Solène Eholié

Mike-Donald Tapi-Nzali

Sandra Bringay

Clement Jonquet

MuEVo, a breast cancer Consumer Health Vocabulary built out of web forums

Keywords: Semantic Web, Biomedical Ontologies, Standard Terminologies, Consumer Health Vocabulary, SKOS, Web Forums, Breast Cancer, BioPortal

Semantically analyze patient-generated text from a biomedical perspective is challenging because of the vocabulary gap between patients and health professionals. The medical expertise and vocabulary is well formalized in standards terminologies and ontologies, which enable semantic analysis of expertgenerated text; however resources which formalize the vocabulary of health consumers (patients and their family, laypersons in general) remain scarce. The situation is even worse if one is interested in another language than English. In previous studies, we attempted to produce a French preliminary Consumer Health Vocabulary (CHV) by mining the language used within online public forums & Facebook groups about breast cancer. In this work, we show our effort to concretely align the vocabulary produced to standard terminologies and to represent its content (terms & mappings) using semantic web languages such as RDF, SKOS and PROV. We used a sample of 173 relations built around 64 expert concepts which have been automatically (89%) or manually (11%) aligned to standard biomedical terminologies, in our case: MeSH, MedDRA and SNOMEDint. The resulting vocabulary, called MuEVo (Multi-Expertise Vocabulary) and the mappings are publicly available in the SIFR BioPortal French biomedical ontology repository.

Introduction

Over the last years, the Web has taken a great importance in the way health consumers access health information. In France, according to a 2013 TNS Sofres survey, 4 one in two people has already used the Web to search for medical information or discuss health topics. Health consumers, i.e., patients, are generally laypersons who do not have the technical or scientific expertise and hence expressions and vocabulary [START_REF] Mccray | Terminology issues in user access to web-based medical information[END_REF]. Often, they seek a first diagnosis, more precise information about a given disease or testimonies of other people facing the same health issues. According to Google's blog, 5one percent, so millions, of searches on Google online search engine are symptomrelated. However, when laypersons express their queries or discuss in social media, they use a vocabulary different from the one of health-care professionals. With the explosion of Web 2.0 and social medias, doctors have definitively realized the enormous potential of data generated by patients [START_REF] Breton | Epimining: Using Web News for Influenza Surveillance[END_REF].

In fact, expert language (terms, wording and expressions) is quite well captured and formalized thanks to miscellaneous semantic resources such as standard terminologies such as MeSH (Medical Subject Headings) or any other terminology from the Unified Medical Language System (UMLS) or ontologies such as the one in the OBO Foundry or the NCBO BioPortal [START_REF] Noy | BioPortal: ontologies and integrated data resources at the click of a mouse[END_REF]. However, capturing the language actually used by patients and formalizing it in to a Consumer Health Vocabulary (CHV) remains a research issue. Laypersons use abbreviations, misspellings, neologisms or existing words that are diverted from their standard professional use. Hence, the classic biomedical naturallanguages-processing-resources do not apply easily to analyze patient-generated text. Many researchers have been working to reduce this vocabulary gap between laypersons and health care professionals by identifying CHV constituents and/or mapping them to their equivalents in the standard biomedical semantic resources [START_REF] Zeng | Term identification methods for consumer health vocabulary development[END_REF][START_REF] Jiang | Discovering consumer health expressions from consumercontributed content[END_REF]. However, this effort does not often result in reusable open access resources. Indeed, one of the only freely available CHV is the (English) Open-Access and Collaborative CHV (itself included in the UMLS Metathesaurus) that was developed by Univ. of Utah and recently updated by mining social network data [START_REF] Doing-Harris | Computer-assisted update of a consumer health vocabulary through mining of social network data[END_REF].

Despite of the lack of publicly available CHV, patients express themselves online more and more every day, for instance on social medias such as web forums. According to a 2011 Health On the Net Foundation survey [START_REF] Pletneva | How Do General Public Search Online Health Information? Results of survey[END_REF], the Web has become the second source of information for patients after consultations with a doctor. 24% of the population uses the Web to find health information at least once per day (and up to 6 times per day) and 25% at least several times per week. While maintaining anonymity, social media allow them to freely discuss with other users, and also with health professionals. They discuss about their medical results and their treatment options, but they also receive moral support. In France, online forums such as Doctissimo.fr (general health) or Lesimpatientes.com (breast cancer) are very successful. Therefore, online social medias are very relevant data sources to help to build CHV. Semantically representing CHVs' content and using them inside forum applications will enhance the patient's access to information by connecting the formal medical expertise to the actual content of the forums, inside the forums. It would also enable to process semantically patient-generated text. For instance, topics discussed will be more easily mined in order to identify what are the principal concerns of the patients [START_REF] Opitz | Breast cancer and quality of life: medical information extraction from health forums[END_REF]; forum providers would be able to connect their users to reference data resources that are indexed with standard medical terminologies but that are targeted for patients e.g., MedLinePlus.

In this paper, using patient-generated text from breast cancer health forums, we will present elements of responses and concrete results to this research question. Building on the previous results presented in [START_REF] Tapi Nzali | Construction d'un vocabulaire patient/médecin dédié au cancer du sein à partir des médias sociaux[END_REF] where we focused on the methodology to extract a preliminary CHV out of forum patient posts, we present is this paper a concrete machine-readable formalization of the extracted vocabulary, the provenance information and the alignment to standard terminologies, using the semantic Web languages: RDF, SKOS and PROV. We focused on breast cancer and French language, but our model is generalizable to other domain or language. As a result we have produced a CHV, called MuEVo (Multi-Expertise Vocabulary) of 64 concepts and 173 lay-expert relations defined in SKOS that has been automatically (89%) or manually (11%) aligned to standard biomedical terminologies, in our case: MeSH, MedDRA and SNOMEDint. Although the size of the current vocabulary is quite small, this is the result of an automatic process that will be reproduced on other datasets to augment it. This paper mainly focuses on the representation of this CHV which is independent from the size.

The rest of the article is organized as follows: we first present the related work and background, second, we propose a SKOS/RDF formalization of a CHV, third we expose the methods we used to map this resource to standard biomedical terminologies. Finally, we present experimental results of the mapping step before concluding.

2 Background and related work

Characterization of a Consumer Health Vocabulary

A Consumer Health Vocabulary (CHV) is a set of terms preferred and used by laypersons to describe medical concepts like symptoms or diseases for examples. See [START_REF] Tse | Exploring medical expressions used by consumers and the media: an emerging view of consumer health vocabularies[END_REF] for a more conceptual characterization of CHV constituents. The research effort to acquire CHV terms and map them to expert terms has been important this last decade. In fact, CHVs are a key element to reduce the communication gap between laypersons and health care professionals. The existence of the gap has been enlightened namely by [START_REF] Mccray | Terminology issues in user access to web-based medical information[END_REF] while studying details of the queries performed by users of the US National Library of Medicine online services. They observed that visitors used a lot of misspelled forms and abbreviations. In their experiments the authors found 84% of query terms would not match directly to UMLS although a deeper analysis showed that 30% of those mismatches could manually be identified. Other works include: [START_REF] Patrick | Evaluation of controlled vocabulary resources for development of a consumer entry vocabulary for diabetes[END_REF] that used email questions. [START_REF] Vydiswaran | Mining consumer health vocabulary from community-generated text[END_REF] used Wikipedia. [START_REF] Doing-Harris | Computer-assisted update of a consumer health vocabulary through mining of social network data[END_REF] that used Patientslikeme.com forum data. Beyond this lexical gap, there is, in part, a problem of comprehension of the medical jargon [START_REF] Patrick | Evaluation of controlled vocabulary resources for development of a consumer entry vocabulary for diabetes[END_REF] or use of popular expressions. For instance, a doctor will talk about a malignant neoplasm with a colleague, to describe the cancer of a patient, whereas the patient will exchange on forums about his/her crab. From this observation, the community raised the need to build lexical resources to mediate between the two worlds. Such a resource should include misspellings, incomplete terms, specific synonyms. Attempts to bridge the gap have been motivated by essentially three reasons: (i) to enable search of professional content by laypersons and vice versa [START_REF] Kogan | Problems and challenges in patient information retrieval: a descriptive study[END_REF], (ii) to vulgarize professional content in a consumer-friendly terminology [START_REF] Zeng | Exploring and developing consumer health vocabularies[END_REF], (iii) to analyze semantically the content produced by laypersons [START_REF] Opitz | Breast cancer and quality of life: medical information extraction from health forums[END_REF]. From the related work, two challenges clearly appear:

-One is to identify the constituents of CHV (that is, to build a controlled vocabulary). For instance, we can cite [START_REF] Zeng | Term identification methods for consumer health vocabulary development[END_REF] that mined MedlinePlus queries logs to extract frequent n-grams not in UMLS with frequency greater than 50. Then they validated 753 terms out of 7967 reviewed. -The other is to explicitly map the constituents to expert vocabulary with the use of lexical or semantic tools. For instance, we can cite [START_REF] Keselman | Consumer health concepts that do not map to the umls: where do they fit?[END_REF] that mapped consumer health concepts extracted from health-focused bulletin boards to UMLS concepts (both manually and using MetaMap).

These are the challenges we attempted to tackle in previous work [START_REF] Tapi Nzali | Construction d'un vocabulaire patient/médecin dédié au cancer du sein à partir des médias sociaux[END_REF] where we proposed a semi-automatic hybrid approach in which the mapping task serves as validation step in the CHV acquisition process. The final output was a set of pairs of layexpert terms. In this method, the candidates terms were extracted automatically from text generated by patients in social media. The validation step was semi-automatic. The mapping was performed based on, first, lexical information such as Carry stemming algorithm [START_REF] Paternostre | Carry, un algorithme de désuffixation pour le français[END_REF] to detect abbreviations and Levenshtein distance for spelling errors and, second, external resources such as Google index and Wikipedia structure. Few mappings were obtained based on the corpus only: 22 given by a custom Jaccard measure. Out of 1900 candidates, we reported a total number of 122 lay-expert pairs. However, the pairs extracted might not be exact synonyms.

Standard formalization of Consumer Health Vocabulary content

Although identification and mapping are crucial, to the best of our knowledge, there has been no proposition on how to represent the content (terms & mappings) of a CHV using a standard format to facilitate semantic interoperability and reuse. Even if this has been sometime pointed by the community [START_REF] Zielstorff | Controlled vocabularies for consumer health[END_REF], existing CHVs are either unavailable for public use or stored in a non-standardized format. In addition of OAC-CHV previously mentioned (available in XLS format) or our French preliminary CHV [START_REF] Tapi Nzali | Construction d'un vocabulaire patient/médecin dédié au cancer du sein à partir des médias sociaux[END_REF] (available in plain text), we can mention the Personal Health Terminology, developed by Intelligent Medical Object, the Mayo Consumer Vocabulary, developed by Mayo Clinic or the MedlinePlus Health Topics dictionary, developed by US National Library of Medicine.

Recently, some ontology designers have taken the initiative to directly populate their ontologies with layperson synonyms, which is probably the best practice (i.e., represent them as any other synonyms). For instance, in [START_REF] Vasilevsky | Enhancing the human phenotype ontology for use by the layperson[END_REF], the authors present how they have involved layusers in describing more than 6000 synonyms for the Human Phenotype Ontology. 6 In our study, this will not be possible as we are not the developers of the targeted ontologies, however, once available in standard format, the lay terms could be considered as candidate synonyms by the developers (e.g., French INSERM organization which is in charge of the French version of MeSH).

