The aim of this paper is to present the last developments made on **Domain** **Decomposition** **Method** inside the **A** POLLO 3
**R** core **solver**, M
INARET . The fundamental idea consists **in** splitting **a** large boundary value problem into several similar but smaller ones. Since each sub-problem can be solved independently, the **Domain** **Decomposition** **Method** is **a** natural candidate to introduce more parallel computing into deterministic schemes. Yet, the real originality of this work does not rest on the well-tried **Domain** **Decomposition** **Method**, but **in** its implementation inside M INARET . The first validation elements show **a** perfect equivalence between the reference and the **Domain** **Decomposition** schemes, **in** terms of both k ef f and flux mapping. These first results are obtained without any parallelization or acceleration.

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ABSTRACT. This paper presents **a** micromechanical modeling strategy for complex multibody **in**-
teractions and the associated numerical framework. The strategy rests on **a** periodic multi- body **method** **in** the framework of the Non Smooth Contact Dynamics approach of Moreau (Moreau, 1988) extended to classical **domain** **decomposition** problems. Many complex inter- actions can be taken into account : interactions between discrete elements, between discrete or rigid bodies, (quasistatic) contact or impact, friction or adhesion, decohesion (cracking), etc. The associated numerical platform, Xper, is composed of three independant libraries with Object Oriented Programming. The libraries are specifically developed for : (1) the solution of systems of partial differential equations (PDEs), (2) the modeling of complex interaction prob- lems and the time discretization, (3) the integration of complex non linear constitutive models. The ability of this computational approach is illustrated by two examples of fracture of hetero- geneous materials.

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This work aims at conducting an energy and power analysis of the simulation of frequency- **domain** electromagnetic wave propagation, as detailed **in** Section 3. Such **a** simulation is represen- tative of **a** numerical simulation involving the usage of **a** complex software stack. **In** the present case, we study the combined HORSE/MaPHyS numerical software stack. The HORSE (High Order **solver** for Radar cross Section Evaluation) simulation software implements an innovative high order finite element type **method** for solving the system of three-dimensional frequency- **domain** Maxwell equations, as presented **in** Section 4. From the computational point of view, the central operation of **a** HORSE simulation is the solution of **a** large sparse and indefinite linear system of equations. High order approximation is particularly interesting for solving high frequency electromagnetic wave problems and, **in** that case, the size of this linear system can easily exceed several million unknowns. **In** this study, we adopt the MaPHyS [4] - [5] hybrid iterative-direct sparse system **solver**, which is based on **domain** **decomposition** principles [15]. MaPHyS is representative of fully-featured adaptive sparse linear solvers [14, 16, 8, 10] involving multiple numerical linear steps combining the usage of dense and sparse direct numerical linear algebra kernels as well as iterative methods, as further discussed **in** Section 5.

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From the above discussion, it is clear that understanding of open flow dy- namics through global modes could greatly benefit from the development of efficient Navier-Stokes solvers devoted to large computational domains and three-dimensionality. **In** this context, matrix-free methods appears to be an appropriate choice. Nevertheless, their main drawback lies **in** two major facts. **In** one hand, it necessits several time-integration of linearized DNS which is time consuming, **in** particular when dealing with low frequency un- steadiness. On the other hand, both the storage of snapshots and Krylov methods related to large computational domains, fine spatial discretization and three-dimensionality may yield difficulties **in** terms of memory require- ment and time spending of the eigenmodes algorithm. To overcome these limitations, **domain** **decomposition** methods **in** which the geometry is de- composed into subdomains, combined with parallel architectures seems to be an appropriate choice. For that purpose, this work is motivated by de- veloping and validating **a** time-stepping global stability **method** based on **a** mutlidomains **solver** according to the linearized DNS **in** combination with an Arnoldi algorithm associated with snapshots of each subdomain which composes the full geometry.

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1.3 Outline
**In** Section 2, we first introduce different solute-solvent boundaries including the VdW surface, the SAS and the SES, which are fundamental and classical concepts of the implicit solvation models, which however are mostly unknown to the applied mathematics community. **In** Section 3, we construct **a** continuous dielectric permittivity function ε(x) of PCM, ensuring that the SES-cavity always has the dielectric constant of vacuum. Then, **in** Section 4, we present the electrostatic problem of the PCM, its equivalent transformation, and **a** global iterative strategy for solving it. **In** Section 5, we introduce the scheme of the **domain** **decomposition** **method** for solving the associated partial differential equations **in** the global strategy. This requires to develop **a** Laplace **solver** and **a** GP-**solver** **in** the ball, which are presented **in** Section 6. After that, **in** Section 7, we give **a** series of numerical experiments on the performance of the proposed **method**. **In** the last section, we draw some conclusions.

