1886 IEEE TRANSACTIONS ON MAGNETICS, VOL. 29, NO. 2, MARCH 1993

Adaptive Refinement Strategies in Three Dimensions

N.A.Golias , T.D.Tsiboukis

Department of Electrical Engineering * Aristotle University of Thessaloniki

Thessaloniki, 54006 Greece

Abstract : 3-D Adaptive mesh refinement seems to be the answer to- wards full automation in the analysis of electromagnetic devices. The computational burden of 3-D problems can be reduced since with an efficient error estimator, optimal tetrahedral meshes are produced and the computational cost is minimized Further, the process of mesh generation, a head-ache for a finite element analyst, is completely automatic, requiring no user intervention. So self-adaptive mesh generation is proved to be an indispensable tool in 3-dimensional fi- nite element analysis. In this paper 3 different criteria for error esti- mation are presented that are able to estimate the error present in the approximation model and the strategy followed in their implementa- tion is analyzed Application to a test problem shows the effectiveness of the self-adaptive mesh generation algorithm

I. INTRODUCTION

Computer aided design and analysis (CAD-A) of elec- tromagnetic devices [l] is now not only a reality but in ad- dition a necessity. The boom in computer engineering poses the need for robust and efficient software that can take advantage of the new computer resources towards full automation in the development stage. The challenge to the software engineer is perhaps the most difficult task, since he must combine in-depth knowledge of different sciences. Computer science, numerical analysis and elec- tromagnetic field theory are three different branches of science that a finite element software engineer needs to possess in order to implement and develop a fully auto- matic CAD-A system.

Self-adaptive schemes of mesh refinement have been employed in 2-D finite element applications [2-131 and re- cently in 3-D [14]. The advantages of a self-adaptive finite element analysis system are well known. The adaptive scheme is able to approximate better the exact solution with a smoother distribution of nodes, especially in re- gions of high error. Aposterion' schemes of error estima- tion are employed so that a reliable adaptive process is implemented.

In this paper three schemes of aposteriori error estima- tion are tested for their efficiency. The strategy followed for each of the presented error estimation criteria is ana- lyzed. Application to a test problem is presented so that a comparative study of the efficiency of each criterion is given. The criteria presented are different in nature but all provide an efficient error estimation. The complete al- gorithm of self-adaptive automatic mesh refinement is analyzed in detail. The topological process of refining and producing tetrahedral meshes with elements of large as- pect ratios (nearly all tetrahedra are equilateral), without the presence of sliver elements, has been presented exten- sively in a previous work of the authors [14] and will not

be mentioned here. Further this topological algorithm has been implemented as an O(N) procedure, and this means that time and computations increase linearly with the problem dimensions, that is the number of nodes or the number of tetrahedra in our case, making it directly ap- plicable in the self-adaptive procedure.

A different approach of error estimation by means of a complementary scheme is not discussed here [5]. Al- though theoretically strong, this approach presents some drawbacks. The existence of a complementary solution is not always known. Further the computational cost is that of obtaining two solutions.

11. A-POSTERIORI ERROR ESTIMATION

Various error estimation procedures have been pro- posed [3-131. In order to estimate the error I 1 fa - fh I 1 a function f must be found that is closer to the exact solu- tion fa than the approximation function fh and the error is estimated as I I f - fh I I . Three criteria that exhibit good error estimation are described in the following. Discontinuity of Vf l lo

Suppose a continuous finite element approximation f with 1st order Lagrangian elements. C1 continuity is not ensured resulting in discontinuity of the normal compo- nent Vf n across element interfaces. Consider two neighboring tetrahedral elements e and f with a common face as is shown in Fig.1. Assuming that no is the unit vec- tor normal to the common face a parameter E1 (Error In- dicator) is defined as follows.

where Vf (e I, Vf (f) are the field gradients of tetrahedra e,f respectively and Sef the face area.