One candidate format for representing CHV as an independent resource is the Simple Knowledge Organization System (SKOS) language. This W3C recommendation is widely used in the semantic Web community. SKOS is a language to develop thesauri, taxonomies or controlled vocabularies [START_REF] Miles | Skos core: simple knowledge organisation for the web[END_REF]. It allows easy knowledge representation in a machine-readable format based on RDF graphs; plus it offers multiple standard properties to represent mappings between concepts (skos:exactMatch, skos:closeMatch, etc.). In biomedicine, OBO and OWL are also common standard used for knowledge formalization with more details than SKOS/RDF. However, when developing CHVs, researchers are usually only interested by the lexical/terminological level i.e., the labels and their relations only. Therefore, SKOS seems the most suitable standard to use and this is the choice made in this paper to formalize both the CHV terms and the mappings to standard terminologies.

SKOS formalization model 3.1 Data used

We propose hereafter a model to formalize a CHV into a SKOS vocabulary and then a protocol to align it to standard terminologies. We have experimented with our preliminary CHV [START_REF] Tapi Nzali | Construction d'un vocabulaire patient/médecin dédié au cancer du sein à partir des médias sociaux[END_REF] and used terminologies and web services offered by the SIFR BioPortal, a repository of French biomedical terminologies (http://bioportal.lirmm.fr) [START_REF] Jonquet | SIFR BioPortal : Un portail ouvert et générique d'ontologies et de terminologies biomédicales francaises au service de l'annotation sémantique[END_REF].

Our CHV, MuEVo, is built out of 173 lay-expert relations extracted from public French breast cancer forums. These relations have been obtained via a mapping between a patient-generated corpus made of posts from Cancerdusein.org and public Facebook groups 7 and a seed expert vocabulary offered by French National Cancer Institute (INCa -www.e-cancer.fr/dictionnaire) [START_REF] Delavigne | Peut-on "traduire" les mots des experts? un dictionnaire pour les patients atteints de cancer[END_REF]. As provenance information, each relation has a type, a discovering method, and a weight which represents the confidence of the relation (cf. table 1).

Representation of CHV terms

The knowledge unit in SKOS is skos:Concept; it is a RDF resource which formalizes an idea, a reality. It can have at most one preferred label (skos:prefLabel) used to denote the concept. Others terms can be associated to the concepts as valid variants (skos:altLabel) or deprecated/hidden variants (skos:hiddenLabel). SKOS's model alone is not enough to capture the provenance metadata describing each lay-expert relation: type, method and weight. To capture that, we used the PROV Ontology, a W3C recommandation to formalize provenance information. The complete model is described in figure 1 and exemplified in the listing hereafter.

<skos:Concept rdf:about =" http: // purl .lirmm. fr / ontology /MuEVo/vpm52"> < rdf:type rdf:resource =" http: // www.w3.org/ns/prov#Entity"/> <skos:inScheme rdf:resource =" http: // purl .lirmm. fr / ontology /MuEVo"/> < skos:prefLabel xml:lang="fr">oncologue</skos:prefLabel> < skos:altLabel xml:lang="fr">onco</ skos:altLabel > <prov:wasDerivedFrom rdf:resource =" http: // purl .lirmm. fr / ontology /MuEVo/provEntity86"/> <skos:broadMatch rdf:resource =" http: // chu-rouen.fr/cismef /SNOMED_int.#J-06120"/> </skos:Concept> < prov:Entity rdf:about =" http: // purl .lirmm. fr / ontology /MuEVo/provEntity86"> <!--carry|onco|oncologue|Abbr|50.0 --> < rdfs:label >onco</ rdfs:label > < isocat:abbreviation rdf:resource =" http: // purl .lirmm. fr / ontology /MuEVo/vpm52"/> < isocat:weight >50.0</ isocat:weight > <prov:wasGeneratedBy rdf:resource =" http: // purl .lirmm. fr / ontology /MuEVo/carry"/> </ prov:Entity > Fig. 1. Model to formalize lay-expert relations using SKOS+PROV Each skos:Concept (in blue) is a formal representation for all the relations found for a given expert term. Relations between the expert term of the concept (represented with skos:prefLabel) to a lay term is formalized via the use of SKOS alternate labels (skos:altLabel or skos:hiddenLabel). The metadata describing the provenance of the relation is represented using a prov:Entity (in yellow). This entity is linked to the concept thanks to a ISOcat property8 selected according to relations in table 2. The weight of the relation is also represented in the prov:Entity, with the property isocat:weight. Additionally, each prov:Entity representing the relation of a lay term relation with the expert term of a concept is linked to the corresponding skos:Concept with a prov:wasDerivedFrom property. Finally, each mapping method is represented by a prov:Activity (in red). Methods are simply described by a label e.g., carry, wikipedia.

Representation of the CHV mappings

Now, we would like to align MuEVo with some standard biomedical terminologies such as the ones that can be found in the NCBO BioPortal [START_REF] Noy | BioPortal: ontologies and integrated data resources at the click of a mouse[END_REF], a repository of biomedical ontologies and terminologies. As part of the SIFR project, our local appliance of BioPortal gives access to some of the French versions formally mapped to the original English ones [START_REF] Jonquet | SIFR BioPortal : Un portail ouvert et générique d'ontologies et de terminologies biomédicales francaises au service de l'annotation sémantique[END_REF]. Via the portal, a user can share an ontology and align it to the ones already available in the repository or any other resources thanks to SKOS mappings. We thus uploaded MuEVo in the SIFR BioPortal and then explicitly linked it to the standard biomedical terminologies also available there. Such formal mappings will enable anyone to benefit from the expanded structured knowledge available in standard terminologies when using MuEVo. It can be useful, for example, to semantically index forum content with MuEVo. In our experiments, we used the SIFR BioPortal ontology recommender (working exactly as the NCBO Recommender originally described in [START_REF] Jonquet | Building a Biomedical Ontology Recommender Web Service[END_REF]) to identify the most appropriate target terminologies and identified MeSH, SNOMEDint and MedDRA which are the set of terminologies that offer the better coverage of the expert terms. The alignment follows two phases:

The direct mapping phase consists in searching each expert term of MuEVo's expert vocabulary via BioPortal REST API search service. 9 The search is restricted to the targeted terminologies. If we find the exact same term (or its plural form) as preferred or alternative label of a concept in one of the targeted terminologies, we set an equivalence mapping, skos:exactMatch, between the current MuEVo concept and the one of the targeted terminology. In figure 2, the expert term vpm:abdomen matches the preferred label of a concept in a standard terminology, thus, a mapping with the standard concept std:Abdomen is created is mapped. The expert term cancer is an alternative label for the standard concept std:Tumeurs so a skos:exactMatch mapping is also created. The indirect mapping phase is necessary for the terms that are not preferred terms or synonyms of any concept in the targeted terminologies. In this case, we hy-pothesis that there exists more general resources that can serve as intermediate between INCa list and the targeted standard terminologies. Thus, for a given MuEVo concept C M uEV o , we will make use of an external resource, such as Wiktionary (www.wiktionary.org) [START_REF] Meyer | Wiktionary: A new rival for expert-built lexicons? exploring the possibilities of collaborative lexicography[END_REF] in our case, to find semantically related terms noted t expert (synonyms, hypernyms (broader terms), hyponyms (narrower terms)) which are themselves used as labels in the targeted terminologies. We automatized the search in Wiktionary using the Java Wiktionary Library API [START_REF] Zesch | Extracting lexical semantic knowledge from wikipedia and wiktionary[END_REF] and customized the extraction of semantic relations for French language. We adopted the following workflow:

1. Given a concept C M uEV o of MuEVo, we search the expert term t expert in Wiktionary using C M uEV o 's expert term. If a corresponding page exits, we fetch all terms t synonyms, hypernyms and hyponyms; 2. For each term t, we try to directly map it as described before; 3. In case of success, we define the following mappings between C M uEV o and the concept C target returned by BioPortal's search API: 3), the expert term cure (cure) has synonym traitement (treatment); oncologue (oncologist) has médecin spécialiste (specialized physician) as hypernym and a hyponym of atome (atom) is ion (ion). 4. If no success, we do a manual mapping.

-if t

MuEVo terms and mappings results

A first version of MuEVo (version 1.3) as described in this paper is available within the SIFR BioPortal: http://bioportal.lirmm.fr/ontologies/MUEVO as well as the mappings created to MeSH, SNOMEDint and MedDRA. It contains 64 skos:Concepts which is the result of a fully automatic approach that will be reused in the future on other datasets to enhance the vocabulary. One feature of the portal is to dereference URIs (for skos:Concepts in the purl.lirmm.fr namespace) to the corresponding web page in the web application, which makes the use of the vocabulary more comprehensible. Table 3 sums up the results of the automatic mapping of the 64 processed expert terms. Although the current version is pretty small, the mapping process has been automatized for scalability when the CHV will grow. The three targeted terminologies cover 84,38% of the expert terms: MeSH (70,31%), SNOMEDint (51,56%) and MedDRA (37,5%). 25% are only in MeSH, 7,81% only SNOMEDint and 4,69% only in Med-DRA. 10 Among the ten missing terms, three have been successfully mapped thanks to hyponyms from Wiktionary. For the seven others 11 , we performed a manual mapping. The validity of each mapping, both manual and automatic has been manually checked. The automatic alignment rates are good because: (i) the union of the three targeted ontologies is quite large, which increases the chance of finding a term using the SIFR BioPortal search web service, especially for the cancer domain that is well cover by MeSH or SNOMEDint; (ii) the seed expert vocabulary provided by INCa already used relevant biomedical terms that could have be found in standard terminologies. Some obvious terms related to breast cancer may actually not be included in the vocabulary if they are not used by laypersons as synonyms within the forum data. For instance, the term 'sein' (breast in English) is not in MuEVo mainly because this is the term actually used by patients online, where as the term 'cancer' (cancer in English) is actually in MuEVo because we have identified the relation with the term 'crabe' (crab in English), familiar term that happens to be used by layusers. In addition, it is important to note that MuEVO is not an ontology as it's goal is not to capture medical information that has already been captured in existing standards terminologies and ontologies. MuEVo is a simple vocabulary which goals is to complete existing knowledge with relevant alignments. It essentially aims at building a bridge, in the context of breast cancer forums, between lay expressions and their expert semantic neighbors.

Conclusion & Perspectives

In this article, we presented a formalization of a lay-expert vocabulary with semantic Web languages as well as our method to map the expert terms to standard biomedical terminologies. This new resource, once extended in future work, could become a key component for multi-expertise (lay-expert) information retrieval or text mining application. For example, health professionals could be interested in identifying recurrent symptoms mentioned in social media. Moreover, it will enable to mediate between these two expertise levels: from expert to lay, e.g., to vulgarize a medical production such as medical records or from lay to expert, e.g., to retrieve expert documents given a lay query. Once the two vocabularies interconnected, it will become possible to use the knowledge formalized in ontologies to semantically search patient data. Or it will be possible to automatically classify forum posts with general medical terms as it was done for instance with scientific literature (e.g., GoPubMed). This feature are very relevant for forum provider such as Doctissimo.com or Lesimpatientes.com. For MuEVo, being hosted in the SIFR BioPortal enables to benefit from all functionalities especially the SIFR/French Annotator that will now be able to use MuEVo to semantically index patient generated text.

The current version of MuEVo is limited to breast cancer but our CHV extraction method and representation model (terms & mappings) are generalizable to other domains or languages. In future work, we consider several perspectives for this work:

to extract more lay-expert pairs of terms and obtain finer type of association. Especially by using ontology terms directly in the seed vocabulary; to process data from other sources (other topic, other domain) and adapt our method to deal with new issues such as disambiguation; to use other resources than Wiktionary in the indirect mapping phase, including other ontologies (that can be aligned to one target ontology); to use MuEVo to semantically index social media data and evaluate the results in terms of semantic search. For instance, to mesure recall increase when including posts with misspelled words in the query responses; to use MuEVo for classification tasks using the hierarchies of the ontologies to which MuEVo is aligned to.

