Gontran Lance
email: lance@ljll.math.upmc.fr

Emmanuel Trélat
email: emmanuel.trelat@sorbonne-universite.fr

Enrique Zuazua
email: enrique.zuazua@fau.de

Keywords: Optimal shape design, turnpike, numerical analysis, optimal control

 where we have established the turnpike property for some optimal shape design problems. Considering linear parabolic partial differential equations where the shapes to be optimized acts as a source term, we want to minimize a quadratic criterion. Existence of optimal shapes is proved under some appropriate assumptions. We prove and provide numerical evidence of the turnpike phenomenon for those optimal shapes, meaning that the extremal time-varying optimal solution remains essentially stationary; actually, it remains essentially close to the optimal solution of an associated static problem.

Introduction

This paper is devoted to studying large-time shape design problems and to providing evidence of the turnpike phenomenon, stating that the optimal time-varying design mostly stays stationary along the time interval. We complete and comment on results in [START_REF] Lance | Shape turnpike for linear parabolic PDE models[END_REF] and we focus on the numerical solving of those large-time time-varying shape design problems, which we illustrate with a number of numerical simulations. We introduce in Section 1 the general framework and theoretical results which are then numerically illustrated along the paper. We use the relaxation method which is classical in optimal shape design. Section 2 focuses on the numerical implementation issues, with an approach following the theoretical study. Some numerical illustrations of the turnpike phenomenon in shape design are given in Section 3. In Section 4, we give some open questions and issues. We illustrate with numerical examples some geometric properties of optimal solutions. A section is particularly set apart for the open issue of the kinetic interpretation of shallow water equations for solving a given shape design problem. We finally provide some proofs in an appendix.

1 Optimal shape design in parabolic case

Context

Throughout the paper, we denote by: -|𝑄| the Lebesgue measure of a measurable subset 𝑄 ⊂ R 𝑑 , 𝑑 ≥ 1; -(𝑝, 𝑞) the scalar product in 𝐿 2 (Ω) of 𝑝, 𝑞 in 𝐿 2 (Ω); -‖𝑦‖ the 𝐿 2 -norm of 𝑦 ∈ 𝐿 2 (Ω); -𝜒𝜔 the indicator (or characteristic) function of 𝜔 ⊂ R 𝑑 ; -𝑑𝜔 the distance function to the set 𝜔 ⊂ R 𝑑 .

Let Ω ⊂ R 𝑑 (𝑑 ≥ 1) be an open bounded Lipschitz domain. We consider the uniformly elliptic second-order differential operator with 𝑎 𝑖𝑗 , 𝑏 𝑖 ∈ 𝐶 1 (Ω), 𝑐 ∈ 𝐿 ∞ (Ω) with 𝑐 ≥ 0. We consider the operator (𝐴, 𝐷(𝐴)) defined on the domain 𝐷(𝐴) encoding Dirichlet conditions 𝑦 |𝜕Ω = 0; when Ω is 𝐶 2 or a convex polytop in R 2 , we have 𝐷(𝐴) = 𝐻 1 0 (Ω) ∩ 𝐻 2 (Ω). The adjoint operator 𝐴 * of 𝐴, also defined on 𝐷(𝐴) with homogeneous Dirichlet conditions, is given by

𝐴 * 𝑣 = - 𝑑 ∑ 𝑖,𝑗=1 𝜕𝑥 𝑖 ( 𝑎 𝑖𝑗 (𝑥)𝜕𝑥 𝑗 𝑣 ) - 𝑑 ∑ 𝑖=1 𝑏 𝑖 (𝑥)𝜕𝑥 𝑖 𝑣 + ( 𝑐 - 𝑑 ∑ 𝑖=1 𝜕𝑥 𝑖 𝑏 𝑖 )

𝑣

and is also uniformly elliptic, see [START_REF] Evans | Partial differential equations[END_REF]Definition Chapter 6]. The operators 𝐴 and 𝐴 * do not depend on 𝑡 and have a constant of ellipticity 𝜃 > 0 (for 𝐴 written in non-divergence form), i.e.:

𝑑

∑

𝑖,𝑗=1

𝑎 𝑖𝑗 (𝑥)𝜉 𝑖 𝜉 𝑗 ≥ 𝜃|𝜉| 2 ∀𝑥 ∈ Ω.

Moreover, we assume that 𝜃 > 𝜃 1 (1) where 𝜃 1 is the largest root of the polynomial

𝑃 (𝑋) = 𝑋 2 4 min(1, 𝐶𝑝) -‖𝑐‖ 𝐿 ∞ (Ω) 𝑋 - ∑ 𝑑 𝑖=1 ‖𝑏 𝑖 ‖ 𝐿 ∞ (Ω) 2
where 𝐶𝑝 is the Poincaré constant of Ω. This assumption is used to ensure that an energy inequality is satisfied with constants not depending on the final time 𝑇 (see Appendix A.2). We assume throughout that the operator 𝐴 satisfies the classical maximum principle (see [8, sec. 6.4]) and that

𝑐 * = 𝑐 - 𝑑 ∑ 𝑖=1 𝜕𝑥 𝑖 𝑏 𝑖 ∈ 𝐶 2 (Ω).
A typical example satisfying all assumptions above is the Dirichlet-Laplacian, which we will consider in some of our numerical simulations.

We recall that the Hausdorff distance between two compact subsets 𝐾 1 , 𝐾 2 of R 𝑑 is defined by

𝑑 ℋ (𝐾 1 , 𝐾 2 ) = sup ( sup 𝑥∈𝐾2 𝑑 𝐾1 (𝑥), sup 𝑥∈𝐾1 𝑑 𝐾2 (𝑥) ) .

Setting

Let 𝐿 ∈ (0, 1). We define the set of admissible shapes

𝒰 𝐿 = {𝜔 ⊂ Ω measurable | |𝜔| ≤ 𝐿|Ω|} (2)
Dynamical optimal shape design problem (DSD T ) Let 𝑦 0 ∈ 𝐿 2 (Ω) and let 𝛾 1 ≥ 0, 𝛾 2 ≥ 0 be arbitrary. We consider the parabolic equation controlled by a (measurable) time-varying map 𝑡 ↦ → 𝜔(𝑡) of subdomains

𝜕 𝑡 𝑦 + 𝐴𝑦 = 𝜒 𝜔(•) , 𝑦 |𝜕Ω = 0, 𝑦(0) = 𝑦 0 (3)
Given 𝑇 > 0 and 𝑦 𝑑 ∈ 𝐿 2 (Ω), we consider the dynamical optimal shape design problem (DSD T ) of determining a measurable path of shapes 𝑡 ↦ → 𝜔(𝑡) ∈ 𝒰 𝐿 that minimizes the cost functional

𝐽 𝑇 (𝜔(•)) = 𝛾 1 2𝑇 𝑇 ∫ 0 ‖𝑦(𝑡) -𝑦 𝑑 ‖ 2 𝑑𝑡 + 𝛾 2 2 ‖𝑦(𝑇 ) -𝑦 𝑑 ‖ 2
where 𝑦 = 𝑦(𝑡, 𝑥) is the solution of (3) corresponding to 𝜔(•).

Static optimal shape design problem (SSD)

For the same target function 𝑦 𝑑 ∈ 𝐿 2 (Ω), we consider the associated static shape design problem min

𝜔∈𝒰 𝐿 𝛾 1 2 ‖𝑦 -𝑦 𝑑 ‖ 2 , 𝐴𝑦 = 𝜒𝜔, 𝑦 |𝜕Ω = 0 (SSD)
1.3 Results

Existence of solutions

Existence and uniqueness of solutions have been established in [START_REF] Lance | Shape turnpike for linear parabolic PDE models[END_REF]. To facilitate the reading, we give a sketch of proof in the static case in Section A.3. In few words, we consider a relaxation of (DSD T ) and (SSD) with the following idea: we replace the source term 𝜒𝜔, the characteristic function of some shape 𝜔 ∈ 𝒰 𝐿 by a function 𝑎 ∈ 𝐿 ∞ (Ω, [0, 1]) (convexification, also called relaxation in classical optimal shape design theory). We can then use classical arguments of optimal control theory such as the use of the adjoint variable and the application of first-order optimality conditions coming from the Pontryagin maximum principle (see [16, Chapter 2, Theorem 1.4; Chapter 3, Theorem 2.1]). Convexification methods can be found in [START_REF] Allaire | Conception optimale de structures[END_REF] and are a particular case of homogenization methods in shape design which are also very powerful for numerical design. In Section 2.1 we introduce the convexified (relaxed) version of the problems (DSD T ) and (SSD). Existence of optimal shapes is ensured under appropriate assumptions (see [START_REF] Lance | Shape turnpike for linear parabolic PDE models[END_REF] for complete proofs).

Theorem 1 ([12]). We distinguish between Lagrange and Mayer cases. 1. 𝛾 1 = 0, 𝛾 2 = 1 (Mayer case): If 𝐴 is analytic hypoelliptic in Ω then there exists a unique optimal shape 𝜔 𝑇 solution of (DSD T ). 2. 𝛾 1 = 1, 𝛾 2 = 0 (Lagrange case): Assuming that 𝑦 0 ∈ 𝐷(𝐴) and that 𝑦 𝑑 ∈ 𝐻 2 (Ω):

(i) If 𝑦 𝑑 < 𝑦 0 or 𝑦 𝑑 > 𝑦 1 then there exist unique optimal shapes ω and 𝜔 𝑇 , respectively solutions of (SSD) and of (DSD T ). (ii) There exists a function 𝛽 such that if 𝐴𝑦 𝑑 ≤ 𝛽, then there exists a unique optimal shape ω, solution of (SSD).

Some illustrations are provided in Section 2.2. Under the assumptions of Theorem 1, we next derive turnpike properties.

Turnpike

The turnpike phenomenon was first observed and investigated by economists for discrete-time optimal control problems (see [START_REF] Dorfman | Linear programming and economic analysis[END_REF][START_REF] Mckenzie | Turnpike theorems for a generalized leontief model[END_REF]). There are several possible notions of turnpike properties, some of them being stronger than the others (see [START_REF] Zaslavski | Turnpike theory of continuous-time linear optimal control problems[END_REF]). Exponential turnpike properties have been established in [START_REF] Grüne | Exponential sensitivity and turnpike analysis for linear quadratic optimal control of general evolution equations[END_REF][START_REF] Porretta | Long time versus steady state optimal control[END_REF][START_REF] Porretta | Remarks on long time versus steady state optimal control[END_REF][START_REF] Trélat | Integral and measure-turnpike properties for infinite-dimensional optimal control systems[END_REF][START_REF] Trélat | Steady-state and periodic exponential turnpike property for optimal control problems in Hilbert spaces[END_REF][START_REF] Trélat | The turnpike property in finite-dimensional nonlinear optimal control[END_REF] for the optimal triple resulting of the application of Pontryagin's maximum principle, ensuring that the extremal solution (state, adjoint and control) remains exponentially close to an optimal solution of the corresponding static controlled problem, except at the beginning and at the end of the time interval, as soon as 𝑇 is large enough. This follows from hyperbolicity properties of the Hamiltonian flow.

In [START_REF] Lance | Shape turnpike for linear parabolic PDE models[END_REF] we establish some results on turnpike property for linear parabolic optimal shape design problem. We denote by (𝑦 𝑇 , 𝑝 𝑇 , 𝜒𝜔 𝑇 ) and (ȳ, p, 𝜒ω) the optimal triple of both problems (DSD T ) and (SSD). We show that the dynamical optimal triple, when 𝑇 is large, remains most of the time "close" to the static optimal triple. In the Mayer case, we show that the Hausdorff distance between the dynamical optimal shape and the static one satisfies an exponential turnpike.

Theorem 2 ([12]

). For 𝛾 1 = 0, 𝛾 2 = 1 (Mayer case), for Ω with 𝐶 2 boundary and 𝑐 = 0 there exist 𝑇 0 > 0, 𝑀 > 0 and 𝜇 > 0 such that, for every 𝑇 ≥ 𝑇 0 ,

𝑑 ℋ ( 𝜔 𝑇 (𝑡), ω) ≤ 𝑀 𝑒 -𝜇(𝑇 -𝑡) ∀𝑡 ∈ (0, 𝑇 ).
Concerning the Lagrange case, an exponential turnpike property is conjectured, supported by several numerical simulations (see Section 3) but for theoretical results, we only establish integral and measure turnpike properties.

