text:
Philippe A Martin
email: philippe.martin@univ-reunion.fr

Jérémy Bénard
email: jeremy.benard@logicells.com

Categorizing or Generating Relation Types and Organizing Ontology Design Patterns

This article proposes an ontology design pattern leading knowledge providers to represent knowledge in more normalized, precise and inter-related ways, hence in ways that help automatic matching and exploitation of knowledge from different sources. This pattern is also a knowledge sharing best practice that is domain and language independent. It can be used as a criteria for measuring the quality of an ontology. This pattern is: "using binary relation types directly derived from concept types, especially role types or process types". The article explains this pattern and relates it to other ones, thereby illustrating ways to organize such patterns. It also provides a top-level ontology for generating relation types from concept types, e.g., those from lexical ontologies such as those derived from the WordNet lexical database. This generation and categorization helps normalizing knowledge, reduces having to introduce new relation types and helps keeping all the types organized.

I. INTRODUCTION

NTOLOGY Design Patterns (ODPs) are "modeling solutions to solve a recurrent ontology design problem" [START_REF] Gangemi | Ontology Design Patterns[END_REF]. A "Conceptual ODP" describes a best practice (BP) for content modelling [START_REF] Gangemi | Ontology Design Patterns[END_REF]. Since we only consider ODPs that represent BPs, we use these two terms interchangeably in this article to ease its reading. Many ODPs have been described. E.g., about 160 are registered in the ODP catalog at http://ontologydesignpatterns.org which, in this article, will now be referred to as ODPC. Despite these ODPs, most of thousands of existing ontologies that exist are still poorly inter-connected and heterogeneous in their design. It is then difficult for people and automated agents to compare or match such independently created knowledge representations (KRs, e.g., types or statements) to know if some KRs are equivalent to others or specializations of others. Thus, it is difficult for people and automated agents to align and aggregate -and thus, relate, infer from, search or exploit -KRs or ontologies.

O

In other words, there is a need for ODPs specifically aimed for knowledge modeling and sharing -as opposed to knowledge exploitation with computational tractability constraints -and, more precisely, specifically aimed for solving the problem of leading knowledge providers to create more matchable and re-usable KRs. As later detailed,  This work was not supported by any organization.

this implies leading them to create more precise, normalized, well related and easy-to-understand KRs. To be adopted, these ODPs should also be easy to follow and easy to use as criteria for automatically measuring the quality of an ontology, to help developing an ontology or selecting ontologies to re-use. Finally, the ODPs -or, at least the knowledge sharing ODPs -should be well inter-related by semantic relations to help people i) know about them and their advantages, and ii) select those they want to commit to. Then, tools can check or enforce these commitments.

This article proposes such a knowledge sharing focused ODP and relates it to other ones, via specialization relations and gradual pattern relations. This BP, which in this article will now be referred to as ABP, is: "using binary relation types directly derived from concept types, especially role types or process types". No ODP catalog appears to include ODPs similar to this one or to any of its parts. Like most BPs, it is domain and language independent. The sections 2, 3 and 4 explain, formalize and illustrate the different parts of ABP. Section 5 relates them to other ODPs and thereby also gives more rationale.

II. USING BINARY RELATIONS

ABP starts by advocating the use of binary relations, i.e., logical statements based on binary predicates. In the RDF model, these statements are called triples and binary relation types are called properties. In this article, types that are not relation types (RTs) are refered to as concept types (CTs), i.e., classes in the RDF model. The expression concept individual will be used for anything that is neither a type nor a relation.

Since ABP is language independent, this article uses a general terminology, one compatible with those for Conceptual Graphs and RIF-FLD [START_REF]RIF Framework for Logic Dialects (2nd edition)[END_REF], the W3C Framework for Logic Dialects of the Rule Interchange Format. For its formal textual examples, this article uses RIF-FLD PS, the Presentation Syntax of RIF-FLD. Indeed, this notation is both expressive and rather intuitive. For clarity purpose too, in the examples, RT names begin by "r__" and function names begin by "f__". Logical rules are used since RIF-FLD is used and since this shows the direction the implications are expected to be used. However, in each case, a logical equivalence could also be used instead.

Following ABP does not prevent using non-binary RTs as long as definitions or rules are also provided to enable the automatic translation of "KRs using non-binary RTs" into "KRs using binary RTs". Table I illustrates such rules for various kinds of use cases but only the third row is also about the focus of Section 3, i.e., deriving a RT from a CT.

One reason why such definitions or rules are useful for knowledge sharing is that binary relations can be compared while two relations of different arities generally cannot. Two types or KRs are comparable if and only if an equivalence or specialization relation between them has been directly stated or can be automatically inferred. Thus, KRs using binary relations can be ordered by generalization relations, typically, implications. This is more difficult with KRs using relations of different arities, thus reducing possibilities for knowledge matching or inferences. E.g., as illustrated by Table I, some relations of different arities can be translated into binary relations using a list as destination . Then, they can be compared.

A related reason why such definitions or rules are useful for knowledge sharing is that they make more information explicit. As detailed it Section 5, normalizing knowledge, enhancing its comparability or adding more information have strong relationships.

In practice, with a KR language (KRL) allowing contexts and sets or lists, it is easy to avoid the use of relations with arity greater than two. A context, i.e., a contextualizing statement, is a meta-statement specifying restrictive conditions for the contextualized statement to be true, e.g., via temporal relations or modalities. Although RIF-FLD is not restricted to first-order logic, it lacks a construct for expressing contextualizations in simple ways, as in KIF [START_REF] Genesereth | Knowledge Interchange Format, Version 3.0, Reference Manual[END_REF] for example. However, the second row of Table I shows how simple contextualizations can still be represented -albeit in a rather cumbersome way -using binary relations. To that end, this example uses an adaptation of the ODP named Context Slices in ODPC [START_REF] Welty | Context Slices[END_REF]. It relies on introducing concept individuals within contexts and relating them to their context as well as to their context-independent counterpart. This is an alternative to the more common approach of reifying a statement and asserting a relation between the reification and the context. With the reification based approach, handling contexts is a bit more difficult when simple KR management tools are re-used and extended. Both approaches lead to rather lengthy statements and are ad-hoc since they require extensions to inference engines to fully handle them correctly. Therefore, for the purpose of knowledge modeling and sharing -as opposed to knowledge exploitation which comes after and may require converting the knowledge into KRLs of reduced expressiveness but which can be handled efficiently -a BP is to i) use a KRL that handles contexts or use more ad-hoc concise constructs, and then ii) provide or use rules for translating into the various ways to represent contexts in other KRLs. The same idea applies for the many ODPs that deal with the problems of translating "KRs using high expressive constructs" into "KRs using lower expressive constructs". E.g., in ODPC, there are many ODP for translations into OWL or from OWL.

To conclude, although formally specifying the semantics of relations of arity greater than two requires at least one primitive ternary relation [START_REF] Correia | The Teridentity and Peircean Algebraic Logic[END_REF], in practice there is no necessity to use such relations for knowledge modelling.

There is no claim here that the idea of "translating nonbinary RTs into binary ones or directly using them" is original. Yet, it should be an ODP for various reasons: i) it is useful, ii) some claims seemingly about the necessity of using non-binary relations are actually claims about the necessity of using constructs supporting different kinds of contexts [START_REF] Zarri | Representation and Management of Narrative Information: Theoretical Principles and Implementation[END_REF], and iii) this best practice is sometimes unknown to users of KRLs allowing non-binary relations.

