Jackson Mark. Machining with Abrasives

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applied to three different case study optimization problems for the surface grinding

process, including minimization of grinding cost, minimization of cycle time, and

process control.

The objective of optimization in process control is to ﬁnd operating conditions to

achieve desirable values of process outputs. In surface grinding, three aspects are

supposed to be concerned at the same time which are desirable surface roughness,

grinding power and G-ratio. In this case, the objective function can be given as

follows:

J ¼

 R



 P



 G



(6.24)

where R

, P

, and G

are the desirable process output values of surface roughness,

grinding power and G-ratio. In practical situation, this idea can be extended by

employing weighting factors w

 w

in the objective function:

Minimize :

J ¼ w

R



þ w

P



þ w

G



Subject to :

0:10

b0:30

0:75

b2:0

12:7

b50:8

0:015

bab0:03

b600

(6.25)

The constraints come out in terms of work speed v

, the crossfeed s

, the dressing

depth a

, the depth of cut aand maximum residual stress s

Xu and Shin [38, 39] developed a multi-level fuzzy controller and implemented

to deal with the real-world nonlinear plants with intrinsic uncertainties and time-

varying parameters, such as a creep-feed grinding process. The proposed fuzzy

control strategy can be used to control a system with an input-output monotonic

relationship or a piecewise monotonic relationship, which is efﬁcient and cost-

effective for practical control applications.

The multi-level fuzzy controller is a two-input single-output controller with a

two-level, hierarchical structure with an adaptation mechani sm embedded in the

lower level to tune the output membership functions (MFs) of the ﬁrst layer fuzzy

controller as shown in Fig. 6.22. The ﬁrst layer fuzzy control rules are generated

based on the experience of human operators. In order to correct the impreciseness of

the ﬁrst layer fuzzy control rules and compensate for the time-varying behavior or

system uncertainty, a self-organizing fuzzy controller is used to tune the output

membership functions (MFs) of the ﬁrst layer fuzzy controller based on the

evaluation of system performance.

6 Traditional and Non-traditional Control Techniques for Grinding Processes 295

For the ﬁrst layer fuzzy controller, the two inputs are the error eðkÞ and the

change of the error DeðkÞ. The output is the change in the control action DuðkÞ:

eðkÞ¼y

ðkÞyðkÞ (6.26)

DeðkÞ¼eðkÞek 1ðÞ (6.27)

where y

ðkÞ is the desired output at time instant k, yðkÞis the actual output, eðkÞ is

the error and Deðk Þ is the change of the error. The change of control action is

calculated based on the two inputs using a fuzzy inference mechanism, which is

described in the fuzzy control rule as the following form:

R : if e

is E

and e

is E

then u

is U

nði;jÞ

(6.28)

where e

is the error eðkÞ at time instant k, e

is the change of the error DeðkÞ, u

the change of the control signal DuðkÞ, E

is the linguistic variable of e

, E

is the

linguistic variable of e

, and U

nði;jÞ

is the linguistic variable of u

. By employing

Mamdani’s minimum fuzzy implication and center average defuzziﬁcation, the

fuzzy control law of the ﬁrst layer is:

i;j

½ðm

ðÞ\m

ðÞÞU

nði;jÞ



i;j

ðm

ðÞ\m

ðÞÞ

(6.29)

Similarly, the self-organizing fuzzy controller (the lower fuzzy layer) has the

control rule in the form of:

R : if e

is E

and e

is E

; then u

is C

mði;jÞ

(6.30)

where C

mði;jÞ

is the linguistic variable of C which is the change of the center of the

output MF of the ﬁrst layer fuzzy controller. Hen ce, using Mamdani’s minimum

First Layer

Fuzzy Logic

Controller

Adaptive Fuzzy

Logic

ZOH Process

−1

e(k)

e(k−1)

Δe(k)

e(k)

u(k)

Fig. 6.22 Multilevel fuzzy control (MLFC) system

296 J. Liu et al.

implication and center average defuz ziﬁcation, the center of output MF of the ﬁrst

layer controller should be tuned as:

nði;jÞ

¼ U

nði;jÞ

i;j

½ðm

ðÞ\m

ðÞÞC

mði;jÞ



i;j

ðm

ðÞ\m

ðÞÞ

(6.31)

This proposed systematic procedure is proved to be capable of controlling

nonlinear systems with uncertainties and time varying parameters to reduce track-

ing error and improve system performance by simulation results conducted on

cargo ship steering and fuzzy cruise control, as well as another control work done

on cutting force control for creep-feed grinding processes. The grinding forc e was

maintained at the maximum allowable level for varying depth of cut of raw work-

pieces. Ex perimental results illustrate that the cycle time was reduced by up to 25%

than that without any controller, and the effect of wheel wear was also compensated

by the adaptation scheme. These indicated the effectiveness of the intelligent

system in improving the productivity and enhancing the reliability of grinding

processes in practical industrial applications.

