Developing a framework for classification and/or recommendation

Ayeley Tchangani1 and Franc ¸ois Pérès2 Abstract-The objective of this communication is to establish a framework for classifying or recommending an object characterized by several attributes into classes or for uses for which a nominal representative is known or some entry conditions are specified. In the case where classes are characterized by entry conditions, the mathematical problem to be solved is typically a constraints satisfaction problem. But in general, constraints are subject to uncertainty, so in this paper, we propose to transform these constraints into functions of membership or non-membership of fuzzy subsets; thus for each class, these functions, given an object to be classified or recommended, can be aggregated in synergy to give two measures: a measure of selectability of the class and a measure of rejectability; the final choice of the class is then made by optimizing an index based on these two measures. When classes are determined by a primary or main representative, the leader to whom the object to be classified should be compared, it seems natural to use measures of similarity or dissimilarity to classify the object in the right class. To do this, given that we consider that classes are characterized by normalized numerical indicators and therefore resemble a probabilistic structure, we propose to use Kullback-Leibler (KL) divergence that compares a given probability distribution to a main one as dissimilarity measure between an object and the representative of a class. The application of the approach developed to a real-world problem shows a certain potentiality.

Index Terms-Classification, Recommendation, Fuzzy Constraints Satisfaction, Similarity, Dissimilarity, KL divergence.

I. INTRODUCTION

The era in which we live today is governed by the personalization of things. Thus, consumers (in the broadest sense) would want products and/or services that are tailormade. It is therefore imperative for the producers of these products or providers of these services, who in reality cannot fully personalize their offers, to at least find a way or procedure to recommend to each consumer what best suits him or her. It is therefore a question of finding, for a given consumer, the product or service to recommend to him; which returns to assigning it to a given class or category. It is therefore imperative for the producers of these products or providers of these services, who in reality cannot fully personalize their offers, to at least find a way or procedure to recommend to each consumer what best suits him or her. It is therefore a question of finding, for a given consumer, the product or service to recommend to him; which amounts to assigning it to a given class or category. In general terms, francois.peres@enit.fr many decision problems rising in different domains such as social, economics or engineering, among others, concern the assignment or classification of objects to classes according their scores for a certain number of criteria or attributes that characterize them; they constitute then a subset of the socalled multicriteria decision making or multicriteria decision analysis (MCDM or MCDA) problems, see for instance [START_REF] Brans | PROMOTHE: A new family of outranking methods in multicriteria analysis[END_REF], [START_REF] Brans | How to select and how to rank projects: the PROMETHE method[END_REF], [START_REF] Rigopoulos | A Decision Support System for Supervised Assignment in Banking Decisions[END_REF], [START_REF] Saaty | The Analytic Hierarchical Process: Planning, Priority, Resource Allocation[END_REF], [START_REF] Saaty | The Analytic Network Process: Decision Making with Dependence and Feedback[END_REF], [START_REF] Steuer | Muticriteria Optimization: Theory, Computation, and Application[END_REF]. The majority of contributions to these problems encountered in literature concern mainly the ordered classification case, classes must be ordered, let say, from most/least desired class to least/most desired one, see for instance [START_REF] Doumpos | Multi-criteria classification methods in financial and banking decisions[END_REF]. The purpose of classification methods or algorithms is then to establish a procedure that linearly rank classes and assign objects to them; one may notice that this is a relative decision making process as objects are finally compared with each other. But it is being shown that the case of non ordered classification where the classes are just defined by some features, conditions or constraints over the attributes or criteria is of great importance in many domains. In finance and banking for instance, decision maker(s) face the problem of classifying customers for a credit or service into classes defined by entrance thresholds with regard to their performance in some of their attributes for instance, see [START_REF] Rigopoulos | A Decision Support System for Supervised Assignment in Banking Decisions[END_REF]; in international finance or commerce, countries are often ranked or classified in different categories in terms of risk to which potential investors will be exposed in these countries (country risk ranking or classification) using a certain number of attributes such as GDP per unit of energy use, telephone mainlines per 1000 people, human development index, percentage of military expenditure of the central government expenditure and others, see [START_REF] Wang | Country risk classification and multicriteria decision-aid[END_REF]; etc. The next section considers formal specification of such problems.

II. FORMAL SPECIFICATION

We place in the case where a fixed number of classes is known and the problem is, given an object, to classify it in the right class; following subsection considers formal specification of the considered problem.

A. Definition of the problem

Formally nominal classification problem considered in this paper is defined by the following materials.

• An object u to be classified is characterized by a set of m attributes or criteria where the evaluation x l of attribute l belongs to a structured set X l so that this object can be designated by its attributes vector

x ∈ X = X 1 × X 2 × ... × X m ; that is x = x 1 x 2 ... x m .
• The former defined object must be assigned to one of the n classes or categories of the set C = {c 1 , c 2 , ..., c n }; each class or category c j is characterized by same p features, conditions, or constraints through functions (to be understood in a broad sense including verbal description) f k j (x) ∈ F k j , k = 1, 2, ..., p of the attributes vector x ∈ X; a feature f k j is a mapping from attributes evaluation space X onto F k j . A class is therefore completely determined by its features vector

f j (x) = f 1 j (x) f 2 j (x) ... f p j (x) ∈ F j = F 1 j × F 2 j × ... × F p j .
In practice, the process of entering a class can be characterized in several ways; in the following subsection, several practical oriented ways of this characterization will be described.

B. Classes entry characterization

The previous description shows that finally a class c j in front of an object characterized by its attributes vector x is also completely characterized by its features vector f j (x). In terms of classification, it returns to establishing a procedure that determines the conditions that the vector f(x) must satisfy in order for the object in question to be included in that class. In order to do this, it is necessary to measure some kind of compatibility degree between the class c j and the vector f j (x); this calls for a good description of the conditions for entry into class c j ; this is the subject of this sub-section.