Theorem 3 ([12]

). For 𝛾 1 = 1, 𝛾 2 = 0 (Lagrange case), there exists 𝑀 > 0 (independent of the final time 𝑇 ) such that

𝑇 ∫ 0 ( ‖𝑦 𝑇 (𝑡) -ȳ‖ 2 + ‖𝑝 𝑇 (𝑡) -p‖ 2 ) 𝑑𝑡 ≤ 𝑀 ∀𝑇 > 0.
Theorem 4 ( [START_REF] Lance | Shape turnpike for linear parabolic PDE models[END_REF]). For 𝛾 1 = 1, 𝛾 2 = 0 (Lagrange case), the state-adjoint pair (𝑦 𝑇 , 𝑝 𝑇 ) satisfies the state-adjoint measure-turnpike property: for every 𝜀 > 0, there exists Λ(𝜀) > 0, independent of 𝑇 , such that

|𝑃 𝜀,𝑇 | < Λ(𝜀) ∀𝑇 > 0 where 𝑃 𝜀,𝑇 = { 𝑡 ∈ [0, 𝑇 ] | ‖𝑦 𝑇 (𝑡) -ȳ‖ + ‖𝑝 𝑇 (𝑡) -p‖ > 𝜀 } .
Proofs are done in [START_REF] Lance | Shape turnpike for linear parabolic PDE models[END_REF] but, for the convenience of the reader, we give a sketch of proof of Theorem 3 in Appendix A.3. Based on the numerical simulations presented in Section 3, we conjecture that the exponential turnpike property is satisfied, i.e., given optimal triples (𝑦 𝑇 , 𝑝 𝑇 , 𝜒𝜔 𝑇 ) and (ȳ, p, ω), there exist 𝐶 1 > 0 and 𝐶 2 > 0 independent of 𝑇 such that

‖𝑦 𝑇 (𝑡) -ȳ‖ + ‖𝑝 𝑇 (𝑡) -p‖ + ‖𝜒 𝜔 𝑇 (𝑡) -𝜒ω‖ ≤ 𝐶 1 ( 𝑒 -𝐶2𝑡 + 𝑒 -𝐶2(𝑇 -𝑡)
)

for a.e. 𝑡 ∈ [0, 𝑇 ].

Numerical implementation

Computation of the cost function is conditioned to solve the PDE (3). We use a variational formulation and a finite element approach. Using FreeFEM++ we consider a mesh of the domain Ω, a finite element space with P1-Lagrange elements and decompose all functions according to this triangulation. To solve numerically (DSD T ) and (SSD), we consider convexification method which introduces a more general problem whose solution is expected to be a shape under the assumptions of Theorem 1 and whose solution is computed via an interior point optimization method (IpOpt, see [START_REF] Wächter | On the Implementation of a Primal-Dual Interior Point Filter Line Search Algorithm for Large-Scale Nonlinear Programming[END_REF]).

Convexification method

The convexification method is a relaxation of the initial problem which allows to use classical tools of PDE optimal control. This is a well known method in shape optimization which has the double benefit to provide strategies to set theoretical results (see Section A.3 and see [START_REF] Privat | Optimal shape and location of sensors for parabolic equations with random initial data[END_REF]) and a framework for numerical solving. Given any measurable subset 𝜔 ⊂ Ω, we identify 𝜔 with its indicator (characteristic) function 𝜒𝜔 ∈ 𝐿 ∞ (Ω; {0, 1}) and, following [START_REF] Allaire | Long time behavior of a two-phase optimal design for the heat equation[END_REF][START_REF] Privat | Optimal shape and location of sensors for parabolic equations with random initial data[END_REF][START_REF] Privat | Optimal observability of the multidimensional wave and Schrödinger equations in quantum ergodic domains[END_REF], we identify 𝒰 𝐿 with a subset of 𝐿 ∞ (Ω). Then, the convex closure of 𝒰 𝐿 in 𝐿 ∞ weak star topology is the set

𝒰 𝐿 = ⎧ ⎨ ⎩ 𝑎 ∈ 𝐿 ∞ ( Ω; [0, 1] ) ⃒ ⃒ ⃒ ∫ Ω 𝑎(𝑥) 𝑑𝑥 ≤ 𝐿|Ω| ⎫ ⎬ ⎭ (4)
which is weak star compact. More generally, we refer the reader to [START_REF] Allaire | Conception optimale de structures[END_REF] for details on convexification and homogenization methods. We define the convexified (or relaxed ) optimal control problem (OCP) T as the problem of determining a control

𝑡 ↦ → 𝑎(𝑡) ∈ 𝒰 𝐿 minimizing the cost 𝐽 𝑇 (𝑎) = 𝛾 1 2𝑇 𝑇 ∫ 0 ‖𝑦(𝑡) -𝑦 𝑑 ‖ 2 𝑑𝑡 + 𝛾 2 2 ‖𝑦(𝑇 ) -𝑦 𝑑 ‖ 2
under the dynamical constraints

𝜕 𝑡 𝑦 + 𝐴𝑦 = 𝑎, 𝑦 |𝜕Ω = 0, 𝑦(0) = 𝑦 0 . (5)
The corresponding convexified static optimization problem is min

𝑎∈𝒰 𝐿 𝛾 1 2 ‖𝑦 -𝑦 𝑑 ‖ 2 , 𝐴𝑦 = 𝑎, 𝑦 |𝜕Ω = 0. (SOP)
Solving (OCP) T and (SOP) under the assumptions of Theorem 1 implies that the optimal solutions of these convexified problems happen to be the upper level sets of the adjoint-state and hence are shapes that are the optimal solutions, respectively, of (DSD T ) and of (SSD). When the assumptions done in Theorem 1 are not satisfied, a relaxation phenomenon might be observed, namely, it may happen that the optimal solution of (OCP) T or of (SOP) is not the characteristic function of some subset, insofar it takes values in (0, 1) on a subset of positive measure. In Section 2.2 we give an example of a target function 𝑦 𝑑 such that the optimal solution ā of (SOP) is not 0 or 1. Since (OCP) T and (SOP) are almost linear quadratic, we generate a mesh of Ω and we introduce the finite element space 𝑉 ℎ of P1-Lagrange element with its basis (𝜔 𝑖 ) ∈ 𝐻 1 (Ω). Introducing the matrices

𝐴 𝑖𝑗 = ∫ Ω ∇𝜔 𝑖 • ∇𝜔 𝑗 𝑑𝑥, 𝐵 𝑖𝑗 = ∫ Ω 𝜔 𝑖 𝜔 𝑗 𝑑𝑥, 𝐿𝑣 𝑖 = ∫ Ω 𝜔 𝑖 𝑑𝑥
we write (SOP) as

min (𝑌,𝑈 )∈R 2𝑁𝑣 (𝑌 -𝑌 𝑑 ) 𝑇 𝐵(𝑌 -𝑌 𝑑 ), 𝐴𝑌 = 𝐵𝑈, 𝐿 𝑇 𝑣 𝑈 ≤ 𝐿|Ω| ( 6
)
where 𝑁𝑣 is the size of the finite element space (for P1-Lagrange elements, 𝑁𝑣 is the number of vertices). For time-discretization, we use an implicit Euler scheme with 𝑁 𝑡 steps. Introducing the matrices

𝐴 𝑡 = 𝐴 + 1 𝛿𝑡 𝐵, 𝐵 𝑡 = 1 𝛿𝑡 𝐵, 𝛿𝑡 = 𝑇 𝑁 𝑡 -1 (7) Ã = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝐼 𝑁𝑣 0 𝑁𝑣 • • • 0 𝑁𝑣 𝐵 𝑡 𝐴 𝑡 . . . . . . . . . . . . . . . 0 𝑁𝑣 0 𝑁𝑣 • • • 𝐵 𝑡 𝐴 𝑡 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , B = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 𝑁𝑣 0 𝑁𝑣 • • • 0 𝑁𝑣 0 𝑁𝑣 𝐵 𝑡 . . . . . . . . . . . . . . . 0 𝑁𝑣 0 𝑁𝑣 • • • 0 𝑁𝑣 𝐵 𝑡 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (8)
we write (OCP) T as min

{ (𝑌 -𝑌 𝑑 ) 𝑇 B(𝑌 -𝑌 𝑑 ) ⃒ ⃒ ⃒ (𝑌, 𝑈 ) ∈ R 2𝑁𝑣𝑁𝑡 , Ã𝑌 = B𝑈, 𝐿 𝑇 𝑣 𝑈 𝑖 ≤ 𝐿|Ω| ∀𝑖 ∈ {1 • • • 𝑁 𝑡 } } (9)
The state variable 𝑌 is seen here as an optimization variable. The discretized convexified problems are linear quadratic optimization problems with inequality and equality constraints. It is then easy to compute the derivatives of the various functions involved. We implement them as well as their derivatives and we call the optimization routine IpOpt via FreeFEM++. The more information on derivatives and Hessians we give to IpOpt the faster solving is. To solve ( 6) and ( 9) we compute the gradient and Hessian of the cost function, the Jacobian of the constraint function and we add (dummy) bound variables to help the solver.

Some numerical examples

We present here some particular cases of the following optimization problem (SSD)

min 𝜔 1 2 ‖𝑦 -𝑦 𝑑 ‖ 2 , 𝐴𝑦 = 𝜒𝜔, 𝑦 |𝜕Ω = 0, |𝜔| ≤ 𝐿|Ω|
for some various operators 𝐴, spaces Ω and target functions 𝑦 𝑑 .

2.2.1 Ω = [-1, 1] 2 , 𝑦 𝑑 = 0.1 and 𝐴 = -△
We solve here the problem for 𝑦 𝑑 = 0.1 and

𝑦 𝑑 = 2-𝑥 2 +𝑦 2 20
. The function 𝑦 𝑑 satisfies the assumptions done in Theorem 1 in the first example and does not in the second one. Convexification method does not require many iterations. This may be due to two facts: first, IpOpt is known to be a very efficient optimization method; second, we have implemented the Hessian of the cost function and we have therefore a Newton method. We notice in the second case, when

𝑦 𝑑 = 2-𝑥 2 +𝑦 2 20
is no more convex that the optimal solution is not 0 or 1 but, in the middle of Ω, takes values into (0, 1).

Ω = [-1, 1]

2 , 𝑦 𝑑 = 0.1 and 𝐴 second-order operator Here, the assumptions of Theorem 1 are not always satisfied. We take various second-order operators 𝐴, of ellipticity constant 𝜃 1 > 0, 𝜃 1 = 0 or 𝜃 1 < 0 with the following operators: When 𝜃 1 > 0, we are in the hypothesis of Theorem 1. When 𝜃 1 = 0, it seems we lose some regularity on the optimal solution. Finally, when 𝜃 1 < 0, we still observe convergence of our algorithm but we can no more infer the theoretical results stated at the beginning.

2.2.3 Ω = [-1, 1] 3 , 𝑦 𝑑 = 0.025 and 𝐴 = -△
FreeFEM++ is also very powerful to solve PDEs in 3D. Except mesh generation, the previously described methods are unchanged. Of course, in 3D the time required to compute the solution is much larger and will be a limiting factor for the solving of the time problem.

Numerical examples illustrating the turnpike phenomenon

In this section, in order to illustrate the turnpike phenomenon, we give numerical simulations for various domains, target functions 𝑦 𝑑 and operators 𝐴.

Case: Ω = [-1, 1] 2 , 𝑦 𝑑 = 0.1 and 𝐴 = -△ We plot on Figure 4 the time solution cylinder and the shape's behavior at some sample times and for several examples. We plot in comparison the time shape at the middle-time of the trajectory and we observe that it is very similar to the optimal static shape.

To observe the exponential turnpike phenomenon we plot the quantity 𝑡 ↦ → ‖𝑦 𝑇 (𝑡) -ȳ‖ + ‖𝑝 𝑇 (𝑡) -p‖ + ‖𝜒 𝜔 𝑇 (𝑡) -𝜒ω‖ for various final times 𝑇 ∈ {1, 3, 5} on Figure 5.

The larger is 𝑇 , the more close to 0 is the residual between the two optimal triples. The behavior of the residual function is typical of the exponential turnpike and supports our conjecture that this phenomenon might be proved. Moreover, we observe that, most of the time, the dynamical shape remains very close to the static one. These numerical observations seem to hold systematically under the assumptions done in Theorem 1. The turnpike phenomenon occurs even when we observe relaxation or when the second-order operator 𝐴 is such that 𝜃 1 = 0.

Further comments

We highlight here some open questions and possible way to pursue the research. Some geometric properties can be conjectured and we address the question of shape design for semi-linear partial differential equations. Finally, we present shortly a shape design problem in the context of shallow water equations.

Symmetries of solutions

On the basis of our numerical examples, we conjecture that if Ω, the operator 𝐴 and the target function 𝑦 𝑑 share the same symmetry properties, then optimal solutions ω and 𝜔 𝑇 (•) share as well symmetry properties.