Other researchers such as Mise et al. [40], Yuan et al. [41] have also used fuzzy

logic technique in grinding processes. Mise’s work proposed an autonomous

control system which can compensate for the sizing points in an in-process control

based on information from a total inspecting mach ine located in the post -processing

end of an online-controlled measuring instrument.

6.7 Hybrid Control Schemes

In the case when the plant model is partially known, the fuzzy technique can be

designed to work with conventional control methods to improve the system perfor-

mance accuracy and efﬁciency. Xiao et al. [42] designed a fuzzy + conventional PI

controller to reduce the multi-axis contour error and promote the dynamic perfor-

mance of a grinding machine based on human knowledge and analytical models.

The hybrid control system was implemented in an open architecture system for

practical applications. This system is capable of optimizing the grinding and

dressing parameters in resp onse to in-process and post-process measurements

which characterize the process and update the process model which has considered

a more complete set of realistic constraints compared with previous ly proposed

systems.

Orchard et al. [43, 44] presented a fuzzy predictive controller for a mineral

concentration grinding plant in order to maximize the ore feed rate and to follow a

predetermined particle size set-point. The process was a typical multi-input multi-

output system, where the inputs were water and fresh ore ﬂows, the outputs were

percentage of solids, particle size and power demand. The proposed hybrid control

scheme was based on a traditional linear multivariable ARX model, and a fuzzy

6 Traditional and Non-traditional Control Techniques for Grinding Processes 297

characterization of the process objective functions and constraint functions. The

process objective function was expressed as multiple fuzzy criteria based on

operators’ knowledge about the process itself and the corresponding requirements.

The overall scheme of the fuzzy predictive control regulator is shown in Fig. 6.23.

Compared with conventional predictive control method, simulation results illu-

strated a better system performance in minimizing the output errors and the input

variations under multiple process input/output constraints and disturbances, such as

different sizes and hardness variations of the in-feeding ore particle. Moreover,

the computational time of the fuzzy predictive cont roller was reduced signiﬁcantly,

which ensured for a real-time operating environment.

Even though many fuzzy control schemes have been investigated and imple-

mented in grinding processes, in most cases, the control rules are predeﬁned

and ﬁxed during the entire operation process. The parameters of such controllers

may not be optimal when the process is not accurately known or time-varying,

and the desired system performance can not be guaranteed. Hence, adaptive fuzzy

controllers would be desirable with the ability to tune the control parameters on-

line. To

nshoff and Walter [11] designed a self-tuning fuzzy controller for

an internal grinding process. The membership functions and the fuzzy rules were

automatically adjusted without human intervention. Tang and Song [45] imple-

mented a real-time fuzzy controller with online self-learning ability for a grinding

process, which could provide better system performance than pure conventional

linear PID controller under the system dynamic variations and process uncertain-

ties, such as the variations of the infeed parts’ materials, sizes and hardness. The

block diagram of the closed-loop control system is illustrated in Fig. 6.24, where the

fuzzy control rules are updated in real time continuously through online rule

modiﬁcation algorithm.

optimization pl an t

Process

ARX model

Fuzzy

Objective

function

Human

knowledge

controller

Fig. 6.23 Fuzzy predictive control scheme

298 J. Liu et al.

6.8 Conclusions

It is well-accepted that model-based control systems can ensure the most satis-

factory system performances as desired . The system mathematical models are

usually set up either from physical laws or from experimental data. (Sometimes

operators’ knowledge can also be used to generate system model, but in often

cases it is not accurate enough.) It is the conclusion of the authors that, in the

future, the most efﬁcient control systems will probably be hybrid in nature.

