NASA Cot_raetor lgtport 198349

ICASE Report No. 96.44

! A

A PDE SENSITIVITY EQUATION' METHOD

FOR OPTIMAL AERODYNAMIC DESIGN

Jeff Borggaard

John Burns

_rtc Q_tjxt_r _'n_' 4

NASA Contract No. NAS1.19480

June 1906

Inatituta for Computer Applications in Science and Engineering

NASA Langley Research Center

Hampton, VA 23681.0001

Operated by Unive_ities Space Research Association

National Aeronautics and

Space Adminiatration

Langlty Research Center

Hampton, $_rginia 23681-0001

t.,..,N= _ba,i !

IXswomm _

19960820 063
A PDE Sensitivity Equation Method for Optimal

Aerodynamic Design

Jd Bor_aard 8nd John Burns"

In1_rdieciplinary Oent_r for Applied Mathemstics

Virginia Tech

Blscksburg, VA 24061-0531

Abata'act

The _ d Sr_l_t _ optim_._in dSodthm, in ia_ d_ip b w_l es_blmhed

a practical approach to MrodyHmJc design. A typic_ pzocedum uses • 6mml_tiom w.heme to

,_a_ the objec_*e _ {lrom the _e st_) and it. uadlest, _es _ this

isfmm•_im to n opti_sl_t_ _. Once t_ sim_ _e_ (C_D |ow s¢l_)

been selected _,d uJed to provide epprazinste _ ,,wih_t/_,., film u_ sev_ld pmeible

• eproJc_n to the problem oLcompu_In_ puJieJt.. One popu_ method i. to d_matiLte

the mnul_tJon 8dteme md ¢_al_te dedp eadtivitJeJ that are then used to obtsJm gradi_ts.

Altb_ih tlm l_,_.b_ _ _ li_ _msCala _ _ olp_ _blm_, on®

¢urpm. _ .u,itlvlcil la o_d_ ¢oco.,.p,,¢, the _ 0mdC/viCy.

In this paper, we pnwnt an Llte_stt_ _ _ t_e PDE ea_l_y equtioa to

d_v_ st_ost_, fox comp_ths$ _ts. This approsck ku th_ _ t]u_ _ een_

_ivl_i_ seed sot be comp_d. Moreover, w]_ it is pomibb to we tke C_D sek_e far bo_h

the forward problem and the sasltl_y eq_, _ _ _ comptta_ioisI sdvmt_ps. An

uppuzmt diMdvamqe of this •pprouch is tkag it does not alwa35 p_odm:e cowl•teat derLva.

five.. Howmm_, _m • pmp_ comblu_ o( _•tl_a s_a, one can s}_w _Vmptofl¢

¢ond_¢emcWuncle= mad, _m, which b _ .dlci_S to patas_ _smr_sce of tke

optimd de._ dSwitkm. In part_ular, we .how tkat wkas uymMotlca_ ,xm.iJum¢ .ct.m.

are comb_ed with• tru_-zqkm op_ _o_Itkm, the res_d_l_s opt_m_ldm_a method

wn_.r|m. We denote _ _pprosch as tile ..n_i_iei_ qv,,_on m_/io_.

The Im_flvtty equation m_bod is presented,¢ou_s'ilcs_e _su_ Im_lltVcu•ud ekeapproach

is fllutr&t_l oa two o574d_/desert pro_ lavolvln 8 _.

'TSb rmmads .l_ mSlppmsedia Ipl'¢ by the Air Fm'_ _ _ _ l_sur_h _ _ F4M_0.93-J.

00'70 mid F4_'J0.M-I.0'/S0, eld by the NMlould Awoum_¢o mid Spice Admlldsttsthm under NASA Coutnug No,

WAJI-1_480 _ die e0eNd au_thorwse• vbitinl scJen_J_ s¢ the Institute foe _mmput_ AppUe._lelu ia Science

_8 (_CA_B), _ASA _AmS_ _ C4att4s',Hampton,VA 'J_881-0001.
1 Introduction

Optinnfl design problems consist of selecting design parameters for s system in order

to optimize a given design objective, usually constrained to satisfy a partial differential

equation. In many of these problems, design parameters describe the shape of an

object. Examples of these shape optimization problems include drag reduction [21],

[22], weight minimization [14], optimal sensor/actuator placement [G], airfoil design

[16], [17], [18], [19] sad the design of wind tunnel elements [15].

Traditionally, approximate wlutions of these problems are found by "cut and try"

methods, combining & designer's engineering experience with repeated experimental

testing. This is often expensive, motivating computational methods which compute

the optima] design directly. These methods require defining an objective functicm

and an @propriat¢ PDE model of the states of the system. A comparison of several

optimal desiOn methods may be found in [13].

M,my popular approeches couple a, gradient-based optimization algorithm with

function evaluations provided by a proven simulation scheme. One of the disadvantages of these approaches is the expense of computing the gradient. Using finite

differences is often too costly, even if appropriate step sizes can be found and the simulation scheme can take advantage of "nearby" solutions (as is the case with iterative

solvers for nonlinear equations).

Two strategies for alleviating the computational expense of gradient evaluations

are adjoint variables [17] and design sensitivities [14]. Adjoint methods are advantageous when either the problem is self-adjoint or there are a large number of design

parameters. However, when there are relatively few deslgn parameters, using design

sensitivities, quantities which describe the influence of the design parameters on the

states of the system, is an attractive alternative. In addition to efficient gradient

computations, they can be used in some problems to construct an effective update of

the approximate Hessian for quasi-Newton optimization algorithms, e.g. [I0].

A standard approach often used to compute the design sensitivities is based on

(imp]icit_) differentiating the simulation scheme (for the states) with respect to the

design variables. Using the chain rule to carry out this calculation, re,tits in in

e_cient numerical scheme for the sensltivities. The ellJciency arises from reusing

many of the quantities computed in the simulation scheme. In fact, the "inversion"

of the system matrix (i.e. the matrix fsctorization) can often be reused.
.............

A disadvantage of this approach is that for shape optimization problems, the discretization is parameter dependent. Thus, derivatives of the discretization (mesh

sensitivities) are required for each shape parameter. Depending on the simulation

scheme used for the states, determining the discretlzation can require the solution of

a partial ditrerential equation (as is the cue for finite diiFerence solutions of viscous

flow problems [26]). Tkis reqmm a strat_y for computing the M smsitivities

[20], or for c.mnputing an approximation to them [24], [25].

Another approach to/_nding design sensitivities relies on approximating the partial di_erential equation, known as the sensitivity equation. This equa,tion is obtained

by implicitly differentiating the (infinite dimemioz.ai) state equation with respect to

each design parameter. As shown in [2], using the same numerical scheme to approxinutte the se_itivlty equation which is used to approximate the states, leads to

an efficient scheme with sir_h_r computational advantages as the design sensitivity

approach described above. Mhe_more, since the disc:reti=ation is applied directly

to the sensitivity equations, no se_sltivlty of the mesh is required. The sensitivity

equation is always linear in the design sensitivity, even if the state equation is nonlinear. Since there is no reqnirement to use the same numerical scheme, it is possible

to gain additional computational savings by using a scheme which takes advantage

of the linearity in the sensitivity equations.

