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With the introduction of renewable energy sources into the power system, transmission system operators need extremely reliable and flexible simulation tools that should also be fast. Sinusoidal predictor method, aims to meet this need by simulating the full-waveform of AC power systems by integrating the arising systems of DAE by trying to free the time step limitation introduced by the frequency of some electrical components. The searched solution is split in a predicted sinusoidal part and a corrector DAE solution part. This paper presents the mathematical convergence of this approach leading to the choice of computing Fourier coefficients by minimizing a function that captures the distance of the desired sinusoidal part of the solution with respect to its oscillating steady state. Using the Dynaωo framework that aims to completely separate the modeling and the solving aspects of power systems, It details latest developments of an add-on in the IDA solver of SUNDIALS and the implementation optimisations. Results on performance speed up of the method computational time using much larger time steps than those used by classical DAE integration methods.

International audience; In this work, a 2D Finite Element Method (FEM) is used to calculate the frequency dependent impedance and admittance matrices of underground and submarine cables. A harmonic magnetodynamic formulation is used to calculate the series impedances, and for the calculation of the parallel admittance an electrodynamic formulation is applied. The frequency response of the cable was obtained by coupling the FEM method with an external electric circuits. This technique allows the physical representation of the nonlinear behavior of the frequency dependence of the cable parameters. The numerical results are compared with those obtained from commonly used analytical formulations and other methods available in the literature. Different surrounding environment cases are treated, and their effect is shown in frequency. Time domain simulations under fault circumstances point out the necessity of an accurate cable model. Current and voltage measurements are the input of protection algorithm and breaking decision depends on these values. Therefore, the use of a precise method is mandatory.

International audience

This paper deals with the representation of the trace of iterative Schwarz solutions at the interfaces of domain decomposition to approximate adaptively the interface error operator. This allows to build a cost-effectively accelerating of the convergence of the iterative method by extending to the vectorial case the Aitken’s accelerating convergence technique. The first representation is based on the building of a nonuniform discrete Fourier transform defined on a non-regular grid. We show how to construct a Fourier basis of dimension N+1 on this grid by building numerically a sesquilinear form, its exact accuracy to represent trigonometric polynomials of degree N / 2, and its spectral approximation property that depends on the continuity of the function to approximate. The decay of Fourier-like modes of the approximation of the trace of the iterative solution at the interfaces provides an estimate to adaptively select the modes involved in the acceleration. The drawback of this approach is to be dependent on the continuity of the trace of the iterated solution at the interfaces. The second representation, purely algebraic, uses a singular value decomposition of the trace of the iterative solution at the interfaces to provide a set of orthogonal singular vectors of which the associated singular values provide an estimate to adapt the acceleration. The resulting Aitken-Schwarz methodology is then applied to large scale computing on 3D linear Darcy flow where the permeability follows a log normal random distribution. Cet acte traite de la représentation des solutions itérées aux interfaces de la méthode de décomposition de domaine de type Schwarz afin d’approximer de manière adaptative son opérateur d’erreur aux interfaces des sous domaines. Ceci permet de construire de manière économique l’accélération de la convergence de la méthode itérative en étendant la technique d’accélération de la convergence de Aitken au cas vectoriel. La première représentation est fondée sur la construction d’une transformée de Fourier discrète non uniforme définie sur un maillage non régulier. Nous montrons comment construire une base de Fourier de dimension N+1 sur ce maillage à partir de la construction numérique d’une forme sesquilinéaire, son exactitude pour les polynômes trigonométriques de degré N/2, et numériquement sa capacité d’approximation spectrale dépendante de la continuité de la fonction à approximer. La décroissance des modes de Fourier de la trace de la solution itérée aux interfaces nous fournis une estimation pour sélectionner de manière adaptive les modes intervenant dans l’accélération. Le défaut de cette approche est d’être dépendant de la continuité de la solution itérée aux interfaces. La deuxième représentation, purement algébrique, utilise une décomposition en valeurs singulières des solutions itérées aux interfaces pour fournir un ensemble de vecteurs singuliers orthogonaux dont les valeurs singulières associées fournissent une estimation pour adapter l’accélération. La méthode Aitken-Schwarz résultante est ensuite appliquée au calcul à grande échelle pour résoudre un écoulement 3D en milieu poreux modélisé par l’équation de Darcy linéaire où la perméabilité suit une distribution aléatoire log-normale.