Fig. 2 .

 2 Fig. 2. Examples of direct mappings (namespaces are taken as examples)

Fig. 3 .

 3 Fig. 3. Examples of indirect mappings (namespaces are taken as examples)

Table 1 .

 1 Examples of lay-expert relations from our preliminary CHV[START_REF] Tapi Nzali | Construction d'un vocabulaire patient/médecin dédié au cancer du sein à partir des médias sociaux[END_REF]

	Lay term	Expert term	Relation type Method	Weight
	nez	pharynx	association	wikipedia	10.0
	abaltion	ablation	misspelling	aspell	100.0
	onco	oncologue	abbreviation	carry	50.0
	traitement hormonal hormonothérapie	association	wikipedia	100.0

Table 2 .

 2 Selection of SKOS labels and the ISOcat property per type of relation in the CHV.

	Relation type	SKOS label	ISOcat property
	abbreviation	skos:altLabel	isocat:abbreviation (DC-331)
	misspelling	skos:hiddenLabel	isocat:variant (DC-330)
	association	skos:hiddenLabel	isocat:relatedTerm (DC-438)

  is a synonym of t expert : C M uEV o skos:exactMatch C target ; if t is a hypernym of t expert : C M uEV o skos:broadMatch C target ; if t is a hyponym of t expert : C M uEV O skos:narrowMatch C target . For example (figure

Table 3 .

 3 Mapping results obtained automatically for 64 input terms at direct mapping phase (1A, 1B) and 10 terms at indirect mapping phase (2).

		Number	Examples
	1A : Singular	51	abdomen → Abdomen (MeSH)
	1B : Plural	17	glucide → Glucides (MeSH)
	1A+1B	54
	2 : Hyponyms	3	atome → ion (SNOMEDint)

http://www.patientsandweb.com/?p=90

googleblog.blogspot.fr/2016/06/im-feeling-yucky-searching-for-symptoms.html (June 2016)

This recent results have not been used in our work.

Cancer du sein, Octobre rose 2014, Cancer du sein -breast cancer, brustkrebs

www.isocat.org -ISOCat identifies properties with ids DC-XX, but for readability with have used here the corresponding name.

http://data.bioportal.lirmm.fr/documentation

16 are only in MeSH, 5 only SNOMEDint and 3 only in MedDRA

cure, guérison, médecin traitant, oncologue, organe, physicien, rémission

Acknowledgements

This work is achieved within the SIFR project funded by the European H2020 Marie Curie actions (grant 701771) and the French National Research Agency (grant ANR-12-JS02-01001) as well as by University of Montpellier and the CNRS. We also acknowledge support of French IReSP (Institut de Recherche en Santé Publique).


Georges Sadaka
email: georges.sadaka@u-picardie.fr

Generation and propagation of a Tsunami wave : a new mesh adaptation technique

, over a flat bottom then over a variable bottom in space and in time and apply this system, first, using a mesh generated using a photo of the Mediterranean sea, second, using a mesh generated using an imported xyz bathymetry for the sea near Java island and then we will consider a realistic example of the Tsunami wave near Java island which happened in 2006. We choose here to use FreeFem ++ [8] software which simplifies the construction of the domain, in particular, one of the advantage of FreeFem ++ is that we can build a mesh using a photo and we can easily export bathymetric data in order to consider more realistic simulations where a special adapt mesh technique applied for these two methods is detailed in the sequel.

Introduction

We consider here the numerical simulation of the sBBM in 2D over a variable bottom in space D(x, y) and in time ζ(x, y, t) :

η t + ∇ • ((D + η + ζ)V ) + ζ t + Ã∇ • D 2 ∇ζ t + ∇ • AD 2 [∇ (∇D • V ) + ∇D∇ • V ] -bD 2 ∇η t = 0, V t + g∇η + 1 2 ∇|V | 2 + BgD [∇ (∇D • ∇η) + ∇D∆η] -dD 2 ∆V t -BD∇ζ tt = 0, (1)
where

a = θ - 1 2 , b = 1 2 (θ -1) 2 - 1 3 , Ã = ν a -(1 -ν) b, A = -b, B = 1 -θ, b = 1 2 θ 2 - 1 3 (1 -ν), d = 1 2
1 -θ 2 (1 -µ) and g = 9.81 is the gravity.

This system is an approximation to the three-dimensional Euler equations describing the irrotational free surface flow of an ideal fluid Ω ⊂ R 3 , which is bounded below by -h(x, y, t) = -D(x, y) -ζ(x, y, t) and above by the free surface elevation η(x, y, t) (cf. Figure 1). The variables in (1) : X = (x, y) ∈ Ω and t > 0 are proportional to position along the channel and time, respec-Figure 1 -The domain Ω.

tively. η = η(X, t) being proportional to the deviation of the free surface departing from its rest position and Remark 1. In our study, we suppose that η = O(a), where here the amplitude a is the difference between the water surface and the zero level. Also we set λ = O( ) be the wave length. In addition, we limit ourselves to the case where η + D > 0 (there is no dry zone), since we are in a big deep water wave regime.

V = V (X, t) = u(X, t) v(X,
This paper is organized as follows: in Section 2, we present the space and time discretization of equation [START_REF] Valentin | Théorie générale des mouvements, qui sont propagés dans un canal rectangulaire horizontal[END_REF]. In Section 3, we present a method to build a mesh using a photo and then using an imported xyz bathymetry, in which, we will also present a special adaptive mesh technique around the initial data used for the generation of a Tsunami wave. In Section 4, we first check the convergence of our code, which establishes the adequacy of the chosen finite element discretization, then we simulate the propagation of a wave, that looks like a Tsunami wave generated by an Earthquake, in the Mediterranean sea over the sBBM system (1) with a flat bottom using the mesh generated from a photo of the Mediterranean sea, then with a variable bottom in space using the mesh generated from the xyz bathymetry of the sea near Java island and finally, with a realistic example of the Tsunami wave near Java island which happened in 2006.

Discretization of the sBBM system

In this section, we present the spatial discretization of (1) using finite element method with P 1 continuous piecewise linear functions and for the time marching scheme an explicit second order Runge-Kutta scheme.

Spatial discretization

We let Ω be a convex, plane domain, and T h be a regular, quasiuniform triangulation of Ω with triangles of maximum size h < 1. Setting V h = {v h ∈ C 0 ( Ω); v h | T ∈ P 1 (T ), ∀T ∈ T h } be a finite-dimensional subspace of H 1 (Ω), where P 1 is the set of all polynomials of degree ≤ 1 with real coefficients and denoting by •; • the L 2 inner product on Ω, we consider the weak formulation of system (1) :

Find η h , u h , v h ∈ V h such that ∀φ η h , φ u h , φ v h ∈ V h , we have :

η ht -b∇ • D 2 ∇η ht + ∇ • ((D + η h + ζ) (u h ; v h )) + ζ t ; φ η h + Ã∇ • D 2 ∇ζ t ; φ η h + ∇ • AD 2 [∇ (∇D • (u h ; v h )) + ∇D∇ • (u h ; v h )] ; φ η h = 0, u ht -dD 2 ∆u ht + gη xh + u h u hx + v h v hx -BDζ xtt ; φ u h + Bg D (∇D • ∇η h ) x + D x ∆η h ; φ u h = 0, v ht -dD 2 ∆v ht + gη yh + u h u hy + v h v hy -BDζ ytt ; φ u h + Bg D (∇D • ∇η h ) y + D y ∆η h ; φ v h = 0.
(2)

For simplicity, we set

φ η h = Φ η , φ u h = Φ u , φ v h = Φ v , η h = E, u h = U, v h = V
, so that system (2) can be rewrite in the following way :

                           ∂ t E -b∇ • D 2 ∇∂ t E ; Φ η = -(D + ζ + E)∇ • (U; V) + (D x + ζ x + E x )U + (D y + ζ y + E y )V + ζ t + Ã∇ • D 2 ∇ζ t + A∇ • D 2 [∇ (∇D • (U; V)) + ∇D∇ • (U; V)] ; Φ η = F (E, U, V, Φ η ) , (I d -dD 2 ∆)∂ t U; Φ u = -gE x + UU x + VV x + BgD (∇D • ∇E) x + D x ∆E -BDζ xtt ; Φ u = G (E, U, V, Φ u ) , (I d -dD 2 ∆)∂ t V; Φ v = -gE y + UU y + VV y + BgD (∇D • ∇E) y + D y ∆E -BDζ ytt ; Φ v = H (E, U, V, Φ u ) . (3)
Next, we discretize system (3). First, integrating by parts, the left hand side in (3) gives :

-b∇ • D 2 ∇∂ t E ; Φ η = b D 2 ∇∂ t E; ∇(Φ η ) - Γn bD 2 Φ η ∂(∂ t E) ∂n ∂γ, -dD 2 ∆∂ t U; Φ u = d D 2 ∇∂ t U; ∇Φ u + d 2D∇D • ∇∂ t U; Φ u - Γn dD 2 Φ u ∂(∂ t U) ∂n ∂γ,
and

-dD 2 ∆∂ t V; Φ v = d D 2 ∇∂ t V; ∇Φ v + d 2D∇D • ∇∂ t V; Φ v - Γn dD 2 Φ v ∂(∂ t V) ∂n ∂γ.
Now, dealing with the right-hand side of the first equation in system (3), we have :

∇ • D 2 ∇ζ t ; Φ η = D 2 ζ xt x + D 2 ζ yt y ; Φ η = 2DD x ζ xt + D 2 ζ xxt + 2DD y ζ yt + D 2 ζ yyt ; Φ η , and ∇ • D 2 [∇ (∇D • (U; V)) + ∇D∇ • (U; V)] ; Φ η = ∇ • D 2 (D x U + D y V) x ; (D x U + D y V) y + (D x ∇ • (U; V); D y ∇ • (U; V)) ; Φ η = ∇ • D 2 D xx U + D 2 D x U x + D 2 D xy V + D 2 D y V x + D 2 D x ∇ • (U; V); D 2 D xy U + D 2 D x U y +D 2 D yy V + D 2 D y V y + D 2 D y ∇ • (U; V) ; Φ η = (2DD x D xx + 2DD y D xy + D 2 D xyy + D 2 D xxx )U + (2DD x D xy + 2DD y D yy + D 2 D yyy +D 2 D xxy )V + (4DD 2 x + 3D 2 D xx + 2DD 2 y + D 2 D yy )U x + 2(D 2 D xy + DD x D y )U y + (4DD 2 y +3D 2 D yy + 2DD 2 x + D 2 D xx )V y + 2(DD x D y + D 2 D xy )V x ; Φ η + 2D 2 D x U xx ; Φ η + D 2 D y U xy ; Φ η + D 2 D x U yy ; Φ η + D 2 D y V xx ; Φ η + D 2 D x V xy ; Φ η + 2D 2 D y V yy ; Φ η ;
On the other hand, we have :

2D 2 D x U xx ; Φ η = -2D 2 D x U x ; Φ η x -(4DD 2 x + 2D 2 D xx )U x ; Φ η + Γn 2D 2 D x Φ η ∂U ∂n ∂γ, D 2 D y U xy ; Φ η = -D 2 D y U x ; Φ η y -(2DD 2 y + D 2 D yy )U x ; Φ η + Γn D 2 D y Φ η ∂U ∂n ∂γ, D 2 D x U yy ; Φ η = -D 2 D x U y ; Φ η y -(2DD x D y + D 2 D xy )U y ; Φ η + Γn D 2 D x Φ η ∂U ∂n ∂γ, D 2 D y V xx ; Φ η = -D 2 D y V x ; Φ η x -(2DD x D y + D 2 D xy )V x ; Φ η + Γn D 2 D y Φ η ∂V ∂n ∂γ, D 2 D x V xy ; Φ η = -D 2 D x V x ; Φ η y -(2DD x D y + D 2 D xy )V x ; Φ η + Γn D 2 D x Φ η ∂V ∂n ∂γ, 2D 2 D y V yy ; Φ η = -2D 2 D y V y ; Φ η y -(4DD 2 y + 2D 2 D yy )V y ; Φ η + Γn 2D 2 D y Φ η ∂V ∂n ∂γ,
and, consequently,