An apparent disadvantage of this approach is that it does not compute ¢onai_ent

der/eativ_. In other words, the scmitivity equation approach does not capture the

sensitivity of the truncation errors in the scheme. Thus, there is a concern that

providing ta optimization algorithm with an approximation of the gradient of the

infinite dimensional objective _mction instead of the gradient of the approximate

objective function woeld cause the algorithm to fail. One might expect, however,

that if the gradi_mts are "close enough" to the true gradients, then the optimization

algorithm should still converge. We show that this convergence can be established if

one combines compatible ,imtttstion and optimizatlon schemes.

'_ust-region optimization algoritluns are consgructed to be globally convergent by

minimizing a model of the objective function in a region where the mode] is "trusted".

Th_ leads to robust algorithms capable of handling inaccuracies in the model. In fact,

convergence results have been given for these algorithms when the model is based on

imw.cttrate gradient information [7], [8]. The results hold provided the gr_:ente satidy

a given error condition. Therefore, it is natural to consider an optimal design method
whichcouplesa trust-regionoptimization algorithm with gradients computed using

the sensitivity equation approach. We denote this combined sendtivity/trust-re#on

algorithm by the ,ens/tiv/ty e_at/on method (SEM).

In this work, we present sad mmlyze the sensitivity equation method. The method

can be applied to a wide class of optimal design problems, including those mentioned

above, however, we focus on the particular example of shape optimization of Eule¢

flows in order to illustrate the method. In the next section, we describe two design

problems. In Section 3, we present the sensitivity equation method including the

trust-region algorithm sad the use of the sensitivity equation to find the design semitivities. Furthermore, we compare various numerical approximations of the sensitivity

equation w':h approaches based on the discretized equstiom. Section 4 discusses a

number of convergence issues sad includes a convergence theore_ for the sensitivity

equation method. In Section 5, we use a one dimemsions] duct design problem to describe the implementation of the sensitivity equation approach. Fimdly, we describe

the implementation and perform shape optimization for a two dimensional forebody

simulator design problem where the steady-state Euler equations are used to model

the state variables.

2 Illustrative Examples

We p_sent two optimal design problems below which are used to illustrate the sensitivity equation method. These problems consist of determining shape parameters

which produce a solution to the Eu]er equations that matches a desired flow "u

closely as po.sible." The first problem is motiwted by the design of a wind tunnel

element in order to produce a desired flow in the test section. We study s two dimenfional analogue of this problem. The second problem consists of prescribing the

crosg-mctional are_ of a one dimensiorusl duct to produce _ duct flow which matches

a desired flow profile. This problem wu used by Frank and Shubiu [13] in their study

of optimal design.

3.1 ¥orebody Simulator Design Problem

The Arnold Engineering Development Center (AEDC) operates a free-jet test facility

which is used for full-scale testin$ of commercial sad military aircrdt engines. En#nes

are evaluated for performance and safety under various free Eight conditions. While
c.,

Figtwe I: Fo_body Simulltor Design Problem

this facility is large enough to house engines, it is not large enough to house an entire

aircraft forebody. Thus, the effect of the aircraft forebody on the engine inlet flow

profile must be simulated. One wLy of doing _his is to replace the actual forelz)dy by

a sr._ai]er object, called a forebod v simulator (FBS). The use of the FBS is illustrated

in Figure I. The FBS design prob]ern is to specify the shape of this FBS so that

it produces an en_ue inlet flow prof_e which is u close to some desired profile as

possible [15]. The desired prone can be determined by performing either • wind

tunnel simulation or • computational simulation of a mode] conflgurztion resembling

• test condition of the aircraft engine.

In order to demonstrate the applicability of the $EM, we consider a two dimensiond analogue of this problem. This problem, depicted in Figure 2, is to find the

shzpe of the curve F, which produces an out,tow that maCchee the outflow Seneratecl

by the origiz_l (longer) _orebody as closely u poe_ble. The i_ow, Q (conslst]ns of the

density p, the momentum p_ + pv_ and the sum of the internal _nd kinetic energy

E) is modeled using the steady state Euler equations,
FIilil/I/i]ill////I)_

where

, , lu

Inflow Outflow

,, IlL IL _e

Jr j" _P_ _t" _ _ _P_ i Jr d_ °m°m°mm°ma'ao°°°°°°°°°°o°qll

_lilliil /11_'[[ l

.... -- W

F_11_iJr1111_ ....... _.r a

Fisure 2: 2D Fombody Simulator Design Problem

aG

0 FO p

Pf.Q+ P o-_Q+ 0 _d Q= _ (2)

0 ' P pv

Pu Pv E

The procure P is related to the elements of Q by

(.,+,,,1],

where "? is the ratio of specific heats (7 " 1.4 for air). Given a forebody simulator

shape P. the flow Q(I') is d_aed by miring the Eulcr equation, (1) in the test

cell domain fl(P) subject to the boundary conditions (for mzpenmnic flow):

Q m Q,_ at the test cell inflow, (4)

(u,v)._ = 0 ,_d (5)

_) ((u,v). _) - 0 at the wa]]s (no flow penetration), (6) an

S
where ,_ and _ are the normal and tangential vectors at the boundary, respectively.

The set of admissible forebody _imulator shapes is

.4 = _r _ Cl(a,b)[ r(.) = r., rib ) = rb and r(.) > r., v. _ (a,b)}. (7)

A statememt of the design problem is given below.

Problem 2.1 (Forebody Simulator Design) Let _ be a d_ired flow at the oatjYow(en_ne _),

s= {(_,u)l_=b, rb _<__<_}. (s)

Define the objective function

_(r) ilQ(r)-oi: ,5, (9>

where Q(F) represents the solution of (I) with boundary conditions (4)-(6) in the teat

_i n(r). ne /orebodysimulator design problem i, to flnd F. E A each that

j(r.) s j(r) Jor_z r e a. (10)

Closed form solutions to (1) with (4)-(6) are a\-tilable only for special domains.

Therefore, we consider approximate solutions of (1) and hence the approximation of

Problem 2.1.

The disoretization is performed by selecting mesh points in the flo_ domain fl(F)

where the flow variables will be approximated. It b desirable to select this mesh

in such a way that the points are more dense in regions where flow gradients are

expected to be "large" (in order to h&ve more sr.c_ste di_erenclng) and more coarse

in regions where the flow is nearly constant (in orde_ to save computer time). Other

issues sttch as selecting points with no sharp changes in density and with sufficient

resolution to treat the boundary conditions, make the mesh generation a science in

and of itself (see e.g. [26]).

Another constraint on the discretization, to simplify the implementation of a finite

difference scheme, is to use a resular mesh, i.e. a mesh where there exists a bijective

map taking the mesh points to a lattice of points in the computational space. For

example, suppose that M is a C 1 mxppiug,

M" (.,y) --. (_,_), (11)

then derivatives in the physical space are easily approximated on the lattice using

the chain rule. Denoting the Jtcobian of the mapping by Y,u, the transformed Euler
equationsbecome,

where

0 0

= u +P3 ' , = vO+ I 03)

U V

Oz O_ "

A standard finitedi_erence scheme, developed by Beam and Warming [I]is used

to approximate the tr_dormed equations.The scheme introducesa time v_risble,t

as a means of iteratinSan initialguess for the solution,to a solutionof the steady

state equations. Second an_ fourth order artificialdiuipation terms are added for

stability, represented by _O) and _(4) respectively. This scheme is implemented

in the PARC2D code [9]. Several implementation issues are discussed briefly below

which are referred to in later sections. Re_ers interested in more code details or the

actual expressions used for _O) and _(4), should consult [9].

