Abdelmalek Aboussoror

Samir Adly
email: samir.adly@unilim.fr

Fatima Ezzahra Saissi

Fatima Ezzahra
email: fatima-ezzahra.saissi@etu.unilim.fr

Strong-Weak Nonlinear Bilevel Problems: Existence of Solutions in a Sequential Setting

Keywords: Bilevel optimization, Convergence of multifunctions, Marginal functions, Variational-convergence, Convex analysis Mathematics Subject Classification (2010) Primary: 91A65, 90C30, 90C31; Secondary: 46N10

The paper deals with a strong-weak nonlinear bilevel problem which generalizes the well-known weak and strong ones. In general, the study of the existence of solutions to such a problem is a difficult task. So that, for a strong-weak nonlinear bilevel problem, we first give a regularization based on the use of strict -solutions of the lower level problem. Then, via this regularization and under sufficient conditions, we show that the problem admits at least one solution. The obtained result is an extension and an improvement of some recent results appeared recently in the literature for both weak nonlinear bilevel programming problems and linear finite dimensional case.

Introduction

Let X and Y be two Hausdorff topological spaces, with Y is first countable. We are concerned with the following bilevel optimization problem, called strong-weak Stackelberg problem (S) :

min x∈A inf y∈M(x) sup z∈M(x)
F (x, y, z)

where M(x) is the solution set of the lower level problem

P(x) : min z∈B f (x, z) F : X × Y × Y → R, f : X × Y → R
are functions, A and B are respectively two nonempty subsets of X and Y . As it is well-known, bilevel optimization problems have applications in many areas; we cite for example, economics, transportation network design and engineering. The class of strong-weak bilevel problems has been first introduced in [START_REF] Aboussoror | Strong-weak Stackelberg problems in finite dimensional spaces[END_REF] (1995). Thereafter, other works have appeared on this class of problems; we cite for example [START_REF] Cao | A partial cooperation model for non-unique linear two-level decision problems[END_REF][START_REF] Jia | New partial cooperation model for bilevel programming problems[END_REF][START_REF] Jia | A new coordination model for ill-posed bilevel programming problem[END_REF][START_REF] Liu | A partial cooperation model for bilevel programming problems with finite reactions follower[END_REF][START_REF] Mallozzi | Hierarchical Systems with Weighted Reaction Set[END_REF] and [START_REF] Zheng | A new method for strong-weak linear bilevel programming problem[END_REF]. However, the majority of them deal with the linear finite dimensional case, for which algorithms and numerical results are given. For illustration of such a class of bilevel problems, let us give the following practical examples [START_REF] Aboussoror | Strong-weak Stackelberg problems in finite dimensional spaces[END_REF].

Example 1 (Stackelberg games) Let us consider a two-player game composed by a leader and a follower where the two players try to minimize their objective functions F and f respectively. Let  and B denote the sets of strategies of the leader and the follower respectively. The leader assumed to have all information about the follower, announces first a strategy x ∈  and the follower reacts optimally by choosing a strategy y(x) ∈ B. If the follower's solution set is not always reduced to a singleton (i.e., there exists x ∈  such that M( x) is not a singleton), then, two extreme cases are possible.

i) The pessimistic case : In this case the game is noncooperative. Then, the leader provides himself against the possible worst choice by the follower in M(x) by minimizing the marginal function sup y∈M(x)

F (x, y).

(

ii) The optimistic case : In this case the follower is in full cooperation with the leader. So that, the leader minimizes the marginal function inf y∈M(x)

F (x, y).

The two above cases lead to a weak and a strong Stackelberg problems respectively. Here, the terms "weak" and "strong" are taken in the sense of [START_REF] Breton | Sequential Stackelberg equilibria in two-person games[END_REF]. However, by taking into account the cooperation degree of the follower, these two cases can be obtained from the following general one. Then, if α ∈ [0, 1] is the cooperation index between the two players, the leader minimizes the function

α inf y∈M(x) F (x, y) + (1 -α) sup z∈M(x) F (x, z).
Set F α (x, y, z) = α F (x, y) + (1α) F (x, z). Then, the leader's problem is formulated as

( Ŝα ) min x∈ Â inf y∈M(x) sup z∈M(x) F α (x, y, z)
which is a strong-weak bilevel problem. Then, weak and strong bilevel problems are particular cases of problem ( Ŝα ) which are respectively obtained for α = 0 and α = 1.

Example 2 (System of nondifferentiable convex inequalities) Let f i : R n → R, i = 1, ...., m, be closed proper convex functions and consider the following system of inequalities f i (x) ≤ 0, i = 1, ..., m.

Set C = x ∈ R n / f i (x) ≤ 0, i = 1, ...m} and θ(x) = (f + 1 (x), ..., f + m (x)), with f + i (x) = max(0, f i (x)), i = 1, ..., m. Auslender and Crouzeix in [START_REF] Auslender | Global regularity theorems[END_REF] have been interested by the best constant denoted by k 2,∞ that satisfies

d 2 (x, C) ≤ k 2,∞ θ(x) ∞ i.e., k 2,∞ = sup x ∈C d 2 (x, C) θ(x) ∞
where d 2 (x, C) = min x∈C xy 2 , . 2 and . ∞ denote respectively the euclidean norm on R n and the Tchebycheff norm on R m . Set f (x) = max i=1,...,m f i (x). Then, under appropriate assumptions, the authors show that the constant k 2,∞ is given by

k 2,∞ = 1 k * 2,∞ with k * 2,∞ = min x∈bdC inf y∈∂ f (x) sup z∈∂ f (x) y y 2 , z
where bdC is the boundary of C and ∂ f (x) is the subdifferential of f at x (see [START_REF] Rockafellar | Convex Analysis[END_REF]). Define the functions

f (x, z) = f * (z) -x, z and F (x, y, z) = y y 2 , z , y = 0
where f * is the conjugate function of f . Set à = bdC and let M(x) denote the solution set of the problem min

z∈R n f (x, z). Then, M(x) = ∂ f (x). Hence k * 2,∞ = min x∈ Ã inf y∈M(x) sup z∈M(x) F (x, y, z)
which is the minimal value of a strong-weak Stackelberg problem.

Note that in the formulation above of problem (S), when the function F does not depend on y, we obtain a weak Stackelberg problem, and when F does not depend on z, we obtain a strong Stackelberg problem. Therefore, strong-weak bilevel problems constitute a wide class of optimization problems which generalizes the class of weak and strong ones.

As it is well-known, weak nonlinear bilevel problems are difficult to solve from both theoretical and numerical aspects. Due to the generalization above, strong-weak bilevel problems present more difficulties. In fact, the problem (S) consists to minimize the marginal function u(x) = inf y∈M(x) sup z∈M(x) F (x, y, z) which has a more complicated expression than the marginal function considered in [START_REF] Aboussoror | Strong-weak Stackelberg problems in finite dimensional spaces[END_REF] for weak bilevel problems. In general, even if the data are convex and differentiable, the marginal function u(.) is neither convex nor differentiable. Furthermore, the multifunction M(.) which represents the constraints of u(.) does not possess enough useful topological properties. This explains the difficulties encountered in particular in the study of the existence of solutions to such a class of problems. For instance, the bilevel problem (S) may fail to have solutions even if the variables x, y and z are in compact sets and the objective functions F and f are continuous (see Example 3 below). This fact results principally from the lack of lower semicontinuity of the multifunction M(.). Then, let us summarize the following studies concerning this subject. The existence of solutions for weak and strong-weak bilevel problems in the linear finite dimensional case was established respectively in [START_REF] Aboussoror | Weak linear bilevel programming problems: existence of solutions via a penalty method[END_REF] and [START_REF] Zheng | A new method for strong-weak linear bilevel programming problem[END_REF]. The studies are based on the use of linear duality and penalty methods. However, the nonlinear case presents more difficulties. For the finite dimensional case, sufficient conditions ensuring the existence of solutions to nonlinear weak bilevel programming problems are given in [START_REF] Aboussoror | Weak bilevel programming problems: existence of solutions[END_REF]. So that, it is interesting to consider the class of infinite dimensional nonlinear strong-weak bilevel problems. In this case, the obtained results shall also be applied to the infinite dimensional nonlinear weak bilevel problems by deleting the variable y in the formulation of problem (S) (that is when F does not depend on y).

