B Delinchant
email: benoit.delinchant@g2elab.grenoble-inp.fr

D Lahaye
email: d.j.p.lahaye@tudelft.nl

F Wurtz

J.-L Coulomb

Manifold mapping optimization with or without true gradients

Keywords: Space Mapping, Manifold Mapping, Optimization, Surrogate model, Gradients, Symbolic derivation, Automatic differentiation

This paper deals with the Space Mapping optimization algorithms in general and with the Manifold Mapping technique in particular. The idea of such algorithms is to optimize a model with a minimum number of calls using a less accurate but faster model. In this optimization procedure, fine and coarse models interact at each iteration in order to adjust themselves in order to converge to the real optimum. The Manifold Mapping technique guarantees mathematically this convergence but requires gradients of both fine and coarse model. Approximated gradients can be used for some cases but are subject to divergence. True gradients can be obtained for many numerical model using adjoint techniques, symbolic or automatic differentiation. In this context, we have tested several Manifold Mapping variants and compared their convergence in the case of real magnetic device optimization.

Introduction

The space-mapping technique [START_REF] Bandler | Space mapping: the state of the art[END_REF] allows computationally expensive simulation based optimization procedures to be speeded up through the use of approximate models. In the space mapping literature the so-called fine and coarse models are conveived as mappings from the design space to the space of model responses. The key element is the space mapping function. It reparametrises the coarse model domain in such a way to minimize the discrepancy between the fine and coarse model responses. The composition of the space-mapping function and the coarse model response defines a surrogate for the fine model. Instead of solving the fine model problem directly, space-mapping solves the surrogate optimization problem through a sequence of approximations of the space-mapping function. This in turn defines a sequence of coarse model optimization problems whose solution by definition converges to the space-mapping solution. The computational efficiency of this procedure stems from the fact that it takes less fine model evaluations to converge than it takes to solve the fine model optimization problem. The drawback is that the space-mapping solution does not necessarily coincide with the fine model optimum.

In the manifold-mapping technique [START_REF] Echeverria | Space Mapping and Defect Correction[END_REF], the surrogate model is constructed in such a way that in a neighbourhood of the fine model optimum, the surrogate model response closely ressembles its fine fine model counterpart. This guarantees that the solutions of the surrogate and fine model optimization problem do coincide. The spacemapping function is replaced by the so-called manifold-mapping function. The latter is an affine transformation between the tangent manifolds of the fine and coarse model image spaces. Manifold-mapping is computationally as efficient as space-mapping. Space mapping techniques have been used in electromagnetic device optimization for several years now [START_REF] Lahaye | Adaptive manifold-mapping using multiquadric interpolation applied to linear actuator design[END_REF] [START_REF] Crevecoeur | Space mapping optimization of the magnetic circuit of electrical machines including local material degradation[END_REF]. Different techniques can be used, but manifold mapping, which is the only one proved to converge to the fine model optimum is always using approximated gradients of the fine model since true gradients are not always available.

This paper details the manifold mapping technique and argues that exact gradients can be available more or less easily nowadays. The computational cost of these gradients is generally small compared with that of the fine model, and the convergence of the manifold mapping algorithm is improved. This property will be required in the future when optimization specifications becomes more and more constrained.

Manifold mapping algorithm

Mathematical background

Let us consider an optimization problem with design variables x in the design space

y z c S x that such X x find k X z mm k mm k - = ∈ ∈ )) ( ( min arg * , * , (2)
)] ( ) ( , ), ( ) ( ), ( ) ( [ ) 0 , max( 2 1 n k k k k k k x c x c x c x c x c x c C - - - - - - = ∆ L )] ( ) ( , ), ( ) ( ), ( ) ( [ ) 0 , max( 2 1 n k k k k k k x f x f x f x f x f x f F - - - - - - = ∆ L
During the first n iterations, these matrices are not fully describing the tangent planes but are enough to define a search direction until k becomes greater than n .

In order to improve robustness of the approximation k S is defined with a complementary term

) ( ) ( ). ( , , * * T c k c k k k k U U I x F x C S - + ∆ ∆ = + where c k U , is provided by the singular value decomposition of T c c c V U C . .Σ = ∆ at each k iteration.
Using k S mapping function, an update objective

( ) ) ( ) ( y x f S x c y k k k k - - =
can be introduced leading to an asymptotically equivalent problem:

k X z mm k mm k y z c x that such X x find - = ∈ ∈ ) ( min arg * , * , (3)
In other words, the mapping is transfer from the coarse model to the optimization objectives which leads to an easier algorithm implementation. And by construction

* * , f mm k x x =
corresponding to the fine model optimum.

A trust region strategy has to be implemented in addition to this algorithm in order to prevent arbitrary large step size

1 - -k k x x
. To prevent manifold mapping from taking such steps, a trust-region stabilization was presented in [START_REF] Hemker | A Trust-Region Strategy for Manifold Mapping Optimization[END_REF].

Validation on a simple test case

A first test case has been computed in order to show that algorithm proposed can find good results. The problem is defined by 3 degrees of freedom and an objective function is a least square functional with 3 components. A coarse model has been defined using minor modifications in some formulas in order to approximate the fine model.

The objective function has been plotted according the optimization iterations for several space mapping strategies:

-OM : Output Mapping, which is a manifold mapping with the identity matrix for the mapping function S.

The objective update is then

) ( ) ( k k k x f x c y y - + = .
-MM Approx: Manifold mapping using a tangent plane approximation for C ∆ and F ∆ .

-MM Approx without SVD: Manifold mapping using a tangent plane approximation without correction :

) ( ). ( * * k k k x F x C S + ∆ ∆ = .
-MM Approx trust: Trust Region manifold mapping in order to ensure convergence.

-MM Exact: Manifold mapping using true gradients.

-MM Exact Trust: Trust Region manifold mapping in order to ensure convergence. Fig. 1 shows that MM with true gradient converge extremely fast to a good solution, while OM and MM with approximated gradients converge slower with some estimations. Trust region leads to slower convergence but in the case of MM with approximated gradient the same solution is reached with only one iteration more. In the case of approximated gradient correction term

T c k c k U U I , ,
in the mapping function, is important. -mobile magnet with x-axis translation degree of freedom, -an iron plate with y-axis rotation degree of freedom.

Optımızatıon specıfıcatıons

The objective is to find mobile magnet dimensions in order to obtain a torque as linear as possible. To do this, a least squares objective functional is defined in Equation 4and Fig. 3:

∑         Γ - Γ = i i i tx tx tx x F 2 max max . ) ( ) ( (4)

Fine modeling

For the mobile magnet, a Coulombian equivalent charge approach is used in order to compute magnetic field applied on the ferromagnetic plate.

For this last one, a steady-state Method of Moments (MoM) is applied. It consists in the meshed of ferromagnetic bodies along the X, Y and Z axes, with uniform induced magnetized elementary blocks [START_REF] Chadebec | A Review of Magnetostatic Moment Method[END_REF]. This method does not require to mesh the air and is particularly efficient for "radiating" systems. The issues with such an integral method are full matrices and computation memory limitation.

Fig. 4. Surface charge method to compute torque when magnetization of each bloc is known.

Fig. 4 shows the magnetization of each block which depends on the external field (H ext produced by field sources such as magnets or conductors) as well as on the field produced by other blocks depending on their own magnetization.

M Q H H ext ⋅ + = (5)
Blocks interaction is defined by the interaction matrix (Q in equation 5) : Q is a square matrix of size (3m)x(3m) (m = number of blocks), composed of 3x3 square matrices which represent the magnetic excitation created by a block to an other.

The ferromagnetic material behavior law is defined by non linear law, parameterized by the saturation induction and initial permeability of the material. The unknown ferromagnetic magnetizations can then be found by using a Newton-Raphson solver with a relaxation method to ensure convergence.

When magnetizations are known, each block can be seen as a set of parallelepiped magnets (Fig. 4), in order to compute the magnetic field or force and torque (equation 6).

[ ]

∑∑ ∫∫ = = × = N i j s s s s ds ij ij ij ij 1 6 1 0 . ) ( . P H OP Γ ext σ µ ( 6
)
where O is the pivot point where the torque is computed, P ij are barycenter of each bloc and σ ij there equivalent surface charge derived from their magnetization.

Computation time of this model depends on the number of blocks (see Fig. 5). Computation is fast, a good accuracy is reached in less than one minute, but we have always to think about increasing model computation and optimization time if our objective is system simulation and design.

Coarse modeling

The iron plate is not discretized and global demagnetisation coefficients in 3D space are computed analytically using classical rectangular shapes formulas. Under the assumption that the plate is saturated along the x-axis, the magnetization along z-axis remains in the linear domain. It is then possible to solve explicitly the 2 magnetizations M x and M z in order to compute torque Γ y by equation 7.

( )

0 0 0 . . . . H Mz H Mx V µ plate y + - = Γ (7)
This model is fully analytical and an optimization using its gradients is less than one second

Modeling comparison

A FEM simulation has been done but can not be considered as a reference (just a good approximation) due to the mesh issues for such thin geometries. Fig. 7. Modeling comparison on torque regarding mobile magnet position for both Fine and Coarse models but also with FEM

Model derivation

Nowadays, many techniques and tools are available in order to compute model Jacobian of analytical model and numerical models. Two kind of derivation techniques exist, the symbolic one which examines computation model in order to simplify as far as possible the derivatives expressions; and the automatic one which is more systematic but less efficient.

Symbolic derivation

First is the symbolic derivation based on mathematical theorems which tries to express derivatives of a model for each kind of modeling methods. For example, if an unknown 'I' is expressed by numerical integration (eq 8) partial derivatives of function I regarding parameters p can be given by (eq 9).

∫ = ) ( ) (
). , ( ) (

p up p lo dx x p f p I (8) )) ( , ( ) ( )) ( , ( ) ( . ) , ( ) ( ) ( )
( p lo p f p p lo p up p f p p up dx p x p f p p I p up p lo ⋅ ∂ ∂ - ⋅ ∂ ∂ + ∂ ∂ = ∂ ∂ ∫ (9)
Other well known applications of such kind of derivation are :

-implicit theorem [START_REF] Jittorntrum | An Implicit Function Theorem[END_REF]: which allows symbolic derivation of unknwons computed using implicit solver such as Newton-Raphson procedure.