Description of a class by its main representative

In many applications, such as in the advertising industry, the targets of an advertising message are often classified in relation to an idol, a leader who thus constitutes a representative of the segment of this class of targets. Thus in terms of features, the representative of the class c j is described by the vector f r j = f 1,r j f 2,r j ... f p,r j ; it will now be a question of being able to position any object described by f j (x) in relation to this representative.

Description of class by an entry score of the features

In many applications, the access to a class j is conditioned by the membership of its characteristics to a subset of F j . By considering, a particular characteristic k of the class j, there will exist a subset F k,C j ⊂ F j such that if f k j (x) belongs to this subset then we can include the considered object in the class in question and a subset F k,R j ⊂ F j for which the decision is the rejection of the class. But in practice, these two subsets may overlap in the sense that the decision to classify or to reject is not certain and usually accompanied by a degree of uncertainty, that is

F j = F k,C j ∪ F k,R j and F k,C j ∩ F k,R j = ∅.
As each feature k characterizing a class is fuzzily described, an object u characterized by vector x will be described with regard to this feature by two membership functions or measures

0 ≤ m F k,C j (x) ≤ 1 and 0 ≤ m F k,R j (x) ≤ 1 respectively.
With regard to a class c j , an object u with attributes vector x will therefore be characterized by two vectors m j C (x) and m j R (x) gathering its membership functions as

m j × (x) = m F 1,× j (x) m F 2,× j (x) ... m F p,× j (x)
where × stands for R or C on a bipolar basis, see [START_REF] Tchangani | Bipolar Fuzzy Nominal Classification (BFNC) framework: Application to risk analysis[END_REF] [14] [START_REF] Tchangani | Fuzzy Nominal Classification using Bipolar Analysis[END_REF].

III. PROPOSED CLASSIFICATION PROCEDURES Classification procedures will depend on how the entry of classes is described; in the following subsection we consider different cases.

A. Classes characterized by bipolar membership functions of features

Bipolar reasoning constitutes a sort of divide to better apprehend paradigm; indeed uncertainty resulting from fuzzy characterization of classes by their features appeals for a flexible classification procedure. To this end, we use the concept of bipolar analysis as the underlying methodology to derive flexible classification algorithms. The stepping stones in bipolar analysis are the classifiability and rejectability measures µ j C (u) and µ j R (u) given a class c j and an object u similar to selectability and rejectability degrees in the case of alternatives selection problems [START_REF] Bouzarour-Amokrane | A bipolar consensus approach for group decision making problems[END_REF] [START_REF] Tchangani | Evaluation Model in Decision Analysis: Bipolar Approach[END_REF] ; so their derivation is an important step towards a sound classification algorithm. These measures must be established considering the performance of the considered object with regard to the considered class. The classifiability and rejectability measures µ j C (u) and µ j R (u) are then obtained as equation ( 1)

µ j C (u) = G(m j C (x)) l G(m l C (x) ; µ j R (u) = G(m j R (x)) l G(m l R (x) (1)
where G is a certain aggregation operator. Given the synergy obtained by considering separately, classifiability and rejectability zone, it is obvious of considering a synergistic aggregation operator. Furthermore, features characterizing classes do not necessarily have the same importance in the classification process and experts may be able to weight them through a normalized vector ω. In this case, Choquet integral associated with a weighted cardinal fuzzy measure (wcfm) [START_REF] Tchangani | Bipolar Aggregation Method for Fuzzy Nominal Classification using Weighted Cardinal Fuzzy Measure (WCFM)[END_REF] is a suitable aggregation operator that permits to overcome difficulties due to the dimension of the vector to aggregate encountered in the literature [START_REF] Grabisch | The application of fuzzy integrals in multicriteria decision making[END_REF] because a straightforward formula does exist. Indeed, given a numerically valued n dimension vector θ with a relative importance vector ω, Choquet integral of x associated to a weighted cardinal fuzzy measure with relative importance vector ω is give by equation ( 2)

C wcf m ω (θ) = n k=1 ϕ (A k ) θ σ(k) -θ σ(k-1) (2)
where ϕ (A k ) is a weighted cardinal fuzzy measure of subset A k given by equation ( 3)

ϕ (A k ) = n -(k -1) n   j∈A k ω j   (3)
and A k is defined as equation ( 4)

A k = {σ(k), σ(k + 1), ..., σ(n)} ( 4
)
where σ is a permutation over x as given by equation ( 5)

θ σ(1) ≤ θ σ(2) ≤ ... ≤ θ σ(n) ; θ σ(0) = 0 (5)
The operator G(.) of equation ( 1) is therefore given by C wcf m ωj (.) where ω j is the relative importance weights vector of features characterizing the class c j .

From classifiability and rejectability measures µ j C (u) and µ j R (u) one can define the classifying set C(u) (classes where u can be classified) of object u as given by equation ( 6)

C(u) = c j : µ j C (u) ≥ q µ j R (u) (6)
where q is a non decreasing function representing the caution or boldness attitude of decision maker. One should notice that many categories or classes may be qualified for inclusion of a given object so that the ultimate class c * (u) in which to include the object can be selected by optimizing (let say maximizing) an index π(u), for instance π(u) =

µ j C (u) µ j R (u)
or other as given by equation [START_REF] Kullback | On information and sufficiency[END_REF].

c * (u) = arg{max v∈C(u) (π(v))} (7)

B. Classes described by main representative

In the case where the classes are described by representatives through the vectors f r j , j = 1, 2, ...n, two questions arise:

• how to determine vectors f r j representing classes? • given an object u with attributes x, how to determine its image vector f j (x) ? If the attributes are evaluated numerically, there are no particular difficulties because many metrics (distance, similarity, divergence, etc.) exist to measure the proximity of each object to the representative of each class. Here we are particularly interested in the case where the characteristics of the classes and/or the attributes of the objects are not evaluated numerically. In this case we can use approaches such as peer comparison; for this we consider two levels: the inter-classes level to determine the evaluation of the characteristics f k,r j and the intra-class level to determine the evaluations of the characteristics f k j (x) given an object of attribute vector x. It should be noted that the determination of f k,r j is an off line process whereas the determination of f k j (x) must be done each time a new object asks to be classified.

