On Numerical Results for Shape Optimization

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ON NUMERICAL RESULTS FOR SHAPE OPTIMIZATION C. E. Pedreira* and R. B. Vinter** *f)('/)(frftllllt'lIio tit' ElIgt'll//(/IHI L/t'trim. j'ollil/frill L'lIh'('nitilldl' (.'aftilim do Rio dt' jllllrilll. RIIII ,\[111'1/111\ dl' Stio I'irt'llil' . 225-22-153 Ri(J tit' jtllll'iro. Bm:iI "" ''E/nirim/ 1:'lIgill('nillg /)1'/Jllrillll' lIi. fill/mill/ CO//I'gl'. ,-(Jilt/Oil SI\,7 2Al, ('/\

Abastract. In this paper we are concerned with numerical solutions for shape optlmlzation problems associated with elliptic partial differential equations. Our main motivation is a thermal problem where one wants to find an optimal boundary such that the heat loss is minimized for a fixed volume of insulation material. Keywords. Optimal shape, elliptic partial

1. INTRODUCTION Problems of shape optimization are attracting considerable attention at the present time, both on account of its theoretical interest and its practical applications. This paper is directed at the numerical implementation of a penalty approach to optimal shape problems associated with an elliptic partial differential equation. Much of the earlier work on shape optimization prob lems associated with elliptic partial differential equation involves a finite

dimensional

parame-

trization of the unknown boundary (e.?,. the parameters are taken to be the nodes of a piecewise linear domain), by contrast we address an infinite dimension minimization problem, in which

no

such

parametrization is employed. Hou et al [1] have proposed a method for solving the problem of shape optimal design for multiplyconnected elastic bars in torsion using the ideas of the "speed method" introduced by Zolesio [2]. A finite dimensional parametrization of the optimal shape is obtained and the derivatives of the cost and constraints function are calculated with respect to the design parameters. They suggested the application of some non linear programming algorithm to optimize the shape interactively. Some nu merical examples usin£ a recursive quadratic pro= gramming algorithm proposed in [3] and [4] are giv en. Begisand Glowinski [5] and Delfour et. al [61

consider design of optimal shape problems widl part of the boundary fixed and part defined as a graph. Begis and Glowinski analise a problem of diffusion in 2 dimensions. Results concerning existence and uniqueness of the optimal solution are given. Quad rilateral finite elements, and gradient methods are employed. Delfour et. al have treated a thermal dif fuser problem with non differentiable constraints~ Their approach uses the techniqu e proposed by Zolesio [2] to derive the gradient of the cost and the minimizing parameters and apply penalty methods to obtain numerical results. Zolesio [7] has applied domain variational formulation to approach a free boundary problem arising from plasma physics. Tepper [8] studied the existence, uniqueness as well as some conditions on the optimal boundary. In the important contribution [9], Acker derived inequalities comparing the rates of heat flow across regions with different boundaries. These inequal-

differential equations, ;inite elements.

volving the Laplace equation. In his two subsequent papers [10] and [11] he analyses free boundary optimization problems concerning the minimization of capacitance, and on [12] he address a minimum flow of current in the context of electrostatics. Gonzalez de Paz [13] applied penalty methods and techniques of convex analysis to a problem where the external boundary is fixed and one seeks the optimal internal one. Some comprehensive surveys are available on the wide subject of optimal structural desif',n. Cea [14] gives a good review on numerical metho.ds of shape optimal design usin?, some models give, in [15]. In H~ug [16] one can find a exten~ive , revie~ of t~e l~terature

Our contribution is to provid e new algorithms for an important class of shape optimization problems and to test their performance. The algorithm employ a reduction of the orieinal problems to ~inimization problems over function spaces, and a penalty technique. This reduction technique has'leen used by Gonzales de Paz in an analytic techh1que in a special case. He examine its computational implications for the first time. 2. TP.E PROBLE11

Our problem is formulated as follows: Find the boundary an i (or an ) for an (or an i) eiven, s uch that the cost function (heat loss) au an

d,

1S minimized, where u is the temperature distribu-

tion. The state equation for this problem is given by:

tu = 0

in

and the boundary conditions are: u

ities are the basis of an existence and uniqueness

theory for free boundary optimization problems in-

on structural opt1m1zat1on. P1ronneau s

book [17] is an excellent application-oriented study of optimal shape design for systems governed by elliptic partial differential equations. Finally, Banichuk' s book [17] is a very good indroduction to the subject.

u

o

on on

474

C. E. I'cdrc ira and R. B. VinICI'

v Di We label the problem with all fixed ' The internal problem' and the one with ani fixed 'The external problem'. It ,,'ould appear that the external problem, in which 've seek the optimal disposition of insulator around a fixed body to minimize heat loss, has more widespread applications. However the free external boundary in the external problem creates certain difficulties, and the external problem is more difficult to analyse and solve numerically.