The difference scheme produces a system of equations for the update of the flow

variables, A0_. Thus, the solution _t the nth iteration, 0" is determined from

0" = 0 "-_ + _0 "-_. (l_)

The system matrix produced by the approximation above is quite large due to differencing in each direction. However, this problem is circumvented using _n appro_mate

f4ctorization into a product of two matricea, each corresponding to differencing in one

of the i_ttice directions. The finL! system has the form:

where

A_ OF O_

(16)

07)
The subscripted terms 6, V and A represent the centrxl, backward and forward

difference operators, respectively, in the latti_ direction indicated by the subscfivt.

The conver_9_d solution is denoted by ON(z,y) m _* (A4(z,V)).

We introduce Bezier curves to parsmeterize the forebody simulator. Bezier polynomials pouem t_veral nice properties when used in approxirrmtions. The most iraportant for the examples presented here are the eonvez hid[ amd endpoint interpoiafion

properties (see e.g. F_rin [12]). For this problem, we consider a set of two psr_,ter,

s = {r _ e'[0,1]I ro)= (r.(_),r,(,; q)),r,(a;q) _>r., a_ [0,1},q_ ee} Os)

where

and

r.(a) = aBo_(a) + O.eB,_(s) + o.sa_,,(a) + bB_(a),

r,o;O = r.Bo,,O) + q_e_(_) + q_B2j(a) + rbB_Ca),

(lO)

(20)

B_"(z)- ( ri ) x_O- z)'-'" (21)

We also a_sume a -- 0.5 and b ffi 1.0. We can now introduce the approximate forebody

simulator design problem.

Problem 2._t (Approximate Forebody Simulator Design) Let {_,}affil be de.

aired flow measurements at $. We assume that the data measurements are given

at the quadrature points, othertaise interpolation must be uaed. Define the objective

function

#

jr(r) = E _ 1_'(,,; r)-0,_, (_)

tmt

u,heee Qlv(zi; r) rcproenta He approzimate solution to (I) in the domain fl(r) at the

quadrature point z_.

r. E B such that

The appro_raate forebedy simulator deaign problem is to _nd

jf(r.) < j,_(r) !o, art r e e. (23)

Let

_l= {(q',q') e IR'I r(.;q',q ') e e}, (24)

then the problem can be equiv_hmtly suited as finding (q._, q,_) $ _ such that

N t

if; (q., q.) __ _ (qt, q_) .{or sli (q', q') e _. (2,5)
2.2 Duct Design Problem

This problem consists of desisting the crou-sectional are8 of s one-dimensional duct

such that, under specitled inlet and outlet conditions, it produces a flow which is as

close to a desired transmit flow ms poaible. The governing conservation laws (steady

state continuity, momentum and energy equations) can be reduced to a single twopoint ]xmnda_'y value problem (BVP) for the velocity. It was shown iu [13] that the

velocity u, is the solution of

_/(_) +9(.,A) = 0,

_(0) = u,., and u(l) = u.,,

(28)

where um and Uo_t are the velocities at the inlet and outlet of the duct, A is the

cross-sectional are_ of the duct,

f(u)=U+u g(u,A)=_ _ {u and q 7-1+ x' (27)

where _ and -f are flow constants taken to be 1.14 and 1.4, respectively. The RankineHugoniot condition yields the speed of sound as u, = V_. Unique solutions of this

BVP are guaranteed for monotone are_ functions, therefore, cro_e-sectional areas, A,

are restricted to

.A= {AECI(O, 1)IA(O)f A_,A(1)f A,,and _A(z)>O, VzE(0,1)) (28)

for fixed inlet and outlet areas of At, and Aunt. We now describe the optimal design

problem.

Problem 2.3 (Duct Design) Let _(.) _ L2(0,1) be a desired transonic flow profile

for the duct and define the objective _nction by

YCA)=/o' [t,(x;A)- _(x)]' d.z (29)

where u(.; A) is the aolution to (t_6) corresponding to A. The optimal design problem

b to find an A. E A such that

Y(A.) _<Y(A) /or ,at A ¢ .4. (30)

While the BVP has a clmed form solution [13], we consider approximstiom of

(26) and consequently of Problem 2.3 in order to study the more general case. We
begin by discretizingthe duct length into N ce.l]s (of length h = _) with centers,

zj - (j - _)h,j - I,... ,N and define u_v to be the average velocity in the jth cell,

i.e.

1/"qu(z;A)d, .

uf(A) = _ s.,-t

A system of nonlinear equations for u/v (A) -- { u_v(A)_,_}__,

(26) over each cell,

h

where it was assumed

tion to u_" is found by

cell center values fj --

type _ methods are the Enquist-Osher scheme

(31)

can be found by integrating

f (U (zJ--_;A))+9(uZ_(A),A(z:_))-O, j ffi I,...,N,

(s2)

that 2. o A _'/A was nearly constant over each ceil. An _pproximaf(u_) and fj+t - f(u_l). Two standard first order "Godunov

{ .fi,, "7."_+1< ".;

f u zj + _ FJ+a/2 ,o: - {33) /(_.) _' < _. < _+,;

and the artificial viscosity scheme

f a:_ "t" _" Fi+,/a "- _ (fi+, "i" It- o(uj+l- uj)), (34)

where a has been selected as 1 for this study. These approximations were used in

[13], but are included above for completeness.

We turn now to the approximation of the cross-sectional area A. The space .4

is replaced by a subset of Bezier quadratic polynomials. The properties of Bezier

polynomials allow us to easily impose both the monotonicity requirement and the

matching of inflow and outflow cross-sectional areas. Consider

e = {A e c'(o, 1)IA(z)= A_,,V..2(z) + q.e,,2(:z) + a..,e2,2(x);

z E (0, 1), q E [A_,,,A_,]}, (3,5)

where B_.v is defined in (21). Thus, B is a one parameter set of curves in .4. We

restrict our optimization problem to this set B.

Our final step in the approximation of Problem 2.3 is replaclng the integral by a

quadrature rule, with the set of quadrature weights and pointa {(c_, zi)}ie=l. We now

state the approximate design problem.

I0
Problem 2.4 (Approximate Duet Design) Let {ui),=l • 8 represent data for a deaired transonic.flow profile in the duet. We a_ume that the data and the approxim,ate

solution are gieen at the quadratere points, other,rise interpol.tion m_ be ned. De-

.fine the objectiee _netion

_ere uS(A) is an approzimate solution to (_6) with the cross-se_tional area A. The

approximate design problem, is to ._nd an A. E 15 s_h that

<_Jt'(A) A e. (37)

Note that we can identify any A E B with the parameter q E Q --- [Am, A_] which

uniquely represents it. Thus we can equivalently state the problem as to find qo e Q

such that

_<jfCq) fo,all q,EQ. (38)

3 Sensitivity Equation Method

3.1 Trust-Region Algorithms

We shall use a trust-region algorithm for the optimization loop. The reason for

selecting this type of scheme will be clear when we discuss the convergence pzoperties

in Section 4. This is a well known algorithm. However, we give a brief description

below in order to prepare for the formulation of the sensitivity equation method.

