at the cylinder centers, so the gap between the conductors and the fictitious boundary is o.14 R, and is filled with two layers of elements (the mesh _was formed by 162 second-order triangular elements with 409 nodes). The problem _was solved with OCI and GMRES-accerelated charge ^iteration with different restarting parameters m (m

=

1, 3, lo). Table I shows the number of iterations

performed (with _an end-iteration tolerance of 0.02$lo), ^{the normalized} computing times for the solution of the system (the normalizing value being the minimum one), and the normalized capacitance _per unit length C/co (calculated by integrating the normal derivative of the _po- tential _on the conductor surface). Note that the _case

=

lo results in _a full GMRES and the

computing time is approximately 2 /3 of that required by OCI Moreover, independently of _m,

the accuracy of the method is very good and _a little bit better than OCI.

z

~. R

/

,

,R~ r

j

~

Fig. 3. Two toroidal conductors of circular

section.

The second example _concerns two identical toroidal conductors sharing the _same axisymme- try axis, _say the z-axis, and having circular cross-sections (see Fig. 3). The radius of the tori is Rt

=

o.4 _m; the radius of their section is R

=

o.1

and their median planes have

distance D

=

o.8 _m. The conductors _are voltaged with opposite unitary potentials. For symmetry

reasons the problem _can be studied in the first quadrant only of the

plane, by imposing

homogeneous Dirichlet and Neumann boundary conditions _on the

and z-axis, respectively.

The fictitious boundary is placed at _a distance of o.02

from the conductor surface with two layers of elements between them, _a total of 194 second-order triangular elements with 484 nodes. This problem _was solved first by GMRES-accelerated charge iteration with

restarting

parameter m

10, which results in _a full GMRES because the solution is achieved in six iterations. The calculated capacitance is C

=

33.72 pF.

The _same problem _was also solved by _means of co-ordinate transfomations [6]. ^The un-

bounded exterior region outside the circle of radius _a

=

1 _m centered at the origin is mapped

onto _a bounded _one by _means of the transformations r'

=

a~/r and z'

=

a~/z. The interior and exterior regions _are discretized with much _more elements than in the previous _case: in fact the mesh is formed by 1616 second-order triangular elements with 3356 nodes. Figures 4 and

5 show the contours of the potential (for the values Vk

k/20, k

=

0,..., 20) in the interior and exterior regions, respectively. The calculated capacitance is C

=

33.39 pF, showing a

good _agreement with the previous result, but the computing time _was much higher than that

required by GMRES-accelerated charge iteration: _{on a} DEC Alpha workstation it is of128 ms, against 35 _ms.

The third example concerns a three-dimensional problem. Two identical circular cylin- ders, having radius R

=

0.5 _m, height H

=

0.6

and parallel _axes separated by _a distance D

1.4 m, are voltaged with opposite unitary potentials (see Fig. 6). Owing to the symme- tries of this system only _one eighth of the total domain needs _to be discretized. The fictitious

boundary is chosen _as two cylindrical surfaces with _a radius of 0.6 _m and _a height of 0.8 m,

so that the _mean distance between it and the conducting cylinders is 0.1 _m. Two layers of second-order tetrahedral _elements

are inserted between the _two cylindrical surfaces la total of 2240 elements with 4277 nodes), _as shown in Figure 7. Using GMRES-accelerated charge iter- ation with _m

=

10, only _seven iterations _are required to get the final solution with _a computing

time of 2.56 _s. In Figure 8 the contours of the potential _are drawn for the values Vk

k /20,

k

=

0,..., 20. The computed capacitance is C

=

46.13 pF.

Fig 4. Contours of the potential

the interior region for the second example

Fig. 5. Contours of the potential in the exterior region for the second example.

6. Conclusions

In this paper the basic charge iteration procedure has been modified with the introduction of the Generalized Minimal Residual (GMRES) algorithm. This procedure has been implemented in ELFIN, _a large finite element code developed by the authors for electromagnetic ^CAD

research [19]. By virtue of this approach the overall computing time reduces to about 50% of

that required by the basic charge iteration procedure with _an optimal relaxation parameter

which, _among other things, is not simply determinable

_priori.

I

< >< >< >

R D R

Fig. 6. Two conducting cylinders.

Fig. 7. Tetrahedral mesh for the third example

Fig. 8. Contours of the potential for the third example.

Of _course the proposed method does _not work well for electrostatic problems only, but it is

expected to be successfully applicable to other electromagnetic problems in which the global algebraic system is similar to that of charge iteration, _as, for example, two-dimensional time-

harmonic skin effect [20] ^and wave scattering [21] problems

_open boundaries.

Acknowledgments

This work _was partially supported by MURST (the Italian Ministry for University and Scientific and Technological Research).

Appendix

The GMRES algorithm builds _an orthonormal base (vi, ..,vm) for the Krylov subspace Km(A, ro) by _means of _a modified Gram-Schmidt orthogonalization:

vi

ro/ I ro ^I for I

=

1 to m do

begin

wi

Avi

for j

1 _{to i} do

begin

h~,~

w(Av~; _w~

=

wi h~,ivi

end

hi+1,~ =ll _{Wi II; v~+i}

=

wi/hi+i,i

end

The coefficients _{h~,~ are} stored in _an upper Hessenberg matrix Hm+i,m and the orthonormal

vectors v~ are stored _as the columns of _a rectangular matrix Vm. The upper bound for _m is

mainly determined by the computer _memory available. The relation between A, Hm+i,m and Vm is

AVm

Vm+iHm+i,m. (16)

After having constructed these matrices _an approximate solution _xm is given by:

x~

xo + yivi +. + Ymvm

xo + VmY~~~ Ii?)

where the vector yl'~J, containing the coefficients j,is determined in such _{a way as to} minimize the residual _norm:

I rm 11=11 b Axm _{11=11 ro} AVmY~~~ ^I l18)

Using (16) ^{and the} fact that Vm has orthonormal columns, expression (18) becomes:

ii rm ii=ii _ro Vm+iHm+i,mY~~~ ii=ii ⁱ _ro

ii ei Hm+i,mY~'~~ ¹ l19)

The minimization problem (18) is easily solved by means of orthogonal transformations, which reduce Hm+i,m to _an upper triangular form. M/hen the memory area reserved for Hm+i,m and Vm is filled,

restart is performed, _{i-e- an} approximate solution is formed with ii?) and is

used _{as a new guess;} the old vectors vi and the Hessenberg matrix _are discarded.

Owing to the structure of GMRES the residual _{norm can} be computed at each step I

I,..., _m ^without actually solving the minimization problem, _so ^~the solution only needs to be found when the residual _norm is small enough _{or a restart} is needed. Once the approximate

solution ii?) is found, the residual _can be computed in two ways. The _most obvious _one is a

direct computation:

rm

b Axm (20)

which requires a further matrix-vector multiplication. Using (16) and ₁₁ 7) we obtain _a cheaper

way ^to compute ^it:

rm

ro Avmy(~)

=

ro Vm+iHm+i.mY~~~

=

Vm+i _{Ill ro} I ei Hm+i,mY~~~). (21)

This expression _can be further manipulated to achieve _a less expensive version of the algorithm,

as described in [11].

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