-Adjoint method [START_REF] Rakotoarison | Formal Sensitivity Computation of Magnetic Moment method[END_REF]: which allows performant computing of multiple calls to gradients using a dual code which depends on the numerical problem solved (linear system, ...) .

Automatic derivation

Defining derivative of each basic mathematical operators can lead to automatic differentiation (AD) tools [START_REF] Enciu | Delinchant Automatic differentiation for electromagnetic models used in optimization[END_REF].

Typically, AD can be implemented using either the operator overloading (ADOL-C, CppAD, etc) or the source transformation technique (ADIFOR, ADiJaC, etc). In operator overloading one overloads the operators which are applied on new variable types, with the routine call performing the actual derivative computation. The source transformation approach examines the source code of the original function and generates new code that computes the desired derivative together with the original function value.

Software implementation

CADES framework1 [START_REF] Delinchant | An optimizer using the software component paradigm for the optimization of engineering systems[END_REF] implements symbolic and automatic differentiation techniques. So it is easy for several kind of models to add Jacobian computation in order to perform optimization. This software will be used in this work in order to produce Jacobians on both coarse and fine model.

Model derivation

As defined in the previous section, several techniques are available in order to compute gradients. For the fine modeling, derivatives have been computed using implicit theorem and an adjoint code [START_REF] Rakotoarison | Formal Sensitivity Computation of Magnetic Moment method[END_REF] leading to a low cost compared to the computation of the magnetization vector itself. This is also due to the fact that no non-linear solving procedure is required. For the coarse model which is essentially based on analytical equations, a simple derivation has been done automatically using CADES framework. The torque gradient with respect to the magnet position is plotted in Fig. 9.

Fig. 9. Fine and coarse torques and their formal derivatives, versus the magnet translation.

Optimization results

Fig. 10. Optimization algorithms convergence on the electromagnetic device optimization.

A direct optimization has been performed using fine model and its gradients in order to get a reference solution.

The initial solution is provided by the coarse model optimal solution. The algorithm used is an Interior Point algorithm. A good solution has been found after 20 iterations, which means 20 fine model computations and 20 Jacobian computations which represent a high cost.

Simple output mapping convergence is good and the algorithm is stopped after 8 iterations. But it might fail to converge to the fine model optimal solution. Indeed, any interaction between variables in this mapping is missing.

Regarding Manifold Mapping, both true gradients and approximated ones need a trust region adaptation. Indeed the problem is very sensible and original algorithms failed. Default trust region parameters given in [START_REF] Echeverría | Two new variants of the manifold-mapping technique[END_REF] have been used.

MM using true gradients converges with the highest speed. The consequence is a bouncing effect because it overcomes the target and has to return slower. This can be improved by trust region parameters modification (default values are used).

MM using approximated gradients failed to converge even with trust region strategy. The examination of k S during a convergence process shows that level of value and generally with the same elements sign. Our approximation is based on the previous steps but if steps are in the same direction, the gradients are not really well identified. This approximation has to be improved in order to have a robust algorithm.

Conclusions

In this work, several variants of the manifold mapping technique have been compared. Results on a realistic test case show that the use of exact gradients allows to convergence to more accurate solutions than reached by gradient approximations. These accurate solutions are reached three times faster than an interior algorithm iterating solely on the fine model. The manifold mapping algorithm with exact gradients therefore opens interesting perspectives on solving more complex optimization problems in the future.

  individual iterands are defined by coarse model optimization:

Fig. 1 .

 1 Fig. 1. Optimization algorithms convergence on a simple test case.

Fig. 2 .

 2 Fig. 2. Magnetic MEMS topologyA MEMS magnetic actuator topology has been defined to reach these requirements Fig.2. It is made of 2 parts:

Fig. 3 .

 3 Fig. 3. Design specification: torque has to be linear 3 torque computations, equally distributed along translation position, have been chosen in order to do the least square minimization.

Fig. 5 .

 5 Fig. 5. Fine model computation time and its derivatives, depending on the discretizing.

Fig. 8 .

 8 Fig. 8. CADES Generator: an automated tool performing symbolic and automatic differentiation composition and produce software component.

Table I .

 I Design variables and constants

	Parameter	Values
	Magnet width	[1;25] mm
	Magnet high	[1;25] mm
	Magnet length	[1;30] mm
	Iron plate width	600 µm
	Iron plate high	8 µm
	Iron plate length	600 µm
	Magnet Polarization	1 T
	Gap between magnet and iron	1 mm

CADES framework : Component Architecture for the Design of Engineering Systems (available : http://forge-mage.g2elab.grenobleinp.fr/project/cadesframework )


Adrien Quintard

Jean Rodriguez

Enantioselective Ir-Catalyzed Bidirectional Reductive Coupling

In the presence of a chiral iridium complex, commercially available 3-chloro-2-chloromethyl-1-propene (1) was selectively activated for various reductive couplings. Depending on the reaction conditions it allows a selective mono-or bidirectional condensation with one or two external aldehydes with excellent enantiocontrol (>90% ee). This approach occurring simply under mild conditions and avoiding premetalated reagents constructs rapidly chiral homoallylic alcohols, key precursors of important molecular fragments such as furans, pyrans, ketodiols, or 1,3,5-polyols.

A ssembling simple building blocks into elaborate complex chiral chemical architectures with good stereocontrol while limiting waste generation represents a priority for modern organic synthesis. 1 Notably, the discovery of greener and easy to use enantioselective catalytic methods should enhance the application potential of organic transformations facilitating their transposition in the synthesis of elaborated drugs or materials. Among crucial synthetic building blocks necessary for the construction of divers chemical architectures, homoallylic alcohols 3 and 4 stand out (Scheme 1.a). They constitute privileged precursors for the synthesis of molecular patterns found in a number of natural products and drugs. For example, 3 provides a direct access to furans and pyrans while preparation of 4 constitutes a straightforward route to keto-diols, 1,3,5-polyols or spiroketals (Scheme 1b). [START_REF] Alonso | For example of synthetic applications of 3 and 4[END_REF] As a result of this pivotal synthetic interest, numerous methods have been developed for their stereoselective preparation 3 mainly relying on stoichiometric metal activation and chiral additives. [START_REF] Barrett | Using stoichiometric boron prepared from transmetalation with the lithiated starting material[END_REF] The only chiral Lewis-acid-catalyzed strategy to access alcohols 3 from 1 involved allylic stannanes obtained after initial Cl-Li exchange. [START_REF] Keck | transmetalation with the lithiated starting material see reference 2c[END_REF] Access to parent ketodiols A can also be based on bidirectional aldolization by condensing two molecules of aldehydes with a ketone equivalent, however the direct enantioselective variant on aliphatic aldehydes still represents a challenge. 6 As a result, it is clear that the development of a modular enantioselective catalytic synthesis of 3 and 4 starting f rom aliphatic aldehydes, occurring without the use of stoichiometric prematallated reagents and under mild conditions is highly desirable.

To circumvent waste and hazards associated with the use of stoichiometric premetalated nucleophiles, the group of Krische has proposed a wide array of coupling reactions based on the concept of alcohol mediated carbonyl addition. 7,[START_REF] Kim | For leading references notably based on iridium complexes[END_REF] In a typical reaction, dehydrogenation of an alcohol by a catalytic metal complex (Ru or Ir) generates a carbonyl and a metal hydride. The formed metal hydride then inserts in the allyl pronucleophile, forming a reactive metal-allyl complex able to stereoselectively add to an electrophilic carbonyl generating the desired chiral alcohol.

This type of reductive coupling is illustrated in Scheme 1c where two allyl donors 5 are condensed to a pivotal 1,3-diol 6, providing enantioenriched diols 7. 9 Interestingly, this reductive coupling chemistry can be performed by starting from an alcohol or from an aldehyde. In the latter case, addition of an environmentally friendly reductant such as 2-propanol initiates the formation of the required catalytic metal-hydride.

Continuing with our interest on bidirectional condensations, we hypothesized that 2-propanol might be used to activate commercially available 3-chloro-2-(chloromethyl)-1-propene (1) as pivotal pronucleophile addition to an aldehyde. This would allow for a rapid and selective generation of 3 and subsequently 4 through a bidirectional condensation with a second external aldehyde (Scheme 1d). However, the development of this bidirectional chemistry using a pivotal bis-donor might be hampered by numerous challenges. First of all, 1 should be selectively monoactivated through the use of greener hydride donors such as 2-propanol, generating 3 after addition to an aldehyde. Moreover, to apply the allylic chloride 3 in other derivatizations, it should be stable enough under the reaction conditions to avoid direct decomposition. Finally, advancing intermediate 3 to diol 4 will require an additional metal hydride insertion, resulting in a particularly sensitive hydroxyfunctionalized Ir-π-allyl complex 10 prompt to evolve by a 5endo cyclization to form a methylene tetrahydrofuran 8. This trimethylene methane (TMM) like-cycloaddition popularized by the group of Trost has, however, no efficient enantioselective version as far as aliphatic aldehydes are concerned. 11 Therefore, of utmost importance for the success of our approach is that the presence of the unprotected alcohol function should neither inhibit the reactivity of the metal-allyl complex through intramolecular coordination 12 nor influence the stereochemical outcome of the second nucleophilic addition allowing for a catalyst-controlled diastereodivergent process.

To initiate our study, we started investigating the reactivity of 1 with isovaleraldehyde (2a) as the electrophile for the selective synthesis of monoaddition product 3a (Table 1). [START_REF] Townsend | When 1,1-disubstituted allyl donors are reacted, use of chloro derivatives are preferred over esters derivatives: (a) Hassan[END_REF] 2-Propanol was chosen as the green activator in the presence of various easily accessible iridium complexes. When the active iridium complex was generated in situ from [Ir(COD)Cl] 2 , phosphine ligand, and the corresponding acid, only a trace amount of the adduct 3a was observed and 1 was mostly recovered (entry 1). [START_REF] Gao | reactivity between isolated and in situ formed complexes see ref 8b and[END_REF] However, use of 5 mol % preformed Binap complex Ir-cat1 restored the reductive coupling providing 3a in promising 38% yield and 90% ee (entry 2).

