ON THE OPTIMALITY SYSTEM FOR A 1 D EULER FLOW PROBLEM

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1 ON THE OPTIMALITY SYSTEM FOR A D EULER FLOW PROBLEM Eugene M. Cliff Matthias Heinkenschloss y Ajit R. Shenoy z Interisciplinary Center for Applie Mathematics Virginia Tech Blacksburg, Virginia 46 Abstract In this paper, a problem of shape esign for a uct with the flow governe by the one-imensional Euler equations is analyze. The flow is assume to be transonic, in the sense that we have a shock embee in the flow. The presence of the shock introuces analytical an numerical ifficulties that are overcome by introucing the shock location as an explicit state variable. We justify the use of ajoint variables by establishing conitions to ensure that the linearize constraint is surjective. These ajoint variables are use to state necessary optimality conitions an to compute graients. In aition, our characterization of the ajoint variables is of interest for comparison to ajoints associate with shock-capturing formulations.. Introuction In this paper we apply the shock-fitting framework evelope in Cliff et al 3 to an optimal uct esign problem governe by the one imensional Euler equations. The objective of this research is the rigorous mathematical justification of ifferentiability of the problem, surjectivity of the linearize constraints, an the accurate formulation of the ajoint equations for optimality. The purposes of this analysis are two-fol: first, as the founation for the efficient numerical solution of the problem by an all at once metho, an; secon, for comparison to the behavior of ajoints arising from more formal treatments, incluing those arising from shock-capturing proceures. Reynols Metal Professor, Aerospace Engineering Department, Associate Fellow AIAA y Assistant Professor, Mathematics Department z Grauate Stuent, Aerospace Engineering Department, Stuent Member AIAA The following notation will be use: = ensity u = velocity e = specific total energy h = height of the channel q = h x =h logarithmic erivative of channel height = specific heat ratio p = ( ; )(e ; u ) pressure: A quasi one imensional invisci steay state flow can be escribe by the steay state Euler equations in conservation form F(U(x)) where F(U) x q(x) Q(U(x)) = x () U U U U 3 A u e A u p u A Q(U) (e p)u u u A : (e p)u See e.g. Anerson. It shoul be note that, that F an Q are homogeneous of egree one, i.e. F(tU) =tf(u) an Q(tU) = tq(u) for all t IR. One can easily euce that F(U) =A(U) U an Q(U) =S(U) U () Copyright c996 by the authors. Publishe by the, Inc. with permission.

2 where an A(U) = F(U) U ;3 u (3 ; ) u ; ;ue ( ; ) u 3 e ; 3 ; u u S(U) = Q(U) U ;u u ;ue ( ; ) u 3 e ; 3 ; u u A A : It can be shown that A(U) has eigenvalues u ; a, u, u a, where a is the spee of soun. Bounary conitions for the system will be specifie below. Subsequently we will often omit the argument x. The problem we are intereste in is the esign of an area function h generating a flow that best approximates a esire pressure istribution p in the least squares sense. We assume that the area function h is fixe at some point, say x =. This allows us to use the logarithmic erivative q of h as the esign parameter. For given q(x) =h (x)=h(x) the area function can be compute from h(x) = exp ln(h()) Z x (3) (4) q(t) t : (5) The mathematical formulation of our esign problem is as follows: Minimize J(U q)= (p(x) ; p (x)) x (6) subject to the steay state Euler equations F(U) q Q(U) = x (7) x The flow is assume to be transonic, with the shock implicitly efine by the Rankine Hugoniot conition, F(U(x s ;)) = F(U(x s )): (8) We have bounary conitions at the inlet B i (U()) u e A in (u) in (e) in A = (9) an bounary conitions at the outlet B o (U) = p ; p out = () = ( ; ) U 3 ; U ; p U out : In aition, we impose bouns on the logarithmic erivative of the area function q low q(x) q upp x [ ]: () Iollo et al 6 have stuie this problem using an ajoint metho to formulate the optimality conitions. The ajoint system erive by them iffers from that in the current paper. The bounary conitions impose on the ajoint variables at the shock by Iollo et al 6 seem to be incorrect, except in a special case. This will be iscusse in greater etail in Section 4. An extension of their results to the two imensional case is consiere in Iollo an Salas. 