Let us consider the following example where the functions F and f are continuous and the variables x, y, and z range over compact sets, but the problem (S) has no solution. Then

Example 3 Let X = R 2 , Y = R, A = 0, 1 × 1, 2 , B = 0, 1 , F and f be the functions defined respestively on R 2 × R × R and R 2 × R by F (x, y, z) = -x 2 yz - 1 2 x 1 x 2 , f (x, z) = 3 2 x 2 - 3 2 z + 1 4 x 2 1 (x 2 + 1) 2 with x = (x 1 , x 2 ). Then M(x) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ {1} if x 2 ∈ 1, 3 2 [0, 1] if x 2 = 3 2 {0} if x 2 ∈ 3 2 ,
w(x, y) = ⎧ ⎨ ⎩ -x 2 y -1 2 x 1 x 2 if x 2 ∈ 1, 3 2 -3 4 x 1 if x 2 = 3 2 -1 2 x 1 x 2 if x 2 ∈ 3 2 , 2 and u(x) = ⎧ ⎨ ⎩ -x 2 -1 2 x 1 x 2 if x 2 ∈ 1, 3 2 -3 4 x 1 if x 2 = 3 2 -1 2 x 1 x 2 if x 2 ∈ 3 2 , 2 . Therefore, the infimal value of problem (S) min x∈A inf y∈M(x) sup y∈M(x) F (x, y, z) is v = -9
4 , but (S) has no solution. Note that the multifunction M(.) is not lower semicontinuous on A.

Before describing the steps of our investigation and giving its advantages, it is useful to give a brief survey on the study given in [START_REF] Aboussoror | Strong-weak Stackelberg problems in finite dimensional spaces[END_REF] considered in the finite dimensional case. For each > 0, the authors first present a regularized problem (S ) of (S). The regularization is based on the use of -approximate solutions of the lower level problem. Under mild assumptions, they show that (S ) admits solutions. Then, approximation results are obtained when the parameter of regularization goes to zero. This procedure of regularization has been inspired from the one given in [START_REF] Loridan | On strict -solutions for a two-level optimization problem[END_REF] for weak bilevel problems. Our goal in this paper is to provide sufficient conditions under which the problem (S) admits solutions. The study is considered in a sequential setting and based on a new regularization procedure and tools from variational convergence. This regularization uses strict -solutions of the lower level problem and is inspired from those given in [START_REF] Aboussoror | Strong-weak Stackelberg problems in finite dimensional spaces[END_REF] and [START_REF] Loridan | On strict -solutions for a two-level optimization problem[END_REF]. Under mild assumptions, we establish the existence of solutions for the regularized problem (S ). Note that for example in the case of real Hausdorff locally convex topological vector spaces, contrary to the regularization used in [START_REF] Aboussoror | Strong-weak Stackelberg problems in finite dimensional spaces[END_REF], this regularization allows us to avoid convexity assumptions. Then, via this regularization and under sufficient conditions, we show that the original bilevel problem (S) has at least one solution. The obtained result can be applied to weak nonlinear bilevel problems in the finite and infinite dimensional cases by deleting the variable y. In particular, it extends and improves the result given in [START_REF] Aboussoror | Weak bilevel programming problems: existence of solutions[END_REF] for weak nonlinear bilevel programming problems. In fact, the result will be obtained under weaker assumptions without using convexity assumptions. Furthermore, our result gives a generalization from the linear finite dimensional case to the nonlinear infinite dimensional one of the result given in [START_REF] Zheng | A new method for strong-weak linear bilevel programming problem[END_REF]. However, since the data are nonlinear, this generalization requires a new approach and other tools than those used in [START_REF] Zheng | A new method for strong-weak linear bilevel programming problem[END_REF]. Note that one of the used sufficient conditions reduces the problem of existence of solutions for (S) to a problem of the existence of a common solution of two parametric one-level optimization problems. Therefore, according to our study, we are led to investigate in one-level optimization instead of two-level optimization.

The outline of the paper is as follows. In Section 2, we recall and establish some basic results concerning the lower level problem. In section 3, we first present our regularization and prove the existence of solutions to the regularized problem under mild assumptions. Finally, we establish our main result on the existence of solutions to the original problem (S).

The Lower Level Problem: Basic Results

In this section, we recall and establish some fundamental results concerning the lower level problem that we need in the sequel.

Definition 1

Let Z be a Hausdorff topological space. We recall the following definitions 1) Let C be a nonempty subset of Z. The set C seq defined by

C seq = a ∈ Z/ ∃ a n ∈ C, a n → a, as n → +∞ is called the sequential closure of C. 2) Let (C n ) n∈N be a sequence of nonempty subsets of Z. The sets lim inf n→+∞ C n and lim sup n→+∞ C n are defined as follows i) lim inf n→+∞ C n = y ∈ Z/ ∃ y n → y, as n → +∞, y n ∈ C n , ∀n ∈ N , ii) lim sup n→+∞ C n = y ∈ Z/ ∃ y n k → y, as k → +∞, y n k ∈ C n k , ∀k ∈ N .
Throughout the rest of the paper, the sets A and B will be equipped with the induced topologies of X and Y respectively. Moreover, we will always assume that B is sequentially compact.

Remark 1 In our case, the topological space Y is first countable. So that, the topological closure and the sequential one coincide. In the sequel, we will work with the sequential closure because this notion will be associated with the sequential compactness of B.

We will use the following assumptions.

(2.1) The function f is sequentially lower semicontinuous on the topological space A×B, (2.2) For any (x, z) ∈ A × B, and any sequence (x n ) converging to x in A, there exists a sequence

(z n ) in B such that lim sup n→+∞ f (x n , z n ) ≤ f (x, z),
(2.3) For any (x, z) ∈ A × B, and any sequence (x n ) converging to x in A, there exists a sequence

(z n ) converging to z in B such that lim sup n→+∞ f (x n , z n ) ≤ f (x, z). For x ∈ A, set v(x) = inf z∈B f (x, z). Remark 2 1) Assumption (2.
2) is weaker than assumption (2.3). Note that assumptions (2.1)-(2.3) are weaker than the continuity of the function f and are used in several papers; we cite for example [1-3, 6, 17, 19-21]. 2) Assume that assumption (2.1) is satisfied. Then, since B is sequentially compact, it follows that for any x ∈ A, the problem P(x) has at least one solution and v(x) is a finite real number.

Let x ∈ A such that v(x) is a finite real number. For > 0, let M( , x) and M s ( , x) denote respectively the sets of -approximate solutions and strict -solutions of the lower level problem P(x), i.e.,

M( , x) = z ∈ B/ f (x, z) ≤ v(x) + and M s ( , x) = z ∈ B/ f (x, z) < v(x) + .
Then, we have the following convergence results.

Proposition 1 Assume that assumptions (2.1) and (2.2) hold. Then, for any sequence

(x n ) converging to x in A, we have i) lim sup n→+∞ M( , x n ) ⊂ M( , x), ∀ ≥ 0, ii) lim sup n→+∞ M( n , x n ) ⊂ M(x), ∀ n 0 + , iii) lim sup n→+∞ M s ( n , x n ) seq ⊂ M(x), ∀ n 0 + .
Proof For i) and ii) see [START_REF] Loridan | On strict -solutions for a two-level optimization problem[END_REF]. We give the proof of iii

M s ( , x) seq ⊂ lim inf n→+∞ M s ( , x n ) seq .
That is the multifunction M s ( , .) seq is sequentially lower semicontinuous on A.

Proof From the sequential compactness of B and assumptions (2.1) and (2.3), we have (see [START_REF] Loridan | On strict -solutions for a two-level optimization problem[END_REF])

M s ( , x) ⊂ lim inf n→+∞ M s ( , x n ).
Then

M s ( , x) seq ⊂ lim inf n→+∞ M s ( , x n ) seq .
Since Y is first countable, then, the set lim inf n→+∞ M s ( ,

x n ) is closed. Therefore M s ( , x) seq ⊂ lim inf n→+∞ M s ( , x n ) ⊂ lim inf n→+∞ M s ( , x n ) seq .
Remark 3 Since the set of strict -solutions of the lower level problem is only defined for > 0, then, the case = 0 can not be considered in Proposition 2. However, for the case of -approximate solutions, the sequential lower semicontinuity of the multifunction M(.) corresponding to the case where = 0 (M(.) = M(0, .)) requires strong assumptions to be satisfied. This property plays a crucial role in the existence of solutions to weak and strong-weak bilevel problems.

The First Level Problem: Existence of Solutions

In this section, we first present our regularization and prove the existence of solutions to the regularized problem under mild assumptions. Then, under sufficient conditions, using the regularization and the notion of variational convergence, we establish the existence of solutions to problem (S). We note that contrary to the study given in [START_REF] Aboussoror | Strong-weak Stackelberg problems in finite dimensional spaces[END_REF], no convexity properties are required for our investigation.