F (E, U, V, Φ η ) = -(D + ζ + E)∇ • (U; V) + (D x + ζ x + E x )U + (D y + ζ y + E y )V + ζ t ; Φ η -Ã 2DD x ζ xt + D 2 ζ xxt + 2DD y ζ yt + D 2 ζ yyt ; Φ η -A (2DD x D xx + 2DD y D xy + D 2 D xyy +D 2 D xxx )U + (2DD x D xy + 2DD y D yy + D 2 D yyy + D 2 D xxy )V + D 2 D xx U x + D 2 D xy U y -2DD x D y V x +(D 2 D yy + 2DD 2 x + D 2 D xx )V y ; Φ η + A 2D 2 D x U x + D 2 D y V x ; Φ η x + D 2 D y U x + D 2 D x U y +D 2 D x V x + 2D 2 D y V y ; Φ η y -A Γn (3D 2 D x + D 2 D y )Φ η ∂U ∂n + (D 2 D x + 3D 2 D y )Φ η ∂V ∂n ∂γ.
For the right-hand side of second equation in system (3), we have :

G (E, U, V, Φ u ) = -gE x + UU x + VV x + BgD (D x E x + D y E y ) x + D x (E xx + E yy ) -BDζ xtt ; Φ u = -gE x + UU x + VV x + Bg (DD xx E x + DD xy E y ) -BDζ xtt ; Φ u -Bg 2DD x E xx + DD y E xy + DD x E yy ; Φ u = -gE x + UU x + VV x + Bg (DD xx E x + DD xy E y ) -BDζ xtt ; Φ u +Bg 2DD x E x ; Φ u x +Bg (2D 2 x + 2DD xx )E x ; Φ u +Bg DD y E x + DD x E y ; Φ u y +Bg (D 2 y + DD yy )E x + (D x D y + DD xy )E y ; Φ u - Γn Bg(3DD x +DD y )Φ u ∂E ∂n ∂γ = -g I d -B DD xx + 2D 2 x + DD yy + D 2 y E x + UU x + VV x -BgD x D y E y -BDζ xtt ; Φ u +Bg 2DD x E x ; Φ u x +Bg DD y E x + DD x E y ; Φ u y - Γn Bg(3DD x + DD y )Φ u ∂E ∂n ∂γ.
Finally, for the right-hand side of the third equation in system (3), we have :

H (E, U, V, Φ v ) = -gE y + UU y + VV y + BgD (D x E x + D y E y ) y + D y (E xx + E yy ) -BDζ ytt ; Φ v = -gE y + UU y + VV y + Bg (DD xy E x + DD yy E y ) -BDζ ytt ; Φ v -Bg DD y E xx + DD x E xy + 2DD y E yy ); Φ v = -gE y + UU y + VV y + Bg (DD xy E x + DD yy E y ) -BDζ ytt ; Φ v +Bg DD y E x ; Φ v x +Bg (D x D y + DD xy )E x ; Φ v +Bg DD x E x + 2DD y E y ; Φ v y +Bg (D x D y + DD xy )E x + (2D 2 y + 2DD yy )E y ; Φ v - Γn Bg(DD x +3DD y )Φ v ∂E ∂n ∂γ = --Bg(2D x D y + DD xy )E x + UU y + VV y + g I d -B DD yy -2D 2 y E y -BDζ ytt ; Φ v + Bg DD y E x ; Φ v x +Bg DD x E x + 2DD y E y ; Φ v y - Γn Bg(DD x + 3DD y )Φ v ∂E ∂n ∂γ.
Remark 2. We note that, during the simulation, when there is a steep gradient, we obtain a blow-ups. In order to avoid this problem, we need to change the bottom (make it smoother) or/and to get rid of the high order derivatives for the bottom as in [START_REF] Mitsotakis | Boussinesq systems in two space dimensions over a variable bottom for the generation and propagation of Tsunami waves[END_REF]. That's why, we take into account in the sequel, the smoothness of the bottom and the fact that the derivatives of the bottom of order greater then one are neglected. Thus, we will now deal with the following system:

               ∂ t E; Φ η + b D 2 ∇∂ t E; ∇(Φ η ) - Γn bD 2 Φ η ∂(∂ t E) ∂n ∂γ = F (E, U, V, Φ η ) ∂ t U; Φ u + d D 2 ∇∂ t U; ∇Φ u + d 2D∇D • ∇∂ t U; Φ u - Γn dD 2 Φ u ∂(∂ t U) ∂n ∂γ = G (E, U, V, Φ u ) ∂ t V; Φ v + d D 2 ∇∂ t V; ∇Φ v + d 2D∇D • ∇∂ t V; Φ v - Γn dD 2 Φ v ∂(∂ t V) ∂n ∂γ = H (E, U, V, Φ v ) (4) with F (E, U, V, Φ η ) = -(D + ζ + E)∇ • (U; V) + (D x + ζ x + E x )U + (D y + ζ y + E y )V + ζ t ; Φ η -Ã 2DD x ζ xt + 2DD y ζ yt ; Φ η -A -2DD x D y V x + 2DD 2 x V y ; Φ η +A 2D 2 D x U x + D 2 D y V x ; Φ η x + D 2 D y U x +D 2 D x U y + D 2 D x V x + 2D 2 D y V y ; Φ η y -A Γn (3D 2 D x + D 2 D y )Φ η ∂U ∂n + (D 2 D x + 3D 2 D y )Φ η ∂V ∂n ∂γ, G (E, U, V, Φ u ) = -g I d -B 2D 2 x + D 2 y E x + UU x + VV x -BgD x D y E y -BDζ xtt ; Φ u + Bg 2DD x E x ; Φ u x +Bg DD y E x + DD x E y ; Φ u y - Γn Bg(3DD x + DD y )Φ u ∂E ∂n ∂γ, and
H (E, U, V, Φ v ) = --2BgD x D y E x + UU y + VV y + g I d -2BD 2 y E y -BDζ ytt ; Φ v + Bg DD y E x ; Φ v x +Bg DD x E x + 2DD y E y ; Φ v y - Γn Bg(DD x + 3DD y )Φ v ∂E ∂n ∂γ.

Time marching scheme

Our method is based on an explicit second order Runge-Kutta scheme. For that, let us denote by (E n+1 , U n+1 , V n+1 ) and (E n , U n , V n ) the approximate values at time t = t n+1 and t = t n , respectively and by δt the time step size. Then, owing to (4), the unknown fields at time t = t n+1 are defined as the solution of the following system:

             E n+1 ; Φ η = E n + E k1 + E k2 2 ; Φ η , U n+1 ; Φ u = U n + U k1 + U k2 2 ; Φ u , V n+1 ; Φ v = V n + V k1 + V k2 2 ; Φ v , (5)
where

E k1 ; Φ η + b D 2 ∇E k1 ; ∇(Φ η ) - Γn bD 2 Φ η ∂(E k1 ) ∂n ∂γ = δt • F (E n , U n , V n , Φ η ) , U k1 + 2dD∇D • ∇U k1 ; Φ u + d D 2 ∇U k1 ; ∇Φ u - Γn dD 2 Φ u ∂(U k1 ) ∂n ∂γ = δt • G (E n , U n , V n , Φ u ) , V k1 + 2dD∇D • ∇V k1 ; Φ v + d D 2 ∇V k1 ; ∇Φ v - Γn dD 2 Φ v ∂(V k1 ) ∂n ∂γ = δt • H (E n , U n , V n , Φ v ) (6)
and

E k2 ; Φ η + b D 2 ∇E k2 ; ∇(Φ η ) - Γn bD 2 Φ η ∂(E k2 ) ∂n ∂γ = δt • F E n + E k1 , U n + U k1 , V n + V k1 , Φ η , U k2 + 2dD∇D • ∇U k2 ; Φ u + d D 2 ∇U k2 ; ∇Φ u - Γn dD 2 Φ u ∂(U k2 ) ∂n ∂γ = δt • G E n + E k1 , U n + U k1 , V n + V k1 , Φ u , V k2 + 2dD∇D • ∇V k2 ; Φ v + d D 2 ∇V k2 ; ∇Φ v - Γn dD 2 Φ v ∂(V k2 ) ∂n ∂γ = δt • H E n + E k1 , U n + U k1 , V n + V k1 , Φ v . (7)
3 Mesh generation and initial data

In this section, we present the technique used in order to generate a mesh using a photo of the Mediterranean sea then using an imported xyz bathymetry. Also, we will present a way in order to obtain the initial data and explain the special adapt mesh technique that we will use in our numerical simulation.

Mesh generated using a photo

We present here the method to build a mesh from a photo inspired from a FreeFem ++ script made by F. Hecht [START_REF] Hecht | [END_REF] and another one made by O. Pantz [12].

Owing to Google Earth and for better resolution, we take severals parts of the Mediterranean sea (cf. Figure 2) that are subsequently assembled using Photoshop to obtain a complete picture of the Mediterranean sea (cf. Figure 3). Using Photoshop , we can also eliminate the land areas that circumvent the Mediterranean sea (cf. Figure 4). We note that we must smoothen the borders in Photoshop and put the black color inside our domain and the white color outside. Then we convert the jpg photo to a pgm photo which can be read by FreeFem ++ using in a terminal window : convert Medit_sea . jpg Medit_sea . pgm

In order to generate the mesh of the Mediterranean sea domain, we read the pgm file using load " ppm2rnm " load " isoline " real [ int , int ] f1 ( " Medit_sea . pgm " ) ; int nx = f1 .n , ny = f1 . m ; mesh Sh = square ( nx -1 , ny -1 ,[( nx -1) *( x ) ,( ny -1) *(1 -y ) ]) ; fespace Vh ( Sh , P1 ) ; Vh fxy ; fxy []= f1 ; fxy =( fxy >=0.5) -( fxy <0.5) ; // to get value between -1 and 1 real [ int , int ] Curves (3 ,1) ; int [ int ] be (1) ; int nc = isoline ( Sh , fxy , iso =0. , close =0 , Curves , beginend = be , smoothing =.005 , ratio ¯=0.1) ;

The function obtained from the pgm file has values between 0 and 1, where the value of 0.5 represents the contour between two different colors. We note that, we can regularize this contour (thanks to O. Pantz [12]), before using the isoline function which computes the number of closed curves of our image, by solving :

Ω 2ε∇(δu)∇(φ) + 4 ε u 2 δuφ + αδuφ + Ω ε∇(u)∇(φ) - 2 ε (1 -u 2 )uφ = Ω α(f xy -u)φ, (8)
where in our example, we take ε = 1, α = 1 and δx = 1.5. We also note that when ε is close to zero, the solution u takes the values equal to 1 and -1 and α sets the balance between length of the curve and fitting the actual interface: As α increase, the approximation become better.

