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International audience; This paper is dedicated to the analysis of the rate of convergence of the classical and quasi-optimal Schwarz waveform relaxation (SWR) method for solving the linear Schrödinger equation with space-dependent potential. The strategy is based on i) the rewriting of the SWR algorithm as a fixed point algorithm in frequency space, and ii) the explicit construction of contraction factors thanks to pseudo-differential calculus. Some numerical experiments illustrating the analysis are also provided.

Modeling fluid flow in three-dimensional (3D) Discrete Fracture Networks (DFNs) is of relevance in many engineering applications, such as shale oil/gas production, geothermal energy extraction, nuclear waste disposal and CO2 sequestration. A new Boundary Element Method (BEM) technique with discontinuous quadratic elements, in conjunction with a parallel Domain Decomposition Method (DDM), is presented for the simulation of the steady-state fluid flow in DFNs, consisting of stochastically generated 3D planar fractures, arbitrarily oriented, and having differing hydraulic properties. Numerical examples characterized by DFNs of increasing complexity are proposed to show the accuracy and the efficiency of the presented technique, that provides good approximations of the fluid flow around domain interfaces, where the solution usually displays sharp gradients, like around intersections between traces (the segments originated by the intersection between two fractures), intersections between traces and fracture boundaries, or intersections between fractures and wellbores. The conjunction with a DDM approach is a promising strategy to speed up the computations, by also exploiting the advantages of parallel computing techniques. The technique is implemented in the code PyDFN3D, available at https://github.com/BinWang0213/PyDFN3D.

International audience; We consider the time-harmonic problem of the diffraction of an incident propagative mode by a localized defect, in an infinite straight isotropic elastic waveguide. We propose several iterative algorithms to compute an approximate solution of the problem, using a classical finite element discretization in a small area around the perturbation, and a modal expansion in unbounded straight parts of the guide. Each algorithm can be related to a so-called domain decomposition method, with or without an overlap between the domains. Specific transmission conditions are used, so that only the sparse finite element matrix has to be inverted, the modal expansion being obtained by a simple projection, using the Fraser bi-orthogonality relation. The benefit of using an overlap between the finite element domain and the modal domain is emphasized, in particular for the extension to the anisotropic case. The transparency of these new boundary conditions is checked for two- and three-dimensional anisotropic waveguides. Finally, in the isotropic case, numerical validation for two- and three-dimensional waveguides illustrates the efficiency of the new approach, compared to other existing methods, in terms of number of iterations and CPU time.

Summary The paper presents a semi analytical method for solution of a statically indeterminate non-uniform bar problem. The solution of such statically indeterminate bar problem becomes more complicated due to the presence of singularity points in domain. Some examples of such one dimensional problems include, (i) stress and deformation analysis of a clamped–clamped (C–C) bar subjected to a concentrated load within domain, (ii) thermal stress analysis of a ‘C–C’ bar with a point heat source within domain, (iii) three point bending analysis of a roller supported curved beam under a concentrated transverse load, etc. In the present bar problem, only one such singularity point arising from the application of a concentrated axial load, is considered. Governing equation of the problem is derived from equilibrium condition and expressed in variational form with assumed displacement field by using direct variational principle. The computational domain is divided into two sub-domains based on the location of singularity point within the domain. An approximate solution of the governing equation is obtained assuming a series expression of the unknown variable by using Galerkin's principle. This approximation is carried out by a linear combination of sets of orthogonal co-ordinate functions which satisfy prescribed conditions at three points. The three conditions comprises of two boundary conditions and another condition at the point of application of concentrated load. The solution algorithm is implemented with the help of MATLAB ® computational simulation software. The problem is also studied by using energy functional based variational method and an identical solution is observed. The present analysis highlights the generalized application of domain decomposition method based on variational principle in solving structural problems having singularities in domain.

Isogeometric collocation methods are very recent and promising numerical schemes that preserve the advantages of isogeometric analysis but often exhibit better performances than their Galerkin counterparts. In the present paper, an additive overlapping Schwarz method for isogeometric collocation discretizations is introduced and studied. The resulting preconditioner, accelerated by GMRES, is shown to be scalable with respect to the number of subdomains and very robust with respect to the isogeometric discretization parameters such as the mesh size and polynomial degree, as well as with respect to the presence of discontinuous elliptic coefficients and domain deformations.