The quasi.Newton optimization algorithm produces a sequence of iterates which

are obtained by minimizing a local quadratic model of the objective function. This

model is constructed using the evaluation of the objective function ,3'_(qt), its graclient VJ'f(qh) and a secant approximation to its Hessian, Hh at the current iterate

qk. The minimization of this model produces the next iterate qh+z, i.e.

l T

Thus the next step is

qk+l B qk -- HflVff_(qk).

(39)

It is well known that for sxtfllciently clc_e initial guesses (and assumptions on the

objective flmction), the iterates converge super]ine_rly to the minimum, q,.

11
However,the initial guessmay not be in this super]inear region. Thus globalization strategies are employed to bring the iterates into the superlinear region. It is

desirable to choose strategies which reduce to the quasi-Newton alsorithm close to

the minimum. One such strategy is a _,rust-reg/on algorithm. In this algorithm, a

quantity 6, known as the trust-reglon radius, is used to measure the region in which

the local quadratic model, ink, is _trusted" as an approximation of the actual ohjective function, ff_. Thus, the next iterate, qt+l, is now found by minimizing the

model in this region, i.e.

-_,(q.+aJ= rain ml,(q,+st). (40) ll,,U<__h

where 6k is the trust-region radius at the L-th iteration.

A heuristic for changing the trust-region radius needs to be developed which increases 6h when the mode] prediction is good and decreases 6_ when the mode" prediction is poor. One such strategy uses the r_tio,

pk = "m_Cq,)- mk(qk+l)

which is the ratio of the computed reductiov to the reduction predicted by the model.

If this ratio is small (or negative), then the model did a poor job of predicting ff_

and the trust-region is decreased. Whereas, if the ratio is near 1, then the model did

very well at predicting ,7_ and the trust-region radius is increased.

We present the resulting trust-region algorithm below.

Algorithm 8.1 (Tru=t-Re64on)

Select an initial guess q0 E Q, an initial trust-region radius 6o aud constants 0 < _ <

72 < 1 tad 0 < 7t < 1 < _/_. Compute Jf(qo), V,Yf(qo) and select or initialize Ho.

Do k = 0,1,..., until _¢onvergence"

I. Determine the approxln_te solution 6k to equation (40). We cho6e the optimally

constrained hook-s_ep method [1I] to do this.

2.If p_< _,, thenset6,,, _ (0,_,_,)andq,,, = q_,j,_(q_,,)= j,_(q,),

vj_(_+,) = vj, N(q_)a,,an,., = H,.

3. If rh < Pk < r/a, then set Jk+t E (0, 6hi and q_,+l -- qk "4"ale. Compute ff_(q_+_,),

V,Tf(q_+t) and the update H_,+t.

12
4. If _3 < pk, then ,st _k+_ E [6h,-1_6,]aad q,+_ = qk + 8k.

VJ_(qk+l) and the update H_.I.

Continue

3.2 Design Sensitivities

In order to ,_pply * gradient based optimi_tion algorithm, such as the trust-region

algorithm described above, we need to consider methods for computing the gradient

of J_. In this discussion, we consider fmdlng the sradlent of J'_ (or a suitable

approximation) with respect t_ the single design parameter q. This dkcussion can be

easily extended to find the gradient of _'_ with respect to multiple design parameters.

A straight forward approada is to use a finite di_erence approximation, e.g.

(42)

Aq

Unfortunately, this approach may not be practical for problems where the approximation of the PDE is computationatly expensive, and is overly c,mplex in shape

optimization prob]ems due to the necessity of computing mesh sensitivities. One way

of alleviating the computationaJ burden is to use design sensitivities, quantities which

describe the influence of the design variables on the flow variables. For example, we

can directly compute the sradient by di_erentiating (36) as

_j; (q)= __c_ _u_(q). (43)

N N

The quaatity _u _ -- {_ u, _,=, is the design sensitivity for the discretized rio," u_.

There are seveza] w_ys to compute this sensitivity. As above, one might use finite

differences, yielding the Lpproximation

_qU (z,;q)= q _q)-uJV(z';q)Aq " (44)

When the dis_retiz_tion is p_rsmeter dependent, it is euier to compute this approximation using,

us(z';q) _ u_(,_+_,_(_,!_q; Aqq+_q_-uN(,,;q)

- _zuN(z,; q)_q/_(:,) (40

13
in order to avoid interpolating back to the unperturbed mesh. This approach has the

advantage that it may be possible to select a step size Aq using error estimates for

u_v. However, it is as computationally expensive ae computing finite differences on

A more efficient appmac.h can be obtained by differentiating the simulation scheme

used to approximate the flow (the discrete sensitivity approach). For example, in

the FBS design problem, the simulation scheme (16) could be differentiated with

respect to q, leading to a numerical scheme for terms like _u s. Since the chain rule

must be used to carry this out, the remdting scheme for the sensitivities contains

terms similar to those found in the simulation scheme. Thus, the sensitivities can

be computed efficiently along with the flow. A disadvantage of this approach is that

when the discretization is parameter dependent, as in shape optimization problems,

then derivatives of the discretization (terms llke _.A4) need to be considered, see e.g.

12o].

An alternative approach is based on differentiating the original flow equation with

respect to the design parameter and then approximat',ng the resulting ser_tiv_ty ¢quc_-

tion. The result is (_u) _v_, where the superscript N Mers to the approximation

of the flow equation and the superscript M rders to the approximation of the sensitivity equation. Since this approach interchanges the order of differentiation and

approximation, no mesh seusitivitim are required. Furthermore, it has been shown

[2] that applying the same approximation ,,daeme to the sensitivity equation leeds to

simila: computational advantages as the discrete approach described above. Moreover, additional computational savings could be obtained by applying a scheme which

takes advantage of the linearity of the _witivity equation. A potential _vantMe

/¢,M

of this approach, however, is that in general _u _' ¢ (_u) ,even if the same

approximation scheme is used for both the tlow sad sensitivity equations.

However, if we consider the gradient of the infinite dimensional objective function,

_--_.7(q)= 2_' [u(z; A)- _(z)] _qu(z; A), (46)

then using the sensitivity equation approach provides an approximation of this gr_-

client, i.e.

' /0 \

Thus, we have reasoz to expect that this approach could produce feasible gradients

14
for the optimization sch,_ne. These two sensitivity approaches are described in detail

in l_ter sections using concrete examples.

S.S Sensitivity Equation Method

The sensitivity equation method couples a trust-region optimiz&tion algorithm with

gradient eva/u_tions provided by approx/m&ting the sensitivity equation. Thus we

consider applying Algorithm S.l with the following quadratic mode],

_,(_,+I) = mln _(_k + 8k) = 'n n,,ll.<6, ll..n_6, (q') + 9_ak + ]_,//,8, C48)

/_'ote that we rep]ace the qeadrttic model mk by St to empheeize the fact that Vfff

(qh).

The intent is to use the robustness of the trust-region optimization algorithm to

compensate for the non-consistent gradieats. The result is an optimal design method

which is often more efficient tnd considerably easier to implement than current me_hods. In the sections below, we discuss convergence issues and describe the implementation of this method.