The role of the 2.2 equiv of K 3 PO 4 was crucial since use of a lower amount (1.1 equiv) or alternatively of Cs 2 CO 3 considerably reduced the reactivity (entries 3 and 4). Turning to the use of Segphos Ir-cat2 dramatically improved the efficiency of the 2-propanol-promoted reductive coupling forming 3a in excellent 70% yield and 98% ee (entry 5). To promote the formation of 3a and avoid its decomposition, an excess of 1 and of 2-propanol is required in the process (entry 5 vs entries 6 and 7). It must be noticed that the use of 3.5 equiv of 2-propanol also provides higher enantiocontrol (entries 5 and 7, 98 vs. 93% ee). Finally, use of more electron-poor aromatic ligand on the iridacycle as in Ir-cat3 reduced the reactivity forming only around 15% of 3a (entry 8). Given previous reports on reductive couplings, the presence of traces amount of inorganic salts in Ir-cat3 reducing the reactivity cannot be excluded. [START_REF] Gao | reactivity between isolated and in situ formed complexes see ref 8b and[END_REF] With the optimized conditions of entry 5 in hand, we then scrutinized the scope of the monodirectional reductive coupling (Scheme 2). The 2-propanol-promoted coupling tolerated aldehydes with various steric substitution patterns. Aliphatic isobutyl, isopropyl, or neopentyl substituents led to the formation of the expected adducts 3a-c in 70-80% yields. Moreover, the Ir-cat2 was able to induce an excellent enantiofacial discrimination during the addition providing these products in very high 96 to 98% ee. Starting from enantiopure (S)-citronellal, the terpenoid chain could be introduced in 3d with 85% yield and excellent diastereocontrol (>95:5 dr). Use of less sterically demanding hydrocinnamaldehyde 2e or trans-2-pentenal 2f decreased the reaction efficiency providing 3e and 3f in 37% and 43% yield, respectively but again with a high level of stereocontrol (97 and 92% ee). [START_REF] Kumar | Use of additional carboxylic acid was required to improve the yield, probably by avoiding undesired iridacycle decomposition[END_REF] Finally, 2-substituted enal 2g or Boc-protected amine 2h were well tolerated, providing 3g and 3h in 71 and 61% yields and >90% ee. Interestingly, demonstrating the synthetic potential of this approach, it must be pointed out that 3b could be prepared starting from 10 mmol of 1 and using only 2 mol % Ir-cat2.

With a practically convenient and highly enantioselective access to hydroxyl functionalized allylic chlorides 3, we next turned our attention to the development of the challenging second functionalization. Preliminary experiments (see the Supporting Information) were conducted mixing 3d under the conditions of Scheme 1 and in the absence of aldehyde. Of

Organic Letters

Letter DOI: 10.1021/acs.orglett.8b03669

Org. Lett. 2019, 21, 453-456 importance, removing any external electrophile, evolution of 3d notably through cyclization was noticed possibly through the formation of the corresponding catalytic Ir-π-allyl complex. This is in agreement with the reactivity observed in Table 1 showing that an excess of 1 was required to improve the yield.

To promote the bidirectional addition process, we thus reduced the temperature to 50 °C while placing 3 in the presence of an excess of aldehydes. Under the reductive conditions, Ir-cat2 efficiently reacted with 3b providing the unprotected diols 4b,d,g in 44-70% yield without racemization of the initial stereocenter (Scheme 3). Gratifyingly, despite the presence of the unprotected alcohol in the starting material, the Ir-cat2 was able to independently control the formation of the newly formed stereogenic center. Indeed, using the appropriate enantiomer of the catalyst, either anti or syn diols could be prepared with more than 80% diastereocontrol.

With a method able to selectively functionalize allylic chlorides 3, we focused on a last challenge, the development of a cascade direct synthesis of symmetric bis-alcohols 4. This bidirectional reductive coupling would constitute an alternative to the ones already reported in the literature such as the condensation of two pro-nucleophiles on a central bis-alcohol (Scheme 1.b). In the present approach, an internal stoichiometric pro-nucleophile would have to react directly with two external electrophiles. As a result, for the success of this cascade, the in situ generated hydroxy functionalized allylic chloride 3, should directly react again with another aldehyde.

Gratifyingly, the selective use of Ir-cat2 allowed for the efficient condensation of structurally different aldehydes (Scheme 4). Use of 2 equiv of aldehydes with respect to each chlorine at 50 °C provided the bidirectional products 4ad in 65-74% yield. Interestingly, the diols could be obtained with almost perfect enantiocontrol and >95:5 dr for 4d. As a result, this cascade rapidly assembles simple substrates into valuable diols key direct precursors of 1,3,5-polyols, 16 easily desymmetrizable substrates. 17 For example, anti4a could be converted through reductive ozonolysis to 1,3,5-poyol 9a, a polyol whose synthesis was only disclosed in the racemic series (Scheme 5.a). 6d To prove the feasibility of a direct catalytic stereoselective TMM like-cycloaddition on aliphatic aldehydes, we performed the one-pot direct preparation of methylene tetrahydrofuran 8d. Directly in situ treating the reductive coupling adduct with NaH, the resulting TMM like-adduct could be formed in 73% overall yield (Scheme 5.b). Of importance, the reductive coupling can also be performed from the alcohol oxidation state. Preliminary experiment condensing 1 with 10d provided the expected adduct 3d in 49% yield and >95:5 dr. Finally, determination of the absolute configuration of compounds 3 and 4 supports the enantioselection model proposed for allylation reactions using (S)-Ir-cat2. 18 In conclusion, activation of commercially available 3-chloro-2-chloromethyl-1-propene (1) by the appropriate use of a catalytic chiral iridium complex provides an innovative enantioselective reductive coupling avoiding the use of premetalated reagents. This constitutes a straightforward access to synthetically valuable bis-functionalized homoallylic alcohols with high levels of stereocontrol. The second allylic chloride functionalization involves a peculiar Ir-π-allyl complex possessing an adjacent unprotected alcohol function. Despite the poor stability of this complex, the second carbonyl addition occurs with perfect catalyst control allowing for an efficient diastereodivergent process. The discovery of an efficient way to selectively generate and react functionalized chloro-allylic alcohols opens strong perspectives for the development of other cascade reactions. It should allow a rapid and environmentally friendly construction of a broad range of valuable building blocks with important implications notably in the synthesis of natural products or drugs. Finally, we are currently investigating the strong potential of using 2-propanol reductive coupling to promote other TMM like-cycloadditions. proposed mechanism, experimental procedures and characterization of compounds, NMR and intensity spectra (PDF)

■ AUTHOR INFORMATION Corresponding Author *E-mail: adrien.quintard@univ-amu.fr.

ORCID

Adrien Quintard: 0000-0003-0193-6524

Scheme 2 .a

 2 Scheme 2. Scope of the Ir-cat2 Condensation of 1 with 2a-h

  Scheme 3. Diastereodivergent Reductive Coupling of 3b

Table 1 .

 1 Optimization of the Iridium-Catalyzed Addition of 1 to 2a

	entry	cat	base (equiv)	1/2/i-PrOH	yield (%) a	ee (%) b
	1	in situ formed Ir-cat	K 3 PO 4 (2.2)	5/1/3.5	trace	nd
	2	Ir-cat1	K 3 PO 4 (2.2)	5/1/3.5	38	90
	3	Ir-cat1	Cs 2 CO 3 (2.2)	5/1/3.5	trace	nd
	4	Ir-cat1	K 3 PO 4 (1.1)	5/1/3.5	15 c	nd
	5	Ir-cat2	K 3 PO 4 (2.2)	5/1/3.5	70	98
	6	Ir-cat2	K 3 PO 4 (2.2)	3/1/3	10 c	nd
	7	Ir-cat2	K 3 PO 4 (2.2)	5/1/2	51	93
	8	Ir-cat3	K 3 PO 4 (2.2)	5/1/3.5	15 c	nd

a Isolated yields. b Enantiomeric excess determined by chiral gas chromatography. c NMR yield. Nd = not determined.

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

Notes

The authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

The Centre National de la Recherche Scientifique (CNRS) and the Aix-Marseille Universitéare gratefully acknowledged for financial support.


Abir Rezgui
email: abir.rezgui@g2elab.grenoble-inp.fr

Laurent Gerbaud

Benoît Delinchant

UNIFIED MODELLING TECHNIQUE USING VHDL-AMS AND SOFTWARE COMPONENTS

Keywords: VHDL-AMS, software components, coupling models, complex multi physics systems

come

Introduction

Mechatronic systems can be defined as complex and heterogeneous systems. Their models deal with different physical aspects: mechanical, electrical, thermal, and magnetic, etc. It is therefore essential to achieve a successful simulation of theses systems; which is a delicate task facing several problems related to models and simulation tools: [START_REF]VHDL Analog and Mixed-Signal Extensions[END_REF] wide choice of models of different natures and described in different formalisms; and (2) multitude of time simulation tools, specific for its domain, which don't have a direct compatibility between them. The modelling level of the components of such systems depends on the aim of the study carried out for them. For example, a electrical generator may be represented by an ideal source for its study in a large power grid simulation, a park model for it study in a wind generator simulation, a finite element model in the analyse of its magnetic behaviour. So, the choice of the abstraction level is a compromise between computations speed, accuracy, model description easiness, or its availability in a model library. However, there is no tool allowing a complete simulation of mechatronic system and its components and be able to meet all the user's needs. To address these problems, specially for modeling electromagnetic devices as a kind of mechatronic system, it is important to adopt interoperability solutions ensuring compatibility with other models and with simulation tools. The IEEE Glossary defines interoperability as: "'the ability of two or more systems or components to exchange information and to use this information that was exchanged". Various approaches can be used:

• The first approach focuses on the possibility to link applications at runtime in order to co-operatively exchange information. This co-simulation strategy allows a more flexibility modeling and simulation than others because the models are simulated separately on their own dedicated software.