Process for determining f k,r j : inter-classes comparison In this case, the evaluation process consists in answering a question such as "how important is characteristic k in the evaluation of class j compared to class l?" by using AHPtype language scales, see [START_REF] Saaty | The Analytic Network Process: Decision Making with Dependence and Feedback[END_REF] or any other scale to obtain the consistent (consistence can be always ensured by using pivot approach [START_REF] Saaty | The Analytic Network Process: Decision Making with Dependence and Feedback[END_REF]) comparison matrix Ψ k (j, l) and the evaluation ψ k,r (j) of the attribute k for class j by equation ( 8)

ψ k,r (j) = 1 n n l=1     Ψ k (j, l) n t=1 Ψ k (t, l)     (8)
and finally the vector f k,r j is obtained by normalizing each characteristic as shown by equation ( 9)

f k,r j = ψ k,r (j) p q=1 ψ q,r (j) (9)
Process for determining f k j (x): intra-classes comparison For the determination of f k j (x), given a x attribute vector object, one must respond to questions such as, "how important is the contribution of x attributes to characteristic k compared to characteristic l of class j? "to obtain the matrix Φ j (k, l), possibly using the same scale as in the previous paragraph. One will then deduce the importance φ k j (x) of the characteristic k for the class j by the equation ( 10)

φ k j (x) = 1 p p l=1     Φ(k, l) p t=1 Φ(t, l)     (10)
and finally the evaluation f k j (x) by normalization as indicated by the equation ( 11)

f k j (x) = φ k j (x) p l=1 φ l j (x) (11)
In the case where classes are described by representatives numerically evaluated with respect to the characteristics of the classes, the first (natural) idea of classification criteria is certainly to use measures of similarity or dissimilarity; many of these measures exist in the literature, see [START_REF] Goshtasby | Similarity and Dissimilarity Measures[END_REF] for instance, and many of which are symmetrical. But in the case which interests us, can one admit symmetry considering the fact that an object must be compared to a reference? Indeed it is the object to resemble the reference and not the other way; in these circumstances the measure of similarity or non similarity should not be symmetrical. Given the probabilistic structure of the measures f r j and f j (x) obtained by construction and the fact that the object finally characterized by the vector f j (x) must be compared with the class representative characterized by f r j , an interesting measure for this is the Kullback-Lieibler divergence (also called relative entropy) [START_REF] Kullback | On information and sufficiency[END_REF] [8] that measures how one probability distribution is different from a second, reference probability distribution in mathematical statistics. Thus, given an object u with attributes vector x, we will calculate its divergence from each class j as indicated by equation [START_REF] Saaty | The Analytic Network Process: Decision Making with Dependence and Feedback[END_REF].

D KL f j (x)//f r j = p k=1 f k j (x) log f k j (x) f k,r j (12)
The optimal class c * in which to include the object u is therefore obtained by minimizing this divergence as given by equation ( 13)

c * (u) = arg min j D KL f j (x)//f r j (13)

A. Description of the application

This application is taken from [START_REF] Rigopoulos | A Decision Support System for Supervised Assignment in Banking Decisions[END_REF] and concern the problem of assigning retailers that use a EFTPoS (Electronic Fund Transfer at Point of Sale) service of a bank to some classes in order for the bank manager to consider their appropriate strategic treatment. Any retailer is characterized by 13 attributes or criteria (G 01 -G 13 ) which signification, scale of evaluation and the relative weight are shown on Figure 1 and the categories with their definition and strategy to apply are given on Figure 2.

Data for 20 retailers to classify, the relative importance of criteria (vector ω), as well as classes entry conditions (entrance threshold vectors b j , j = 1 : 4) are given on Table I. We should mention that the intention here is just to show how the approach developed can be applied in practice and not to show any superiority to procedure based on concordance/discordance indexes used by [START_REF] Rigopoulos | A Decision Support System for Supervised Assignment in Banking Decisions[END_REF]. As defined, to conform the approach developed in this paper, we transform data as following.

• In the case of Classes characterized by bipolar membership functions of features, we define memberships functions for each criterion to enter a class to be given by Figure 3 where the defined low entrance threshold Fig. 2. Classes with their significance and strategy to apply is given by 0.9b k j . Criteria weighting vector ω has been normalized to fulfill requirements of using Choquet integral associated a weighted cardinal fuzzy measure.

• To use the approach of Classes described by main representative, we consider the representative of each class j to have as criterion k value the center of interval between b k j and

G k max = 100 that is G k,r j = b k j + 100-b k j 2 so that f k,r j = x k,r j
is given by equation ( 14)

f k,r j = x k,r j = ω k G k,r j 13 l=1 ω l G l,r j (14)
and f k j (x) for each retailer by equation ( 15)

f k j (x) = x k j = ω k G k j 13 l=1 ω l G l j (15)

B. Results

By applying the approaches developed previously to the modeling adopted in this application we obtain the results reported in Table II both in the case of a characterization of the classes by a bipolar fuzzy membership function that we referred to as MF method and in the case where the III together with the classification result of an empirical or traditional method (Trad. Proc.) reported in the publication [START_REF] Rigopoulos | A Decision Support System for Supervised Assignment in Banking Decisions[END_REF] for comparison purposes. We can notice that the classification result obtained by the method built around a nominal representative of the class NR method is almost identical to the result of the traditional method (Trad. Proc.) of [START_REF] Rigopoulos | A Decision Support System for Supervised Assignment in Banking Decisions[END_REF]; only the classification of R 4 differs between the two methods. The result of the MF method, by maximizing the index

G 01 G 02 G 03 G 04 G 05 G 06 G 07 G 08 G 09 G 10 G 11 G 12 G 13 R
µ j C (u) µ j R (u)
, is a little bit different from the traditional method as well as from our NR method; this is certainly due to the sensitivity of discretization made when determining fuzzy membership functions. This proves that the approaches developed in this paper are promising in various ways: they require little computational effort and the various parameters for their use can be obtained from various data: data collected by companies, exchanges through social networks, information on sites of professional societies, etc. This challenge constitutes the road-map for our future research on this topic of nominal classification.