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generated, for example, using the MSMS [13] or the NanoShaper [14] etc. The BEM is efficient to solve the LPB equation and some techniques can be used to accelerate the BEM solvers, including the fast multipole **method** [15] and the hierarchical “treecode” technique [8]. For instance, the PAFMPB **solver** [16, 15] developed by Lu et al. provides **a** fast calculation of the solvation energy, which uses the adaptive fast multipole **method** and achieves linear complexity with respect to (w.r.t.) the number of mesh elements. Another interesting BEM **solver**, called TABI-PB [17], has been developed **in** the past several years, which uses the “treecode” technique. However, the BEM has **a** limitation that it can not be easily generalized to solve the NPB equation.

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At this stage, only **a** sequential implementation of the **method** was performed **in** MATLAB. It is subsequently not possible to provide **a** relevant estimate of the speed-up. However, the average normalised CPU time required to compute one iteration of local FE-DIC (including image interpo- lation and FE system resolution) is plotted **in** Figure 9 versus average subdomain size. It appears that the numerical complexity seems to be **a** bit less that O.N 2 /. **In** other words, the CPU time of one iteration is divided by almost 100 when the size of the subdomains is divided by 10. With **a** sequential implementation of the **method**, 65% of the CPU time is devoted to compute the ini- tialisation (first block of Figure 2), whereas the remaining 35% is used to perform the eight extra iterations with the DD **solver** (second block of Figure 2). Because the first 65% corresponds to com- putations that are independent by subdomain (it requires no data exchange between subdomains), this first initialisation is thus highly parallelisable, and its computational cost, with **a** parallel imple- mentation, is expected to be divided by the number of subdomains. The second part (35%) is also highly parallelisable but requires some data exchange between subdomains. The results presented **in** Figures 8 and 9 exemplify that the proposed approach combines the advantages of FE-based DIC and subset-based DIC methods.

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so that the power of these commercial numerical libraries can be used **in** the multiphysics framework.
Typically when solving large scale numerical simulations, signicant amounts of parallelization are sought **in** the codes used. Parallel codes make use of the largest and fastest computers currently available. **A** signicant weakness of the simulations presented **in** this work is the lack of parallel methods. **A** large amount of work could be devoted to adding parallel capability to the underlying physics component codes and the multiphysics framework. **In** the neutron transport model, **a** parallel sweeping algorithm could improve the size of transport prob- lems able to be tractably solved. **Domain** **decomposition** methods can allow larger problems to be solved using large parallel machines. **In** the thermal hydraulic model, subchannels can be split among several processes which communicate to evaluate mixing among channels. On the level of the multiphysics framework, sig- nicant amounts of parallelization are possible. Chapter 5 demonstrated that **a** signicant amount of time was devoted to evaluating the delayed neutron precur- sor residual, which involves several manipulations of the ssion source. Splitting these manipulations over several processes has the potential to signicantly re- duce the time spent evaluating this residual. The evaluation of other residuals can also be performed **in** parallel, especially if the underlying physics components have parallel capabilities. The linear solvers can be made to use parallel capabil- ities through manipulations of matrix vector products **in** parallel. Furthermore, the physics-based preconditioner used **in** Chapter 5, which is **a** block diagonal ma- trix, can be inverted **in** parallel by simultaneously inverting each block diagonal matrix. Implementing such parallel methods will be essential for the continued used of the present multiphysics framework.

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‡ UCA, CNRS, Geoazur, France; § Sorbonne University, CNRS, France; SUMMARY
Frequency-**domain** full-waveform inversion (FWI) is suitable for long-offset stationary-recording acquisition, since reliable subsurface models can be reconstructed with **a** few frequen- cies and attenuation is easily implemented without computa- tional overhead. **In** the frequency **domain**, wave modeling is **a** Helmholtz-type boundary-value problem which requires to solve **a** large and sparse system of linear equations per fre- quency with multiple right-hand sides (sources). This system can be solved with direct or iterative methods. While the for- mer are suitable for FWI application on 3D dense OBC ac- quisitions covering spatial domains of moderate size, the later should be the approach of choice for sparse node acquisitions covering large domains (more than 50 millions of unknowns). Fast convergence of iterative solvers for Helmholtz problems remains however challenging **in** high frequency regime due to the non definiteness of the Helmholtz operator, on one side and on the discretization constraints **in** order to minimize the dis- persion error for **a** given frequency, on the other side, hence requiring efficient preconditioners. **In** this study, we use the Krylov subspace GMRES iterative **solver** combined with **a** two-level **domain**-**decomposition** preconditioner. Discretiza- tion relies on continuous Lagrange finite elements of order 3 on unstructured tetrahedral meshes to comply with complex geometries and adapt the size of the elements to the local wavelength (h-adaptivity). We assess the accuracy, the con- vergence and the scalability of our **method** with the acoustic 3D SEG/EAGE Overthrust model up to **a** frequency of 20 Hz.