III. DERIVING RELATION TYPES FROM CONCEPT TYPES

ABP advocates the use of -or specifications of translations into -binary RTs directly derived from CT". A CT may have multiple directly derived RTs if they have uncomparable signatures, i.e., if none specializes another one. The third row of Table I illustrated a way to directly derive an RT from a CT using a rule and a relation of type r__directly_derived_relation. The first two rows illustrate the definitions of non-binary RTs mainly with respect to binary RTs. This is useful as an intermediary step: the final step -deriving these last binary RTs from a CT, e.g., one named List_of_surrounding_entities -was not illustrated in Table I.

Manually or automatically defining each RT with respect to a CT makes additional information explicit and ensures that every distinction in the (subtype) hierarchy of RTs is also included in the CT hierarchy. This last point is important for two reasons. First, it prevents some knowledge providers to develop distinctions only in the RT hierarchy while others develop distinctions only in the CT hierarchy, thus leading to undetected redundancies within a shared knowledge base or in different ontologies. Second, it ensures that any distinction can be used -without losing possibilities of knowledge representation and matchingwith both its CT form and its RT form. More possibilities come from the CT form since i) unlike RTs, CTs can be quantified in many different ways (e.g., "3 landings", "all landings" or "8% of landings" can only be described via the CT "Landing", not the RT r__landing), ii) CTs are easier to organize by subtype relations than RTs, and iii) the number of used or re-usable existing CTs is much greater than the number of used or re-usable RTs. Thus, both cases lead to better categorizations in the concept and relation hiearchies.

These advantages of using defined RTs come for free when RTs are automatically derived from CTs and hence defined with respect to them. Furthermore, such derivations permits a system to display fewer types in the RT hierarchy which is then easier to read and grasp. Indeed, the derived RTs may be left hidden or may not have to be created at all. This last option was used in the knowledge server Ontoseek [START_REF] Guarino | Ontoseek: Content-based Access to the Web[END_REF] and is used in the knowledge base server WebKB (www.webkb.org; [START_REF] Martin | Collaborative knowledge sharing and editing[END_REF]). In Ontoseek, any type derived from the noun-related part of the lexical ontology Sensus could be re-used as a CT or a RT. WebKB also re-uses a lexical ontology derived from WordNet. However, unlike Ontoseek, WebKB only allows the subtypes of certain types to be reused as RTs. This is defined by specifications that users can adapt. More precisely, this is defined by relation signatures which are directly associated to certain top-level CTs. Table II illustrates the approach and then gives rules that would actually generate the derived RTs. The next section complements this framework by giving an ontology of the CTs these rules can be applied to. These RT generation rules permit to formalize the framework. They rely on the functions f__type_name and f__denotation_of_type_name which are identical to the KIF functions name and denotation formalized in the documentation of KIF [START_REF] Genesereth | Knowledge Interchange Format, Version 3.0, Reference Manual[END_REF]. In WebKB, such rules are not actually executed but a more efficient process relying on the same idea is used. Indeed, during the parsing of statements, whenever a CT is used where a RT is expected, WebKB simply checks that one of the signatures associated to the CT is respected and acts as if the relevant derived RT was actually used. Thus, in WebKB, there is no need to use the actual names of the virtually derived RTs: the CT names can be used directly. As in the framework described by Table II, signatures are inherited along subtype relations between CTs and an error is generated if a CT is associated to two signatures that are comparable. This approach and ODP seem original.

IV. DERIVING FROM ROLE TYPES OR PROCESS TYPES

ABP advocates the derivation of RTs from CTs, "especially role types or process types". The third row of Table I illustrated this for processes. In this article, a process refers to a situation that is not a state, and hence that makes a change. A situation is something that occurs in a real/imaginary region of time and space. These conceptual distinctions come from the Situation Semantics [START_REF] Barwise | Situation Theory and its Applications[END_REF] and are the basis of John Sowa's first top-level ontology [START_REF] Sowa | Conceptual Graphs Summary[END_REF]. There are re-used in this article for at least the following reasons:

• They are rather intuitive and generalize other well known types. E.g., Perdurant from Dolce [START_REF] Borgo | Theory and Applications of Ontology: Computer Applications[END_REF] is subtype of Process. Forall ?process ?agent ?object And ( r__agent (?process ?agent) r__ object (?process ?object) ) :-And ( ?process # ?processType ?r (?agent ?object) ) ) :-?processType ## Process //this rule derives the expected RT for each subtype of Process

TABLE II. RULES FOR AUTOMATICALLY DERIVING A BINARY RT FROM A CT (AND, IF NEEDED,DOING SO FOR ALL ITS SUBTYPES) BASED ON A KIND OF SIGNATURE ASSOCIATED TO THIS CT (NOTE: IN THESE EXAMPLES, THE TYPES CREATED BY THE AUTHORS OF THIS ARTICLE HAVE NO PREFIX TO INDICATE THEIR NAMESPACE).

Table I gave examples of how a rule can define a RT with respect to a CT. This had to be done for each RT. Here, the approach is simpler. The derived RT does not have to be explicitly defined. Its signature is directly associated to the CT via a relation of type r__signature_for_derived_binary_relation or a function of type f__derived_binary_relation.

Thanks to the definitions given in the next row of this

Furthermore, the derived RTs have the same subtype relations as the CTs they derive from. However, to keep things simple, it is here assumed that no RT with the same name as the derived RT has previously been manually created. The RT name is created by taking the CT name, lowering its initial and prefixing it with "r__". The functions f__denotation_of_type_name, f__type_name, f__cons, f__cdr, f__lowercase used below are identical to their counterparts (without the prefix "f__") in KIF.

Forall ?t ?r__t ?t_domain ?t_range ?t_supertype ?r__t_supertype ?t_sup_domain ?t_sup_range ( And ( rdfs:domain (?r__t ?t_domain ) rdfs:range (?r__t ?t_range ) ?r__t = f__denotation_of_type_name ( f__cons ( f__lowercase ( f__car ( f__type_name ( ?t ) ) ) f__cdr ( f__name ( ?t ) ) ) ?r__t ## ?r__t_supertype //"##" means "is subtype of" :-And ( ?t ## ?t_supertype ?r__t_supertype = f__ derived_binary_relation ( ?t_supertype List ( ?t_sup_domain ?t_sup_range ) ) ) ) :-?r__t = f__derived_binary_relation ( ?t List ( ?t_domain ?t_range ) ) ) Forall ?t ?t_domain ?t_range ( Exists ?r__t ( ?r__t = f__ derived_binary_relation ( ?t List ( ?t_domain ?t_range ) ) ) :-r__signature_for_derived_binary_relation ( ?t List ( ?t_domain ?t_range ) ) )

Other rules can be built upon these last ones, e.g., this rule for deriving functional binary relations:

Forall ?t ?t_domain ?t_range Exists ?r__t ( And ( ?r__t = f__ derived_binary_relation( ?t List ( ?t_domain ?t_range ) ) ?r__t # owl:FunctionalProperty ) //"#" means "is instance of"; owl:FunctionalProperty is a 2nd-order type :-r__signature_for_derived_functional_binary_relation ( ?t List ( ?t_domain ?t_range ) )

• They are very adequate for the signatures of thematic relations [START_REF] Carnie | Syntax: A Generative introduction[END_REF], e.g., r__agent, r__recipient, r__cause, r__instrument. Such types are top-level types of relations from a process. • In this article, a role type (e.g., Agent, Experiencer, Recipient, Cause, Instrument) is a CT which is defined -or could be defined -as being the range of a thematic RT. This informal definition of a role is a bit more general than what is usually thought to be a role type [START_REF] Mizoguchi | Ontological Analyses of Roles[END_REF] but here it is sufficient: as defined in this article, processes and role types can be used for deriving CTs into binary RTs.