V. CONCLUSION

The issue of deriving a framework that permit to classifying or recommending objects (potential customers, candidates for a job, suppliers engaged in an auction, etc.) or activities (investment projects, industrial activities, marketing projects, etc.) for their appropriate treatment toward some stakes such as risks has been considered in this paper. Classification and/or recommendation is based on a certain adequacy between the attributes of the object to be recommended or classified and the characteristics of the classes to be considered. The contribution of this paper is therefore relative to the characterization of how a class is chosen. In this paper, we have therefore considered two ways of characterization, namely: similarity (to be maximized) or dissimilarity (to be minimized) with respect to a reference individual of the class, or else the classes are characterized by fuzzy indicators. In the case of classes characterized by fuzzy indicators, two measures have been established that measure the adequacy or inadequacy of an object in relation to a class, and the classification procedure seeks [START_REF] Rigopoulos | A Decision Support System for Supervised Assignment in Banking Decisions[END_REF] to maximize or minimize this adequacy or inadequacy. In the case of classes represented by a reference individual, a divergence measure between an object to be classified and this reference individual has been established which allows the most appropriate class to be chosen. Finally, the results obtained by applying the approaches developed here to a real case encourage us to deepen them.

MF method NR method µ S µ R D KL R i //f r j c 1 c 2 c 3 c 4 c 1 c 2 c 3 c 4 c 1 c 2 c 3 c 4 R 1 0.

Fig. 1 .

 1 Fig. 1. Criteria, their definition, their range of evaluation, and their weights

  R 2 , R 3 , R 6 , R 16 , R 2 , R 3 , R 6 , R 16 R 2 , R 3 , R 6 , R 16 c 2 R 1 , R 9 , R 11 , R 13 , R 14 , R 17 , R 18 R 11 , R 14 , R 17 , R 18 R 11 , R 14 , R 17 , R 18 c 3 R 10 , R 15 , R 19 R 1 , R 4 , R 8 , R 10 , R 13 , R 15 , R 19 R 1 , R 8 , R 10 , R 13 , R 15 , R 19 c 4 R 4 , R 5 , R 7 , R 8 , R 12 , R 20 R 5 , R 7 , R 9 , R 12 , R 20 R 4 , R 5 , R 7 , R 9 , R 12 , R 20

		2264	0.3460	0.1530	0.2746	0.2633	0.0853	0.4759	0.1755	0.1391	0.0998	0.0917	0.0945
	R 2	0.4233	0.1762	0.1719	0.2286	0.0000	0.3819	0.4073	0.2108	0.0369	0.1064	0.1097	0.1308
	R 3	0.4219	0.1511	0.1833	0.2437	0.0047	0.4625	0.3510	0.1817	0.0379	0.1287	0.1361	0.1416
	R 4	0.2480	0.2277	0.2204	0.3039	0.2470	0.2787	0.3088	0.1656	0.1183	0.0874	0.0604	0.0814
	R 5	0.1804	0.3247	0.1546	0.3402	0.3197	0.1563	0.3730	0.1510	0.1399	0.1932	0.1591	0.1062
	R 6	0.4948	0.1212	0.2077	0.1763	0.0016	0.4646	0.2351	0.2987	0.0617	0.1571	0.1804	0.1904
	R 7	0.2521	0.2311	0.1169	0.3999	0.2199	0.2369	0.4331	0.1101	0.1898	0.1745	0.1440	0.0843
	R 8	0.1622	0.2602	0.3631	0.2145	0.4493	0.1888	0.0657	0.2962	0.1086	0.0628	0.0356	0.0777
	R 9	0.2444	0.2939	0.1652	0.2965	0.2422	0.1584	0.4379	0.1615	0.0766	0.0816	0.0710	0.0574
	R 10	0.1719	0.2067	0.4071	0.2142	0.4407	0.2881	0.0000	0.2712	0.1491	0.0825	0.0480	0.1227
	R 11	0.2224	0.3906	0.2524	0.1346	0.2281	0.0000	0.1404	0.6316	0.1223	0.0411	0.0769	0.1630
	R 12	0.2154	0.2827	0.1346	0.3674	0.2689	0.1820	0.4343	0.1148	0.1297	0.1288	0.0988	0.0455
	R 13	0.1633	0.2807	0.2880	0.2680	0.4304	0.1689	0.2020	0.1987	0.1416	0.0956	0.0636	0.0994
	R 14	0.2304	0.4047	0.2254	0.1395	0.2056	0.0000	0.2250	0.5694	0.1580	0.0614	0.1019	0.1968
	R 15	0.1922	0.2310	0.3864	0.1904	0.3662	0.2394	0.0188	0.3756	0.1349	0.0693	0.0455	0.1061
	R 16	0.4735	0.1710	0.1923	0.1632	0.0000	0.3574	0.2830	0.3596	0.0575	0.1490	0.1700	0.1829
	R 17	0.2338	0.4107	0.2141	0.1415	0.1978	0.0000	0.2543	0.5479	0.1967	0.0869	0.1328	0.2401
	R 18	0.2346	0.4122	0.2112	0.1420	0.1960	0.0000	0.2613	0.5427	0.2158	0.1022	0.1505	0.2642
	R 19	0.1405	0.2162	0.4258	0.2175	0.5279	0.2325	0.0000	0.2396	0.1941	0.1112	0.0753	0.1653
	R 20	0.1963	0.2577	0.1472	0.3988	0.2995	0.2028	0.4217	0.0760	0.1006	0.1087	0.0712	0.0287
							TABLE II
						COMPUTED INDICATORS
							Included retailers
		Classes		MF method		NR method			Trad Proc [9]
		c 1