The above error indicator is referred to faces of te- trahedra. This error indicator must be weighted on an ele- ment basis. An element error indicator is obtained as EEI = max( EI1, EI2, EI3, E L ) , where the indices 1, 2, 3, 4 denote the four faces of a tetrahedron. In the sequel ele- ments with greater error indicators are refined as is de- scribed in the self-adaptive refinement algorithm. Energy perturbation

The perturbation of some parameter between 1st and 2nd order solutions seems to be a simple and clear cut way of error estimation. Clearly the 2nd order solution is

0018-9464/93$03.00 0 1993 IEEE

LEEE? TRANSACTIONS ON MAGNETICS, VOL. 29, NO. 2, MARCH 1993 1887

A

B C

Fig.1 Discontinuity of Vf n o across element interface

more accurate than the 1st order one. The larger the dif- ference between the two solutions the larger the error in the approximation. The parameter chosen is the value of the energy functional on each tetrahedron F. The dif- ference between the 1st and 2nd order solutions is an Ele- ment Error Indicator.

€E/ = F7 - F2

The strategy proposed is as follows. First the solution is obtained on the 1st order mesh. Then new nodes are added on the middle of tetrahedron edges and the ele- ments are transformed to 2nd degree. Solution with 2nd degree elements follows and error estimation takes place. Nodal perturbation

When employing perturbation schemes old values of the approximation function corresponding to the coarser mesh fold are calculated on the refined mesh by interpola- tion and compared with new valuesfnew obtained after solution. By comparing the difference between the two solutions an error estimation can be deduced [11,12]. An element error indicator is obtained as follows

E N = ( l# iew-dIdl+ I$ew-6 ld I+ I 6 e w - 6 l d I

(2)

where the indices 1, 2, 3, 4 refer to the four nodes of the tetrahedron.

When finite element nodes retain their positions this is a straightforward procedure. New values off are com- pared with old values. A problem arises when nodes are moved during the refinement process and this is the case in three-dimensional meshes [14]. Nodes have to be moved during the mesh refinement procedure in order elements’ aspect ratio is kept high, sliver elements are avoided and no ill-conditioning occurs.

To apply the nodal perturbation scheme f must be cal- culated by interpolation, which means a lot of searching and hence a high computational cost. It is understood that when this is the case the computational burden increases

more than linearly with the number of elements of the sys- tem. Another way to proceed is to solve the problem with- out node relaxation (movement of nodes) and obtain the new solution, then stabilize the mesh so that good quality tetrahedra are formed, calculate the element error indica- tors and fiially refine the mesh. The effectiveness is main- tained since the difference 1 ford - fnew / for a node should remain the same although the node is moved a small dis- tance. The regions for refinement should be identified and these regions stay invariable with node movement.

111. REFINEMENT. STRATEGY By the application of either of the above criteria, error

indicators are obtained for each tetrahedron. Elements with greater error indicators must be refined. Although this is easily understood it is not directly obvious which is the better way to proceed. One way is to find the element with greater error, refiie it and so on. Although this is a rigorous approach since in general elements with local error contribute the most to the global error, applicable in a self-adaptive algorithm because it would require a lot of refinement steps to reduce the error to ac- ceptable levels. In each refinement step a sufficient num- ber of elements should be refiied or equivalently a sufficient number of degrees of freedom should be in- serted in the system, so that the solution is attained in a few steps, i.e. at minimum computational cost.

the mean value has been calculated all elements with EEI > =a*EEImean must be refined, a being a constant usually in the interval 1.0-1.2. In this way an efficient n u -

Number 04 IThnehold Value l I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . .

. .. .......... ......... ........I

Error ERROR DISTRI8UTION

Fig.2 Selective Refinement above threshold

ber of elements are refined in each step resulting in a mesh with about 3-4 the number of elements of the pre- vious mesh.