The Regularized Problem : Existence of Solutions

First, note that it is not difficult to show that the bilevel problem (S) admits solutions under the lower semicontinuity of the multifunction M(.) and additional mild assumptions. Unfortunately, as it is well-known in the literature, the property of lower semicontinuity of M(.) is not satisfied in general. So that, the first step in our investigation is to substitute the multifunction M(.) with M s ( , .) seq which is lower semicontinuous under mild assumptions (Proposition 2). This substitution will be as follows.

For > 0, we consider the following regularized problem of (S) (S ) :

min x∈A inf y∈M(x) sup z∈M s ( ,x) seq F (x, y, z).
As mentioned above, such a regularization is new and is inspired from those given in [START_REF] Aboussoror | Strong-weak Stackelberg problems in finite dimensional spaces[END_REF] and [START_REF] Loridan | On strict -solutions for a two-level optimization problem[END_REF]. Define the following marginal functions on A × B and A respectively by

w (x, y) = sup z∈M s ( ,x) seq F (x, y, z) u (x) = inf y∈M(x)
w (x, y).

In order to prove the existence of solutions to problem (S ), we will show under sufficient conditions that the marginal function u (.) is sequentially lower semicontinuous on A sequentially compact.

The following assumptions will be used. For (x, y) ∈ A × B, set f x (.) = f (x, .) and F x,y (.) = F (x, y, .).

Remark 4

1) Assumption (3.3) generalizes the following one which was used in [START_REF] Aboussoror | Weak bilevel programming problems: existence of solutions[END_REF] for weak nonlinear bilevel programming problems, namely :

• For any x ∈ A, the following problems admit a common solution max y∈B f 1 (x, y) and min

y∈B f 2 (x, y)
where f 1 and f 2 are respectively the objective functions of the leader and the follower.

2) Assumption (3.3) is satisfied for example when F and f have the following decompositions

F (x, y, z) = h(x, y) -ĥ(x, y)K(x, z) and f (x, z) = g(x) + ĝ(x)K(x, z)
where h, ĥ : X × Y → R, g, ĝ : X → R and K : X × Y → R are functions, with ĥ(x, y) > 0 and ĝ(x) > 0, ∀ (x, y) ∈ A × B.

3) Consider the case where Y is a Banach space and assume that the following assumptions are satisfied i) The set B is convex, ii) For every x ∈ A, the function f x (.) is convex and lower semicontinuous on Y , iii) For every (x, y) ∈ A × B, the function F x,y (.) is concave and upper semi continuous on Y .

Then, assumption (3.3) can be written as

• For any (x, y) ∈ A × B, there exists z ∈ B, such that 0 ∈ ∂f x (z) + N B (z) ∩ ∂(-F x,y (z)) + N B (z)
where N B (z) is the normal cone to B at z (see for example [START_REF] Azé | Elements d'analyse convexe et variationnelle[END_REF]).

We begin by the sequential lower semicontinuity of the marginal function w (., .). The result can be deduced from the results given in [START_REF] Lignola | Semicontinuities of marginal functions in a sequential setting[END_REF] in a general case. For the convenience of the reader, we give a short proof adapted to our case.

Proof Let (x, y) ∈ A × B and (x n , y n ) be a sequence converging to (x, y) in A × B. Let us show that lim inf n→+∞ w (x n , y n ) ≥ w (x, y).

Let z ∈ M s ( , x) seq . From Proposition 2, there exists z n → z, as n → +∞ and

z n ∈ M s ( , x n ) seq , ∀n ∈ N. Then w (x n , y n ) = sup t∈M s ( ,x n ) seq F (x n , y n , t) ≥ F (x n , y n , z n ).
It follows that

lim inf n→+∞ w (x n , y n ) ≥ lim inf n→+∞ F (x n , y n , z n ) ≥ F (x, y, z). Since z is arbitrary in M s ( , x) seq , it follows that lim inf n→+∞ w (x n , y n ) ≥ sup z∈M s ( ,x) seq F (x, y, z) = w (x, y).
Hence, the marginal function w (., .) is lower semicontinuous on A × B.

The following result shows the sequential lower semicontinuity on A of the marginal function u (.). Assume the contrary. Then, there exists α ∈ R such that

lim inf n→+∞ u (x n ) < α < u (x).
(

) Set lim inf n→+∞ u (x n ) = lim n→+∞ n∈N u (x n ) 2
where N is an infinite subset of N. Then, there exists n 0 ∈ N , such that

u (x n ) < α ∀ n ≥ n 0 , n ∈ N .
So that, for all n ≥ n 0 , n ∈ N , there exists

y n ∈ M(x n ) verifying w (x n , y n ) < α.
Since B is sequentially compact, then, there exists N ⊂ N such that y n → ȳ, as n → +∞, n ∈ N . Then, Proposition 1 implies that ȳ ∈ M(x). Therefore, by Proposition 3, we obtain w (x, ȳ) ≤ lim inf n→+∞ w (x n , y n ) ≤ α.

So that u (x) ≤ α

which gives a contradiction with the second strict inequality in [START_REF] Aboussoror | Existence of solutions to two-level optimization problems with nonunique lower-level solutions[END_REF].

Then, we obtain the following result on the existence of solutions to the regularized problem.

Theorem 1 Let > 0. Assume that assumptions (2.1), (2.3) and (3.1) are satisfied. If moreover, A is sequentially compact, then, the regularized problem (S ) has at least one solution.

Proof From Proposition 4 the marginal function u (.) is sequentially lower semicontinuous on A. Therefore, the result follows from the sequential compactness of A.

Existence of Solutions to Problem (S)

In this section, under sufficient conditions, we will establish the existence of solutions to problem (S). First, for > 0 and x ∈ A, consider the following auxiliary problem

S (x) : min y∈M(x) sup z∈M s ( ,x) seq F (x, y, z).
Then, we have the following result on the existence of solutions to problem S (x).

Proposition 5 Let > 0 and x ∈ A. Let assumptions (2.1), (2.3) and (3.1) hold. Then, the problem S (x) has at least one solution.

Proof From Proposition 3 the marginal function w (., .) is sequentially lower semicontinuous on A × B. So that, the marginal function w (x, .) is sequentially lower semicontinuous on B. On the other hand, the set M(x) is sequentially closed in B sequentially compact. Then, M(x) is sequentially compact and the result follows.

As mentioned in the introduction, the notion of variational convergence will play an important role in our investigation. So that, we recall some results related to this notion. Throughout the rest of the paper, for given problems of the form

(P) : min x∈ A f (x) and (Q) : max x∈ B ĝ(x)
we shall denote by ArgminP and ArgmaxQ their solution sets respectively, where f , ĝ : Then, we have the following well-known important convergence result concerning the solution sets of problems (P ϕ n ) and (P ϕ ).

Z → R
Theorem 2 [START_REF] Attouch | Variational Convergence of Functions and Operators[END_REF][START_REF] Dontchev | Well-Posed Optimization Problems[END_REF] Assume that there exists z * ∈ B such that

F (t, s, z) < F (t, s, z * ).
Then, the sequential continuity of F implies that

F (t n , s n , z n ) < F (
+ ψ M(t) (z)
in the sense that they have the same solution sets and opposite optimal values. Define the functions h and h n on B by

h n (z) = -F (t n , s n , z) + ψ M s ( n ,t n ) seq (z) and h(z) = -F (t, s, z) + ψ M(t) (z)
. Then, we have the following convergence result.

Proposition 6 Assume that assumptions (2.1), (2.2) and (3.1)-(3.3) are satisfied. Then, for any (t, s) ∈ A × B and any sequence (t n , s n ) n converging to (t, s) in

A × B, we have lim sup n→+∞ Argmax R n (t n , s n ) ⊂ Argmax R(t, s).
Proof We will show that the sequence (h n ) converges variationally to h in B.

1) Let z ∈ B and (z n ) be a sequence converging to z in B. Let us show that lim inf n→+∞ h n (z n ) ≥ h(z).

We distinguish the following cases. i) If z ∈ M(t), then, h(z) = -F (t, s, z). On the other hand, assumptions (3.1) and (3.2) imply that lim

n→+∞ F (t n , s n , z n ) = F (t, s, z).
So that, for η > 0, there exists n 0 ∈ N such that

|F (t n , s n , z n ) -F (t, s, z)| < η ∀n ≥ n 0 , n ∈ N. It follows that -F (t, s, z) -η < -F (t n , s n , z n ) ≤ -F (t n , s n , z n ) + ψ M s ( n ,t n ) seq (z n ). Hence h(z) -η < h n (z n ) ∀n ≥ n 0 , n ∈ N. Then lim inf n→+∞ h n (z n ) ≥ h(z) -η.
Since η is arbitrary, we deduce that

lim inf n→+∞ h n (z n ) ≥ h(z). ii) If z ∈ M(t), then, h(z) = +∞. Moreover, z n ∈ M s ( n , t n ) seq for large n ∈ N.
Otherwise, there exists an infinite subset N ⊂ N such that z n ∈ M s ( n , t n ) seq for all n ∈ N . Then, z ∈ lim sup n→+∞ M s ( n , t n ) seq ⊂ M(t) (Proposition 1), which gives a contradiction. We deduce that

h n (z n ) = +∞ for large n ∈ N
and the result is obvious.