• -The second approach is achieved on the level of the models, which describes mechanical, electrical, magnetic and other physical processes. It consists on extracting models developed in a specialized tool for externally reuse in industrial mechatronic simulators (e.g. Portunus, Simplorer, Smash, etc) in order to benefit from the expertise behind. This is done by: (1) describing models with a standardized multi-physics modeling languages like VHDL-AMS [START_REF]VHDL Analog and Mixed-Signal Extensions[END_REF], Modelica [www.modelica.org], system-C [www.systemc.org] and Verilog-AMS [www.designers-guide.org]; or (2) exchanging and reuse of existing components models at the source code level or through a dynamic link library.

The paper focuses on the system modeling level for dynamic simulation, applied to electromagnetic devices and for interoperability needs in order to develop a unified architecture for the multilevel modeling of theses devices, based on a standard for model description. The most difficult task of the proposed approach is to simulate accurately the magnetic component in VHDL-AMS. However, what to do for modelling aspects that are not explicitly supported by VHDL-AMS (ex: partial differential equations) or difficult to describe in such a language (e.g. due to numerous symbolic treatment, like for the force computation in the application presented in the last section)? To avoid the limits of the language, the paper proposes for that an approach based on the transfer of models from dedicated magnetic tools of these components like RelucTool software [START_REF] Du Peloux | A method and a tool for fast transient simulation of electromechanical devices: application to linear actuators[END_REF], developed at G2ELAB, for designing reluctant networks [START_REF] Turowski | Reluctance networks, Computational Magnetics[END_REF]; to a general mechatronics simulation environment like Smash, thus allowing the simulation of the component coupled to its electro-mechanical environment. RelucTool software generates automatically a black box model as a software component from a graphical description of the reluctant network. This standardized software component has been defined in our laboratory and they are named ICAr (for: Interface for Component Architecture). So, the proposed approach have to export this specific software components into a VHDL-AMS models, ensuring then the interoperability between models and offer to designers the possibility to use VHDL-AMS as a unified description language for many modeling aspects.

In this way, section 2 deals with VHDL-AMS aspects and the software component paradigms, focusing on the ICAr standard developed in our lab. Section 3 presents the need of the proposed approach (of exchanging models) for designing an electromagnetic actuator. Then, section 4 discusses the way to carry out the coupling between VHDL-AMS and ICAr software. In this section, a generic implementation of this coupling will be presented discussing the data exchange between the two formalisms. Finally, in section 5, this approach is applied on the linear actuator. It shows how to mix such formalisms to perform the dynamic simulation of the actuator.

Modeling techniques and tools

VHDL-AMS Language

The VHDL-AMS, an IEEE standard 1076.1, is an extension of VHDL hardware description language, which is used for the description and simulation of event-driven systems [START_REF]VHDL Analog and Mixed-Signal Extensions[END_REF]. VHDL-AMS propose to support the description and the simulation of analog and mixed signal circuits and systems. It allows to define and to deal with the modeling of the behavior of multi-physical and multi-technological systems [START_REF] Christen | VHDL-AMSA Hardware Description Language for Analog and Mixed-Signal Applications[END_REF]. Its purpose is to provide formalism for the hierarchical description and simulation of physical systems. Designers can create and use models that integrate high-level behavioral descriptions, as well as structural and physical descriptions. A VHDL-AMS model consists of an entity and one or more architectures [START_REF] Wilson | Multiple domain behavioral modeling using VHDL-AMS, Circuits and Systems[END_REF]. The entity specifies the extern view of the model. It includes the description of its physical ports and the definition of its generic parameters. The architecture contains the implementation of the model. It may be coded using a structural or behavioral description, or both. The structural description specify the connectivity between the physical components; which can be carried out with the Architectural Description Language (ADL [START_REF] Medvidovic | A Classification and Comparison Framework for Software Architecture Description Languages[END_REF]),and the behavioral description is made by concurrent statements to describe eventdriven behaviors and simultaneous statements for continuous ones. Fig. 1

is a VHDL-AMS description of an electrical resistor.

This language specifies the possibility to describe the behavior of a complex continuous system by formulating its corresponding differential algebraic equations (DAE). Thanks to its standard well-defined in terms of syntax and semantics, it ensures portable descriptions among different industrial simulators (e.g. Portunus, Simplorer, Smash, etc.). Due to their complexity, partial differential equations (PDE) were left out in VHDL-AMS [START_REF] Nikitin | Modeling partial differential equations in VHDL-AMS[END_REF]. This limits the accuracy of system block modeling which includes local physical effects. Because of the importance of PDE in electromagnetism, different approaches filling this lack in the design flow using VHDL-AMS are proposed. A reduction of FEM models is presented in [START_REF] Schlegel | Analyzing and Simulation of MEMS in VHDL-AMS Based on Reduced Order FE-Models[END_REF], which is an export of reduced order modeling (ROM) to VHDL-AMS with an interface matching the system structure. In paper [START_REF] Nikitin | Modeling partial differential equations in VHDL-AMS[END_REF], two modelling approaches were used: the first consist in using a spatial discretization and the second consists in using an equivalent circuit approach. Then, an extension of VHDL-AMS has been also proposed in order to take into account this aspects [START_REF] Nikitin | Modeling partial differential equations in VHDL-AMS[END_REF], but it is not yet integrated on the standardized language.

Software components

The "software component" modeling approach has been extensively studied to design models on several applications and domains, as electromagnetism, electronics, microsystems and so on [START_REF] Du Peloux | A method and a tool for fast transient simulation of electromechanical devices: application to linear actuators[END_REF] [9] [10] [START_REF] Enciu | AD for Optimization in Electromagnetism Applied to Semi-Analytical Models Combining Composed Functions[END_REF] [START_REF] Delinchant | Mixing of FEM and Analytical Modeling for the Preliminary Design of a Transformer, Optimization and Inverse Problems in Electromagnetism OIPE[END_REF]. A software component standard is defined as an autonomous deployment entity, encapsulating a software code and described by its interfaces. A specific pattern has been defined in our laboratory for the standardization of such software components [START_REF] Delinchant | An Optimizer using the Software Component Paradigm for the Optimization of Engineering Systems[END_REF]. They are named ICAr (for: Interface for Component Architecture). The ICAr component is an executable Java code defined by its input/output variables and the services that it can bring. It is characterized by an evolutionary architecture and can be seen as a multi-services component. Services are available via different facets that are accessible via the interface "component" (Fig. 2). Each facet may represent behavioral model, sensitivity computation, time simulation, model accuracy. Facets mayalso represent a part of the system to be simulated (electromagnetical, mechanical, thermal, economical).

The "component" interface presents the way to instantiate the component, defines its inputs and outputs, and how to interact with all its features, as shown in Fig. 3 This paradigm of software component in general and specifically the ICar component assures sharing, exchanging, capitalizing and reusing models. It can also support complex models and their solvers whatever their physical nature is.

The need of an exchange model approach on electromagnetic design

Fig. 4 show the studied device, which is a dynamic moving core actuator with a coil bounded to its central leg [START_REF] Atienza | A methodology for the sizing and the optimization of an electromagnetic release[END_REF]. The dynamic motion of a core is driven by a magnetic system, and it is dealing with electric, magnetic and mechanical modeling. It is very challenging to design such a system accurately because its operating is based on several physics and complexity levels. Indeed, it is a multi-domain device including the magnetic circuit, interfacing the electrical domain and the mechanical loads. The electrical driver circuit controls the current in the coil, and the mechanical subsystem computes the influence of damping sources on the displacement of the moving part damping behaviour. Also, many physical effects in this system can be mathematically formulated by a coupling between PDE and DAE. In this way, how to design this system by invoking the multi physical modeling using VHDL-AMS as unified language modeling?

The objective of this work is to be able to design and simulate this system with VHDL AMS language to solve interoperability problems. In this electromagnetic actuator: • the state system modeling of the electrical circuit feeding the coil was defined by two differential equations; the inductance value is provided by the magnetic model.

• the state variable of the mechanical part of this system are the core position z, and its velocity v; the state system is defined by two differential equations depending on the value of applied residual forces.

• the magnetic part of the actuator which is represented by reluctant networks (RNM [START_REF] Turowski | Reluctance networks, Computational Magnetics[END_REF]).

The complete device model can be carried out in several ways, among them:

• a VHDL-AMS modelling of the complete device using the model of paper [START_REF] Atienza | A methodology for the sizing and the optimization of an electromagnetic release[END_REF]; however, the magnetic force calculation is very approximated. This difficulty is more presented in paper [START_REF] Bodrich | A Magnetic Library for Modelica, International Modelica Conference[END_REF] proposing a Modelica description of reluctance networks.

• an accurate model by finite element for the magnetic part and a VHDL-AMS modelling for the electrical and mechanical parts; however, the numerical method is computing time consuming [START_REF] Rezgui | VHDL-AMS to Support DAE-PDE Coupling and Multilevel Modeling[END_REF] • a reluctance network model for the magnetic part with the magnetic force obtained from the derivative of the co-energy according to the movement variable (z in figure 4) [START_REF] Mazauric | From thermostatistics to Maxwells equations: A variational approach of electromagnetism[END_REF], and a VHDL-AMS modelling for the electrical and mechanical parts.

The third is more suitable for system level modeling of the dynamic actuator. However, how to render easy the modelling of magnetic part, in particular as said previously VHDL-AMS language don't support non temporal derivative? The solution proposed is then to apply the approach based on exchanging model from a dedicated tool for designing magnetic circuit and coupling the VHDL-AMS model to the electro-mechanical one on Smash simulator.

ReluctTool is proposed as a dedicated tool for the fast simulation and presizing of electromechanical actuators [START_REF] Delinchant | Mixing of FEM and Analytical Modeling for the Preliminary Design of a Transformer, Optimization and Inverse Problems in Electromagnetism OIPE[END_REF]. In which, the reluctance network is described graphically. The magnetic force expression is obtained automatically from the derivative of the co-energy in the reluctances according to the movement variable [START_REF] Mazauric | From thermostatistics to Maxwells equations: A variational approach of electromagnetism[END_REF]. The model is automatically generated in a software component as an ICAr component. As a result, for the approach, the behavior model of the magnetic part of the actuator is a RNM model, implemented in RelucTool tool (Fig. 5). It generates, then, an ICAr component encapsulating this model. It is the software component, that it be included into a system simulation of the whole dynamic actuator on Smash simulator using VHDL-AMS language. How can the coupling between VHDL-AMS and ICAr software can be achieved?