After regularizing, we update u = 0 till u = u + δu, and we adapt the mesh around the contour (cf. Figure 5), by using Sh = adaptmesh ( Sh ,u , nbvx =1 e7 ) ; then, we solve again the regularizing problem, then we update the function u = u + δu and finally we interpolate the solution on the initial mesh (cf. Figure 6) before using isoline : mesh Shinit = Sh ; fespace Vhinit ( Shinit , P1 ) ; Vhinit fxyinit ; fxyinit = u ; int nc = isoline ( Shinit , fxyinit , iso =0. , close =0 , Curves , beginend = be , smoothing ¯=.005 , ratio =0.1) ;

Now, we show in Figure 7, the mesh created around Crete island, where we see in the top left, the mesh after first regularization and in the top right, the function f xy after interpolation, in the down left, three isoline level -1, 0 and 1 of the f xy function, where here the mesh generated at the 0 level is shown in the down right. The complete script is written in [START_REF] Sadaka | FreeFem++, a tool to solve PDEs numerically[END_REF]. We note that, we take into account the area of the Mediterranean sea, which is almost 2.5 million km 2 and we did a scale for our final mesh : real Areasea =2.5 e6 ; // area of the Mediterranean sea in Km ^2 real scale = sqrt ( Areasea / Th . area ) ; Th = movemesh ( Th ,[ x * scale , y * scale ]) ;

Mesh generated using an imported xyz bathymetry

In order to consider more realistic case, this means that, we now take into account the bathymetry near Java island which can be downloaded from the NOAA 3 website, (cf. Figure 8), we also use FreeFem ++ to generate the mesh of the area where the amplitude is zero.

We can read the xyz file (cf. Figure 9, left), using this script : We can smoothen the bathymetric data by solving :

Ω (βuφ + ∇(u)∇(φ)) + Γ ∂u ∂n ∂γ = Ω βf xyφ, ∂u ∂n = 0. (9)
In our code, we take β = 5.e3, in order to build the mesh to get rid of smallest islands, then, in order to obtain the mesh only around the sea, which is limited by the zero level of the amplitude, we use : fxy =( fxy >=0.5) -( fxy <0.5) ; // to get value between -1 and 1 and then proceeding similarly, as above for the mesh generation using a photo, in order to obtain the mesh of our domain, by taking ε = 1.e -2 and α = 1.e2 for the regularization phenomena (8).

Remark 3. We note that our method takes into account the different label for each part of the boundary, which facilitates the use of different types of boundary condition.

Remark 4. For all simulation with bathymetry, we use β = 2.e1 in [START_REF] Mitsotakis | Boussinesq systems in two space dimensions over a variable bottom for the generation and propagation of Tsunami waves[END_REF] to smoothen the initial bathymetry after generation of the mesh (cf. Figure 9, right) in order to ensure the stability of the numerical method, we also note that in order to be in a big deep water wave regime for sBBM system we change the depth close to the shoreline to 100 m. Remark 5. The bathymetry data downloaded from the NOAA website are in degree coordinate and we need to convert them to meter. So, on the first hand, we must know the degree of Latitude (south and north) and of Longitude (west and east) of our domain where we can deduce the Latitude lat0 = .5(lat south + lat north ) and the Longitude long0 = .5(long west + long east ). On the other hand, we must take into account the spherical shape of the Earth, even if it does not play significant role because of the small spatial scale of the experiments. So, we know that the radius of the Earth near the equator is R equator = 6378, 137 Km, and near to the pole R pole = 6356, 752 Km, thus the radius of our domain equals to:

R = R 2 equator cos(lat0 • π/180) 2 + R 2 pole sin(lat0 • π/180) 2 R equator cos(lat0 • π/180) 2 + R pole sin(lat0 • π/180) 2 .
So, we move the mesh of our domain using the following translation (coefl0 = πR/180):

[x; y] -→ [(x -lon0) cos(πy/180)coefl0; (y -lat0)coefl0].
Therefore, we need to move the bottom from the original reference downloaded to the new mesh, which is easy to do in FreeFem ++ by writing this script : // set the value of u without any mesh update

Vh

Mesh adaptation technique

We introduce here a special mesh adaptation technique since some computation domains are huge as in the case of the Mediterranean sea with 1.7e6 triangles (8.7e5 degree of freedom) and our initial solution is concentrated in a small domain, a circle C(O, R) or a rectangle [aa, bb] × [cc, dd]. So we build the mesh Th of the small rectangle or the circle by doing a trunk to the initial full mesh Thinit respecting that the function equals to 1 inside the rectangle and 0 outside as in : then we compute the initial solution uold=uadapt in this domain. Using the keyword boundingbox in FreeFem ++ , we obtain the limit min max of Thold=Th on x and y direction, in which we add epsadapt from each side in order to build the new rectangle Thnew that contains Thold, then using the keyword interpolate in FreeFem ++ , the old FEspace in P 1 Vhold and the new FEspace in P 1 Vhnew, we interpolate uold to unew in the new domain. We smooth the function obtained from abs(unew)>=erradapt using : [START_REF] Okada | Surface deformation due to shear and tensile faults in a half space[END_REF] with zero Dirichlet boundary condition on the boundary label 0 of Thnew in order to obtain usmadapt. And, at the end, we trunk Thnew respecting the function usmadapt>=isoadapt. We can put all the previous detail for the special mesh adaption in a macro such as : ¯usmadapt " ) ; if ( output ) { Vhnew unewp =( usmadapt >= isoadapt ) ; plot ( unewp , fill = true , dim ¯=2 , value = true , wait = wv , cmm = " zone to cut " ) ;} Th = trunc ( Thnew , usmadapt >= isoadapt , label =0) ; // Remark 6. We trunk always from the initial full mesh, in this case, we keep the original vertices of the mesh throughout the simulation, and also we keep the original label of the boundary and we put 0 for the label of the rest of the boundary domain. We also note that we need to interpolate all the variables of any kind of FEspace from the old mesh to the newest one using the keyword interpolate in FreeFem ++ . We remark also that we must choose the parameters for the AdaptGS in order to obtain the value of erradapt on the boundary of Th for the function uadapt. In addition, we use a reflective boundary condition (BC) on label 0, i.e. zero Neumann BC for η and zero Dirichlet BC for V, cause our sBBM system gives artificial numerical explosion on the boundary if we do not use any BC or if we use only Neumann BC for η and V.

Ω (smoothadapt • usmadapt • φ + ∇(usmadapt)∇(φ)) = Ω smoothadapt • (|unew| ≥ erradapt) • φ,

Initial data

Inspiring from [START_REF] Dutykh | On the use of finite fault solution for tsunami generation problems[END_REF][START_REF] Mitsotakis | Boussinesq systems in two space dimensions over a variable bottom for the generation and propagation of Tsunami waves[END_REF], Tsunami waves are generated by a deformation of the bottom due to an Earthquake, which may be approximated by Okada's formula's [START_REF] Okada | Surface deformation due to shear and tensile faults in a half space[END_REF][START_REF] Okada | Internal deformation due to shear and tensile faults in a half-space[END_REF] in the case of the so called dip-slip dislocation, where the vertical component of displacement vector O(x, y), is given by the following formulas in Chinnery's notation, cf [2, 10]

f (ξ, η) ||= f (ξ, p) -f (ξ, p -W ) -f (ξ -L, p) + f (ξ -L, p -W ) , O (
ỹ = η cos δ + q sin δ, d = η sin δ -q cos δ, R 2 = ξ 2 + η 2 + q 2 = ξ 2 + ỹ2 + d2 , X 2 = ξ 2 + q 2 and I =      µ λ + µ 2 cos δ arctan η(X + q cos δ) + X(R + X) sin δ ξ(R + X) cos δ if cos δ = 0, µ λ + µ ξ sin δ R + d if cos δ = 0.
Here, W and L are the width and the length of the rectangular fault, (x, y) are the points where we computes displacements, (x 0 , y 0 ) is the epicenter, d = fault depth(x 0 , y 0 ) + W sin δ, δ is the dip angle, θ is the rake angle, D is the Burger's vector, U = |D| sin θ is the slip on the fault, φ is the strike angle which is measured conventionally in the counter-clockwise direction from the North (cf. Figure 10 (left)), µ, λ are the Lamé constants derived from elastic-wave velocities :

λ = ρ c V 2 P -V 2 S and µ = ρ c V 2 S
, where ρ c is the crust density, V P is the compressional-wave (P-wave) velocity, V S is the shear-wave (S-wave) velocity.

Remark 7. We can download the script which computes co-seismic displacements according to the classical Okada solution O(x, y) from the following link http://www.denys-dutykh.com/downloads.php.

We will distinguish here the two cases for the mechanisms of the dynamics of Tsunami wave generation as in [START_REF] Mitsotakis | Boussinesq systems in two space dimensions over a variable bottom for the generation and propagation of Tsunami waves[END_REF] : the passive generation and the active generation.

Passive generation :

In order to compute the initial data for η(x, y, 0) = O(x, y) in meters (cf. Figure 10 (right)), V (x, y, 0) = 0 which is referred to as a passive generation of a Tsunami wave near Java island, using our mesh adaptive technique, we will use the fact that the solution is concentrated in the small rectangle [x 0 -3.2W ; x 0 +1.2W ]×[y 0 -L; y 0 +L] where L = 100 Km, W = 50 Km, δ = 10.35 • , φ = 288.94 • , θ = 95 • , U = 2 m, ρ c = 2700 Kg/m 3 , V P = 6000 m/s, V S = 3400 m/s, (x 0 ; y 0 ) = (107.345 • , -9.295 • ) and the fault depth(x 0 ; y 0 ) = 10 Km. All these geophysical parameters can be downloaded from this website https://Earthquake.usgs.gov.

Therefore, we build the mesh of the small rectangle by doing a trunc to the initial full mesh respecting that the function equals to 1 inside the rectangle and 0 outside as in :

Active generation :

For a more realistic case as in the Java 2006 event, we use the active generation in order to model the generation of a Tsunami wave as in [START_REF] Dutykh | On the use of finite fault solution for tsunami generation problems[END_REF][START_REF] Dutykh | On the contribution of the horizontal sea-bed displacements into the tsunami generation process[END_REF]. In this case we consider zero initial conditions for both the free surface elevation and the velocity field, and assume that the bottom is moving in time. This case may be described by considering the bottom motion formula : -h(x, y, t) = -D(x, y) -ζ(x, y, t) with

ζ(x, y, t) = N x•N y i=1 H(t -t i ) • 1 -e -α(t-ti) • O i (x, y),
where N x sub-faults along strike and N y sub-faults down the dip angle, H(t) is the Heaviside step function and α = log (3) tr , where t r = 8 s is the rise time. We choose here an exponential scenario, but in practice, various scenarios could be used (instantaneous, linear, trigonometric, etc) and could be found in [START_REF] Dutykh | On the use of finite fault solution for tsunami generation problems[END_REF][START_REF] Dutykh | On the contribution of the horizontal sea-bed displacements into the tsunami generation process[END_REF][START_REF] Hammack | Tsunamis -A Model of Their Generation and Propagation[END_REF]. Remark 8. Parameters such as sub-fault location (x i , y i ), depth d i , slip U and rake angle θ for each segment, given in Table 3 of the paper [START_REF] Dutykh | On the use of finite fault solution for tsunami generation problems[END_REF], can be downloaded from this website https://Earthquake.usgs.gov. In this file, we remark that the fault's plane is conventionally divided into N x = 21 sub-faults along strike and N y = 7 sub-faults down the dip angle, leading to a total number of N x × N y = 147 equal segments.

For our special adapt mesh technique, since the fault plane is considered to be the rectangle with vertices located at (109.20508 • (Lon),-10.37387 • (Lat)), (106.50434 • (Lon),-9.45925 • (Lat)), (106.72382 • (Lon),-8.82807 • (Lat)) and (109.42455 • (Lon),-9.74269 • (Lat)), we will consider that our bottom displacement is concentrated on the big rectangle which is equidistant of 1 • from each side of the initial fault plane as in Figure 11 (left), then we compute each Okada solution O i on a circle of center (x i -10m, y i -10m) and of radius 6 max(L, W ) and at then end all the Okada solution will be interpolated on the big rectangle before starting to compute the vertical displacement of the bottom ζ(x, y, t), in Figure 11 (right) we plot O 14 . For the computation of ζ(x, y, t), we start the mesh by a circle of center (x c -5m, y c -5m) and of radius 4 max(L, W ) and we adapt the mesh each 3 iterations i.e. each 6 s by using the following value for the adapt mesh uadapt= ζ, isoadapt=5e-2, erradapt=1e-4, smoothadapt=5e-9, epsadapt=50e3.