2) Let z ∈ B. Let us show that there exists a sequence

(z n ) in B such that lim sup n→+∞ h n (z n ) ≤ h(z).
As above, we distinguish the following cases.

i) If z ∈ M(t), then, h(z) = +∞, and the result is obvious.

ii We have

) If z ∈ M(t), then h(z) = -F (t,
z * n ∈ M(t n ) ⊂ M s ( n , t n ) seq . Then, h n (z * n ) = -F (t n , s n , z * n ). Let N be an infinite subset of N such that lim sup n→+∞ h n (z * n ) = lim sup n→+∞ -F (t n , s n , z * n ) = lim n→+∞ n∈N -F (t n , s n , z * n ) .
Using the sequential compactness of B, there exists N ⊂ N such that z * n → z * as n → +∞, n ∈ N . That is z * is an accumulation point of the sequence (z * n ) n∈N . According to Lemma 1, z * solves the problems max z∈B F (t, s, z) and min z∈B f (t, z).

So that lim sup

n→+∞ h n (z * n ) = lim n→+∞ n∈N {-F (t n , s n , z * n )} = lim n→+∞ n∈N {-F (t n , s n , z * n )} = -F (t, s, z * ) ≤ -F (t, s, z) = h(z).
We conclude that the sequence (h n ) converges variationally to h. Then, using the result of Theorem 2, we obtain

lim sup n→+∞ Argmax R n (t n , s n ) ⊂ Argmax R(t, s).
Let n 0 + and t be an arbitrary point in A. Let us consider the following parametric minimization problems

(Q t,

That is sup

z∈M( x) F ( x, ȳ, z) = F ( x, ȳ, z). (9)
On the other hand, for n ∈ N 1 , ȳn solves the problem S n ( xn ). So that, ȳn ∈ M( xn ). Since ( ȳn ) n∈N 1 converges to ȳ, then it follows from Proposition 1 that ȳ ∈ lim sup n→+∞ M( xn ) ⊂ M( x). Then, from ( 7

For n ∈ N 1 , let y x * ,n be a solution of problem

(Q x * ,n ) considered in Proposition 7 ((Q x * ,n ) is obtained for t = x * ). That is u n (x * ) = inf y∈M(x * ) sup z∈M s ( n ,x * ) seq F (x * , y, z) = sup z∈M s ( n ,x * ) seq F (x * , y x * ,n , z). ( 11
)
From the sequential compactness of B, there exists an infinite subset N 2 of N 1 such that (y x * ,n ) n∈N 2 converges to y x * . Therefore, y x * solves the problem (see Proposition 7)

(Q x * ) : min y∈M(x * ) sup z∈M(x * ) F (x * , y, z).

That is min

y∈M(x * ) sup z∈M(x * ) F (x * , y, z) = sup z∈M(x * ) F (x * , y x * , z). It follows from (10) that sup z∈M(x * ) F (x * , y x * , z) < F ( x, ȳ, z). (12)
In [START_REF] Cao | A partial cooperation model for non-unique linear two-level decision problems[END_REF], for n ∈ N 2 , let z * n ∈ M s ( n , x * ) seq such that

u n (x * ) = F (x * , y x * ,n , z * n ). ( 13
)
Using the sequential compactness of B, we can find N 3 ⊂ N 2 such that z * n → z * which belongs to M(x * ) according to Proposition 1. Since z * ∈ M(x * ), then from (12), we get F (x * , y x * , z * ) < F ( x, ȳ, z).

Hence F (x * , y x * ,n , z * n ) < F ( xn , ȳn , zn ) for large n ∈ N 3 .

Using respectively ( 13) and ( 8), we obtain

u n (x * ) < u n ( xn ) for large n ∈ N 3
which contradicts the optimality of xn to the problem (S n ), for large n ∈ N 3 .

Remark 5 Theorem 3 is also valid for weak nonlinear bilevel problems in the finite and infinite dimensional cases by deleting the variable y. Then, by application of this Theorem, we obtain under weaker assumptions without assuming convexity properties the result given in [START_REF] Aboussoror | Weak bilevel programming problems: existence of solutions[END_REF] for weak nonlinear bilevel programming problems. So that, our result constitutes an extension and an improvement of the result given in [START_REF] Aboussoror | Weak bilevel programming problems: existence of solutions[END_REF]. Moreover, it gives a generalization from the linear finite dimensional case to the nonlinear infinite dimensional one of the result given in [START_REF] Zheng | A new method for strong-weak linear bilevel programming problem[END_REF]. However, according to our investigation, this generalization required a new approach and tools different from those used in [START_REF] Zheng | A new method for strong-weak linear bilevel programming problem[END_REF]. Furthermore, by means of the property in assumption (3.3), we are led to investigate in one-level optimization instead of two-level optimization (see the remark below).

Remark 6

The property considered in assumption (3.3) concerns two parameterized problems in one-level optimization. In order to verify this property, we are led to verify if the solution sets of these two problems have a nonempty intersection. So that, by means of Theorem 3, the problem of the existence of solutions to (S) is reduced principally to the problem of the existence of a common solution of two parameterized one-level optimization problems.

For illustration, let us consider the following example where all assumptions of Theorem 3 are satisfied. F (x, y, z) which admit respectively the following solution sets

Example 4 Let X = R 2 , Y = R, A = [1, 2] × [1, 2] and B = [1, 2]. Let F and f be the functions defined on R 2 × R × R and R 2 × R respectively by F (x, y, z) = -(x 1 + 1 2 x 2 2 )yz 2 f (x, z) = (x 1 -1)x 3 2 z 2 + 1 2 (1 -x 1 ) 2 x 2 with x = (x 1 , x 2
M(x) = [1, 2] if x 1 = 1 {1} if x 1 ∈ ]1, 2]
and M(x, y) = {1}

Then, z = 1 is a common solution to the problems P(x) and Q(x, y). Hence, assumption (3.3) is satisfied. So that, according to Theorem 3 the solution set of the bilevel problem (S) : min

x i ≥1 x i ≤2 i=1,2 inf y∈M(x) sup z∈M(x) {-(x 1 + 1 2 x 2 2 )yz 2 }
is nonempty.

The following example shows that assumption (3.3) is not necessary for the existence of solutions. So that, assumption (3.3) is not necessary.

Conclusion

As we have mentioned, the study of the existence of solutions to strong-weak nonlinear bilevel problems is a difficult task. For illustration, the Example 3 presents a case where the objective functions are continuous and the decision variables range over compact sets, but the considered bilevel problem has no solutions. Then, our main result in Theorem 3 provides sufficient conditions ensuring the existence of solutions to such problems. For that, we have first given a regularization for problem (S) which consists to substitute M(.) with a lower semicontinuous multifunction. Then, the main result is established via this regularization and the notion of variational convergence. The use of such a regularization allowed us to avoid convexity properties contrary to the one used in [START_REF] Aboussoror | Strong-weak Stackelberg problems in finite dimensional spaces[END_REF]. The obtained result can be applied to weak nonlinear bilevel problems in the finite and infinite dimensional cases by deleting the variable y. It is an extension and an improvement of the result given for weak nonlinear bilevel programming problems in [START_REF] Aboussoror | Weak bilevel programming problems: existence of solutions[END_REF]. It is also a generalization of the result given in [START_REF] Zheng | A new method for strong-weak linear bilevel programming problem[END_REF] dealing with strong-weak bilevel problems. The generalization is from the linear finite dimensional case to the nonlinear infinite dimensional one. Another advantage of our result is that we are led to investigate in one-level optimization instead of two-level optimization. On the other hand, we note that the numerical study for such a class of problems is still in its infancy. The linear finite dimensional case was studied in [START_REF] Zheng | A new method for strong-weak linear bilevel programming problem[END_REF], where numerical results are given. Another interesting subject is the numerical experiments of the nonlinear strong-weak bilevel problems. This is out of the scope of this paper and will be the subject of another paper.

  2 which is a compact set for all x ∈ A. Set w(x, y) = sup z∈M(x) F (x, y, z), u(x) = inf y∈M(x) w(x, y) and v = inf x∈A u(x).