International audience; This paper is devoted to the acceleration by Aitken's technique of the convergence of the Schwarz domain decomposition method applied to large scale 3D problems with non-separable linear operators. These operators come from the discretization of groundwater flow problems modeled by the linear Darcy equation, where the permeability field is highly heterogeneous and randomly generated. To be computationally efficient, a low-rank approximation of the Aitken's formula is computed from the singular value decomposition of successive iterated solutions on subdomains interfaces. Numerical results explore the efficiency of the solver with respect to the random distribution parameters, and specific implementations of the acceleration are compared for large scale 3D problems. These results confirm the numerical behavior of the methodology obtained on 2D Darcy problems (Tromeur-Dervout D. Meshfree adaptive Aitken-Schwarz domain decomposition with application to Darcy flow. Comput Sci Eng Technol 2009;21:217-50).

International audience; We developed a parallel time domain decomposition method to solve systems of ODEs based on the Aitken-Schwarz domain decomposition method only in time (Linel P, Tromeur-Dervout D. Aitken-Schwarz and schur complement methods for time domain decomposition. In: Parallel computing: from multicores and GPU's to Petascale. Advances in parallel computing, vol. 19; 2010. p. 75-82). The method transforms the initial time value problem into a time boundary values problem. This paper details the proof of the pure linear convergence of this DDM in case of linear system of ODEs and provides an optimized MPI parallel implementation when a regular size splitting of the time interval is performed. Then an extension of the method to the linear heat partial derivative equation is provided with a special care on boundary conditions to impose. A Fourier convergence analysis of this Schwarz DDM in time for this problem allows to limit the acceleration by Aitken's technique to Fourier low modes only.

The Schwarz domain decomposition method [1] is a very attractive numerical method for parallel computing as it needs only to update the boundary conditions on the artificial interfaces generated by domain decomposition. Thus only local communications between the neighbouring sub-domains are required. Nevertheless, the main drawback of this method is its slow rate of convergence which depends of the partial differential problem, the geometry of the sub-domains, and the size of the overlap when overlap is present. The idea of using Aitken acceleration [2] on the classical additive Schwarz DD method was introduced in [3]. These authors have called the corresponding method the Aitken-Schwarz (AS) method. This review paper is on the Aitken's acceleration of the convergence technique applied to the Schwarz domain decomposition method. It gives the two salient features of the methodology: first the pure linear convergence of the Schwarz domain decomposition method when it applies to a linear system of equations. Second, the building of an approximation space in order to represent the Schwarz iterate solution at the artificial interfaces generated by the domain decomposition. Some properties such as the decrease in absolute value of the solution's coefficients in the approximation space are searched in order to approximate the error operator and to apply the acceleration on a reduced space for saving computing.In [4] the author extends the methodology with an Aitken acceleration based on the singular value decomposition of the solution at the artificial boundary. Then this method becomes totally mesh non dependant, on some a priori criterion based on the singular values decreasing and gives a tool to select the singular vectors involved in the Aitken operator approximation. This allows three-dimensional computation on the linear Darcy equation to be achieved where the permeability field follows a random log normal distribution law [5].

International audience; It is of interest to solve large scale sparse linear systems on distributed computers, using Krylov subspace methods along with domain decomposition methods. If accurate subdomain solutions are used, the restricted additive Schwarz preconditioner allows a reduction to the interface via the Schur complement, which leads to an unpreconditioned reduced operator for the interface unknowns. Our purpose is to form a preconditioner for this interface operator by approximating it as a low-rank correction of the identity matrix. To this end, we use a sequence of orthogonal vectors and their image under the interface operator, which are both available after some iterations of the generalized minimal residual method.

