fuzzy connective based hierarchical aggregation networks for decision making abstract decision making is usually based on the aggregation of several criteria although the aggregation can be performed at one level in simple cases in more complex situations it becomes necessary that it be performed in a hierarchical manner in a multilevel framework this is particularly true in many engineering applications in this paper we investigate the use of fuzzy connectives in multilayer networks to arrive at appropriate overall aggregation functions the characteristics of suitable fuzzy connectives are studied and an iterative algorithm to determine the type of aggregation function and its parameters at each node in the network is proposed we believe that our aggregation scheme is powerful and flexible since we use several types of aggregation connectives and since the parameters of the network can be determined using a simple algorithm the behavior of the network can also be modified depending on the desired attitude conditions for unique convergence of the algorithm in simple cases are discussed and examples of applications of the method are also presented

dual primal domain decomposition method for uncertainty quantification abstract the spectral stochastic finite element method ssfem may offer an efficient alternative to the traditional monte carlo simulations mcs for uncertainty quantification of large scale numerical simulations in the framework of the intrusive ssfem the main computational challenge involves solving a coupled set of deterministic linear systems for large scale numerical models the computational efficiency of the intrusive ssfem primarily depends on the solution techniques employed to tackle the resulting coupled linear systems in this paper we report a probabilistic version of the dual primal domain decomposition method for the intrusive ssfem in order to exploit high performance computing platforms for uncertainty quantification in particular we formulate a probabilistic version of the dual primal finite element tearing and interconnect feti dp technique to solve the large scale linear systems in the intrusive ssfem in the probabilistic setting the operator of the dual interface system in the dual primal approach contains a coarse problem the introduction of the coarse problem in the probabilistic setting leads to a scalable performance of the dual primal iterative substructuring method for uncertainty quantification of large scale computational models the convergence properties numerical and parallel scalabilities of the probabilistic feti dp method and the recently developed probabilistic version of the balancing domain decomposition by constraints bddc method are contrasted for numerical illustrations we consider flow through porous media and linear elasticity problems with spatially varying system parameters modelled as non gaussian random processes the algorithms are implemented on a linux cluster using mpi and petsc parallel libraries

a feti dp preconditioner with a special scaling for mortar discretization of elliptic problems with discontinuous coefficients we consider two dimensional elliptic problems with discontinuous coefficients discretized by the finite element method on geometrically conforming nonmatching triangulations across the interface using the mortar technique the resulting discrete problem is solved by a dual primal feti method in this paper we introduce and analyze a preconditioner with a special scaling of coefficients and step parameters and establish convergence bounds we show that the preconditioner is almost optimal with constants independent of the jumps of coefficients and step parameters extensive computational evidence is presented that illustrates an almost optimal convergence for a variety of situations distribution of subregions grid assignment grid ratios number of subregions for both continuous and discontinuous problems

robust feti dp methods for heterogeneous three dimensional elasticity problems iterative substructuring methods with lagrange multipliers are considered for heterogeneous linear elasticity problems with large discontinuities in the material stiffnesses in particular results for algorithms belonging to the family of dual primal feti methods are presented the core issue of these algorithms is the construction of an appropriate global problem in order to obtain a robust method which converges independently of the material discontinuities in this article several necessary and sufficient conditions arising from the theory are numerically tested and confirmed furthermore results of numerical experiments are presented for situations which are not covered by the theory such as curved edges and material discontinuities not aligned with the interface and an attempt is made to develop rules for these cases

fuzzy modeling based on generalized conjunction operations an approach to fuzzy modeling based on the tuning of parametric conjunction operations is proposed first some methods for the construction of parametric generalized conjunction operations simpler than the known parametric classes of conjunctions are considered and discussed second several examples of function approximation by fuzzy models based on the tuning of the parameters of the new conjunction operations are given and their approximation performances are compared with the approaches based on a tuning of membership functions and other approaches proposed in the literature it is seen that the tuning of the conjunction operations can be used for obtaining fuzzy models with a sufficiently good performance when the tuning of membership functions is not possible or not desirable

a hybrid discontinuous galerkin method for computing the ground state solution of bose einstein condensates a numerical method for computing the ground state solution of bose einstein condensates modeled by the gross pitaevskii equation is presented in this method the three dimensional computational domain is divided into hexahedral elements in which the solution is approximated by a sum of basis functions both polynomial and plane wave bases are considered for this purpose and lagrange multipliers are introduced to weakly enforce the interelement continuity of the solution the ground state is computed by an iterative procedure for minimizing the energy the performance results obtained for several numerical experiments demonstrate that the proposed method is more computationally efficient than similar solution approaches based on the standard higher order finite element method

overlapping and non overlapping domain decomposition methods for large scale meshless efg simulations abstract meshless methods have a number of virtues in problems concerning crack growth and propagation strain localization dynamic shear band propagation projectile penetration among others the main drawback of such methods is that the resulting matrices are more densely populated and the computational cost for the formulation and solution of the problem is much higher than the conventional fem this is the reason for their limited application to academic and small scale problems until now in this paper we introduce a novel approach for reducing the computational cost of meshless element free galerkin efg methods by employing domain decomposition techniques on the physical as well as on the algebraic domains specifically we propose the implementation of the dual domain decomposition feti family of methods that have been successfully used with fem in many problems in computational mechanics we address the issues of dividing an overlapping meshfree efg domain to several overlapping subdomains following the algebraic decomposition of the feti method and alternatively on non overlapping subdomains by modifying the required displacement compatibility conditions and by improving the accuracy of the solution through a stress update procedure

convergence of a substructuring method with lagrange multipliers we analyze the convergence of a substructing iterative method with lagrange multipliers propose recently by farhat and roux the method decomposes finite element discretization of an elliptic boundary value problem into neumann problems on the subdomains plus a coarse problem for the subdomain nullspace components for linear conforming elements and preconditioning by the dirichlet problems on the subdomains we prove the asymptotic bound on the condition number c 1 log h h alpha alpha 2 or 3 where h is the characteristic element size h subdomain size and c is independent of the number of subdomains

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

3 d electromagnetic scattering computation in free space with the feti fdp2 method the electromagnetic dual primal finite element tearing and interconnecting feti dpem method is a nonoverlapping domain decomposition method developed for the finite element analysis of large scale electromagnetic problems where the corner edges are globally numbered this paper presents an extension of the feti dpem2 method named feti full dual primal feti fdp2 where more flexible robin type boundary conditions are imposed on the inner interfaces between subdomains as well as on the corner edges leading to a new interface problem its capacities are tested in the framework of a three dimensional 3 d free space scattering problem with a scattered field formulation and a computational domain truncated by perfectly mathed layers pml first we compare its accuracy with respect to other feti dpem2 methods and to a complete resolution of the fem problem thanks to a direct sparse solver we show that the convergence of iterative solvers is affected by the presence of the pml and can be accelerated by means of a more accurate approximation between adjacent subdomains of the dirichlet to neumann dtn operator the effectiveness of the iterative solvers are also considered for different test cases the advantages of the proposed feti fdp2 method combined with the associated dtn approximation is numerically demonstrated regardless the chosen working frequency or the iterative solvers

a domain decomposition approach for full field measurements based identification of local elastic parameters abstract we propose a domain decomposition formalism specifically designed for the identification of local elastic parameters based on full field measurements this technique is made possible by a multi scale implementation of the constitutive compatibility method contrary to classical approaches the constitutive compatibility method resolves first some eigenmodes of the stress field over the structure rather than directly trying to recover the material properties a two steps micro macro reconstruction of the stress field is performed a dirichlet identification problem is solved first over every subdomain the macroscopic equilibrium is then ensured between the subdomains in a second step we apply the method to large linear elastic 2d identification problems to efficiently produce estimates of the material properties at a much lower computational cost than classical approaches