Model interchange using VHDL-AMS language

Interchange porcess

As it has been indicated, the approach followed for model interchange between the ReluTool software et SMASH tools consists in transferring the models using VHDL-AMS for model description. Fig. 6 shows the overall process that starts by creating the model of the magnetic component using the proprietary languages of ReluTool software. Then this tool exports its models in VHDL-AMS format, using the language subset described previously. This architecture allows the transfer of models from RelucTool to other VHDL-AMS native simulators. This independent entity can be imported in SMASH as a VHDL-AMS model via external functions. For that, the VHDL-AMS export function is added to Reluctool software have to deal with the coupling between VHDL-AMS models and others described in software components.

Export an ICAr component in VHDL-AMS

The export functionality consists in defining an interface or overlapping layer between the VHDL-AMS models and an external code (written in C or encapsulated software component), for data exchanges. The final model obtained is a VHDL-AMS model with external functions, as one single ENTITY of VHDL-AMS, with its dynamic behaviour represented by ARCHITECTURE. It requires as a first step to encapsulate the whole magnetic model as a black box that behaves externally as any dynamic system. VHDL-AMS offers this opportunity to extend his functionalities by including external functions written in another programming language or and foreign subprograms in its models. That is possible thanks to its pre-defined attribute called "Foreign" [START_REF]VHDL Analog and Mixed-Signal Extensions[END_REF], which allows the user to transfer additional information to the simulator or other external entities.

The "FOREIGN" attribute is used then to complement architectures, functions or subprograms for VHDL-AMS models. Although this attribute is defined in the VHDL-AMS its use is not specified, so it depends on its implementation in simulation software. A foreign subprogram is a program in other formalism than VHDL-AMS, but call in VHDL-AMS thanks to FOREIGN' attribute, as shown in Fig. 7. This call is defined by a string characterizing the external program as a specific function. This string value depends on the implementation of both the "Foreign" attribute in the time simulator and the external program implementation. However, the VHDL-AMS standard does not specify the mechanism for the implementation the FOREIGN attribute. Consequently, the exact interpretation of this attribute depends on its implementation in a particular simulator. At the present time, the only simulator which implements this attribute is Smash from Dolphin integration [www.dolphin.fr]. This implementation requires only the integrating of C functions. So, the export functionality of RelucTool relies basically on the use of native code (C/C++) to perform this coupling [START_REF] Smash | Foreign C functions in VHDL-AMS and SPICE models[END_REF].

As a result, to import of the external models on various formats into VHDL-AMS systems or to make it able to communicate with external environements/solvers, the authors have developped a generic C-communication interface for this coupling. This interface makes able easily (Fig. 8) the VHDL-AMS model to include C/C++ models or dynamic linked libraries (DLL) and allows exchanging data between them. However, ICAr components are developed in Java language, because it is an independent platform language and so it offers more portability for component. That is why, to be able to load and manipulate these software components from a program or a software tool developed primarily in C/C++, Java data-structures have to be converted into programs compatible with C data-structures. These conversion routines are based on the Java Native Interface (JNI), i.e. the standardized C-interface of the Java Virtual Machine (JVM).

So, to import an ICAr component from a C program, this last should be able to understand the component services via its various facets. For that, the paper proposes an "Interface Adapter" to implement this functionality proposing generic services for introspection of theses components (Fig. 8). A C/C++ dll project was built gathering the different C functions for JVM launching, JNI adaptation routines, ICAr loading, facets and interfaces introspection and methods calling. Once the adaptation between the ICAr component and the C code is made, each C function compiled in a library can be after that called in VHDL-AMS using Foreign attribute defined by C/++ external functions. Finally, a VHDL-AMS -skeleton, corresponding to the architecture of the ICAr model, is generated automatically by a Java program. This program introspects the component in order to determine its facets, different inputs and outputs and generates the corresponding VHDL-AMS block description. This block defines its inputs/outputs and parameters. It also describes relations between them, in equation section, defining the model. These relations are defined through a well defined sequence of C external functions.

Example of coupling Strategy on electromagnetic actuator

n order to validate the methodology of "data coupling" between VHDL-AMS model and ICAr component, in order to validate the export functionality added to Reluctool, an electromechanical actuator is designed in VHDL-AMS language and simulated (Fig. 5). As said in section 3, the dynamic motion of a core is driven by a magnetic system, and the model consists of electric, magnetic and mechanical modeling. The dynamic model of the device represents a coupling between the electrical circuit feeding the coil and the mechanical part of the moving part. Only the linear vertical movement of the moving part is possible, which is subject to different damping sources due to translational linear spring.

This electromagnetic device is designed and simulated using the approach previously presented:

• the electrical circuit was described by its two differential equations (ODE) on VHDL-AMS with the inductance value as an external inputs;

• the mechanical part was also described by its two differential equations (ODE) on VHDL-AMS to compute position z and velocity v, depending on the value of applied residual forces;

• -the magnetic circuit of the actuator is developed on RelucTool as RNM model (Fig. 7), providing an ICar component exported into VHDL-AMS model. This component compute the magnetic force from the derivative of the co-energy according to the movement variable (z) [START_REF] Mazauric | From thermostatistics to Maxwells equations: A variational approach of electromagnetism[END_REF];

The ICAr component of the RNM model was generated with the Reluctool tool as a static component (no time differential expressions). Once the component is generated, the "interface adapter" based on the JNI is implemented. The last program must respect the "FOREIGN" mechanism as implemented on SMASH software, as shown in Fig. 8. Then, the VHDL-AMS package is programmed to be able to call the "foreign subprogram" on VHDL AMS model by specifying the name of the called function, its path and its arguments. This "interface adapter" is an automatic interface which allows to load an ICar component, and makes it able to communicate with its exterior environment.

This component is then inserted into the global dynamic actuator with its environment circuit model described in VHDL-AMS language. The full VHDL-AMS model is shown in Fig. 9.

Fig. 10 shows the exchange protocol between a VHDL-AMS model and an ICAr software component in a time simulation process. The protocol is implemented into SMASH software, considering the actuator as a macro VHDL-AMS model calling external functions which exchange inputs/outputs through a software component. So, when the simulator starts, all the modeling objects are loaded and connected, and initial values are set. During this stage, all foreign shared libraries are loaded and initialization functions of all foreign architectures are called.

A first analysis was performed with the moving core actuator system. The main dynamic behaviour is analysed. Fig. 11 shows simulation results for the evolution of all state variables of the actuator happening during the simulation:

• the position of the moving core (z) and the speed of the moving core (v).

• the residual forces (Ftot) applied to core and the magnetic force (Fmag) generated by magnetic component

• the current generated by the RLC circuit (I) and the variable inductance.

The aim of the paper is the coupling of VHDL-AMS and component software for the dynamic modelling of a electromechanical actuator. So, results are not discussed. It can be say that they are similar to the result given in [START_REF] Atienza | A methodology for the sizing and the optimization of an electromagnetic release[END_REF] where the force is more approximated, and where a finite element simulation is used as a reference.

Conclusions

The approach proposed in the paper aims to couple models described in VHDL-AMS models and in ICAr software component, allowing the addition of functions which are not in VHDL-AMS standard like numerical algorithm used in electromagnetism. This approach provides also capabilities to import structural models implemented in ICAr components that describe physical properties of the system (e.g. finite element models), in order to be integrated into the global system described in VHDL-AMS. This strategy offers the possibility to deal with complex This approach is valid for any time simulator which provides mechanisms for the coupling of VHDL-AMS models with any C/C++ interfaces (or other language) thanks to its 'Foreign' attribute, which is not yet implemented in all VHDL-AMS compatible simulators, and is often specific to each simulator. So, some adjustments must be made from a 'Foreign' attribute implementation to another. Since both ICAr components and VHDL-AMS have been defined for purposes of portability and reusability, models developed under different formalisms must be interconnected to build global system models.It an be also used for any systems can be designed as an ICar software component such magnetic microsystem, and others.

Figure 1 :

 1 Figure 1: VHDL-AMS model of a resistor

Figure 2 :

 2 Figure 2: Multi-facets component: (a)multi information, (b) multi-physics.

Figure 3 :

 3 Figure 3: ICAr interfaces: component and facet (Java implementation)

Figure 4 :

 4 Figure 4: Simplified moving core actuator description

Figure 5 :

 5 Figure 5: The reluctance representation of the magnetic circuit

Figure 6 :

 6 Figure 6: Process and Model interchange using VHDL-AMS

Figure 7 :

 7 Figure 7: Package defining the 'Foreign' attribute of VHDL-AMS

Figure 8 :

 8 Figure 8: Coupling between the AMS and VHDL model component ICar

Figure 10 :

 10 Figure 10: Exchanging of data between a VHDL-AMS model and an ICAr software component controlled by SMASH

Very High Speed Integrated Circuit Hardware Description Language -Analog and Mixed-Signal Preprint submitted to Elsevier July 17,

2012

Acknowledgements

The authors thank the French National Research Agency (ANR) for their support throughout the project Mocosymec, Yannick Herve from the INESS for its advices, and Dolphin Integration for their specific development to improve the FOREIGN implementation in Smash.


Anh Hoang
email: hoang-anh.dang@g2elab.grenoble-inp.fr

Benoit Dang

Sana Delinchant

Frederic Gaaloul

Wurtz

ELECTRICAL PERFORMANCE OPTIMIZATION OF AN HVAC DUAL FLOW

Keywords: HVAC dual flow, Electronic driver, Electrical motor, Optimization

This work highlight potential gains linked to the control of HVAC system used in buildings. For this, the modelling of electrical system is realized in order to optimize its behaviour. This study is based on the calculation of losses generated in ventilator motors and variable speed drivers to determine the optimal control mode and to evaluate the energy saving.

INTRODUCTION

Inside the building, HVAC dual flow assures the exchange air flow and the temperature stabilization following the user request. Based on the operation, it is divided into two groups: the first includes the electrical and air handling part, ensuring the flow of air in desired normal operation and maximum flow in incident and the second is the thermal part, ensuring the desired temperature.