( 3 . 1 )

 31 The function F is sequentially lower semicontinuous on A × B × B, (3.2) The function F is sequentially upper semicontinuous on A × B × B, (3.3) For any (x, y) ∈ A × B, the problems max z∈B F (x, y, z) and min z∈B f (x, z) admit a common solution.

Proposition 4

 4 Let > 0. Assume that assumptions (2.1), (2.3) and (3.1) are satisfied. Then, the marginal function u (.) is sequentially lower semicontinuous on A. Proof Let x ∈ A and (x n ) be a sequence converging to x in A. Let us show that lim inf n→+∞ u (x n ) ≥ u (x).

  ), we have inf y∈M(x * ) sup z∈M(x * ) F (x * , y, z) < sup z∈M( x) F ( x, ȳ, z). It follows from (9) that inf y∈M(x * ) sup z∈M(x * ) F (x * , y, z) < F ( x, ȳ, z).

Example 5

 5 Let X = Y = R, A = [2, 3] and B = [-5, 5]. Let F and f be the functions defined on R × R × R and R × R respectively byF (x, y, z) = xyz f (x, z) = 0 i fz 2x 2 ≤ 0 z 2x 2 if z 2x 2 > 0. For (x, y) ∈ A × B,consider the problems P(x) : min z∈[-5,5] f (x, z) and Q(x, y) : max z∈[-5,5] F (x, y, z). So that, assumptions (2.1), (2.3), (3.1) and (3.2) are satisfied. Moreover, A and B are compact sets. With the same notations as above in Example 4, we have M(x) = [-x, x] and M(x, y) = {-5}.If follows that assumption (3.3) is not satisfied. On the other hand, we can easily check that x = 2 solves the problem (yz).

Proposition 3

 3 Let > 0. Assume that assumptions (2.1),(2.3) and (3.1) are satisfied. Then, the marginal function w (., .) is sequentially lower semicontinuous on A × B.

  are functions, Z is a Hausdorff topological space, A and B are nonempty subsets of Z.

	Definition 2 [8, 12] Let ϕ Consider the problems (P ϕ n ) : min y∈Z	ϕ n (y)	and	(P ϕ ) : min y∈Z	ϕ(y).

n , ϕ : Z → R, n ∈ N be functions. We say that the sequence (ϕ n ) converges variationally to ϕ in Z if the following properties are satisfied i) For any y ∈ Z and any sequence (y n ) converging to y in Z, we have

lim inf n→+∞ ϕ n (y n ) ≥ ϕ(y).

ii) For any y ∈ Z, there exists a sequence (y n ) in Z such that lim sup n→+∞ ϕ n (y n ) ≤ ϕ(y).

  Assume that the sequence (ϕ n ) converges variationally to ϕ in Z. Then Argmin P ϕ n ⊂ Argmin P ϕ , ∀ n 0 + , where n -Argmin P ϕ n denotes the set of

	i) lim sup n→+∞	Argmin P ϕ n ⊂ Argmin P ϕ ,
	ii) lim sup	n -
	n→+∞

n -approximate solutions of problem (P ϕ n ). Lemma 1 Assume that assumptions (2.1), (2.2), (3.1) and (3.2) are satisfied. Let (t, s) ∈ A × B and (t n , s n ) be a sequence converging to (t, s) in A × B. For n ∈ N, let z n be a common solution of problems max z∈B F (t n , s n , z) and min z∈B f (t n , z). Then, any accumulation point of the sequence (z n ) n∈N solves the problems max z∈B F (t, s, z) and min z∈B f (t, z). Proof Let z be an accumulation point of the sequence (z n ) n∈N . Since Y is first countable, then, there exists N ⊂ N such that z n → z as n → +∞, n ∈ N . Since z n ∈ M(t n ), for n ∈ N , it follows that z ∈ lim sup n→+∞ M(t n ) ⊂ M(t) (Proposition 1). That is z solves the problem min z∈B f (t, z). Let us show that z solves the problem max z∈B F (t, s, z).

  + . Let (t, s) ∈ A × B and (t n , s n ) n be a sequence converging to (t, s) in A × B and consider the following auxiliary problems

	Let n	0 R n (t n , s n ) :	max z∈M s ( n ,t n ) seq F (t n , s n , z)
	and
			R(t, s) :	max z∈M(t)	F (t, s, z).
	Then, R n (t n , s n ) and R(t, s) are respectively equivalent to the following problems
		min z∈B	-F (t n , s n , z) + ψ M s ( n ,t n ) seq (z)
	and
			min z∈B	-F (t, s, z)
				t n , s n , z * )	for large n ∈ N
	which contradicts the optimality of z n for large n ∈ N to the problem
				max z∈Y	F (t n , s n , z).

In the sequel, for a subset B of B, ψ B (.) denotes the indicator function of the set B, i.e.,

ψ B (x) = 0 ify ∈ B +∞ if y ∈ B.

  s, z). According to assumption (3.3), for n ∈ N,

	let z * n be a common solution of problems
	max z∈B	F (t n , s n , z)	and	min

z∈B f (t n , z).

  For n ∈ N, let y t,n be a solution of problem (Q t,n ) and y t be an accumulation point of the sequence (y t,n ) n∈N . Then, y t solves the problem (Q t ), i.e., is sequentially compact). From the sequential compactness of B, there exists an infinite subset N 2 of N 1 such that the sequence (z t,n ) n∈N 2 converges to z t ∈ B.Let N 3 be an infinite subset of N 2 such that (z * n ) n∈N 3 converges to z * ∈ B. Proposition 1 implies that z * ∈ M(t). Then, from (3), we have F (t, y * , z * ) < w(t, y t ). Now, we are able to state our main result on the existence of solutions to problem (S). + . According to Theorem 1, for n ∈ N, let xn be a solution of the regularized problem (S n ). For n ∈ N, let ȳn be a solution of problem S n ( xn ) (Proposition 5). Since A × B is sequentially compact, then, there exists an infinite subset N of N such that the subsequence ( xn , ȳn ) n∈N converges to ( x, ȳ) ∈ A × B. Let us show that x solves the problem (S). Argmax R n ( xn , ȳn ), n ∈ N (such a point exists from the sequential compactness of the set M s ( n , xn ) seq and the sequential continuity of the function F ( xn , ȳn , .) on B). Using again the sequential compactness of the set B, we deduce that there exists an infinite subset N 1 of N such that the sequence (z n ) n∈N 1 converges to z.

	and i.e.,
	(Q t ) : u n ( xn ) = inf min y∈M(t) y∈M( xn ) z∈M s ( n , xn ) sup z∈M(t) F (t, y, z). sup F ( xn , y, z) seq For (x, y) ∈ A × B and n ∈ N, set w(x, y) = sup z∈M(x) F (x, y, z) = sup F ( xn , ȳn , z). Using (5), we get seq z∈M s ( n , xn ) F (t, y * , z * ) < F (t, y t , z t ). Let zn ∈ M s ( n , xn ) seq such that So that, the sequential continuity of F implies that and w n (x, y) = w n (x, y) F (t, y * , z * n ) < F (t, y t,n , z t,n ) for large n ∈ N 3 . z∈M s ( n , xn ) seq u n ( xn ) = sup F ( xn , ȳn , z) = F ( xn , ȳn , zn )	(8)
	= Then, using respectively (6) and (4), we obtain sup z∈M s ( n ,x) seq w n (t, y * ) < w n (t, y t,n ) for large n ∈ N 3 F (x, y, z). which contradicts the optimality of y t,n to the problem (Q t,n ) for large n ∈ N 3 . i.e., zn ∈ Proposition Proposition 7 Assume that assumptions (2.1), (2.2) and (3.1)-(3.3) are satisfied. lim sup 0 + . For x ∈ A and n ∈ N, set 6 implies that Let n u(x) = inf z∈M(x) y∈M(x) sup F (x, y, z) and u n (x) = u n (x). z ∈ lim sup
	Recall that
	u n (x) = inf y∈M(x)	w n (x, y)
	z∈M(t) = inf y∈M(x) z∈M s ( n ,x) sup seq F (t, y So that Theorem 3 Let assumptions (2.1), (2.3) and (3.1)-(3.3) hold. If moreover, A is sequentially F (x, y, z). compact, then, the original strong-weak nonlinear bilevel problem (S) has at least one F (t, y t,n be solution.
	a solution of the problem
	R n (t, y t,n ) : w n (t, y t,n ) = F (t, y t,n , z t,n ) max z∈M s ( n ,t) seq F (t, y t,n , z) (such a solution exists since the function F (t, y t,n , .) is sequentially continuous and i.e., (4) M s ( n , t) Proof Let n 0 Feasibility: We obviously have x ∈ A. Then, Proposition 6 implies that Optimality: Assume that there exists x * ∈ A such that
	z t ∈ lim sup	u(x * ) < u( x)
	i.e.,
	That is z t solves the problem R(t, y t ) : inf y∈M(x * ) sup z∈M(x * ) F (x * , y, z) < inf max z∈M(t) y∈M( x) F (t, y t , z). sup z∈M( x) Hence	F ( x, y, z).
	n ) : F (x * , y, z) < sup min y∈M(t) sup z∈M s ( n ,t) seq F ( x, y, z) F (t, y, z) z∈M(x * ) sup z∈M( x) For n ∈ N , we have ȳn is a solution of problem So that inf y∈M(x * ) w(t, y (6) (7) ∀ y ∈ M( x). S n ( xn ) : min z∈M s ( n , xn ) seq y∈M( xn ) sup F ( xn , y, z)

n→+∞ Argmin Q t,n ⊂ Argmin Q t .