1888 IEEE TRANSACTIONS ON MAGNETICS, VOL. 29, NO. 2, MARCH 1993

CONSTRAINTS I

DELAUNAY TRANSFORMATIONS NODE RELAXATION

* t

ERROR ESTIMATION

Satisfied

Yes

POST-PROCESSING

Fig3 Self-Adaptive Refinement Algorithm

IV. SELF ADAPTIVE REFINEMENT ALGORITHM

Employment of the self-adaptive algorithm in a finite element analysis system has as a result full automation during the analysis procedure. The characterization of the self-adaptive refinement algorithm as an expert system can be considered appropriate. Indeed the procedure is completely automatic and further the system is capable of recognizing the error and fixing it completely automat- ically. As a result this gives the optimum solution of a problem, a solution that even the most experienced engin- eer cannot get. The flow diagram of the self-adaptive re- finement algorithm is shown in Fig.3.

An initial coarse mesh is formed, covering the domain under question, by some automatic mesh generation scheme. Then problem definition and boundary condi- tions are specified for the solution of the problem. The system begins an iterative cycle of solution and selective self-adaptive refinement until some termination condi- tions are satisfied, resulting in the optimal solution of the problem for the time and computations involved.

V. APPLICATION

Application to a test problem shows the effectiveness of the proposed technique and the error estimation criteria in the self-adaptive refinement algorithm. Solution of the Laplace equation of a scalar functionf

V2f = 0 (4)

with the corresponding functional

is obtained for the domain presented in Fig.4a. An in- itial coarse mesh is generated with 18 tetrahedra and 16 nodes.

As can be seen in FigAe where the equipotentials of the approximation function f are presented abrupt field vari- ation occurs near the bend of the domain. The result of the adaptive procedure is to refine the mesh in this area thus minimizing the error as can be seen in F i g 5 All the three criteria of error estimation succeed in refining the mesh near the bend, minimizing thus the error in the ap-

Fig.4 a) Initial mesh with 18 tetrahedra b) Adaptive mesh with 1,314 tetrahedra c) Adaptive mesh with 12,314 tetrahedra d) Adaptive mesh with 29,774 tetrahedra

e) Equipotentials

IEEE TRANSACTIONS ON MAGNETICS, VOL. 29, NO. 2, MARCH 1993 1889

log(error) - 1

0 2 4 8 8 10 12 14 18 Nodes x10=

+ Unlform + Grad Dlscontlnulty

-B Nobd Parturbstlon * Energy Perturbillon

Fig.5 Error in functional of T-conductor

proximation. A refined mesh with 1,314, 12,314 and 29,774 tetrahedra is shown in Fig.4b, 4c, 4d respectivelly.

As exact value of the functional is considered a solution with 9,171 2nd order elements and 13,478 nodes. Based on this almost exact value error is calculated as

From the results presented in Fig.5, where the loga- rithm of the error of the three adaptive procedures is presented with a uniform refinement, we observe that all the three different error estimation criteria managed to assess the error sufficiently. The fact that one criterion may give results better than the others is not very crucial since we refer to the solution of a specific problem.

What is most important is the effectiveness in the im- plementation of each error estimation criterion. The first criterion using the discontinuity of the normal component of gradient seems to be the most efficient since its im- plementation is direct and general. The energy perturba- tion criterion works very well although more complex than the first since it requires a solution with 2nd order ele- ments. Finally the nodal perturbation criterion requires old values of the solution to be retained so that error can be estimated.

VI. CONCLUSION

A comparative study of various error estimation strategies has been attempted in this paper. The proposed strategies of self-adaptive mesh generation proved suc- cessful resulting in full automation during the analysis of electromagnetic devices. The process takes the burden from the finite element engineer-analyst of generating a valid tetrahedral mesh, since it produces the mesh com- pletely automatically. Further it results in the formation of an optimal mesh, a scope that even the more experienced user cannot achieve with so much accuracy. The aposte- riori error schemes presented are simple, exhibit good error estimation, and are directly employed in the adap- tive procedure. All three error estimation criteria presented have estimated with accuracy the error of the

approximation procedure. As regards the efficiency of the three error estimation criteria the discontinuity of the field gradient seems to be the more efficient error estima- tor since it is directly applicable in the self-adaptive pro- cedure.