pictorial representations of fuzzy connectives part ii cases of compensatory operators and self dual operators abstract as the continuation of part i in which pictorial representations of t norms t conorms and averaging operators were made with the aid of a computer this paper discusses compensatory operators and proposes generalized compensatory operators which can be obtained from averaging operators self dual operators are discussed which can be defined by using t norms t conorms and averaging operators finally symmetric sums are reviewed which are also self dual operators the pictorial representations of these operators are also made

a feti dp formulation of three dimensional elasticity problems with mortar discretization in this paper a feti dp formulation for three dimensional elasticity on nonmatching grids over geometrically nonconforming subdomain partitions is considered to resolve the nonconformity of the finite elements a mortar matching condition is imposed on the subdomain interfaces faces a feti dp algorithm is then built by enforcing the mortar matching condition in dual and primal ways in order to make the feti dp algorithm scalable a set of primal constraints which include average and momentum constraints over interfaces are selected from the mortar matching condition a condition number bound c 1 log h h 2 is then proved for the feti dp formulation for the elasticity problems with discontinuous material parameters only some faces need to be chosen as primal faces on which the average and momentum constraints are imposed numerical results are included

construction of fuzzy σ algebras using triangular norms abstract generalizing the definitions given by the author fuzzy sets and systems4 1980 83 93 we introduce and study t fuzzy algebras t being any triangular norm the main result is that for a large class of triangular norms each t fuzzy algebra is generated i e consists of all functions x 0 1 being measurable with respect to some algebra on x

on the use of inexact subdomain solvers for bddc algorithms the standard bddc balancing domain decomposition by constraints preconditioner is shown to be equivalent to a preconditioner built from a partially subassembled finite element model this results in a system of linear algebraic equations which is much easier to solve in parallel than the fully assembled model the cost is then often dominated by that of the problems on the subdomains an important role is also played both in theory and practice by an averaging operator and in addition exact dirichlet solvers are used on the subdomains in order to eliminate the residual in the interior of the subdomains the use of inexact solvers for these problems and even the replacement of the dirichlet solvers by a trivial extension are considered it is established that one of the resulting algorithms has the same eigenvalues as the standard bddc algorithm and the connection of another with the feti dp algorithm with a lumped preconditioner is also considered multigrid methods are used in the experimental work and under certain assumptions it is established that the iteration count essentially remains the same as when exact solvers are used while considerable gains in the speed of the algorithm can be realized since the cost of the exact solvers grows superlinearly with the size of the subdomain problems while the multigrid methods are linear

a new parametric family of fuzzy connectives and their application to fuzzy control abstract it is of practical importance to have parametric families of fuzzy connectives which are easy to handle and which are rich enough in the situation under consideration this paper introduces a new axiomatic description to conjunctive and compensative connection of fuzzy sets the new parametric family of connectives is developed through the use of logical valuation for the use of logical valuation the dependence of the correcting variable described by linguistic rules is compared with the dependence of the correcting variable calculated by means of an analytical description of the fuzzy controller the use of the new parameterised fuzzy connective is demonstrated for a hydraulic servo drive system the control of the process shows that with the new parameterised fuzzy connective we reach the best result of control compared with the conventional connectives e g t norms t or averaging operators d

dual primal feti methods for three dimensional elliptic problems with heterogeneous coefficients in this paper certain iterative substructuring methods with lagrange multipliers are considered for elliptic problems in three dimensions the algorithms belong to the family of dual primal finite element tearing and interconnecting feti methods which recently have been introduced and analyzed successfully for elliptic problems in the plane the family of algorithms for three dimensions is extended and a full analysis is provided for the new algorithms particular attention is paid to finding algorithms with a small primal subspace since that subspace represents the only global part of the dual primal preconditioner it is shown that the condition numbers of several of the dual primal feti methods can be bounded polylogarithmically as a function of the dimension of the individual subregion problems and that the bounds are otherwise independent of the number of subdomains the mesh size and jumps in the coefficients these results closely parallel those of other successful iterative substructuring methods of primal as well as dual type

structure preserving stability and accuracy properties of the energy conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models summary the computational efficiency of a typical projection based nonlinear model reduction method hinges on the efficient approximation for explicit computations of the scalar projections onto a subspace of a residual vector for implicit computations it also hinges on the additional efficient approximation of similar projections of the jacobian of this residual with respect to the solution the computation of both approximations is often referred to in the literature as hyper reduction to this effect this paper focuses on the analysis and comparative performance study for nonlinear finite element reduced order models of solids and structures of the recently developed energy conserving mesh sampling and weighting ecsw hyper reduction method unlike most alternative approaches this method approximates the scalar projections of residuals and or jacobians directly instead of approximating first these vectors and matrices then projecting the resulting approximations onto the subspaces of interest in this paper it is shown that ecsw distinguishes itself furthermore from other hyper reduction methods through its preservation of the lagrangian structure associated with hamilton s principle for second order dynamical systems this enables it to also preserve the numerical stability properties of the discrete system to which it is applied it is also shown that for a fixed set of parameter values the approximation error committed online by ecsw is bounded by its counterpart error committed off line during the training of this method therefore this error can be estimated in this case a priori and controlled the performance of ecsw is demonstrated first for two academic but nevertheless interesting nonlinear dynamic response problems for both of them ecsw is shown to preserve numerical stability and deliver the desired level of accuracy whereas the discrete empirical interpolation method and its recently introduced unassembled variant are shown to be susceptible to failure because of numerical instability the potential of ecsw for enabling the effective reduction of nonlinear finite element dynamic models of solids and structures is also highlighted with the realistic simulation of the nonlinear transient dynamic response of a complete car engine to thermal and combustion pressure loads using an implicit scheme for this simulation ecsw is reported to enable the reduction of the cpu time required by the high dimensional nonlinear finite element dynamic analysis by more than four orders of magnitude while achieving a very good level of accuracy copyright 2015 john wiley sons ltd

incorporation of multipoint constraints into the balancing domain decomposition method and its parallel implementation the domain decomposition method ddm is a major solution algorithm that is used to parallelize the finite element method in the case of implicit structural analysis using the ddm the substructuring based iterative linear solver is a powerful tool when an effective preconditioner such as the balancing domain decomposition bdd method is used in the present study a method by which to incorporate a set of linear multipoint constraints mpc into the bdd method is proposed in this method when an mpc is enforced on the internal degrees of freedom dofs in some subdomains the dofs are converted into interface dofs that is all of the dofs constrained by mpcs become interface dofs then the interface problem with the set of mpcs for the interface dofs is solved by the conjugate projected gradient method in order to combine the above procedure with the preconditioner used in the bdd method the effect of the mpcs for the interface dofs is also imposed on the coarse grid problem of the bdd method using the penalty method a parallel implementation of the present method is also described some illustrative examples are solved and good convergence and parallel performance of the present method are demonstrated copyright 2006 john wiley sons ltd

cholesky decomposition of a positive semidefinite matrix with known kernel the cholesky decomposition of a symmetric positive semidefinite matrix a is a useful tool for solving the related consistent system of linear equations or evaluating the action of a generalized inverse especially when a is relatively large and sparse to use the cholesky decomposition effectively it is necessary to identify reliably the positions of zero rows or columns of the factors and to choose these positions so that the nonsingular submatrix of a of the maximal rank is reasonably conditioned the point of this note is to show how to exploit information about the kernel of a to accomplish both tasks the results are illustrated by numerical experiments