For instance, Figure 10

Semilinear PDEs and numerics

An open and interesting issue is to deal with semilinear PDEs

𝜕 𝑡 𝑦 + 𝐴𝑦 + 𝑓 (𝑦) = 𝜒𝜔, 𝑦 |𝜕Ω = 0, 𝑦(0) = 𝑦 0 .
Theoretical results are not established in this context but we refer to [START_REF] Henrot | Shape optimization of a Dirichlet type energy for semilinear elliptic partial differential equations[END_REF] where authors are interested in semilinear shape optimization problems and whose existence proofs involve shape optimization problems close to the one we deal with in this section. From our side, we provide hereafter several first numerical simulations. We still use the convexification approach combining IpOpt and FreeFEM++ for numerical solving. Now, anyway, systems cannot be written as linear quadratic optimization problem as ( 6) and ( 9). We use a fixed point method, so that the solution of the static problem

𝐴𝑦 + 𝑓 (𝑦) = 𝜒𝜔, 𝑦 |𝜕Ω = 0 (10)
is sought with an iteration process. We show on Figure 11 some examples of solutions for several functions 𝑓 and for 𝐴 = -△.

We observe that the optimal solutions are quite similar to the solutions obtained in the linear case. For theoretical results, due to the nonlinearity, new assumptions should be made in order to ensure existence and/or uniqueness of solutions. We still observe the turnpike phenomenon for the optimal shape design in 1D. We take Ω = [0, 2], 𝐴 = -△, 𝑦 𝑑 = 0.1 and 𝑇 = 10. We plot on Figure 12 the dynamical optimal shape and we observe that it remains most of the time stationary (which is the solution of the corresponding static optimal shape problem). Moreover, we plot the error between both optimal triples and we still observe an exponential turnpike phenomenon. We consider the shallow water system in space dimension one, a specific case of shallow water equations, which is a usual way to understand the behavior of a fluid when the height of the water level is much smaller than the other spatial dimensions of the problem, for instance flows in river or in maritime coast. Using several assumptions (small height of water, hydrostatic pressure, vertical homogeneity of horizontal velocities, etc), we can derive these from the Euler system. We describe the behavior of the fluid at time 𝑡 > 0 and at a position 𝑥 ∈ R characterized by the water level ℎ(𝑡, 𝑥) and the flow ℎ(𝑡, 𝑥)𝑢(𝑡, 𝑥) and we take into account the variation of the topography 𝑧(𝑡, 𝑥) and some viscous effects 𝑆 𝑓 . We consider the bottom shape 𝑧 as the control. For 𝑇 > 0, (ℎ 0 , 𝑢 0 ) initial conditions and (ℎ 𝑑 , 𝑢 𝑑 ) given as a static wave profile, we search an optimal solution minimizing the functional

𝐽 𝑆𝑊 (𝑧) = 1 2 𝑇 ∫ 0 ‖ℎ(𝑡) -ℎ 𝑑 ‖ 2 + ‖𝑢(𝑡) -𝑢 𝑑 ‖ 2 𝑑𝑡 (11)
subject to the shallow water equations

𝜕 𝑡 ℎ + 𝜕𝑥(ℎ𝑢) = 0 𝜕 𝑡 (ℎ𝑢) + 𝜕𝑥(ℎ𝑢 2 + 𝑔 ℎ 2 2 ) = -𝑔ℎ𝜕𝑥𝑧 + 𝑆 𝑓 ℎ(0) = ℎ 0 , 𝑢(0) = 𝑢 0 (12)
Since the target (ℎ 𝑑 , 𝑢 𝑑 ) is static, having in mind the phenomenon of static wave occuring in the nature (Eisbach Wave in München or wavemaker in [1]), we would expect that a solution, if it exists, should remain most of the time close to a stationary state solution of a static problem.

Kinetic interpretation of shallow water equations

To lead this study, we propose a relaxation of the problem via kinetic equations (see [START_REF] Audusse | Kinetic entropy inequality and hydrostatic reconstruction scheme for the Saint-Venant system[END_REF][START_REF] Bristeau | Boundary Conditions for the Shallow Water Equations solved by Kinetic Schemes[END_REF][START_REF] Perthame | A kinetic scheme for the Saint-Venant system with a source term[END_REF]. We consider a real-valued function 𝜒 of class 𝐶 1 , compactly supported on R, such that

𝜒(𝜔) = 𝜒(-𝜔), ∫ R 𝜒(𝜔) 𝑑𝜔 = 1, ∫ R 𝜔 2 𝜒(𝜔) 𝑑𝜔 = 𝑔 2 2 .
Defining 𝑐 = √

) = ℎ(𝑡, 𝑥) 𝑐(𝑡, 𝑥) 𝜒 ( 𝜉 -𝑞(𝑡, 𝑥) ℎ(𝑡, 𝑥)𝑐(𝑡, 𝑥) ) (13)
we get

⎛ ⎜ ⎝ ℎ 𝑞 𝑔 2 ℎ 2 + 𝑞 2 ℎ ⎞ ⎟ ⎠ = ∫ R ⎛ ⎜ ⎝ 1 𝜉 𝜉 2 ⎞ ⎟ ⎠ 𝑀 (𝜉) 𝑑𝜉. (14)
Following [START_REF] Perthame | A kinetic scheme for the Saint-Venant system with a source term[END_REF], (ℎ, 𝑢) is a weak solution of (12) if and only if 𝑀 (defined by ( 13)) satisfies

𝑀 𝑡 + 𝜉.𝑀𝑥 -𝑔𝑧𝑥.𝑀 𝜉 = 𝑄 (15)
for some 𝑄, a collision factor such that

∫ R 𝑄 𝑑𝜉 = ∫ R 𝜉𝑄 𝑑𝜉 = 0.
An open question is to write the initial problem as the minimization of some cost functional

min 𝑧 1 2 𝑇 ∫ 0 ∫ R×Ω 𝑓 0 ( 𝑥, 𝜉, 𝑀 (𝑡, 𝑥, 𝜉) )

𝑑𝜉 𝑑𝑥 𝑑𝑡

with respect to the kinetic equations

𝑀 𝑡 + 𝜉.𝑀𝑥 -𝑔𝑧𝑥.𝑀 𝜉 = 𝑄, 𝑀 (0) = ℎ 0 𝑐 0 𝜒 ( 𝜉 -𝑢 0 𝑐 0 ) (16)
This would have the double benefit to obtain in an easier way theoretical and numerical results since the nonlinear system ( 12) is replaced by the linear kinetic equation [START_REF] Lieb | Analysis[END_REF].

A Appendix

A.1 Optimality conditions

In order to show some theoretical results introduced before and in the framework provided by the convexification (see Section 2.1) we first give necessary optimality conditions to optimal solutions of the convexified problems stated in [16, Chapters 2 and 3] or [14, Chapter 4] and infer from these necessary conditions that, under appropriate assumptions, the optimal controls are indeed characteristic functions.

Necessary optimality conditions for (OCP) T According to the Pontryagin maximum principle (see [16, Chapter 3, Theorem 2.1], see also [START_REF] Li | Optimal control theory for infinite-dimensional systems[END_REF]), for any optimal solution (𝑦 𝑇 , 𝑎 𝑇 ) of (OCP) T there exists an adjoint state 𝑝 𝑇 ∈ 𝐿 2 (0, 𝑇 ; Ω) such that

𝜕 𝑡 𝑦 𝑇 + 𝐴𝑦 𝑇 = 𝑎 𝑇 , 𝑦 𝑇 |𝜕Ω = 0, 𝑦 𝑇 (0) = 𝑦 0 𝜕 𝑡 𝑝 𝑇 -𝐴 * 𝑝 𝑇 = 𝛾 1 (𝑦 𝑇 -𝑦 𝑑 ), 𝑝 𝑇 |𝜕Ω = 0, 𝑝 𝑇 (𝑇 ) = 𝛾 2 ( 𝑦 𝑇 (𝑇 )-𝑦 𝑑 ) (17)
∀𝑎 ∈ 𝒰 𝐿 , for a.e. 𝑡 ∈ [0, 𝑇 ] :

( 𝑝 𝑇 (𝑡), 𝑎 𝑇 (𝑡) -𝑎 ) ≥ 0. (18)
Necessary optimality conditions for (SOP) Similarly, applying [16, Chapter 2, Theorem 1.4], for any optimal solution (ȳ, ā) of (SOP) there exists an adjoint state p ∈ 𝐿 2 (Ω) such that 𝐴ȳ = ā, ȳ|𝜕Ω = 0

-𝐴 * p = 𝛾 1 (ȳ -𝑦 𝑑 ), p|𝜕Ω = 0 (19)
∀𝑎 ∈ 𝒰 𝐿 : (p, ā -𝑎) ≥ 0.

Using the bathtub principle (see, e.g., [START_REF] Lieb | Analysis[END_REF]Theorem 1.14]), [START_REF] Perthame | A kinetic scheme for the Saint-Venant system with a source term[END_REF] and [START_REF] Porretta | Remarks on long time versus steady state optimal control[END_REF] give

𝑎 𝑇 (•) = 𝜒 {𝑝 𝑇 (•)>𝑠 𝑇 (•)} + 𝑐 𝑇 (•)𝜒 {𝑝 𝑇 (•)=𝑠 𝑇 (•)} (21)
ā = 𝜒 { p>s} + c𝜒 { p=s} (22)
with, for a.e. 𝑡 ∈ [0, 𝑇 ],

𝑐 𝑇 (𝑡) ∈ 𝐿 ∞ (Ω; [0, 1]) and c ∈ 𝐿 ∞ (Ω; [0, 1]) (23)
𝑠 𝑇 (•) = inf { 𝜎 ∈ R | |{𝑝 𝑇 (•) > 𝜎}| ≤ 𝐿|Ω| } (24)
s = inf { 𝜎 ∈ R | |{p > 𝜎}| ≤ 𝐿|Ω| } . (25)

A.2 Energy inequalities

We recall some useful inequalities to study existence and turnpike. Since 𝜃 satisfies (1), we can find 𝛽 > 0, 𝛾 ≥ 0 such that 𝛽 ≥ 𝛾 and

(𝐴𝑢, 𝑢) ≥ 𝛽‖𝑢‖ 2 𝐻 1 0 (Ω) -𝛾‖𝑢‖ 2 𝐿 2 (Ω) . (26)
From this follows the energy inequality (see [START_REF] Evans | Partial differential equations[END_REF]Chapter 7, Theorem 2]): there exists 𝐶 > 0 such that, for any solution 𝑦 of (5), for almost every 𝑡 ∈ [0, 𝑇 ],

‖𝑦(𝑡)‖ 2 + 𝑡 ∫ 0 ‖𝑦(𝑠)‖ 2 𝐻 1 0 (Ω) 𝑑𝑠 ≤ 𝐶 ⎛ ⎝ ‖𝑦 0 ‖ 2 + 𝑡 ∫ 0 ‖𝑎(𝑠)‖ 2 𝑑𝑠 ⎞ ⎠ . (27)
Let us improve this inequality. Having in mind [START_REF] Wächter | On the Implementation of a Primal-Dual Interior Point Filter Line Search Algorithm for Large-Scale Nonlinear Programming[END_REF], the Poincaré inequality and that 𝑦 verifies (5), we find two constants 𝐶 1 , 𝐶 2 > 0 such that 𝑑 𝑑𝑡 ‖𝑦(𝑡)‖ 2 + 𝐶 1 ‖𝑦(𝑡)‖ 2 = 𝑓 (𝑡) ≤ 𝐶 2 ‖𝑎(𝑡)‖ 2 . We solve this differential equation to get ‖𝑦(𝑡)‖ 2 = ‖𝑦 0 ‖ 2 𝑒 -𝐶1𝑡 + ∫ 𝑡 0 𝑒 -𝐶1(𝑡-𝑠) 𝑓 (𝑠) 𝑑𝑠. Since for all 𝑡 ∈ (0, 𝑇 ), 𝑓 (𝑡) ≤ 𝐶 2 ‖𝑎(𝑡)‖ 2 , we obtain that

‖𝑦(𝑡)‖ 2 ≤ ‖𝑦 0 ‖ 2 𝑒 -𝐶1𝑡 + 𝐶 2 𝑡 ∫ 0 𝑒 -𝐶1(𝑡-𝑠) ‖𝑎(𝑠)‖ 2 𝑑𝑠 (28)
for almost every 𝑡 ∈ (0, 𝑇 ). The constants 𝐶, 𝐶 1 , 𝐶 2 depend only on the domain Ω (Poincaré inequality) and on the operator 𝐴 but not on final time 𝑇 since ( 26) is satisfied with 𝛽 ≥ 𝛾.