Proof Feasibility: We have (y t,n ) n∈N ⊂ M(t) which is sequentially closed. Then, y t ∈ M(t). Optimality: Assume that there exists y * ∈ M(t) such that

w(t, y * ) = sup * , z) < w(t, y t ). * , z) < w(t, y t ) ∀ z ∈ M(t). (

3

)

There exists

N 1 ⊂ N such that y t,n → y t as n → +∞, n ∈ N 1 . For n ∈ N 1 , let z seq n→+∞ Argmax R n (t, y t,n ) ⊂ Argmax R(t, y t ). t ) = F (t, y t , z t ).

(5)

For

n ∈ N 2 , let z * n ∈ M s ( n ,

t) seq such that w n (t, y * ) = sup z∈M s ( n ,t) seq F (t, y * , z) = F (t, y * , z * n ). n→+∞ Argmax R n ( xn , ȳn ) ⊂ Argmax R( x, ȳ).


Yangwei Liu
email: liuyangwei@126.com

Baojie Liu

Lipeng Lu

An Exponential Decay Model for the Deterministic Correlations in Axial Compressors

Keywords:

The average-passage equation system (APES) provides a rigorous mathematical framework to account for the UBRI in steady state environment by introducing the deterministic correlations (DC). How to model the DC is the key in APES method. The primary purpose of this study is to develop a DC model for compressor routine design. A 3D viscous unsteady and time-averaging CFD flow solver is developed to investigate the APES technique. Steady, unsteady and time-averaging simulations are conducted on the investigation of the UBRI in the first stage of NASA 67 compressor. Based on DC characteristics and its effects on time-averaged flow, an exponential decay DC model is proposed and implemented into the time-averaging solver. Based on the unsteady simulation, the proposed model is validated by comparing DC distributions and mean flow fields. The comparison indicates that the proposed model can take into account the major part of UBRI and provide significant improvements for predicting spanwise distributions of flow properties in axial compressors, compared with the steady mixing plane method.

INTRODUCTION

The flow within turbomachinery is inherently unsteady and the unsteady blade row interaction (UBRI) has a large impact on turbomachinery performance. However, unsteady calculations are still too time consuming for use in routine design and so steady mixing plane method that neglects UBRI is widely used [START_REF] Denton | Some Limitations of Turbomachinery CFD[END_REF]. A steady simulation taking the unsteadiness into account may be a solution of this tricky problem. By applying three averaging operators (ensemble, time and passage averaging operator) to Navier-Stokes equations, Adamczyk proposed the average-passage equation system (APES) [START_REF] Adamczyk | Model equations for simulating flows in multistage turbomachinery[END_REF], which could provide a rigorous mathematical framework for accounting for the UBRI in steady state calculation by introducing deterministic correlations (DC). The DC in momentum equation are called deterministic stresses (DS) and the DC in energy equation are called deterministic enthalpy fluxes (DEF). How to model the DC to close the equation system is the key in the APES.

In recent years, many efforts have been made and some DC models have been proposed [START_REF] Adamczyk | A Model for closing the invisid form of the average passage equation system[END_REF][START_REF] Rhie | Development and application of a multistage Navier-Stokes solver-Part I: Multistage modeling using bodyforces and deterministic stresses[END_REF][START_REF] Li | Influence of deterministic stresses on flow prediction of a low-speed axial flow compressor rear stage[END_REF][START_REF] Sondak | Modeling turbomachinery unsteadiness with lumped deterministic stresses[END_REF][START_REF] He | Modeling issues for computation of unsteady turbomachinery flows[END_REF][START_REF] Hall | Aerodynamic modeling of multistage compressor flow fields. Part II: modeling deterministic stresses[END_REF][START_REF] Van De Wall | A transport model for the deterministic stresses associated with turbomachinery blade row interactions[END_REF][START_REF] Charbonnier | Steady flow simulation of rotor-stator interactions with a new unsteady flow model[END_REF][START_REF] Stollenwerkl | Deterministic stress modeling for multistage compressor flowfields[END_REF]. Some DC models [START_REF] Adamczyk | A Model for closing the invisid form of the average passage equation system[END_REF][START_REF] Rhie | Development and application of a multistage Navier-Stokes solver-Part I: Multistage modeling using bodyforces and deterministic stresses[END_REF][START_REF] Li | Influence of deterministic stresses on flow prediction of a low-speed axial flow compressor rear stage[END_REF][START_REF] Hall | Aerodynamic modeling of multistage compressor flow fields. Part II: modeling deterministic stresses[END_REF] are axisymmetric based on a hypothesis that the purely unsteady fluctuations are negligible, though the DC is not uniform in circumferential direction and the gradients are large in some regions in the blade passage [START_REF] Liu | Study of modeling unsteady blade row in-teraction in a transonic compressor stage part 1: code development and deterministic correlation analysis[END_REF]. Adamczyk et al. [START_REF] Adamczyk | A Model for closing the invisid form of the average passage equation system[END_REF] proposed the first model using overlapping grids for all blade rows without the interface planes. Rhie et al. [START_REF] Rhie | Development and application of a multistage Navier-Stokes solver-Part I: Multistage modeling using bodyforces and deterministic stresses[END_REF] developed a continuous interface plane approach using overlapping grids for downstream row only. Based on an enhanced wake decay concept, Hall [START_REF] Hall | Aerodynamic modeling of multistage compressor flow fields. Part II: modeling deterministic stresses[END_REF] proposed a deterministic mixing-stress model within the framework of "mixing-plane"-type prediction scheme and could be easily implemented.

Some DC models could present circumferential distributions [START_REF] Sondak | Modeling turbomachinery unsteadiness with lumped deterministic stresses[END_REF][START_REF] He | Modeling issues for computation of unsteady turbomachinery flows[END_REF][START_REF] Van De Wall | A transport model for the deterministic stresses associated with turbomachinery blade row interactions[END_REF][START_REF] Charbonnier | Steady flow simulation of rotor-stator interactions with a new unsteady flow model[END_REF][START_REF] Stollenwerkl | Deterministic stress modeling for multistage compressor flowfields[END_REF]. Some reduced order models by simplified unsteady calculations were proposed [START_REF] Sondak | Modeling turbomachinery unsteadiness with lumped deterministic stresses[END_REF][START_REF] He | Modeling issues for computation of unsteady turbomachinery flows[END_REF], and are still time consuming in comparison with the mixing plane method. Sondak et al. [START_REF] Sondak | Modeling turbomachinery unsteadiness with lumped deterministic stresses[END_REF] developed a "lumped" DC model based on lower-order (inviscid) time-dependent simulations and the model saved 63 percent the CPU time compared with unsteady viscous simulation. He [START_REF] He | Modeling issues for computation of unsteady turbomachinery flows[END_REF] developed a nonlinear harmonic method that is much more efficient than the unsteady time-marching methods while still modeling the dominant nonlinear effects. Van de wall [START_REF] Van De Wall | A transport model for the deterministic stresses associated with turbomachinery blade row interactions[END_REF] proposed a transport model that can take into account the effects related to slicing of the upstream wakes by a downstream blade row. Charbonnier & Leboeuf [START_REF] Charbonnier | Steady flow simulation of rotor-stator interactions with a new unsteady flow model[END_REF] found that the velocity-pressure gradient correlation is a key parameter for the transport model and proposed a closure for this correlation term. Only DS in momentum equations are modeled in the transport models [START_REF] Hall | Aerodynamic modeling of multistage compressor flow fields. Part II: modeling deterministic stresses[END_REF][START_REF] Van De Wall | A transport model for the deterministic stresses associated with turbomachinery blade row interactions[END_REF][START_REF] Charbonnier | Steady flow simulation of rotor-stator interactions with a new unsteady flow model[END_REF][START_REF] Stollenwerkl | Deterministic stress modeling for multistage compressor flowfields[END_REF], though there are also DEF in energy equation.