constraint local principal curve existing principal curve algorithms have some drawbacks such as time consuming and narrow application scope in practice since these algorithms are mainly based on global optimization in this paper we present the concept of constraint local principal curve clpc which uses local optimization methods and restricts the principal curve with two fixed endpoints to reduce the computational complexity in addition we propose three clpc algorithms by local optimization and adaptive radius to expand the range of applications and increase the solution quality the first algorithm i e clpcg is based on greedy thinking the second algorithm i e clpcs uses one dimensional search and the last algorithm clpcc combines the greedy thinking and one dimensional search then we define six performance indices to evaluate the performance of the clpc algorithms finally we present some numerical experiments with three simulation data sets and two gps measured data sets in both highway and railway the results indicate that all of the three clpc algorithms can obtain high accuracy data from multiple low accuracy data efficiently the clpc algorithms can improve the accuracy and computational speed compared with the existing k segment principal curve kpc algorithm in addition clpcc outperforms clpcg and clpcs according to the comprehensive experiments while clpcg runs much faster than other ones we present the concept of constraint local principal curve clpc to reduce the computational complexity we propose three clpc algorithms that combine local optimization and adaptive radius to expand the range of applications and increase the solution quality we present some numerical experiments with three simulation data sets and two measured gps data sets in highway and railway the clpc algorithms can improve the accuracy and computational speed compared with the existing kpc algorithms the features of the each clpc algorithm are analyzed according to the comprehensive experiments

incorporation of linear multipoint constraints in domain decomposition based iterative solvers part ii blending feti dp and mortar methods and assembling floating substructures two approaches are presented for extending the feti dp domain decomposition based iterative method to the solution of problems with linear multipoint constraints lmpcs both cases of arbitrary lmpcs and lmpcs generated by the mortar method for enforcing a weak continuity of the solution across non matching finite element interfaces are considered in the first approach the lmpcs are addressed during the preconditioning step but enforced only at convergence in the second approach the lmpcs are enforced at each iteration through the solution of an auxiliary coarse problem both extended feti dp solvers are benchmarked on a parallel processor for a wide range of constrained structural mechanics problems the obtained results reveal that both solvers are numerically scalable and complement each other from the memory and cpu performance viewpoints

high performance domainwise parallel direct solver for large scale structural analysis large scale structural analysis using a finite element method requires a high performance and efficient parallel algorithm that is scalable in terms of both performance and storage most of the research for large scale parallel structural analysis has focused on iterative solution methods because they are much easier to parallelize in contrast direct solution methods have generally been considered in adequate for large scale finite element computations because of many difficulties and disadvantages for carrying out large scale problems however direct solution methods are still generally preferred due to their ease of use and the numerical robustness that guarantees the solution to be obtained within an estimated time without failure therefore for the general application of large scale structural analysis to a wide range of problems an efficient parallel direct solution method that has scalability comparable to that of iterative methods is proposed a new parallel direct solver for large scale finite element analysis the domainwise multifrontal solver is proposed by realizing domainwise parallelism for a direct solver by the use of the proposed solver with our own structural analysis code good scalability can be shown as a direct method and can solve the largest problem ever known to be solved by direct solvers

a parallel implementation of dual primal feti methods for three dimensional linear elasticity using a transformation of basis dual primal feti methods for linear elasticity problems in three dimensions are considered these are nonoverlapping domain decomposition methods where some primal continuity constraints across subdomain boundaries are required to hold throughout the iterations whereas most of the constraints are enforced by lagrange multipliers an algorithmic framework for dual primal feti methods is described together with a transformation of basis to implement the primal constraints numerical results obtained from a parallel implementation of these algorithms applied to a model benchmark problem with structured meshes and to problems with more complicated geometries from industrial and biological applications using unstructured meshes are provided these results show that the presented dual primal feti algorithms are numerical and parallel scalable

the schwarz alternating method in solid mechanics abstract we advance the schwarz alternating method as a means for concurrent multiscale coupling in finite deformation solid mechanics we prove that the schwarz alternating method converges to the solution of the problem on the entire domain and that the convergence rate is geometric provided that each of the subdomain problems is well posed i e their corresponding energy density functions are quasi convex it is shown that the use of a newton type method for the solution of the resultant nonlinear system leads to two kinds of block linearized systems depending on the treatment of the dirichlet boundary conditions the first kind is a symmetric block diagonal linear system in which each diagonal block is the tangent stiffness of each subdomain i e the off diagonal blocks are all zero and the coupling terms appear only on the right hand side the second kind is a nonsymmetric block system with off diagonal coupling terms several variants of the schwarz alternating method are proposed for the first kind of linear system including one in which the schwarz alternating iterations and the newton iterations are combined into a single scheme this version of the method is particularly attractive as it lends itself to a minimally intrusive implementation into existing finite element codes finally we demonstrate the performance of the proposed variants of the schwarz alternating method on several one dimensional and three dimensional examples

optimal convergence properties of the feti domain decomposition method abstract the finite element tearing and interconnecting feti method is a practical and efficient domain decomposition dd algorithm for the solution of self adjoint elliptic partial differential equations for large scale structural problems discretized with shell and beam elements this method was found to outperform popular iterative algorithms and direct solvers on both serial and parallel computers and to compare favorably with leading dd methods in this paper we discuss some numerical properties of the feti method that were not addressed before in particular we show that the mathematical treatment of the floating subdomains and the specific conjugate projected gradient algorithm that characterize the feti method are equivalent to the construction and solution of a coarse problem that propagates the error globally accelerates convergence and ensures a performance that is independent of the number of subdomains we also show that when the interface problem is optimally preconditioned and the mesh is partitioned into well structured subdomains with good aspect ratios the performance of the feti method is also independent of the mesh size however we also argue that the feti and other leading dd methods for unstructured problems lose in practice these scalability properties when the mesh contains junctures with rotational degrees of freedom or the decomposition is irregular and characterized by arbitrary subdomain aspect ratios finally we report that for realistic problems optimal preconditioners are not necessarily computationally efficient and can be outperformed by non optimal ones

a scalable substructuring method by lagrange multipliers for plate bending problems we present a new lagrange multiplier based domain decomposition method for solving iteratively systems of equations arising from the finite element discretization of plate bending problems the proposed method is essentially an extension of the finite element tearing and interconnecting substructuring algorithm to the biharmonic equation the main idea is to enforce continuity of the transversal displacement field at the subdomain crosspoints throughout the preconditioned conjugate gradient iterations the resulting method is proved to have a condition number that does not grow with the number of subdomains but rather grows at most polylogarithmically with the number of elements per subdomain these optimal properties hold for numerous plate bending elements that are used in practice including the hsieh clough tocher element the discrete kirchhoff triangle and a class of nonlocking elements for the reissner mindlin plate models computational experiments are reported and shown to confirm the theoretical optimal convergence properties of the new domain decomposition method computational efficiency is also demonstrated with the numerical solution in 45 iterations and 105 seconds on a 64 processor ibm sp2 of a plate bending problem with almost one million degrees of freedom