A.3 Proofs

A.3.1 Proof of Theorem 1

Here we first establish existence of optimal solutions for (OCP) T and similarly for (SOP). Then we focus on the static case in order to highlight a key observation to derive existence of an optimal shape for (SSD) as well as the relaxation phenomenon.

We first see that the infimum exists. Let us take a minimizing sequence

(𝑦𝑛, 𝑎𝑛) ∈ 𝐿 2 (0, 𝑇 ; 𝐻 1 0 (Ω)) × 𝐿 ∞ ( 0, 𝑇 ; 𝐿 2 ( Ω, [0, 1] ))
such that, for 𝑛 ∈ N, for 𝑎.𝑒. 𝑡 ∈ [0, 𝑇 ], 𝑎𝑛(𝑡) ∈ 𝒰 𝐿 , the pair (𝑦𝑛, 𝑎𝑛) satisfies ( 5) and

𝐽 𝑇 (𝑎𝑛) → 𝐽 𝑇 . The sequence (𝑎𝑛) is bounded in 𝐿 ∞ ( 0, 𝑇 ; 𝐿 2 ( Ω, [0, 1]
))

, so using ( 27) and (28), the sequence (𝑦𝑛) is bounded in 𝐿 ∞ (0, 𝑇 ; 𝐿 2 (Ω)) ∩ 𝐿 2 (0, 𝑇 ; 𝐻 1 0 (Ω)). We show then, using [START_REF] Audusse | Kinetic entropy inequality and hydrostatic reconstruction scheme for the Saint-Venant system[END_REF], that the sequence ( 𝜕𝑦𝑛 𝜕𝑡 ) is bounded in 𝐿 2 (0, 𝑇 ; 𝐻 -1 (Ω)). We subtract a sequence still denoted by (𝑦𝑛, 𝑎𝑛) such that one can find a pair

(𝑦, 𝑎) ∈ 𝐿 2 (0, 𝑇 ; 𝐻 1 0 (Ω)) × 𝐿 ∞ ( 0, 𝑇 ; 𝐿 2 ( Ω, [0, 1] )) with 𝑦𝑛 ⇀ 𝑦 weakly in 𝐿 2 (0, 𝑇 ; 𝐻 1 0 (Ω)) 𝜕 𝑡 𝑦𝑛 ⇀ 𝜕 𝑡 𝑦 weakly in 𝐿 2 (0, 𝑇 ; 𝐻 -1 (Ω)) 𝑎𝑛 ⇀ 𝑎 weakly * in 𝐿 ∞ ( 0, 𝑇 ; 𝐿 2 ( Ω, [0, 1] )) . ( 29
)
We deduce that

𝜕 𝑡 𝑦𝑛 + 𝐴𝑦𝑛 -𝑎𝑛 → 𝜕 𝑡 𝑦 + 𝐴𝑦 -𝑎 in 𝒟 ′ ( (0, 𝑇 ) × Ω ) 𝑦𝑛(0) ⇀ 𝑦(0) weakly in 𝐿 2 (Ω). (30)
We get using (30) that (𝑦, 𝑎) is a weak solution of (5). Moreover, since

𝐿 ∞ ( 0, 𝑇 ; 𝐿 2 ( Ω, [0, 1] )) = ( 𝐿 1 ( 0, 𝑇 ; 𝐿 2 ( Ω, [0, 1] )) ) ′
(see [START_REF] Hytönen | I. Martingales and Littlewood-Paley theory, volume 63 of Ergebnisse der Mathematik und ihrer Grenzgebiete[END_REF]Corollary 1.3.22]) the convergence (29) implies that for every 𝑣 ∈ 𝐿 1 (0, 𝑇 ) satisfying 𝑣 ≥ 0 and ‖𝑣‖ 𝐿 1 (0,𝑇 ) = 1, we have

𝑇 ∫ 0 ( ∫ Ω 𝑎(𝑡, 𝑥) 𝑑𝑥 ) 𝑣(𝑡) 𝑑𝑡 ≤ 𝐿|Ω|.
Since the function 𝑓𝑎 defined by

𝑓𝑎(𝑡) = ∫ Ω 𝑎(𝑡, 𝑥) 𝑑𝑥 belongs to 𝐿 ∞ (0, 𝑇 ), we have ‖𝑓𝑎‖ 𝐿 ∞ (0,𝑇 ) = sup ⎧ ⎨ ⎩ 𝑇 ∫ 0 ( ∫ Ω 𝑎(𝑡, 𝑥) 𝑑𝑥 ) 𝑣(𝑡) 𝑑𝑡 ⃒ ⃒ ⃒ 𝑣 ∈ 𝐿 1 (0, 𝑇 ), ‖𝑣‖ 𝐿 1 (0,𝑇 ) = 1 ⎫ ⎬ ⎭ Therefore ‖𝑓𝑎‖ 𝐿 ∞ (0,𝑇 ) ≤ 𝐿|Ω| and

∫

Ω 𝑎(𝑡, 𝑥) 𝑑𝑥 ≤ 𝐿|Ω| for a.e. 𝑡 ∈ (0, 𝑇 ). This shows that the pair (𝑦, 𝑎) is admissible. Since 𝐻 1 0 (Ω) is compactly embedded in 𝐿 2 (Ω) and by using the Aubin-Lions compactness Lemma (see [START_REF] Aubin | Un théorème de compacité[END_REF]), we obtain 𝑦𝑛 → 𝑦 strongly in 𝐿 2 (0, 𝑇 ; 𝐿 2 (Ω)).

We get then by weak lower semi-continuity of 𝐽 𝑇 and by Fatou Lemma that

𝐽 𝑇 (𝑎) ≤ lim inf 𝐽 𝑇 (𝑎𝑛).
Hence 𝑎 is an optimal control for (OCP) T , that we rather denote by 𝑎 𝑇 (and ā for (SOP)).

Let us now focus on the part 2-(ii) of Theorem 1. We focus on 𝛾 1 = 1, 𝛾 2 = 0 (Lagrange case) and (SSD). Since ā is a s solution of (SOP), it verifies the optimality conditions stated in ( 19)- [START_REF] Privat | Optimal observability of the multidimensional wave and Schrödinger equations in quantum ergodic domains[END_REF]. One key observation is to note that, if | { p = s} | = 0, then it follows from ( 22) that the static optimal control ā is actually the characteristic function of a shape ω ∈ 𝒰 𝐿 . With this in mind, we give a useful lemma.

Lemma 5 ([13, Theorem 3.2]). Given any 𝑝 ∈ [1, +∞) and any 𝑢 ∈ 𝑊 1,𝑝 (Ω) such that |{𝑢 = 0}| > 0, we have ∇𝑢 = 0 𝑎.𝑒. on {𝑢 = 0}.

2-(ii)

We assume that 𝐴𝑦 𝑑 ≤ 𝛽 in Ω with 𝛽 = s𝐴𝑐 * . Having in mind ( 19) and ( 22), we assume by contradiction that |{p = s}| > 0. Since 𝐴 and 𝐴 * are differential operators, applying 𝐴 * to p on {p = s}, we obtain by Lemma 5 that 𝐴 * p = 𝑐 * s on {p = s}.

Since (ȳ, p) verifies ( 19) we get 𝑦 𝑑 -ȳ = 𝑐 * s on {p = s}.

We apply then 𝐴 to this equation to get that 𝐴𝑦 𝑑 -s𝐴𝑐 * = 𝐴ȳ = ā on {p = s}.

Therefore

𝐴𝑦 𝑑 -s𝐴𝑐 * ∈ (0, 1) on {p = s} which contradicts 𝐴𝑦 𝑑 ≤ 𝛽. Hence |{p = s}| = 0 and thus [START_REF] Privat | Optimal observability of the multidimensional wave and Schrödinger equations in quantum ergodic domains[END_REF] implies ā = 𝜒ω for some ω ∈ 𝒰 𝐿 . Existence of solution for (SSD) is proved. Uniqueness of optimal controls comes from the strict convexity of the cost functionals. Indeed, in the dynamical case, whatever (𝛾 1 , 𝛾 2 ) ̸ = (0, 0) may be, 𝐽 𝑇 is strictly convex with respect to variable 𝑦. The injectivity of the control-tostate mapping gives the strict convexity with respect to the variable 𝑎. In addition, uniqueness of (ȳ, p) follows by application of the Poincaré inequality and uniqueness of (𝑦 𝑇 , 𝑝 𝑇 ) follows from Gronwall inequality (28) in the appendix. Remark 6. Condition in Theorem 1: 2-(ii) is a necessary condition. We can construct example, where 𝐴𝑦 𝑑 ≤ 𝛽 is not satisfied and where we observe relaxation, which is closely related to the fact |{p = s}| > 0.

Indeed, we plot on Figure 14 the adjoint state p for the static problem in 1D. At the left-hand side, p is assumed to be analytic: in this case, all level sets of p have zero Lebesgue measure (there is no subset of positive measure on which p would remain constant). When p is not analytic and remains constant on a subset of positive measure (see Figure 14 in red), we do not have necessarily zero Lebesgue measure level sets, and on {p = s}, ā can take values in (0, 1).

Proof of Theorem 3

For 𝛾 1 = 1, 𝛾 2 = 0 (Lagrange case), the cost is

𝐽 𝑇 (𝜔) = 1 2𝑇 𝑇 ∫ 0 ‖𝑦(𝑡) -𝑦 𝑑 ‖ 2 𝑑𝑡.
We consider the triples (𝑦 𝑇 , 𝑝 𝑇 , 𝜒𝜔 𝑇 ) and (ȳ, p, 𝜒ω) satisfying the optimality conditions [START_REF] Mckenzie | Turnpike theorems for a generalized leontief model[END_REF] and [START_REF] Porretta | Long time versus steady state optimal control[END_REF]. Since 𝜒 𝜔 𝑇 (𝑡) is bounded at each time 𝑡 ∈ [0, 𝑇 ] and by application of Gronwall inequality (28) in the appendix to 𝑦 𝑇 and 𝑝 𝑇 we can find a constant 𝐶 > 0 depending only on 𝐴, 𝑦 0 , 𝑦 𝑑 , Ω, 𝐿 such that ∀𝑇 > 0 ‖𝑦 𝑇 (𝑇 )‖ 2 ≤ 𝐶 and ‖𝑝 𝑇 (0)‖ 2 ≤ 𝐶. The two terms at the left-hand side are positive and using the inequality [START_REF] Zaslavski | Turnpike theory of continuous-time linear optimal control problems[END_REF] with 𝜁(𝑡) = p(𝑇 -𝑡), we finally obtain 𝑀 > 0 independent of 𝑇 such that 1 𝑇 𝑇 ∫ 0 ( ‖𝑦 𝑇 (𝑡) -ȳ‖ 2 + ‖𝑝 𝑇 (𝑡) -p‖ 2 ) 𝑑𝑡 ≤ 𝑀 𝑇 .

𝑏

  𝑖 (𝑥)𝜕𝑥 𝑖 𝑦 + 𝑐(𝑥)𝑦

Fig. 1 : 20 .Fig. 2 :

 1202 Fig. 1: Optimal static shape: (a) 𝑦 𝑑 = 0.1; (b) 𝑦 𝑑 = 2-𝑥 2 +𝑦 2 20 .

Fig. 3 :

 3 Fig. 3: Optimal static shape 3D

Fig. 4 :

 4 Fig. 4: Optimal dynamical shape: (a) Time shape; (b) 𝑡 = 0; (c) 𝑡 = 0.5; (c) 𝑡 ∈ (0.5, 1.5); (e) 𝑡 = 1.5; (f) 𝑡 = 𝑇 ; (g) Static shape.

Fig. 5 :Fig. 6 :

 56 Fig. 5: 𝑡 ↦ → ‖𝑦 𝑇 (𝑡) -ȳ‖ + ‖𝑝 𝑇 (𝑡) -p‖ + ‖𝜒 𝜔 𝑇 (𝑡) -𝜒ω‖ for 𝑇 ∈ {1, 3, 5}

  (a) highlights four axes of symmetry (𝑥 = 0, 𝑦 = 0, 𝑦 = 𝑥, 𝑦 = -𝑥), Figure 10(b) two axes (𝑦 = 𝑥, 𝑦 = -𝑥) and Figure 10(c) shows up a central symmetry property (of center (0, 0) and with angle 𝜋).