The PREDIS platform of Grenoble Electrical Engineering Laboratory is one of the places of experimentation in order to make research on building energy saving in France. In this platform, all electrical and thermal equipments are modelled for a global modelling in order to find the intelligent control strategies.

Focus on the HVAC dual flow modelling, this paper examines its electrical part and optimal control mode.

MODELLING

This part is dedicated to the modelling of electrical parts (motor and speed drive). Models are made for the objective of power consumption evaluation; they are parameterized by variables (voltage, frequency, load torque …) but not for design needs (geometrical parameterization). One of the goals is also to build fast analytical model in order to embed them in real time controller for future works.

INDUCTION MACHINE

In order to simulate the permanent operation of induction machine (asynchronous motor), this model is constructed by using the equivalent circuit. Based on the electrical expressions, all permanent performances will be easy to calculate [START_REF] Bakshi | Asynchronous machine[END_REF]- [START_REF] Shepherd | Power electronics and motor control, Book, The press syndicate of the University of Cambridge[END_REF]- [START_REF] Wurtz Frédéric | A new approach for the design of electrical machines under constraints[END_REF]. Thanks to motor datasheet from manufacturer, this model can be used to determine the motor parameters by solving this equation system:

n n n µ µ R S S I s f L R N R R I = ) , , , , , , ( n n n µ µ R S s f L R N R R φ φ cos ) , , , , , , ( cos = n r n n µ µ R S u C f s f L R N R R C = ) , , , , , , , ( start n µ µ R S S I f L R N R R I = ) 1 , , , , , , ( start r n µ µ R S u C f f L R N R R C = ) , 1 , , , , , , ( (1)
R S 13.9 Ω R R 7.1 Ω N 41 mH R µ 1.71×10 4 Ω L µ 795 mH
By applying these parameters, motor characteristic curves in permanent operation can be plotted. Each operating point from datasheet are marked by a cross in Fig. 4, Fig. 5, and Fig. 6

VARIABLE SPEED DRIVER

In this component, losses of control circuit are lower than global losses (about 1000 times) and then not included in our modelling. So, for studying the performance of this device, only the power electronic circuit is considered. The structure of this circuit is the following:

A three-phase rectifier is connected to the building supply network and generates a DC voltage ripple.

To limit the discontinuous of the DC voltage, a capacitor is integrated in the intermediate circuit.

For the motor power supply, a three-phase inverter generates the power signal voltage and frequency variables. We can conclude that switching losses are the most important in the operation of variable speed driver and are enough great anough to make good approximation of losses for our modelling.

CONTROL -COMMAND

CONTROL MODE: VARIABLE TORQUE

The type of HVAC Load is an aerodynamics one.

According to the aerodynamics theory, the motor rotation speed follows a linear law with the flow and a quadratic law with the torque.

In the case of the PREDIS platform, this load was determined by measurement (dots on Fig. 9) and a law was identified in order to measures. The variable speed driver can allow controlling the motor in different mode. A classical control is the control law V/f = constant. In the case of fan or pump, the torque is quadratic depending on the rotation speed; a specific control mode for variable torque is also available.

To identify this control mode, we made measurements in the platform PREDIS. Using an identification method, the relationship between voltage and frequency is determined by this function:

CONTROL MODE: OPTIMAL RESEARCH

Our objective is to optimize overall performance of system {motor + variable speed driver}. Thanks to our models, and using the CADES software which is developed for modelling and optimization by G2Elab , both motor and speed driver was composed in order to make a global model. CADES computes gradients of the model automatically in order to make fast constrained optimization. The optimal parameters for the variable torque control law were found and reported in equation (9).

Fig. 1 .

 1 Fig. 1. Structure of the PREDIS HVAC dual flow.

Fig. 2 .

 2 Fig. 2. Siemens induction machine and Schneider Electric variable speed driver.

Fig. 3 .

 3 Fig. 3. Asynchronous motor equivalent circuit in permanent operation.

Fig. 4 .Fig. 5 .Fig. 6 .

 456 Fig. 4. Asynchronous motor useful torque.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Power electronic circuit of variable speed driver In this electronic circuit, because of high switching frequency (12000 Hz) in mode PWM, almost all the losses are the losses of the three-phase inverter. Inside the inverter, there are six switches which one comprises a transistor (IGBT) and an antiparallel diode. Their losses are also divided into two types of losses: conduction losses and switching losses [2]-[5]. Conduction losses: IGBT: 2 0 _ IGBTeff IGBT IGBTmoy CE cond IGBT I R I V P + = ∆

Fig. 9 .

 9 Fig. 9. HVAC fan torque measurement and approximated identified law

Fig. 10 .

 10 Fig. 10. V(f) law for variable torque control

Fig. 11 .

 11 Fig. 11. Optimal parameters for the variable torque control law found with an optimization approach

Table I .

 I Motor parameters identification

	Datasheet
	Nominal current (I n )	1.73 A
	Nominal power factor (cosφ n )	0.86
	Nominal speed	2855 rpm
	Nominal frequency (f n ) 50 Hz
	Nominal torque (C n )	2.5 Nm
	Starting current (I srart )	9.69 A
	Starting torque (C start )	5.75 Nm
	Motor parameters

COMPARISON RESULTS

On these figures below, the optimal control mode is always better than others, for the entire speed range. It is possible to quantify the efficiency gains of this optimal control mode by considering an actual operation of the HVAC. The average flow rates measured in the PREDIS platform during several days (Fig. 15 in last page) is 600 m 3 /h while the maximum flow is 1500 m 3 /h. Indeed, the HVAC has been designed to provide cooling during summer nights; it must support a maximum flow rate more than the average flow (about 10 volumes per hour). However, the system variable speed drive + motor has been optimized to run optimally at a speed of 3000 rpm (corresponding to 1500 m 3 / h) and a load torque of 2.5 Nm which is double of the measure.

Meanwhile, the optimal control mode improves essentially the performance for operating points lower than the rated operation. This allows saving of 14% energy in comparison to the control mode installed.

Table III V(f) optimal 0.9 14 93

CONCLUSIONS AND PROSPECTS

In order to optimize the electrical operation of a HVAC, induction motor and speed drive of the PREDIS dual flow HVAC were modelled.

The desired control mode improves the performance of 14% relative to the initial control mode. Indeed, we found that the average operation of the HVAC is lower than the rated operation, which leaves considerable room for improvement, achieved by our control mode.

We founded that if technician, in charge of the HVAC installation, forgot to switch from the V/f law to the variable torque law, a lack of 23% of energy saving can be reached.

In term of prospects, the models developed will be used for the global modelling of HAVC including aerodynamics and thermal part. Then, the HVAC will be coupled with models of buildings. The models developed will also be used to optimize overall building and especially playing on the instructions of HVAC (depending on the desired temperatures and measured the number of occupants and their activities ...).

Minimal losses


Design study on magnetic springs with low resonance frequency J.L.G. Janssen 1 , J.J.H. Paulides 1 , E.A. Lomonova 1 , B. Delinchant 2 1 Eindhoven University of Technology, P.O. Box 513, Eindhoven, N-Brabant, 5600MB,The Netherlands 2 Laboratoire dElectrotechnique de Grenoble (LEG), Domaine Universitaire, Saint Martin d'Hères Cedex, France email: j.l.g.janssen@tue.nl ABSTRACT Passive magnetic gravity compensation may be a very feasible alternative to commonly used mechanical or pneumatic solutions for vibration isolation systems. The final paper discusses a study on the optimal design of a magnetic spring with passive gravity compensation. For vibration isolation systems it is often preferred to combine a certain force level with very low spring stiffness to achieve a low resonance frequency. Several optimization methods are used in the paper to optimize the dimensions of planar magnet structures to exhibit a certain vertical force with low stiffness and the optimized results are compared. The results from this study give insight in the influence of topology choice on the achievable vibration isolation performance of permanent-magnet based springs.

INTRODUCTION

Many industrial and academic applications require an advanced form of vibration isolation. In such applications a table or platform needs to be isolated from all vibration sources, viz. floor vibrations, acoustical noise or forces acting on the isolated platform itself. These applications are often characterized by a heavy mass which remains vibration-free in a predefined position, with minimized energy consumption, minimal movement and a vibration rejection up to hundreds of Hertz. As a result, the suspension should exhibit a low stroke, a high static force with minimized energy consumption, a low intrinsic spring stiffness and a high isolation bandwidth.

As an alternative to the commonly used pneumatic actuators, this paper proposes electromagnetic suspension to perform these tasks. In the electromagnetic mount the isolated platform is passively supported by a magnetic spring which is based on interaction between permanent magnets and as such provides a high passive vertical force. Active actuators, which are not considered in this paper, firstly stabilize this inherently unstable system and secondly provide the advanced vibration isolation. In order to prevent a mechanical path for floor vibrations to travel to the isolated platform a contactless magnetic solution is chosen.

Permanent magnet devices are often designed to exert a very high force density as for example is seen in magnetic couplings [START_REF] Furlani | Analysis and optimization of synchronous magnetic couplings[END_REF][START_REF] Ravaud | Permanent magnet couplings: Field and torque threedimensional expressions based on the coulombian model[END_REF] or in the magnetic springs shown in [START_REF] Janssen | Passive magnetic suspension limitations for gravity compensation[END_REF][START_REF] Will Robertson | Parameters for optimizing the forces between linear multipole magnet arrays[END_REF]. In such designs a small relative movement often leads to a significant change in reaction force, i.e. a high stiffness. This disadvantageous in terms of vibration isolation because such systems are designed to exhibit resonance frequencies as low as 0.5 Hz. To achieve such a low resonance frequency, a compromise in terms of reduced force and force density is eminent. This paper researches the relationship between this force and the achievable resonance frequency.

THE PASSIVE MAGNETIC SPRING

An impression of the magnetic spring that is considered in this paper is shown in Fig. 1. It comprises of two horizontal permanent magnet arrays which both have a vertical checkerboard magnetization. The orientation of both arrays is in this case such, that the upper array is repelled by the lower array. The number of magnets along the horizontal directions, n, is chosen odd because of symmetry reasons. With such odd number of magnets the responses when moving in positive or negative direction along one of the horizontal axes are equal.