TABLE III RESULTS

 III OF CLASSIFIED USING PROPOSED AND THAT OF

Ayeley Tchangani is with Laboratoire Génie de Production, Université de Toulouse, 47 Avenue d'Azereix, 65016 Tarbes, France ayeley.tchangani@iut-tarbes.fr

[START_REF] Brans | PROMOTHE: A new family of outranking methods in multicriteria analysis[END_REF] Franc ¸ois Pérès is with Laboratoire Génie de Production, Université de Toulouse, 47 Avenue d'Azereix, 65016 Tarbes, France


Neetu Kumari
email: neetu.kumari@femto-st.fr

Micky Rakotondrabe

Ph D Researcher

A Hybrid vibrational-thermal harvesting using a closed system based on lead free piezoelectric materials for constant temperature condition

Keywords: Energy Harvesting, Piezoelectric, Pyroelectric, Finite element analysis, COMSOL

Within the ENHANCE European project, hybrid energy harvesters have to be designed and developed to supply small and autonomous sensors in automobiles. The focus of this work is particularly related to the harvesting of energy from the vibration of the vehicle and from the high temperature (in fluctuation or not) around the engine. To this aim, an ideal hybrid energy harvester should be able to harvest vibrational energy at lower frequency ranges (>500 Hz) and pyroelectric energy at a higher temperature range. However, pyroelectric effect work with a time-varying temperature. Furthermore, the use of lead based piezoelectric materials as usual creates an environmental concern due to lead-toxicity. In this paper, a hybrid energy harvester has been designed using Lithium Niobate piezoelectric material, shape memory alloy, polyvinyl difluoride, piezoelectric polymer and helium gas, and its performance has been investigated using COMSOL Multiphysics® finite element method (FEM) software for rapid validation of the proof of concept and for visualization of the transducer functioning. The proposed hybrid harvester is capable of converting both vibrational mechanical energy and constant temperature energy into electrical energy. The principle behind the constant temperature conversion is based on the Joule-Thomson physical principle. Different transducer design and material configurations have been evaluated to identify the optimum conditions, materials, and design of the hybrid transducer. For the FEM analysis, Solid mechanics, electrostatics for piezoelectricity, heat transfer in solids, heat transfer in fluids, electric currents chemistry modules have been used for pyro electricity and associated thermodynamic properties in global definitions as well as for the vibrational piezoelectricity. The simulation demonstrates that the hybrid harvester was able to generate energy the constant temperature, as predicted. The harvester has a volume of 12.21 cm 3 and can work at a constant temperature between 20 and220 °C, and surrounding mechanical vibration 0-500 Hz of frequency. On the one hand, the harvester was able to work for up to 8 hours under constant temperature. On the other hand, the calculated stored energy was 8.92 µJ at 261Hz of mechanical vibrational excitation. This lead-free piezo-pyro energy harvester is a promising device, which can be used for environmental purposes, especially in automobiles. Its high-temperature performance stability, energy harvesting at constant external temperature, piezoelectric energy harvesting at lower frequencies, and lead-free nature make it an ideal to be used as energy harvester and supplier for small appliances.

Introduction

We can say that Energy harvesting is promising and upcoming replacement for batteries [1]. In our surrounding, we can find various forms of energies are present and among them the most common one is the waste heat and vibrational one, which many processes produce and therefore, which makes them a promising source for energy harvesting technology, and various methods have been proposed. If we talk about the waste heat and how we can use it then we have to mention the thermoelectric generators, which are based on the See beck effect, have been widely researched [2]. Nevertheless, the more people studied more they explored about it and they discovered some drawbacks related to the efficiency of these generators as figure of merit was one of the limiting factor for the thermoelectric materials, ZT. Due to this reason, pyroelectric materials gained all the attention, which has high conversion efficiency, moreover pyroelectric materials have the pyroelectric effect with spontaneous polarization [3,[START_REF] Kang | Yeatman Thermal Energy Harvesting Using Pyroelectric and Piezoelectric Effect[END_REF].However working with pyroelectric materials is also quite challenging, as the pyroelectric energy harvesting works on the temporal temperature oscillation which means that you have to establish a time varying temperature in the system. In the past, few years many people have worked on thermal systems like S. K. T. Ravindran et al. developed an engine chamber with sealed air and a bistable membrane [3]. Miwon Kang et al. studied the thermal Energy Harvesting Using Pyroelectric and Piezoelectric Effect based on a bimetallic beam structure [START_REF] Kang | Yeatman Thermal Energy Harvesting Using Pyroelectric and Piezoelectric Effect[END_REF], and Marcel Gueltig et al. adapted magnetic shape memory alloy films [5]. A. Ibragimov et.al developed a micro machined membrane based heat engine as an electrostatic power generator where he encapsulated a liquid-gas mixture inside a closed chamber and the thermal energy flowing through the chamber causes the pressure changes according the cycles of the heat engine and thus cyclically moves the flexible chamber's wall [6] In this paper, we tried to implement a similar concept of closed system like A. Ibragimov but instead the used of liquid-gas mixture we have used helium gas the reason for using it is illustrated in the coming section the closed system or we can say a throttling device which works on the concept of joule-Thompson effect where we can see the change in temperature occurs when a gas is encapsulated inside a closed boundaries with the opening is closed with a flexible membrane due to which the throttling will appear. Joule-Thomson cooling is the name given to the drop in temperature that occurs when a real gas such as CO2 or N2 expands from high pressure to low pressure at constant enthalpy (i.e., adiabatic expansion) the gas exceptions to this cooling upon expansion effect are Hydrogen, Helium and Neon, which actually heat up. Here, a temperature change is necessarily desirable. As this system will be working as a hybrid system where we would like to harvest both the vibrational and thermal energies, we decided to use lithium Niobate as it is a lead free materials, which is primary aim of this work, moreover it shows both the pyroelectric and piezoelectric effects, which makes it a perfect choice for our work Figure 1 illustrates the proposed concept. It is assumed that this closed system will be vibrating at 1 g of acceleration, for the Thermal gradient we are going to follow the joules Thompson effect where the temperature of the gas will simultaneously changes over time. The structure consist of a hemisphere made up of copper and a circular ring consists six cantilevers are attached on the top and sealed with a PVDF film.