7 Jameson 8 has also use an ajoint metho to formulate a similar class of optimal esign problems. A similar problem has been stuie by Frank an Shubin. 5 They point out that solutions of this simple moel exhibit phenomena quite similar to those in two imensional invisci flow over an airfoil. They have shown that uner given assumptions, the continuity an energy equations can be integrate out reucing the Euler system to a single ODE. In their version of the problem, a esire velocity profile is set as a target instea of a pressure istribution. They iscuss numerical solutions for this problem, employing stanar iscretization schemes, an using both a black box metho an a Sequential Quaratic Programming (SQP) metho. They have reporte that the SQP scheme is not very robust, as compare to the black box scheme, though when it oes converge, it proves to be the superior metho. Similar results, using the SQP metho, have been reporte in Sachikova et al. The failure of the SQP metho stems from the existence of the shock in the flow. In particular, when Gounov type schemes with low continuity properties are use to capture the shock, we fin that the SQP metho, which is esigne for smooth problems, performs poorly. In previous research 3 we iscusse an approach to overcome this problem by treating the shock location as an explicit state variable an enforcing the shock conition as an aitional state constraint. This was one in the context of the single ODE moel escribe by Frank an Shubin. 5 We foun that this approach yiels better smoothness properties for the constraints. We provie a rigorous mathematical analysis of the existence an

3 nature of optimal solutions for the infinite imensional problem, an stuie the relation between the infinite imensional an the iscrete problem, which reveale valuable insight into improving the performance of the optimization algorithm. Other aspects of this esign problem an other methos for its solution have been stuie by Borggaar, by Shenoy an Cliff an by Narucci, Grossman, an Haftka. In Borggaar the flow variables are viewe as functions of the esign variables an a sensitivity equation approach is use to compute the graient. An optimal control approach is use by Shenoy an Cliff. They assume that the shock location is known. The area an the flow velocity are the state variables. The esign problem is then formulate as an optimal control problem governe by a system of ODEs an solve using a multiple shooting metho for the two point bounary value problem which is obtaine from the necessary optimality conitions. In Narucci et al sensitivities for various (semi ) iscretizations are stuie. Although the shock location is treate implicitly, it enters the iscretization scheme for the objective function. The esign problem is solve numerically using the black box approach with a steepest escent metho.. The Design Problem in the Transforme Domain For a rigorous mathematical treatment of the optimal control problem we transform the esign problem presente in the introuction. In this transformation, the shock location is introuce as an explicit variable an the regions left an right of the shock are transforme onto the fixe interval [ ]: an [ x s ] ;! [ ] U(x) ;! U(x s ) [ x s ] ;! [ ] U(x) ;! U( ; ( ; x s )): we fin that the Euler equations (7), the Rankine Hugoniot shock conitions (8) an the bounary conitions (9), () are transforme into the following system of equations for U l an U r : F(U l ) x sq l Q(U l ) = [ ] F(U r ) (3) (x