Since the observed values of security returns in real-world problems are sometimes imprecise or vague, an increasing effort in research is devoted to study the properties of risk measures in fuzzy portfolio optimization problems. In this paper, a new risk measure is suggested to gauge the risk resulted from fuzzy uncertainty. For this purpose, the absolute deviation and absolute semi-deviation are first defined for fuzzy variable by nonlinear fuzzy integrals. To compute effectively the absolute semi-deviations of single fuzzy variable as well as its functions, this paper discusses the methods of computing the absolute semi-deviation by classical Lebesgue–Stieltjes (L–S) integral. After that, several useful absolute deviation and absolute semi-deviation formulas are established for common triangular, trapezoidal and normal fuzzy variables. Applying the absolute semi-deviation as a new risk measure in portfolio optimization, three classes of fuzzy portfolio optimization models are developed by combining the absolute semi-deviation with expected value operator and credibility measure. Based on the analytical representation of absolute semi-deviations, the established fuzzy portfolio selection models can be turned into their equivalent piecewise linear or fractional programming problems. Since the absolute semi-deviation is a piecewise fractional function and pseudo-convex on the feasible subregions of deterministic programming models, we take advantage of the structural characteristics to design a domain decomposition method to separate a deterministic programming problem into three convex subproblems, which can be solved by conventional solution methods or general-purpose software. Finally, some numerical experiments are performed to demonstrate the new modeling idea and the effectiveness of the solution method.

This paper presents the extension of the global description approach of a discontinuous function, which is proposed in the previous paper, to a spectral domain decomposition method. This multi-domain spectral immersed interface method(IIM) divides the whole computation domain into the smooth and discontinuous parts. Fewer points on the smooth domains are used via taking advantage of the high accuracy property of the spectral method, but more points on the discontinuous domains are employed to enhance the resolution of the calculation. Two discontinuous problems are tested to verify the present method. The results show that the domain decomposition technique can reduce the error of the spectral IIM, especially when more collocation points are placed around the discontinuity. The present method is favorable for the reason that the same level of the accuracy can be reached, in spite of the enlarged computational domain.

International audience; 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 exhibits geometrical and structural heterogeneities, such as plate and stiffener assemblies in presence of structural details. The problem of the structural heterogeneities 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.

In this paper, we analyze the BEM-FEM non-overlapping domain decomposition method introduced in Boubendir [Techniques de Décomposition de Domaine et Méthode d’Equations Intégrales, Ph.D. Thesis, INSA, Toulouse, France, 2002] and improved in Boubendir et al. [A coupling BEM-FEM method using a FETI-LIKE domain decomposition method, in: Proceedings of Waves 2005, Providence, RI, 2005, pp. 188–190] and Bendali et al. [A FETI-like domain decomposition method for coupling FEM and BEM in large-size problems of acoustic scattering, to appear.] The transmission conditions used in this method introduce a quantity that prevents the approach of Després [Méthodes de décomposition de domaine pour les problèmes de propagation d’ondes en régime harmonique, Le théorème de Borg pour l’équation de Hill vectorielle, Ph.D. Thesis, Paris VI University, France, 1991] to establish convergence to be adapted. However, we show that convergence can be established when the geometry allows for a decomposition of the solution into propagating and evanescent portions with a methodology based on modal analysis. Here, we exemplify this in the case of circular cylindrical geometries where the derivations can be based on properties of Bessel functions.

When some rows of the system matrix and a preconditioner coincide, preconditioned iterations can be reduced to a sparse subspace. Taking advantage of this property can lead to considerable memory and computational savings. This is particularly useful with the GMRES method. We consider the iterative solution of a discretized partial differential equation on this sparse subspace. With a domain decomposition method and a fictitious domain method the subspace corresponds a small neighborhood of an interface. As numerical examples we solve the Helmholtz equation using a fictitious domain method and an elliptic equation with a jump in the diffusion coefficient using a separable preconditioner.

