on the application of a continuous method introduction · pdf filethat arise in some...

On the application of a continuous method

to 3-D shape sensitivity analysis: a BEM

approach for electric design

E. Schaeidt, J. Unzueta, A. Longo, J.J. Anza

Analysis & Design Department, LA'BEIN, Technological

Research Centre, Cuesta de Olabeaga, 16, 4 013 Bilbao,

Spain

INTRODUCTION

This work is motivated by the need to solve shape optimization design problems

that arise in some industrial applications, particularly in electric and magnetic

design. Modern electrical plants and machinery have to be designed to operate at

minimum cost, optimum performance and high reliability grade. These strict

requirements need an accurate prediction of the performance at the design stage.

For example, some of these devices have to satisfy, at the same time, very strict

criteria on electromagnetical performance while occupying as little as possible.

The shape optimization problem for such a device consists in finding a geometry

which minimizes a given functional (such as the volume) and yet simultaneously

satisfies specific constraints (design bounds, electric field, tangential field). The

geometry of the component can be considered as a given domain in the three-

dimensional Euclidean space. In general the cost function takes the form of an

integral over the domain or its boundary, where the integrand depends smoothly

on the solution of a boundary value problem.

Both the automatic as well as the interactive design shape optimization involve the

shape sensitivity analysis, which is considered as a valuable tool for evaluating the

dependency of the technological specifications (cost function and constraints) on

the geometry defined by means of a set of design variables. Therefore, the shape

sensitivity analysis has to comply with the following requirements:

Transactions on Modelling and Simulation vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X

204 Boundary Element Technology

- consistency in terms of accuracy of results, which conditions the convergence

of the optimization algorithms, and flexibility to be able to treat a wide

variety of design criteria functional as well as the type of domains and

boundaries over which they are defined.

- practical applicability by means of calculation speed, a determinant factor for

the use of the tool in the industrial environment.

An extensive literature survey can be found in Unzueta [1,2], where a

classification of the most widely used approaches is described according to the

discretization-differentiation order - discrete and continuous approach -,

dependency on total or partial shape derivatives in the obtained expressions -

domain and boundary formulation -, and procedure for evaluating the shape

derivatives of the state variables - direct and adjoint method-.

The discrete approach is more expensive in CPU time than the continuous one

(higher number of operations to perform). In addition to this, the discrete

approach is mesh distortion dependent and requires a deep knowledge of the

analysis code due to the different formulations needed depending upon the analysis

method used. Moreover, it requires a significant programming effort to develop

the numerical integration of sensitivity kernels when BEM is used as analysis tool

(Defourny [3], Kane [4], Saigal [5]).

For these reasons, the selected approach for this paper is the continuous approach.

BEM will be used as analysis tool to avoid inherent numerical difficulties in FEM

regarding lack of accuracy for the response over the boundaries. Taking this into

account, the boundary formulation is the sensible choice for this development.

Thus, only the design velocity along the varied boundaries is required. This

represents a considerable saving compared to the domain approach, in which the

design velocity field over the entire domain needs to be specified, specially if 3-D

domains are considered, as it is done in this paper.

Concerning the procedure for evaluating the shape derivatives of the state

variables, although the formal adaptation of the adjoint technique is conceptually


Boundary Element Technology 205

straightforward, major computational difficulties arise when evaluating sensitivities

of functional at discrete boundary points, or sensitivities of boundary functional

involving spatial gradients of state variables. The problem stems from the fact that

the adjoint solution can not be expressed in terms of the boundary element

formulation, because they give rise to infinite integrals. This problem is solved for

two dimensional cases in references Mota Scares [6], Zhao [7] and Unzueta [1].

For three dimensional cases, the problem becomes more important and difficult.

Moreover, a longer computational time is required for the adjoint method, even

if the number of functionals is less than the number of design variables for 3-D

problems. These drawbacks lead to the choice of the direct technique for

evaluating the 3-D shape sensitivity.

NOTATION AND BASIC FORMULATION

In order to treat the surface functionals in a clear and easy way, it is convenient

to use certain concepts of Differential Geometry and the main relations between

them:

Notation:

(.),; partial derivative with respect to spatial cartesian coordinates

(.),;i derivative in the direction of the unitary vector \i

(.),„ partial derivative with respect to curvilinear surface coordinates

(.);« surface covariant derivative

g surface metric tensor

g*p covariant components of the metric tensor g

g^ contravariant components of the metric tensor g

b*p components of the second fundamental surface form

The contravariant components of g satisfy the following relation:

g^%u,Xjj, = Sy-n., % (U

A spatial vector over a surface can be decomposed as follows:

V; = V^tt; + V"X;«

where v. = V;n= and v" = g^ V: x,,,



If a surface vector field v is generated by a differentiate spatial vector field V,

then, by using the previous decomposition, the Gauss-Weingarten formula and

relation (1), the following expression is obtained:

where H is the surface mean curvature, defined by:

H = 1/2 b^ g^

The Green theorem for a continuous and continuously differentiate surface vector

field v specified on a regular surface S bounded by a piecewise smooth closed

curve L reads:

/a . r ry\ da — (p V LL dJ. — (p V dJL C\\

S ' J L " J L **

MATERIAL DERIVATIVE CONCEPT FOR SHAPE SENSITIVITY

The material derivative concept is the tool which relates the shape variation of a

domain with the resulting variations in vector fields, functions and functional

defined over it. The considered design variables are the ones which determine the

shape of a domain, and the shape variation can be treated as a deformation of a

continuous medium, thinking of each design variable as a "time like" parameter.

The velocity field defining the transformation direction from a point x in the

original or reference domain 0 to a point \ in the deformed domain ty is given

by the transformation x< = x 4- tV. All the quantities refer ing to the deformed

domain will be denoted by the subindex t.

Thus, if u is the solution of a boundary value problem, its variation with respect

to shape variation is expressed as the material or total derivative of u at t = 0,

= lim tc = ,/U) +Vu7 (4)c-o

being u*(xj the solution of the same boundary value problem on Q,, evaluated at

a point X( that moves with t, and



u'(x) = lim— = 4^ (x)c-o t at

named local or partial derivative, is the variation of u at a fixed point x

The material derivatives of some vector fields or real quantities are needed for

obtaining the desired sensitivities:

Let J be the jacobian matrix of the transformation and |J| its determinant, then:

^=^Zdt= DV(x) , with DV(x) = I-

c=o "j

and

/i i-i\ _ d\J\ = divV(x)

If n is the normal vector field to a surface S, its material derivative in terms of the

normal and tangential velocities can be obtained after a lengthy manipulation:

n\ = -g V x,,, 4- V-n.,. (6)

and

n\ = -g V x,,, - VA.. (7)

MATERIAL DERIVATIVE OF DOMAIN AND SURFACE FUNCTIONALS

The starting point of the sensitivity analysis is to obtain the expressions of the

derivatives of the functional defined as design criteria in an interactive design

process or cost function and constraints in an optimization process. Generally, the

functional arising in industrial applications involve the state variable and

frequently also its spatial gradient (e.g. electric fields in electrostatics and

magnetic induction in magnetostatics). In spite of their great importance in the

applications, functional involving the spatial gradient have not been considered

in the developments of continuous approaches to shape sensitivity until 1991, when

Unzueta [1,2] includes them for the two dimensional case.

The functional treated in this paper are the following:



where f is a real function and h a t dependent vector function defined over fy, and

I, = f ft(u,Vu,h) da, (9)•* &t

where S^ is a surface composed by regular surface sections R<, f is a C~ function

independently defined over each regular surface R^ for x belonging to a

neighborhood of R^ and h any vector function defined over a neighborhood of

each regular surface. A very frequent case is when h represents an extension to

a neighborhood of R< of a vector field associated to the geometry, e.g. normal and

tangential vector fields to the surface, u and Vu are in both cases the state variable

and its spatial gradient.

The material derivative of functional (8) is obtained applying (5), (4) and the

divergence theorem:

f dQ + f^ da (10)

where f is given by:

fi - df j _, df i ^ df ,/r - -ir-u' + — - u + — -— /]j /j j\du du ' dh ^^

Let us now consider functional (9) for each regular surface R. Its material

derivative can be written using relation (1) as:

[f + f . (6 .-n )] da (12)

In shape optimization problems, the velocity field over the edges is available, but

only the normal velocities are known over the surfaces (Longo [8]). Therefore, it

is convenient to rewrite the previous derivative involving only the normal

velocities over the surface. Taking (2) and (4) into account, it follows that



j = f [ft + y f + y*f 4- fvfg - 2fVnH] da (13)

where H is the surface mean curvature. Applying Green's theorem (3), the final

expression is obtained:

£^-2fVnH] da + <£ £Vp dl (14)

where 3R is the piecewise regular closed boundary of R and V^ = V°X = V/x is

the velocity vector component in the direction of the unitary vector /-t, which is

normal to 6R and tangent to R, pointed towards the outside of R.

Finally, the material derivative of (9) for the surface S is obtained, adding the

material derivatives (14) for each of the regular surfaces R:

1 da + Y, fgt <** (15)

The sum is extended to all the closed boundaries dR of the regular sections R of

the surface S. If S is closed, the sum can also be expressed as follows:

E /, (16)

In this case, the sum is extended over all the edges L of S and the signs ( + ) and

(-) refer to the quantities evaluated on each of the two regular surfaces intesecting

along L. By means of geometric considerations, expression (16) can be written in

terms of the normal velocities, provided that the intersection angle is different than

0 or TT.