Fig. 7 :Fig. 8 :Fig. 9 :

 789 Fig. 7: (a) Time shape; (b) 𝑡 = 0; (c) 𝑡 = 0.5; (c) 𝑡 ∈ (0.5, 1.5); (e) 𝑡 = 1.5; (f) 𝑡 = 𝑇 ; (g) Static shape.

Fig. 10 :Fig. 11 :

 1011 Fig. 10: Symmetries of static optimal shapes : (a) 𝐴 = -△, 𝑦 𝑑 = 0.1; (b) 𝐴 = -△, 𝑦 𝑑 = 𝑥𝑦+1 20 ; (c) 𝐴𝑢 = -𝜕𝑥((𝑥 -𝑦) 2 𝜕𝑥𝑢) -𝜕𝑦((𝑥 + 𝑦) 2 𝜕𝑦𝑢), 𝑦 𝑑 = 1 20 (2 sin(2(𝑥 2 + 𝑦 2 )) + 1)

Fig. 12 :Fig. 13 :

 1213 Fig. 12: Semilinear optimal designs: (a) Dynamical optimal shape; (b) 𝑡 ↦ → ‖𝑦 𝑇 (𝑡) -ȳ‖ + ‖𝑝 𝑇 (𝑡) -p‖ + ‖𝜒 𝜔 𝑇 )(𝑡) -𝜒ω‖.

𝑔ℎ 2 and

 2 𝑀 (𝑡, 𝑥, 𝜉) = ℎ(𝑡, 𝑥) 𝑐(𝑡, 𝑥) 𝜒 ( 𝜉 -𝑢(𝑡, 𝑥) 𝑐(𝑡, 𝑥)

≥ 0

 0 Setting ỹ = 𝑦 𝑇 -ȳ, p = 𝑝 𝑇 -p, ã = 𝜒𝜔 𝑇 -𝜒ω, we have𝜕 𝑡 ỹ + 𝐴ỹ = ã, ỹ|𝜕Ω = 0, ỹ(0) = 𝑦 0 -ȳ(31) 𝜕 𝑡 p -𝐴 * p = ỹ, p|𝜕Ω = 0, p(𝑇 ) = -p. for almost every 𝑡 ∈ [0, 𝑇 ]. Multiplying (31) by p, (32) by ỹ and then adding them, one can use the fact that ( ȳ -𝑦 0 , p(0) )‖ 2 𝑑𝑡.

Fig. 14 :

 14 Fig. 14: Optimal shape design existence and relaxation: (a) No relaxation: Shape existence; (b) Relaxation.

Acknowledgment: This project has received funding from the Grants ICON-ANR-16-ACHN-0014 and Finite4SoS ANR-15-CE23-0007-01 of the French ANR, the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement 694126-DyCon), the Alexander von Humboldt-Professorship program, the Air Force Office of Scientific Research under Award NO: FA9550-18-1-0242, Grant MTM2017-92996-C2-1-R COSNET of MINECO (Spain) and by the ELKARTEK project KK-2018/00083 ROAD2DC of the Basque Government, Transregio 154 Project *Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks* of the German DFG, the European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement 765579-ConFlex.


Shervin Kadkhoda-Ahmadi

Alaa Hassan
email: alaa.hassan@univ-lorraine.fr

Elnaz Asadollahi-Yazdi

Process and Resource Selection Methodology in Design for Additive Manufacturing

Keywords: Additive Manufacturing, Design for Additive Manufacturing, Multi-criteria decision making, Manufacturability, Process selection

This paper deals with an integrated design approach for Additive Manufacturing (AM), as a Design for Additive Manufacturing (DfAM) approach. This DfAM approach focuses on two important activities of manufacturability analysis, as well as material and process selection in the concept of Concurrent Engineering and Integrated Design approach (CE/ID). It could be considered as a guideline for the researchers in the early phase of the product development process. For this purpose, a novel process and resource selection problem for AM is formalized as an Additive Manufacturing Process and Resource Selection Problem (AMPRSP). This problem is defined to investigate simultaneously the manufacturability, and process and resource selection issues for AM. A Multi-Criteria Evaluation System (MCES) is proposed to solve this problem by evaluating the manufacturability of the product, and selecting the AM resources. Firstly, the AM process, machine, and material are explored and selected regarding technical and economic evaluation criteria. Then, the most appropriate alternative for fabrication with AM is selected by considering the sub-criteria including build time, accuracy performance, and cost. To solve this multi-criteria

Introduction

Nowadays, Additive Manufacturing (AM) is a popular manufacturing system which brings a new approach to producing the different versions of complex products with a material range [START_REF] Asadollahi-Yazdi | Toward integrated design of additive manufacturing through a process development model and multi-objective optimization[END_REF]. The main advantages of AM include producing complex shapes without any additional cost, operations or tooling; and reducing product development cycles, as well as increasing the demand for customized and personalized products, are the main reasons for using the additive technologies [START_REF] Gao | The status, challenges, and future of additive manufacturing in engineering[END_REF][START_REF] Whitney | Manufacturing by design[END_REF][START_REF] Ahuja | Additive manufacturing in production: challenges and opportunities[END_REF]. This manufacturing method is derived from Rapid Prototyping and it makes the products directly from the digital model based on a layer-by-layer manufacturing process [START_REF] Asadollahi-Yazdi | Integrated design for additive manufacturing based on skin-skeleton approach[END_REF][START_REF] Mueller | Additive Manufacturing Technologies -Rapid Prototyping to Direct Digital Manufacturing[END_REF]. AM includes various technologies, such as Fused Deposition Modeling (FDM), Selective Laser Sintering (SLS), Selective Laser Melting (SLM), etc. which differ in terms of process, material, power source, and print ink [START_REF] Gao | The status, challenges, and future of additive manufacturing in engineering[END_REF]. Recently, different industries such as aerospace, automotive, engineering, medicine, and biological systems have been using these different AM technologies to fabricate their end-user customized products [START_REF] Gao | The status, challenges, and future of additive manufacturing in engineering[END_REF].

It is necessary to provide new design and manufacturing tools and guidelines for AM regarding its unique properties to manage the design and manufacturing process simultaneously during the product development process steps [START_REF] Asadollahi-Yazdi | Toward integrated design of additive manufacturing through a process development model and multi-objective optimization[END_REF]. Despite the existence of several advantages for AM, some disadvantages also exist, such as dependency between the material and physical process, material selection limitation, low efficiency for serial production, and machine capacity limitation, as well as surface finishing and quality problems [START_REF] Asadollahi-Yazdi | Toward integrated design of additive manufacturing through a process development model and multi-objective optimization[END_REF]. Therefore, these constraints must be considered during the product development process. Design for Manufacturing (DfM), which is derived from Concurrent Engineering and Integrated Design (CE/ID) approaches, is one of the basic concepts for helping the designer and manufacturer to investigate the constraints and attributes of the manufacturing process during the design step. DfM can reduce manufacturing costs and improve product quality [START_REF] Chakladar | A combined TOPSIS-AHP-method-based approach for non-traditional machining processes selection. Proceedings of the Institution of Mechanical Engineers[END_REF]. Thus, the concept of DfAM has emerged as an integrated design approach for AM to consider its constraints and attributes during the design stage [START_REF] Kathryn Thompson | Design for Additive Manufacturing: Trends, opportunities, considerations, and constraints[END_REF]. Systematic DfM consists of two main activities of material and process selection, as well as manufacturability evaluation [START_REF] Van Vliet | State-of-the-art report on design for manufacturing[END_REF][START_REF] Asadollahi-Yazdi | Integrated design in additive manufacturing based on design for manufacturing[END_REF]. As shown in Fig. 1, this evaluation must be performed in three levels of verification, quantification, and optimization. Providing these activities for AM involves analysis of the conflicting attributes and different specific needs of organizations, making them complex tasks in CE/ID. Moreover, AM properties and its standards, guidelines, tools, and rules which, are not sufficiently developed, increase this complexity. The main objective of this study is to provide a DfM-based approach adapted to AM that could be integrated into a DfAM process. This approach is an estimation for designer and manufacturers that helps them to find the suitable process and material for production with AM technologies due to manufacturability evaluation. The remainder of the paper is organized as follows: Section 2 provides an overview of the DfAM approaches and it reviews the related literature. This analysis per-Fig. 1: Systematic DfM [START_REF] Van Vliet | State-of-the-art report on design for manufacturing[END_REF][START_REF] Asadollahi-Yazdi | Integrated design in additive manufacturing based on design for manufacturing[END_REF] mits us to define the proposed approach. Section 3 describes the applied methodology and the components of the proposed MCES for AM Process and Resource Selection Problem (AMPRSP). Section 4 is devoted to a case study for illustration of the proposed methodology. Finally, Section 5 contains conclusions and perspective works.

Literature review

Until now, different AM processes have been reviewed and classified based on applied technology, application, material, etc. by researchers [START_REF] Bikas | Additive manufacturing methods and modelling approaches: a critical review[END_REF][START_REF] Frazier | Metal Additive Manufacturing: A Review[END_REF][START_REF] Huang | Additive Manufacturing: Current State, Future Potential, Gaps and Needs, and Recommendations[END_REF][START_REF] Shah | Comparison of different additive manufacturing methods using computed tomography[END_REF]. Recently, considerable attention has been paid to providing an integrated design approach for AM in order to investigate its different constraints and attributes. Thus, the DfM approach for AM is proposed as a DfAM approach.

As mentioned in the previous section, DfM consists of two main activities of manufacturability evaluation, and material and process selection [START_REF] Van Vliet | State-of-the-art report on design for manufacturing[END_REF].

Manufacturability was evaluated in some studies: A DfAM approach is proposed to consider the manufacturing constraints in the design process [START_REF] Hu | Optimal design for additive manufacturing of heterogeneous objects using ultrasonic consolidation[END_REF]. Also, a new numerical chain on a DfAM approach was proposed for optimizing a three-step design process by investigating the manufacturing constraints simultaneously [START_REF] Ponche | A novel methodology of design for Additive Manufacturing applied to Additive Laser Manufacturing process[END_REF]. Moreover, manufacturing complexity is also estimated by Hascoet et al. [START_REF] Hascoet | Manufacturability analysis to combine additive and subtractive processes[END_REF]. In this study, the manufacturability indexes were used to calculate a value for AM complexity based on the CAD model and geometrical features, material, and product specifications [START_REF] Hascoet | Manufacturability analysis to combine additive and subtractive processes[END_REF]. Manufacturability is also optimized through presenting an optimization problem to consider the manufacturing constraints and criteria simultaneously [START_REF] Asadollahi-Yazdi | Toward integrated design of additive manufacturing through a process development model and multi-objective optimization[END_REF]. It can also be analyzed by providing special design rules in the study of Pradel et al. [START_REF] Pradel | Investigation of design for additive manufacturing in professional design practice[END_REF]. For assessing the potential quality of the products made by AM, visual design for AM worksheet was utilized to identify the AM mistakes [START_REF] Booth | The Design for Additive Manufacturing Worksheet[END_REF].

A multi-criteria decision-making method was presented due to lightweight design, cost, strength, displacement, and surface quality to find the best design solution without a manufacturability analysis [START_REF] Salonitis | Redesign optimization for manufacturing using additive layer techniques[END_REF]. Mechanical behavior of AM products is also an important criterion that is analyzed by some researchers [START_REF] Cantrell | Experimental characterization of the mechanical properties of 3d printed abs and polycarbonate parts[END_REF][START_REF] Croccolo | Experimental characterization and analytical modelling of the mechanical behaviour of fused deposition processed parts made of abs-m30[END_REF][START_REF] Asadollahi-Yazdi | Multi-objective optimization of additive manufacturing process[END_REF]. A comprehensive review of the pore formation, mechanical properties, and application of metallic porous material is presented by Zhao et al. [START_REF] Zhao | A review on metallic porous materials: pore formation, mechanical properties, and their applications[END_REF]. The primary mechanical properties, such as compressive strength, elastic modulus, fatigue properties, and flexural strength of metallic porous materials depending on pore morphology and porosity are investigated in this study [START_REF] Zhao | A review on metallic porous materials: pore formation, mechanical properties, and their applications[END_REF]. The fatigue behavior of SLM Ti-6Al-4V components is also analyzed by Fotovvati et al. [START_REF] Fotovvati | Fatigue performance of selective laser melted ti6al4v components: state of the art[END_REF] in order to improve this behavior and look into this mechanical property of AM parts under cyclic loading.

Several design rules were presented to optimize topology, manufacturing time, and mechanical properties of the product [START_REF] Qattawi Ala'aldin Alafaghani | mad Ali Ablat. Design consideration for additive manufacturing: Fused deposition modelling[END_REF]. The design guideline is provided through featuring a graph-based design improvement technique to analyze the manufacturability while reducing costs. Moreover, DfAM-constrained topology optimization-based design was used to integrate DfAM design rules with topology optimization (TO) [START_REF] Ranjan | Integration of design for manufacturing methods with topology optimization in additive manufacturing[END_REF].