It was shown in [START_REF] Janssen | Passive magnetic suspension limitations for gravity compensation[END_REF] that high force densities can be achieved with such topology, up to 6000 kN/m 3 . Because further research of these topologies has shown that the vertical spring stiffness in such case is extremely high, the focus in the article is on a significant reduction of this vertical spring stiffness. In order to achieve this goal, a variable pole pitch and a variable magnet pitch are introduced in the magnet arrays. This is shown schematically in Fig. 2.

Two optimization methods are used to optimize the respective dimensions of the permanent magnet arrays. The first method optimizes analytical models of the interaction force obtained by the surface charge model [START_REF] Akound | 3D analytical calculation of the forces exerted between two cuboidal magnets[END_REF] with nonlinear constrained multivariable optimization fmincon in Matlab. Although the force computation is fully analytical, this gradient-based optimization SQP algorithm computes a quasi-Newton approximation to obtain the Hessian. The main risk of such a deterministic algorithm is that it may become trapped in local minima. On the other hand, metaheuristic algorithms such as genetic algorithms or particle swarm may have troubles converging to an optimum with the given set of constraints. The second optimization method which is used is therefore a hybrid method which combines both types of algorithms. This method uses semi-analytical force computation together with a true gradient optimization [START_REF] Rakotoarison | Methodology and tool for generating semi-analytical models used to pre-design electromagnetic mems[END_REF][START_REF] Delinchant | An optimizer using the software component paradigm for the optimization of engineering systems[END_REF].

OPTIMIZATION GOALS

Although in Fig. 2 the number of magnets in horizontal direction, n, equals 3, this integer number is varied in the final paper and always kept odd because of symmetry reasons. Each optimization is now carried out in two steps: The first step maximizes the vertical force exhibited by the magnetic spring within the set of constraints. The second optimization step then tries to bring the spring stiffness as close to 0.5 Hz as possible while maintaining a minimum vertical force level. This minimum force is given as a nonlinear constraint to the optimization routine and is varied between 100 N and 10 kN. The resonance frequency is obtained with the assumption that the vertical spring force is in equilibrium with the vertical gravity force, or

f res = 1 2π √ k m = 1 2π √ k F ver /g = 1 2π √ gk F ver , (1)
where g is the gravitational acceleration of 9.81 m/s 2 and k is the vertical spring stiffness. The final publication will discuss the optimization methods in more detail.

RESULTS

The optimization of the various topologies shows that the number of magnets is of significant influence on the spring stiffness. Figure 3 clearly shows a relationship between number of magnets in the arrays, force levels and achievable resonance frequencies. The optimization results are further elaborated on in the final paper.

CONCLUSION

This paper investigates the feasibility of permanentmagnet based topologies for vibration isolation applications. For such applications a low resonance frequency combined with a high vertical spring force is an advantageous property and therefore the research is focused on achieving these goals.

Several optimization methodologies are utilized to optimize the dimensions of two arrays with rectangular magnets which together form a magnetic spring as a result of repulsion force. The optimum number of permanent magnets in these arrays is also part of the research. With these results this paper presents some novel topologies which exhibit large vertical forces and low stiffness based on passive permanent magnet interaction.

FFigure 1 :

 1 Figure 1: Schematic representation of a 3 × 3 permanent-magnet spring with checkerboard magnetization. The magnets are oriented such that the top array is repelled.

1 gFigure 2 :

 12 Figure 2: Dimensions used in the optimization shown for a topology where the number of magnets along the sides, n, is equal to 3.

Figure 3 :

 3 Figure3: Relation between the vertical force and the achievable resonance frequency of the magnetic spring. Each line represents a different number of magnets (n ∈[START_REF] Furlani | Analysis and optimization of synchronous magnetic couplings[END_REF][START_REF] Janssen | Passive magnetic suspension limitations for gravity compensation[END_REF][START_REF] Akound | 3D analytical calculation of the forces exerted between two cuboidal magnets[END_REF][START_REF] Delinchant | An optimizer using the software component paradigm for the optimization of engineering systems[END_REF]).


Phuong Pham Quang
email: phuong.pham-quang@g2elab.grenoble-inp.fr

Benoît Delinchant

C Llie

E Slusanschi

Jean-Louis L Coulomb

Bertrand Du Peloux

P Pham-Quang

C Ilie

B Du Peloux

Mixing Techniques to Compute Derivatives of semi-numerical models: Application to Magnetic Nano Switch Optimization

published or not. The documents may come   L'archive ouverte pluridisciplinaire

I. INTRODUCTION

Sizing by optimization is nowadays of major interest of the design process of electromagnetic device. Our goal is to use optimization algorithms in conjunction with semianalytical model in order to solve the sizing problem. Among the deterministic algorithms, the constrained gradient based optimization using Sequential Quadratic Programming (SQP) algorithms is very efficient. Such algorithms require accurate values of the objective function and constraint derivatives. Unfortunately, it is often difficult to obtain the symbolic expression of the derivatives of the sizing model. But several techniques as Automatic Differentiation (AD) can be used to compute these derivatives.

The paper highlights a generic framework of model composition using different derivation techniques. This approach is shown through the example of a magnetic nano switch optimization.

II. DERIVATION TECHNIQUES

The finite difference approximation leads to an approximation suffering from both truncation and cancellation errors. It is very easy to set-up, but it is very difficult to settle the adjustment parameter.

The derivatives can be computed symbolically. This method is reliable and the faster that can be found, not only with symbolic expression but also with functions resulting from numerical integration or from implicit solver (thanks to implicit theorem). The main drawback of this method is the impossibility to differentiate the computing code (with conditional instructions, loops, etc.).

Automatic differentiation (AD) exploits the natural process of source code compilation and makes use of the intermediate representation of implemented functions in different programming languages. The derivatives of each elementary operation can be obtained in a straightforward way and can be combined according to the chain rule of differential calculus in order to obtain the derivative of the initial function. Thus, depending on the chosen accumulation strategy, functions of arbitrary complexity can be differentiated in two ways, namely forward and reverse.

In the forward mode one propagates derivatives of intermediate variables with respect to the independent variables, while in reverse, derivatives of the function with respect to intermediate variablesalso called adjointsare propagated.

Typically, AD can be implemented using either the operator overloading (ADOL-C [START_REF] Enciu | Automatic differentiation for electromagnetic models used in optimization[END_REF], CppAD, etc) or the source transformation technique (ADIFOR, ADiJaC [START_REF] Slusanschi | Algorithmic differentiation of Java programs[END_REF], etc). In operator overloading one overloads the operators which are applied on new variable types, with the routine call performing the actual derivative computation. The source transformation approach examines the source code of the original function and generates new code that computes the desired derivatives in the same time that the original function.

III. SENSITIVITY PROPAGATION OF COMPOUND MODEL As studied in [START_REF] Delinchant | Reducing sensitivity analysis times-cost of compound model[END_REF] , and as shown in Fig. 1 and Fig. 2, two kinds of sensitivity propagation mechanism can be considered to build global sensitivities.

IV. DERIVATIVES OF MAGNETIC NANO SWITCH MODEL

The semi-numerical model of magnetic nano switch was presented in [START_REF] Pham-Quang | Semi-analytical magneto-mechanic coupling with contact analysis for MEMS/NEMS[END_REF]. It involves a coupled magneto-mechanical deformation of a cantilever beam, and allows evaluating the contact length and the contact force. The following part details how the derivatives of each model blocks are computed.

A. Magnetic model

For specific configuration of MEMS/NEMS, the Coulombian approach is used for the modeling of magnet, leading to a full analytical magnetic field model. Forces and magnetic torques applied on the beam (in magnetic parts) are computed by adaptive numerical integrations. The derivatives of this magnetic model can be computed symbolically with accuracy and rapidity [START_REF] Delinchant | Gradient based optimization of semi-numerical models with symbolic sensitivity[END_REF].

B. Mechanical deformation model

The model used to compute deformation in the presence of contact of a cantilever beam has inputs and outputs as shown in Fig. 3 Fig. 3. Inputs and outputs of mechanical deformation model ADiJaC [START_REF] Slusanschi | Algorithmic differentiation of Java programs[END_REF] was used in order to manage the derivation of this complex algorithm. ADiJaC is a source transformation AD tool that implements both the forward and reverse modes on Java codes. To date it is the first useable AD tool for the Java programming language. Differentiating the NEMS model with ADiJaC was improved to deal with special cases of matrix operations, nested method calls, special functions (e.g. "length"), and special array initializationsall intra-procedural transformations. During this work, ADiJaC's interprocedural analysis was also enhanced, so as to allow the tool to differentiate functions returning arrays, and receiving any number of vector and scalar parameters.

C. Magneto-Mechanic coupling

Sequential magnetic-mechanical coupling

The sequential magneto-mechanical coupling described in Fig. 4. has been solved by an implicit solver with Gauss method. Its derivatives are computed symbolically thanks to a classical implicit theorem. All these models are given with their own Jacobian using different techniques (symbolic or AD). CADES framework defines a standard to make the composition possible. Each model implementing this standard can be integrated into the framework. Then the overall is treated automatically by another AD tool (ADOL-C) to build the global model (Fig. 5).

D. Composition in the CADES framework

V. OPTIMIZATION RESULTS

Working principle of magnetic nano switch was presented in [START_REF] Pham-Quang | Semi-analytical magneto-mechanic coupling with contact analysis for MEMS/NEMS[END_REF]. Geometry and parameters to be optimized are given in Fig. 6. The optimization aims to determine dimensions of both fixed and mobile magnets and their positions on the beam. The objective is to minimize the volume of magnets, while respecting constraints such as length of contact and contact force, in order to ensure the quality of contact or the contact resistances are below a desired value. Fig. 6. Geometry and parameters to be optimized A genetic algorithm (without gradients) and the SQP algorithm (using gradients) were used. Both gave the same results but the first algorithm needed 3 hours computation time while the second only 92 seconds.

VI. CONCLUSIONS Deterministic optimization using gradient based algorithms is very efficient for sizing problems with constraints. However, such algorithms meet a major lack of usage because of the difficulty to obtain exactly and rapidly the derivatives values. This paper highlights the composition of different derivation techniques in a compound model through the example of magnetic nano switch.