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[3] A.M.Pinchuk and P.P.SiIvester,"Ekror estimation for automatic adap- tive finite element mesh generation", IEEE Trans. M a p , Vol. 21,110.6,

[4] A.Raizer,G.Meunier and J.L.Coulomb,"An approach for automatic adaptive mesh refinement in finite element computation of magnetic fields", IEEE Pam. Map., Vol-25, No.4, pp.2965-2967,1989. [SI J. Penman, M.D. Grieve, "Self-adaptive mesh generation technique for the finite element method", IEE Proceedings Vol-134, Pt.A, No.8,

[6] I. Babuska, W.C. Rheinboldt, "A-posteriori error estimates for the fi- nite element method", In& J. Numer: Meth. Eng., 12, pp.1597-1615,1978. [7] D.W. Kelly, J.P. De, S.R. Gago, O.C. Zienkiewicz, I. Babuska, "A- posteriori error analysis and adaptive processes in the finite element method : Part I--Error analysis", Int. J. N u " Meth. Eng., 19, pp.1593- 1619,1983. [8] W.C. Rheinboldt, "Adaptive mesh refinement processes for finite ele- ment solutions", Int. .l Numer: Meth. Eng, 17, pp.649-662,1981. [9] S.R.H.Hoole, S. Yoganathan, S. Jayakumaran, "Implementing the smoothness Criterion in Adaptive Meshes", IEEE Trans. Mu@, Vol. 22, No.lO,808-810, Sept. 1986. [lo] S.R.H.Hoole, "Relevant, Purpose Based Error Criteria for Adaptive Mesh Refinement", J. Electromag. Waves and Applications, vo1.3, No.3, pp. 167-177, 1989. [ l l ] S.RH.Hoole, S. Jayakumaran, N.R.G.Hoole, "Flux Density and En- ergy Perturbations in Adaptive FE Mesh Generation*, IEEE D m . Magn., Vo1.24, pp. 322-325, Jan. 1988. [12] S.RH. Hoole, "Nodal Perturbations in Adaptive Expert Finite Ele- ment Mesh Generation", IEEE Trans. Ma@, Vo1.23, pp.2635-2637,1987. [13] N.A.Golias, T.D.Tsiboukis, "Adaptive mesh refinement in 2-D finite element applications", Intemational Journal of Numerical Modeling Elec- tronic Networks, Devices and Fieldr, Vo1.4, pp.81-95,1991. [14] N.A.Golias, T.D.Tsiboukis, "Three-dimensional Autom tive Mesh Generation", IEEE Pam. Map. Vo1.28, N0.2, pp 1992.

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BIOGRAPHIES Nikolaos A. Golias was born in Veria, Greece, April 29, 1966. He re- ceived his DipLEng. degree from the Department of Electrical Engineer- ing at the Aristotle University of Thessaloniki, Thessaloniki, Greece in 1989. Since 1989 he has been working towards his Ph.D degree at the De- partment of Electrical Engineering o His current interests are focused on method to the field solution of vario velopment of adaptive techniques in of the Technical Chamber of Greece.

Theodoros D. Tsiboukis was born in Larissa, Greece, on February 25, 1948. He received the Dipl. Eng. degree from the National Technical University of Athens, Athens, Greece, in 1971 and the Ph.D degree from the Aristotle University of Thessaloniki, Thessaloniki, Greece, in 1981. Since 1981 he has been working at the department of Electrical Engin- eering of the Aristotle University of Thessaloniki, where he is now an Associate Professor. He is the author of several books and papers. research activities include electromagnetic field analysis by energy met ods with emphasis on the development of special finite element tech- niques to the field solution of various engineering applications. Dr. Tsiboukis is a member of the Technical Chamber of Greece.