• Thematic RTs or their subtypes can also be used for defining most RTs. Thus, doing so normalizes KRs. • Most statements implicitly or explicitly refer to a process. Representing it, either directly or via RTs directly derived from a process, strongly normalizes KRs. Not doing so, which unfortunately is the case in many ontologies, amounts to losing precisions and many KR comparison possibilities. Fig. 1 Legend: i) the arrow " " represents a supertype (subClassOf) relation, ii) by default, each subclass set is here a subclass partition, hence its "{disjoint, complete}" UML annotation is left implicit, ii) for name-spaces, XML shortcuts are used but type names created by the authors of this article have no prefix, iv) "wn" refers to WordNet, v) "(*)" is the RT signature for any set of arguments, vi) comments are delimited by "/*" and "*/", vii) for readability purposes, the boxes around classes (concept types) are not drawn. includes types for processes and roles. Fig. 2 illustrates subtype relations between such derived RTs. Fig. 3 displays common top-level types for relations from a process, most of which are thematic RTs. Fig. 3 re-uses top-level types shown in Fig. 1. All the types in these figures are part of the Multi-Source Ontology (MSO [START_REF] Ph | Correction and Extension of WordNet 1.7[END_REF]) which is accessible and cooperatively updatable via WebKB. Hence, the names in these figures are names accessible via this Web server. However, these figures have not previously been published.

The MSO includes more than 75,000 categories and relates them by more than 100,000 relations, mainly subtype relations. It categorizes WordNet types as well as types from various top-level ontologies (DOLCE included) with respect to the types shown in Fig. 1 or specializations of them. More precisely, about a hundred of top-level WordNet types and some more specialized WordNet types were manually set as subtypes of those in Fig. 1 or specializations of them. Thus, in the subtype hierarchy of the MSO for things usable for directly deriving a binary relation, there are currently more than 4800 process types, 2900 role types (for things playing some role), 650 types of attributes or qualities or measures and 240 types of description content/medium/container. This makes more than 8600 types usable for creating relations without having to declare new RTs. The 4800 process types can also be used directly with relations from a process. Finally, the types shown in Fig. 3 for these relations can implicitly or explicitly be specialized by types derived from the 2900 role types. To sum up, the proposed approach and the MSO permit people and automated agents to create KRs that are well normalized, inter-related and comparable. Furthermore re-using the approach and content of the MSO to extend other ontologies is eased by the fact that i) the MSO relates, generalizes and specializes types from various other ontologies, and ii) the MSO can be complemented online via WebKB.

In Fig. 1, the types named Relative_thing and Mediating_thing come from John Sowa's second top-level ontology [START_REF] Sowa | Knowledge Representation: Logical, Philosophical, and Computational Foundations[END_REF].

To show how rules can be used to associate a signature to a CT and thereby to a derived RT, examples in Table II used a process type and the type of things usable for deriving a binary relation with it as destination. Similar rules can be used for other types of things usable for deriving a binary relation". Fig. 2 shows how the various relations types -derived or not from CTs -can be related by subtype relations. Organizing relations of different arities is permitted by the use of "*" in the relation signatures: it refers to any number of arguments. In Fig. 2, a signature is shown as an ordered list of comma-separated arguments, within parenthesis. Both KIF and RIF-FLD allow relations with a variable number of arguments. However, unlike in KIF, there is no special construct in RIF-FLD for definitions, hence for signatures.

ODPC includes the DOLCE+DnS-Ultralite ontology [START_REF] Gangemi | DOLCE+DnS-Ultralite, RDF+OWL ontology at[END_REF] and categorizes it as Content ODP. ODPC also includes related but smaller content ODPs such those named ActingFor and Agent-Role. Its DnS (Descriptions and Situations) part includes some types which can be seen as subtypes of those in Fig. 3. ODPC proposes many RTs which could be -but, it seems, are not -derived from process types, e.g., RTs with names such as actsFor, conceptualizes or defines. Yet, some of its CTs have been aligned with OntoWordNet [START_REF] Gangemi | Restructuring Wordnet's Top-Level[END_REF]. Thus, the ontology and approach proposed in this section and the previous one could be used to extend and normalize DOLCE+DnS-Ultralite. This would support more KR comparison possibilities. Legend: same as in Fig. 1; each RT signature is delimited by parenthesis; "*" refers to 0 or more arguments of any type.

V. RELATING TO OTHER ODPS

To be adopted, knowledge sharing ODPs should be well inter-related by semantic relations to help people know about them and the criteria or advantages they fulfill. Thus, people can search and select ODPs to commit to. Then, tools can check or enforce these commitments, or retrieve ontologies satisfying them.

Thus, ideally, ODPs should at least be organized into categories related by specializations and exclusion relations, as in the hierarchy presented in Fig. 1. However, this is not easy. The most organized of current ODPC or BP repositories [START_REF] Poveda-Villalón | Reusing Ontology Design Patterns in a Context Ontology Network[END_REF] seems to be ODPC. It organizes its ODPs into a specialization hierarchy with a first level of six categories. Each of them has 0 to 3 sub-levels. These six categories and their current content are:

• Content ODP: 101 ontologies, some having only a few types. • Reasoning ODP: no ODP has yet been submitted in this category. • Structural ODP: 1 ODP in the Architectural ODP category -BPs about the structure of an ontology, e.g., the use of subtype partitions, i.e., unions of disjoint types as in Fig. 1 -and 13 in the Logical ODP category -translations between constructs from KRLs of different expressiveness. Legend: same as in Fig. 1 plus i) arrows with dashed lines are relations like UML associations, i.e., the source is universally quantified and a cardinality (or multiplicity) is associated to the destination; here, each cardinality is either "0 to many" or "1 to many", and is left implicit; and ii) comments are enclosed within "/*" and "*/"; "e.g.," is used for introducing subtypes. • Lexico-Syntactic ODP: 20 linguistic structures for extracting KRs or displaying them, as with a controlled language. • Presentation ODP: no submission of ODP has yet been submitted in this category about the usability and readability of ontologies. It has two subcategories: Annotation ODP and Naming ODP. All these categories are not exclusive. An ODP can be placed in several of them. E.g., the ODPs listed in the sections 2, 3 and 4 seem to be architectural ODPs as well as logical ODPs and, for some of them, also Content ODP, e.g., the DOLCE+DnS-Ultralite. The ODPs we gave in Section 5 are Naming ODPs but are also related to Structural ODPs.

Since there are multiple categorization possibilities, different persons will search or add a same ODP in different categories, thus leading to less relations between the ODPs and more undetected redundancies, as noted in the previous sections. This structure also does not lead ODP providers to collaboratively build a finely organized hierarchy or graph of ODPs. Such a structure could be obtained by formally representing each ODP as a process, using a same base ontology, e.g., the MSO, hence with the types shown in Fig. 1 and Fig. 3 as top-level types. Most of the subtype relations between ODPs could then be automatically calculated. Although this approach would scale well, such a formal and homogenous representation would be a huge work and would require quite motivated ODP providers.

Furthermore, relations to criteria and advantages would still probably not be sufficient since relating ODPs to criteria -or processes representing these criteria -is difficult. Therefore, for the ODPs advocated in this article, another approach has been adopted: i) manually setting subtype relations between ODPs or BPs represented as process types, and ii) using positive gradual pattern relations. Fig. 4 is the result.

These last relations represent rules of the form "the more X, the more Y". [START_REF] Ayouni | Mining closed gradual patterns[END_REF] gives a formalization for such relations. Arrows with dashed lines are positive gradual pattern relations. E.g., the dashed arrow from "keeping the types organized" to "avoiding undetected redundancies" can be read "the more `keeping the types organized´ is achieved, the more `avoiding undetected redundancies' is achieved´ ".