s ; )q r Q(U r ) = [ ] (4) with Rankine Hugoniot shock conition F(U l ()) = F(U r ()) (5) an bounary conitions B i (U l ()) = (6) at the inlet an with bounary conitions B o (U r ()) = (7) at the outlet. For convenience, we summarize these conitions in a composite constraint function C(U l U r x s q l q r )= F(U l ) x sq l Q(U l ) F(U r ) (x s ; )q r Q(U r ) F(U l ()) ; F(U r ()) B i (U l ()) B o (U r ()) : C A (8) With this formulation the system (3 7) can be written as: C(U l U r x s q l q r )=: The systems (7 ) an (3 7) are equivalent. If U is a solution of the original system (7 ) with shock location x s, then the solution of the transforme system (3 7) can be compute from (). On the other han, if (U l U r x s ) is a solution to the transforme system (3 7), then the solution U to the original system can be compute from 8 < x U l U(x) = : ;x U r x s x [ x s ) ;x s x [x s ]: (9) With To formulate the control problem for the transforme problem, we efine the esire pressures U l () = U(x s ) U r () = U( ; ( ; x s )) () p l ( x s) = p (x s ) p r ( x s) = p ( ; ( ; x s )): () 3

4 With the transformations (), () the objective function (6) is given by (p(x) ; p (x)) x = Z xs (p(x) ; p (x)) x x s (p(x) ; p (x)) x = x s (p l () ; p l ( x s)) ( ; x s ) (p r () ; p r ( x s)) : The control problem we have to solve is given as follows: Minimize J(U l U r x s q l q r ) = x s ; x s subject to the equality constraints (p l () ; p l ( x s)) () (p r () ; p r ( x s)) C(U l U r x s q l q r )= () an to the inequality constraints x s (3) q low q l () q r () q upp [ ]: (4) The states are given by the triple (U l U r x s ) an the controls are (q l q r ). The equations (3), (4), (5), (6), (7) are the state equations. The flow solutions (U l U r ) an the esign functions (q l q r ) o not epen on the shock location explicitly; they are couple to shock location x s only through the state equations (3), (4), (5). For a complete escription of the control problem, we have to specify the control space an the state space. It will be seen that an appropriate choice for the control space is Q = L ( ) L ( ): (5) The set of amissible parameters is efine by Q a = f(q l q r ) Qjq low q l () q r () q upp g (6) for [ ]. Recall that q low. A typical control vector is enote by q (q l q r ): (7) The state space is chosen to be V = W ( ) W ( ) IR (8) where W ( ) = (W ( )) 3 an a typical vector is enote as Defining the space v (U l U r x s ): (9) C = L ( ) L ( ) IR 7 (3) with L ( ) = (L ( )) 3, the constraint, (8), is viewe as a map from VQinto C: C : VQ!C: For later purposes we also efine the ajoint space = W ( ) W ( ) IR 7 : (3) As usual, for a function f :[ ]! IR we set kfk = ess sup [ ] jf ()j kfk = kfk kf k : For a vector value function F we efine kfk kfk analogously. 3. Differentiability an Surjectivity The transformation of the esign problem performe in Section allows us to establish the Fréchet ifferentiability of the constraint function C an of the objective function J. The proofs for Fréchet ifferentiability are very similar to those presente in Cliff et al 3 for the Frank an Shubin moel problem. In the following we make use of the notation (3), (4) an (9). Theorem 3. The nonlinear operator C : VQ!C is Fréchet ifferentiable at all (v q) VQ a with l r >. The partial Fréchet erivatives are given by C v (v q)(ev) = [A(U l )eu l ] x sq l S(U l )eu l ex sq l Q(U l ) [A(U r)eu r] (x s;)q rs(u r)eu rex sq rq(u r) A(U l ())eu l ();A(U r())eu r() U Bi (U l ())eu l () U Bo (U r())eu r() C A (3) 4