We developed parallel time domain decomposition methods to solve systems of linear ordinary differential equations (ODEs) based on the Aitken-Schwarz [5] or primal Schur complement domain decomposition methods [4]. The methods require the transformation of the initial value problem in time defined on ]0, T] into a time boundary values problem. Let f(t, y(t)) be a function belonging to \(\mathcal{C}^{1}(\mathbb{R}^{+}, \mathbb{R}^{d})\) and consider the Cauchy problem for the first order ODE: $$\displaystyle{ \left \{\begin{array}{@{}l@{\quad }l@{}} \dot{y} = f(t,y(t)),\,t \in ]0,T],\;y(0) =\alpha \in \mathbb{R}^{d}.\quad \end{array} \right. }$$ (1)

International audience; A heterogeneous domain decomposition with non-matching grids is developed, extending the Aitken-Schwarz method [M. Garbey, D. Tromeur-Dervout, On some Aitken like acceleration of the Schwarz method, Internat. J. Numer. Methods Fluids 40 (2002) 1493- 1513] which has proved to be efficient on metacomputing architectures. This novel numerical technique for the approximate solution of boundary value problems applies when the solution is assumed to possess many features in several subdomains such as in underground environmental problems with different geological layers. The numerical technique involves methods for the approximation and the solution of arising linear systems, as well as for parallel computing issues. We consider a natural coupling between the mixed finite element and spectral element approximations as well as the efficient solution of the coupled discrete systems on remote parallel computers with different architectures connected via a low speed network.

International audience; This Note is concerned with the severely ill-posed Cauchy-Helmholtz problem. This Cauchy problem being rephrased through an "interfacial" equation, we resort to an Aitken-Schwarz method for solving this equation. Numerical trials highlight the efficiency of the present method.

Time domain decompositions to solve ODEs/DAEs have been numerically investigated by introducing adaptivity in the definition of the refinement of the time grid, time domain splitting. We show that the parareal method [Lions J-L, Maday Y, Turinici G. Resolution d’EDP par un schema en temps “parareel”, CRAS Ser I Math 2000;332(7):661–8] is a particular case of the multiple shooting method of Deuflhard. Numerical evidences of the limitation of this method to solve very stiff problems are exhibited, leading us to propose an adaptive parallel extrapolation method.

International audience; This paper presents a method that speeds up the solution of unsteady nonlinear problems. When a Newton–Krylov method is used, the most time-consuming part of the numerical simulation is the solution of the successive linear systems. Generally, there are only slight changes between two consecutive Jacobian matrices. Thus, in order to save computations, it is possible to use the same preconditioning matrix for a few successive linear systems, but the convergence of the Krylov method is slowed down. In the case of a preconditioner based on domain decomposition, one can update only the parts of the preconditioner associated to certain subdomains, keeping the other ones constant. In order to achieve an efficient parallel implementation, we propose to add processes dedicated to the asynchronous update of some parts of the preconditioner. Numerical results are provided for a reaction–diffusion problem and for a model problem based on the lid-driven cavity. They show that this addition of processes can speed up the computation and improve the robustness.

Metacomputing is a method of using the GRID, which originated in the US and quickly was also picked up by European and Japanese researchers where a number of challenging projects were aiming at the exploitation of such distributed resources. Especially the GLOBUS project (The globus project: a status report, Proceedings IPPS/SPDP'98 Heterogeneous Computing Workshop, 1998, pp. 4-18) has contributed to the success of metacomputing substantially. However, high latency and low bandwidth have made people doubt the feasibility of this concept for big simulation codes. In this paper we present new numerical methods that help to exploit such configurations and overcome the problems of low network performance. To proof the feasibility of our approach we show results of simulations in an innovative GRID environment of supercomputers.

We present a family of domain decomposition based on Aitken-like acceleration of the Schwarz method seen as an iterative procedure with a linear rate of convergence. We first present the so-called Aitken-Schwarz procedure for linear differential operators. The solver can be a direct solver when applied to the Helmholtz problem with five-point finite difference scheme on regular grids. We then introduce the Steffensen-Schwarz variant which is an iterative domain decomposition solver that can be applied to linear and nonlinear problems. These solvers have reasonable numerical efficiency compared to classical fast solvers for the Poisson problem or multigrids for more general linear and nonlinear elliptic problems. However, the salient feature of our method is that our algorithm has high tolerance to slow network in the context of distributed parallel computing and is attractive, generally speaking, to use with computer architecture for which performance is limited by the memory bandwidth rather than the flop performance of the CPU. This is nowadays the case for most parallel. computer using the RISC processor architecture