The manufacturability issues are analyzed by Hassan et al. [START_REF] Hassan | Conceptual process planningan improvement approach using QFD, FMEA, and ABC methods[END_REF] from different perspectives. A new approach is proposed to develop quality/cost-based conceptual process planning (QCCPP). They used QFD, FMEA, and ABC methods to improve the conceptual process planning.

Some researchers [START_REF] Asadollahi-Yazdi | Integrated design for additive manufacturing based on skin-skeleton approach[END_REF][START_REF] Walton | Design and development of an additive manufactured component by topology optimisation[END_REF] utilized TO in order to suggest design solutions for AM by optimizing the mass and structure of the product in terms of its mechanical behavior. Creating a lattice structure is also another approach that is addressed in some studies [START_REF] Primo | Additive manufacturing integration with topology optimization methodology for innovative product design[END_REF][START_REF] Salonitis | Design for additive manufacturing based on the axiomatic design method[END_REF]. Filtering infeasible and unprintable overhanging sections are also considered to obtain the design solutions through TO [START_REF] Langelaar | An additive manufacturing filter for topology optimization of print-ready designs[END_REF]. TO is used to analyze multi-functional and multimaterial parts of AM [START_REF] Panesar | Design framework for multifunctional additive manufacturing: coupled optimization strategy for structures with embedded functional systems[END_REF], as well as self-supporting structures in another study [START_REF] Xu Guo | Self-supporting structure design in additive manufacturing through explicit topology optimization[END_REF].

The importance of process selection in CE/ID as another main activity of DfM encourages the researchers to investigate it. Several studies existed for traditional manufacturing processes but the study on process selection for AM is still lacking and it is a large domain that must be continued. Different approaches like the flexible decision-making technique [START_REF] Desai | Material and process selection in product design using decision-making technique (AHP)[END_REF], TOPSIS-AHPbased method [START_REF] Chakladar | A combined TOPSIS-AHP-method-based approach for non-traditional machining processes selection. Proceedings of the Institution of Mechanical Engineers[END_REF], exploratory analysis about existence and nature of manufacturing methods [START_REF] Souza | Materials & Manufacturing Methods selection in product design: Experiences in undergraduate programs[END_REF], and characteristics of key parts [START_REF] Lu | A selection methodology of key parts based on the characteristic of carbon emissions for low-carbon design[END_REF] have been used for traditional manufacturing processes.

Different methods are used to analyze and select the best manufacturing method between AM and traditional processes like analysis of complexity and production volume by a neural networks approach, considering environ-mental aspects including energy consumption for three manufacturing categories of bulk-forming, subtractive, and additive processes [START_REF] Fera | A new perspective for production process analysis using additive manufacturing-complexity vs production volume[END_REF].

The AM process selection is another approach that is used for AM based on a test-part comparison approach. Byun and Lee [START_REF] Byun | A decision support system for the selection of a rapid prototyping process using the modified TOPSIS method[END_REF] proposed a Decision Support System (DSS) through analysis of part costs, build time, accuracy, roughness, strength, and elongation as evaluation factors. To evaluate these parameters they applied linguistic values, described by means of fuzzy numbers and a test-part approach. In another study [START_REF] Venkata Rao | Rapid prototyping process selection using graph theory and matrix approach[END_REF], a test-partbased methodology is presented by using graph theory and matrix approach. Moreover, the AMRPSP approach is also addressed by Baumers et al. [START_REF] Baumers | The cost of additive manufacturing: machine productivity, economies of scale and technology-push[END_REF] for two different AM technologies, Electron Beam Melting (EBM) and Direct Metal Laser Sintering (DMLS). A cost-estimation model is provided by considering two direct costs of manufacturing, energy consumption, and raw material cost, as well as total indirect costs which depend on the build time. Build time is also another factor that affects the manufacturing cost [START_REF] Alexander | Part orientation and build cost determination in layered manufacturing[END_REF][START_REF] Atzeni | Economics of additive manufacturing for end-usable metal parts[END_REF][START_REF] Byun | Determination of the optimal build direction for different rapid prototyping processes using multicriterion decision making[END_REF][START_REF] Di | A neural network-based build time estimator for layer manufactured objects[END_REF][START_REF] Yim | Build Time and Cost Models for Additive Manufacturing Process Selection[END_REF]. Technical specifications of the part are selected as other criteria to select the appropriate technologies and machines through an AHP method [START_REF] Cauê | Additive manufacturing process selection based on parts' selection criteria[END_REF].

A generic decision-making methodology is introduced in the context of material and Manufacturing Process Selection (MPS) [START_REF] Khaleeq Uz Zaman | Integrated product-process design: Material and manufacturing process selection for additive manufacturing using multi-criteria decision making[END_REF]. In another study [START_REF] Khaleeq Uz Zaman | Integrated product-process design to suggest appropriate manufacturing technology: a review[END_REF], a concerned literature review is provided for manufacturing technology selection and related criteria. A generic methodology is proposed to take into account the AM applications, DfAM guideline, three design criteria, such as cost, function, and environment. Moreover, an integrated design-oriented framework is proposed by Zaman et al. [START_REF] Zaman | Integrated design-oriented framework for resource selection in additive manufacturing[END_REF] to define a structure for design knowledge pertaining in conceptual and embodiment design. This framework is a guideline for design and feasible materialmachine-process combinations by analysis environmental axioms, like avoid toxic and harmful material, as well as using renewable material.

A brief survey of the literature shows that most of the studies, that addressed the process selection for the AM technology investigate only one evaluation criterion, or the key parameters as qualitative parameters, and use linguistic measurement approaches for evaluating them. Since there are also several studies that consider some key parameters as the quantitative parameters and propose models to estimate them, it will be interesting to investigate them as quantitative parameters in one practical general model. Therefore, this paper aims to evaluate concurrently two evaluation criteria: economic and performance for AMPRSP in order to select the most suitable technology and machine as the resources in the product development cycle. This paper aims to develop a DfAM approach by focusing on the material and process selection, as well as manufacturability evaluation concurrently. For this purpose, a new kind of process selection problem for AM called Additive Manufacturing Process and Resource Selection Problem (AMPRSP) is presented. This approach helps the designer and manufacturer to reduce the product development cost and time, as well as to modify the design by integrating different steps of the total product development cycle presented by Wright [START_REF] Wright | Design methods in engineering and product design[END_REF] (see Fig. 2).

AMPRSP allows us to find and select the most suitable process and resources among the AM technologies and machines based on the different selection criteria. Furthermore, most of the AM process selection studies are based on the test-part comparing method, which cannot be applicable to the problems with large-scale alternatives. Moreover, a Multi-Criteria Evaluation System is developed for the AMPRSP by considering manufacturability tools and two evaluation criteria of economics and performance. The conflicting key parameters of the evaluation criteria encourage the use of Multi-Criteria Decision-Making (MCDM) methods to deal with this problem. The main contributions of this paper that differentiate it from other studies are as follows:

-Designing a new multi-criteria evaluation system for AMPRSP. -Evaluation of the manufacturability of the product in verification and quantification levels. -Investigation of the economic and performance evaluation criteria simultaneously for the proposed MCES. -Identification of the key selection parameters and development of an assessment approach for each evaluation criteria in the proposed MCES. -Considering manufacturer preference and experts opinions to specify the importance of different evaluation criteria by weighting them. -Applying a developed AHP method for solving and providing a suitable AM process and resources.

Overall, a DfAM approach is proposed, which helps the designers and manufacturers to evaluate the manufacturability of the product with AM technologies, as well as, to find the suitable material and process for fabrication with AM by analysis of important AM criteria, like build time, cost, and accuracy.

Table 1 illustrates a summary of the investigated literature related to material and process selection strategies in comparison with the proposed methodology in this paper. As shown in this table, manufacturability evaluation is one of the important activity in DfM approach that is still lacking in the AM domain. Quantitative analysis of the significant AM criteria and constraints is another important issue that must be per-Fig. 2: The status of AMPRSP in the Product Development Process presented by [START_REF] Wright | Design methods in engineering and product design[END_REF] formed in a DfAM approach. For this purpose, it is necessary to provide the assessment models for AM criteria.

The assessment models are presented in study of Byun et al. and Rao et al [START_REF] Byun | A decision support system for the selection of a rapid prototyping process using the modified TOPSIS method[END_REF][START_REF] Venkata Rao | Rapid prototyping process selection using graph theory and matrix approach[END_REF] for some AM techniques. Fera et al. [START_REF] Fera | A new perspective for production process analysis using additive manufacturing-complexity vs production volume[END_REF] compared generally the AM and subtractive manufacturing process. Accuracy is analyzed qualitatively in the study of Mancanares et al [START_REF] Cauê | Additive manufacturing process selection based on parts' selection criteria[END_REF]. Zaman et al. [START_REF] Khaleeq Uz Zaman | Integrated product-process design: Material and manufacturing process selection for additive manufacturing using multi-criteria decision making[END_REF] provides the quantitative analysis for cost, but the analyses on the other criteria are qualitative. Environment aspects are also considered by Zaman et al. [START_REF] Khaleeq Uz Zaman | Integrated product-process design to suggest appropriate manufacturing technology: a review[END_REF][START_REF] Zaman | Integrated design-oriented framework for resource selection in additive manufacturing[END_REF].

In this study, all key parameters have been investigated as quantitative models. These models cover the most important affecting issues for each of the key parameters with a maximum range of application in AM processes. Also, quantitative analysis of significant AM criteria like cost, accuracy, and time permits to define the AM manufacturability in quantification level and providing a precondition to entering into optimization as the next level. Finally, these analyses help to define an integrated approach that helps designer and manufacturers in material and process selection as the first DfM activity (see Fig. 1). Therefore, an integrated DfAM approach will be presented by focusing on the AM material and process selection, as well as manufacturability concurrently.

This methodology will be presented comprehensively in the next section.

Proposed methodology for Additive Manufacturing Process and Resource Selection Problem (AMPRSP)

The importance of the selection of resources like materials and machines, as well as process selection in DfAM approaches, illustrates the necessity of providing a multicriteria evaluation system to deal with this AMPRSP. It Fig. 3: Methodology for solving AMPRSP consists of two major steps; manufacturability analysis and selection of the most appropriate alternative in AM technology. The goal is to select the most suitable AM process and machine as the resources. This methodology is formalized by a flowchart in Fig. 3 and each stage is described below:

Step 1: Manufacturability analysis

Manufacturability analysis is a key activity in DfM approaches to investigate product feasibility for manufac- turing. This analysis determines whether or not it is possible to manufacture the product with its known design specifications through AM. To achieve this goal, manufacturability is assessed based on the selected productdesign specifications, provided by the designer, and the characteristics of the different AM techniques such as geometrical accuracy, surface quality, and material class, as shown in Fig. 3. Therefore, the designer can assess the potential geometrical, functional and mechanical product characteristics in terms of their feasibility by an AM process. To perform this assessment, the DfAM worksheet [START_REF] Booth | The Design for Additive Manufacturing Worksheet[END_REF] and IT-classes table [START_REF] Lieneke | Systematical determination of tolerances for additive manufacturing by measuring linear dimensions[END_REF] (see Figs. 4 and5) are employed as manufacturability tools. This worksheet helps the designers to assess the potential quality of an AM part. It reduces print failures by filtering out undesirable designs before printing. The IT-classes table, which was defined initially in DIN EN ISO 286-1, represents the size of the ISO-tolerances by IT-numbers [START_REF] Asadollahi-Yazdi | Toward integrated design of additive manufacturing through a process development model and multi-objective optimization[END_REF][START_REF] Gao | The status, challenges, and future of additive manufacturing in engineering[END_REF][START_REF] Whitney | Manufacturing by design[END_REF][START_REF] Ahuja | Additive manufacturing in production: challenges and opportunities[END_REF][START_REF] Asadollahi-Yazdi | Integrated design for additive manufacturing based on skin-skeleton approach[END_REF][START_REF] Mueller | Additive Manufacturing Technologies -Rapid Prototyping to Direct Digital Manufacturing[END_REF][START_REF] Chakladar | A combined TOPSIS-AHP-method-based approach for non-traditional machining processes selection. Proceedings of the Institution of Mechanical Engineers[END_REF][START_REF] Kathryn Thompson | Design for Additive Manufacturing: Trends, opportunities, considerations, and constraints[END_REF][START_REF] Van Vliet | State-of-the-art report on design for manufacturing[END_REF][START_REF] Asadollahi-Yazdi | Integrated design in additive manufacturing based on design for manufacturing[END_REF][START_REF] Bikas | Additive manufacturing methods and modelling approaches: a critical review[END_REF][START_REF] Frazier | Metal Additive Manufacturing: A Review[END_REF][START_REF] Huang | Additive Manufacturing: Current State, Future Potential, Gaps and Needs, and Recommendations[END_REF][START_REF] Shah | Comparison of different additive manufacturing methods using computed tomography[END_REF][START_REF] Hu | Optimal design for additive manufacturing of heterogeneous objects using ultrasonic consolidation[END_REF][START_REF] Ponche | A novel methodology of design for Additive Manufacturing applied to Additive Laser Manufacturing process[END_REF][START_REF] Hascoet | Manufacturability analysis to combine additive and subtractive processes[END_REF][START_REF] Pradel | Investigation of design for additive manufacturing in professional design practice[END_REF]. It allows providing a qualitative analysis for the product accuracy. The relevant influential factors are identified based on the geometrical accuracy for the material extrusion and laser-based processes of AM technology. Overall, this step is completed through some activities:

1. The manufacturability of the part using the AM process is assessed by the DfAM worksheet (Fig. 4). 2. Different AM processes and technologies are filtered via two steps in order to obtain the available AM processes and techniques for fabricating the part.

