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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:
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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
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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,,,
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206 Boundary Element Technology
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
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Boundary Element Technology 207
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:
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208 Boundary Element Technology
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
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Boundary Element Technology 209
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.
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210 Boundary Element Technology
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
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Boundary Element Technology 211
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)
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212 Boundary Element Technology
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:
Transactions on Modelling and Simulation vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X
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
Transactions on Modelling and Simulation vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X
214 Boundary Element Technology
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
Transactions on Modelling and Simulation vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X
Boundary Element Technology 215
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.
Transactions on Modelling and Simulation vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X
216 Boundary Element Technology
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.
Transactions on Modelling and Simulation vol 3, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X
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