Qussay Jarrar

Farouk Belkadi

Alain Bernard

A Knowledge-Based Approach for Decision Support System in Additive Manufacturing

Keywords: Knowledge-based approach, Decision support system, Automatic Cost Estimation, Additive manufacturing

The large amount and different types of data and knowledge generated within the Additive Manufacturing (AM) value chain are highly challenging in terms of management and organization. Understanding the interconnections between all these immaterial corpuses is important for decision making and process optimization issues. Moreover, AM has more parameters than conventional manufacturing processes, and many of these parameters are difficult to assess and monitor. Therefore, it becomes important to develop computer-based solutions that are able to aid the decision maker and to support the management of all information along the AM value chain. In this paper, a knowledge-based decision support framework using ontological models and mechanisms is proposed for the above objective. Cost estimation is conducted as an application of the proposed framework.

Introduction

Even though the history of Additive Manufacturing (AM) technology is relatively short, it has shown expeditiously growing interest in various domains and significant improvement on the quality of the produced parts as well as the efficiency of the related processes [START_REF] Negi | Basics, applications and future of additive manufacturing technologies: A review[END_REF]. Despite of these benefits and opportunities, many issues and challenges such as traceability, accuracy, and parameters' inconsistency still need to be solved [START_REF] Al-Meslemi | Modeling key characteristics in the value chain of additive manufacturing[END_REF].

With the growing complexity of AM process due to multiple interconnections between heterogeneous parameters, changing one parameter could affect multiple criteria of the produced parts [START_REF] Wang | A decision support system for additive manufacturing process selection using a hybrid multiple criteria decision-making method[END_REF]. Even for expert users, it is difficult to select the suitable values of process parameters for successful AM building results [START_REF] Liu | Ontology based knowledge modeling and reuse approach of supporting process planning in layer-based additive manufacturing[END_REF]. Moreover, different types of data generated along the product transformation cycle are creating new opportunities for knowledge discovery [START_REF] Witherell | Emerging Datasets and Analytics Opportunities in Metals Additive Manufacturing[END_REF]. So, understanding these data and knowledge definition and correlation is essential for decision making and optimization from the earliest stages of AM process.

In this topic, it becomes important to develop the right architecture of decision-making system that correctly uses and manages all this corpus of information [START_REF] Kim | Streamlining the additive manufacturing digital spectrum: A systems approach[END_REF], while paying particular attention to the key performance indicators and process parameters management [START_REF] Belkadi | Towards an Unified Additive Manufacturing Product-Process Model for Digital Chain Management Purpose[END_REF]. Thus, the main objective of this paper is to propose a knowledgebased system (KBS), using an AM dedicated ontology, to support the expert decision along the AM value chain. Section 2 presents a literature review on KBS and decisionaid systems for AM. Section 3 proposes the main foundations of the proposed KBS. Then, an application of this framework to support the decision making in cost estimation is presented in Section 4.

Knowledge-Based Decision Support solutions for AM

Decision Support Systems (DSS) have proven an important role to support the experts in various fields related to AM value chain. For instance, process planning support aims to identify and adjust all parameters necessary to optimize the build and post-processing steps [START_REF] Gibson | Additive manufacturing technologies[END_REF]. Even though, the decision process is still complicated in terms of data management due to huge amount of data and heterogeneity of knowledge. The use of Artificial Intelligence (AI) leads to improvement of DSS through better exploitation of expert knowledge (rule inferences) as well as smart data mining and machine learning [START_REF] Merkert | A Survey of the Application of Machine Learning in Decision Support Systems[END_REF]. KBS are kind of computer systems that imitate human reasoning for problem-solving by reusing previous experiences and inferring business rules [START_REF] Sanfilippo | Ontology-based knowledge representation for additive manufacturing[END_REF]. These systems are currently coupled with Knowledge Management (KM) practices to capture and formalize tacit knowledge [START_REF] Li | Recent development of knowledge-based systems, methods and tools for one-of-a-kind production[END_REF]. This kind of system consists of three main components; Inference engine, User Interface, and Knowledge Base (KB) [START_REF] Ghazy | Development of an additive manufacturing decision support system (AMDSS)[END_REF]. In this type of approaches, knowledge modeling is a critical issue to satisfy the consistency of the KB. With regard, ontology is one of the most effective tools to formalize the domain knowledge, while supporting the automatic reasoning and information retrieval based on business rules [START_REF] Meski | A generic knowledge management approach towards the development of a decision support system[END_REF][START_REF] Eddy | Knowledge management with an intelligent tool for additive manufacturing[END_REF]. Lately, attempts have been made to develop a knowledge base in a form of ontology as a decision support for AM. Some works have contributed to propose a knowledge base that formalize Design for AM (DfAM) knowledge to be retrieved and reused to support the designers in process planning, [START_REF] Kim | A design for additive manufacturing ontology to support manufacturability analysis[END_REF][START_REF] Hagedorn | A knowledge-based method for innovative design for additive manufacturing supported by modular ontologies[END_REF]. For instance, Kim et al. [START_REF] Kim | A design for additive manufacturing ontology to support manufacturability analysis[END_REF] use Ontology Web Language OWL to categorize DFAM knowledge into: Part design, Process planning, and Manufacturing features. SQWRL queries are applied to formalize design rules and retrieve DFAM knowledge to be reused for solving new problems. Using the same mechanisms, Hagedorn et al. [START_REF] Hagedorn | A knowledge-based method for innovative design for additive manufacturing supported by modular ontologies[END_REF] aim to support innovative design in AM, by capturing information related to various fabrication capabilities of AM, and their application in past innovation solutions. Samely, Liu et al. [START_REF] Liu | Ontology based knowledge modeling and reuse approach of supporting process planning in layer-based additive manufacturing[END_REF] propose a KB that uses previous process planning cases and manufacturing rules. Eddy et al. [START_REF] Eddy | Knowledge management with an intelligent tool for additive manufacturing[END_REF] introduce Rules-Based algorithm to select the best process plan as a combination of feasible processes allowing the lowest total cost of the resulted AM part.

In summary, most of the developed approaches are specific and dedicated to solve particular aspects of AM process, so they need high labor intensive to operate in general context. In previous work, Sanfilippo et al. [START_REF] Sanfilippo | Ontology-based knowledge representation for additive manufacturing[END_REF] propose an ontology as a generic reference model with the ambition to represent all main characteristics of AM value chain. The proposed knowledge-based approach is realized as a part of this research work by extending the scope to the preparation and post-processing steps, and taking into account the link between customer specification and process characteristics, for effective cost estimation at the earliest stage of the AM project.

3

The Proposed KBS framework

The computational ontology proposed by [START_REF] Sanfilippo | Ontology-based knowledge representation for additive manufacturing[END_REF] is developed in OWL by means of Protégé tool, and uses the concepts of "template" and "reusable case" as generic containers to structure and reuse useful knowledge for decision-aid perspective. Templates can be seen as a semantic dictionary that organizes the collection of specific types of AM data and knowledge. A "case" is used for the traceability of a given AM project (successful or failed) through the instantiation of a collection of templates.