The problem is to minimize

r J

aD i

au an

are:

u(x) = 1

g(x)u 2 dx

J

g(X)

dx

Do-Di

q

u

0

in

D - 0·1

U

q

on

3Di

on

g(x) dx

V

where B is a suitably large penalty constant. This last problem is posed over a large domain (the previous problem considered in effect only functions g which where characteristic functions), but it can be shown [18] that in the limit, i.e. when B ~ 00 the infimum is the same, and a minimizing g is actually the characteristic function of a mi-

and

m

f IVuI 2 +BU2

subject to the constraints

L1U

measure (Do -

g(x)dx =V}

We follow a penalty approach, and replace this pro~ lem by a partly unconstrained one. The new problem becomes

f

in

f

Do- Di

0 0 -0i

a.e

where

0,

g E G: = {g E Loo (Do -m: g(x) E [0,1]

Subject to

= 0

and

ani,

Minimize JB(u,g)

dS

n

over sets 0 , Di D Do and u E Hl(Do-Di), where au/an denotes the gradient in the direction of the outward normal, and n is the unit outward normal.

u

on

Do-Di

The external problem Let Di and Do be bounded open ~ets such that Di 0 0 , Let V be a positive number such that rreas(Q{>
J(u,m

It can be proved [ 18] that the two problems are equivalent. It can also be proved that the problem can be replaced by a equivalent one where one performs the minimization over functions g (and u) instead of sets no. In this case the constraints

Do - n

V

We remark that 0 0 is usually an artificial set, say a large cube or ball, chosen to contain the Insulated body and insulation, and one which is not ex pected to affect the shape of the optimal insula~ tion. We ~ust introduce the set DQ in order to li~ It attentIon to spaces of functIons on compact sets, and also for purpose of numerical implement~ tion.

nimizing set.

3. THE ALGORITHM AND

~JUMERICAL

RESULTS

In this section we shall describe the algorithm which we propose for solving the penalty problems. We use a finite element technique in order to discretize the domain and solve these problems.

The internal problem

(i)

Let D be a bounded open set, and V be a positive number such that meas D > V. The problem is to mi nimize

Construct a triagular (other geometricscould be used) mesh over the domain Do-Di ·

(ii)

Choose a initial domain

au an n dS over sets 0i

o

and u

E

H~ (m.

(iii) Calculate the initial go (x) such that if

go (x) go(x)

Subject to L1u

o

in

0 - 0i

(iv)

u

q

in

Di

(v)

meas(Q - Di)

J

=

f

0

u(x)

q

a.e. x E ani

u(x)

0

a.e. x

E

no - D

and

measurable

sets

function

_ nO

a il = V

and take Di = {x : u(x) > a i } (vi)

Take gi(x) to be

gi (x)

over elements u E Hl (Qo-Di) D Do' subject to

no

Di'

Calculate the level ai such that

if

I

Do- Di

E

indicator

r.o

Calculate the temp e rature distribution.

g. (x)

IVul2 dx

x

such that 00

otherwise

meas(x: u(x) >

V

We replace these problems by variational formulations: for the external problem, we minimize the functional

rP

o

u(x)

otherwise

(vii) Go to (iv). It is easy to see that in the algorithm we alternately minimize the cost function over u and g.

475

Numerical Results for Shape Optimization REFERENCES

3.1 The Numerical Results

In this section we present the numerical results produced by our algorithm. ~e provide examples for both, the internal and external problems.

[1)

Hou, J.W.; Haug, E.J. and Benedict, C.. "Shape Optimization of Elastic Bars in Torsion". J. Mech. Trans. (1984).

Each figure shows four shapesillustratinr the evo lution in one, three, five and ten iterations. For the external problem examples each figure conrnins also the fixed internal shape, for the internal problem examples the fixed external boundary is also shown.

(2)

Zolesio, J.P .. "The Material Derivative (or Speed) Hethod for Shape Optimization". Optimi zation of Distributed Parameter Structures-;Bol. n, Ed. Haug, E.J. and Cea, J., 1089-1151

[3)

Choi et al. "Pshemichny's Linearization Method for Mechanical System Optimization". Trans.ASME J. Mech. Design (1984).

(4)

Dems, K. "Multiparameter Shape Optimization of Elastic Bars in Torsion". Int. Journal for Num. Hath. In. Eng., Vol. 15, 1517-1539 (1980).