SHAPE SENSITIVITY FOR POTENTIAL PROBLEMS

Most of the functions and fields appearing in the shape derivative of a domain or

surface functional ((10) or (15)) are known or easy to evaluate with the available

results, after having solved by means of BEM the boundary value problem. Only

the fields u' and u' ; are unknown, which states the need for an associated problem

to calculate them.



The direct method differentiates the continuum equations leading to a similar

boundary value problem as the original one, called associated problem or system

of equations. Following the boundary formulation, the unknown of this system is

if, but this new problem doesn't have any physical interpretation.

Poissons equation will be considered as the governing equation due to its

generality for potential problems:

u.ii = f in Q

f doesn't depend explicitly on the design variable, but only implicitly through the

dependency on the position x. It follows that f = 0 and thus, f = fjVj. Taking the

material derivative at both hands of the equation and using (4),

(u.ij)' = 0 in Q (17)

Considering a constant Dirichlet condition over S^, its tangential derivatives u „ are

zero over S^, so that

u' = - V.u.,, overS, (18)

The Neumann condition for the associated problem is obtained in a similar way

u'.n — - u A* - V,,(u J,,

and substituting (7) for n-/, after operating:

u\ = g"* V^u.p-V.n.,u,j n^ (19)

On an interface surface S,^ separating two materials Q" and 1? with material

characteristics (permitivities or permeabilities) k, and ky, differentiating the

potential continuity equation u* — u^ over S,^ and the normal flux continuity

equation k. (u J" = - k (u$ over S, , and following the same steps as for the

Dirichlet and Neumann conditions, the interface conditions for the associated

problem are:

ir" = u*' - (V, u „)" + (V.ujh over S,""* (20)

and

Kn)' = - *b KJ" + E* ( " n..".|> - VnW.lj"]) (21)a,b



where the sum is extended to both regions f? and & sharing the interface.

Finally, the float boundary conditions characterizing high permitivity screens are

considered, that is, constant but not fixed potential over a closed surface Sp (u«

= 0 a =1,2 over Sp) and

f u^ da = 0** S

Taking into account that (Zolesio [9])

(u,)' = u", - Vj, u j

and (X; J* = Vg Xj , it follows that u*« = 0 and in terms of the partial derivative:

u' + V,,u.n = constant over Sp (22)

The second float condition is a functional of type (9). Therefore, applying (15),

the corresponding equation for the associated problem is:

which can be transformed in

-'n <** - - E L <" (23)

by applying relation (2) to u ; and considering the nullity of the laplacian in this

material as well as the constant potential over its boundary.

Summarizing the obtained expressions (17)-(23), the system associated to the

direct method by means of the boundary formulation is as follows:

IT.-,; = 0 in Q

u' = - Va u. over S,

u'.n = g^ V«.« u^ - V. nj u.ij HJ over S,

(24)

u»' = u"' -f [V,, u.J

over S/"

k,(uJ = - k,(uJ' + E k(^ V u - V n- u n)



u' = cte - Vn u.n

over

/,«.<*» = -E L«^

where [f]J* = f - f and the sum is extended to the regions 1? and sharing the

interface S \

Comparing this system with the original one, it can be observed that only the right

hand side term is different. This implies a great computational time saving if the

decomposed matrix has been kept on disk. In this case, the time employed for the

sensitivity is negligible versus the analysis time. After solving the associated

system, the obtained fields u' and u\, are substituted into the shape derivative

expressions of the design criteria functionals, obtaining the desired sensitivities.

EXAMPLES

Two examples are used to show the accuracy and applicability to shape

optimization problems of the presented approach.

As a first example, a simple problem with analytical solution has been chosen, in

order to be able to compare the numerical values for the sensitivities with the

exact ones. Figure 1 shows the geometry of the model, which is a cylindrical ring

with constant Dirichlet boundary conditions over the inner (u = 100) and outer

(u = 0) surfaces and Neumann boundary conditions (q = 0) over the upper and

lower surfaces. The inner and outer radius denoted by Vj and V? are the design

variables. Three types of functionals are considered: volume functional (f,),

punctual functionals representing the modulus of the electric field at a point

located on the inner surface (r%) and on the outer surface (fj, and distributed

functionals for the mean deviance of the electric field modulus or tangential

electric field modulus from a reference value over a surface % - Q. Denoting by

Su, S|, S,, and S, the upper, lower, outer and inner surfaces of the solid, and by m%

the area of the surface S, functionals are defined as follows:


Boundary Element Technology

/• = la «*

213

r *. *So

J_ ,'* m J^ Js, 25

f$, fy and fg, fy are defined over surfaces S^ and S; respectively, analogous to t^,

tV Table 1 shows a comparison between the numerical values for the sensitivities

obtained by means of the described approach and the exact or analytical values.