This last particular rule refers to the idea that was mentioned again two paragraphs ago and which could be rephrased as: "the more a KR (type or statement) has a `unique place´ [START_REF] Dromey | Scaleable Formalization of Imperfect Knowledge[END_REF] in a hierarchy of KRs, the less chances there are that another person will add an equivalent KR in another place". E.g., as opposed to subtype hierarchies, taxonomies relate objects (terms, documents, ...) with relations which are neither typed nor formal. Thus, people use these relations for representing subtypes, parts, instances, agents, etc. This leads to hierarchies that are difficult to search and that often have redundancies. When subtype partitions are used, this is far less the case. This is also far less the case when the hierarchy is automatically built based on the definition of each type. Like subtype relations, gradual pattern relations are typed and transitive. Hence, if used correctly, each KR can have a unique place [START_REF] Dromey | Scaleable Formalization of Imperfect Knowledge[END_REF] in the graph formed by these transitive relations. However, gradual pattern relations do not enable as many automatic checking possibilities as subtype partitions.

Given the explanations provided in the previous sections, the relations in Fig. 4 should now be understandable. The use of gradual pattern relations between ODPs or BPs is original. The direct setting of subtype relations between them also seems original.

"using RTs directly derived from CTs" "keeping the RT "using precise statements" hierarchy small" "using binary RTs" "avoiding undetected redundancies" "keeping "using binary "using types "using normalized RTs directly primitive organized" statements" derived "using (hence from CTs" process binary) "using "using types"

RTs precise and easytounderstand "using binary normalized statements" RTs directly "following the statements" derived from CTs, graphbased "using well related and especially role types reading easytocompare statements" or types of process" convention" Legend: same as in Fig. 3 except that arrows with dashed lines now represent positive gradual pattern relations; relations inherited via subtype relations are left implicit, e.g., like those inherited by "using precise and normalized statements". VI. CONCLUSION Knowledge sharing is difficult. It implies satisfying many criteria -and following various BPs -which, as Fig. 4 showed, are inter-related. To provide such BPs and ways to follow them, this article has focused on the idea of deriving RTs from CTs and has shown the relationships between this ODP to other ones for knowledge modeling and sharing. Some of these ODPs were already known, several were original.

This article also provided various kinds of ODPs. According to the categories of ODPC, these are architectural, logical, content and naming ODPs. However, given their inter-relations and the focus on derivation mechanisms, it is also true that this article focused on one ODP -the one named ABP -composed of simpler ODPs.

The ODPs we proposed are applied to -and supported by -the MSO which includes more than 75,000 categories and which is accessible and updatable online via the WebKB shared knowledge base server. Together, these resources and tools help people and automated agents create KRs that are more normalized, inter-related, comparable and understandable. Furthermore, the multi-source nature of the MSO would help applying the proposed content ODPs to other ones such as DOLCE+DnS-Ultralite.

Finally, the following of the proposed ODPs can easily be tested, e.g., via SPARQL queries on an ontology or, interactively, within WebKB. For example, it is easy to test if each RT is defined with respect to one CT. This makes these BPs usable as criteria for selecting ontologies. This work will be extended by relating knowledge sharing techniques, BPs and criteria, via specialization relations and gradual pattern relations. Negative gradual pattern relations -"the more X, the less Y" -will also be used. The focus will be on representing various approaches to knowledge sharing, e.g., those based on formal documents, those based on collaborative editing within a shared ontology server and those based on knowledge exchange between ontology servers. Thanks to their organization by specialization relations and gradual pattern relations, the various kinds of ways to share knowledge and their respective advantages and drawbacks should be clearer.

Fig. 1 .

 1 Fig. 1. Slightly adapted UML representation of a subtype hierarchy to compare the type Thing_usable_for_directly_deriving_a_binary_relation with other types.

Fig. 2 .

 2 Fig. 2. Subtype hierarchy of some relation types derived from subtypes of the concept type Thing_usable_for_directly_deriving_a_binary_relation.

Fig. 3 .

 3 Fig. 3. Examples of common types of relations from a process; most of them are thematic RTs.

Fig. 4 .

 4 Fig. 4. Supertype relations and gradual pattern relations between ABP (the BP advocated in this article; see the BP name in italic bold characters at the bottom left of the figure) and related BPs.

TABLE I .

 I EXAMPLES OF HOW TO DEFINE A GIVEN RT WITH RESPECT TO OTHER TYPES (THE RIFFLD PS NOTATION IS USED IN THE NONHIGHLIGHTED PARTS; VARIABLES BEGIN BY "?"; " :" MEANS "<=")

If you wish to (re-)use non-binary RTs, as in

	r__spatial_entity_between_3_other_ones ( Jack Joe John Mary )
	Exists ?X ( r__spatial_entity_between_2_other_ones (?X Joe John) )
	instead

of using binary RTs as in r__list_of_surrounding_entities

( Jack List( Joe John Mary ) ) Exists ?X ( r__list_of_surrounding_entities ( ?X List( Joe John ) ) )

then provide ways to translate the 1st ones into the 2nd ones, e.g., Forall

?A ?B ?C ?D ( r__list_of_surrounding_entities ( ?A List( ?B ?C ?D ) ) :-r__spatial_entity_between_3_other_ones ( ?A ?B ?C ?D ) )

since it is then much easier to make inferences, e.g., ?X = Jack and the above 3rd statement specializes (hence implies) the 4th The above approach also works for contextualizations, e.g.,

	r__list-of-surrounding-entities_at-time ( Jack Joe John D-Day )

can automatically be translated into the binary relation

	r__list_of_surrounding-entities ( Jack_at_D-Day List (Joe_at_D-Day John_at_D-Day) )

This cannot be specified in RIF PS but something similar can be:

	Forall ?A ?B ?C ?time_T (
	Exists ?A_at_time_T ?B_at_time_T ?C_at_time_T (
	And ( r__list_of_surrounding_entities (?A_at_time_T List (?B_at_time_T ?C_at_time_T) )
	r__extended_specialization (?A ?A_at_time_T) r__time ( ?A_at_time_T ?time_T )
	r__extended_specialization (?B ?B_at_time_T) r__time ( ?B_at_time_T ?time_T ) )
	:-r__list-of-surrounding-entities_at-time (?A ?B ?C ?time_T) ) )

Similarly, if you wish to use RTs representing types of processes, as in r__landing ( Joe Omaha_Beach D-Day ) r__defining (Joe Square) instead of using classic primitive binary RTs as in

	Exists ?landing ( And ( ?landing # landing	// "?i # ?t" <=> instanceOf (?i ?t)
	r__agent(?landing Joe)	r__place(?landing Omaha_Beach)
	r__time(?landing D-Day) ) )
	Exists ?defining ( And ( ?defining # defining	r__agent (? defining Joe)
	r__object (?defining "square") ) )

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Tian Long He

Philippe Karamian

Daniel Choï

A Fictitious Domain Method for Numerical Homogenization

Keywords: Finite Element Method, Homogenization, Fictitious Domain Method

Numerical Homogenization gives numerical approximations of the effective properties of a composite material. In the case of periodic composite, multi-scale modeling allows to consider only a Representative Volume Element (RVE). In this paper, we propose an original finite element method based on fictitious domain principle, in which the RVE is represented by a structured mesh and the inclusions are represented by independent and non matching meshes. The integral computations on inclusions meshes are substituted into the structured mesh of the RVE, with the help of a connection matrix.

Introduction

A composite material is an heterogeneous material, composed of at least two different constituents : a matrix, usually serving as a binder, and inclusions made of one or more materials. The inclusions are often called reinforcements as the aim of manufacturing composite materials is to combine the properties of the matrix/inclusions to produce material with different properties from the individual constituents. To efficiently model such materials, numerical homogenization techniques allows the computation of the effectives properties.