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c
Principia Marine, 1 rue de la No¨ e, BP 22112, F-44321 Nantes CEDEX 3, FRANCE
Abstract
**In** this article, we extend **a** **domain** **decomposition** **method**, based on the FETI- DP linear **solver**, to applications such as passenger ship analysis. More generally, the **method** is designed for large-scale elastic analysis of **a** structure which ex- hibits geometrical and structural heterogeneities, such as plate and stiffener assemblies **in** presence of structural details. The problem of the structural het- erogeneities on the subdomain interfaces, arising from the presence of stiffeners or elastic joints on these interfaces, is addressed. **A** suited interface connection between subdomains modeled with plate elements **in** the case of **a** 3D assembling is proposed and tested. The selection of an efficient preconditioner is presented, and the performances and results are discussed **in** terms of convergence rate for several examples.

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The goal of this paper is to report on further developments of the HDG **method** proposed **in** 9 aiming at improving its accu-
racy and scalability for the simulation of large-scale three-dimensional problems. As such, this paper does not present some new HDG formulation or theoretical results on the formulation initially proposed **in** 9 , and is rather meant to offer an updated picture of
the development of this **method** towards its adoption for large scale simulations. Section 2 defines the considered boundary value problem for the three-dimensiobal time-harmonic Maxwell equations and introduces some notations. Section 3 presents the prin- ciples and general formulation of the HDG **method**. The implementation of the HDG **method** is the subject of section 4. Though the HDG **method** results **in** **a** smaller linear system than the one associated with **a** classical upwind flux-based DG **method**, the size of this system is often too large to be solved by **a** sparse direct **solver** as soon as one consider realistic three-dimensional problems. **In** addition, for very large-scale propagation problems, exploiting **a** multi-processor system is **a** mandatory path to reduce the time to solution and have access to the required memory capacity. **In** section 5, we briefly discuss about the solution strategies that we have considered **in** this work, **in** particular the PDE-based and algebraic **domain** **decomposition** solvers that have been initially developed **in** other contexts. Numerical examples are given **in** section 6 with two objectives **in** mind: first, using **a** simple (model) problem, the proposed DD-HDG solution strategy is validated and its convergence properties are assessed; second, by considering more com- plex problems, we study the overall efficiency of the DD-HDG solution strategy. Finally, we draw some conclusions and state future works **in** section 7.

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We observe that, despite the extra-memory traffic due to indirections, the fact that it induces much fewer floating point operations than the dense variant leads to **a** lower overall energy consumption. The high energy required by the dense+CSC preconditioner is mainly due to the setup of the CSC, which is both memory and CPU demanding. Because the dense and sparse preconditioners do not implement any global coupling numerical mechanisms, the number of iterations is expected to grow as the number of subdomains for the 1D **decomposition** of the **domain** on the Poisson test example. This poor numerical behavior can be observed **in** Figure 4c, while it can be seen that the coarse space correction (dense+CSC variant) plays its role and ensures **a** number of iterations independent from the number of domains (see [6] for further insights on the numerical properties of the **method**). This nice numerical behavior translates **in** terms of time to solution for the iterative part where the dense+CSC **method** outperforms the two other variants. However, the overhead of the setup phase for the construction of the coarse space, which requires the solution of generalized eigenproblems, is very high and cannot be amortized at that intermediate scale if only **a** single right-hand side has to be solved (which would not be the case for, e.g., radar cross section evaluation as considered **in** Section V-C, where multiple right-hand sides must be solved **in** real-life test cases). Nonetheless, the relative ranking **in** terms of power requirements are different. Through **a** simple linear regression, we can observe that the average power requirements are about 328, 326 and 321 W/node for the dense, sparse and dense+CSC preconditioners, respectively. C. Scattering of **a** plane wave by **a** PEC sphere

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The problem is discretized and solved with the open-source parallel finite element soft- ware FreeFEM [17]. FreeFEM is **domain** specific language (DSL) where the problem to be solved is defined **in** terms of its variational formulation. Then the local matrices pA N eu i q 1ďiďN (see eq. (4)) and p ˜ C i q 1ďiďN (see eq. (11)) are easily obtained by restricting the corresponding variational formulations to adequate local subdomains. Note that these matrices are different from the restriction of the global matrices **A** and C to the local degrees of freedom. The **domain** **decomposition** algorithm presented **in** this paper is implemented on top of the ffddm framework, **a** set of parallel FreeFEM scripts implementing Schwarz **domain** **decomposition** methods. ffddm already implements the GenEO **method** [32] for SPD problems, and its building blocks are designed to simplify the implementation and prototyping of new **domain** **decomposition** methods such as the saddle point **solver** pre- sented **in** this paper. The ffddm documentation is available on the FreeFem.org web page, see [35].