We show in the Figure 12, the bottom displacement ζ(x, y, t) at time t = 100s and t = 270s using our adapt mesh technique. Remark 9. We note that after building the Okada solution O(x, y) in the passive generation or O i (x, y) in the Active generation, we can remark that this solution is non-local and decays slowly to zero, that why in our adapt mesh technique we put 0 where the absolute value of the solution is less then min(|min (O i (x, y))| , |max (O i (x, y))|)/9.2 meter, we make the same thing without adapt mesh in order to compare the solution using the same initial data.

Numerical simulations

In this section, we study first the rate of convergence of our codes for the sBBM (4) with non-dimensional and unscaled variable i.e., with g = 1 over a variable bottom in space, which establishes the adequacy of the chosen finite element discretization and the used time marching scheme, for the flat bottom case, we refer to [START_REF] Sadaka | Solution of 2D Boussinesq systems with FreeFem++ : the flat bottom case[END_REF], where we use the same technique as in this paper. Then we simulate the propagation of a wave, that looks like a Tsunami wave generated by an Earthquake, in the Mediterranean sea over the sBBM with a flat bottom, near Java island over a variable bottom in space and at the end near Java island over a variable bottom in space and in time.

In all numerical simulations we used P 1 continuous piecewise linear functions for η, u, v, D and ζ.

Rate of convergence

We prove in the figure below, that the RK2 time scheme considered for the sBBM variable bottom in space system is of order 2, we note that ζ(x, y, t) is only used for the generation of the Tsunami wave and will not be taken into account in the convergence rate test. In this example, we took Bi-Periodic Boundary Conditions for In order to compare the results between adaptmesh of FreeFem ++ , our new adapt mesh technique and without using mesh adaptation, we plot in addition to the free surface elevation η in the Figures 14 → 15, the variation of η vs time in Figure 16 at two wave 'gauges' placed at the positions represented by in Figure 7 (top, right) and the mass of the water η. Specifically, gauges were placed at the points (i) : (2350. * scale, 550. * scale), (ii) : (2104. * scale, 665. * scale). In the Figure 18, we represent the comparison between the three methods : Full, Adapt FF and Adapt GS of the maximum of the propagation of the solution at time t = 6800sec. We also plot the computation time for each adapt mesh, the computation time of the simulation, the number of degree of freedom in Figure 17. We can see in Figures 16 and17 that the adaptmesh of FreeFem ++ with err=1.e-2 is the fastest method but unfortunately it does not preserve the mass invariant η. On the other hand, our new adapt mesh technique preserves the mass invariant throughout the simulation with an error of order 2.1e -3 and an important time computation difference with the one without mesh adaptation which is very good method for the Tsunami wave propagation.

For the adaptmesh of FreeFem ++ with err=1.e-7, we also get an almost a mass conservation with an error of order 9.5e -4, but we obtain some difference in wave gauge with the Full method which is due to the refinement mesh adaptation and the interpolation of the solution, although the computation time is almost the double of the new adapt mesh technique.

We note also that we can go faster with our new mesh adaptation technique if we can also trunk the mass matrix after trunking the mesh, of course if the mass matrix is a constant along the simulation of the Full mesh, this is an outgoing project. 4.3 Propagation of a Tsunami wave near Java island : passive generation .

In this section, we will take the same initial data as defined above in passive generation part of Section 3, we take δt = 1 s as the time step size and we note that, we adapt the mesh after computing the initial data for η and then every 50 s by using the following value for the adapt mesh uadapt= η + u + v, isoadapt=3e-2, erradapt=1e-4, smoothadapt=5e-9, epsadapt=30e3. We compare here the results between our new adapt mesh technique and without using mesh adaptation. To this end, we plot the free surface elevation η in the Figures 19 and20. We also plot, the variation of η vs time (in Figure 21) at four numerical wave gauges placed at the following locations: (i) (107.345 • , -9.295 • ), (ii) (106.5 • , -8 • ), (iii) (105.9 • , -10.35 • ) and (iv) (107.7 • , -11 • ) (see Figure 11 (left)) where (i) is the position of the epicenter. However, because of the large variations of the bottom, shorter waves were generated, especially around Christmas Island (southwest of Java) and around the undersea canyon near the Earthquake's epicenter.

Finally, we present a comparison of the Kinetic, Potential and Total energy with the Full mesh (in Figure 22, top left) and with the Adapt GS method (in Figure 22, top right) defined in [START_REF] Dutykh | Energy of Tsunami waves generated by bottom motion[END_REF] as: where ρ w = 1027 Kg/m 3 is the ocean water density, the number of degree of freedom (in Figure 22, down left) and the computation time of the simulation (in Figure 22, down right). We obtain here an error of order 2.6e -4 between the Total Energy with Adapt GS and without adaptation. For a more realistic case as in the Java 2006 event, we use the active generation in order to model the generation of a Tsunami wave as in [START_REF] Dutykh | On the use of finite fault solution for tsunami generation problems[END_REF][START_REF] Dutykh | On the contribution of the horizontal sea-bed displacements into the tsunami generation process[END_REF]. In this case we consider zero initial conditions for both the surface elevation and the velocity field, we take δt = 2 s as the time step size, we assume that the bottom described in the Section 3 is moving in time and we note that we adapt the mesh, before the end of the generation time t = 270 s, each 3 iterations i.e. each 6 s by using the following value for the adapt mesh uadapt= η + u + v, isoadapt=5e-2, erradapt=1e-4, smoothadapt=5e-9, epsadapt=50e3 and then for t > 270 s each 25 iterations i.e. each 50 s. We compare here only the results between our new adapt mesh technique and without using mesh adaptation. To this end, we plot the free surface elevation η in the Figures 24 → 26. However, because of the large variations of the bottom, shorter waves were generated, especially around Christmas Island (southwest of Java) and around the undersea canyon near the Earthquake's epicenter. We also plot, the variation of η vs time (in Figure 27) at four numerical wave gauges placed at the following locations: (i) (107.345 • , -9.295 • ), (ii) (106.5 • , -8 • ), (iii) (105.9 • , -10.35 • ) and (iv) (107.7 • , -11 • ) (see Figure 11 (left)) where (i) is the position of the epicenter. At the end, we present a comparison of the Kinetic, Potential and Total energy with the Full mesh (in Figure 28, top left) and with the Adapt GS method (in Figure 28, top right) defined in [START_REF] Dutykh | Energy of Tsunami waves generated by bottom motion[END_REF] as:

E c = 1 2 ρ w Ω η -D(x,y) |V | 2 dz dxdy, E p = 1 2 ρ w • g Ω η 2 dxdy,
E c = 1 2 ρ w Ω η -D(x,y)
|V | 2 dz dxdy,

E p = 1 2 ρ w • g Ω η 2 dxdy,
the number of degree of freedom (in Figure 28, down left) and the computation time of the simulation (in Figure 28, down right). We obtain here an error of order 2e -5 between the Total Energy with Adapt GS and without adaptation.

We present in the Figure 29 the comparison of the maximum of the propagation of the solution between the Full and the Adapt GS method at t = 1750 s.

Conclusion and Outlook

We show in this paper, the usefulness of FreeFem ++ for the simplified Boussinesq system of BBM type by building the domain, on the one hand using a photo taken from Google Earth and on the other hand through an xyz bathymetry downloaded from the NOAA website. For the simulation of a Tsunami wave near Java island, the digital computing environment that we developed allows the integration of realistic data (bathymetry and geography) in a relatively simple framework. Another work concerning the generation, propagation and inundation of a Tsunami wave will be discussed in the case of the Shallow Water equations in [START_REF] Sadaka | Solving Shallow Water flows in 2D with FreeFem++ on structured mesh[END_REF], where in this case, we are not constraint by the smoothness of the bathymetry to avoid blow-up and where the same special adapt technique with a parallel version of the code will be introduced.

  t) = (u, v) T = (u; v) being proportional to the horizontal velocity of the fluid at someheight. ∇• = ∂ x • ∂ y • is the gradient, ∇•( ; • ) = ∂ x +∂ y • is the divergence and ∆• = ∂ xx •+∂ yy • is the laplacian.

Figure 2 -

 2 Figure 2 -The pictures of two parts of the Mediterranean sea.

Figure 3 -

 3 Figure 3 -The Mediterranean sea after assembly with Photoshop .

Figure 4 -

 4 Figure 4 -The Mediterranean sea.

Figure 5 -

 5 Figure 5 -The adapted mesh around the contour of the Mediterranean sea.

Figure 6 -

 6 Figure 6 -The function f xy of the contour of Mediterranean sea after interpolation.

Figure 7 -

 7 Figure 7 -Creation of the mesh around Crete island in the Mediterranean sea (top left: mesh after first regularization, top right: solution after interpolation, : wave gauge, : epicenter, down left: isoline of the contour, down right: mesh generated).

3 .

 3 https://maps.ngdc.noaa.gov/viewers/wcs-client/

Figure 8 -

 8 Figure 8 -Importation of bathymetry datas through the NOAA website.

Figure 9 -

 9 Figure 9 -Left : Bathymetry downloaded from the NOAA website, (min = -7239m and max = 3002m). Right : smoothed bathymetry with β = 2.e1 in (9), (min = -6207m and max = -100m).

  D = fxy ; real coefl0 = pi * R /180.; Sh = movemesh ( Sh ,[( x -lon0 ) * cos ( y * pi /180.) * coefl0 ,( y -lat0 ) * coefl0 ]) ; Vh tmp ; // define a temporary function tmp = D []; // save the value D =0; // to change the FEspace and mesh associated with u D []= tmp ;

  Th = trunc ( Thinit ,(1.*( x <= bb & x >= aa ) *( y <= dd & y >= cc ) ) >.5 , label =0) ; or Th = trunc ( Thinit ,(1.*(( x -xO ) ^2+( y -yO ) ^2 <= R ^2) ) >.5 , label =0) ;

macro

  AdaptGS ( Th , uadapt , erradapt , isoadapt , smoothadapt , epsadapt , Thinit , output ¯, wv ) mesh Thold = Th ; fespace Vhold ( Thold , P1 ) ; Vhold uold = uadapt ; real [ int ] bbmM (4) ; boundingbox ( Thold , bbmM ) ; mesh Thnew = trunc ( Thinit ,(1.*( x <= ( bbmM [1]+ epsadapt ) & x >= ( bbmM [0] -¯epsadapt ) ) *( y <= ( bbmM [3]+ epsadapt ) & y >= ( bbmM [2] -epsadapt ) ) ) > ¯erradapt , label =0) ; fespace Vhnew ( Thnew , P1 ) ; Vhnew usmadapt , vsmadapt , unew ; matrix BSinterp = interpolate ( Vhnew , Vhold , inside = true ) ; unew []= BSinterp * uold []; solve smoother ( usmadapt , vsmadapt ) = int2d ( Thnew ) ( smoothadapt * usmadapt * ¯vsmadapt + grad ( usmadapt ) '* grad ( vsmadapt ) ) -int2d ( Thnew ) ( smoothadapt *( ¯abs ( unew ) >= erradapt ) * vsmadapt ) + on (0 , usmadapt =0.) ; if ( output ) plot ( usmadapt , fill = true , dim =2 , value = true , wait = wv , cmm = "

  x, y) = -U 2π dq R(R + ξ) + sin δ arctan ξη qR -I sin δ cos δ . where ξ = (x -x 0 ) cos φ + (y -y 0 ) sin φ, Y = -(x -x 0 ) sin φ + (y -y 0 ) cos φ, p = Y cos δ + d sin δ, q = Y sin δ -d cos δ,

Figure 10 -

 10 Figure 10 -Geometry of the source model (left) and the initial solution for η (right, min= -0.46 m, max= 0.73 m ).

Figure 11 -

 11 Figure 11 -Left : Surface projection of the fault's plane and the mesh around, : wave gauge, : epicenter. Right : the 14-th Okada solution (min= -0.09 m, max= 0.17 m ).