For instance, the template of machine will be used to create library of existing machine families by defining the main attributes for each category. The first instantiation permits the declaration of a given reference of machine (or process, product, material, etc.) with interval of possible values for the predefined parameters (Ex. Machine XX has heating temperature between 300 and 500 degrees). Then, the creation of reusable case is achieved by combining a set of templates with exact parameters values (ex. Case project ZZ uses the processes PP to transform the material MM in the machine XX with the parameters heating temperature 450°, material density is 4.4 g/cm 3 , etc. It results on the product AA with characteristics: surface roughness: 5-18 µm).

Relaying on that conceptual framework, first the ontology has been enriched with high level of granularity and details axioms covering a wider range of control, and post processes, among others. Five types of templates are initially distinguished: Product, machine, feature, material, and process. Then, templates contents have been updated and other types are added to cover project characteristics, customer specifications, and finally, cost model as an example of decision-aid perspective.

By doing so, the proposed KB is structured and formalized around five layers as shown in fig. 1. The first layer forms the generic knowledge model representing all AM related concepts (i.e. process, Machine, Material, resource, etc.), and their relationships, following the meta-model proposed in [START_REF] Sanfilippo | Ontology-based knowledge representation for additive manufacturing[END_REF] based on the Descriptive Ontology for Linguistic and Cognitive Engineering (DOLCE).

The second layer includes the meta-definition of the different types of templates as well as the semantic definition of case as concepts connected through constraints and the relation HasTemplate. Concretely, Project template allows the description of main categories of projects through their classification by customer activity sector, result type (study, prototype, realization, etc.), complexity, etc. Product template classifies the main families of products according to some invariants (i.e. category of size, form of the support, generic shape, etc.). Feature template completes the product template by classifying main elementary geometric shapes, so that a product can be identified as a combination of elementary shapes and other free forms with specific dimensions. Process Template classifies possible standard processes connected to the AM value chain (support optimization, build, control, post-process, etc.), and identify for every type of process the main parameters to be adjusted in and the constraints to be respected (compatible resources and materials, etc.). Material template helps to distinguish all material types based on their physical properties, as well as compatibility and treatment constraints. Finally, Machine template describes the nominal characteristics of existing AM machines and related equipment (manufacture name, max. build dimensions, temperature range, etc.).

Based on these definitions, the real basic templates are defined in the fourth layer as individuals of the types of templates defined in layer 2, with range of values for every parameter. The fifth layer contains the second instantiation where a collection of templates conducts to a specific reusable case. The exact values of these individuals are automatically extracted from the database. This latest contains all the necessary data collected from customer specification, 3D models and other technical documents. By means, the ontology contains only useful information while the real data is still stored in the database to avoid redundancy. Finally, the third layer gathers the needed business rules and similarity constraints to support the decision-making process through linking the current project data with similar cases, on one side, and the generation of new knowledge by inference, on the other side.

Fig. 1. Knowledge Base structure and layer interconnection

In practice, the execution of decision support starts with the analysis of the customer requirements and additional specifications. New knowledge could be defined and parameters values refined through the application of some business rules. Then, the related templates are identified and business rules are again used to classify the new project in the right category (project complexity, product size, etc.) Based on that, similar cases are searched and the right decision model is identified for application. The results are saved in the database after validation of the expert. However, if similar cases or templates are not found, the enrichment process of the knowledge base is trigged in collaboration with industrial experts to identify new rules and/or cases. The next section illustrates the execution of the decision-aid mechanism for cost assessment, as an important step to help expert for defining cost quotations.

4

Framework Application: (Cost Estimation DSS)

Cost is one of the major performance indicators for the AM manufacturer to be competitive. Its estimation is a challenging task that requires a vast amount of manufacturing knowledge about the whole process in several cases, part of this knowledge is not available in the beginning of the project and the expert should base his quotation on his own expertise with different hypothesizes [START_REF] Kadir | Additive manufacturing cost estimation models-A classification review[END_REF]. Therefore, the proposed solution aims to provide a support for the cost estimation of AM products to help expert when answering specific client demands or to answer call for tenders. For the decision model, the proposed approach is based on Activity-Based Costing (ABC) method for providing appropriate cost structure of the final cost for metal and laser-based Powder Bed Fusion (PBF). The model helps identifying the main cost drivers within AM value chain and their correlations with the available inputs and parameters at early stages when receiving customer specifications. The ABC model is prepared in a form of mathematical equations, coupling the total cost of each activity with the related cost-center resources (machines, labor, tools, etc.).

Fig. 2. ABC Meta-Model

Fig. 2 illustrates the ABC meta-model for the integration of their useful knowledge as a specific template in the knowledge repository (ontology). For the simplicity of reading, UML graphical language is used. The mathematical equations are integrated as constraints and business rules, while the requested parameters to evaluate these equations are available in different templates to support their automatic evaluation. By means, the global equation of the cost model is given bellow. The detailed parameters are out of the scope of this paper because of the page size restrictions.

Final Product cost = Total material cost + Build Platform cost + ∑Activity cost + consumable costs For the sake of example, we assume that we want to estimate the cost of fabricating a specific part using Titanium Ti64 material, and manufactured on an EOS290 DMLS (Direct Metal Laser Sintering) machine (example from [START_REF] Barclift | Cost modeling and depreciation for reused powder feedstocks in powder bed fusion additive manufacturing[END_REF]). In this example only AM process and machine templates are illustrated due to the paper size limitation, and only the costs related to build job, machine setup, and machine output activities are estimated. At first, the required basic templates for the DMLS process are selected from the ontology. Then, the machine type and the resource needed for this process are coupled (machine operator with its cost rate which is equal to 110 euro/hour) as shown in fig. 3. Fig. 4 shows the characteristics of this machine type including the cost rate for running this machine as a cost center rate for build job activity. However, the estimation of build time obtained from several software packages exist for estimating the build time like (Materialise-Magics, EOSprint, etc.). Based on the values stored in the templates, the cost for build job, machine setup, and machine output activities are inferred using the SWRL rules in the ontology and built-in reasoner Pellet that represent the cost estimation equations for these activities, as show in the two rules below R1 and R2. The mapping with the identified templates conducts to the categorization of the project as medium complexity for massive product with medium size. The supporting strategy is standard category.

