Haut PDF A domain decomposition method in APOLLO3$^R$ solver, MINARET

A domain decomposition method in APOLLO3$^R$ solver, MINARET

A domain decomposition method in APOLLO3$^R$ solver, MINARET

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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A NonSmooth Contact Dynamics-based multi-domain solver

A NonSmooth Contact Dynamics-based multi-domain solver

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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Energy analysis of a solver stack for frequency-domain electromagnetics

Energy analysis of a solver stack for frequency-domain electromagnetics

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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A domain decomposition matrix-free method for global linear stability

A domain decomposition matrix-free method for global linear stability

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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A Domain Decomposition Method for the Polarizable Continuum Model based on the Solvent Excluded Surface

A Domain Decomposition Method for the Polarizable Continuum Model based on the Solvent Excluded Surface

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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A Domain Decomposition Method for the Poisson-Boltzmann Solvation Models

A Domain Decomposition Method for the Poisson-Boltzmann Solvation Models

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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A dual domain decomposition method for finite element digital image correlation

A dual domain decomposition method for finite element digital image correlation

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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Developing a Multiphysics Solver in APOLLO3 and Applications to Cross Section Homogenization

Developing a Multiphysics Solver in APOLLO3 and Applications to Cross Section Homogenization

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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Large-scale frequency-domain seismic wave modeling on h-adaptive tetrahedral meshes with iterative solver and multi-level domain-decomposition preconditioners

Large-scale frequency-domain seismic wave modeling on h-adaptive tetrahedral meshes with iterative solver and multi-level domain-decomposition preconditioners

‡ 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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A Domain Decomposition Method for problems with structural heterogeneities on the interface: Application to a passenger ship

A Domain Decomposition Method for problems with structural heterogeneities on the interface: Application to a passenger ship

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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High order HDG method and domain decomposition solvers for frequency‐domain electromagnetics

High order HDG method and domain decomposition solvers for frequency‐domain electromagnetics

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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Energy Analysis of a Solver Stack for Frequency-Domain Electromagnetics

Energy Analysis of a Solver Stack for Frequency-Domain Electromagnetics

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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A GenEO Domain Decomposition method for Saddle Point problems

A GenEO Domain Decomposition method for Saddle Point problems

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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Performance study of a parallel domain decomposition method

Performance study of a parallel domain decomposition method

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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Deriving a new domain decomposition method for the Stokes equations using the Smith factorization

Deriving a new domain decomposition method for the Stokes equations using the Smith factorization

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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The First Zianide Minaret "minaret Of Agadir"

The First Zianide Minaret "minaret Of Agadir"

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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A parallel non-invasive multiscale strategy for a mixed domain decomposition method with frictional contact

A parallel non-invasive multiscale strategy for a mixed domain decomposition method with frictional contact

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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A Posteriori error estimates combined with the asymptotic behavior for a generalized domain decomposition method in evolutionary PDEs

A Posteriori error estimates combined with the asymptotic behavior for a generalized domain decomposition method in evolutionary PDEs

In general, the a priori estimate for stationary problems is not suitable for assessing the quality of the approximate solutions on subdomains, since it depends mainly on the exact solution itself, which is unknown. An alternative approach is to use an approximate solution itself in order to …nd such an estimate. This approach, known as a posteriori estimate, became very popular in the 1990s with …nite element methods; see the monographs [ 1 ], [ 39 ]. In [ 39 ], an algorithm for a nonoverlapping domain decomposition has been given. An a posteriori error analysis for the elliptic case has also been used by [ 1 ] to determine an optimal value of the penalty parameter for penalty domain decomposition methods for constructing fast solvers. In [ 4 ], the authors derived a posteriori error estimates for the Generalized Overlapping Do- main Decomposition Method (GODDM) with Robin boundary conditions on the boundaries for second order boundary value problems; they have shown that the error estimate in the continuous case depends on the di¤erences of the traces of the subdomain solutions on the boundaries after a discretization of the domain by …nite elements method. Also they used the techniques of the residual a posteriori error analysis to get an a posteriori error estimate for the discrete solutions on subdomains.
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ETUDE D’UNE MOSQUEE A DEUX NIVEAUX « R+1 et L’INFLUENCE DE LA HAUTEUR TOTALE DE MINARET SUR LA  STABILITÉ AU VENT

ETUDE D’UNE MOSQUEE A DEUX NIVEAUX « R+1 et L’INFLUENCE DE LA HAUTEUR TOTALE DE MINARET SUR LA STABILITÉ AU VENT

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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Deriving a new domain decomposition method for the Stokes equations using the Smith factorization

Deriving a new domain decomposition method for the Stokes equations using the Smith factorization

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].

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