The full paper will detail the propagation mechanism of derivatives. Each derivation techniques will be detailed, highlighting advantages and drawbacks. More details will be given on the optimization procedure and results.

Fig. 1 .

 1 Fig. 1. A compound model based on partial derivative propagation

Fig. 5 .

 5 Fig. 5. Sensitivity propagation of magnetic nano switch in CADES

Magnetic Mechanic


Benjamin Nottelet
email: benjamin.nottelet@umontpellier.fr

Anita Schulz

Vincent Darcos

Hussein Awada

Edouard Girard

Grégory Chagnon

Danielle Laurencin

Yannick Guari

Joulia Larionova

Xavier Garric

Jean Coudane

Functional Degradable Polymers for Advanced Applications in Clinical Imaging

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Medical devices and tissue engineering applications are largely relying on polymeric biomaterials that are intrinsically not visible under the clinical imaging modalities currently in use. In this contribution, we report on various advanced functional polymers, especially polyesters, that have been designed and synthesized to make them visible under scanner (X-ray) or medical resonance imaging (MRI).

Experimental Methods

Various strategies have been followed to yield polymers for clinical imaging. These include functional monomers synthesis and copolymerization (lactones, acrylates), direct polyester post-polymerization modification in solution or surface modification of polymer substrates. All these strategies have been applied to X-ray or MRI contrast agents including iodine, triiodobenzoic acid (TIB), gadolinium complexes (Gd-DTPA or Gd-DOTA) or superparamagnetic iron oxide nanoparticles (SPIONs). Imaging tests have been performed both in vitro and in vivo to evaluate the visibility, stability and perennity of the vizualisation.

Results and Discussion

In a first strategy, iodo-poly(-caprolactone) (PCL-I) and triiodobenzoate-PCL (PCL-TIB) have been successfully prepared via an anionic activation of PCL with LDA as a strong base; followed by nucleophilic substitution with I2 or TIB. The resulting radiopaque copolymers have been used as additives to prepare implantable degradable biomaterials. 1 A good and stable visibility under micro-computed tomography analyses was confirmed in vivo in a rat model (Fig. 1a). In a second strategy, clickable polyester or poly(meth)acrylates have been synthesized via copolymerization of alkyne functional monomers followed by CuAAc reaction with azido-functional Gd ligands. These MRI-visible copolymers have been used as coatings for surgical meshes and demonstrated their high potential as T1 contrast agents in vivo. [2][3][4] Finally, in a last approach, polymer surfaces, including PLA electrospun fibers, PLA-Plu ® -PLA threads or PP surgical meshes were efficiently modified with photoreactive Gd ligands (Fig. 1b) or photoreactive SPIONs. The absence of bulk properties modification as well as the ability of these functional surfaces to be imaged under T1 or T2 MRI modalities both in vitro and in vivo have been assessed.

5

 5

Figure 1 .

 1 Figure 1. Example of radiopaque copolyester (in vivo, µCT) (a) and of MRI visible PLA-Plu ® -PLA threads (b)


N Galopin

P Pham-Quang

B Delinchant

J L Coulomb

COUPLED MULTI-PHYSICS PROBLEMS

This work focuses on the modeling of mechanical contact applied to Nano-Electro-Mechanical-Systems (NEMS).

A magneto-elastic formulation associated to the definition of unilateral contact in static case is presented. This model is discretised by finite element method and a numerical study of a magnetic nano switch is performed. FEM results are compared to those of a semi-analytical model developed for optimization processes.

I. INTRODUCTION

Magnetic NEMS based instruments and devices are at the forefront of nanotechnology, enabling fundamental measurements and standards and new types of devices related to magnetism. This is given by low voltage and power consumption with large actuation distance which provides a number of advantages compared to electrostatic NEMS [START_REF] Cugat | Magnetic Micro-Actuators and Systems (MAGMAS)[END_REF]. In NEMS technology, magnetic nano switches have many applications as nanomechanical memory, power switches,. . . Their working principle is based on the deflection of a beam submitted to the influence of a magnetic field. The mechanical contact between beam and substrate is an important parameter for the behavior of these nano switches [START_REF] Zhao | Mechanics of adhesion in MEMS -a review[END_REF]. Indeed, contact quality strongly depends on roughness of contact area and on surface and contact forces. Nevertheless, their modelings are often strongly approximated [START_REF] Gray | Modeling and Performance of a Magnetic MEMS Wiping Actuator[END_REF]. We propose in this paper a magnetoelastic model included unilateral contact formulation and its discretisation by the finite element method. The modelisation is applied to a magnetic nano switch, and the numerical results are compared with a semi-analytical model realizing the magnetic structural coupling considering contact analysis.

II. FINITE ELEMENT MODELING

The problem under consideration involves the contact of an elastic body, submitted to magnetic forces, with a rigid (or elastic) body in static case (Fig. 1).

A. Contact problem description

Consider two elastic bodies occupying a bounded domain, Ω 1 and Ω 2 (Fig. 1), of R d with d = 2 or 3. The bodies are subjected to body forces f and to prescribed tractions t and displacements u 0 on the part Γ σ and Γ u of the boundary Γ respectively. For the two bodies, the potential contact boundaries are Γ c1 and Γ c2 . For a classical linear elasticity

Ω1 Ω1 Ω2 Ω2 Γc 1 Γc 1 Γc 2 Γc 2 Ω3 Γ3 Br Js Fig. 1. Study domain of magneto-mechanical contact problem (Ω = Ωm ∪ Ω 3 and Ωm = Ω 1 ∪ Ω 2 )
problem, the equations are:

         ∇ • σ + f = 0 in Ω m σ = C : s in Ω m σ(u) • n = t on Γ σ u = u 0 on Γ u ( 1
)
where n is the outward unit normal vector, C the stiffness tensor and s the strain tensor defined in the assumption of small displacement by:

s(u) = 1 2 ∇u + ∇u t (2)
To state the laws of contact and friction, the displacement and tractions vectors are decomposed on Γ c into normal and tangential components:

σ n = σ ij n i n j σ ti = σ ij n j -σ n n i u n = u i n i u ti = u i -u n n i (3)
For a frictionless contact, the unilateral contact conditions on the boundaries Γ c , which implies that no boundary point of the first body may penetrate the other, are defined by :

     g n = (u 2 -u 1 ) • n ≥ 0 σ n = σ n1 = -σ n2 ≤ 0 σ n • g n = 0 (4)
with g n the signed normal distance. The first condition states that no penetration may occur. Hence, this is the form in which the impenetrability constraint is cast. Using this, the normal traction can be characterised. The second condition states that the contact normal traction should be compressive. Finally, the third condition states a complementarity condition. If there is no contact, then no compressive tractions can occur. Alternatively, if there are no compressive stresses, then the distance must be positive.

B. Magneto-elastic contact problem formulation

In the static case, the formulation of the magneto-elastic contact problem, without magnetostrictive phenomena [START_REF] Besbes | A generalized finite element model of magnetostriction phenomena[END_REF], can be established from a minimisation of the functional energy F in terms of magnetic flux density b and strain s:

F(b, s) = W (b, s) -T (5)
where W (b, s) and T are respectively the magneto-elastic energy and the work of magnetic and mechanical sources, defined by:

W (b, s) = Ω   b 0 h(b ′ )db ′ + s 0 σ(s ′ )ds ′   dΩ (6)
T = Ω a • j dΩ + Γ h a • (h × n) dΓ h + Ωm u • f dΩ + Γσ u • (σ • n) dΓ σ (7)
with a the magnetic vector potential and j the current density. Application of variational principles, and taking into account the contact conditions (4), gives the following magnetic and mechanical formulations associated to arbitrary variations δa and δu:

Ω ∇× δ a ν ∇× a dΩ+ Γ h δ a •(h× n) dΓ = Ω δ a •j dΩ (8) Ωm s(δu)Cs-δu•f dΩ- Γσ δu•tdΓ-δu (W (b, 0)) ≥ 0 (9)
The inequality ( 9) is due to the contact definition and define the variational inequality of the magneto-elastic contact problem. The last term of the right-hand side of (9) are the nodal magnetic forces determined by the local derivative of magnetic energy [START_REF] Ren | Local force computation in deformable bodies using edge elements[END_REF]. Resolution are realizing by a penalty method, the Lagrange multiplier method or an augmented lagrangian method [START_REF] Wriggers | Nonlinear finite element methods[END_REF].

III. SEMI-ANALYTICAL MODELING

Thanks to basic geometries of NEMS, magnetic fields radiated by permanent magnets and conductors are computed by using pure algebraic equations through Coulombian equivalent surface charge approach and Biot-Savart law. Surface and volume numerical integrations are used to compute magnetic forces and torques.

A semi-analytical structure model compute the deflection, due to magnetic forces and torques, of the cantilever beam in the presence of contact. It is based on assumptions of small displacement, which allows reducing the cantilever beam 3D to 1D problem, on linear and isotropic materials and on considering only one direction of deformation. To compute the deformation, the 1D-beam is split into a set of segments, and deformation of each segments is calculated by the following equation:

E I ∂ 2 s ∂x 2 = M ( 10
)
where E is the Young's modulus, I the second moment of area and M the bending moment. Adequate boundary conditions and connection conditions are used to ensure continuous deformation. In application of the superposition principle, the total deformation is the sum of the deformations of each segments created by each forces and torques. Model of contact is inherently coupled with the structural model by replacing the contact by a distribution of force on contact. The contact force F c is computed by:

F c = F t -F r (11)
where F t is the total force applied on the beam and F r the reaction force on fixed side of the beam. As the distribution of forces, which replaces the contact, is unknown, an iterative method is used. Figure 2 shows the variation of the contact length and contact force as function of forces applied on the beam.

IV. CONCLUSION In this paper, the formulation of a magneto-mechanical contact problem for a finite element analysis has been presented. In the full paper, this model will be developed and particularly the finite element discretisation, the numerical resolution thanks to regularization methods and the procedure to improve the contact management. Comparisons with the semi-analytical model will be made for a magnetic nano switch.

Fig. 2 .

 2 Fig. 2. Results of the semi-analytical structural contact model