R1: ProcessTemplate(?x) ^ hasBuildDurationInHour(?x, ?y) ^ specifiesMachineType(?x, ?z) ^ hasCostRateEu-roPerHour(?z, ?a) ^ swrlb:multiply(?I, ?y, ?a) -> has-BuildingCostInEuro(?x, ?I) R2: ProcessTemplate(?x) ^ specifiesMachineType(?x, ?y) ^ hasMachineSetupDurationInHour(?y, ?z) ^ hasMachine-OutputDurationInHour(?y, ?a) ^ hasActorBusinessRole(?x, ?t) ^ hasCostRateEuroPerHour(?t, ?i) ^ swrlb:multiply(?u, ?z, ?i) ^ swrlb:multiply(?p, ?a, ?i) -> hasMachineSetupCostInEuro(?x, ?u) ^ hasMachineOut-putCostInEuro(?x, ?p)

Conclusion and Future work

The main goal of the proposed knowledge-based framework is to improve the performance of additive manufacturing value chain, by supporting data sharing and tracing along AM value chain, and support the experts' decision making in different cases of AM field. This work details the methodological framework in terms of the conceptual approach, and framework layer's structure. Ontology has been used to constitute the knowledge base with its ability to capture heterogeneous knowledge, along with rulebased approach in a form of SWRL in the ontology to integrate the amount of knowledge provided by experts (know-how, decisions, etc.) in order to deduce new knowledge. One application of this framework is exposed to support the decision making in cost estimation which aims to aid the manufacturing products cost estimation. Using Activity-Based Costing (ABC) method for providing appropriate cost structure of the final cost, identifying the main cost drivers within AM process chain and their correlations to the available inputs and parameters at early stages, considering customer specifications, and product characteristics. The project is currently under progress. At one hand, the approach needs to be complemented with other techniques beside the rule-based approach such as similarity measures, in order to foster the use of this framework in real world setting. On the other hand, the implementation of this framework in software application requests the creation of specific connectors and parsers to automatically extract data from documents and 3D models. Moreover, future work will be important to define other decision-making scenarios focusing on qualitative performance indicators.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Individual of AM Process Template (DMLS)

Acknowledgement. The presented results were conducted within the French national project "SOFIA" (SOlution pour la Fabrication Industrielle Additive métallique). This project has received the support from the French Public Investment Bank (Bpifrance) and the French National Center for Scientific Research (CNRS). The authors would like to thank all industrial and academic partners for their involvement in this research.


Emilie Bechard

Jamie Bride

Jean-Yves Le Guennec

Fabien Brette
email: fabien.brette@inserm.frmariedemion

Marie Demion
email: marie.demion@umontpellier.fr

TREK-1 in the heart: Potential physiological and pathophysiological roles

Keywords: TREK-1, K2P2.1, atrial fibrillation, myocardial infarction, RVOT

Introduction

In cardiomyocytes, excitation-contraction coupling, the process that links electrical activity to cell contraction, is well established. On the contrary, the mechano-electric feedback (MEF) mechanism that can be summarized by how mechanical stimulation can modulate action potential (AP) shape is less understood. Such a feed-back mechanism must involve mechanosensitive signals, mainly stretch, especially via mechano-gated ionic channels. K2P channels are modulators of cardiac repolarization and within this family of channels, TREK-1 is the most abundant channel expressed in the human heart [START_REF] Wiedmann | Mechanosensitive TREK-1 two-pore-domain potassium (K2P) channels in the cardiovascular system[END_REF]. The mechano-gated TREK-1 channel is differentially expressed in regions of the heart participating in electrical heterogeneity within specific areas [START_REF] Tan | Differential expression of the mechanosensitive potassium channel TREK-1 in epicardial and endocardial myocytes in rat ventricle[END_REF]. TREK-1 is 17 time more express at the mRNA level in the endocardium when compared to epicardium, and 3-fold at the functional (current) level [START_REF] Kelly | Gene expression of stretch-activated channels and mechanoelectric feedback in the heart[END_REF].

TREK-1 and mechano-electrical feedback

As well as TREK-1 channel expression is heterogenous in ventricle, MEF differs regionally. In 1999, it was demonstrated that, in response to stretch, canine AP was shortened at the endocardial level but not in the epicardium leading to ventricular arrhythmias [START_REF] Takagi | Gadolinium suppresses stretch-induced increases in the differences in epicardial and endocardial monophasic action potential durations and ventricular arrhythmias in dogs[END_REF]. Such stretch-induced arrhythmias were sensitive to gadolinium, a non-specific stretchactivated channel blocker. In 2006, Kelly and collaborators demonstrated that in Langendorff perfused rat isolated heart, increasing intraventricular pressure by inflation of a balloon, induced a shortening of the AP both in subepicardial and subendocardial zones [START_REF] Kelly | Gene expression of stretch-activated channels and mechanoelectric feedback in the heart[END_REF]. However, AP shortening was more pronounced in the endocardium than in the epicardium at all repolarization levels. This suggests that the spatial variation of TREK-1 channel expression may influence regionally the MEF altering in turn the dispersion of AP repolarization and induces disturbances in repolarization in several models including human [START_REF] Sarubbi | Electrophysiological changes following balloon valvuloplasty and angioplasty for aortic stenosis and coartaction of aorta: Clinical evidence for mechanoelectrical feedback in humans[END_REF]. Such repolarization dispersion (i.e. AP at different part of the ventricle repolarizing at different times) is arrhythmogenic [START_REF] Roden | QT prolongation and arrhythmia suppression[END_REF] and is normally reduced under physiological conditions. The different AP durations (APD) in the myocardium finally synchronize their repolarization along the wall despite an asynchronous depolarization time [START_REF] Volders | Progress in the understanding of cardiac early afterdepolarizations and torsades de pointes: Time to revise current concepts[END_REF].

However, in several cardiac pathologies, both TREK-1 channel expression and MEF are modified. In pathological conditions where mechanical properties are altered, such as fibrosis, a slowed AP propagation may induce a dispersion of repolarization and thus arrhythmias, altering both MEF and APD regulation.

For example, in rat, TREK-1 channel expression is increased by cardiac hypertrophy induced by isoproterenol and this is associated with an amplified transmural gradient (endocardium vs. epicardium) of TREK-1 channel [START_REF] Wang | An increased TREK-1-like potassium current in ventricular myocytes during rat cardiac hypertrophy[END_REF]. This upregulation might be linked to reexpression of fetal genes, characteristic finding of pressure overload-induced cardiac hypertrophy, however to the best of our knowledge there is no information about a possible developmentally regulated expression of TREK-1 gene. In mammals, global deletion of TREK-1 induces an exaggerated concentric hypertrophy in response to pressure overload, but protects from the development of systolic and diastolic dysfunction. As TREK-1 is present in cardiac fibroblasts and modulates their function, its specific deletion in fibroblasts reduces fibrosis development. Thus, cardioprotective effects of TREK-1 loss of function are driven by cardiac fibroblasts [START_REF] Abraham | The two-pore domain potassium channel TREK-1 mediates cardiac fibrosis and diastolic dysfunction[END_REF]. TREK-1 activity can also be changed during pathology since the activation of TREK-1 by Polyunsaturated fatty acids like arachidonic acid might occur in pathophysiological situations like ischemia when ATP cause the release of this fatty acid [START_REF] Aimond | Simultaneous activation of p38 MAPK and p42/44 MAPK by ATP stimulates the K+ current ITREK in cardiomyocytes[END_REF].