simultaneous feti and block feti robust domain decomposition with multiple search directions domain decomposition methods often exhibit very poor performance when applied to engineering problems with large heterogeneities in particular for heterogeneities along domain interfaces the iterative techniques to solve the interface problem are lacking an efficient preconditioner recently a robust approach named feti geneo was proposed where troublesome modes are precomputed and deflated from the interface problem the cost of the feti geneo is however high we propose in this paper techniques that share similar ideas with feti geneo but where no pre processing is needed and that can be easily and efficiently implemented as an alternative to standard domain decomposition methods in the block iterative approaches presented in this paper the search space at every iteration on the interface problem contains as many directions as there are domains in the decomposition those search directions originate either from the domain wise preconditioner in the simultaneous feti method or from the block structure of the right hand side of the interface problem block feti we show on 2d structural examples that both methods are robust and provide good convergence in the presence of high heterogeneities even when the interface is jagged or when the domains have a bad aspect ratio the simultaneous feti was also efficiently implemented in an optimized parallel code and exhibited excellent performance compared to the regular feti method

on the convergence of a dual primal substructuring method in the dual primal feti method introduced by farhat et al 1999 the domain is decomposed into non overlapping subdomains but the degrees of freedom on crosspoints remain common to all subdomains adjacent to the crosspoint the continuity of the remaining degrees of freedom on subdomain interfaces is enforced by lagrange multipliers and all degrees of freedom are eliminated the resulting dual problem is solved by preconditioned conjugate gradients we give an algebraic bound on the condition number assuming only a single inequality in discrete norms and use the algebraic bound to show that the condition number is bounded by c 1 log 2 h h for both second and fourth order elliptic selfadjoint problems discretized by conforming finite elements as well as for a wide class of finite elements for the reissner mindlin plate model

an additive schwarz preconditioner for the feti method a new additive schwarz preconditioner for the finite element tearing and interconnecting feti method is analyzed in this paper this preconditioner has the unique feature that the coefficient matrix of its coarse grid problem is mesh independent for a model second order heterogeneous elliptic boundary value problem in two dimensions the condition number of the preconditioned system is shown to be bounded by c 1 ln h h 2 where h is the mesh size h is the typical diameter of the subdomains and the constant c is independent of h h the number of subdomains and the coefficients of the boundary value problem

an extension of the feti domain decomposition method for incompressible and nearly incompressible problems incompressible and nearly incompressible problems are treated herein with a mixed finite element formulation in order to avoid ill conditioning that prevents accuracy in pressure estimation and can lead to a lack of convergence for iterative solution algorithms a multilevel dual domain decomposition method is then chosen as an iterative algorithm the original feti and feti dp methods are extended to deal with such problems when the discretization of the pressure field is discontinuous throughout the elements a dedicated augmentation of the algorithms is proposed and the different methods are compared with several preconditioners for bidimensional test cases the resulting approaches are both optimal and numerically scalable and their costs are estimated with a complexity analysis

high scalable non overlapping domain decomposition method using a direct method for finite element analysis abstract in this paper a novel domain decomposition method is introduced to solve a large finite element model in this method the decomposed domains do not overlap and are connected using a simple connective finite element which influences the nodal point equilibrium between adjacent finite elements this approach has the advantage that it allows use of a direct method such as gauss elimination even in a singular problem the singular stiffness matrices from the floating domain without the dirichlet boundary conditions are changed into invertible stiffness matrices by assembling the connective elements another advantage is that computational time and storage can be reduced by using a banded matrix in the direct solver in order to describe this proposed method we first review the feti method which is the most popular domain decomposition method then the proposed method is introduced with a technical approach in a distributed computer system finally the high scalability and computational efficiency of the proposed method are verified by comparing with the traditional feti method for 2d and 3d finite element examples which have floating subdomains in the result we demonstrate high scalability for a large finite element model

the two level feti method part ii extension to shell problems parallel implementation and performance results the two level feti method presented in part i of this work is a mathematically optimal and scalable substructuring algorithm for solving iteratively large scale systems of equations arising from the finite element discretization of static and dynamic plate bending problems in this paper we extend the two level feti method to shell problems describe a variant approach revisit the preconditioning problem address the efficient implementation of the corresponding iterative solvers on massively parallel processors highlight the computational price of mathematical optimality and discuss its consequences and report some impressive performance results on two paragon xp s and ibm sp2 massively parallel processors for several realistic plate and shell problems

a feti method with a mesh independent condition number for the iteration matrix abstract we introduce a framework for feti methods using ideas from the decomposition via lagrange multipliers of h 0 1 derived by raviart thomas p a raviart j m thomas primal hybrid finite element methods for second order elliptic equations math comput 31 138 1977 391 413 and complemented with the detailed work on polygonal domains developed by grisvard p grisvard singularities in boundary value problems rech math appl 22 1992 we compute the action of the lagrange multipliers using the natural h 00 1 2 scalar product therefore no consistency error appears our analysis allows to deal with cross points and floating subdomains in a natural manner as a byproduct we obtain that the condition number for the iteration matrix is independent of the mesh size and there is no need for preconditioning this result improves the standard asymptotic bound for this condition number given by 1 log h h 2 where h and h are the characteristic subdomain size and element size respectively shown by mandel tezaur in j mandel r tezaur convergence of a substructuring method with lagrange multipliers numer math 73 1996 473 487 numerical results that confirm our theoretical analysis are presented

nonlinear feti dp and bddc methods new nonlinear feti dp dual primal finite element tearing and interconnecting and bddc balancing domain decomposition by constraints domain decomposition methods are introduced in all these methods in each iteration local nonlinear problems are solved on the subdomains the new approaches can significantly reduce communication and show a significantly improved performance especially for problems with localized nonlinearities compared to a standard newton krylov feti dp or bddc approach moreover the coarse space of the nonlinear feti dp methods can be used to accelerate the newton convergence it is also found that the new nonlinear feti dp and nonlinear bddc methods are not as closely related as in the linear context numerical results for the p laplace operator are presented

pictorial representations of fuzzy connectives part i cases of t norms t conorms and averaging operators part i of this paper shows a number of examples of existing t norms and t conorms which are known as good models of fuzzy set theoretic intersection and union and then the pictorial representations of t norms and t conorms are made with the aid of a computer several new examples of t norms and t conorms are also given averaging operators between min and max are summarized and their pictorial representations are also made

feti and neumann neumann iterative substructuring methods connections and new results the feti and neumann neumann families of algorithms are among the best know and most severely tested domain decomposition methods for elliptic partial differential equations they are iterative substructuring methods and have many algorithmic components in common but there are also differences the purpose of this paper is to further unify the theory for these two families of methods and to introduce a new family of feti algorithms bounds on the rate of convergence which are uniform with respect to the coefficients of a family of elliptic problems with heterogeneous coefficients are established for these new algorithms the theory for a variant of the neumann neumann algorithm is also redeveloped stressing similarities to that for the feti methods

deflated preconditioned conjugate gradient solvers for linear elasticity extensions of deflation techniques previously developed for the poisson equation to static elasticity are presented compared to the scalar poisson equation j comput phys 2008 227 24 10196 10208 int j numer meth engng 2010 doi 10 1002 nme 2932 int j numer meth biomed engng 2010 26 1 73 85 the elasticity equations represent a system of equations giving rise to more complex low frequency modes multigrid elsevier amsterdam 2000 in particular the straightforward extension from the scalar case does not provide generally satisfactory convergence however a simple modification allows to recover the remarkable acceleration in convergence and cpu time reached in the scalar case numerous examples and timings are provided in a serial and a parallel context and show the dramatic improvements of up to two orders of magnitude in cpu time for grids with moderate graph depths compared to the non deflated version furthermore a monotonic decrease of iterations with increasing subdomains as well as a remarkable acceleration for very few subdomains are also observed if all the rigid body modes are included copyright 2011 john wiley sons ltd