[5)

Begis, D. and Glowinski, R.R .. "Application de la Methode des Elements Finis a 1 'approximation d'un probleme de Domain Optimal". Methode de Resolution des Problemes Approaches". Applied and Op. Vol. 2, 2 (1975).

(6)

Delfour, M.; Payre, G. and Zolesio, J.P .. "De sign of a Mass-Optimal Thermal Diffuser". Op= timization of Distributed Parameter Structures Ed. Haug, E. J. and Cea, J. (1981).

(7)

Zolesio, J.P .. "Domaine Variational Formulation to Free Boundary Problems". Optimization of Distributed Parameter Structures (1981).

[8)

Tepper, D.. "Free Boundary Problem". SIAM Hath. Anal. Vol. 5, 5 (1974).

(9)

Acker, A.. "Heat Flow Inequalities with Applications to Heat Flow Optimization Problems'. SIAM J. Math. Anal. Vol. 8, 4 (1977).

After ten iterations the shapes we obtain are very similar to what we expect to be the optimal ones. Very small changes are observed after this point. Our first numerical result is shown in figure 2. This is an example of the internal problem. The external fixed boundary is a circle centered in the point with coordinates (7,7) and radius equal to 6. We choose as initial internal shape (iteration 1) a square. The volume of insulation material i.e the area between the internal and external boundaries, is fixed to be equal to 60. And the penalty constant is taken to be equal to 15. This is an important example because the solution is known to be a circle, and it thereby provides a good test of our algorithm. We see that the algorithm gives a shape very close to a circle after ten iterations. Figure 3 shows an example of the internal probler.. In this case the external shape is a square and the initial internal one is chosen to be a circle. Figure 4 is an example of the external ·problem. Here the fixed internal boundary is a circle centered in (7,7) with radius 5. The initial boundar~ i.e at the first iteration is taken to be a square. The volume of insulation material is fixed equal to 60. This is important example since the solution is known in advance to be a circle centered in (7,7). The penalty constant was chosen to be equal to 15.

(1981).

[10] Acker, A.. !lA Free Boundary Optimization Prob lem". SIAM J. Hath. Anal. Vol. 9, , (1978). [11) Acker, A.. "A Free Boundary Optimization Prob lem 11". SIAM J. !1ath. Anal. Vol. 11, 1 (1980). (12) Acker, A.. "An External Problem Involving Cur

rent Flow Though Distributed Resistence". SI&'1 J. Math. Anal. Vol. 12,2 (1981). (13) Gonzalez De Paz, R.B .. "Sur un Probleme

4. FINAL REMARKS Although we have directed our research at the prob lem of determining optimal shapes in order to mi~ nimize the heat loss, given a fixed geometry (internal or external), the methods we develop can be applied to a variety of other shape optimization problems, e.g design of elastic bars in torsion. We adopted a variational formulation of the heat loss minimization problem, involving a penalty term to take account of the volume constraint on the area enclosed by the free boundary. One attractive feature of the formulation we employed is that the minimization is conducted over a linear function space rather than the mush less numerically tractable space of subsets of Rn. The minimizing shape emerges in the limit, as the sup port of the minimizin g function. We propose an al gorithm for solving both, the internal and the ex ternal, penalty problems. We use the finite ele= ment technique in order to obtain the numericru so lution for our problem over a discretized domain~ \-le present the numerical results, i.e the shapes obtained by our algorithm for both the internal and external problems. The results, appear to be very satisfact ory. Eve n better results can be obtained if one refines the finite element mesh.

J.

d'Op

timisation de Domaine". Numeric. Ftinct. Ana 1-:and Opt. 5, 22 (1982). (14) Cea, J .. "Numerical Methods of Shape

Optimal Design". Optimization of Distributed Parameter Structures (1981). \

[15) Cea, J .. "Problems of Shape Opti~

Optimization of Distributed Parameter tures (1981).

Design". Struc-

[16) Pironneau, 0 .. "Optimal Shape Design for liptic Systems". Springer-Verlag (1981).

El-

[17) Banichuk, N. V.. "Prob lems and 11ethods of

OpPress

timal Structural Design". Ed.

Plenum

(1983) . [18) Pedreira, C.E. "Theory of Shape

Optimization with Applications to Heat Loss Minimization" Ph.D. Thesis, Imperial College, University of London, February 1987.

476

C. E. Pcdreira and R. B. Vinle r

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FIOURE 2

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:'>Jumcrical Results for Shape

()ptimil~lli()1l

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