-Initially, AM processes, illustrated in Table 2, are filtered based on the material class used in the part. The density, tensile strength, flexural modulus, elongation at break, impact strength, hardness, and heat deflection temperature are supposed as the important material properties that are considered for filtering the AM processes having similar material classes as the material to be used for the part. -The resultant alternatives are filtered by a developed IT-classes table (Fig. 5) and expert opinions for satisfying required interval tolerance and surface roughness of the part.

A detailed view of the manufacturability issues are developed to help the decision-makers, for example, the process-design department, the head engineer, the group leader, etc. This step makes possible to determine the available AM techniques to fabricate the product with AM technology.

Step 2: Selection of the most appropriate AM process and resource through a Multi-Criteria Evaluation System (MCES)

The second step is developed to select the most appropriate AM machine as the resource among the available alternatives of the AM technique. This step is devoted to MCES that will be completed in four main sub-steps, including alternative screening, economic analysis, performance analysis, and finally ranking the alternatives. Cost for economic analysis, as well as build time and accuracy for the performance analysis, are considered as Fig. 5: Achievable tolerances of the selected traditional and AM processes [START_REF] Lieneke | Systematical determination of tolerances for additive manufacturing by measuring linear dimensions[END_REF] the significant criteria of this production system for the proposed MCES. In the following, this MCES system will be discussed comprehensively.

Alternative screening

Primary screening is performed on the available alternatives in this step. For this purpose, an AM database has been presented which contains the AM processes, materials, and specifications. This database is developed based on the machine applicability to the different evaluation models of AM processes. To create this database, this approach focuses on several models that exist in the literature for evaluation of the AM process, and the machine specifications are introduced regarding their required information as listed by the vendors. In this study, the AM machine specifications will be used in the economic and performance analysis as listed in Table 3.

It must be mentioned that this database covers more than 100 different types of AM machines, such as 3D Systems, EOS GMbH, Stratasys, etc. These screened alternatives must meet the geometrical specifications of the product derived from the first step.

Economic analysis

Cost is defined as one of the key parameters for economic analysis in this proposed approach. The main focus of this step is evaluating the total cost of different AM machines as resources for the manufacturing process.

Regarding the reviewed literature and the application range in AM processes, the cost model developed by Yim and Rosen [START_REF] Yim | Build Time and Cost Models for Additive Manufacturing Process Selection[END_REF] was selected to make a preliminary estimation of the overall cost of manufacturing a product with an AM machine. The applied cost model consists of four sub-costs: machine purchase cost (P ), material cost (M ), machine operation cost (O), and labor cost (L), as shown in Equation 1. Machine operation cost is build time cost and it can be calculated based on the recoating time, material processing time, and delay time that will be explained in build time (T b ) estimation section. Therefore, the cost model applied in this paper is elaborated as follows:

Cost T otal = P + M + O + L (1)
Where:

P = Purchase Price × T b 0.95 × 24 × 365 × Y lif e (2)
M = K S .K r .N.v.c m .ρ (3)
O = T b .C O (4) L = C L .T L (5)
Equation 2 illustrates how the machine purchase cost is calculated. It was computed based on the purchase cost per hour during build time (T b ). Also, the useful life (Y lif e ) was used for calculating the purchase cost per hour, so that it has been assumed to have 95% of the machine up-time during the useful time of a machine. Equation 3 presents the detail computation of the material cost (M ) which was computed based on the support structure factor (K s ), recycling factor (K r ), material rate (C m ), number of parts (N ), part volume (v), and material density (ρ), where K s is captured in order to consider the cost of additional material consumption. It ranges from 1 to 1.5 and is supposed to equal 1 for the cases without the support structure. K r is presented to capture the cost related to the contribution of wasting loose powder, which is not recycled after the building and ranges from 1 to 7. In Equation 4, machine operation cost (O) was computed by the build time (T b ) as a parameter, and operation rate of the product (C O ) as an empirically determined constant. The operation rate is related to factory overhead. Finally, Equation 5calculates the labor cost based on the labor rate (C L ) and the labor time (T L ).

Performance analysis

In this approach, the performance of the AM process is evaluated based on the two parameters, build time and accuracy performance, which were previously used in several studies [START_REF] Khaleeq Uz Zaman | Integrated product-process design: Material and manufacturing process selection for additive manufacturing using multi-criteria decision making[END_REF][START_REF] Khorram | Selection and Implementation of Additive Manufacturing[END_REF][START_REF] Alberto | Towards a standard experimental protocol for open source additive manufacturing[END_REF]. In the previous related studies, authors considered only one aspect or merely used expert opinions for evaluating performance. However, it is important to analyze simultaneously all effective parameters for providing a comprehensive evaluation.

For each parameter, the validated approach was applied and explained as follows:

Build time estimation: Build time is a crucial parameter for AM which affects the strategy of machine selection, cost, and production scheduling processes. For computing the build time, most of the existing models are process-or machine-specific. In this study, the build time estimation model proposed by Yim and Rosen [START_REF] Yim | Build Time and Cost Models for Additive Manufacturing Process Selection[END_REF] is chosen, as it can be applied to most of the different types of AM processes. The applied build time estimation model is explained as follows:

T b = T r + T s + T d (6)
where:

T r = L p .T rp + L s .T rs (7)
T d = L p .(T pre + T post ) + T startup (8)
T s = N n × Sl 3600 × ss avg (9) n = ( P L x + g x -g e bb x + g x )( P L y + g y -g e bb y + g y ) (10) sl = (bb x × bb y × A f n )( N × L p hr × d + K s × L s d ) = A avg ( N × L p hr × d + K s × L s d ) (11)
T s = T s-layer × (L P + L S ) (12)
Equation 6 illustrates the total build time (T b ), which is estimated by the accumulation of recoating time (T r ), material processing time (T s ), and delay time (T d ). Recoating time is defined as the process of laying or depositing material on each layer for further processing. Also, some AM processes, such as extrusion, do not have the recoating process; therefore, T r is 0 in such cases. In Equation 7, the detailed computation of the recoating time is shown, where L p and L s represent the number of layers of the part and support. Also, T rp and T sp are part recoat time and support recoat time per layer.

The detailed computation of delay time was presented in Equation 8, where T pre and T post are the delay time before and after scanning each layer respectively, and T startup is start-up time. Equation 9calculates the material processing time for different AM processes applied in this study, except laser-polymerization, based on average scan speed or nozzle (ss avg ) and scan length (s l ). Equations 10 and 11 represent the material processing time calculation. For laser-polymerization AM processes, Equations 9 to 11 are not directly applicable. Hence, the material processing time for this type of AM processes is computed based on scan time per layer (T s-layer ) and the number of total layers, as shown in Equation 12.

Accuracy performance: For the evaluation of accuracy performance, the approach proposed by Brajlih and Igor [START_REF] Brajlih | Speed and accuracy evaluation of additive manufacturing machines[END_REF] is used. The accuracy performance is evaluated by the average of the maximum expected deviation of the dimensions X, Y, and Z (D Maxx , D Maxy , D Maxz ) for each alternative, as shown in Equation 13. This evaluation is based on the typical accuracy (A) of machine and CAD dimensions of the part (X, Y , and Z). It is crucial to separate the deviations due to the nature of the AM machine, according to the coordinate directions of the machine. Therefore, Equations 14 to 16 represent the maximum deviation for each coordinate.

Accuracy = D Maxx + D Maxy + D Maxz 3 (13)
D Maxx = X.A (14)
D Maxy = Y.A (15)
D Maxz = Z.A (16)

Ranking alternatives

The main goal of this proposed approach is to determine the most appropriate AM technology and machine for a company to fabricate a product. Alternatives were pre-evaluated in the previous steps based on economic and performance analysis. In this step, the AHP method is used to define the most appropriate alternative. In particular, a developed AHP as an MCDM method is applied to solve the proposed MCES for AMPRSP. The applied method is defined as follows:

Constructing hierarchical structure of the problem: The three main parts of a decision-making system include the main objective, evaluation criteria or sub-criteria, and available alternatives. A multi-level hierarchical structure is constructed based on these three main parts. Fig. 6 presents the multi-level hierarchical structure proposed in this paper. It consists of four levels:

-Objective: Selecting the most appropriate AM machine and technology for AMPRSP. -Criteria: Considering two evaluation criteria: economic and performance. -Sub-criteria: Investigating the three key selection parameters, such as build time, accuracy performance, and cost. -Alternatives: Evaluating the different alternatives.

Pair-wise comparison: AHP is a pair-wise comparisonbased methodology. In this study, the pair-wise comparisons are carried out based on the manufacturer preferences and experts opinions to determine the relative importance of the main criteria and sub-criteria respectively. Manufacturer preferences translate a strategic plan of the company, which are specified by the administrator or any strategic decision-maker of the company. Two evaluation criteria, which include economics and performance are supposed for this decision-making approach. Also, expert opinions are used for determining the relative importance of three proposed key parameters. The applied scales for performing the pair-wise comparisons are shown in Table 4 [START_REF] Thomas | Decision making with the analytic hierarchy process[END_REF].

Definition of the relative importance of criteria and subcriteria: In this step, double-entry matrices are made from the collected data in the previous step in order to compute the priority vector of criteria (weights). Then, the relative importance of criteria and sub-criteria are obtained from the eigenvector of the double-entry matrices normalized to 1.

Consistency of judgment:

The consistency of judgment is a review of judgment which aims to ensure a reasonable level of consistency of expert opinion and manufacturer preferences in the pair-wise comparison. Thus, the consistency index proposed by Satty [START_REF] Thomas | Decision making with the analytic hierarchy process[END_REF] is applied for evaluation of the consistency. Also, an inconsistency problem is supposed if this index value exceeds 0.1.

Ranking alternatives based on prioritization: Until now, the importance of the criteria for machines and technologies have been defined and validated. The prioritization of the alternatives must be specified by evaluating the alternatives by giving the normalized results, related to economic and performance analysis due to the importance of each criterion. Overall, this proposed approach helps to develop a DfAM approach which focuses on the process and resource selection as AMPRSP process. An MCES is developed to find the most appropriate process and resource for AM regarding two evaluation criteria: economic and performance. This approach considers three main AM criteria simultaneously, including build time, accuracy performance, and cost. Finally, the most appropriate process and machine are selected for fabrication. In the next section, this approach will be illustrated and verified through a case study.

Case study: Fog light bezel

In this section, the methodology described in the previous section will be explained through an industrial case study which is used in the automotive industry as shown in Fig. 7. The most appropriate process and resource will be selected for the manufacturing of this part. This part is a fog light bezel which is made of Polypropylene TD30 with relatively high accuracy and surface finishing level. This part is manufactured through a traditional molding process. Therefore, the company decided to investigate the possibility of manufacturing this part by using AM technology regarding its benefits. Thereupon, the objective is initially to assess the manufacturability of this part by AM technologies and then to select the most appropriate process and resource of AM for fabricating this part. Table 5 shows the required properties of this part as its design specifications. In the following, the proposed approach will be implemented in the case study in two steps:

Step 1: Manufacturability analysis of the fog light bezel

This step aims to investigate the feasibility of manufacturing the part through AM technology. Initially, manufacturability of the part is assessed based on the AM work-sheet as shown in Fig. 8; its rate is 23, which shows that the design of the part studied has a moderate likelihood of success. Then, regarding the material class of the part, plastic-based processes are filtered based on similarity in the material properties of Polypropylene TD30. Further, for satisfying the needed IT-class, the manufacturability tools defined in step 1 are applied. The final set of available AM processes are listed due to the selected product design specifications, i.e. material properties, IT-class, and average surface quality of the molding process, as shown in Fig. 9.