a preconditioned dual primal finite element tearing and interconnecting method for solving three dimensional time harmonic maxwell s equations a new preconditioned dual primal nonoverlapping domain decomposition method is proposed for the finite element solution of three dimensional large scale electromagnetic problems with the aid of two lagrange multipliers the new method converts the original volumetric problem to a surface problem by using a higher order transmission condition at the subdomain interfaces to significantly improve the convergence of the iterative solution of the global interface equation similar to the previous version a global coarse problem related to the degrees of freedom at the subdomain corner edges is formulated to propagate the residual error to the whole computational domain at each iteration which further increases the rate of convergence in addition a fully algebraic preconditioner based on matrix splitting is constructed to make the proposed domain decomposition method even more robust and scalable perfectly matched layers pmls are considered for the boundary truncation when solving open region problems the influence of the pml truncation on the convergence performance is investigated by examining the convergence of the transmission condition for an interface inside the pml numerical examples including wave propagation and antenna radiation problems truncated with pmls are presented to demonstrate the validity and the capability of this method

incorporation of linear multipoint constraints in substructure based iterative solvers part 1 a numerically scalable algorithm we consider the iterative solution by a class of substructuring methods of the large scale systems of equations arising from the nite element discretization of structural models with an arbitrary set of linear multipoint constraints we present a methodology for generalizing to such problems numerically scalable substructure based iterative solvers without interfering with their formulations and their well established local and global preconditioners we apply this methodology to the feti method and show that the resulting algorithm is numerically scalable with respect to both the substructure and problem sizes 1998 john wiley sons ltd

extending substructure based iterative solvers to multiple load and repeated analyses direct solvers currently dominate commercial finite element structural software but do not scale well in the fine granularity regime targeted by emerging parallel processors substructure based iterative solvers often called also domain decomposition algorithms lend themselves better to parallel processing but must overcome several obstacles before earning their place in general purpose structural analysis programs one such obstacle is the solution of systems with many or repeated right hand sides such systems arise for example in multiple load static analyses and in implicit linear dynamics computations direct solvers are well suited for these problems because after the system matrix has been factored the multiple or repeated solutions can be obtained through relatively inexpensive forward and backward substitutions on the other hand iterative solvers in general are ill suited for these problems because they often must restart from scratch for every different right hand side in this paper we present a methodology for extending the range of applications of domain decomposition methods to problems with multiple or repeated right hand sides basically we formulate the overall problem as a series of minimization problems over k orthogonal and supplementary subspaces and tailor the preconditioned conjugate gradient algorithm to solve them efficiently the resulting solution method is scalable whereas direct factorization schemes and forward and backward substitution algorithms are not we illustrate the proposed methodology with the solution of static and dynamic structural problems and highlight its potential to outperform forward and backward substitutions on parallel computers as an example we show that for a linear structural dynamics problem with 11640 degrees of freedom every time step beyond time step 15 is solved in a single iteration and consumes 1 0 second on a 32 processor ipsc 860 system for the same problem and the same parallel processor a pair of forward backward substitutions at each step consumes 15 0 seconds

an interface strip preconditioner for domain decomposition methods application to hydrology in this paper the efficiency of a parallelizable preconditioner for domain decomposition methods in the context of the solution of non symmetric linear equations arising from discretization of the saint venant equations is investigated the proposed interface strip preconditioner is is based on solving a problem in a narrow strip around the interface it requires much less memory and computing time than classical neumann neumann preconditioner and handles correctly the flux splitting among sub domains that share the interface the performance of this preconditioner is assessed with an analytical study of schur complement matrix eigenvalues and numerical experiments conducted in a parallel computational environment consisting of a beowulf cluster of 20 nodes copyright 2005 john wiley sons ltd

a bddc algorithm for mortar discretization of elasticity problems a balancing domain decomposition by constraints bddc algorithm is developed for compressible elasticity problems in three dimensions with mortar discretization on geometrically nonconforming subdomain partitions material parameters of the elasticity problems may have jump across the subdomain interface coarse basis functions in the bddc algorithm are constructed from primal constraints on faces which are similar to the average matching condition and the moment matching condition considered in a klawonn and o b widlund comm pure appl math 59 2006 pp 1523 1572 and h h kim a feti dp formulation of three dimensional elasticity problems with mortar discretization technical report new york university 2005 a condition number bound is proved to be c 1 log h h 3 for geometrically nonconforming partitions and to be c 1 log h h 2 for geometrically conforming partitions the bound is not affected by the jump of the material parameters across the subdomain interface numerical results are included

robust bddc preconditioners for reissner mindlin plate bending problems and mitc elements a balancing domain decomposition method by constraints bddc is constructed and analyzed for the reissner mindlin plate bending problem discretized with mixed interpolation of tensorial components mitc finite elements this bddc algorithm is based on selecting the plate rotations and deflection degrees of freedom at the subdomain vertices as primal continuity constraints after the implicit elimination of the interior degrees of freedom in each subdomain the resulting plate schur complement is solved by the preconditioned conjugate gradient method the preconditioner is based on the solution of local reissner mindlin plate problems on each subdomain with clamping conditions at the primal degrees of freedom and on the solution of a coarse reissner mindlin plate problem for the primal degrees of freedom the main results of the paper are the proof and numerical verification that the proposed bddc plate algorithm is scalable quasi optimal and most important robust with respect to the plate thickness while this result is due to an underlying mixed formulation of the problem both the interface plate problem and the preconditioner are positive definite the numerical results also show that the proposed algorithm is robust with respect to discontinuities of the material properties

a deluxe feti dp algorithm for a hybrid staggered discontinuous galerkin method for h curl elliptic problems summary convergence theories and a deluxe dual and primal finite element tearing and interconnecting algorithm are developed for a hybrid staggered dg finite element approximation of h curl elliptic problems in two dimensions in addition to the advantages of staggered dg methods the basis functions of the new hybrid staggered dg method are all locally supported in the triangular elements and a lagrange multiplier approach is applied to enforce the global connections of these basis functions the interface problem on the lagrange multipliers is further reduced to the resulting problem on the subdomain interfaces and a dual and primal finite element tearing and interconnecting algorithm with an enriched weight factor is then applied to the resulting problem our algorithm is shown to give a condition number bound ofc 1c log h h 2 independent of the two parameters whereh his the number of triangles across each subdomain numerical results are included to confirm our theoretical bounds copyright 2013 john wiley sons ltd received 12 september 2013 accepted 11 november 2013 in this paper a dual and primal finite element tearing and interconnecting feti dp algorithm is developed for a fast and stable solution of a new hybrid staggered dg hsdg method applied to h curl problems staggered dg sdg methods were first introduced in 1 3 for wave propagation problems the idea was subsequently applied to other problems such as convection diffusion equations 4 electromagnetic problems 5 9 stokes equations 10 and multiscale wave simulations 11 12 similar to 5 the advantages of using sdg for h curl problems are the preservation of the structures of the differential operators the local conservation property and the optimal convergence in particular the discretizations of the two curl operators by the sdg method are adjoint to each other and the null space of the discrete curl operator is exactly the gradients moreover the divergence condition is automatically satisfied in an appropriate weak sense despite the aforementioned advantages the implementation of sdg methods requires careful numbering of the supports of the basis functions to allow an easier implementation of the sdg method the hsdg method is introduced in this paper one distinctive advantage of the hsdg method is that the basis functions are all locally supported in the triangular elements hence the numbering can be easily performed the fact that the basis functions are totally discontinuous requires some global couplings in particular we will use the lagrange multiplier approach to enforce the global connection of the basis functions this idea is also related to the hybridized dg methods 13 16 in fact it is shown that the sdg method is the limit of a hybridized dg method