However, the transmural variation of TREK-1 expression cannot explain by itself all the MEF variability. Indeed, the cellular strain and forces are locally different from the endocardial to epicardial cells when the ventricle is deformed. This was first shown at the cellular level [START_REF] Cazorla | Length-tension relationships of subepicardial and sub-endocardial single ventricular myocytes from rat and ferret hearts[END_REF] and more recently in living individuals cardiac layers [START_REF] Pitoulis | Intact myocardial preparations reveal intrinsic transmural heterogeneity in cardiac mechanics[END_REF]. For example, during end-diastolic relaxation, significant circumferential stretch, wall thinning, and in-plane and transverse shear were observed. Left ventricular mechanics during the early relaxation phase involves substantial deformation of fiber and sheet structures with significant transmural heterogeneity. A significant epicardial stretch along myofibers was observed during early relaxation when compared to endocardial fibers [START_REF] Ashikaga | Transmural left ventricular mechanics underlying torsional recoil during relaxation[END_REF].

Physiological role of TREK-1

While some changes in the expression of TREK-1 might appear in pathological circumstances, a putative physiological role of TREK-1 has never been shown. Knowing its electrophysiological properties, channel open at rest and outwardly rectifying current, led to the inference that this channel must hyperpolarize the resting membrane potential and/or shorten the APD of cardiac cells but this has never been shown [START_REF] Benoist | Systems approach to the study of stretch and arrhythmias in right ventricular failure induced in rats by monocrotaline[END_REF][START_REF] Kelly | Gene expression of stretch-activated channels and mechanoelectric feedback in the heart[END_REF]. The reason for this is probably that there is no specific pharmacology for TREK-1 and that TREK-1 blockers are often other potassium conductance blockers [START_REF] Wiedmann | Mechanosensitive TREK-1 two-pore-domain potassium (K2P) channels in the cardiovascular system[END_REF]. It has to be underlined that the co-expression of KIR2.1, the molecular identity responsible for the current responsible of the cardiac resting membrane potential IK1, and TREK-1 shows that TREK-1 influences slightly the resting membrane potential (Decher et al., 2017b). Also, there are many cases of mutations in channels carrying the rapid and slow delayed-rectifier and basal inward-rectifier potassium currents (IKr, IKs and IK1, respectively), involved in the reserve of repolarization, that lead to long QT syndrome [START_REF] Crotti | Heritable arrhythmias associated with abnormal function of cardiac potassium channels[END_REF]. To the best of our knowledge, such mutations have never been described for TREK-1. This does not mean that ITREK-1 is not involved in ventricular repolarization, since such mutations can be lethal or induce other kind of adaptation.

Interestingly, it has been observed that TREK-1 channel is mainly expressed in the more mechanical constrained area of the heart. For example, TREK-1 is more abundant in endocardial cells than in epicardial cells [START_REF] Tan | Differential expression of the mechanosensitive potassium channel TREK-1 in epicardial and endocardial myocytes in rat ventricle[END_REF][START_REF] Kelly | Gene expression of stretch-activated channels and mechanoelectric feedback in the heart[END_REF]. Hence, the effects on AP shortening of inflating a balloon in the left ventricle are more pronounced in the endocardium than in the epicardium. Besides these observations, endocardial cells are known to be more rigid than epicardial cells [START_REF] Cazorla | Resting tension participates in the modulation of active tension in isolated Guinea pig ventricular myocytes[END_REF][START_REF] Cazorla | Modulation of ions channels and membrane receptors activities by mechanical interventions in cardiomyocytes: Possible mechanisms for mechanosensitivity[END_REF]. In addition, [START_REF] Gannier | Streptomycin reverses a large stretch induced increases in [Ca2+]i in isolated Guinea pig ventricular myocytes[END_REF], observed that stretching single cells can induce a large increase in diastolic calcium. This phenomenon can occur for small sarcomere lengthening that is coherent with stiff endocardial cells. This phenomenon is accompanied by a stable membrane depolarization that is at the origin of the diastolic calcium increase [START_REF] White | The effects of increasing cell length on auxotonic contractions; membrane potential and intracellular calcium transients in single Guinea-pig ventricular myocytes[END_REF][START_REF] Gannier | A possible mechanism for large stretch-induced increase in [Ca2+]i in isolated Guinea-pig ventricular myocytes[END_REF]. Thus, this phenomenon, linked to the presence of inward stretch-activated current, must be prevented by the presence of TREK-1 that would compensate the inward current, especially in cells more prone to present this phenomenon, i.e., stiffer cells, endocardial cells.

FIGURE 1 Schematic representation of TREK-1 involvement in cardiac physiology and pathology.

Left, in the heart, there is a gradient of TREK-1 expression (hence current) between endocardium (endo) and epicardium (epi). Right, TREK-1 is a potassium channel and this physiological role is unclear. It may be involved in resting membrane potential (RMP) stabilization, action potential duration (APD) shortening. Decrease expression has been observed in several pathologies: pulmonary arterial hypertension (PAH), myocardial infarction (MI) and atrial fibrillation (AF). TREK-1 can be linked to arrhythmia via mutations. See text for details.

The right ventricular outflow tract (RVOT) is also a region of the heart with high physical strain. This might explain, in part at least, that the RVOT is a region of the heart where arrhythmic syndromes and the majority of idiopathic ventricular arrhythmias are found in the absence of any structural modifications and without any known etiology [START_REF] Srivathsan | Ventricular tachycardia in the absence of structural heart disease[END_REF]. It has been noted that in rat ventricle, the RVOT APD was shorter than in the other part of the right ventricle even though there was no evidence that it was due to an overexpression or overactivity of TREK-1 [START_REF] Benoist | Systems approach to the study of stretch and arrhythmias in right ventricular failure induced in rats by monocrotaline[END_REF].

However, there are some evidences of the role of TREK-1 remodeling in the development of some arrhythmias [START_REF] Benoist | Systems approach to the study of stretch and arrhythmias in right ventricular failure induced in rats by monocrotaline[END_REF][START_REF] Kamatham | TREK-1 protects the heart against ischemia-reperfusion-induced injury and from adverse remodeling after myocardial infarction[END_REF].

TREK-1 and pathological condition

While the physiological role of TREK-1 has not been proven formally, some cardiac diseases have been shown to be associated with changes in TREK-1 expression and the implication of TREK-1 can be inferred based on its electrophysiological properties or even proved by restoration of the initial expression [START_REF] Zhao | Regional differential expression of TREK-1 at left ventricle in myocardial infarction[END_REF][START_REF] Benoist | Systems approach to the study of stretch and arrhythmias in right ventricular failure induced in rats by monocrotaline[END_REF][START_REF] Lugenbiel | TREK-1 (K2P2.1) K+ channels are suppressed in patients with atrial fibrillation and heart failure and provide therapeutic targets for rhythm control[END_REF][START_REF] Kamatham | TREK-1 protects the heart against ischemia-reperfusion-induced injury and from adverse remodeling after myocardial infarction[END_REF][START_REF] Dragasis | Modern mapping and ablation of idiopathic outflow tract ventricular arrhythmias[END_REF].