Figure 12 -

 12 Figure 12 -Bottom displacement at t = 100 s (left, min= -0.18 m, max= 0.38 m) and at t = 270 s (right, min= -0.18 m, max= 0.45 m).

Figure 16 -

 16 Figure 16 -Comparison between the three methods : Full, Adapt FF and Adapt GS of the free surface elevations (in meters) vs time (in seconds), computed numerically at two wave gauges and of the mass conservation.

Figure 17 -

 17 Figure 17 -Comparison between the three methods : Full, Adapt FF and Adapt GS of the computation time of each adaptemsh, the number of degree of freedom and the computation time of the simulation.

Figure 18 -

 18 Figure 18 -Comparison between the three method Full (up), Adapt GS (middle) and Adapt FF (down) of the maximum of the propagation of the solution of a Tsunami wave in the Mediterranean sea for t = 6800 s.

Figure 19 -

 19 Figure 19 -Passive generation : The bottom and the solution at t = 250 s (left, solution (min= -0.36 m, max= 0.38 m), bottom (min= -6207 m, max= -2096 m)), t = 500 s (center, solution (min= -0.26 m, max= 0.35 m), bottom (min= -6207 m, max= -243 m)) and t = 1000 s (right, solution (min= -0.21 m, max= 0.29 m), bottom (min= -6207 m, max= -100 m)), with the Adapt GS method.

Figure 20 -

 20 Figure 20 -Passive generation : Comparison between the bottom and the solution at t = 1500 s, with the Adapt GS method (left, solution (min= -0.38 m, max= 1.07 m), bottom (min= -6207 m, max= -100 m)) and with the Full one (right, solution (min= -0.38 m, max= 1.07 m), bottom (min= -6207 m, max= -100 m)).

Figure 21 -

 21 Figure 21 -passive generation : Comparison between the two methods Full and Adapt GS of the free surface elevations (in meters) vs time (in seconds), computed numerically at four wave gauges where the gauge (i) correspond to the epicenter.

Figure 22 -

 22 Figure 22 -passive generation : Comparison between the two methods Full and Adapt GS of the Kinetic, Potential and Total energy, the number of degree of freedom and the computation time of the simulation.

Figure 23 -

 23 Figure 23 -Passive generation : Comparison between the maximum of the solution at t = 1750 s, with the Adapt GS method (left) and with the Full one (right).

4. 4

 4 Propagation of a Tsunami wave near Java island : active generation .

Figure 24 -

 24 Figure 24 -Active generation : The bottom and the solution at t = 100 s (left, solution (min= -0.17 m, max= 0.07 m), bottom (min= -6207 m, max= -2589 m)), t = 200 s (center, solution (min= -0.19 m, max= 0.08 m), bottom (min= -6207 m, max= -2285 m)) and t = 270 s (right, solution (min= -0.14 m, max= 0.10 m), bottom (min= -6207 m, max= -2084 m)), with the Adapt GS method.

Figure 25 -

 25 Figure 25 -Active generation : The bottom and the solution at t = 500 s (left, solution (min= -0.15 m, max= 0.10 m), bottom (min= -6207 m, max= -260 m)) and t = 1000 s (right, solution (min= -0.14 m, max= 0.09 m), bottom (min= -6207 m, max= -100 m)), with the Adapt GS method.

Figure 26 -

 26 Figure 26 -Active generation : Comparison between the bottom and the solution at t = 1500 s, with the Adapt GS method (left, solution (min= -0.29 m, max= 0.13 m), bottom (min= -6207 m, max= -100 m)) and with the Full one (right, solution (min= -0.29 m, max= 0.13 m), bottom (min= -6207 m, max= -100 m)).

Figure 27 -

 27 Figure 27 -Active generation : Comparison between the two methods Full and Adapt GS of the free surface elevations (in meters) vs time (in seconds), computed numerically at four wave gauges where the gauge (i) correspond to the epicenter.

Figure 28 -

 28 Figure 28 -Active generation : Comparison between the two methods Full and Adapt GS of the Kinetic, Potential and Total energy, the number of degree of freedom and the computation time of the simulation.

Figure 29 -

 29 Figure 29 -Active generation : Comparison between the maximum of the solution at t = 1750 s, with the Adapt GS method (left) and with the Full one (right).

Acknowledgements : This work would not be done without a lot of advice and special care from Denys Dutykh, Frédéric Hecht, Dimitrios Mitsotakis, and Olivier Pantz, to whom I am grateful for all their help, their fruitful discussions and their remarks.

All the videos of the simulations of a Tsunami wave for the results presents in this paper are given in the following links : http://www.lamfa.u-picardie.fr/sadaka/movies/Tsu_Medit_Full.mov http://www.lamfa.u-picardie.fr/sadaka/movies/Tsu_Medit_Adapt_GS.mov http://www.lamfa.u-picardie.fr/sadaka/movies/Tsu_Medit_Adapt_FF_1EM2_sol.mov http://www.lamfa.u-picardie.fr/sadaka/movies/Tsu_Medit_Adapt_FF_1EM2_mesh.mov http://www.lamfa.u-picardie.fr/sadaka/movies/Tsu_Medit_Adapt_FF_1EM7_sol.mov http://www.lamfa.u-picardie.fr/sadaka/movies/Tsu_Medit_Adapt_FF_1EM7_mesh.mov http://www.lamfa.u-picardie.fr/sadaka/movies/Okada_Java_Dynamic.gif http://www.lamfa.u-picardie.fr/sadaka/movies/Java_Bottom_Displacement.gif http://www.lamfa.u-picardie.fr/sadaka/movies/Tsu_Java_sBBM_Pas_Adapt_GS_Full.mov http://www.lamfa.u-picardie.fr/sadaka/movies/Tsu_Java_sBBM_Act_Adapt_GS_Full.mov

η h , u h and v h on the whole boundary of the square [0, 2L] × [0, 2L], where L = 50 and we consider the following exact solutions: η ex = .2 cos(2πx/L -t) cos(2πy/L -t), u ex = .5 sin(2πx/L -t) cos(2πy/L -t), v ex = .5 cos(2πx/L -t) sin(2πy/L -t), D(x, y) = 1 -.5 cos(2πx/L) cos(2πy/L), adding an appropriate right-hand side function.

We measure at time T = 1 and for θ 2 = 2 3 , δt = .01

and we end up with the following results: So, the L 2 (Ω×]0, T [) 2 norm slope for η and V is of order ∼ 2 and the L 2 (0, T ; H 1 (Ω) 2 ) slope for η and V is of order ∼ 1 as shown in the Figure 13 and which confirms the convergence of the second-order Runge-Kutta scheme in time for the sBBM system with variable bottom in space.

Propagation of a

Tsunami wave in the Mediterranean sea with a flat bottom.

In this section, the non-dimensional and unscaled variables in (4) i.e. g = 1. We simulate here, the propagation of a wave that looks like a Tsunami wave generated by an Earthquake in the Mediterranean sea over the sBBM (4) with a flat bottom -D(x, y) = -1, 5 Km which is the average depth of the Mediterranean sea. This wave was defined above in the passive generation part of the Section 3 where, in this case, the initial solution is concentrated in the small rectangle [x 0 -5W ; x 0 + 4W ] × [y 0 -1.5L; y 0 + 2.5L] and we take these following values : L = 20 Km, W = 10 Km, δ = 7 • , φ = 0 • , θ = 90 • , E = 9, 5 GPA is the Young's modulus, ν = 0, 27 is the Poisson's ratio, U = 2, 5 m, (x 0 ; y 0 ) = (2390. * scale, 590. * scale) and the fault depth(x 0 ; y 0 ) = 10 Km. In this example, we will take the fact that the Lamé constants µ and λ are given by the formulas µ = E/2(1 + ν) and λ = Eν/(1 + ν)(1 -2ν). We also use the following settings : for the step time δt = 0.1 = 1 s, a reflective BC for all the boundary, for the adaptmesh of FreeFem ++ : We note that, we adapt the mesh around the solution each 100 iterations i.e. each 10 s by using the following value for the adapt mesh uadapt= η + u + v, isoadapt=5e-2, erradapt=1e-7, smoothadapt=5e-3, epsadapt=2e-2.

Figure 14 -The mesh and the solution at t = 1000 s, with the Full method at left, the Adapt GS at the center and the Adapt FF with err=1.e-7 at the right.

Figure 15 -The solution at t = 3000 s, with the Full method at left, the Adapt GS at the center and the Adapt FF with err=1.e-7 at the right.


ASSEMBLING BENEVOLENCE -THE TECHNO-POLITICAL PATTERNING OF AUTOMATIC WATER DISPENSERS

published or not. The documents may come

Introduction

Concerned about "providing reliable and sustainable water supply in the developing world," a multinational water pump manufacturing company identified the inability to: collect and manage revenue, reduce "non-revenue-water" and lack of money for service and maintenance as the main challenges for "sustainable water supply" in the Global South. To provide a solution to these supposed problems, the company developed "automatic water dispensers". The dispensers are meant to replace kiosk attendants in already existing water kiosks in "slums" and rural areas. The dispenser can be described as a foursquare unit with a display, an integrated touch screen and a slot on the front side for the blue customer cards that come with the dispenser. Above the slot, the display shows the balance on the card, the amount of the water fetched, and the price per litre. According to the product brochure, the prepayment feature that is integrated in the water cards shall "remove uncertainty in revenue collection". Combined with an internet connected water management system (WMS), the dispensers promise to improve the efficiency of the respective tap water system by facilitating transparent revenue collection. Through these features the water system promises to become sustainable and respectively costcovering. In the cartoon-like promotion film of the dispenser, the protagonist -a person labelled "water provider" -will by the end of the video have transformed into a "water manger".

The emergence of automatic water dispensers can be associated with what Schnitzler (2016, p. 130) identifies as an "unprecedented investment" in creating technologies that are supposed to improve the lives of "the poor" -for example, water filtering (Redfield 2012) or biometric (Donovan 2015) devices. They are therefore being framed as neutral solutions for supposed managerial deficiencies. According to Law and Ruppert however, what "devices are doing is not necessarily written on the package" (2013, p. 230, emphasize in original).

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I FRA

Nairobi

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Nairobi

Volume XIV, (4) 2017

Christiane TRISTL

Lifting jerry cans full of water on a donkey's back after fetching water from an automatic water dispenser in a Kenyan village

CO-CONSTITUTING COST-RECOVERY, ASSEMBLING BENEVOLENCE -THE TECHNO-POLITICAL PATTERNING OF AUTOMATIC WATER DISPENSERS

on how automatic water dispensers play out, is a village in rural Kenya where 30 water dispensers had been implemented by a well know international NGO and funded by a foundation from the UK. Here, typical water sources are rivers, dams, rain water and boreholes. One borehole usually supplies several water kiosks which are connected with pipes. Similarly, as the narrative in the advertisement of the dispenser goes, the problem formulated was that, due to money-mismanagement through the water committee -civilians managing the borehole -and fraud by the kiosk attendants, there was not enough revenue to cover the costs for maintenance. The implementation of the dispenser thus went in line with the efforts to achieve neoliberal policies of full cost recovery that had been brought underway in the kenyan water sector with the Water Act from 2002, and globally since the late 1990s/early 2000s.