mechanism free domain decomposition the simulation of three dimensional 3d structural dynamics on massively parallel platforms places stringent requirements on the existing software infrastructure a constrained and nonlinear graph partitioning problem that arises in scalable iterative substructuring methods such as finite element tearing and interconnecting feti methods is identified new sufficient criteria on a partition are presented that ensure the applicability of feti methods and improve the associated preconditioner one dimensional finite elements in 3d structures are treated by an encapsulation method the techniques are demonstrated on complex finite element model problems

analysis of feti methods for multiscale pdes in this paper we study a variant of the finite element tearing and interconnecting feti method which is suitable for elliptic pdes with highly heterogeneous multiscale coefficients x in particular coefficients with strong variation within subdomains and or jumps that are not aligned with the subdomain interfaces using energy minimisation and cut off arguments we can show rigorously that for an arbitrary positive coefficient function alpha in l infty omega the condition number of the preconditioned feti system can be bounded by c 1 log h h 2 where h is the subdomain diameter and h is the mesh size and where the function c depends only on the coefficient variation in the vicinity of subdomain interfaces in particular if alpha omega i varies only mildly in a layer i of width near the boundary of each of the subdomains i then c alpha mathcal o h eta 2 independent of the variation of in the remainder i i of each subdomain and independent of any jumps of across subdomain interfaces the quadratic dependence of c on h can be relaxed to a linear dependence under stronger assumptions on the behaviour of in the interior of the subdomains our theoretical findings are confirmed in numerical tests

on large scale generalized inverses in solving two by two block linear systems abstract the goal is to analyze a role of generalized inverses in the solution of two by two block linear systems by means of an algorithm combining the schur complement reduction with the null space method the efficient procedures for computing actions of generalized inversions to rectangular large scale matrices are presented with a special attention to the moore penrose inverse a regularization technique for matrices with the known null space basis is introduced that enables to get generalized inverses without necessity to recognize zero pivots the theoretical results are confirmed by numerical examples

newton krylov bddc solvers for nonlinear cardiac mechanics abstract the aim of this work is to design and study a balancing domain decomposition by constraints bddc solver for the nonlinear elasticity system modeling the mechanical deformation of cardiac tissue the contraction relaxation process in the myocardium is induced by the generation and spread of the bioelectrical excitation throughout the tissue and it is mathematically described by the coupling of cardiac electro mechanical models consisting of systems of partial and ordinary differential equations in this study the discretization of the electro mechanical models is performed by q1 finite elements in space and semi implicit finite difference schemes in time leading to the solution of a large scale linear system for the bioelectrical potentials and a nonlinear system for the mechanical deformation at each time step of the simulation the parallel mechanical solver proposed in this paper consists in solving the nonlinear system with a newton krylov bddc method based on the parallel solution of local mechanical problems and a coarse problem for the so called primal unknowns three dimensional parallel numerical tests on different machines show that the proposed parallel solver is scalable in the number of subdomains quasi optimal in the ratio of subdomain to mesh sizes and robust with respect to tissue anisotropy

the two level feti method for static and dynamic plate problems part i an optimal iterative solver for biharmonic systems abstract we present a lagrange multiplier based substructuring method for solving iteratively large scale systems of equations arising from the finite element discretization of static and dynamic plate bending problems the proposed method is essentially an extension of the feti domain decomposition algorithm to fourth order problems the main idea is to enforce exactly the continuity of the transverse displacement field at the substructure corners throughout the preconditioned conjugate projected gradient iterations this results in a two level feti substructuring method where the condition number of the preconditioned interface problem does not grow with the number of substructures and grows at most polylogarithmically with the number of elements per substructure these theoretically proven optimal convergence properties of the new feti method are numerically demonstrated for several finite element plate bending static and transient problems the two level iterative solver presented in this paper is applicable to a large family of biharmonic time independent as well as time dependent systems it is also extendible to shell problems

a simple and efficient extension of a class of substructure based preconditioners to heterogeneous structural mechanics problems several domain decomposition methods with lagrange multipliers have been recently designed for solving iteratively large scale systems of finite element equations while these methods differ typically by implementational details they share in most cases the same substructure based preconditioners that were originally developed for the feti method the success of these preconditioners is due to the fact that for homogeneous structural mechanics problems they ensure a computational performance that scales with the problem size in this paper we address the suboptimal behaviour of these preconditioners in the presence of material and or discretization heterogeneities we propose a simple and virtually no cost extension of these preconditioners that exhibits scalability even for highly heterogeneous systems of equations we consider several intricate structural analysis problems and demonstrate numerically the optimal performance delivered by the new preconditioners for problems with discontinuities copyright 1999 john wiley sons ltd

spectral element feti dp and bddc preconditioners with multi element subdomains abstract the discrete systems generated by spectral or hp version finite elements are much more ill conditioned than the ones generated by standard low order finite elements or finite differences this paper focuses on spectral elements based on gauss lobatto legendre gll quadrature and the construction of primal and dual non overlapping domain decomposition methods belonging to the family of balancing domain decomposition methods by constraints bddc and dual primal finite element tearing and interconnecting feti dp algorithms new results are presented for the spectral multi element case and also for inexact feti dp methods for spectral elements in the plane theoretical convergence estimates show that these methods have a convergence rate independent of the number of subdomains and coefficient jumps of the elliptic operator while there is only a polylogarithmic dependence on the spectral degree p and the ratio h h of subdomain and element sizes parallel numerical experiments on a linux cluster confirm these results for tests with spectral degree up to p 32 thousands of subdomains and coefficient jumps up to 8 orders of magnitude

the mosaic of high performance domain decomposition methods for structural mechanics formulation interrelation and numerical efficiency of primal and dual methods a multitude of domain decomposition methods ddm for structural mechanics is available in the literature today a unified framework for formulating primal and dual ddm is thus presented in this paper aiming at providing a mathematical platform for a uniform treatment of high performance ddm in structural mechanics a novel approach for developing new ddm from existing methods is also proposed and is applied to dual and primal methods in the field of the feti methods this approach leads to a new category of methods derived from existing feti variants furthermore two alternative formulations of the balancing domain decomposition method are described while interrelations between the introduced and existing methods are established finally comparative numerical tests demonstrate the differences in the computational performance of the methods in question

feti dp a dual primal unified feti method part i a faster alternative to the two level feti method the feti method and its two level extension feti 2 are two numerically scalable domain decomposition methods with lagrange multipliers for the iterative solution of second order solid mechanics and fourth order beam plate and shell structural problems respectively the feti 2 method distinguishes itself from the basic or one level feti method by a second set of lagrange multipliers that are introduced at the subdomain cross points to enforce at each iteration the exact continuity of a subset of the displacement field at these specific locations in this paper we present a dual primal formulation of the feti 2 concept that eliminates the need for that second set of lagrange multipliers and unifies all previously developed one level and two level feti algorithms into a single dual primal feti dp method we show that this new feti dp method is numerically scalable for both second order and fourth order problems we also show that it is more robust and more computationally efficient than existing feti solvers particularly when the number of subdomains and or processors is very large copyright 2001 john wiley sons ltd