We can distinguish two kinds of numerical homogenization techniques : mean field methods and full field methods [START_REF] Valeriy | Micromechanics of Heterogeneous Materials Springer[END_REF][START_REF] Mura | Micromecanics of Defects in Solids[END_REF]. The mean field methods are analytic or semianalytic methods, such as the Mori-Tanaka method or the self consistent method [START_REF] Hatta | Equivalent inclusion method for steady state heat conduction in composites[END_REF][START_REF] Benveniste | A new approach to the application of Mori-Tanaka's theory in composite materials[END_REF]. They are used with an incomplete knowledge of the composite: only the volume fraction and the specific geometry of the inclusions are known. As they are based on the study of the Eschelby's tensor, see [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion and related problems[END_REF], they are usually restricted to spherical inclusions with low volume fraction. Full field methods consider more general composites, but require the exact knowledge of the geometry. Effective properties can be computed with multi-scale modelling [START_REF] Sanchez-Palencia | Introduction aux méthodes asymptotiques et à l'homogénéisation -Application à la mécanique des milieux continus[END_REF], such as the finite element method [START_REF] Debordes | Homogenization Computations in the Elastic or Plastic Range. Applications to Unidirectional Composite and Perforated Sheets Computational Mechanics[END_REF] or iteratives methods based on Fast Fourier Transfom technique [START_REF] Moulinec | A fast numerical method for computing the linear and nonlinear mechanical properties of composites[END_REF][START_REF] Moulinec | A FFT based-numerical method for computing the mechanical properties of composites from images of their microstucture[END_REF]. These methods are valid for a periodic composite material. The material period is then identified as the Representative Volume Element (RVE).

As the RVE becomes more complex, especially in the cases with very thin or tiny inclusions or inclusions coated with a thin pellicle, some difficulties arise : to accurately model such RVEs, we need a huge number of meshes (especially in 3D) or a high resolution image. Although some numerical advances allow the study of very high resolution 3D images, the need to efficiently compute the effective properties of more and more complex RVE subsists.

In the case of finite element methods, one difficulty, due to the heterogenity, is to accurately approximate integrals defining the energy bilinear form involved in the weak formulation of the homogenization modeling see [START_REF] Sanchez-Palencia | Introduction aux méthodes asymptotiques et à l'homogénéisation -Application à la mécanique des milieux continus[END_REF][START_REF] Debordes | Homogenization Computations in the Elastic or Plastic Range. Applications to Unidirectional Composite and Perforated Sheets Computational Mechanics[END_REF][START_REF] Bensoussan | Asymptotic analysis for periodic structures[END_REF]. It usually requires a conforming mesh but the building such a conforming and periodic mesh with complex geometries is not trivial, see for instance [START_REF] Schneider | Automatic three-dimensional geometry and mesh generation of periodic representative volume elements for matrixinclusion composites[END_REF].

To circumvent the meshing difficulty, several approaches have been considered, such as eXtended Finite Element Method [START_REF] Sukumar | Modeling holes and inclusions by level sets in the extended finite-element method[END_REF][START_REF] Moës | A finite element method for crack growth without remeshing[END_REF][START_REF] Legrain | Prediction of apparent properties with uncertain material parameters using high-order fictitious domain methods and PGD model reduction[END_REF] (XFEM), or Finite Cell Methods [START_REF] Düster | Ernst Rank Numerical homogenization of heterogeneous and cellular materials utilizing the Finite Cell Method[END_REF] (FCM). The XFEM is combined with the Level Set Method to define the inclusions and enriched shape functions [START_REF] Yvonnet | An XFEM/level set approach to modelling surface/interface effects and to computing the size-dependent effective properties of nanocomposites[END_REF]. The FCM considers only one structured mesh the elements of which are then called cells. The FCM uses cell subdivisions to ensure a better approximation of the integrals, see [START_REF] Düster | Ernst Rank Numerical homogenization of heterogeneous and cellular materials utilizing the Finite Cell Method[END_REF][START_REF] Legrain | Prediction of apparent properties with uncertain material parameters using high-order fictitious domain methods and PGD model reduction[END_REF].

Although considering periodic RVE based on multi-scale analysis rather than numerical homogenization with Kinetic or Stress Uniform Boundary Condition, the method presented in this paper has the same goal in the accurate approximation of integrals defining the energy bilinear form, but relies on computations with independent meshes based a Fictitious Domain principle: We use one structured mesh, for the entire RVE, including the matrix and the inclusions and non-structured independent meshes for the inclusions. We then compute integrals defined both on the whole RVE and integral defined on the inclusions. The inclusion meshes will be related to the structured mesh through a Connection Matrix. The Fictitious Domain Method for Homogenization (FDMH), proposed in this paper, can be directly applied to inclusions of any kind of shape or geometry, as long as a mesh is available.

The paper is organized as follows, after this short introduction we recall the periodic homogenization theory, based on a multiscale modeling and asymptotic developement. This periodic homogenization lead to variational problems that can be solved with the help of a classical finite element method. In section 3, we then introduce our new method based on fictitious domain principle to represent the inclusions. We describe the construction of a connection matrix which allows to connect the meshes and compute the energy forms involved in the periodic homogenization technique. In section 4, we show some numerical tests with different inclusion geometries, in order to evaluate and validate our method. We discuss the performance of our fictitious domain method for homogenization.

Notations

In this paper, scalar values are typed in italic whereas the vector values are typed in bold face. Let us consider a εP periodic material. P constitutes the Representative Volume Element (RVE), ε denotes the characteristic length of the RVE. To fix the idea, we set the RVE P = [0, 1] 3 . The multiscale modeling consider macrocospic and local space variables. To the macroscopic space variables x ∈ R 3 , we associate the local (microscopic) space variables:

y = 1 ε x ∈ P.
The divergence and gradient differential operators are respectively denoted as

div = div x + 1 ε div y , ∇ = ∇ x + 1 ε ∇ y ,
where div x and ∇ x (resp. div y and ∇ y ) denote the divergence and the gradient differential operator with respect to the macroscopic variables x (resp. the local variables y).

Periodic Homogenization

In order to present our original method, we briefly recall here the periodic homogenization theory [START_REF] Bensoussan | Asymptotic analysis for periodic structures[END_REF][START_REF] Sanchez-Palencia | Introduction aux méthodes asymptotiques et à l'homogénéisation -Application à la mécanique des milieux continus[END_REF] and its numerical implementation with the finite element method [START_REF] Debordes | Homogenization Computations in the Elastic or Plastic Range. Applications to Unidirectional Composite and Perforated Sheets Computational Mechanics[END_REF]. For the sake of simplicity, we present the theory and our method in the case of thermal properties. With a periodic heterogeneous media, the thermal constitutive law writes:

q = Λ∇u, (1)
where u denotes the temperature and q the heat flow, the conductivity tensor Λ is εP periodic.

Under the action of external heat source f , the thermal equilibrium writes:

div q + f = 0. ( 2
)
We set the asymptotic expansion:

u = u 0 (x) + εu 1 (y) + . . . (3)
∇u = e(u) = e 0 (u) + ε e 1 (u) + . . .

q = q 0 + εq 1 + . . .