RVOT

The RVOT is the area of the right ventricle through which blood flows in to be expulsed in the pulmonary artery. The RVOT is under a high physical strain and is known to express TREK-1. A significant proportion of ventricular extrasystoles originate from the outflow tracts, left or right [START_REF] Dragasis | Modern mapping and ablation of idiopathic outflow tract ventricular arrhythmias[END_REF]. There are at least two kinds of tachycardia that have the RVOT as the locus for arrhythmias.

Mutated leaky TREK-1 channel

RVOT ventricular tachycardia (RVOT-TC) is a common form of monomorphic ventricular tachycardia without any structural disease and without any explanation. 10% of ventricular tachycardia are idiopathic and interestingly 70% of these arrhythmias originate in the RVOT [START_REF] Goldstein | Sodium leak through K2P potassium channels and cardiac arrhythmia, an emerging theme[END_REF]. While they are considered as benign, it has been shown that they can degenerate in ventricular fibrillation, often preceded by episodes of syncope [START_REF] Noda | Malignant entity of idiopathic ventricular fibrillation and polymorphic ventricular tachycardia initiated by premature extrasystoles originating from the right ventricular outflow tract[END_REF]. Decher et al. (2017b) identified a patient with a point mutation in TREK-1, I267T, located directly before the selectivity filter of the second pore loop. This mutation, that might be dominant-negative, makes TREK-1 permeable to sodium. Also, TREK-1 has an increased sensitivity to stretch accompanied by a reduced desensitization of the channel. Episodes of RVOT-TC are generally observed in situations where the sympathetic system is activated and thus normally TREK-1 is more or less inhibited. Decher et al. (2017b) showed that the activation of ß1-receptors, co-expressed in oocytes with TREK-1I267T, induces a larger depolarizing inward sodium current and thus a larger depolarization that must participate as such to the development of arrhythmias. Indeed, the increased entry of sodium leads to an increased entry of calcium through the sodium-calcium exchanger that in turn produces arrhythmias due to the transient inward current ITI [START_REF] Lederer | Transient inward current underlying arrhythmogenic effects of cardiotonic steroids in Purkinje fibres[END_REF]. It has to be noted that another gene modification, an alternative translation initiation, leads to an increased sodium permeability [START_REF] Thomas | Alternative translation initiation in rat brain yields K2P2.1 potassium channels permeable to sodium[END_REF]. However, the normal and pathological expression of this variant is not known as well as the physiological sodium permeability (Decher et al., 2017b;2017a). Interestingly, the I267T mutant can recover its potassium selectivity thanks to BL-1249, a TREK-1 channel activator, opening hopes in the treatment of such a rare disease.

Pulmonary arterial hypertension

Pulmonary arterial hypertension (PAH) is a cardiac disease resulting from to a vasoconstriction and/or cell proliferation in the pulmonary circulation. The main consequences of PAH are right ventricular hypertrophy, arrhythmias and failure. A rat model of PAH has been developed by injecting monocrotaline, a plant alkaloid. [START_REF] Benoist | Systems approach to the study of stretch and arrhythmias in right ventricular failure induced in rats by monocrotaline[END_REF] used that model to understand the evolution of remodeling associated with heart failure and the occurrence of arrhythmias. They found that while the APD in the RVOT is shorter than in the rest of the right ventricle in healthy animals, this APD increased more in the RVOT with PAH. The authors showed thus that remodeling is more important in the RVOT than in the other areas of the right ventricle and that such changes in APD in adjacent part of the ventricle can be responsible of arrhythmias due the dispersion of repolarization. While there was no functional analysis performed to explain such a lengthening of the AP, [START_REF] Benoist | Systems approach to the study of stretch and arrhythmias in right ventricular failure induced in rats by monocrotaline[END_REF], showed that the level of mRNA coding for TREK-1 decreased only in the right ventricle, even though there was no particular focus on the RVOT. By correlating this decrease in TREK-1 gene expression, and the associated decrease in TRPC1 and increase in TRPC6 expression [START_REF] Benoist | Systems approach to the study of stretch and arrhythmias in right ventricular failure induced in rats by monocrotaline[END_REF], to the channel conductance's, they showed that the increase in APD can be explained.

These findings are of great interest since they were counterintuitive knowing that an acute stretch induces an increase of the TREK-1 protein [START_REF] Zhao | Effects of acute mechanical stretch on the expression of mechanosensitive potassium channel TREK-1 in rat left ventricle[END_REF]. A possible explanation might be that initially the acute strain induces an increase of TREK-1 to compensate and, with time, there is the installation of a chronic strain that induces a decompensation linked to, at least, a remodeling of TREK-1 expression.

Atrial fibrillation

Atrial fibrillation (AF), the most common arrhythmia, is associated with both structural and electrical remodeling that cause disturbance in repolarization and conduction and contribute to arrhythmogenesis [START_REF] Schmidt | Stretch-activated two-pore-domain (K2P) potassium channels in the heart: Focus on atrial fibrillation and heart failure[END_REF][START_REF] Wakili | Recent advances in the molecular pathophysiology of atrial fibrillation[END_REF][START_REF] Dobrev | Novel molecular targets for atrial fibrillation therapy[END_REF]. TREK-1 expression in human atrium displays a relevant level that suggests functional and potential therapeutic roles [START_REF] Schmidt | Stretch-activated two-pore-domain (K2P) potassium channels in the heart: Focus on atrial fibrillation and heart failure[END_REF]. Indeed, a down-regulation of TREK-1 mRNA expression and protein level in atrium were observed in patients with AF complicated by heart failure [START_REF] Schmidt | Class I antiarrhythmic drugs inhibit human cardiac two-poredomain K(+) (K2 ₂p) channels[END_REF][START_REF] Schmidt | Stretch-activated two-pore-domain (K2P) potassium channels in the heart: Focus on atrial fibrillation and heart failure[END_REF][START_REF] Lugenbiel | TREK-1 (K2P2.1) K+ channels are suppressed in patients with atrial fibrillation and heart failure and provide therapeutic targets for rhythm control[END_REF]. Down-regulation of TREK-1 was similarly observed in a tachypacing pig model of AF with reduced left ventricular ejection fraction. This downregulation of TREK-1 expression is consistent with a prolonged APD and effective refractory periods (ERP) but also with an increased susceptibility to experience extrasystoles. The downregulation of TREK-1 mRNA in neonate ventricular cardiomyocytes from rat under mechanical stretch suggests that dilation of atria observed in AF patients may directly contribute to TREK-1 remodeling [START_REF] Lugenbiel | TREK-1 (K2P2.1) K+ channels are suppressed in patients with atrial fibrillation and heart failure and provide therapeutic targets for rhythm control[END_REF]. Then, restoring TREK-1 expression in pig with AF using gene therapy was followed by a recovery of sinus rhythm and prolongation of the ERP was attenuated [START_REF] Lugenbiel | TREK-1 (K2P2.1) K+ channels are suppressed in patients with atrial fibrillation and heart failure and provide therapeutic targets for rhythm control[END_REF]. Moreover, overexpressing TREK-1 by gene therapy enhances TREK-1 current and results in APD shortening in mouse cardiomyocytes [START_REF] Lugenbiel | TREK-1 (K2P2.1) K+ channels are suppressed in patients with atrial fibrillation and heart failure and provide therapeutic targets for rhythm control[END_REF]. TREK-1 channel downregulation is clearly involved in the development of AF and this suggests a potential therapeutic significance of TREK-1 gene therapy to manage AF complicated by heart failure.