unconditionally stable operator splitting algorithms for the incompressible magnetohydrodynamics system discretized by a stabilized finite element formulation based on projections summary in this article we propose different splitting procedures for the transient incompressible magnetohydrodynamics mhd system that are unconditionally stable we consider two levels of splitting on one side we perform the segregation of the fluid pressure and magnetic pseudo pressure from the vectorial fields computation at the second level the fluid velocity and induction fields are also decoupled this way we transform a fully coupled indefinite multi physics system into a set of smaller definite ones clearly reducing the cpu cost with regard to the finite element approximation we stick to an unconditionally convergent stabilized finite element formulation because it introduces convection stabilization allows to circumvent inf sup conditions clearly simplifying implementation issues and is able to capture non smooth solutions of the magnetic subproblem however residual based finite element formulations are not suitable for segregation because they lose the skew symmetry of the off diagonal blocks therefore in this work we have proposed a novel term by term stabilization of the mhd system based on projections that is still unconditionally convergent copyright 2012 john wiley sons ltd

the second generation feti methods and their application to the parallel solution of large scale linear and geometrically non linear structural analysis problems the feti algorithms are a family of numerically scalable domain decomposition methods they have been designed in the early 1990s for solving iteratively and on parallel machines large scale systems of equations arising from the finite element discretization of solid mechanics structural engineering structural dynamics and acoustic scattering problems and for analyzing complex structures obtained from the assembly of substructures with incompatible discrete interfaces in this paper we present the second generation of these methods that operate more efficiently on large numbers of subdomains offer greater robustness better performance and more flexibility for implementation on a wider variety of computational platforms we also report on the application and performance of these methods for the solution of geometrically non linear structural analysis problems we discuss key aspects of their implementation on shared and distributed memory parallel processors benchmark them against optimized direct sparse solvers and highlight their potential with the solution of large scale structural mechanics problems with several million degrees of freedom

a scalable feti dp algorithm with non penetration mortar conditions on contact interface by combining feti algorithms of dual primal type with recent results for bound constrained quadratic programming problems we develop an optimal algorithm for the numerical solution of coercive variational inequalities the model problem is discretized using non penetration conditions of mortar type across the potential contact interface and a feti dp algorithm is formulated the resulting quadratic programming problem with bound constraints is solved by a scalable algorithm with a known rate of convergence given in terms of the spectral condition number of the quadratic problem numerical experiments for non matching meshes across the contact interface confirm the theoretical scalability of the algorithm

a two level nonoverlapping schwarz algorithm for the stokes problem numerical study abstract a general framework of a two level nonoverlapping schwarz algorithm for the stokes problem is developed by relaxing average zero condition on pressure unknowns this framework allows both discontinuous and continuous pressure finite element spaces the coarse problem is built by algebraic manipulation after selecting appropriate primal unknowns just like in bddc algorithms performance of the suggested algorithm is presented depending on the selection of finite elements and primal unknowns under the same set of primal unknowns the algorithm for the case with discontinuous pressure functions outperforms one with continuous pressure functions for the two dimensional stokes problem the algorithm with a set of primal unknowns consisting of velocity unknowns at corners averages of velocity components over common edges and pressure unknowns at corners presents good scalability when continuous pressure test functions are used in both two and three dimensional stokes problems an improvement can be made for the case with continuous pressure test functions by applying the suggested algorithm to the interface problem which is obtained by eliminating velocity unknowns and pressure unknowns interior to each subdomains

a scalable lagrange multiplier based domain decomposition method for time dependent problems we present a new efficient and scalable domain decomposition method for solving implicitly linear and non linear time dependent problems in computational mechanics the method is derived by adding a coarse problem to the recently proposed transient feti substructuring algorithm in order to propagate the error globally and accelerate convergence it is proved that in the limit for large time steps the new method converges toward the feti algorithm for time independent problems computational results confirm that the optimal convergence properties of the time independent feti method are preserved in the time dependent case we employ an iterative scheme for solving efficiently the coarse problem on massively parallel processors and demonstrate the effective scalability of the new transient feti method with the large scale finite element dynamic analysis on the paragon xp s and ibm sp2 systems of several diffraction grating finite element structural models we also show that this new domain decomposition method outperforms the popular direct skyline solver the coarse problem presented herein is applicable and beneficial to a large class of lagrange multiplier based substructuring algorithms for time dependent problems including the fictitious domain decomposition method

a study on the lumped preconditioner and memory requirements of feti and related primal domain decomposition methods in recent years domain decomposition methods ddms have emerged as advanced solvers in several areas of computational mechanics in particular during the last decade in the area of solid and structural mechanics they reached a considerable level of advancement and were shown to be more efficient than popular solvers like advanced sparse direct solvers the present contribution follows the lines of a series of recent publications on the relationship between primal and dual formulations of ddms in some of these papers the effort to unify primal and dual methods led to a family of ddms that was shown to be more efficient than the previous methods the present paper extends this work presenting a new family of related ddms thus enriching the theory of the relations between primal and dual methods with the primal methods which correspond to the dual ddm that uses the lumped preconditioner the paper also compares the numerical performance of the new methods with that of the previous ones and focuses particularly on memory requirement issues related to the use of the lumped preconditioner suggesting a particularly memory efficient formulation

bddc preconditioners for spectral element discretizations of almost incompressible elasticity in three dimensions balancing domain decomposition by constraints bddc algorithms are constructed and analyzed for the system of almost incompressible elasticity discretized with gauss lobatto legendre spectral elements in three dimensions initially mixed spectral elements are employed to discretize the almost incompressible elasticity system but a positive definite reformulation is obtained by eliminating all pressure degrees of freedom interior to each subdomain into which the spectral elements have been grouped appropriate sets of primal constraints can be associated with the subdomain vertices edges and faces so that the resulting bddc methods have a fast convergence rate independent of the almost incompressibility of the material in particular the condition number of the bddc preconditioned operator is shown to depend only weakly on the polynomial degree n the ratio h h of subdomain and element diameters and the inverse of the inf sup constants of the subdomains and the underlying mixed formulation while being scalable i e independent of the number of subdomains and robust i e independent of the poisson ratio and young s modulus of the material considered these results also apply to the related dual primal finite element tearing and interconnect feti dp algorithms defined by the same set of primal constraints numerical experiments carried out on parallel computing systems confirm these results

a two level domain decomposition method for the iterative solution of high frequency exterior helmholtz problems summary we present a lagrange multiplier based two level domain decomposition method for solving iteratively large scale systems of equations arising from the finite element discretization of high frequency exterior helmholtz problems the proposed method is essentially an extension of the regularized feti finite element tearing and interconnecting method to indefinite problems its two key ingredients are the regularization of each subdomain matrix by a complex interface lumped mass matrix and the preconditioning of the interface problem by an auxiliary coarse problem constructed to enforce at each iteration the orthogonality of the residual to a set of carefully chosen planar waves we show numerically that the proposed method is scalable with respect to the mesh size the subdomain size and the wavenumber we report performance results for a submarine application that highlight the efficiency of the proposed method for the solution of high frequency acoustic scattering problems discretized by finite elements