As we develop:

∇u = (∇ x + 1 ε ∇ y )(u 0 + εu 1 + ε 2 u 2 + . . . ) = 1 ε ∇ y u 0 + (∇ x u 0 + ∇ y u 1 ) + ε(∇ x u 1 + ∇ y u 2 ) + . . . ,
we obtain by identification:

e 0 (u) = ∇ x u 0 + ∇ y u 1 e 1 (u) = ∇ x u 1 + ∇ y u 2 q 0 =Λ(∇ x u 0 + ∇ y u 1 ).
Writing asymptotic expansion of the heat flow q in the equilibrium equation (2) gives, by identification, two distinct equations, a local equation and a macroscopic equation, see [START_REF] Sanchez-Palencia | Introduction aux méthodes asymptotiques et à l'homogénéisation -Application à la mécanique des milieux continus[END_REF]. It follows that u 0 depends only on x and u 1 must be P-periodic. The local equation writes : div y (q 0 ) = 0 in P,

We write [START_REF] Geuzaine | Gmsh: a three-dimensional finite element mesh generator with built-in pre-and post-processing facilities[END_REF] in a weak or variational form :

P q 0 ∇ y (u * ) = 0 ∀u * ∈ H 1 per (P) (7)
Where H 1 per (P) is the P-periodic Sobolev space H 1 . Let E 0 = ∇ x u 0 , then q 0 = Λ(E 0 + ∇ y u 1 ).

Averaging q 0 on the RVE P, we then have :

q 0 = P Λ(E 0 + ∇ y u 1 ) = Λ h ∇u 0 . (8)
Λ h denotes the homogenized conductivity tensor.

Furthermore, with [START_REF] Glowinski | Periaux A fictitious domain method for Dirichlet problem and applications[END_REF], ∀E * ∈ R 3 and ∀u * ∈ H 1 per (P) :

P q 0 (E * + ∇ y u * ) = P q 0 E * + P q 0 ∇ y u * = q 0 E * = Λ h ∇u 0 .E *
In other words, we obtain the variational problems:

P Λ(E 0 + ∇ y u 1 ).(E * + ∇ y u * ) = Λ h E 0 .E * ∀u * ∈ H 1 per (P), ∀E * ∈ R 3 . (9)
The resolution of the three problems (9) as Λ h E 0 span R 3 , gives the homogenized conductivity tensor Λ h . The variational problems [START_REF] Hill | A self-consistent mechanics of composite materials[END_REF] are well-posed [START_REF] Sanchez-Palencia | Introduction aux méthodes asymptotiques et à l'homogénéisation -Application à la mécanique des milieux continus[END_REF] and their solutions can be approximated by the finite element method [START_REF] Debordes | Homogenization Computations in the Elastic or Plastic Range. Applications to Unidirectional Composite and Perforated Sheets Computational Mechanics[END_REF].

Fictitious Domain Method

The fictitious domain method is a technique used to solve elliptic boundary value problems in domain with complex boundaries or interfaces for which the meshing or the repetition of the meshing can induces difficulties [START_REF] Glowinski | Periaux A fictitious domain method for Dirichlet problem and applications[END_REF]. To remedy this, distinct meshes are considered, one main mesh, usually structured, and meshes of the interfaces or inclusions. The fictitious domain method is then equivalent to a quadratic optimization problem with a linear constraint : the relations between the meshes or the interface or boundary conditions are taken in account with the help of Lagrange multipliers, see [START_REF] Glowinski | Periaux A fictitious domain method for Dirichlet problem and applications[END_REF]. The method presented here differs since we shall not introduce Lagrange multipliers and we employ a true mesh of the inclusion instead of refining the computation within the element (or cell) that are being cut by the boundary of an inclusion see [START_REF] Düster | Ernst Rank Numerical homogenization of heterogeneous and cellular materials utilizing the Finite Cell Method[END_REF][START_REF] Legrain | Prediction of apparent properties with uncertain material parameters using high-order fictitious domain methods and PGD model reduction[END_REF]. Actually, the use of isoparametric element on a structured mesh of the RVE allows us to connect the different meshes in a very simple way with the help of a Connection Matrix.

Energy form

We present our method in the case of thermal properties, the extension for other cases such as linear elasticity is straightforward.

From the variational problem ( 9) let us consider the quadratic energy functional:

J(u, E) = P Λ(E + ∇ y u).(E + ∇ y u). (10)
To fix the idea, we suppose the RVE P to be constituted by two homogeneous and isotropic media : the matrix P \ S and one inclusion S . The conductivity matrix writes:

Λ(y) =      Λ m in P\S Λ s in S .
We have :

J(u, E) = P\S Λ m (E + ∇ y u).(E + ∇ y u) + S Λ s (E + ∇ y u).(E + ∇ y u). (11)
We then rewrite :

J(u, E) = P Λ m (E + ∇ y u).(E + ∇ y u) J m + S (Λ s -Λ m )(E + ∇ y u).(E + ∇ y u) J s (12)
The main idea of the FDMH consists in defining two distinct and a priori incompatible and independent meshes representing the RVE P and the inclusion S . The consideration of these meshes allow independent numerical computations of J m and J s : Along the structured mesh of the entire domain P, let us define the interpolation space V h of dimension n such that any u h ∈ V h can be represented by û ∈ R n . The functional J m can then be defined with the help of a (n + 3) × (n + 3) matrix K m :

J m (u h , E) = P Λ m (E + ∇ y u h ).(E + ∇ y u h ) = û E K m û E . ( 13
)
On another hand, let us consider a conforming mesh of the inclusion S and let us define the interpolation space W l . Let p be the dimension of W l , any v l ∈ W l can be represented by a vector v ∈ R p . In order to simplify the presentation, the meshes will be denoted respectively as P and S . The functional J s can be defined in W l with a (p + 3) × (p + 3) matrix K s :

J s (v l , E) = S (Λ s -Λ m )(E + ∇ y v l ).(E + ∇ y v l ) = v E K s v E . ( 14
)
Both the matrices K m and K s can be computed using standard finite element procedures.

A Connection Matrix

As we defined the matrices K m and K s , the main difficulty is to connect the Degrees of Freedom (DOFs) v of the inclusion S to the DOFs û of the structured mesh P. With isoparametric elements, this can easily be achieved: For each component v i of v, we have v i = v(N i ) where N i is a node of S . Let E l , be an element of the RVE P in which the node N i is included, see Figure 1. Let x, ŷ, Figure 1: A node N i from a triangular mesh of an inclusion in the quadrangular element E l of the RVE ẑ be the coordinates of the nodes constituting the element E l . With an isoparametric element, we usually define a reference element and its corresponding shape functions Φ. Let (x, y, z) be the coordinates of the node N i , and let (r, s, t) be its coordinates in the reference element associated to E l . By definition, we have :

x = Φ(r, s, t) x y = Φ(r, s, t) ŷ z = Φ(r, s, t) ẑ
Let v i be the value of v l at node N i :

v i = v(x, y, z) (15)
As we work with isoparametric elements, we also have :

v(x, y, z) = Φ(r, s, t) û, (16)
In other words, we have the relation from the value v i at a node of the inclusion mesh to the values û on the mesh of the element E l :

v i = Φ(r, s, t) û. ( 17
)
We fill up (completing with zeroes) the relation [START_REF] Moulinec | A FFT based-numerical method for computing the mechanical properties of composites from images of their microstucture[END_REF] to all the values of u in the mesh of the RVE P and let us denote with C i the 1 × n line matrix such that :

v i = v(x, y, z) = Φ(r, s, t) û = C i û,
(û abusively represents the nodal values of u in an element E l but also in the whole RVE P). Considering every nodes of S , we finally obtain a connection matrix C constructed line by line, which connects v, the DOF associated to the mesh of the inclusions, to û, the dof associated to the mesh of the RVE :

v = C û. ( 18
)
We are then able to rewrite the energy quadratic form computed on the inclusion J s :

J s = v E K s v E = C û E K s C û E . (19)
Finally, we obtain the main energy quadratic form defined only on û :

J(u, E) = û E K m + C 0 0 1 K s C 0 0 1 û E . (20)
In other words, with the matrix defined in [START_REF] Sanchez-Palencia | Introduction aux méthodes asymptotiques et à l'homogénéisation -Application à la mécanique des milieux continus[END_REF], we are applying a finite element method to the problems of ( 9) with the interpolation space defined on the structured mesh of the RVE P.