Figure 1. Illustration the proposed Hybrid concept

This paper is organized as follows. Section I presents the research motivation for hybrid energy harvesting devices. Section II describes the theoretical operation of closed hybrid energy harvesting device and the design of the proposed energy harvester. The simulation of the proposed energy harvester for closed system working at a constant temperature is explained in Section III. Simulation results and analysis results are presented and discussed in Section IV. Finally, concluding remarks are made in Section V.

Hybrid Energy Harvesting Design

A model of the proposed Hybrid energy-harvesting device is presented; along with the equations for the piezoelectric effect here, we have described the pyroelectric effect as well along with that, we have presented a detailed version of joules Thompson effect. This model is also used to estimate the resonant frequency. The system is modeled as a 1 degree-offreedom (DOF), spring-mass damper system. This model is configured as a base excitation, since it represents a device with a solid outer structure (black rectangle) that is shaken by an external acceleration (𝑋(𝑡)). According to modal analysis theory, the governing equations for such a piezoelectric EH system can be written as [START_REF] Zhu | Analyses of Power Output of Piezoelectric Energy-Harvesting Devices Directly Connected to a Load Resistor Using a Coupled Piezoelectric-Circuit Finite Element Method[END_REF]:

𝑚𝑥̈1(𝑡) + 𝐶𝑋 ̈1(𝑡) + 𝐾𝑋 1 (𝑡) + 𝑓 𝑒 (𝑡) = 𝑚𝑋(𝑡) ̈ (1)
Where m, C, and K are the effective mass, mechanical damping, and stiffness, respectively fe is the effective force due to the piezoelectric element; 𝑋 (𝑡) is the acceleration applied to the housing due to external ambient vibration from the environment, and X1(t) is the mechanical displacement of the proof mass. The effective transducer force fe can be presented as [START_REF] Zhu | Analyses of Power Output of Piezoelectric Energy-Harvesting Devices Directly Connected to a Load Resistor Using a Coupled Piezoelectric-Circuit Finite Element Method[END_REF]:

𝑓 𝑒 = 𝑘 𝑚 𝑉 𝑝 (𝑡) (2) 𝑘 𝑚 𝑋 1 (1) -𝐶 𝑝 𝑉 𝑝 (𝑡) = 𝑄 𝑝 (𝑡) ( 3
)
Where 𝑘𝑚 is the effective electromechanical coupling coefficient of a piezoelectric structure; VP (t) and QP (t) are the voltage across the piezoelectric electrodes, and the charge generated on the electrodes; Cp is the capacitance of the piezoelectric material. The main goal here in piezoelectric part is to convert the ambient vibration energy into electrical power. Several design iterations were done whereby the design was simulated by COMSOL. The cantilever-beam structure with proof-mass was found to be the simplest oscillator design that could meet the objectives for our design. A cross-sectional diagram of the proposed configuration is illustrated in In the model for the pyro part, we can say that Pyro electricity is a coupled effect that relates a change in temperature to a change in electrical displacement D (with units of C/m2),

𝑑𝐷 = 𝑝𝑑𝜃 ( 4
)
Where p is the pyroelectric coefficient (with units of C/m2.K).

Both displacement and the pyroelectric coefficient are vectors or first-rank tensors. The pyroelectric coefficient is defined by

𝑝 = (𝜕𝑃 𝑆 /𝜕𝜃) σ, E. ( 5
)
Where the constraints are constant electric field E, and constant elastic stress σ.

When talking about energy harvesting from heat, we need first to establish the equations of pyroelectric materials [START_REF] Sebald | Pyroelectric energy conversion: optimization principles[END_REF]. It is assumed here idealized pyroelectric materials exhibiting no losses and purely linear properties.

𝑑𝐷 = 𝜀 33 𝜃 𝑑𝐸 + 𝑝𝑑𝜃 (6)
𝑑Ӷ = 𝑝𝑑𝐸 + 𝐶 𝐸 𝑑𝜃 𝜃 (7)
Where D, E, θ, and Γ are electric displacement, electric field, temperature, and entropy, respectively. The coefficients are defined as:

𝜀 33 𝜃 = ( 𝑑𝐷 𝑑𝐸 ) 𝜃, 𝑃 = 𝑑𝐷 𝑑𝜃 = 𝑑Ӷ 𝑑𝐸 , 𝐶 𝐸 = ( 𝑑𝑈 𝑑𝜃 )𝐸 (8)
Where U is the internal energy. ε θ 33, p, CE, are the dielectric permittivity, pyroelectric coefficient and calorific capacity, respectively.