4.2

Step 2: Selection of the most appropriate AM method and resource for the fog light bezel This step is performed in several activities:

-Primary screening: After obtaining available AM processes, some alternatives related to these AM processes will be available based on AM database alternatives. Primary screening is performed based on a comparison between geometrical specifications Fig. 9: Selection of the available AM process for the fog light bezel of the product and machine specifications. Table 6 presents the list of screened alternatives of available AM processes as resources for fabricating the part. -Assessment of the screened alternatives: Every screened alternative is evaluated and assessed based on two evaluation criteria: economic and performance. This evaluation enables having a detailed view of economic and performance issues for alternatives and it helps us to select the most appropriate one in the AMPRSP. Three models were defined in the previous section in order to estimate the values of key parameters (total cost, build time, and accuracy performance) for each of the screened alternatives. Several parameters and variables are investigated as inputs for these models, which are related to the machine, part, and either machine or part (machine-part). The information of the parameters related to part and machine is presented in Tables 5 and6 respectively. Moreover, Table 7 also demonstrates the information of the machine-part parameters, provided by AM experts and manufacturers. The results obtained correspond to the assessment of the screened alternatives, as shown in Table 8. In this table, total cost and time are calculated for N parts. -Ranking and decision-making: The AHP method must be applied to select the most appropriate alternative as a final result of the proposed MCES for AMPRSP. Tables 9 and10 represent the double entry matrices obtained from pairwise comparisons of criteria and sub-criteria by experts. These priorities are devoted to these criteria due to the decision makers and experts opinions. The relative importance of the criteria and sub-criteria are shown by the weights (see Fig. 10). As shown in this Figure, the economic factors are supposed more important than the performance criteria in this case study. These weights could be modified according to the company's strat- egy. The consistency of the expert opinion has been tested, obtaining an index of 0.009, which represents that the calculated weights are consistent. Consequently, the most appropriate alternative for this case is Fortus 900mc with 35% priority, as illustrated in Table 11. Sigmax is ranked second with 33% priority. In addition, Fig. 11 summarizes these results and shows the portion of each parameter in total priority for each of the screened parameters. Moreover, the selection of alternative Fortus 900mc can be explained as it presents the best performance for most of the criteria. This paper focuses on AM Process and Resource Selection Problem which is an improvement effort in the field of CE/ID which takes inspiration from process selection for rapid prototyping in order to improve the DfAM approaches. In spite of this inspiration, the application area of AMPRSP is more extensive than AM process selection and covers its main limitations, testpart comparison, and large-scale alternatives. Due to the complexity of AM technologies, the selection of process and resources depends on different functional, technological and economic parameters. The industry needs a pragmatic multi-criteria decision support approach to be able to choose the most appropriate technology for their needs. Our proposed approach integrates these factors into a new systemic framework based on assessment models that have been approved in the literature. Nevertheless, the integration of these models into a single tooled framework clearly addresses the need for a return on investment in AM.

The AMPRSP selects the most suitable AM processes and resources for fabricating a part based on different evaluation criteria and key selection parameters. An MCES was proposed in this paper for the AMPRSP. The proposed MCES simultaneously investigates the manufacturability and two evaluation criteria by considering manufacturability tools and key selection parameters respectively. It has been applied as a step-by-step methodology and it is easy to employ procedures such as screening, comparative assessment, and ranking processes to select the most suitable alternative for AMPRSP. As pre-evaluation, the proposed MCES has adopted different models to estimate the key selection parameters, including total cost, build time, and accuracy performance, in order to provide a detailed view of the evaluation criteria. Then, the most suitable alternative is selected by using the AHP method. A case study is presented to illustrate and verify the proposed approach and its DfAM guidelines for researchers in the product development process.

Despite the passage of more than 30 years since the emergence of AM technology, and the achievement of significant improvements during these years, there is still a major lack in regard to the in-comprehensiveness of methods, tools, and rules in this technology. As reviewed in the literature, most of the AM methods and models are process-specific. Due to this fact, the scope of this study does not end here and it can be extended either in the scope of this methodology or in the application of new models and tools with greater precision and comprehensiveness for this MCES. the other evaluation criteria, such as sustainability and innovation, the introduction of key parameters for each new criterion, and analysis of the dependency and independency of key parameters, these can be investigated as multiple future works for the scope of methodology. Applying more comprehensive models for the assessment of the key parameters and developing a software tool to effectively support this MCES are other points that can be developed in future prospects.

Fig. 4 :

 4 Fig. 4: DfAM worksheet [19]

Fig. 6 :

 6 Fig. 6: Hierarchical structure for the proposed MCES

Fig. 7 :

 7 Fig. 7: Soild model of the fog light bezel

Fig. 8 :

 8 Fig. 8: Manufacturability verification through AM worksheet for the fog light bezel

Fig. 10 :

 10 Fig. 10: Weighted hierarchical structure for the proposed MCES

Fig. 11 :

 11 Fig. 11: Priority of the screened alternatives

Table 1 :

 1 Analysis of AM process selection studies

			Manufacturing domain			Evaluation criteria
				AM process type		Economic		Performance
										Cost	Accuracy Build Time
	Researchers	Manufacturability	Laser Melting	Laser Polymerization	Extrusion	Material adhesion	Material Jetting	Electron beam	Quantitative	Qualitative	Quantitative	Qualitative	Quantitative	Qualitative
	Byun and Lee [39]		*	*	*		*		*		*		*
	Rao and Padmanabhan [40]		*	*	*		*		*		*		*
	Yim et al. [46]		*	*	*	*	*		*				*
	Mancanares et al. [47]		*	*	*	*	*	*				*
	Baumers et al. [41]		*					*	*
	Fera et al. [38]	*							*		*
	Zamen et al. [48]		*	*	*	*	*	*	*			*		*
	Zamen et al. [49]		*	*	*	*	*	*	*
	Zamen et al. [50]		*	*	*	*	*	*	*
	This paper	*	*	*	*	*	*	*	*		*	*	*

Table 2 :

 2 Classification of different AM processes based on ASTM International standard (modified by the authors)

	Process	AM technology	Material class	Materials
	Vat photo-polymerization	Stereolithography (SLA)	Plastic	UV curable resins, waxes, ceramics
	Material jetting	Multi-Jet Modeling (MJM) Metal Injection Molding (MIM)	Plastic Metal	UV curable resins, waxes
	Binder jetting	Three-Dimensional Printing (3DP)	Plastic Metal	Composites, polymers, ceramics, sand
	Material extrusion	Fused Deposition Modeling (FDM) Arburg Plastic Free forming (APF)	Plastic Plastic	Thermoplastics, waxes Thermoplastics
		Selective Laser Sintering (SLS)	Plastic	Thermoplastics
		Selective Laser Melting (SLM)	Metal
	Powder bed fusion	Electron Beam Melting (EBM) Direct Metal Laser Sintering (DMLS)	Metal Metal
		Selective Laser Cladding (SLC)	Metal
		Laser Consolidation (LC)	Metal
	Sheet lamination	Laminated Object Manufacturing (LOM)	Plastic Paper	Thermoplastics
	Direct energy deposition	Laser Engineered Net Shaping (LENS) Electron Beam Welding (EBW)	Metal Metal

Table 3 :

 3 AM machine specifications (proposed by the authors)

	Machine specifications	Unit
	Machine Name	-
	Technology	-
	Manufacturer	-
	Dimensions (W × D × H)	cm × cm × cm
	Build Envelope Capacity (P l x , P l y , P l z )	mm × mm × mm
	Purchase Price	$
	Min Layer Thickness	mm
	Min Resolution	µm
	Scan Speed/Nozzle Speed (Min, Max, Avg)	mm s -1
	Typical Accuracy (X, Y, Z)	mm
	Laser or Nozzle Diameter	mm
	Laser Power	W CO 2
	Electrical Requirement	VAC
	Energy Consumption	MJ h -1
	Useful Time	year
	Part Recoat Time	s
	Start-up Time (Warm-up Time)	h
	Cool Down Time	s
	Materials list	-

Table 4 :

 4 Applied scales for the intensity of importance[START_REF] Thomas | Decision making with the analytic hierarchy process[END_REF]

	Intensity importance Scale
	Equal importance	1
	Weak plus	2
	Moderate importance	3
	Moderate plus	4
	Strong importance	5
	Strong plus	6
	Very strong importance	7
	Very, very strong	8
	Extreme importance	9

Table 5 :

 5 Part design specifications

	Factor	Descriptions
	Name	Fog Bezel Front Light
	Application	Automotive industry
	Material Specifications	Polypropylene TD30
	Geometrical specifications
	Build envelope	174.65 × 171 × 62.5 mm
	Volume	161 730 mm 3
	Tolerance	0.2 mm
	IT class	11
	Surface roughness	N9

Table 6 :

 6 Screened alternatives

	Process	Machines	Manufacturer Price ($US) Build Envelope (mm 3 )	Material
		Fortus 360 mc	Stratasys	100000	355 × 254 × 254	ABS-M30, PC-ABS, ASA, PC, Nylon 12
		Fortus 450 mc	Stratasys	185000	406 × 355 × 406	ABS-M30, ABS-M30i, ABS-ESD7, ASA, PC-ISO, PC, Nylon 12, ULTEM 9085
	FDM	Fortus 900 mc	Stratasys	250000	914 × 609 × 914	ABSi, ABS-M30, ABS-M30i, ABS-ESD7, PC-ABS, ASA, PC-ISO PC, Nylon 6, Nylon 12, ULTEM 9085, ULTEM 1010, PPSF
		Sigma R17	BCN3D	3100	210 × 297 × 210	PLA, ABS, PVA, TPU, Nylon, HIPS, Specials
		Sigmax	BCN3D	5225	420 × 297 × 210	PLA, ABS, PVA, TPU, Nylon, HIPS, Specials
		Replicator Z18	MakerBot	6500	300 × 305 × 457	Tough PLA, PLA, ABS

Table 7 :

 7 Machine-part parameters related to screened alternatives for fog light bezel

	Parameters	Fortus 360mc	Fortus 450mc Fortus 360mc Sigma R17 Sigma x ReplicatorZ18
	K s	1.3	1.3	1.3	1.3	1.3	1.3
	K r	1	1	1	1	1	1
	C m ($/g)	0.15	0.15	0.15	0.15	0.15	0.12
	ρ(g/mm 3 )	0.001	0.001	0.001	0.001	0.001	0.0012
	C o ($)	3	3	3	3	3	3
	C L ($)	23	23	23	23	23	23
	T L (s)	3600	3600	3600	3600	3600	3600
	L P	48	48	48	24	24	120
	L s	48	48	48	24	24	120
	T pre (s)	0	0	0	0	0	0
	T post (s)	60	60	60	60	60	60
	bb x (mm)	36	36	36	36	36	36
	bb y (mm)	30	30	30	30	30	30
	g x (mm) 100	100	100	100	100	100
	n	13	13	13	13	13	13
	A f n	0.54	0.54	0.54	0.54	0.54	0.54
	A avg	579.75	579.75	579.75	579.75	579.75	579.75
	sl(s)	6936453	6967312	15854750	3523718	2349145	39636876

Table 8 :

 8 Evaluation of screened alternatives according to the three key parameters for N part

	Alternatives	AM process Total cost ($) Total build time (h) Accuracy (mm)
	Fortus 360 mc	FDM	24923	3807	0.21
	Fortus 450 mc	FDM	15314	1531	0.21
	Fortus 900 mc	FDM	4140	353	0.20
	Sigma R17	FDM	11081	2757	1.56
	Sigmax	FDM	3418	623	1.56
	Replicator Z18	FDM	32379	9501	1.66

Table 9 :

 9 Double entry matrix of two evaluation criteria

		Economic Performance
	Economic	1	2
	Performance	1 2	1

Table 10 :

 10 Double entry matrix of key parameters as sub-criteria

		Build Time Accuracy
	Build Time	1	3
	Accuracy	1 3	1

Table 11 :

 11 Results of the ranking alternatives

			Normalized Value of
		Cost	Build time Accuracy Priority Ranking
	Fortus 360mc	0.137	0.093	0.939	0.076	5
	Fortus 450mc	0.223	0.230	0.939	0.113	3
	Fortus 900mc	0.826	1.000	1	0.351	1
	Sigma R17	0.308	0.128	0.125	0.099	4
	Sigmax	1.00	0.567	0.125	0.326	2
	Replicator Z18 0.106	0.037	0.118	0.036	6

Acknowledgements The authors would like to thank the Lorraine Fab-living-Lab and the ENSGSI ( École Nationale Supérieure en Génie des Systèmes et de l'innovation)-University of Lorraine, for their collaboration and participation in conducting the case study and the presented research work.