Therefore, multiple possible information sources can be used to effectively

control the entire grinding process. In addition, a particular control technique

will be developed as a modular, which should be designed as an appropriate way

for a speciﬁc grinding process.

General grinding processes are typical complicated nonlinear and time-varying

multivariable systems, which have intrinsic interactions and inevitable uncertain-

ties duri ng the operation. Based on the models obtained from the system input/

output data, hybrid control techniques possess the intelligence, which does not

require a good mathemati cal model of the process. The optimal grinding conditions

can be automatically determin ed within certain operation constraints, such as the

grinding burning and chatter constraints. The appropriate control actions, wheel

speed, table speed, depth of cut and width of cut, will be obtained by the intelligent

inferencing without resorting to trial and error. Moreover, some process variables

are often not measurable. Thus, an effective controller with an observer design will

be beneﬁcial to extend the current application range.

fuzzification

plant

Self-learning

algorithm

Control

Rule base

Decision

maker

power meter

d/dt

−

Fig. 6.24 A fuzzy controller with online self-learning scheme

6 Traditional and Non-traditional Control Techniques for Grinding Processes 299

References

1. To

nshoff, H. K., Zinngrebe, M., and Kemmerling, M., 1986, “Optimization of Internal Grinding

by Microcomputer-Based Force Control”, Annals of the CIRP, Vol. 35, No. 1, pp. 293–296.

2. To

nshoff, H. K., Zinngrebe, M., and Kemmerling, M., 1991, “Optimization of Internal

Grinding by Microcomputer-Based Force Control”, Control of Manufacturing Processes,

ASME Winter Annual Meeting, DSC Division, Vol. 28, pp. 67–77.

3. Kaliszer, H., Mishina, O., and Webster, J., 1979, “Adaptively Controlled Surface Roughness

and Roundness during Grinding”, Proceedings of the 20th International MTDR Conference,

Birmingham, pp. 471–478.

4. Amitay, G., Malkin, S., and Koren, Y., 1981, “Adaptive Control Optimization of Grinding”,

Transactions of the ASME, Journal of Engineering for Industry, Vol. 103, pp. 103–108.

5. Malkin, S., 1981, “Grinding Cycle Optimization”, Annals of the CIRP, Vol. 30, pp. 223–226.

6. Furukawa, Y. and Ohishi, S., 1984, “Adaptive Control of Creep Feed Grinding to Avoid

Workpiece Burn”, Proceedings of the Fifth International Conference on Production Engineer-

ing, Tokyo, Japan, pp. 64–69.

7. Elbestawi, M. A., Yuen, K. M., Srivastava, A. K., and Dai, H., 1991, “Adaptive Force Control

for Robotic Disk Grinding”, CIRP Annals – Manufacturing Technology, Vol. 401, pp.

391–394.

8. Bhattacharyya, S. K., Fowell, B., and Wallbank, J., 1984, “Control of Workpiece Surface

Quality in Grinding”, ASME Winter Annual Meeting on Production Engineering Division,

New Orleans, LA, Vol. 12, pp. 425–444.

9. Ulrich, B. J., Srivastava, A. K., Elbestawi, M. A., and Veldhuis, S., 1989, “Force Modelling of

the Robotic Disk Grinding Process”, ASME Winter Annual Meeting, PED – Vol. 39, San

Francisco, pp. 105–130

10. Srivastava, A. K., Rogers, D. B., and Elbestawi, M. A., 1993, “Optimal Planning of an

Adaptively Controlled Robotic Disk Grinding Process”, International Journal of Machine

Tools & Manufacturing, Vol. 33, No. 6, pp. 809–825.

11. To

nshoff, H. K. and Walter, A., 1994, “Self-tuning Fuzzy Controller for Process Control in

Internal Grinding”, Fuzzy Sets and Systems, Vol. 63, No. 3, pp. 359–373.

12. Hahn, R., 1964, “Controlled-Force Grinding – A New Technique for Precision Internal

Grinding”, Transactions of the ASME, Journal of Engineering for Industry, pp. 287–293.

13. Hahn, R., 1965, “Some Characteristics of Controlled Force Grinding”, Proceedings of the

Sixth International Machine Tool Design Research Conference, pp. 597–609.

14. Zhao, Y. W. and Webster, J., 1990, “Fuzzy Pattern Recognition and Automatic Steady Control

in Roller Grinding”, Proceedings of Second IEEE International Conference on Computer

Integrated Manufacturing, Troy, NY, pp. 395–401.