5 where U Bi (U l ()) = I 33 is equal to the ientity, ev = ( e U l e U r ex s ) an an U Bo (U r ()) = p r() U = ; ; u r () ;( ; )u r() ; C q (v q)(eq) = where eq =(eq l eq r ). x s Q(U l )eq l (x s ; )Q(U r )eq r (33) (34) C A Proof: The proof of this result is similar to the corresponing one in Cliff et al 3 an is therefore omitte. Theorem 3. If the esire pressure p is ifferentiable with absolutely continuous erivative, then the objective function J is Fréchet ifferentiable. The partial Fréchet erivatives are given by J Ul (v q) e U l = x s (p l ; p l ) p l U l eu l (35) J Ur (v q) e U r = (36) where p i (33), an ( ; x s ) U i =( ; (p r ; p r ) p r U r eu r u i ;( ; )u i ; ), i = l r, cf. J xs (v q) (37) = ; ; (p l() ; p l ( x s)) x s (p l () ; p l ( x s)) (p l ) x() (p r() ; p r ( x s)) ( ; x s )(p r () ; p r ( x s)) (p r ) x() : In the following, we interpret p r() U as a row or a column, as appropriate. The form shoul be apparent from the context. Proof: The assertion follows from the efinition of J using stanar estimates. The proof is therefore omitte. The final result of this section concerns the surjectivity of the linearize constraints. This property is important for the formulation of optimality conitions, see e.g. Maurer an Zowe. 9 Due to the partitioning of variables into states an controls, the surjectivity of the linearization of C is guarantee if C v (v q) is surjective. Recall that a l an a r enote the spee of soun in the regions left an right of the shock, respectively. Theorem 3.3 Let (U l U r x s ) be given with u l () >, u l () 6= a l (), u r () >, u r () 6= a r () for all [ ] an x s [ ]. If where A(U r ()) ;T p r() U T [T r q r Q(U r ) ;T r ()T l () ; T l q l Q(U l ) 6= T l () = exp T r() = exp ;R x s q l (t) S(U l (t))a(u l (t)) ; t ;R (x s;) q r(t) S(U ; r(t))a(u r(t)) t (38) then for all r =(r l r r r s r in r out ) Cthe linear system C v (v q)(ev) =r (39) amits a unique solution ev. Proof: Uner the assumptions of the theorem, the matrices A(U l ) an A(U r ) are invertible for all [ ]. Therefore, the first equation in (39) is equivalent to [A(U l ) e U l ] x s q l S(U l )A(U l ) ; A(U l ) e U l ex s q l Q(U l )=r l : Using the integrating factor T l the solution can shown to be eu l () =A(U l ()) ; T l () ; A(U l ()) e U l () Z T l () rl () ; ex s q l ()Q(U l ()) : (4) Similarly, the solution to the secon equation [A(U r ) U e r ] (x s ; )q r S(U r ) U e r ex s q r Q(U r )=r r 5

6 can be compute using the integrating factor T r an is given by eu r () =A(U r ()) ; T r () ; A(U r ()) e U r () Z T r () rr () ; ex s q r ()Q(U r ()) : (4) Inserting (4) an (4) into the thir equation A(U l ()) U e l () ; A(U r ()) U e r () = r s yiels ex s T r () ; T r ()q r ()Q(U r ()) ;T l () ; T l ()q l ()Q(U l ()) = r s ; T l () ; A(U l ()) e U l () T l ()r l () T r () ; A(U r ()) e U r () T r ()r r () : (4) Next, we have to incorporate the equations arising from the bounary conitions. The equation U Bi (U l ()) U e l () = r in implies U e l () = r in. If w w are linearly inepenent vectors orthogonal to p r() U, then U Bo (U l ()) U e r () = r out implies that eu r () = r out p r() k p r() U! w! w : U k Notice that, since 6=, the inequality k p r() U k > hols true. Inserting these expressions for U e l () an eu r () into (4) shows that the existence of a unique solution of (39), i.e. the existence of a unique solution (ex s!! ), can be guarantee if the 3 3 matrix [T r () ; T r q r Q(U r ) ;T l () ; T l q l Q(U l )] ;T r () ; A(U r ())w ;T r () ; A(U r ())w is nonsingular. This is the case if an only if A(U r ()) ; [T r q r Q(U r ) ;T r ()T l () ; T l q l Q(U l )] ;w ;w is nonsingular. Since w w are linearly inepenent vectors orthogonal to p r(), the latter conition is U equivalent to (38). We efine the vector an the Lagrange function L(v ) = x s 4. Optimality Conitions ( l r s i o ) ; x s (p l ; p l ) (p r ; p r ) T l [(F(U l)) x s q l Q(U l )] T r [(F(U r)) (x s ; ) q r Q(U r )] T s [F(U l()) ; F(U r ())] T i B i (U l ()) o B o (U r ()): (43) If the shock location at the optimum obeys x s ( ), then the first orer necessary optimality conitions are = L v (v q )(ev) (44) = L (v q )(e) (45) L q (v q )(eq) (46) for all ev C, for all e, an for all eq with (q eq) Q a. The secon equation (45) in the optimality system yiels the state system (3 7). Using integration by parts an appropriate test functions U e l U e r ex s we fin that the first equation (44) in the optimality system yiels the ajoint equations T A(U l ) ( l ) 6