We present a nonoverlapping domain decomposition method with local Fourier basis applied to a model problem in liquid flames. The introduction of domain decomposition techniques in this paper is for numerical and parallel efficiency purposes when one requires a large number of grid points to catch complex structures. We obtain then a high-order accurate domain decomposition method that allows us to generalize our previous work on the use of local Fourier basis to solve combustion problems with nonperiodic boundary conditions (M. Garbey and D. Tromeur-Dervout, J. Comput. Phys. 145 , 316 (1998)). Local Fourier basis methodology fully uses the superposition principle to split the searched solution in a numerically computed part and an analytically computed part. Our present methodology generalizes the Israeli et al. (1993, J. Sci. Comput. 8, 135) method, which applies domain decomposition with local Fourier basis to the Helmholtz's problem. In the present work, several new difficulties occur. First, the problem is unsteady and nonlinear, which makes the periodic extension delicate to construct in terms of stability and accuracy. Second, we use a streamfunction biharmonic formulation of the incompressible Navier–Stokes equation in two space dimensions: The application of domain decomposition with local Fourier basis to a fourth-order operator is more difficult to achieve than for a second-order operator. A systematic investigation of the influence of the method's parameters on the accuracy is done. A detail parallel MIMD implementation is given. We give an a priori estimate that allows the relaxation of the communication between processors for the interface problem treatment. Results on nonquasi-planar complex frontal polymerization illustrate the capability of the method.

The main object of this paper is to demonstrate the feasibility of coupling parallel codes in the framework of large scale scientific computing. The second aim is to provide a numerical tool to solve 3D frontal polymerization in liquid problems where the multiple physical scales require intensive computation on MIMD architecture. We extend our 2D modelization [14] of such phenomena by coupling a 3D reaction diffusion system well known in solid combustion [22] with the 3D Navier–Stokes equations written in Boussinesq approximation. We develop two separate parallel codes for each physical model, each code running on its own topological network of processors with its own data structures. The non-blocking communications, manage the interaction terms of the two physical models on each code, are performed through a Portable Inter Program Communication Library ( PIPCL ). This communication library computes array-based communication schedules creating a one-to-one implicit mapping between each data representation. The data distribution of each code is hidden to the others, allowing an easy parallel program inter-operability on heterogeneous computers.

Gas combustion, solid combustion as well as frontal polymerization are characterized by stiff fronts that propagate with nonlinear dynamics. The multiple-scale phenomena under consideration lead to very intense computations that require parallel computing in order to reduce the elapsed time of the computation. We develop a methodology to build on the MIMD architecture a parallel numerical method based on the property of the solution, i.e. a stiff quasi-planar two-dimensional combustion front. We illustrate our methodology using two models of the combustion process. The first is a thermo-diffusive model of a two-step chemical reaction exhibiting two transition layers. The second is a thermo-diffusive model of a one-step chemical reaction coupled with a hydrodynamical model using the stream function - vorticity formulation of the Navier - Stokes equations written in the Boussinesq approximation. This methodology makes use of efficient domain decomposition methods, combined with asymptotic analytical qualita...

International audience; This paper is devoted to the solution of nonlinear time-dependant partial differential equations arising in CFD using Broyden's method in the parallel computing framework. We first use Broyden's method in the context of the domain decomposition: we propose to update the Restricted Additive Schwarz preconditioner from one Newton iteration to another when a Newton-Krylov method is used. We also investigate a time-pipelining method where Broyden's method is used as a solver of the nonlinear problem of each time step.

This paper is devoted to the parallel preconditioning of linear systems arising from the discretization of nonlinear time-dependent partial differential equations using implicit methods. During the Newton iterations, the Restricted Additive Schwarz (RAS) preconditioner of the Jacobian matrix can be used to accelerate the convergence of Krylov subspace methods. We propose to extend the secant preconditioner method to update RAS preconditioner from one Newton’s iteration to another. Then, we show that updating the RAS preconditioner can be more efficient than freezing it. Results on numerical efficiency are given.

Colloque avec actes et comité de lecture. Internationale.; International audience; Nous présentons une méthode parallèle pour résoudre l'équation de Darcy 3D où le champ de perméabilité varie aléatoirement suivant une distribution log normale et avec de fortes amplitudes dans des domaines discrétisés de 10^8 à 10^9 inconnues. Cette technique de décomposition de domaine de type Aitken-Schwarz conduit à un parallélisme à deux niveaux où les problèmes locaux sont résolus par multigrille algébrique parallèle (AGMG de Y. Notay). L'influence sur la construction de l'accélération d' Aitken de l'espace d'approximation pour représenter la solution sur les interfaces sera discuté.