The above studies showed that the models employed in the APES obviously improved the numerical results in comparison with the mixing plane method. However, none of these DC models has been widely used in the multistage compressor routine design because these models are either too complex, too temperamental or still too time consuming for routine use. Hence, more research work for modeling the DC should be conducted. In our previous studies, the distribution characteristics of DC and the effects of DC on time-averaging simulation were conducted [START_REF] Liu | Study of modeling unsteady blade row in-teraction in a transonic compressor stage part 1: code development and deterministic correlation analysis[END_REF][START_REF] Liu | Investigation of unsteady impeller-diffuser interaction in a transonic centrifugal compressor stage[END_REF][START_REF] Liu | Study of modeling unsteady blade row in-teraction in a transonic compressor stage part 2: influence of deter-ministic correlations on time-averaged flow prediction[END_REF][START_REF] Liu | Numerical investigations of impeller-diffuser interactions in a transonic centrifugal compressor stage using nonlinear harmonic method[END_REF][START_REF] Liu | Investigation of model development for deterministic correlations associated with impeller-diffuser interactions in centrifugal compressors[END_REF]. We found that the timeaveraging simulation with axisymmetric DC can take into account the major part of the unsteady effects on spanwise redistribution. The time-averaging simulation with either DS or DEF can only reproduce a partial influence of unsteady blade row interactions on time-averaged flow. Hence, both DS and DEF should be modeled in the APES approach.

In this paper, the purpose is to propose a DC model that fits compressor routine design. Firstly, a 3D viscous unsteady and time-averaging CFD flow solver is developed to investigate the APES technique. Secondly, steady, unsteady and time-averaging simulations are conducted on the investigation of the UBRI, DC distribution characteristics, and the influence of DC on the time-averaged flow field in axial compressors. Lastly, an exponential decay DC model is proposed and validated.

METHODS

A 3D viscous unsteady and time-averaging CFD flow solver is developed to investigate the APES technique. Then steady, unsteady and time-averaging simulations are conducted on the investigation of the UBRI in the first transonic stage of NASA 67 at design condition. Based on the studies of DC distribution characteristics, and the influence of DC on the time-averaged flow field, an exponential decay DC model is proposed and validated.

Numerical Solver

The unsteady and time-averaging CFD flow solver was developed based on the Denton steady code [START_REF] Denton | The calculation of 3d viscous flow through multistage turbomachines[END_REF]. The time-averaging CFD flow solver, adopting APES, was developed to account for the unsteady blade row interactions within the framework of a continuous interface plane approach. The numerical methods were improved to enhance the accuracy, convergence and flexibility, including grid generations, spatial/temporal discretisation of governing equations, boundary conditions and acceleration techniques [START_REF] Liu | Investigations of steady state numerical techniques for predicting flows through multistage compressors[END_REF].

The form of the APES used in this study was developed through the consecutive application of ensemble averaging operator and time averaging operator (see Adamczyk [START_REF] Adamczyk | Model equations for simulating flows in multistage turbomachinery[END_REF]) applied to the Navier-Stokes equation, the same form as that used by Hall [START_REF] Hall | Aerodynamic modeling of multistage compressor flow fields. Part II: modeling deterministic stresses[END_REF]. The integral form of the time-averaging equations in cylindrical coordinates are:

  ( )( )( ) () cvd cvd cvd d Ud FFF i GGGj H H H kd S QQd                      F F F i G G G j H H H k dS ( ) ( F F F i G G G j H H H k dS ( ) ( ) ( F F F i G G G j H H H k dS ) ( F F F i G G G j H H H k dS           F F F i G G G j H H H k dS ( ) ( F F F i G G G j H H H k dS ( ) (           ( ) ( F F F i G G G j H H H k dS ( ) ( ) ( F F F i G G G j H H H k dS ) (           ) ( F F F i G G G j H H H k dS ) ( ( ) ( F F F i G G G j H H H k dS ( ) ( F F F i G G G j H H H k dS ( ) ( ( ) ( c v d F F F i G G G j H H H k dS ( ) ( F F F i G G G j H H H k dS ( ) ( c v d F F F i G G G j H H H k dS c v d ( ) ( c v d ( ) ( F F F i G G G j H H H k dS ( ) ( c v d ( ) (                      ( ) (           ( ) ( ( ) ( F F F i G G G j H H H k dS ( ) (           ( ) ( F F F i G G G j H H H k dS ( ) ( ( ) ( c v d ( ) ( F F F i G G G j H H H k dS ( ) ( c v d ( ) (           ( ) ( c v d ( ) ( F F F i G G G j H H H k dS ( ) ( c v d ( ) (                               (1)
where [, , , ] ()

U , c F , c G ,
xr x xx rx x cc c r xr rr r xr VV W V VV p VV WV U F G H V VV VV p WV rV rV V rV V r W V p E                         x r r V H VH WV r p              (2)
+ + + xr rVV V He V He V He                                   (6)
2 22 1 1 ( )( ) 2 2 v xr xx rr E cT V V V V V V V V V                (9) 2 22 11 ( )( ) 22 p xr xx rr p HE cT V V V V V VV V V                    ( 10
) VW r     (11)
In above equations,  is the rotation speed. And ij  is the Reynolds stress tensor as followings:

2 2 () 3 ij ij ij SV              2 ( )   2 ( ) 2 ( ) S V 2 ( )   2 ( ) S V 2 ( )   2 ( )   2 ( ) 2 ( ) S V 2 ( )   2 ( ) S V 2 ( )       2 ( )   2 ( )   2 ( )   2 ( ) 2 ( ) S V 2 ( )   2 ( ) S V 2 ( )   2 ( ) S V 2 ( )   2 ( ) S V 2 ( ) (12)
1 ( )

0 ( ) ij ij ij        (13) 11 22 1 1 2 2 11 22 x x x r x r r r ij x r V V VV V x rx r x V V VV V r S rx r r rr VV V V r r x r rr                                                            r V V rr                          ( 14
) l t    , lp tp r rt cc pp     (15)
The laminar flow viscosity l  is obtained from Sutherland's law and the turbulent flow viscosity t  is obtained from the mixing length turbulence model [START_REF] Denton | The use of body force to simulate viscous effects in 3D flow calculations[END_REF].

For unsteady simulation, the dual time stepping approach is used. The second order 3-point Euler backward scheme is used for physical time discretization and the explicit SCREE scheme is employed for pseudo temporal discretization. Local time stepping, implicit residual smoothing and multigrid are employed to accelerate the convergence. The second order central scheme with addition of second and fourth order artificial dissipation is used for space discretisation. The mixing plane method [START_REF] Denton | The calculation of 3d viscous flow through multistage turbomachines[END_REF] is employed in the steady simulation and the continuous interface plane method [START_REF] Rhie | Development and application of a multistage Navier-Stokes solver-Part I: Multistage modeling using bodyforces and deterministic stresses[END_REF] in the time-averaging simulation. The domain scaling method is employed in the unsteady simulation. More details about the solution method can be found in the reference [START_REF] Liu | Investigations of steady state numerical techniques for predicting flows through multistage compressors[END_REF].

Exponential Decay DC Model

In our previous studies [START_REF] Liu | Study of modeling unsteady blade row in-teraction in a transonic compressor stage part 1: code development and deterministic correlation analysis[END_REF][START_REF] Liu | Study of modeling unsteady blade row in-teraction in a transonic compressor stage part 2: influence of deter-ministic correlations on time-averaged flow prediction[END_REF], the distribution characteristics of DC and the effects of DC on timeaveraging simulation were conducted in axial compressors.The main conclusions related to DC modelling are as followings: 1)The DC has large magnitudes at the rotor-stator interface and decreases rapidly in flow direction from the interface. 2) The time-averaging simulation with axisymmetric DC can take into account the major part of the unsteady effects on spanwise redistribution. 3) Either DS or DEF can only reproduce a partial influence of unsteady blade row interactions. Hence, both DS and DEF should be modeled in the APES approach.