15. Guo, L., Scho

ne, A., and Ding, X., 1992, “A Comprehensive Approach to Nonlinear Adaptive

Control and its Application to Form Grinding Processes”, 31st IEEE Conference on Decision

and Control, Tucson, AZ, Vol. 2, pp. 1267–1272.

16. Guo, L., Scho

ne, A., and Ding, X., 1993, “Grinding Force Control Using Nonlinear Adaptive

Strategy”, 12th World Congress IFAC, Sydney, Australia, Vol. 5, pp. 459–462.

17. Jenkins, H. E., Kurfess, T. R., and Dorf, R. C., 1996, “Design of A Robust Controller for A

Grinding System”, IEEE Transactions on Control Systems Technology, Vol. 4, No. 1, pp.

40–49.

18. Jenkins, H. E. and Kurfess, T. R., 1999, “Adaptive Pole-Zero Cancellation in Grinding Force

Control”, IEEE Transactions on Control Systems Technology, Vol. 7, No. 3, pp. 363–370.

19. Brinksmeier, E. and Popp, C., 1991, “A Self-tuning Adaptive Control System for Grinding

Processes”, Annuals of the CIRP, Vol. 40, No. 1, pp. 355–358.

20. Wada, R. and Kodama, H., 1977, “Adaptive Control in Grinding”, Japanese Society of

Precision Engineering, Vol. 11, No. 1, pp. 1–10.

300 J. Liu et al.

21. Ko

nig, W. and Werner, G., 1974, “Adaptive Control Optimization of High Efﬁciency External

Grinding – Concept, Technological Basics and Application”, Annals of the CIRP, Vol. 23, No.

1, pp. 101–102.

22. Kelly, S., Rowe, W. B., and Moruzzi, J. L., 1989, “Adaptive Grinding Control”, Advanced

Manufacturing Engineering, Vol. 1, No. 5, pp. 287–295.

23. Xiao, G., Malkin, S., and Danai, K., 1992, “Intelligent Control of Cylindrical Plunge

Grinding”, Proceedings of the 1992 American Control Conference, Chicago, IL, pp. 391–398.

24. Xiao, G., Malkin, S., and Danai, K., 1993, “An Autonomous System for Cylindrical Plunge

Grinding”, Proceedings of NSF Design and Manufacturing Systems Conference, pp. 399–403.

25. Xiao, G. and Malkin, S., 1996, “On-Line Optimization for Internal Plunge Grinding”, Annals

of the CIRP, Vol. 45, pp. 287–292.

26. Zhu, J. Y., Shumsheruddin, A. A., and Bollinger, J. G., 1982, “Control of Machine Tools

Using the Fuzzy Control Technique”, Annals of the CIRP, Vol. 31, No. 1, pp. 347–352.

27. Li, G. F., Wang, L. S., and Yang, L. B., 2002, “Multi-Parameter Optimization and Control of

the Cylindrical Grinding Process”, Journal of Materials Processing Technology, Vol. 129, pp.

232–236.

28. Dong, S., Danai, K., Malkin, S., and Deshmukh, A., 2004, “Continuous Optimal Infeed

Control for Cylindrical Plunge Grinding, Part I: Methodology”, Transactions of the ASME,

Journal of Manufacturing Science and Engineering, Vol. 126, pp. 327–333.

29. Dong, S., Danai, K., Malkin, S., and Deshmukh, A., 2004, “Continuous Optimal Infeed

Control for Cylindrical Plunge Grinding, Part I: Controller Design and Implementation”,

Transactions of the ASME, Journal of Manufacturing Science and Engineering, Vol. 126,

pp. 334–340.

30. Srivastava, A. K., Ulrich, B. J., and Elbestawi, M. A., 1990, “Analysis of Rigid-Disk Wear

During Robotic Grinding”, International Journal of Machining Tool and Manufacturing, Vol.

30, No. 4, pp. 521–534.

31. Malkin, S. and Koren, Y., 1984, “Optimal Infeed Control for accelerated Spark-Out in Plunge

Grinding”, Transactions of the ASME, Journal of Engineering for Industry, Vol. 106, pp.

70–74.