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platform.
**A** parallel one-group block Jacobi algorithm and **a** parallel multigroup block Jacobi algorithm have been efficiently implemented **in** **Minaret**. Various parallelism strategies have been tested for the refer- ence and the two **domain** **decomposition** algorithms, depending on the variable parallelized (angular direction or subdomain) and the programming standard used (MPI or OpenMP). **A** performance study is given, showing how important it is to reduce the amount of MPI communications. On the other hand, OpenMP efficiency is very satisfying. At last, it is of the utter interest to couple MPI and OpenMP **in** **a** two layers ‘hybrid’ parallelism, suitable for HPC. Doing so, the number of independent processes is potentially unlimited. Especially, the larger the problem to solve is, the most valuable this ‘hybrid’ computation will be.

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Let us give **a** short outline of the paper. **In** Section 2 we introduce the Stokes equa- tions. Concentrating on the two-dimensional case, these equations are transformed to **a** bi-harmonic operator with the help of the Smith factorization. Then, **in** Section 3 we first introduce an iterative **domain** **decomposition** **method** for the bi-harmonic equa- tions and we show how it can be used for the Stokes equations. Moreover, **in** Section 4, we will discuss briefly, how this approach can be extended to the linearized Navier- Stokes equations (Oseen equations). **In** the case of two subdomains we were able to derive an algorithm which converges indepedently of the Reynolds number **in** two iterations. Most likely, ongoing research will show that we will retrieve this behavior for more general decompositions. Then, **in** section 5 the algorithm is extended to the three-dimensional Stokes problem. **A** finite volume discretization is discussed **in** Section 6. Section 7 is dedicated to numerical results for the two-dimensional Stokes problem. Finally, we give some concluding remarks.

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6 Decor of the Lantern:
The lantern is composed on all sides of **a** rectangular frame whose height is less than the width; it contains **a** horseshoe arch followed by **a** lobed bow nine lobes, and above **a** lozenge network with diamonds with five lobes. It is arranged with **a** row of **a** rhombus **in** the middle and **a** half rhombus on each side, as to the bottom of these lozenges it is garnished with small mosaic tiles of green faience. The lantern frame is surrounded by **a** zellige border that occupies one third of the width of the frame. It is U-shaped which stops at the fall of the bow, decorated by **a** row of eight-pointed stars enamelled **in** green and glued to each other. There are thirty-four stars **in** each face of the lantern. These stars are arranged on **a** white background and bordered on all their sides by **a** narrow band composed of small green rectangles; the same band surrounds the upper part of the **minaret**. This lantern is crowned with **a** hemispherical couplet surmounted by **a** metal rod.

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1 Introduction
The industrial simulations of assemblies require to manage complex models with large numbers of degrees of freedom that provokes memory and time consuming limitations. Nonlinearities coming from the complex behaviors, such as plasticity, damage or contact, amplify increasingly the problems of memory or time consumption. Since 80s, **domain** **decomposition** methods have been developed to face these issues : Schwarz methods [9], Balancing **Domain** **Decomposition** [25], Finite Element Tearing and Interconnecting [8, 7, 6]. They distribute the computation on parallel architecture of hardwares and stretch the memory limitation at the same time they reduce time computation.

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Figure. II.1 : Dessin d’un plancher en corps creux…………………………………………..19 Figure. II.2 : Dimension de la poutrelle……………………………………………………..21 Figure. II.3 : schéma d’un escalier…………………………………………………………..23 Figure. II.4 : L’escalier à volées droites avec paliers intermédiaires………………………..23 Figure. II.5 : Escalier balancé à double quartier tournant…………………………………..24 Figure. II.6 : Escaliers à volées droits avec palier intermédiaire…………………………….26 Figure. II.7 : schéma de l’acrotère…………………………………………………………...27 Figure. II.8 : Dimensions de la poutre……………………………………………………….28 Figure. II.9 : Détail de plancher corps creux terrasse inaccessible………………………….30 Figure. II.10 : Détail des constituants du plancher étage courant…………………………...32 Figure. II.11 : Mur extérieur **a** double cloison……………………………………………….35 Figure. II.12 : Coupes de voiles en élévation………………………………………………..36 Figure. II.13 : Coupes de voiles en plan……………………………………………………..37 Figure. II.14 : section réduit de poteau………………………………………………………38 Figure. II.15 : schéma de poteau Intermédiaire bloc B……………………………………...41 Chapitre III :

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1. Introduction
The last decade has shown, that Neumann-Neumann type algorithms, FETI, and BDDC methods are very efficient **domain** **decomposition** methods. Most of the early theoretical and numerical work has been carried out for scalar symmetric positive definite second order problems, see for example [6, 12–14, 22]. Then, the **method** was extended to different other problems, like the advection-diffusion equations [1, 7], plate and shell problems [26] or the Stokes equations [21, 25].