4 Convergence Issues

Definition 4.1 A numerical echeme is Jaid to produce consistent derieatiee_, with

re_ct to approximations N (/or the states) and M (.[or the sensitivities) iy

_,aq )t (') (49)

This is ezaetll/ the case .[or the d/serete _e_itivib,/ appreaeh, mince one aetuelly de, nee

(computu) N,u

Definition 4.2 A numerical scheme/e sa/d to produce a, ltmpiolica[i ¥ con4tlJtent

derleatiem, with respect to appro:imationa N (/or the state.8) and M (.[or the sen_dtivitiu/i!

[_qJff(q)- ( O J_N'u(q)[ /, "*O,_q vq¢ _o. (bo)

ia satis.t_ed a_ _he approzimat_on_ N and M are r_ned.

We now consider the convergence of the sensitivity equation method. To begin

with, we aesume that the fo|}owing hypotheses hold,

16
(H1) For a given qo in the design space Q, let Qo be an open convex m_bset contsining

the level set of Jf at qo, i.e.

(51)

(H2) Jf is bounded below

(s3) yf is F_.bet _ff=enti_ble on _o

(H4) The F_echet derivative of J_, dGmoted by V Jr, is Lipschitz continuous on Qo

with Lipschitz constant L, i.e.

(HS) The spproximate gradient, g, is asymptotically consistent to VJ_(qh).

(H6) There exists a constant cl E (0,1] such that

c_llg,,lllls,,ll < (-m,,s_,><IIm,lllls,II v,_= 1,2,... (S3)

(H7) There exist constants c_,cs E (0,co) tuch that

-_<d,d) < <_,,,_,d)< c_<d,d) vk ffi1,2 ... (54)

The following discu, sion parallels the proof given in [7] which treats the use of

tnmt-_,_ioa aJgorithms with inexact gradient tad function vltues. Tiffs discussion

mikes use of the fact tb_t we seek the minimum of Jf tad h&ve uymptotic_ly

consistent derivatives.

Lemma 4.1 Under auumptione (H6) and (HT), Algo_thm 3.1 p_duees iterates

whichs_b

I

_'(qk) - _'(q'+') >- 2 e'"g'"min {$" c"lg'" } 'c. (55)

16
Proof Note that since _bt(qk) = ,5"_(qk),

1

CJ(_k)- _k(qk+l)= -- (g,,ak)-- _ (Hksk,,_). (5_)

Now,let 84 = IIs,ll_ - ad,, then_..olva

rain a(gk, dk)+ ! a(Hkdk,dk) (57)

__.<.6, _a .

We can break this up into two razes, whm (Hkd,, A) >_ 0 and when (Hhdh, d,) < 0.

Case f: Assume (HkJk, A) >_ 6, then either

(g_,d.)

a. = (HJk,d_)'

in which cue

_k(qk) -- t/)lc(qk4.l) _" (.q,.d,) (g,,dn,)- 1 (g,,d_)*

(H:.dk,dk) -__ (Hhdl,,d.)

z (gh,,_j)z 1 ,llgkll_

(_) > [,_

using hypotheses (H6) snd (HT), or

in wblch cue

implies

'Pk(qk)-¢'k(_,+O

a. u_

Oi,_k)

6, < --(//ha,, da,)

= --6,.(gj.dj) - _6g(Hh_j,,&)

lcz6,1lghll > -_.O_.a_) ÷.{_ (_,,d_,)>

by hypothesis (H6).

C'o,¢ _. Auume (H_J_, d_) < 0, them a, = _. Therefore

,_,(q,)- ,/,.(q.+,) = -_,, (g_.,_,) - {_ (H._k,d.)

1

> -6k (gk.dk)> ¢,6,,}1g,,11 > _cs$_llg_,ll.

A

17
Lemma 4.2 Assume (If') holda, then

liminfl[g_ u > 0 and lira 6k = 0 (58)

k-_o k--*oe

/reply

• (,,..e,) = x.

#m-ll,httlloJlt (so)

Proof It was shown [11] that, if I{*b{t-- 6h, then the solution to (48) h given by

*(t_t), where

,(t,) = - (u, + _,z)-' Sk

and S', is the _miquered amber that urges IIJ(m)ll- 6h. Therefore, if 6h --* 0,

then _k _ 00 (since Hk is bounded, by (H7)). Thus 'h -* -_;tgh. A

I_mma 4.S Let J_ ,atis]V (I'15), (tt_) and (117), then the iterate. 8atiffV

1

t_,(q_) -¢,cq,.,)l- [_;(q,)-J._cq,.,)] _<_(_+L),.,,_- (g,-w:(q_).,,).

Proof Using the Cauchy-Schwartz inequality and (H3), we obtain

j;(,,._)-_:(,.) - ]*(v_;(_, +_,,_.,,)d_

-- <v<_,,)..,>+_<v<l,,+ _.,)-vj;_.).,,),_

< (v°,_('}'''} + ]o'[v°,_(q'+ _'') -v*(")a tt,.Ita_.

By the Liptehitz hypothesis (H4),

Thu,, truing (HT),

(vJ,_(q')' °')+fo'_,_..,,..,.a

<_J;_,.),.,)+_,.,,'.

[,_lq,)- ¢,w,+,)l- [J:(q,) - J;(q.,)]

! ILII,,II_ -< - _,,,,) - _I_......) + (vj.'%)...) +

< -<,.- v_;_,,_,,,>+[l_+,),.,,'

which completes the proof. A

15
Lemma 4.4 Aasume JeN satss.R_ (H_,), (H$) and (H4), and o,nsume (HT) hoIEa,

:Proof Let c be L co_t suchtb_ J_(q) > c,¥q ¢ _o (as_=*,,teed by (H2)).

Assume to the contrary that there exists a point _ G £o such that

Iv.7.,c )l' > (J.'c,o)-o).

Define J - -_VJ_(q), where we choose o small enough so that q + ._ G Q0. Then

N 2

> "(,)n'(,-;),

This b positive for e e (0,2), thus J'_(_) > J'_(_ + _), which implies _÷ 3 G £0. In

addition

j,*(q) -j[(_+_)>Jf(_)-¢

holds, but this is a contradiction since # and # + _ L_e in/-o. A

Th, ,rein 4.1 Au_me J_ _._flu ('H._), (H$) and (H4). l_eAe..o_, _me the

cpprocimcfe _ient s_bflea ¢o_ifio_ (H_) and (tt6) and that the update in con-

_r_,_d #o that (HT) ko_. Then, .for a r_.O_cientJIt _ne diecretization, _e een_i_/fl/

equation method pro_uce_ a _e._e_ of ifer_es _uc_ that

u_n.._i_f Ilg_,ll"o. (61)

Proof Assume to the contre,-y that lim inf_...eo l_g*I{> 0 and define _ such that

(-g_,,._,)

co,(e.)= Ilg*]l'll'*II

and u_kG _ such that

0 .in(e_,)= 0 "'" =t_(¢, +=(_.1¢,) ,_,,(_.),_o

Then (g,, w_,) = 0 by construction, and

(vJT(q,.),_,).
If sin(0t) _ O,then Ilw_ll-- z sad

•, - II,.11(- co,(e,)_ +,in(e,)w.) •

Let/_ denote the set of succesdul iter&tions, then

J_(g,)- J,_(¢k+,)

i

t

I

(62)

¢,(q,)- ¢,(_,÷,)- (j,_(q,)- j,_(q,÷,))

z- pb= ;_'_(q,)- ¢_(qk+,) '

by Lem_ 4.3 and the definition of _'h, we get

] P, <

- (-Oh,,h)-- i (H,J,. _h) "

Using hypothesis (H7),

1 -p_, < ...... .