In case of multiple inclusions, the procedure is repeated independently for each inclusion; all computations can be performed in parallel. The principle is very general and flexible, the inclusion meshes do not have to be of identical types, we can mix different types of mesh.

A priori, for unstructured meshes, the determination of the connection matrix C is not trivial. This difficulty is completely relieved, in the case of structured mesh for the RVE, with the use of isoparametric elements. In such cases, a simple formula [START_REF] Moulinec | A FFT based-numerical method for computing the mechanical properties of composites from images of their microstucture[END_REF] gives every components of the matrix C based on the definition of the shape functions of the reference element. Furthermore, with a regular structured mesh of P the periodic boundary condition are easily imposed by a simple substitution procedure.

It should be noted that since the material are heterogeneous, the discontinuity of the constitutive law implies that our FDMH solution, as it is not enriched, does not possess the correct weak continuity at the interfaces, except in the case where the distinct meshes are matching, see the test with a cubic inclusion in section 4.2. In the general case (non matching meshes) our method can be seen as conforming the discontinuity but for an approximated inclusion geometry. The approximation of the geometry is equivalent to a pixelization of the inclusion. Thus, the convergence of the method is ensured since a finer RVE mesh induces a better approximation of the inclusion.

Numerical tests

In order to validate the FDMH, we present here some numerical tests. We study thermal and linear elastic cases. For the sake of simplicity, we consider isotropic constitutive laws. We define a contrast parameter as the ratio between the characteristic coefficient of inclusions and that of RVE :

• In the thermal case c thermal = λ inc λ rve
• In the linear elasticity case

c elastic = E inc E rve
where λ inc and E inc (resp. λ rve and E rve ) are the conductivity and the Young's modulus of the inclusion (resp. the matrix). In all cases presented in this paper, we chose to set the contrasts c thermal and c elastic to 100. We consider periodic RVE defined with inclusions of elementary geometry such as sphere, cube, and ellipsoid, To fix the ideas, we chose the 8-nodes hexaedric element (cub8) both for the inclusions and the RVE P, see Figure 2. The structured mesh of the RVE is characterized by its resolution. A resolution n corresponds to a n × n × n subdivision in all three space dimension. For instance, a resolution 10, represents a 10 × 10 × 10, 8-nodes hexaedric mesh of the RVE, which contains 1000 elements, 1331 nodes, 1334 DOFs in the thermal case, and 3999 DOFs in the elasticity case. In the following, the RVE mesh resolution shall be denoted as N rve . Associated to the resolution, we define the parameter h rve = 1/N rve , representing the characteristic length of an element of the mesh.

Besides the RVE resolution N rve , we define another parameter N inc to set the element size of the mesh of the inclusions. As h rve and h inc are respectively the characteristic length of the mesh of the RVE and the inclusion, we define the ratio η as :

η = h rve h inc . (21)
A value of η larger than 1 means a finer mesh of the inclusion than the RVE's. A serial of numerical tests have been performed to evaluate the influence of η on the results. We found out that, as we increase the value of η, the relative error is decreasing to 0. However, the evolution of relative error becomes negligible from the point η = 1, we illustrate this in Figure 3 in the case of spherical inclusion. On the other hand, a large value of η means a finer inclusion mesh and thus a longer time to complete the computations.

As the geometry can be complex, we also imposed a minimal resolution for the inclusions meshes. Hence for low RVE resolution, the parameter η can actually be greater than 1, in order to ensure a correct approximation of the geometry. Moreover, the meshing of the inclusion must be refined enough compared to the RVE meshing : a finer inclusion mesh avoids holes in the connections. This is a standard requirement in Fictitious Domain Methods. In other words, the mesh of the inclusion must be sufficiently refined compared to the resolution of the RVE, but need not to be too fine since that does not improve the approximation beyond a certain value. In all of our following numerical studies, the ratio η is set to the "optimal" value η = 1. A sphere is defined by the coordinates of its center and its diameter d. In this example, we set d = 0.6, the volume fraction is thus ρ = 0.1131, see Figure 4. Since ρ is small enough, the analytical Mori-Tanaka model, based on the Eschelby's tensor is used as the reference solution [START_REF] Hatta | Equivalent inclusion method for steady state heat conduction in composites[END_REF], both in thermal and linear elasticity cases, see [START_REF] Stransky | Tanaka based estimates of effective thermal conductivity of various engineering materials[END_REF][START_REF] Mori | Average stress in matrix and average elastic energy of materials with misfitting inclusions[END_REF]. In Figure 5, we draw the relative error of homogenized conductivity coefficient λ between the FDMH and the reference solution, the contrast c thermal = 100. Linear convergence is observed with respect to the RVE resolution. With a resolution N rve = 100 for the RVE, the relative error is about 1%. In the linear elasticity case, the contrast c elastic is also set to 100. The Poisson's ratio is fixed as 1 3 for both the inclusion and the matrix. As the composite material is isotropic, we use the bulk modulus k and the shear modulus µ to represent the homogenized results. In Figure 6, the relative error of the homogenized elastic coefficient k and µ are plotted versus the RVE resolution. We notice a linear convergence with respect to the RVE resolution for both homogenized coefficients k and µ, which is similar to the thermal case.

We end this section with the case of one spherical inclusion of various size, the contrasts remain fixed at c thermal = c elastic = 100. We plot the comparative results of the normalized conductivity (thermal case), see Fig. 7, and normalized bulk and shear modulii (elasticity case), see Fig. 8 between the Mori-Tanaka analytic model, the FDMH and the FFT method with different resolutions (50 and 100). We note that for greater volumic fraction the results given by the FDMH method deviate from analytical references (Mori-Tanaka and self-Consistent Model) but remain close to the results given by the FFT method.

Cubic inclusion

In this section, we consider a cubic inclusion. A cube of side l = 0.5 is placed at the center of RVE, as shown in Figure 9. The volume fraction of the inclusion is ρ = 0.125.

The iterative methods based on the FFT (Fast Fourier Transform) give in this case an almost exact solution since the geometry is exactly reproduced by the voxelisation, see [START_REF] Moulinec | A fast numerical method for computing the linear and nonlinear mechanical properties of composites[END_REF]. For example, if we consider a centered cubic inclusion of side 0.5, the voxelizations should be made in multiple of 4 subdivisions in order to match the geometry. In such cases, the FFT calculation (with a voxelization resolution 100) is then considered as a reference solution. We emphasize also that in such resolution, the FDMH will coincide exactly with a standard Finite Element Method (FEM) with conform and compatible mesh for the inclusion, see Figure 10, where the inclusion is represented in green and RVE in yellow. In Figure 11, we have plotted the relative error of the homogenized thermal coefficient λ against the RVE resolution. In both matching and non-matching case, we notice a linear convergence. A noticeable difference in the order of magnitude for the relative errors can be observed in these two cases. This difference corresponds to the error made by the FDMH compared to a pure finite element calculation.

In Figure 12, the relative error of homogenized coefficient k and µ is plotted versus the RVE resolution. The same observation is found as in the thermal case, with a similar linear convergence.