32. Ljung, L., 1987, “System Identiﬁcation: Theory for the user”, Englewood Cliffs, NJ: Prentice-

Hall.

33. Rowe, W. B., Li, Y., Mills, B., and Allanson, D. R., 1996, “Applications of Intelligent CNC in

Grinding”, Computers in Industry, Vol. 31, pp. 45–60.

34. Rowe, W. B., Chen, Y., Moruzzi, J. L., and Mills, B., 1997, “A Generic Intelligent Control

System for Grinding”, Computers Integrated Manufacturing Systems, Vol. 10, No. 3, pp.

231–241.

35. Nakajima, T., Tsukamoto, S., Murakami, D., and Yasuda, H., 1993, “Neuro & Fuzzy In-

Process Control Grinding Techniques – Study on Intelligent Automation of Grinding Pro-

cess”, Journal of the Japan Society for Precision Engineering, Vol. 59, No. 8, pp. 1313–1318.

36. Chen, Y. T. and Shin, Y. C., 1991, “A Surface Grinding Process Advisory System With Fuzzy

Logic”, Control of Manufacturing Processes, ASME Winter Annual Meeting, DSC Division,

Vol. 28, pp. 67–77.

37. Lee, C. W., Choi, T., and Shin, Y. C., 2003, “Intelligent Model-based Optimization of the

Surface Grinding Process for Heat-Treated 4140 Steel Alloys with Aluminum Oxide Grinding

Wheels”, Transactions of the ASME, Journal of Manufacturing Science and Engineering, Vol.

125, pp. 65–76.

38. Xu, C. and Shin, Y. C., 2005, “Design of a Multi-level Fuzzy Controller for Nonlinear

Systems and Stability Analysis”, IEEE Transactions on Fuzzy Systems, Vol. 13, No. 6, pp.

761–778.

39. Xu, C. and Shin, Y. C., 2007, “Control of Cutting Force for Creep-feed Grinding Processes

using a Multi-level Fuzzy Controller”, ASME Transaction, Journal of Dynamic Systems,

Measurement and Control. Vol. 129, No. 4, pp. 480–492.

6 Traditional and Non-traditional Control Techniques for Grinding Processes 301

40. Mise, R., Itoga, K., and Kato, C., 1994, “Applying Fuzzy Logic for Hybrid Control of

Grinding Work”, IEEE Proceedings of the Tenth Anniversary Advanced Technologies in

Instrumentation & Measurement Technology, Hamamatsu, Japan, Vol. 2, pp. 615–618.

41. Yuan, L., J

€

arvenp

€

a, V. M., and Keskinen, E., 2004, “Design of Fuzzy Logic-Based Controller

in Roll Grinding System with Double Regenerative Chatter”, Proceedings of IMECE, ASME

International Mechanical Engineering Congress and Exposition, Anaheim, CA, pp. 463–469.

42. Xiao, B. X., Xia, M., and Zhao C. M., 1996, “The Main Control Mode and Fuzzy Control

Strategy of CNC System for Gear Hobbing and Grinding Machine”, Proceedings of the IEEE

International Conference on Industrial Technology, Shanghai, China, pp. 643–646.

43. Orchard, M., Flores, A., Mun

oz, C., and Cipriano, A., 2001, “Predictive Control with Fuzzy

Characterization of Percentage of Solids, Particle Size and Power Demand for Minerals

Grinding”, Proceedings of the 2001 IEEE International Conference on Control Applications,

Mexico City, pp. 600–605.

44. Orchard, M., Flores, A., Mun

oz, C., and Cipriano, A., 2001b, “Model-based Predictive

Control with Fuzzy Characterization of Goals and Constraints, Applied to the Dynamic

Optimization of Grinding Plants”, The Tenth IEEE International Conference on Fuzzy

Systems, Melbourne, Australia, Vol. 2, pp. 916–919.

45. Tang, Y.-G. and Song, G., 2002, “The Mill Load Control for Grinding Plant Based on Fuzzy

Logic”, Proceedings of the First International Conference on Machine Learning and Cyber-

netics, Beijing, China, pp. 416–419.

302 J. Liu et al.

Chapter 7

Nanogrinding

Mark J. Jackson and J. Paulo Davim

Abstract Nanogrinding is an aspect of advanced manufacturing that has been

growing rapidly in recent years. For instance, lenses and mirrors are manufactured

to precise and ultraprecise standards using nanogrinding tec hniques. Most of these

applications require a crack-free surface. Generally, hard and brittle materials such

as glass, silicon, and germanium are commonly used for making these products.