inexact data sparse boundary element tearing and interconnecting methods the boundary element tearing and interconnecting beti methods have recently been introduced as boundary element counterparts of the well established finite element tearing and interconnecting feti methods in this paper we present inexact data sparse versions of the beti methods which avoid the elimination of the primal unknowns and dense matrices however instead of symmetric and positive definite systems we finally have to solve twofold saddle point problems the proposed iterative solvers and preconditioners result in almost optimal solvers whose complexity is proportional to the number of unknowns on the skeleton up to some polylogarithmical factor moreover the solvers are robust with respect to large coefficient jumps

a fast implementation of the feti dp method feti dp rbs lna and applications on large scale problems with localized non linearities as parallel and distributed computing gradually becomes the computing standard for large scale problems the domain decomposition method dd has received growing attention since it provides a natural basis for splitting a large problem into many small problems which can be submitted to individual computing nodes and processed in a parallel fashion this approach not only provides a method to solve large scale problems that are not solvable on a single computer by using direct sparse solvers but also gives a flexible solution to deal with large scale problems with localized non linearities when some parts of the structure are modified only the corresponding subdomains and the interface equation that connects all the subdomains need to be recomputed in this paper the dual primal finite element tearing and interconnecting method feti dp is carefully investigated and a reduced back substitution rbs algorithm is proposed to accelerate the time consuming preconditioned conjugate gradient pcg iterations involved in the interface problems linear non linear analysis lna is also adopted for large scale problems with localized non linearities based on subdomain linear non linear identification criteria this combined approach is named as the feti dp rbs lna algorithm and demonstrated on the mechanical analyses of a welding problem serial cpu costs of this algorithm are measured at each solution stage and compared with that from the ibm watson direct sparse solver and the feti dp method the results demonstrate the effectiveness of the proposed computational approach for simulating welding problems which is representative of a large class of three dimensional large scale problems with localized non linearities

on the general solution by a direct method of a large scale singular system of linear equations application to the analysis of floating structures finding the general solution of a singular system of linear equations requires computing a particular solution and a basis of the null space of the corresponding singular matrix in this paper we consider the case where the singular matrix is large and sparse and the application calls for a direct solution method we highlight the dependence of straightforward factorization algorithms on an arbitrary constant that can influence the correctness of the computed solution and describe a family of improved direct solution methods that alleviate this problem for structural mechanics applications we propose a hybrid geometric algebraic method that is more robust than the purely algebraic direct methods that are currently used for solving singular sparse systems of equations we illustrate the potential of our proposed solution algorithms with examples from structural mechanics and domain decomposition based iterative solvers 1998 john wiley sons ltd

variational theory and domain decomposition for nonlocal problems abstract in this article we present the first results on domain decomposition methods for nonlocal operators we present a nonlocal variational formulation for these operators and establish the well posedness of associated boundary value problems proving a nonlocal poincare inequality to determine the conditioning of the discretized operator we prove a spectral equivalence which leads to a mesh size independent upper bound for the condition number of the stiffness matrix we then introduce a nonlocal two domain variational formulation utilizing nonlocal transmission conditions and prove equivalence with the single domain formulation a nonlocal schur complement is introduced we establish condition number bounds for the nonlocal stiffness and schur complement matrices supporting numerical experiments demonstrating the conditioning of the nonlocal one and two domain problems are presented

a feti preconditioned conjugate gradient method for large scale stochastic finite element problems in the spectral stochastic finite element method for analyzing an uncertain system the uncertainty is represented by a set of random variables and a quantity of interest such as the system response is considered as a function of these random variables consequently the underlying galerkin projection yields a block system of deterministic equations where the blocks are sparse but coupled the solution of this algebraic system of equations becomes rapidly challenging when the size of the physical system and or the level of uncertainty is increased this paper addresses this challenge by presenting a preconditioned conjugate gradient method for such block systems where the preconditioning step is based on the dual primal finite element tearing and interconnecting method equipped with a krylov subspace reusage technique for accelerating the iterative solution of systems with multiple and repeated right hand sides preliminary performance results on a linux cluster suggest that the proposed solution method is numerically scalable and demonstrate its potential for making the uncertainty quantification of realistic systems tractable

mesh partitioning for implicit computations via iterative domain decomposition impact and optimization of the subdomain aspect ratio optimal domain decomposition methods have emerged as powerful iterative algorithms for parallel implicit computations their key preprocessing step is mesh partitioning where research has focused so far on the automatic generation of load balanced subdomains with minimum interface nodes in this paper we emphasize the importance of the subdomain aspect ratio as a mesh partitioning factor and highlight its impact on the convergence rate of an optimal domain decomposition based iterative method we also present a fast optimization algorithm for improving the aspect ratio of existing mesh partitions and illustrate it with several examples from fluid dynamics and structural mechanics applications for a stiffened shell problem decomposed by the optimal recursive spectral bisection scheme and solved by the feti method this optimization algorithm is shown to improve the solution time by a factor equal to 1 54 and to restore numerical scalability

a domain decomposition method for problems with structural heterogeneities on the interface application to a passenger ship 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

a dual primal feti method for solving a class of fluid structure interaction problems in the frequency domain summary the dual primal finite element tearing and interconnecting method feti dp is extended to systems of linear equations arising from a finite element discretization for a class of fluid structure interaction problems in the frequency domain a preconditioned generalized minimal residual method is used to solve the linear equations for the lagrange multipliers introduced on the subdomain boundaries to enforce continuity of the solution the coupling between the fluid and the structure on the fluid structure interface requires an appropriate choice of coarse level degrees of freedom in the feti dp algorithm to achieve fast convergence several choices are proposed and tested by numerical experiments on three dimensional fluid structure interaction problems in the mid frequency regime that demonstrate the greatly improved performance of the proposed algorithm over the standard feti dp method copyright 2011 john wiley sons ltd

bddc and feti dp preconditioners with adaptive coarse spaces for three dimensional elliptic problems with oscillatory and high contrast coefficients abstract bddc and feti dp algorithms are developed for three dimensional elliptic problems with adaptively enriched coarse components it is known that these enriched components are necessary in the development of robust preconditioners to form the adaptive coarse components carefully designed generalized eigenvalue problems are introduced for each faces and edges and the coarse components are formed by using eigenvectors with their corresponding eigenvalues larger than a given tolerance t o l upper bounds for condition numbers of the preconditioned systems are shown to be c t o l with the constant c depending only on the maximum number of edges and faces per subdomain and the maximum number of subdomains sharing an edge numerical results are presented to test the robustness of the proposed approach

bridging multiple structural scales with a generalized finite element method summary this paper presents a generalized fem based on the solution of interdependent coarse scale global and fine scale local problems in order to resolve multiscale effects due to fine scale heterogeneities overall structural behavior is captured by the global problem while local problems focus on the resolution of fine scale solution features in regions where material heterogeneities may govern the structural response fine scale problems are accurately solved in parallel and to address the intrinsic coupling of scales these solutions are embedded into the global solution space using a partition of unity approach this method is demonstrated on representative heat transfer examples in order to examine its accuracy efficiency and flexibility copyright 2014 john wiley sons ltd

feti dp bddc and block cholesky methods the feti dp and bddc algorithms are reformulated using block cholesky factorizations an approach which can provide a useful framework for the design of domain decomposition algorithms for solving symmetric positive definite linear system of equations instead of introducing lagrange multipliers to enforce the coarse level primal continuity constraints in these algorithms a change of variables is used such that each primal constraint corresponds to an explicit degree of freedom with the new formulation of these algorithms a simplified proof is provided that the spectra of a pair of feti dp and bddc algorithms with the same set of primal constraints are essentially the same numerical experiments for a two dimensional laplace s equation also confirm this result copyright 2005 john wiley sons ltd