7 h T = x s q l S(U l ) l x s (p l ; p pl l ) U l T A(U r ) ( r ) = (x s ; )q r S(U r ) T r h ( ; x s )(p r ; p pr i r ) U r with conitions i (47) (48) l () = ; s r () = s (49) at the shock, the bounary conitions T i o U Bi (U l ()) = l () T A(U l ()) U Bo (U r ()) = r () T A(U r ()) (5) an the conition q l T l Q(U l )q r T r Q(U r ) J xs (v q) =: (5) Using the efinitions (9) an () one can see that the first equation in (5) is equivalent to T A(U l ()) l () = i (5) which implies that the Lagrange multipliers l are free at the left en point =. If we enote the components of r by () r () r (3) r, then the secon equation in (5) is equivalent to ;3 u r ()() r () ;ur ()e r () ( ; ) u 3 r () (3) r () = o ; u r () () r ()(3; ) u r() () r () er () ;3 ; u r () (3) r () = ; o( ; )u r () ( ; ) () r () u r() (3) r () = o ( ; ) (53) The thir equation (46) in the optimality system is equivalent to 8 < if q l () =q low x s l () T Q(U l ) = if q l () (q low q upp ) : if q l () =q upp (54) an (x s ;) r () T Q(U r ) 8 < : if q r () =q low = if q r () (q low q upp ): if q r () =q upp : (55) Note the shock conition (49) for the co state. From the examination of the other equations it can be seen that l () = ; r () = can be guarantee only if the esire pressures can be matche exactly, i.e. the source terms in (47), (48), an the term J xs (U l U r x s q l q r ) in (5) vanish. This homogeneity of the co states at the shock was assume generally in Iollo et al. 6 However, the rigorous presentation shows that the ajoints ( l r ) will in general be nonzero at the shock with opposite signs left an right of the shock. The fact that the co states are nonzero at the shock if the optimal value of the objective function is nonzero can also be observe in the numerical experiments for the Frank an Shubin problem reporte in Cliff et al Conclusions an Outlook We have provie a careful erivation of an optimality system for a shape-esign problem base on -D Euler flows. The analysis shows that the ajoints are smooth (inee in W ) in regions where the flow is smooth, but just as with the flow variables, there is a shock in the ajoint variables. The conitions erive in Section 4 can be use for numerical implementation in several ways. The statemoel (3 7) efines a bounary-value problem to be solve for the flow variables an the shock-location given the area-istribution. With the state/control in-han, the ajoint system (47 49, 5, 53) efines the ajoint variables ( l r ). Finally, the optimality conitions are given as (54,55). A simple graient metho woul procee by solving the first of these for the state, the secon for the ajoint an then using an increment in the control as erive from the optimality conitions. As a globalization strategy, one might use the control increment as a search irection in the control space Q an perform a line-search for improvement of the cost function. This woul require solving the system () for the variables (U l U r x s ) for each trial control increment. As an alternative approach one might use an SQP framework, wherein the states an controls are treate as inepenent variables an the state moel () is treate as an explicit constraint. In this all-at-once framework iterates procee towar optimality an feasibility at the same time. The rigorous analysis of ifferentiability an 7