Comparison between matching and non-matching meshes allows us to measure the error made by the FDMH compared to a conventional finite element method. We propose here a specific way to measure this error: in a fixed RVE mesh, from a matching position of two meshes, we move the cubic inclusion slightly in different directions (following x/y, in the xy plane or in the xyz space), until another matching position. In Figure 13, we plot the FDMH / FEM comparison of the homogenized coefficients λ against the relative distance, which is the ratio between the moved distance and h rve , the characteristic length of the mesh of the RVE. Compared to FEM, the error made by our method is null in the matching position, and reach its maximum with a relative distance equal to 0.5. For a resolution 100, the error of the method might be significant but is less than 2%.

Ellipsoidal inclusion

Let us consider an ellipsoidal inclusion. It is defined by its center coordinates and its principal axis or it can be defined as a volume the boundary of which can defined by a quadric :

x 2 a + y 2 b + z 2 c = 1.
We consider here an ellipsoid centered in the middle of the RVE P with a = 0. 13: FDMH / FEM comparison between matching and non-matching meshes, for RVE with a cubic inclusion, thermal case with such RVE are not isotropic, especially if the contrast is large. Thus, for the thermal case, we analyze the diagonal terms of the homogenized conductivity tensor Λ. For the elastic case, since we have a transversely isotropic material in the plane (x, y) and in the longitudinal axis (z), the computation of the compliance tensor S in the Bechterew base [1], makes it possible to identify Young's modulus E 1 , E 3 and shear modulus G 31 . Since a Mori-Tanaka solution is available [START_REF] Benveniste | A new approach to the application of Mori-Tanaka's theory in composite materials[END_REF], we use it as a reference solution. The linear convergence can be found in both cases for the coefficients in the plane (x, y), see Figure 15. The non-linear convergence of the coefficient in the longitudinal axis (z) is possibly caused by the inaccuracy of the referenced Mori-Tanaka solution, in the case of which, the inclusion is too elongated in the longitudinal direction compared to the size of RVE. With a RVE resolution of 100 3 , the relative error is in order of magnitude 10 -2 , see Figure 15

Multiple ellipsoidal inclusions

To illustrate a case of mutiple inclusions, we consider another case of eight ellipsoidal inclusions with the same size a = 0.05, b = 0.05, c = 0.15, which are randomly distributed, see Figure 16. The volume fraction is ρ = 0.01257. We recall that the FDMH procedure, see section 3, is repeated for each inclusion. As we have several inclusions in this case, the reference solution is given by the analytical Self-Consistent Model (SCM) [START_REF] Hill | A self-consistent mechanics of composite materials[END_REF], based on the Eschelby's tensor [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion and related problems[END_REF].

We also notice a linear convergence for both thermal and elastic cases, see Figure 17. This observation proves that FDMH responds properly to the cases with numbers of inclusions.

Pellicle / Hollow sphere inclusion

We consider a pellicle sphere inclusion, centered with d = 0.6, as shown in Figure 18. The reference solution is given by a three phases Mori-Tanaka model based on the Eshelby inclusions [START_REF] Quang | Size-dependent Eshelby tensor fields and effective conductivity of composites made of anisotropic phases with highly conducting imperfect interfaces[END_REF]. Numerical tests are performed with different relative In Figure 19, the homogenized coefficients λ are normalized and plotted versus the relative thickness for both FDMH and Mori-Tanaka solution. The results computed by these two methods are close. With a relative thickness equal to 0.01, the relative difference is about 1%. We conclude that the FDMH can be used to compute effective properties of periodic material with very thin pellicle inclusions. We also note that for the inclusions computation, a 2 dimensional model can be used. This ability to consider very thin inclusions is an interesting advantage of the FDMH over other numerical homogenization methods.

Computing resources

The computations are performed by a finite element library code, named CFEM, developed by the authors, in Python/Fortran. We use the PETSc1 library to solve the linear systems. The meshes are generated with the Gmsh software [START_REF] Geuzaine | Gmsh: a three-dimensional finite element mesh generator with built-in pre-and post-processing facilities[END_REF].

In some cases, as the RVE resolution may involves millions of DOFs, we used Myria, a HPC cluster located in CRIANN 2 . However, all of the computations presented in this paper can be performed on personal computers (We used a 2014 macbook pro laptop or a laptop PC).

Conclusion

We have presented an original alternative method to evaluate the effective properties of periodic heterogeneous materials, using a fictitious domain principle. With the help of a connection matrix between the distinct meshes of inclusions and RVE, we are able to implement a finite element in the framework of a structured mesh of the RVE.

Numerical tests, with inclusions of elementary geometry such as sphere, cube and ellipsoid, have shown a linear convergence of relative errors with respect to reference solutions. The convergence rates are observed as expected, regarding the use of linear interpolation.

One interesting advantage of the FDMH method presented in this paper is the total flexibility concerning the inclusions meshes. Complex geometries of any kind can be considered, as long as a mesh is available. We have shown an example with very thin spherical pellicle, but we can also imagine a case of a very high resolution 3d image of a composite material sample, obtained through imagery techniques, such as tomography. We can obtain the (fine) meshes of the inclusions by segmentation techniques and then consider a coarser mesh of the sample, making the computation possible.

Figure 2 :

 2 Figure 2: Spherical (left) and cubic (right) inclusions in hexaedric mesh

Figure 3 :

 3 Figure 3: Relative error of the homogenized bulk modulus against the value of η for different RVE resolutions, in the case of a spherical inclusion

Figure 4 :

 4 Figure 4: RVE with one spherical inclusion of diameter d = 0.6, volume fraction ρ = 0.1131

Figure 5 :

 5 Figure 5: Relative error of computed homogenized conductivity coefficient λ for one spherical inclusion

Figure 6 :

 6 Figure 6: Relative error of computed homogenized elasticity coefficient k and µ for one spherical inclusion

Figure 8 :

 8 Figure 8: Comparison of normalized effective elastic properties computed by Mori-Tanaka analytic Model (MT), Self Consistent semi-analytic model (SCM), FDMH, FFT at resolution 50 and 100 in the case of one spherical inclusion of different size; bulk modulus (left) and shear modulus (right).

Figure 9 :

 9 Figure 9: RVE with one cubic inclusion of side l = 0.5, volume fraction ρ = 0.125

Figure 10 :

 10 Figure 10: A cubic inclusion with matching (left, N RV E = 4) and non-matching (right, N RV E = 5) meshes.

Figure 11 :Figure 12 :

 1112 Figure 11: Relative error of the homogenized coefficients λ for one cubic inclusion, the mesh of inclusion and RVE are matching (left) or not matching (right)

  15, b = 0.15, c = 0.4, see in Figure 14. The volume fraction is ρ = 0.0377. Material constituted

Figure

  Figure 13: FDMH / FEM comparison between matching and non-matching meshes, for RVE with a cubic inclusion, thermal case

Figure 14 :

 14 Figure 14: RVE with one ellipsoidal inclusion, a = 0.15, b = 0.15, c = 0.4, volume fraction ρ = 0.0377

  .

Figure 15 :

 15 Figure 15: Relative error of the homogenized conductivity tensor (left) and stiffness coefficients (right) for one ellipsoidal inclusion case

Figure 16 :

 16 Figure 16: RVE with eight ellipsoidal inclusions, a = 0.05, b = 0.05, c = 0.15, volume fraction ρ = 0.01257

Figure 17 :

 17 Figure 17: Relative error of the homogenized conductivity tensor (left) and stiffness coefficients (right) for eight ellipsoidal inclusions case

Figure 18 :

 18 Figure 18: RVE with one pellicle spherical inclusion with d = 0.6, relative thickness = 0.01

Figure 19 :

 19 Figure 19: Relative error of computed homogenized conductivity coefficient λ against the relative thickness for one pellicle sphere inclusion

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