Traditionally, optical glasses require grinding and ﬁnishing by a polishing process

in order to remove the damage caused by the previous machining operation and to

obtain a ﬂat surface. The conventional ground surface of glass results in a ﬁne ﬁnish

due to brittle fracture during the removal process and this surface needs to be

polished to 20 nm R

max

to reduce absorption and scattering of light on the glass

surface. However, advances in advanced nanogrinding of brittle materials have led

to the discovery of a ductile regime in which material removal is by performed by

plastic deformation. Fracture mechanics predicts that even brittle solids can be

machined by the action of plastic ﬂow, as is the case in metal, leaving crack free

surfaces when the removal process is performed at less than a critical depth of cut.

This means that under certain controlled conditions, it is possible to machine brittle

materials such as ceramics and glass using specially developed abrasive tools so

that material is removed by plastic ﬂow, leaving a smooth and crack-free surface.

This chapter describes the recent advances in nanogrinding from computational

analysis to the development of an ultra precise machine tool that is used to grind and

polish ﬂat, crack-free surfaces.

Keywords Nanogrinding  Microgrinding  Precision grinding  Brittle materials

M.J. Jackson (*)

MET, College of Technology, Purdue University, 401 North Grant Street, West Lafayette,

IN 47907, USA

e-mail: jacksomj@purdue.edu

M.J. Jackson and J.P. Davim (eds.), Machining with Abrasives,

DOI 10.1007/978-1-4419-7302-3_7,

Springer ScienceþBusiness Media, LLC 2011

303

7.1 Introduction

For the analysis of the molecular dynamic (MD) process simulations, micro

mechanical state variables of all atoms and thermodynamic state variables of the

system as a whole are accessible throughout the simulation. While some state

variables are directly available, such as co-ordinates and velocities, others are

available by statistical mechanics’ analyses that need to be calculated as system

quantities such as thermodynamic state variables: heat; temperature; and pressure.

Some of the capabilities of MD for analyzing nanoscale cutting processes by

simulation are demonstrated on the basis of application examples. For this purpose,

results of orthogonal cutting process simulations of ductile, single-crystal copper is

presented, although MD cutting and machining simulations results of brittle as well

as ductile poly-crystalline materials can be found in the literature. The following

ﬁgures show results of nanogrinding process simulations where hard tool tips move

on {001} surfaces along <110> directions of fcc crystals. The following computa-

tional analysis is described in detail by Rentsch in the classic textbook on nano and

micromachining edited by Jackson and Davim [1].

7.2 Analysis of Microstructural Deformation

The availability of the co-ordinates of the atoms in MD calculations allows one to

visualize the instantaneous positions and to track the motion of the atoms individu-

ally as well in sequence. This information is not only useful for so-called snapshots

of atomic arrangements, through which the progress of process simulations can be

monitored, but it can be used for dynamic analyses on the basis of animations as

well. For this purpos e often the radii of the atoms are chosen in order to enhance a

certain aspect or view of a speciﬁc arrangement than representing realistic atomic

sizes. Using instantaneous co-ordinates rather than averaged positions, due to the

vibration of the atoms, does not cause a signiﬁcant error, because the maximum

vibration amplitude of an atom in a stable MD calculation is less than 1% of the

minimum bonding length.

The model in Fig. 7.1 was employed to study the chip formation process and the

surface generation at a cutting edge in nanoscale cutting. By employing one-

dimensional PBC along the y-axis, the orthogonal cutting condition reduced the

model to a quasi-2D type with a small width, that allows one to correctly model the

3D fcc crystal structure of the copper workpiece. The model contained 71,000 work

atoms and 11,000 tool atoms. Atoms at the bottom and to the left and right hand side

of the model represent ﬁxed boundaries, but were shifted with cutting speed to

create the relative motion between workpiece and tool. Atoms in the layers next to

these hard boundary atoms h ad thermostatic properties controlling the temperature

at the workpiece boundary.

For diamond cutting of copper, no plastic and no signiﬁcant elastic deformation

of the tool was expected, calculating its internal structure was not of special

304 M.J. Jackson and J. P. Davim