Myocardial infarction

Post-myocardial infarct arrhythmias are a major risk factor of sudden cardiac death. After myocardial infarction (MI), an electrical remodeling occurs, increasing the electrical instability of the post-MI heart [START_REF] Huang | Spatial alterations of Kv channels expression and K(+) currents in post-MI remodeled rat heart[END_REF]. In 2002, [START_REF] Barrabés | Ventricular fibrillation during acute coronary occlusion is related to the dilation of the ischemic region[END_REF] demonstrated that MEF can be enhanced in postinfarcted myocardium when compared with normal cardiac muscle, leading to mechano-induced ventricular arrhythmias and consequently ventricular fibrillation [START_REF] Barrabés | Ventricular fibrillation during acute coronary occlusion is related to the dilation of the ischemic region[END_REF]. [START_REF] Zhao | Regional differential expression of TREK-1 at left ventricle in myocardial infarction[END_REF] showed that TREK-1 channel remodeling occurs after MI in both endocardium and epicardium. TREK-1 expression decreased in the infarcted region while it is increased in the infarcted border region as a compensatory mechanism. This remodeling of TREK-1 expression in infarcted border region, positively correlated with outward TREK-1 current, could be responsible of the post-MI arrhythmogenesis by inducing a dispersion of effective refractory periods in adjacent cardiac areas. A more recent study, using TREK-1 invalidated model, demonstrated that following coronary artery ligation, the infarct size was increased in TREK-1 knock-out mice, associated with higher systolic dysfunction when compared to wild-type mice [START_REF] Kamatham | TREK-1 protects the heart against ischemia-reperfusion-induced injury and from adverse remodeling after myocardial infarction[END_REF]. This study suggests that TREK-1 expression could be protective during ventricular remodeling by regulating membrane potential and thus calcium homeostasis. During the process of cardiac work after an infarction, the effects of mechanical stretch on myocardium in different regions may produce an altered electrical activity, inducing disturbances in repolarization and electrical instability that is a critical factor for arrhythmogenesis.

In conclusion, the ventricle preload and wall thickness modifications are able to alter the electrical signal induced by the MEF. This signal provides the trigger and/or the substrate to ventricular arrhythmias.

Future directions

To summarize our knowledge about the functional role of TREK-1 (Figure 1), it can be: 1/putative physiological roles are possible based on the electrophysiological properties of ITREK-1. 2/there are some changes in the expression of TREK-1 with some chronic disease that are involved in the development of the disease but the mechanisms are not understood. Different hypothesis can be put forward to explain this situation. The first is the lack of a selective blocker for TREK-1. Spadin is given as such a selective blocker, but its practical use is limited since the channel first has to be activated by arachidonic acid to be sensitive to spadin [START_REF] Hivelin | Potentiation of calcium influx and insulin secretion in pancreatic beta cell by the specific TREK-1 blocker spadin[END_REF][START_REF] Djillani | Shortened spadin analogs display better TREK-1 inhibition, in vivo stability and antidepressant activity[END_REF][START_REF] Djillani | Fighting against depression with TREK-1 blockers: Past and future. A focus on spadin[END_REF][START_REF] Ma | Spadin selectively antagonizes arachidonic acid activation of TREK-1 channels[END_REF]. The second aspect is the pronounced species differences in K+ channel expression which provide an additional challenge for the elucidation of the role of TREK-1 in the heart. Finally, another aspect that has not been explored is the potential non-conducting properties of TREK-1. Indeed, in pathologies like breast cancer, such non-ionic properties of ion channels are well-known [START_REF] Forzisi | Non-conducting functions of ion channels: The case of integrin-ion channel complexes[END_REF]. Whatever the explanation, it appears more and more obvious that TREK-1 plays a role in cardiac physiology and it represents a new target in cardiology.

Funding

FB is financially supported by the French National Funding Agency for Research (ANR) (ANR-20-CE17-0010). MD, J-YL-G and EB are financially supported by the French National Funding Agency for Research (ANR) (ANR-20-CE44-001). JB is financially supported by the French CNRS labelling PRIME DO-iT and by the Region Occitanie.

Author contributions

J-YLG wrote first draft. EB, JB, FB, and MD contributed to manuscript formulation and revision. All authors have read and approved the final manuscript.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


In this paper, a coupling of the Finite Element Method (FEM) and the Boundary Element Method (BEM) is used to model the behavior of magnetoelectric effects in composite structures. This coupling of numerical methods makes it possible not to have to consider a free space domain, and thus to use a single mesh for the magnetic, mechanical and electrical problems. This results in a consequent reduction of the number of unknowns which is accompanied by shorter computation times compared to a classical FEM approach. A mixed magnetic vector potential, reduced magnetic scalar potential formulation is used for the magnetic problem, and classical FEM formulations are used for electrical and mechanical problems. The resulting global algebraic system is solved by a block Gauss-Seidel solver.

Index Terms-FEM-BEM coupling, magnetoelectric, piezoelectricity, piezomagnetism, multiphysics, Gauss-Seidel method the electrical and mechanical quantities [START_REF] Tiersten | Hamilton's principle for linear piezoelectric media[END_REF]. This is not the case for magnetostrictive materials, which generally exhibit non-linear behavior. However, it is possible to induce a polarization state by applying a magnetic field and/or a mechanical prestress and thus define a set of linear relationships [START_REF]Modeling of Giant Magnetostrictive Materials[END_REF]. In this framework, the mechanical, electrical and magnetic behavior relations are written:

     T = c E,B : S -t e • E -t h • B, (1)
D = e : S + ε S • E, (2)
H = -h : S + ν S • B, (3)
where T , S, D, E, B and H are, respectively, the stress tensor, the linear strain tensor, the electric flux density, the electric field, the magnetic flux density and the magnetic field. c E,B is the stiffness tensor at constant electric field and magnetic flux density, ε S is the electrical permittivity tensor at constant strain, ν S is the magnetic reluctivity tensor at constant strain, e is the piezoelectric tensor and h the piezomagnetic tensor. As we are describing heterogeneous composite structures, the piezoelectric tensor e is taken null in the piezomagnetic phase and, reciprocally, piezomagnetic tensor h is taken equal to zero in the piezoelectric phase.