8 existence of Lagrange multipliers is expecte to be important here, as it was in Cliff et al. 3 The stuy in Frank an Shubin 5 showe that formal evelopment of an SQP algorithm i not lea to a satisfactory computational proceure. This is because the shock-capturing flow-moel use oes not possess the necessary smoothness properties to justify Lagrangian formulation. In Cliff et al 3 we utilize the same shock-fitting formulation employe here, but for the single ODE moel of the flow. 5 In the present paper we have applie this approach to the full -D Euler system. Aitionally, it is worth noting that care is neee in the formulation of the finite-imensional approximation to preserve consistency with the continous problem. In Cliff et al 3 we observe that a moification of the finite-imensional state/ajoint inner-prouct ha a substantial impact on numerical efficiency of the overall proceure. The analysis leas to a characterization of optimal shapes an can be numerically implemente in several ways. The flow formulation can be interprete as a shock-fitting scheme an so is of questionable practical utiltity as a computational proceure for 3-imensional flows. On the other han, the analysis clearly reveals the nature of a shock in the ajoint system an this is of value for comparison to shock-capturing methos. There is no reason to believe that a iscretization that leas to accurate shock moeling for the flow system is necessarily appropriate for moeling the shock in the ajoint system. The results in this paper can be use to construct test examples since we have a characterization of the ajoint system without numerical artifacts, such as artificial viscosity. Acknowlegement This research was supporte by the Air Force Office of Scientific Research uner grant F an by the NSF uner Grant DMS References [] J. D. Anerson. Moern Compressible Flow with Historical Prespective. McGraw Hill Series in Aeronautical an Aerospace Engineering. McGraw Hill, New York, 99. [] Borggaar, J. T., The Sensitivity Equation Metho for Optimal Design, PhD thesis, Virginia Polytechnic Institute an State University, Department of Mathematics, Blacksburg, VA 46 3, USA, 994. [3] Cliff, E.M., Heinkenschloss, M., an Shenoy, A., Optimal Control for Flows with Discontinuities, Technical Report 95 9, Interisciplinary Center for Applie Mathematics, Virginia Polytechnic Institute an State University, Blacksburg, VA 46, 995, also Journal of Optimization, Theory an Application, to appear. [4] Dennis, J. E., Heinkenschloss, M., an Vicente, L. N., Trust region interior point algorithms for a class of nonlinear programming problems, Technical Report 94, Interisciplinary Center for Applie Mathematics, Virginia Polytechnic Institute an State University, Blacksburg, VA 46, 994. [5] Frank, P. D. an Shubin, G. Y., A comparison of optimization base approaches for a moel computational aeroynamics esign problem, J. Comput. Physics, vol. 98, 99, pp [6] Iollo, A., Salas, M. D. an Ta asan, S., Shape Optimization governe by the Euler Equations using an Ajoint Metho, ICASE Report No , also NASA CR 9555, November 993. [7] Iollo, A. an Salas, M. D., Contribution to the Optimal Shape Design of Two-Dimensional Internal Flows with Embee Shocks, ICASE Report No. 95-, also NASA CR 956, March 995. [8] Jameson, A., Optimum Aeroynamic Design Using CFD an Control Theory, In AIAA Paper 95-79, presente at the th Computational Flui Dynamics Conference, San Diego, California, June 995. [9] Maurer, H. an Zowe, J. First an Secon orer necessary an sufficient optimality conitions for infinite imensional programming problems, Mathematical Programming, vol. 6, 979, pp 98. [] Narucci, R., Grossman, B., an Haftka, R. T., Sensitivity algorithms for an inverse esign problem involving a shock wave, In AIAA Paper 94-96, presente at the 3n Aerospace Sciences Meeting & Exhibit, Reno, Nevaa, January -3, 994. [] Sachikova, E., Shenoy, A. an Cliff, E.M., Computational Issues in Optimization Base Design, In Proceeings from the 34th IEEE Conference on Decision an Control, New Orleans, December 995, pp

9 [] Shenoy, A. an Cliff, E.M., An optimal control formulation for a flow matching problem, In Proceeings from the 5th AIAA/USAF/NASA/ISSMO Symposium On Multiisciplinray Analysis An Optimization, Panama City Beach, September 7-9, 994, pp