(-O,,'k)

Substituting expression (62) sad using Ilsh_ < ilk, we set

l-p, _ -. Hukllco.eh

By Lemm_ 4.4 sad the Czuchy-Schwm-z inequality, <VJ_(qh),w,_ is bounded sad

we consider the limit u k --, oo,

,m I,,- *'- ' Ilfbll* < IIo,ll '

20

fo: each/c _ JC. Lemmz 4.1 implies

Since _v is bounded below, by (H2), the above condition implies lim,-_o_,_e_c _ = 0.

Therefore, as _ is decreased in unsu_ itetstions, lin_.._ 6_ = 0. We now have

the conditions for Lemm_ 4.2, and

lira (-g_'"_*) = 1.

_-" Ilskllll'*il

Thxxs limt.e, cos(O_,) = 1 and lim_,..,,, sin(O_,) = O.

Consider the expression
Sinceliminfk..oo][gh[[> 0 sad gk is asymptotic_ly consistent, we c._ select a sui_-

ciently fine discretization such that

llm 1 - p_ < 1 - _=.

k--.0o

Hence, p_ > _ which implies 6k+l > 6k, a contradiction.

5 Duct Design Problem

In this section, we use the duct design problem to illustrate the implementation of the

sensitivity equation method. To begin with, we will int_xluce the discrete approach

for finding design sensitivities in order to compare it with the sensitivity equation

&pprosch.

5.I Discrete Sensitiviti_m

To obtain an algorithm for the sensitivities _u_(q) N _v

nonlinear equations (32) is differentiated, yielding

h + _ u_, u_,A(zj), A(zj) --0. (63)

where _j+]/2 is deterwJ.-,e_i by the scheme used to compute the flow. If the EnquistOshe_ scheme was used,

a_J+l/a o{ +1 ""

uj+ 1 _< u,;

UN _ Uo _ Uj+ I,

< <

or if the srtlt_cial viscosity scheme was used,

.here

and

(64)

(65)

(_)

(67)

21
This differentiated scheme can now be used to compute _u N.

5.2 Sensitivity Equation

We now present the implementstion for the _msitivity equation approach. We begin

by differentiating the flow equation (26) with respect to the parameter q. Thus

(68)

(69)

is the Nnsitivity equation for this problem. Note that the sensitivity equLtion is a

linear eqm_tion with variable coe_cic_ts (determined by u). Th_=¢ has been little

analysis d numerical schemm to appro:dn_te equations of t_is type. Howevez, for

this two point boundary value problem, the same numerical schemes (Enquist.Osher

and artificial viscosity) _rovide convergent algorithms. As in the approximation of

(26), we consider (_u)_ to be. the aversge sensitivity in the jth cell. A system of

I_o__

nonlinear equations for _,_u/_ (q)ffi _u), (q) can be found by integrating

1

(68) over each _U,

j = 1,..., N, where we assume A and _,A are nearly constant oyez each cell. As

before, the terms l (u(=_ + _), _,u(z,. + _)) are replaced by the ceil center values/._

and j_+1, Using the Enquist-Osher scheme, we obtain

and obtain

]_+I .v(,,.,

+ +/(u.,

U_ ,9j÷ 1 _ U.;

uj , uj+ 1 _.;

U_ < u. < U_l;

(71)

Pj_v/2 $(_+x/_ _ a( ° _t O

(72)

92
for the artificial viscosity scheme. It is obvious that the approximation of the sensitivity equations depends on the approximation of the flow equations. As described

earlier, we use the notation (_u) N_ to repres_mt using sche_ne N to approximate

the flow equatioc and scheme M to approximate the sensitivity equation.

5.8 Convergence _esu]ts

The convergence result provided ir Theorem 4.1 can be proved for the case when

the artificial viscosity achew.e is used to approximate the flow and the Enqulst-Osher

scheme is used to approximate the sensitivities in Algorithm 3.1. For this problem,

we assume (the (HI) in Theorem 4.1) that

Q = Qo=

and

The objective function 3"_A v giv_ above is obviously bounded below (by zero if all

of the quadrature weights axe aonnegative) _tisrfying (H2). The hypothesis (H3), the

differentiability of

l

J2""Cq)=

i-I

on 12o mind hypothesis (H4), the Lipschitz continuity of the derivative, follow from the

following

Lemma ILl The approziraate solution t_Jv.*v is diOerentiabl¢ and the derieafiee b

Lipechitz continuous on 0.o.

Proof The approximate solution, u_r is the root of the nonlinear equations

W (uJv_v,q) = [Fj+,/, (u_v_v,q) sv

where _4+v:/n and 9 are C _=functions of their arguments (for uJv_v > 0). Then by the

implicit function theorem, the map

V"" _"v" (V)

i, Lipschitz continuously differentiable. LX

We point out that the differentiability of the approximate objective functional is

strongly dependemt on the discretization scheme used in the approximation. For

2S
example, the objective tunctional associated with a G_dunov approximation of the

flow is not differentiable, a result of matching a parameter dependent discontinuity on

a discrete set of points [4]. Finding feasible optimi_.ation strategies for this problem

ha_ been the focus of recent work, see e.g. [4], [19] sad [23]. However, for the

purpose of this discussion, the artificial viscosity scheme provides a smooth enough

approximate objective function.

The hypothesis (HS) is guaranteed (for some discretization level) by the asymptotic

consistency shown below.

_ -- __ NAV ,MZO

Theorem 5.1 For the one dimen6ioncd Euler equations, the derivative __'J) s

where the flow i_ approzim,,ted using the artiflcicd viscosity approzimetion and the

sensitivities are approzim_ted usin 9 the Enquist-Odter scheme,/J a_j1npt0_/_ly consistent to _ Y_"v .

Proof Consider the norm used in the definition of asymptotic consistency:

The first term on the right head side veaishes since using the artificial viscosity

i_:heme for approximating both the flow and sensitivity equations leads to consistent

derivatives. The _,ast two terms go to zero as the approximations NAy, M_t, and M_;o

are refined, since the artificial vbcosity tad Enquist-Osher schemes converge when

used to approximate the sensitivity equation, (_,u) N'v'ut" is the exact solution to

the sensitivity equation given ujv_V. A

The hypothesis (HS) can be enforced by the optimization algorithm by rejecting

steps which violate this condition and shrinking the trust-region radius. This procedure eventually creates a step which satisfies (H6), since the limit of "his procedure

would produce a step in the steepest descent direction.

Finally, (H?) can be enforced by the secant update strategy. Therefore, we have

shown that these approximation schemes satisfy the conditions of Theorem 4.1. Nume:i¢_| computations using these sensitivity schemes Lre provided below.