III. ELECTRO-MAGNETO-MECHANICAL MODELING

We consider an open boundary domain Ω = Ω 0 ∪ Ω pe ∪ Ω pm where Ω 0 , Ω pe and Ω pm represent the free space, the magnetostrictive and the piezoelectric domains, respectively, as shown in Figure [START_REF] Cheng | Recent development and status of magnetoelectric materials and devices[END_REF]. We will also call Ω m = Ω pe ∪ Ω pm the domain of active materials. The magnetoelectric problem consists in solving Maxwell's equations in the framework of quasistatic regime [START_REF] Jackson | Classical Electrodynamics[END_REF]:

∇×E = 0, ∇•D = 0, ∇×H = J s , ∇•B = 0 in Ω, (4
) as well as the balance of linear momentum for the mechanical equilibrium without body force:

∇ • T = 0 in Ω m .
(5)

Modeling of Magnetoelectric Effects in Composite structures by FEM-BEM coupling

Alberto Urdaneta-Calzadilla, Olivier Chadebec, Nicolas Galopin, Innocent Niyonzima, Gerard Meunier, Bertrand Bannwarth Univ. Grenoble Alpes, CNRS, Grenoble INP, G2Elab, F-38000 Grenoble, France

I. INTRODUCTION

Without relying on induction, electromagnetic coupling can be obtained through the mechanical association of piezoelectric and magnetostrictive materials. The resulting composite structure displays an extrinsic magnetoelectric effect [START_REF] Cheng | Recent development and status of magnetoelectric materials and devices[END_REF] and can allow for magnetic to electric energy conversion or viceversa. For the modelling of magnetoelectric composites, the FEM allows solving for the behavior of materials with nontrivial structures [START_REF] Nguyen | Finite element modeling of magnetic field sensors based on nonlinear magnetoelectric effect[END_REF]. Nevertheless, a classical FEM approach has several disadvantages. A free space domain has to be considered to properly model the decay of magnetic fields at infinity, and the extent of this domain is dependent on the distance between the field sources and the active materials. A FEM-BEM coupling for the magnetic phenomena based on a mixed formulation of the magnetic problem [START_REF] Phan | BEM-FEM formulation based on magnetic vector and scalar potentials for eddy current problems[END_REF] can be applied to the modelling of the multiphysics phenomena arising in magnetoelectric composite structures, thus allowing not to explicitly consider a free space region. The magnetoelectric problem is then decomposed into three sub-problems: a magnetic problem solved using the FEM-BEM approach, and an electrical and a mechanical problem solved using classical FEM formulations. This coupling of numerical methods introduces resolution difficulties that can be overcome by using a block Gauss-Seidel type solving algorithm.

After introducing the used behavioral laws and the chosen formulation for each single-physics problem, we will compare our magneto-mechanical formulation to an analytical solution. We will then introduce our multiphysics solver and study its convergence for a simple test case.

II. BEHAVIORAL LAWS

Magnetoelectic

composites involve piezoelectric and magnetostrictive materials characterized by strong electro(magneto)-mechanical coupling. Piezoelectric materials can be described by a set of linear relationships between To derive a variational formulation inside the active material domain Ω m in terms of the magnetic vector potential a (B = ∇ × a), considering the absence of electric current in Ω m , we multiply (4), Maxwell-Ampere equation ( 4), by the test function δa. After integration by parts, it follows:

A. Magnetic problem ∂Ω m J s Ω 0 Ω pe Ω pm ∂Ω ∞ = ∂Ω 0
Ωm ∇ × δa • H dΩ - ∂Ωm (δa × H) • n d∂Ω = 0 ∀ δa, (6)
where ∂Ω m is the exterior surface of Ω m and n the unit outward normal vector to ∂Ω m . Given the magnetic behavior law (3), we can rewrite the volume term in (6) as:

Ωm ∇ × δa • HdΩ = Ωm ∇×δa • ν S • ∇×a dΩ m - Ωpm ∇×δa • h : S dΩ m . (7)
Furthermore, H = H 0 -∇φ red , with H 0 the field created by external currents J s , calculated by the Biot-Savart law, and φ red the magnetic reduced scalar potential. After the application of some vector identities applied to the surface term of (6), similarly to [START_REF] Phan | BEM-FEM formulation based on magnetic vector and scalar potentials for eddy current problems[END_REF], the weak form reads: find (a, φ red ) such that:

Ωm ∇×δa • ν S • ∇×a dΩ m + ∂Ωm (∇ × δa) • n φ red d∂Ω m - Ωpm ∇×δa • h : S dΩ m = ∂Ωm (δa × n) • H 0 d∂Ω m ∀ δa. (8)
The discretization of ( 8) is performed using edge elements for the vector potential a and 0-order surface elements for the reduced scalar potential φ red . The previous discretization can take account of a nonlinearity in ν S .

The behavior of the magnetic field in the free space domain is taken into account by adding an equation to the system. In ∂Ω m , the potential φ red verifies the Laplace equation ∆φ red = 0 [START_REF] Bastos | Electromagnetic Modeling by Finite Element Methods[END_REF]. Green's third identity applied to φ red in ∂Ω m leads to the following equation [START_REF] Brebbia | Boundary Element Techniques in Engineering[END_REF]:

1 2 φ red - ∂Ωm φ red ∂G ∂n d∂Ω m + ∂Ωm G B n µ 0 d∂Ω m = ∂Ωm G H 0n d∂Ω m , (9)
where G is the Green function of the 3D Laplacian operator.

Using a Galerkin approach. Equation ( 9) is discretized using 0-order surface elements. As B n dS = a surf • dl, after discretization of (9) the magnetic vector potential in edge elements is introduced in the discretized equation from the flux of the magnetic induction through each facet element.

Ω pm ∂Ω t m ∂Ω u m f T u s Ω pe (a) Mechanical domain Ω pm ∂Ω d m -Q s n ∂Ω ϕ m ϕ 0 Ω pe (b) Electrical domain

B. Electric and mechanical problems

The electrical and mechanical problems are limited to the active material's domain Ω m (Fig. 2). Indeed, even if a FEM-BEM coupling can also be formulated for the electrical problem, we can neglect the leaks of the electric field for two reasons: the high permittivity of the considered piezoelectric materials channels the fields inside the active domain, and the electrodes exciting the piezoelectric material are in direct contact with the active domain. Thus treating the electric problem with the FEM and only meshing the active region is a reasonable approximation for the modeled phenomena.

From ( 1) and ( 5), together with the small strains assumption and neglecting the magnetic forces, we obtain the mechanical weak form [START_REF] Galopin | Finite Element Modeling of Magnetoelectric Sensors[END_REF]: find u such that:

Ω δS : c : SdΩ - Ω δS : e t • EdΩ - Ω δS : h t • BdΩ = 0 ∀δS (10
) with u the displacement vector, ϕ the electric scalar potential and S = sym(∇u), E = -∇ϕ, δS = sym(∇δu) together with the Dirichlet boundary condition on ∂Ω u m (Fig. 2a). We do not impose charges on the electrodes, Q s = 0 on ∂Ω d , but potentials ϕ = ϕ 0 on ∂Ω ϕ m , one floating and the other in reference. In these conditions, the electric weak form is: find ϕ such that:

Ωm ∇δϕ•ε S •E dΩ m + Ωpe ∇δϕ•e•S dΩ m = 0 ∀ δϕ. (11)

IV. MATRIX SYSTEM AND RESOLUTION METHOD

The resulting global matrix system (12) is composed of sparse FEM matrices (in light colors) and full BEM matrices (in dark blue), and has a big scaling difference between coefficients. Usually, a suitable solver is used for each singlephysics problem: a preconditioned GMRES iterative solver is appropriate for problems involving full matrices (here the magnetic problem) while the MUMPS direct solver is better suited for problems with sparse matrices (here the electrical and mechanical problems). With these considerations, the system (12) can be solved iteratively using a block Gauss-Seidel scheme (13). Contrary to a direct resolution, the block Gauss-Seidel scheme thus allows using an optimized solver for each sub-system.