Cost-recovery is a contested achievement however, and needs to deal with material negotiations and configurations. Due to often rather far distances between a person's home the next water point, in these rural areas people usually go with their donkeys to fetch water. Before leaving home towards the water point, four empty jerry cans are fixed on one donkey. To accomplish this, two jerry cans are tied together at their handle on top and then lifted on the donkey's back, where they are dangling to the left and right side of her belly (see picture below). Essential for this system of fetching water is to find a hose pipe fixed to the water tap of the kiosk, in order to fill up the jerry cans while they are on the donkey's back. The kiosk attendant used to take care of fixing the hose pipe to the tap every day in the morning, and locking it inside the kiosk in the evening to avoid theft. At many kiosks now, the pipe stayed locked inside the kiosk.

Having no pipe meant there was no way of getting the fluid and fugitive substance all the way from the tab, usually about 50 cm above ground level, to the jerry can on the donkey's back. The customers had to remove the jerry cans to fill them up. Since two jerry cans are always tied together, taking them down means after filling those up, 40 litres have to be put back onto a mostly stubborn donkey. An impossible task for most of the people : "I went to the kiosk sometimes and I filled the jerry cans up. I then had to wait for someone to come by and help me. At times, the next person that comes is not in a position to help you. Maybe it is a child or a Mzee 2 . So, I stopped going there and I now go to the private borehole instead, which is more expensive", a former customer explained. Donkeys seem not to have been taken into account in the scripts of the designers, which meant that they stood in the way of "efficient revenue collection". In other cases, the device managed to pattern different arrangements, like a "collective user" (Akrich 1992, p. 208): people began to organize in a way that they would only go in groups to fetch water, in order to be able to help each other to lift the jerry cans.

The water users that managed to use the dispenser were still confronted with a prepaid component, that patterned the Instead, they are proposing to conceptualize devices as "patterned teleological arrangements". The reference to teleology points towards purpose and functionality which then allows us to say that devices do things. Namely, they "assemble and arrange the world in specific social and material patterns". By hinting towards the social and the material, they are stressing that those patterned arrangements are always heterogeneous. They may include high-tech to bits and pieces, nuclear reactors, but also kit, and they may include people. Devices contribute to the patterning and (re)configuring of the relations between those heterogeneous elements. By so doing, they are constitutive of what is called social order.

Going back to the quote of Law and Ruppert however, how determinate they are is up for empirical investigation. Devices might not always pattern the world according to the scripts that had been imagined by their designers 1 . They may have quite some ordering capacities but the question if they have politics [START_REF] Langdon | Do Artifacts Have Politics?[END_REF], and then which kind of politics they co-constitute, has to be investigated empirically [START_REF] Law | Notes on the Theory of the Actor-Network: Ordering, Strategy and Heterogeneity[END_REF]Akrich 1992). [START_REF] Schnitzler | Democracy's infrastructure. Techno-politics and protest after apartheid[END_REF] argues that technologies can re-stabilize in new contexts and even become constitutive not only of social order, but more specifically of new "ethico-political assemblages". Therefore, the first point is that their effects may change when devices are located in different networks of relations. They might enrol new purposes and elements along the way. The, within the confines of the "same" set of relations, a second point is that these devices might have multiple effects that are often not intentional. Water dispensers might dispense water, but they may also do a range of other things. They might embody a variety of different (political) agendas, which had originally not been envisaged. This again is always subject to change. Devices are contingent, precarious achievements that are never autonomous or final and that may not manage to keep together bits and pieces.

By drawing on ethnographies with automatic water dispensers from two different settings in Kenya, from one settlement within the confines of Nairobi and from a village some hundred kilometres eastwards of Nairobi, I want to discuss how automatic water dispensers pattern heterogeneous materials and reconfigure the relations between them. By doing so I want to show no matter how determinate that patterning is, it is up for empirical investigation, especially if it is political. By drawing on two different settings, I want to engage with how effects change in new sets of relations. For example, how seemingly neutral devices may co-constitute "ethico-political-assemblages" in what one may call techno-political patterning. But the dispensers might also enrol other unforeseen purposes along the way and do different things at the same time.

A disciplinary device for co-constituting cost-recovery

One of the sites where I spent a couple of months researching ] just understands a water card with credits". The word "temptation" can be literally translated in the desire to do something appealing that is against social norms. The dispenser seemed to embody and co-constitute the objectification of some new regime of rational morality, while dismissing behaviour (like giving water to a thirsty person) that others might conceive of as a moral obligation.

Donkeys at times were challenging cost-recovery and led either to absence of people or to new arrangements like "collective users". In many cases however, the patterning of the dispenser seemed to be quite determinate though. Many people were coming forward to play their roles as paying users that even by themselves developed modes of budgeting and thus organizing themselves. At the same time the dispensers failed to assemble budget where there simply was none. Besides its incompatibility with donkeys, the dispenser's own determination undermined its task of dispensing water and cost recovery. The existence of some dispensers was threatened through absence, and they seemed to be in constant danger of "becoming a monument" as the NGO workers said, referring to the numerous water projects implemented in rural areas that are coming to a standstill on a regular basis.

Assembling benevolence, patterning techno-politics

In other sets of relations, the dispensers seemed to play out quite differently. In order to test them in a "peri-urban" area the manufacturer agreed with the Nairobi City Water and Sewerage Company, short Nairobi Water, to install four dispensers in water kiosks in a so called "informal settlement" in the city. The organization of the water supply in this area result in a situation where there are a few water kiosks owned by Nairobi Water and operated by officially registered groups from the area, which usually signed some sort of contract with Nairobi Water. To make some gain, these groups usually take higher prices (around 5 to 10 Kenyan Shilling (KES) per 20 litres) than customers pay when they purchase water from Nairobi Water via household connections. Most importantly however, the area is pervaded by a dense web of "illegal" PVC water pipes that inhabitants connected to the main pipes of Nairobi Water, and that supply taps that are run by informal water vendors. The connections are prone to water rationing -Nairobi Water closing the water for the area frequently, usually without announcement of the time and the duration. In both ways, the inhabitants mostly go to fetch water with yellow 20 litres jerry cans. As a result of shorter distances and rather donkey-unfriendly environments, in this case hose pipes are obsolete for non-commercial water practices at the household level.

"The governor wants the project. The elections are near and the governor wants to say that he has offered cheap, clean water for the poor!", an employee of Nairobi Water named John3 explained. As this quotes describes quite well, the governor, who even made it to the inauguration of the dispensers in the area, strategically used the technology for political goals [START_REF] Hecht | The radiance of France. Nuclear power and national identity after World War II[END_REF]. This techno-political arrangement needed some further patterning however. As opposed to the first setting were cost recovery was set as the priority, here the price for water was set on the low level of 0.50 KES, so that some users even wanted confirmation that Nairobi Water really did mean 50 cents and not 50 Shillings. To circumvent water rationing and provide water 24/7, the dispensers furthermore assembled water tanks on the roof of the kiosks and additionally big blue lorries with some white lettering saying "CLEAN WATER", that had to navigate the narrow paths of the area to fill the tanks up every time when they were empty. To coordinate between empty tanks, the lorries and the inhabitants, Nairobi Water ordered John to be present within the area seven days a week.

Despite the efforts of providing water 24/7 from the dispensers, they were by far not enough to serve the whole area during rationing, which meant that the residents had to walk for long distances to get water from elsewhere. After a period with frequent rationing, there was increasingly unrest. In December 2015, the residents staged a protest blocking the main road to town after lacking water in the area for about three weeks. At this point, the dispensers enrolled a further political agenda since they were now mobilized as a tool of pressure. "They chased us away and they were threatening to burn the dispensers and block the lorries", John said. Indeed, "after the protest, we had to make sure that there was water in the settlement all the time, even in the illegal connections", John continued. Therefore, the dispensers became involved in the resident's struggles for water and contributed to an improvement of water supply in the area in unforeseen ways.

The precarity of the dispenser continued however. Every time when there was no water in the settlement, the inhabitants now come to complain to John. They would threaten to kidnap the lorries and burn the dispensers. When the atmosphere was becoming too tense, John would put pressure on Nairobi Water to provide water for the area. Being present throughout, he would also find out when some protests were being planned. "Now every time we are planning a protest they switch on the water. They try to calm us down", a resident told me. The dispenser and everything it assembled, i.e. the lorries and water tanks, became an arena of political negotiation that furthermore put the existence of the dispenser up for constant negotiation. When I came back to the area after a while, a tank on top of one kiosk had been stolen. Without its water connection, the dispenser became as idle as the previous kiosk operators used to describe themselves, since they had been expelled from the kiosk by the former. In other cases, the dispenser itself was just circumvented by people climbing onto the kiosk and taking out water directly from the tank at night, when John had left the area.

There were many inhabitants coming forward to play their role as users, the dispenser was dispensing water, also the transactions were being tracked by the WMS. People, especially the direct neighbours, were glad about the low prices and about water 24/7. Far from being neutral however, the dispenser co-constituted a political agenda and even enrolled a countering one. It was doing several things at the same time, and it was patterning relations between diverse elements in unforeseen ways. The structures it helped to co-constitute however remained precarious and always prone to breakdown.

Conclusion

Regarding an unprecedented investment in seemingly neutral devices that are supposed to solve the assumed problems of "the poor" and by taking a device to produce "water managers" as an example, I wanted to discuss this assumed neutrality. Instead, I'm prosing to analyse devices as such as what Law and Ruppert term "patterned teleological arrangements". Beyond neutrality, this perspective allows to foreground that the effects of devices change in different sets of relations and that they may also do many things at once, some that were written on the package and some that were unplanned.

In the first setting, the water dispenser co-constituted an assemblage of cost-recovery that set out to discipline and to objectify new regimes of rational morality by dismissing human temptation. In the second setting however, cost-recovery did not play a big role. Other than being determinate, the same mechanism of excluding the human factor mobilized a narrative of benevolence, of providing cheap water for the poor by excluding the cartels and thus became part of the governor's strategy to achieve political goals. Whereas in the rural setting, donkeys and its own determination led to new arrangements or threatened the dispenser's existence through absence, in

  's water practices in different ways, especially regarding the until then, common practice of paying later for water. The more well off of them, like business owners, explained: "It is very good. My guys used to fetch water in the morning and then pay in the evening. Sometimes they would forget, and then all of a sudden you have a chain of enemies. Now I do budgeting for water in the evening, like for everything else". The ones without a budget did not manage to adapt in this way.

	The prepaid
	component seeming quite determinate, they were forced to other
	arrangements: "Now, when there is no money, we go to the dams
	and rivers. Before, we would have talked to the kiosk attendant
	and if you were a trustworthy person, you also could pay the next
	day. But now there is no attendant to listen to you". Opposed
	to this statement, the NGO workers praised the rational virtues
	of the dispenser, by contrasting them to the behaviour of the
	attendant: "She (the attendant) could also give some free water.
	If your mother in law comes or your sister or your father, or a
	friend, who says 'Oh I don't have money, give me some water, I
	will bring money tomorrow.' Maybe you are tempted to give. The
	dispenser […

1 Lifting jerry cans full of water on a donkey's back after fetching water from an automatic water dispenser in a Kenyan village 2 A Kiswahili word for an old person. people

I changed the name due to reasons of anonymization.

Biography

Christiane Tristl is a PhD candidate in Human Geography at Frankfurt University, Germany. She conducts ethnographic research on experiments between philanthrocapitalism and marketization in South-North relations. From a Science and Technology Studies perspective, she is focusing of the coconstitution of these processes with technological devices. The article draws on fieldwork with automatic water dispensers in one settlement within Nairobi, Kenya and in a village on the Kenyan countryside. This allowed her to carve out the relational character of such technological devices. Contact : tristl@em.uni-frankfurt.de