24
OA

0.3

02

0.1

j°-0.1

-.0,l

.-0,_

-0.4

i • m H ! wl ll_ s • ; J

/ io---

i I l i I I I ' i

0.1 0.2 0.3 0.4 0.5 O_ 03 0.8 O.S

Figure 3: I)ai_ Sensitivity Approximations Using Enquist-O_er Scheme

5.4 Numerical Results

The sensitivity of the velocity with respect to the Bezie_ pazameter, q, is presented usins the numerical schemes described above. For this computation, the crou-sectional

area corresponds to an dement of B (,ee (35)) with q = 1.37125. The interval [0, I]

is divided into 45 czJ/,. In Figure 3, the masitivity solution using the Enquist-Osher

scheme to compute both the llow u'vz° end the sensitivity (_u) "vm°'_w'°- is compared

with the dosed form sensitivity solution. In addition, the sensitivities compute_

via t_rdte differene._ of Eaquist-Osher solutions using s finite difference step size of

Aq w (1 × 10-e) q are also provided. Excellent _ment is seen for both of these

methods. The only discrepancy is in the ceU to the ]eft of the shock, where numerical

dissipation appears in the flow solution.

The corresponding design sensitivi*_ies which are computed using only the artificial

viscosity schemes are shown in Figur_ 4. As above, the agreement is excellent except

where dissipation errors appear in the flow approximations. In this c_,e, these erron

_ppea: over moxe ce_ near the shock.

Note that the computation of these sensitivities were performed efficiently, rela25
0A

o.a

0,9

0.1

_-0.t

._A

.-0.6

,.| | .

• i . w i • .

_ ._- Sem_l_yr:mmon

- i o.l:ir_h

• i

7 i

0.t' 02' _s 0.4' o; 0;- 0.7' 0.8' O.g'

NozzbI.m_

Figure 4: Design Sensitivity Appro0dmation. Using Artificial Viscosity Scheme

tive to the cost of s flow spp_ximation. The flow approximation requires solving

a system of non]inear equations. Thr. _e,sitivity approximation, on the other hsad,

only tequilas solving & linear system dace the sensitivity appears only linearly in the

definition of j' sad _. Moreover, if the Newton method is used to solve the nonlineaz system, them the linear system is already svailable in factored form. Therefore,

the sensitivities can be computed uming less computational time than required for

one Newton step. Computations] e_ciencies such am Skis csa be missed if the flow

algorithm is simply di/_erentiated.

,._ote that u long as (_u),, is bounded,

_U (, =o,

.ince ,Q - u,2. Thus, one observes that the numerical algorithms to comI_ut¢ eiEnqui|t-Osher sc_terve to approximate both the ]low sad sensltlvity equations produo. consistent sradients. In _ddition, it is eui]y seen that using the artificial

viscosity scheme to approximate both equations ebo produce| consistent gradients.

26
T_ble I: A Comparison of Gradkmts st the Optimum for Various Mesh SJzee

m

N f,

!15 1.U98

48 1.3437

135 1.3525

1235 1.3543

315 1._5_

[_os_._,.

0.011707

0.004800

0.002,1_5

0.009470

O.OO2645

0.009810

-0.001568

-0.000012

0.007802

0.0_731

0.001584

However, if the artificia3 viscos/ty scheme is used to approximate the flow and the

Enquist-Osher scheme is used to approximate the sensitivity equations, the _'adients

are not co_rJste_t b_tt asymptotically consistent.

Numerical results for this asymptocicaX[y cona_tent case &e provided in Table I.

6 Forebcdy Simulator Design Problem

We now describe the implementation of the sensitivity equation method for the forebody simulator design problem described in Section 2. As in the duct design problem,

we begin by presenting the equations which comprJ_ the Mete sensitivity scheme

in order to compare amd contrast the two methods. UnLike the duct problem, we

have no theoretical conver&ence results for the FBS des/&u problem. However, the

numerical experiments below show that the SEM still converges.

6.1 Discrete SensJtivJti_

DiEorentiAtin_ the numerical scheme (16) with _ to a design para_neter, represented by q, leads ¢o the fo]/owing scheme:

Z¢.._-

[z+ a_a.D"- v,(_ _)+ _(.'>)a_] aO"

27
- At6_P" - A_6,,_

+_tvc(_,7 _ _ t_,_v_),_(_O_) )

+ _tv,,(_,)- ,_,)_v,,),_,,_(a_O_). (74)

The eq_u_tion representing the boundary conditions axe also diferentiated. Note

that the above sensitivity scheme requires de$ivatives of the mapping, _jt4 (denoted

as mesh sensitivities) and the dissipation terms, _/d2) and _4(4). Evaluation of

_.h4 is given by differentiating the scheme which determines .b/, see e.g. [20]. Other

methods for approximating _4 have also been investignted, see e.g. [25]. We see

from (74) that terms containing these expremfions represent a dgnificant portion of

the computational eft'oft, aside from the fact that _,jI4, _(_) and _(4) themselves

need to be deter_ned.

6.2 Sensitivity Equation

The sea_tiviW equation approach to computing design mmsitivities is presented below. To begin with, we differentiate the Euler equatlom and ueociated boundary

conditions with respect to the design parameter q, which leads to:

where

a:,, av

d)

d_

a, = _,Q + ,,Q,+

0

0

0

_P

0

_Pu + P_u

, Q_ =,

_(p,,) '

211
and where

_ = _(P") - _P7/p and_ = (_)- _p. j/p,

since p # 0.

We are now free to apply any appropriate schen_e to solve (75). In particular, it is

possible to use a method which takes advantage af the ]inetrity of the Je_itivity equ_-

tion. However, in this work, the same scheme used to aolve the flow equa_ons b used

to approximate the sensitivity eqattions, which leads to a_ efficient computations]

scheme as in the discrete approach [2]. This scheme is described below.

This equation may now be transformed to generadized coordinates, so that the fufite

differencing can be done more easily. It makes sense to use the same transformation

(which is eq_ve.]ent to using the same mesh) that was used in the solution of the

Euler equations. Thus the resulting system is

o [o] o

u uo

0 0

O, = VO, + gO + _qPJ_ + PJ_ ,

v

0 0

V = V_. (,,,,.,) and V, = V_. (_,..,,_,_),

oq oq -

0 0

v=v,7.(_,,,,) and V,=v,7.(_,,,_,,)

uq oq -

It can be shown '_hat

where

wh_e

and

OI_ OP., aO oO.

29
so that the _etizstion has the s_me factored form _ the Euler equ_ions, thus

- -

-F _tV,(_ (') - *(4)_V,)_,(J.uO)" (77)

Since the left hand side matrices are the _une, a ri&ht hand side vector needs to

be formed for ea_ design sensitivity. I- addition, the boundary condition type is

the same fox both the Eu]er and sensitivity equations. The boundary conditions are

determined using implicit ditferentiatlon.

Note that this scheme is simliar to the discrete sensitivity approach. However, since

the approximation is applied after the differentiagion, there are no mes]_ sensitivity

or dissipation sensitivity terms. The other obvious difference is that the boundary

condition on the parameter dependent bmmdary is different.

6.$ Boundary Conditions

The boundm7 conditioms for the ,en,itivlty equation (75) are provided below for the

case where the f_nebody simulator is described by a two parameter Bezier carve (18)-

(20). Extensions to other fo_ebody descriptions wilt be obvious. The apl_Opriate

conditions are obtained by differentiating the corresponding boundary conditions for

the Eu]er equations. For example, at the inlet, _e flow Oil is prescribed and will not

vary as the forebody parameters q = (qs,q_) aw changed, thus

O, =0

st the test cell inflow. The walls are treated in a similar/uhion. However, the

boundary condition at the forebody simulator surface requires mote _ttention. This

is bec_u,e the points where the condition k evaluated are parameter dependent.