             0 K aφ 0 K G K ∂G ∂n 0 0 K uu K K ϕu K ϕϕ K aa K au 0 K ua 0                       a φ red u ϕ          =          Sh Ωm mag Sh ∂Ωm mag Sh meca Sh elec          (12) [K ii ]{x n+1 i } = {Sh i }- i-1 j=0 [K ij ]{x n+1 j }- n j=i+1 [K ij ]{x n j }.
(13) The block Gauss-Seidel scheme also allows for solving well conditioned sub-systems, as extra-diagonal sub-matrices are scaled by the current solutions and introduced as second hand terms (13). The stopping criteria for the multiphysics resolution algorithm was the relative convergence of the singlephysics solution between each iteration n.

V. VALIDATION

The magneto-mechanical formulation is validated with respect to an analytical formula for the field and strains, considering an unconstrained sphere under a uniform source field. The magnetic field inside a sphere under a uniform source field is given by:

H = 3 µ r + 2 H 0 . ( 14
)
To obtain a coupled analytical solution for an unconstrained piezomagnetic sphere, we solve for T = 0 and (14) with µ r = B/(µ 0 H). We tested our magneto-mechanical formulation with first-order finite elements, with coefficient similar to [START_REF] Do | Homogenization of Magnetoelectric 0-3 Type Composites by 3-D Multiphysics Finite-Element Modeling[END_REF] and increasingly finely meshed spheres imposing no surface forces and a uniform source field. We obtained the convergence curves presented in Figure 3. We observe the convergence up to 10 -4 of the FEM-BEM solution towards the analytical solution. Having validated the magneto-mechanical coupling, we will focus on solving the full problem.

VI. RESULTS

The full proposed model was tested on a composite structure with the geometry presented in Figure 4. It is composed of a PZT piezoelectric layer in between two Terfenol-D magnetostrictive layers and dimensions 3×6×14 mm. The mesh has 5656 elements. The device functions as follows: the source field drives the deformation of the magnetostrictive layers. As both phases are mechanically bonded together, the piezoelectric layer will deform and an electric voltage will appear between the electrodes, as shown in Figure 5. We considered a coil crossed by 100 A as a source for the magnetic excitation and positioned as shown in Figure 6, also, all linear coupling tensors with coefficients from [START_REF] Do | Homogenization of Magnetoelectric 0-3 Type Composites by 3-D Multiphysics Finite-Element Modeling[END_REF]. We obtained, as shown in Figure 5, a deformation of the global structure along its length, and the presence of a static electric potential difference between the electrodes.

VII. SOLVER BEHAVIOR

The resolution order of the global matrix at each block Gauss-Seidel iteration remains an open question, and many possibilities arise. In problems driven by a particular physics, and in order to have non-null solutions for the first iterations, a natural resolution order can sometimes be found. For the device in question, the magnetic field drives the deformation of the piezoelectric layer, cf Figure 4, so a natural order of resolution would be to solve at each Gauss-Seidel iteration for the magnetic problem followed by the mechanical and then the electric problem. Figure 7 gives the convergence ratios, as defined in Figure [START_REF] Bastos | Electromagnetic Modeling by Finite Element Methods[END_REF], of the different problems per resolution number. The resolution pattern of multiphysics problems is an active research subject, for example for the study of fluid-solid interactions, where multiple resolutions in subdomains are performed in order to obtain a coupled solution. In [START_REF] Uekermann | Coupling Algorithms for Partitioned Multi-Physics Simulations[END_REF], multiple resolution patterns are proposed, and the different physics solved in a previously defined and fixed order within a black-box multiphysics solver.

Another approach to solving the multiphysics problem is to have no imposed resolution order, but after each resolution, to compare the convergence ratios of all single-physics problems and solve for the one with the worst convergence ratio, that has not immediately been solved before. For our particular application, as the magnetic and electric problem are uncoupled, the mechanical problem must be solved at least every two resolutions. This resolution algorithm was implemented, and we obtained the convergence curves presented in Figure 8. They can be compared to the classical Gauss-Seidel presented in Figure 7. As shown in Table I, for this particular application, the adaptative algorithm solves the multiphysics problem in 37 iterations vs 42 resolutions for the classic algorithm with a tolerance of 10 -7 . But in our particular implementation, it translated into a total resolution time of 133 s vs 188 s (-29 %) on a personal computer (Intel© Core™ i7-10610U CPU @ 1.80GHz 2.30 GHz), the magnetic problem is indeed particularly difficult to solve as it contains both FEM and BEM matrices. The relative difference between single-physics solutions of the two tested resolution algorithm remained inferior to the tolerance of the solver.

VIII. CONCLUSION

In this paper, we proposed a FEM-BEM formulation of the magneto-electric effect in composite structures. A mixed formulation of the magnetic phenomena allowed us to use vector shape functions and not to explicitly consider and mesh an air region. The magneto-mechanical model was successfully validated against an analytical solution. We then solved the full magneto-electric problem on an example of composite structure and studied two variations of the block Gauss-Seidel resolution algorithm, one with a fixed resolution pattern and an adaptative one leading to a lesser number of resolutions and a lower resolution time in the studied test case.

Fig. 1 .

 1 Fig. 1. Domains of the magnetic problem. With Js a prescribed solenoidal current density and Ω 0 an open boundary domain.

Fig. 2 .

 2 Fig. 2. Mechanical and electrical domains. With f T the surface forces, u s the imposed displacement, ϕ 0 the imposed electric potential and Q s the surface charge density.

Fig. 3 .

 3 Fig. 3. Convergence of averaged H(z) ('.'), S 11 ('×'), S 22 ('o') and S 33 ('+') on meshed spheres vs analytical solutions.

Fig. 4 .

 4 Fig. 4. Three layers magnetoelectric heterostructure, the colored arrows represent the poling direction of the materials.

Fig. 5 .

 5 Fig. 5. Amplified displacement field and nodal voltage (V ).

Fig. 6 .

 6 Fig. 6. Coil positioning.

Fig. 7 .

 7 Fig.7. Convergence ratio of magnetic (o), mechanical (x) and electric (+) solutions after given n°of resolutions for the block Gauss-Seidel algorithm with a 1-2-3-1... resolution pattern for the given mesh. x n Relates the nth solution of the given problem.

Fig. 8 .

 8 Fig.8. Convergence ratio of magnetic (o), mechanical (x) and electric (+) solutions after given n°of resolutions with an adaptative resolution pattern for the given mesh.