When the coefficients of a problem have jumps of several orders of magnitude and are anisotropic, many preconditioners and domain decomposition methods (DDM) suffer from plateaus in the convergence due to the presence of very small isolated eigenvalues in the spectrum of the preconditioned linear system. One way to improve the preconditioner is to use a linear algebra technique called deflation, or very similarly coarse grid corrections. In both cases, it is necessary to identify and compute, at least approximately, all the eigenvectors corresponding to the “bad” eigenvalues. In the framework of DDM, we propose a way to design interface conditions so that convergence is fast and does not have any plateau. The method relies only on the knowledge of the smallest and largest eigenvalues of an auxiliary matrix. The eigenvectors are not used. The method relies on van der Sluis’ result on a quasi-optimal diagonal preconditioner for a symmetric positive definite matrix. It is then possible to design Robin interface conditions using only one real parameter for the entire interface. By adding a second real parameter and more general interface conditions, it is possible to take into account highly heterogeneous and anisotropic media. Numerical results are given and compared with other approaches.

The paper deals with a parallel implementation of a mixed finite element method of solution of general fourth-order elliptic partial differential equations which in the case of aniso-/ortho-/isotropic plate bending problems simultaneously approximates the bending moment tensor field Psi = (psiij) (1less than or equal toi.jless than or equal to 2) and the displacement field u. The experiments on the performance of this method have been done in two parallel environments namely, Silicon Graphics Origin 3800 and Cray T3E. The details of the parallel computer implementation procedures for this mixed method with time, speed-up comparisons for a few illustrative examples are given. (C) 2004

This paper deals with an iterative algorithm for multidomain decomposition applied to the solution of a singularly perturbed convection–diffusion problem. Uniform convergent properties of the algorithm are established. Numerical results are presented.

This paper deals with iterative algorithms for domain decomposition applied to the solution of a singularly perturbed convection-diffusion problem with regular boundary layers. These algorithms are based on a finite-difference domain decomposition approach and on a modified Schwarz alternating method. Convergence properties of the algorithm are established, and their parallel implementations are discussed. Numerical results for a test singularly perturbed problem are presented.

In this paper we propose a new domain decompostion method for solving a Hamilton–Jacobi–Bellman equation of second order. The basic idea is to solve an equivalent quasivariational inequality instead of the original discretized HJB equation.

International audience; We present flow simulations in the Steam Generator of a pressurised water nuclear reactor using Domain Decomposition (DDM) and local zoom methods on workstation cluster. Concerning the DDM, we use a Dirichlet-Neumann approach jointly with FEM for averaged mixture balance equations. The algorithm, based on parallel or sequential iteration-by-subdomain method, works with overlapping or nonoverlapping subdomains and with conforming or nonconforming meshing. With DDM, the computational problem size is easily enhanced to about 100,000 mesh cells and the CPU time is strongly reduced. Concerning the Local Zoom computations, the used Local Defect Correction Method (LDC), in 3D local hierarchical multigrid context, is shown. The LDC computation results are compared with the classical full domain computation results (with high or low spatial resolution). We conclude to an improvement of the accuracy on the full domain with a high coherence between the zoom and the full domain.

This paper deals with an iterative algorithm for domain decomposition applied to the solution of a singularly perturbed parabolic problem with a convection-dominated term. Convergence properties of the algorithm are established. Numerical results are presented.

The domain decomposition method proposed by Schwarz [1] is gaining significance in view of the possibilities for parallel implementation. In this paper, we apply the domain decomposition method to the biharmonic equation in two overlapping discs. In each subdomain, a boundary integral formula is used to obtain the solution. Unlike the finite element or finite difference methods where the iterations have to be performed on the entire domain, the iterations of the boundary integral formulation need to be done only at the discretized points of the boundaries. Another aspect of the paper is that, in order to handle the operations with dense matrices arising from the boundary integral formulation, the discrete wavelet transform is used to compress the matrices which results in reduced computations without loss of accuracy. Examples involving a Laplace equation in an L-shaped region and a domain bounded by a cardioid and a circle are also illustrated.