We study the treatment of condition (5) in detail. The normal vector to the

forebody sm'f_..e is

( ....)

8O
Thus, the boundary condition (5) can be written as

-.(r.(,),r,(,;q);q)_r,(,;_)+_(r.(,),r,(,;q);q),_r.(,) = o. (Tg)

q./O _

The corresponding Nmdtivity equation boundary condition for the fuss parameter,

qZ, can be obtained via differentiation, i.e.,

-_ (r.(,),r,(,;q);q)yr,(,;q)+_ (r,(,;q),r.(,); q) =

0

_,

o _,_ r,r ,; q),y,r,(,; q) (r.(,).r.(o;q)_ o_

+.(r.(,),r,(,;q); 0' q) q) Osaq_r_(s_

0 8 . 0

- _(r.(o). r,(,; O;_)_r,(,.q)yr.(o).

This is simply a nonhomogeneous version d condition (5), namely,

a o o 8 2 8 o o

(_., _). _= _._r, Kr, +.a-i_q,r, _r,_r.,

Using the same technique*, the boundary conditions corresponding to (6) are:

The analogous boundary conditions for q_ are obviou.

(8o)

8.4 Numerical Results

The sensitivity equation appmac.h, which compute, design Icnsitivitie* for the two

dimmsional Euler equation is illustrated below. In this implem_tation, a right hand

side vector for each design sensitivity is formed along with the ¢ozre*pondin_g vector

for the flow appraximatio_. The update, for the flow and sensitivity variables are

obtained simultaneously, exploiting the fact that the left band side matrices are the

|&lTle.

The dmign ,_asitivities with respect to the first Bezier paramete_ qz were computed

for a forebody described by the curve

t' = (_(,),_(,)), , _ [o,1],
where

= o.o ojC,)+ o.la,j(,) + + l.Oeu(,),

ql __ 0.1, _ - 0.15, F, - 0 and F, -- 0.2. This curve is twice as long in the z-diroction

as the admisdble foxebody dmul_tors Siven in B (see (18)). Under a uniform inlet

flow profile described by the inlet Msch number, M, -- 2.0, the approximate flow

variables and sensitivities are computed on a 43 x 40 mesh. The sensitivity of the

z-component of momentum with respect to the Bezier parameter ql, computed using

the sensitivity equation approach and the finite difference approach (for 4 different

step sizes) are plotted along the outflow plane in Figure 5. The corresponding plots

for the Energy sensitivity are provided in Figure 6. Observe that the step size of

0.00001 produces noisy sensitivity values close to the forebody (presumably due to

round-off errors). A larf_er step size of 0.01 sives the best results (when compared

to the sensitivity equ_ion approach) near the ahock loc_tion. The best qualitative

behavior appears wben the step dze b 0.001. These _ures demonstrate the d_culty

of obtaining a s_hJfactory step size at all resolution levek in the flow dom_n.

A model forebody simulator dedl;n problem is discussed below. To begin witu,

we seek the optimum value of the inlet Mach number and two Bezier parameters

( (qt q=), describing a shortened forebody simulator in the admissible _t B) which

minimize the approximate ccet functional ,TaN (siven in equation (25)). The low

data 0 to be matched is 10vea by the flow _" azcmpondinl| to the forebody shape

f' described above. We point out that the arti_Icisd disdpation in the flow solver

produces a "smearing" effect on th_ flow variables. Therefore, based on the results

for the duct desisn problem, we expect a sumdently smooth approxin_te cost functionsd. Furthermore, the comparison of the sensitivities in Figures 5 and 6 lead us to

believe that the sendtivity equation approach may produce uymptotic431y consistent

deriv_ives.

The sensitivity equation method was applied to the FBS design problem with

initial values of the parameters: M, = 2.0, qs = 0.I0 sad q_ - 0.15. These parameters

correspond to thoee used to generate 0 (even thou_ that forebody is longer). We

present the iteration history in Table II. Observe that there is • drutic reduction

in the approximate cost functional in the first three iterations. The iteration history

for the z.¢omponent of momentum i, given in Fisuxe 7. Note that the front end of

a2
IMipIMIo0_

0 1 2 $

SNtP_m ¢omdnm

8Up Ize 0.0001

L . H

"0 I i $

F.J PlsM Om_m,

2

-I 0

I1_ 0.001

1 i

1 2

lxlt PWo _oedbulo

i

Figure 6: Comparison of Sensitivities at Out_ow: x-Component of Momm_um

lira 0.01

0 1 t 8

kl PlmuOoedl_o

811 _b0_001

0 1 | s

Ixt _ q_oevdnmo

tllp 8ill 0_0!

o 1 I $

Ixt Pllne _e_l_l

lie pIIN 0.C_004

o 8 |

Ixll PlIM Oeedlull

Filur, 6: Comp_ of Sen_sivitiu st Outflow: Enerl_

a3
I_,tim

0

)

2

3

4

5

6

?

8

9

I0

I!

12

T_ble 11:Shortaued Forebody Optimisation

2.00000

2.00108

2.01054

2.00897

2.01027

2.01307

2.01070

2.01980

2.01940

2.019N

2.02008

0.10000

0.14808

0.28848

0.30?65

0.S01S9

0.29N7

0_8031

0.29011

0.'_378

0.29420

0.29439

0.29417

0.29415

t2 i

0.15000

0.17177

0.14152

0.13871

O.14007

0.14787

0,15584

0.15921

O.1M21

O.1H6P

0.15404

0.16403

0.15(109

Cmt Fuction,d

3.2339

1.6000

0._182

0.2334

0.2306

0.2289

0._471

0.2249

O.2237

0.2233

0.3230

0.2229

0.2220

GrKlimnt

27.1283

11.6285

3.7955

0.4821

0.5983

0.8881

0.5009

0.1513

0.0575

0.0371

0.0275

0.0173

0.0153

the forebody simulztor becomes more blunt during the fin, t two iter_tlons in which

a sta_ation region is set up in front of the FBS. This has the effect of moving the

shock forward, which comes clole to the shock location crated by the Ions fo:ebody.

The remadning iterations are used to _me tune" the solution near the FB$. The

comparison of the optimal forebody simu]_r to the flow i_ne:&ted by the long

forebody is displayed in Fibre 8. Notice thst the shock locstion is the same in both

4:iows.

In the optirniz_tion above, the initial Hemdau was computed using forward d/fferences. This adds some initial expense in the hope for fewer itorations. However,

without this technique, using the identity m_rix u the initial Hessian, the iteration

converip_d in fifteen iters¢ions. Therefore, neither technique showed an advantage,

8,5 Conclu,ion,

While no rigorous proof of asymptotically consistent itradients hu been ,hown _or

Elder equ,tions, numerJcld evidence in [3} suIHJosts that the ipr_tients may indeed

be asymptotically consistent. Similar numeric_l evidence exist, for finite ele_nent

app:oxlmations of the Ni,vler-StokeJ equstion, [5].

S4
Iteration 0 Iteration 1 Itetalion 2 Iteration 3

Iteration 4 Iteration 6 l_ation 8 Iteration 12

Figure 7: $e_itivity Equatioa Method Itmst/ons

3b
F/hal Converged Solution Original Ions Forebody (data to be matched)

Fi_ue 8: Compsdson of Optimsl Solution with Ds_s

SO
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