The second example shows the application of the sensitivity approach to an

optimization problem, which consists of minimizing the electric field modulus over

the surface of a sphere at 100 Volts, placed inside a grounded box (figure 2). The

design variables are the position of the sphere centre inside the box and the radius

of the sphere. In figure 3 the evolution of cost function versus iterations can be

observed. Finally, figure 4 shows the initial design (boundary elements mesh),

where the sphere is located near a corner of the cube and the final or optimum

design. During the optimization process, the sphere moves to the centre of the box

and reduces its radius to the minimum allowed, as expected.

CONCLUSIONS

In this paper, a general formulation for calculating shape derivatives of punctual,

volume and surface functionals involving the spatial gradient of the state variable



has been presented. The approach allows to treat a wide variety of design criteria

functionals. The considered surfaces can be piecewise regular and the integrand

of the functional can be independently defined over each regular piece or section

of .the surface. There are no restrictions on the movements of the boundaries

(Dirichlet, Neumann, interfaces and float) and functionals can be applied over

whichever varied boundary.

The accuracy of the presented method stated in the previous examples leads to a

very quick convergence of the optimization algorithms. Moreover, it requires a

very short CPU time, negligible compared to the analysis. Thus, requirements for

consistency, accuracy, flexibility and practical applicability have been fulfiled,

providing this way a valuable tool to be used in an industrial environment.

REFERENCES

[1] Unzueta J., Schaeidt E., Longo A., Anza J.J. A General Adjoint Approachto Shape Design Sensitivity Analysis. Boundary Elements Technology VI, p.279 - 292, Computational Mechanics Publications, Elsevier Applied Sciece1991.

[2] Unzueta J., Schaeidt E., Longo A., Anza J.J. A General Related VariationalApproach to Shape Design Sensitivity Analysis. Optimization of StructuralSystems and Industrial Applications, p. 323 - 335, Computational MechanicsPublications, Elsevier Applied Science 1991.

[3] Defourny M. Optimization Techniques and Boundary Element Method. Proc.Boundary Elements X. Vol 1: Mathematical and Computational Aspects.Springer-Verlag, 1988.

[4] Kane J.H., Saigal S. Design Sensitivity Analysis of Solids Using BEM.Journal of Engineering Mechanics. Vol 114, p. 1703 - 1722, 1988.

[5] Saigal S., Borggaard J.T., Kane J.H. Boundary Element ImplicitDifferentiation Equations for Design Sensitivity of Axisymmetric Structures.International Journal for Solids and Structures. Vol 25, No 5, p. 527 - 538,1989.

[6] Mota Soares C.A., Choi K.K. Boundary Elements in Shape Optimal Designof Structures. The Optimum Shape: Automated Structural Design, Plenum,New York, p. 199 - 231, 1986.

[7] Zhao Z., Adey R.A. The Accuracy of the Variational Approach to ShapeDesign Sensitivity Analysis. Boundary Elements in Mechanical and ElectricalEngineering, Springer Verlag,1990.

[8] Zolesio J.P. The Material Derivative (or Speed) Method for ShapeOptimization. Optimization of Distributed Parameter Structures, Sijthoff &Noorhoff, Alphen aan den Rijn, Netherlands, p. 1089 - 1153,1981.

[9] Longo A., Unzueta J., Schaeidt E., Alvarez A., Anza J.J. A General Related



Variational Approach to Shape Optimum Design. Advances in EngineeringSoftware and Workstations. Vol 16, No.2, p. 135 - 142. Elsevier AppliedScience, 1993.

fl

*z

fa

t*

**

f,

f.

^

fa

VAR 1

VAR 2

VAR 1

VAR 2

VAR 1

VAR 2

VAR i

VAR 2

VAR i

VAR 2

VAR 1

VAR 2

VAR i

VAR 2

VAR i

VAR 2

VAR i

NUMERICALVALUES-62.8312

125.663

0.0265543

-0.0417338

0.0415587

-0.0347904

-0.0818417

0.0719

-0.185805

0.161272

-0.140805

0.119583

0.000000

0.000000

A:V,-

0

-0

0

-

-

-0

0

-0

0

0

0.031512 | 0

-0.0513353 -0

0.000000 0

\ALYTICALauEs62.8318

125.664

.025546

.041627

.041627

0.03524

0.08175

0.07165

.184818

.161186

.140839

0.11923

.000000

.000000

.031505

.051336

.000000VAR 2 0.000000 0.000000 I

Table 1.- Example 1: Numerical versus analytical sensitivity values

Figure 1.- Example 1: Geometry and design variables.



Figure 2.- Example 2:Geometry and design variables

NUMBER Of ITERATION

Figure 3.- Example 2: Cost functionversus iterations

Figure 4.- Example 2: Boundary Elements mesh. Initial and final design.


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