Based on above results and other studies [START_REF] Hathaway | Measurements of the Unsteady Flow Field within the Stator Row of a Transonic Axial-Flow Fan-Results and Discussion[END_REF][START_REF] Uzol | Average Passage Flow Field and Deterministic Stresses in the Tip and. Hub Regions of a Multi-Stage Turbomachine[END_REF][START_REF] Poensgen | Three-dimensional Wake Decay inside of a Compressor Cascade and its Influence on the Downstream Unsteady Flow Field-Part 1: Wake Decay Characteristics in the Flow Passage[END_REF], an exponential decay DC model for downstream blade row is proposed as following. upstream blade row by using spatial-temporal transformation [START_REF] Hall | Aerodynamic modeling of multistage compressor flow fields. Part II: modeling deterministic stresses[END_REF]. The DC in the downstream passage can be according to the exponential decay model. Though the model is proposed for DC in downstream blade row to account for the unsteady effects of upstream blade row, it could give reasonable DC in upstream blade row. For upstream blade row, s is the meridional distance upstream from the downstream blade row leading edge and reference point is set at interface plane or 1-3 grid points downstream from interface plane. Then the DC model was added into the developed timeaveraging CFD flow solver. There is only a little increase (less than 1%) in CPU time owing to the addition of the DC model.

(, ) (, ) 1 2 ref s ij ij ref s W f f s r f f s r fun fun W          ( , ) ( , ) 1 2 f f s r f f s r fun fun ( , ) ( , ) 1 2 f f s r f f s r fun fun ( , ) ( , ) 1 2                                                     (16)

Test Case

In this paper, the first transonic compressor stage of NASA 67 was used to validate the proposed model [START_REF] Urasek | Performance of two-stage fan having low-aspectratio first-stage rotor blading[END_REF]. The design rotation speed is 16042.8 rpm, and the total pressure ratio is 1.59. It consists of 22 rotor blades and 34 stator blades. More design details can be seen in the reference [START_REF] Urasek | Performance of two-stage fan having low-aspectratio first-stage rotor blading[END_REF]. In this study, stator blade row was moved upstream close to the rotor blade, resulting that the axial gap between rotor and stator is 10 mm at hub, in order to highlight the UBRI.

Steady, unsteady and time-averaging simulations, employing the developed flow solver, were conducted. The unsteady simulation was simplified by modifying the original blade row airfoil counts from 22: 34 to 22: 33. Then the computational domain consists of 2 rotor blade passages and 3 stator blade passages. The dual time stepping approach was used for unsteady simulation. The number of physical time steps was set to 72 for each cycle and the maximum number of inner iterations was set 150 for each physical time step. The steady and time-averaging simulations were conducted with one rotor passage and one stator passage, and were about 80 times faster than the unsteady simulation with two rotor passages and three stator passages. Two kinds of time-averaging simulations were carried out using exponential decay DC model and DC derived from the unsteady simulation, respectively.

The same grid distribution was given based on the experience employing the Denton's code using mixing length turbulence model. As shown in Figure 1, the grid is 37, 41 and 141 in pitchwise, spanwise and streamwise direction respectively for one rotor or stator passage. The grid density is similar to the reference [START_REF] Chen | Analysis of unsteady blade row interaction using nonlinear harmonic approach[END_REF] for investigating the DLR compressor stage by steady, unsteady and nonlinear harmonic methods.

Total pressure, total temperature and inlet flow angles were specified for the inlet boundary, whereas the static pressure was imposed according to the simple radial equilibrium equation for the outlet boundary. Adiabatic conditions were adopted for all the solid walls.

RESULTS AND DISCUSSION

The results from steady and unsteady simulations are firstly compared. It is found that the steady simulation with mixing plane method can predict the total performance of the single axial compressor well in comparison with the unsteady result for design condition. However, the spanwise distribution of total pressure, total temperature, entropy and Mach number predicted by steady mixing plane method has some discrepancies with the unsteady results (shown later).

In the paper, DesIJ and QesHI represent DS and DEF. And I, J could be R, T or X (R, T and X stand for spanwise, pitchwise and streamwise). For example, DesRT and QesHT represents r VV     and HV     respectively.

DC Analysis

As discussed above, the DC could present the effect of UBRI on time-averaged flow, and how to model the DC is the key in the APES method. The DC can be derived from unsteady simulations directly.

UBRI is a result of potential interaction, vortical interaction (associated with wakes, tip leakage vortex, passage vortices) and shockwave interaction. Figure 2 presents an instantaneous entropy contours at mid span from unsteady simulation. It indicates clearly the rotor wake chopping, stretching and migrating in the downstream stator passage. The circumferential and spanwise transport of wake leads to a circumferential and spanwise redistribution of total temperature and momentum. And accounting for spanwise redistribution is very important in predicting the aerodynamic performance of multistage axial flow compressors.

Figure 3 shows one dimensional distributions of area averaged DC for the whole s3 plane along flow direction in the rotor and stator passage respectively. It is found that DC in the stator is much larger than that in the rotor, which also indicates that the unsteady interaction has much larger effect on the stator than on the rotor for this transonic fan. Hence, later analysis is focused on stator. DC has large magnitudes at the rotor-stator interface and decreases rapidly in flow direction from the interface, especially in the rotor-stator gap. It presents exponential decay characteristic that will be shown later.

DC Comparisons

In this section, the DC distributions resulted from the DC model are compared in detail with the ones from unsteady simulation. We should note that all these DC terms are set zero in steady simulation with mixing plane method.

  c H and Q are conservative variable and inviscid flux vectors. v F , v G and v H are viscid flux vectors. d F , d G , d H and d Q are deterministic flux vectors, which are extra terms compared to the steady flow equations.

2

 2

  the reference point. L is the meridional distance from upstream blade row trailing edge to downstream blade row leading edge. And s is the meridional distance downstream from the upstream blade row trailing edge. s W and ref s W is axisymmetric averaged velocity at local and reference points to account for pressure gradient. 1 c , 2 c and 3 c are empirical coefficients and set to 0.5, 1.0 and 0.25 respectively based on the test case. Usually reference point is set at interface plane or 1-3 grid points upstream from interface plane. Then flow at (, ) sr in the

Figure 1 .

 1 The flow was assumed to be fully turbulent in the computations. (a) S1 flow plane at 50% span (b) Meridional plane (c) Leading and trailing edge of rotor in (a) Computational grid

Figure 2 .

 2 Figure 2. Instantaneous entropy contours at mid span

Figure. 4

 4 and Figure. 5 present circumferential averaged xx VV    and circumferential averaged x VH    . The results show that the DC in stator passage are much larger than the DC in rotor passage. The DC decay quickly from the rotorstator interface plane in the downstream stator. Compared with unsteady results, the proposed exponential decay DC model could predict reasonable DC distribution qualitatively.

Figure. 6

 6 and Figure. 7 present circumferential averaged xx VV    and circumferential averaged x VH    at 10%, 50% and 90% span along chord in the stator passage. The comparisons indicate that the DC model could predict reasonable DC distribution quantitatively.

Figure 4 .

 4 Figure 4. Circumferential averaged xx VV   

Figure 5 .

 5 Figure 5. Circumferential averaged x VH   

Figure 6 . 7 .

 67 Circumferential averaged xx VV    along chord in the stator passage (a)10% span (b) 50% span (c) 90% span Figure Circumferential averaged x VH    along chord in the stator passage2.1.2 Mean Flow FieldsTo further validate the proposed exponential decay DC model, the results from steady simulation (named st), timeaveraging simulation with the DC model (named model) and time-averaging simulation with the DC derived from the unsteady simulation (named dqe) are compared.

Figure. 8 8 .

 88 present spanwise distributions of circumferential averaged total pressure, total temperature and Ma at the stator exit. Though there are still some discrepancies near hub region, the exponential decay model could predict the spanwise distributions of flow parameters well. This indicates that the model can take into account the major part of UBRI and provide significant improvements for predicting spanwise distributions of flow properties in axial compressors, compared with the steady mixing plane method. (a) Total pressure (b) Total temperature (c) Ma Figure Comparison of total pressure, total temperature and Ma at the stator exit

ACKNOWLEDGMENTS

This work is supported by the National Natural Science Foundation of China (No. 51376001, No. 51676007, No. 51420105008), the National Basic Research Program of China (No. 2014CB046405). Yangwei Liu is supported by the China Scholarship Council (CSC) for a one-year visit to the University of Cambridge.

CONCLUSIONS

To develop a DC model for compressor routine design, a 3D viscous unsteady and time-averaging CFD flow solver is developed. Steady, unsteady and time-averaging simulations are conducted on the investigation of the UBRI in the first stage of NASA 67 compressor.

Based on DC characteristics and its effects on timeaveraged flow, an exponential decay DC model is proposed and implemented into the time-averaging solver. There is only a little increase (less than 1%) in CPU time owing to the addition of the DC model.

Based on the unsteady simulation, the proposed model is validated by comparing DC distributions and mean flow fields. The comparisons indicate that the DC model could predict reasonable circumferential averaged DC distribution quantitatively. Furthermore, the model can take into account the major part of UBRI and provide significant improvements for predicting spanwise distributions of flow properties in axial compressors.