This paper deals with an iterative algorithm for domain decomposition applied to solving of a singularly perturbed parabolic problem. This algorithm is based on an implicit scheme inside boundary layers and on an explicit scheme outside them, and suitable for parallel computing. Convergence properties of the algorithm are established, and numerical experiments are presented.

Parallel algorithms combining a time discretization and overlapping domain decomposition methods are applied to the solution of singularly perturbed parabolic problems. Two domain decomposition methods based on the Schwarz alternating procedure are proposed: the method with two-colour ordering of subdomains and the method with additional “correcting” problems. Moreover, modifications of the methods are considered using time extrapolation on subdomain interfaces. Convergence properties of the algorithms are established at the differential level. Numerical results for a test singularly perturbed problem are presented to confirm the convergence theory and to evaluate the parallel arithmetic complexity of the algorithms.

Parallel algorithms combining a time discretization and finite difference domain decomposition methods are applied to solution of a singularly perturbed parabolic problem. Different time discretizations are applied in different subdomains. Convergence results for the algorithms are established.

A frequency-stepping algorithm for solving multifrequency (acoustic) wave propagation is considered. A two-grid method is employed for the problems of single frequency. For high frequency applications, the coarse grid problem is still huge, since one has to choose at least six to eight grid points per wavelength for a stability reason. The coarse grid problem is solved by a nonoverlapping domain decomposition (DD) method. The solution of the former frequency problem is used as the initial guess for the solution of the next larger frequency problem. Such an algorithm turns out to be efficient for multifrequency, as well as single-frequency problems, as shown in numerical results. Also, it is easily parallelizable with a high efficiency.

This paper deals with iterative algorithms for domain decomposition applied to the solution of a quasilinear elliptic problem. Two iterative algorithms are examined: the first one is the Schwarz alternating procedure and the second algorithm is suitable for parallel computing. Convergence results are established in the two-domain and multidomain decomposition cases. Some issues of parallel implementation of these algorithms are discussed.

A combination of a time discretization and domain decomposition methods for the solution of singularly perturbed semilinear parabolic problems is considered. On this basis, iterative algorithms suitable for parallelization are constructed. Convergence properties of these algorithms are established. The implementation of the algorithms on a shared memory multiprocessor is examined. Numerical results for a test singularly perturbed problem are presented.

Accurate numerical simulations of flows around wind turbines play an important role in rotor blades design and operation control. The computation is very challenging because of the large size of the blades, the large fluids domain, the complex moving geometry, and the high Reynolds number. A popular method is the blade element momentum method which is relatively easy to implement, but has low fidelity. More recently, some high fidelity methods have been developed for the wind turbine simulations, such as Reynolds averaged Navier-Stokes methods, large-eddy simulations and direct numerical simulations (DNS). In this paper, we study DNS for flows passing 3D wind turbine rotors. The flow in the moving domain is modeled by the 3D unsteady incompressible Navier-Stokes equations in the arbitrary Lagrangian-Eulerian form. A stabilized bi-linear moving mesh finite element method, based on the unstructured tetrahedron mesh, is introduced to discretize the problem in space, and an implicit scheme is used to discretize the temporal variable. A parallel Newton-Krylov method with a domain decomposition type preconditioner is applied to solve the fully coupled nonlinear algebraic system at each timestep. We mainly focus on the performance of the domain decomposition preconditioner. To understand the efficiency of the algorithm, we test the software on a supercomputer for the simulation of a NREL 5MW wind turbine rotor. The numerical results show that the newly developed algorithm is scalable with thousands of processors for problems with tens of millions of unknowns.

A conservative Galerkin domain decomposition method for time-dependent problems is given and analyzed. This method allows one to apply different domain decompositions at different time levels when necessary, in order to capture time-changing local phenomena, such as, propagating fronts or moving layers. Error estimates in the energy and L 2 norms are established. Numerical results are presented.

Iterative domain decomposition algorithms are applied to the solution of two-dimensional eddy current problem. The system of differential equations describing this problem is considered as singularly perturbed problem. Convergence of these algorithms is established. Several numerical experiments are presented.

We present two domain decomposition methods for 1-D fourth order elliptic equations, and show that, in both cases, the iterative procedure converges in one step only for any partition of the domain into nonintersecting subdomains.