Analogizing to electromechanical case in the work of Lefeuvre, the harvested power in the pyroelectric case can be defined as;

𝑃 𝑚𝑎𝑥 = 𝑝 2 𝜃 2 2𝜋𝜀 33 𝜃 (9)
To understand better, how the joule Thompson effect will take we did mathematical modelling for that as well, We can say we assumed that it would be an adiabatic throttling process by this we imply that there will be no exchange or heat or mechanical work will take place with the environment. As we are assuming that there will be no change in temperature is happening in the surroundings i.e. ΔT = 0. However keeping in mind that there will be always vibration is present the system. After this, we focused on the fundamental thermodynamics to develop an energy balance inside the closed system. Keeping in mind the thermodynamics is the study of systems involving energy in the form of heat and work. A good example of a thermodynamic system is gas confined by a piston here we have a flexible membrane in the closed system. If the gas is heated, it will expand, doing work on the piston; this is one example of how a thermodynamic system can do work. Thermal equilibrium is an important concept in thermodynamics. When two systems are in thermal equilibrium, there is no net heat transfer between them. This occurs when the systems are at the same temperature. In other words, systems at the same temperature will be in thermal equilibrium with each other. The first law of thermodynamics relates changes in internal energy to heat added to a system and the work done by a system. The first law is simply a conservation of energy equation:

dU = dQ -dW (10)
The first law of thermodynamics relates the change in internal energy dU to the work dW done by a system and the heat dQ added to it. By definition, in an adiabatic process, the heat exchanged dQ = 0. Substituting this in the first law and rearranging gives 0 = dQ = dU + dW [START_REF] Sebald | Electrocaloric and pyroelectric properties of 0.75Pb (Mg1/3Nb2/3)O3-0.25PbTiO3 single crystals[END_REF] The second term is easy: the work done dW when a system changes its volume V by dV is PdV.

The first term may be related to the specific heat, which is defined as the heat added per unit temperature change per mole of substance. If we add heat at constant volume, then the gas does not expand and so does not do work. Therefore, the added heat increases the internal energy U. The specific heat at constant volume can be written as [cv = (( 𝑑𝑈 𝑑𝑡 ) 𝑣 ) ] and n is the number of moles. Hence

𝟎 = 𝒏𝒄 𝒗 𝒅𝑻 + 𝑷𝒅𝑽 (12)
Now the equation of state of an ideal gas is

nRT = PV ( 13
)
Where R is the gas constant. Taking derivatives gives

𝒏𝑹𝒅𝒕 = 𝑷𝒅𝑽 + 𝑽𝒅𝑷 (14)
We may now combine equations ( 12) and ( 13) to eliminate T. ( 12) and (13) give respectively these expressions for ncv dT:

-𝑷𝒅𝑽 = 𝒏𝒄 𝒗 𝒅𝑻 = 𝒄 𝒗 𝑹 (𝑷𝒅𝑽 + 𝑽𝒅𝑷) (15)
Now if the gas is ideal, then dU = n cv dt And if some work is done by a gas adiabatically, then heat change will be zero then in this situation we can have two cases: 1) Expansion occurs at the cost of internal energy of the gas and so gas temperature falls as the gas expands. So, internal energy decreases. Or, 2) As the temperature falls, the velocity of the gas molecules deceases too which then consecutively lowers the kinetic energy of the gas. From the equation of state of an ideal gas [START_REF] Badr | Design of a Low Frequency Piezoelectric Energy Harvester for Rodent Telemetry[END_REF], we can calculate the work dW done at constant pressure: it is just PdV = nRdT. So the specific heat of an ideal gas at constant pressure is just cP = cv + R. and the ratio of specific heats is given a standard symbol 𝜸 = 𝒄 𝒑 /𝒄 𝒗 . Therefore, we have

𝟎 = 𝜸 ( 𝒅𝑽 𝑽 ) + ( 𝒅𝑷 𝑷 ) (18)
Hence for an adiabatic process in an ideal gas,

𝑷𝑽 𝜸 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 (19)
While keeping in mind that cP > cV : At constant volume, all the heat you put in goes to increasing the internal energy and thus raising the temperature. At constant pressure, you need to put in not only the heat that raises the internal energy, but also an amount of heat equal to the work done when the system undergoes thermal expansion. For gas filled inside the system, helium γ is about 1.3.

Of course, we may now substitute into (19) from the equation of state (13) to see how P and T or T and V are related for an adiabatic process.

Now focusing on the joules Thompson effects

The classic Joule Thomson expansion consists of a thermally insulated system with gas on one side initially at p1, V1, and T1 that flows through a porous plug and out the other side at p2, V2, and T2. In the absence of heat exchange with the surroundings, the total work done by the expanding gas equals the change in internal energy which is given by

𝒘 = ∆𝑼 = 𝑷 𝟏 𝑽 𝟏 -𝑷 𝟐 𝑽 𝟐 (20)
For an ideal gas, p1V1 = p2V2 and the Joule-Thomson Expansion would be at constant internal energy. However, Our interest is in helium as it's not an ideal gas but the closest one among the other gases when it comes to show the throttling effects. The change in enthalpy of the system in the Joule-Thomson expansion is given by

∆𝑯 = ∆𝑼 + ∆(𝒑𝑽) (21)
Upon substitution of Eq. ( 20) into Eq. ( 21), we have For doing the modelling we have assumed that µJT is negative (i.e., expansion leads to heating). However, the sign of µJT can also be positive (i.e., meaning the gas cools down upon expansion). The so-called Joule-Thomson inversion temperature for a particular gas, gas will heat upon expansion, whereas below the inversion temperature it will cool. At room temperature, common gases as CO2, N2, and O2 cool upon expansion while He and Ne warm upon expansion. That is also another reason that made such choose helium. Table 1 lists the final design parameters of the proposed closed system.

∆𝑯 = 𝟎 (

Use of COMSOL for closed system

A computer model is developed to simulate the performance of the closed system design as specified in Figure 1 and Table 1. The COMSOL Finite Element Analysis (FEA) package is used to estimate the closed system performance, using four different studies. A vast simulation is carried out; the model is simulated with 3-D (three-dimensional) analysis. In order to guarantee mesh integrity, the aspect ratio of individual mesh elements should not be greater than three [START_REF] Badr | Design of a Low Frequency Piezoelectric Energy Harvester for Rodent Telemetry[END_REF]. Figure 2(a), (b) illustrates the plot of the quality mesh of the closed system, internal COMSOL mesh quality analysis showed the mesh to be of high quality. With the help of developed FEA model, three different studies are done to investigate the performance: (1) An Eigen frequency Study to determine the first Eigen frequency (resonant) and its mode shape for all the six cantilevers,

Eigen frequency Study

The closed system design is analyzed to determine its resonant frequencies by using FEA to find the Eigen frequencies and mode shapes of the structure. This analysis is extensively used during the geometrical design of the closed system. The first Eigen frequencies of all the six cantilevers are calculated and represented in Figure 3

Stationary Study

A static FEA analysis is done to determine the distribution of temperature inside the closed system. Figure 4 Where ρ is the density (kg/m 3 ), Cp is the heat capacity (J/(kg•K)), T is the absolute temperature (K), u is the velocity (m/s), Q is the power density (W/m 3 ), Qted and the thermoelastic damping. q Represents the heat flux vector. Figure 5 shows the Simulation of temperature with change in time.

Results and Discussion

The simulation results of the proposed closed system design are analyzed and discussed. The simulation demonstrates the plausibility of the introduced Heat Engine's concept in energy harvesting by combining the Piezoelectric effect with pyroelectric effect where the pyroelectric materials itself work on the principle of the heat engine. We combined different modules for our study like the solid mechanics and electro statics with the heat transfer module with solid and fluid along with the laminar flow we assumed the behavior of the gas to be compressible and after that to study the coupling effect of theses we used the Multiphysics modules where we used the Cantilever temperature

Piezoelectric effect with the non-isothermal flow, we did the study of the frequency, Eigen frequency and the time dependent, by combining these modules and defining the heat flux conditions in the solid and fluid and making the outer surface temperature as constant we can see in Figure 6,that the temperature of the gas is keep on fluctuating even when there is no heat source. By comparing the temperature in Figure 3. A) From the outer surface to the inner surface (Gas), in red is the gas temperature shows the maximum temperature change in temperature. After that, we also calculated the stored energy by the closed system, which we can say, was mainly done with the lead free material Lithium Niobate. Figure 7. Shows the total stored with respect to frequency. The simulations were carried out at 1 g of acceleration, as there will be continuous vibration in the system. Moreover we were able to calculate the fluctuation of the temperature of gas and able to conclude that the gas temperature will fluctuate for eight hours even when there is no natural conduction or external heating source is present.

Conclusions

The proposed closed system is designed to harvest two different sources of energy together as a hybrid system, but the main aim was to harvest the thermal energy at constant temperature as all the pyroelectric materials needs a thermal gradient for them to generate electricity. But keeping in mind that the fabrication of such devices are very difficult as encapsulating gas inside and sealing it itself is a big challenge so for future work we would be focusing more on the experimental part and will try to validate the results we got from the comsol simulation with the experimental one.

Figure 2 .

 2 Figure 2. Models of single DOF vibration based energy harvester.

  Collecting the PdV and VdP terms gives

  22)i.e., the Joule-Thomson expansion is at constant enthalpy. It can be observed that when the Joule-Thomson expansion is carried out for a number of different pressure drops (dP) across the porous plug or in our case the flexible membrane different dT's are measured, that a plot of dT vs. dP is approximately linear with a slope equal to the Joule-Thomson coefficient (µJT):

Figure 2 :

 2 Figure 2: Mesh quality of the closed system (A) Side view (B) Top view (2) A Stationary Study to check the total displacement and corresponding stress of the system due to gravity alone, in a static analysis, and (3) A Time Dependent Study Temperature distribution is simulated using Comsol Multiphysics since heat transfer and distribution are a critical factor in this study .

Figure 3 .

 3 Figure 3. Eigen frequency study of all the six cantilevers.

  shows the result when the temperature increases from 19 •C of the heat reservoir to 25 •C of through the gas compression and joules Thompson effect at a stationary condition.

Figure 4 .

 4 Figure 4. Simulation of temperature distribution 3.3 Time depending study

2 3

 2 𝑞 = -𝑑 𝑧 𝑘∇𝑇 (25)Where k is the thermal conductivity (W/(m • K)). This is Fourier's law of heat conduction. The laminar flow module is based on Navier-Stokes equations as follows:𝑢2. ∇)𝑢2 = ∇. [-𝑝𝐼 + 𝜇(∇𝑢2 + (∇𝑢2) 𝑇 )) -𝜇(∇. 𝑢2)𝐼] + 𝐹 + 𝜌𝑔 (27)Where u2 is the velocity (m/s), p is pressure (Pa), I is the unity tensor, µ is the dynamic viscosity (Pa•s), F is the volume force vector (N/m3), g is the acceleration of gravity. Equation (26) is the continuity equation and represents the conservation of mass, and Equation (27) represents the conservation of momentum. Figure5shows the Simulation of temperature distribution with respect to time in order to calculate the change in temperature over the cantilevers.

Figure 6 .

 6 Figure 6. Shows the change in temperature of the gas inside the closed system.

Figure 7 .

 7 Figure 7. Shows the total stored with respect to frequency.

Table 1 : Design parameters for the closed system

 1

	Design	Description	Design
	Parameter		Value [units]
	Radius of the	Material: shape	cm
	hemisphere	memory alloy
		1.8
	Thickness of the	Material: shape	cm
	Hemisphere	memory alloy
		0.05
	Dimension of	Material:	cm
	the middle two	lithium Niobate
	Cantilever	1.3×0.05× 0.035
		[𝐿 × 𝑊 × 𝐻]

Acknowledgements

This project has been supported by the European ITN Marie Curie ENHANCE project. Members of MACS research group at AS2M/FEMTO-ST are also acknowledged for their kind support as well as the Mechanical department of FEMTO-ST for some experiments. The authors are also grateful to the members of PRIMES center of technology and LGP laboratory, both of ENIT-INPT, for the support to extensive experimental works a part of which was also supported by the national ANR CODE-TRACK project.

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