the nsbeti method an extension of the feti method to non symmetrical bem fem coupled problems summary the finite element tearing and interconnecting feti method is recognized as an effective domain decomposition tool to achieve scalability in the solution of partitioned second order elasticity problems in the boundary element tearing and interconnecting beti method a direct extension of the feti algorithm to the bem the symmetric galerkin bem formulation is used to obtain symmetric system matrices making possible to apply the same feti conjugate gradient solver in this work we propose a new beti variant labeled nsbeti that allows to couple substructures modeled with the fem and or non symmetrical bem formulations the method connects non matching bem and fem subdomains using localized lagrange multipliers and solves the associated non symmetrical flexibility equations with a bi cgstab iterative algorithm scalability issues of nsbeti in bem bem and combined bem fem coupled problems are also investigated copyright 2012 john wiley sons ltd

reduction of substructural interface degrees of freedom in flexibility based component mode synthesis a flexibility based component mode synthesis cms is proposed for reduced order modelling of dynamic behaviour of large structures the approach employs partitioning via the localized lagrange multiplier method the use of the localized lagrange multipliers leads to unlike the classical lagrange multipliers a linearly independent set of interface forces without any redundancies at multiply connected interface nodes the flexibility based cms method has shown significant advantages over the classical craig bampton method a key feature of the method is its substructural mode selection criterion that is independent of loading conditions unlike the majority of available cms approaches where one retains the full dimension of partition boundary degrees of freedom dofs the flexibility based method allows to reduce significantly the interface dofs the reduction of interface dofs represents the major contribution of the present communication the efficiency of the proposed approach is demonstrated on an analysis of a simple plate partitioned and of a more complex 3d structure both partitioned into several substructures copyright 2006 john wiley sons ltd

homoclinic loop and multiple limit cycle bifurcation surfaces this paper establishes the existence and analyticity of homoclinic loop bifurcation surfaces z and multiplicity two limit cycle bifurcation surfaces f for planar systems depending on two or more parameters it determines the side of or f on which limit cycles occur and it shows that if x and intersect then typically they do so at a flat contact

an algebraically partitioned feti method for parallel structural analysis algorithm description an algebraically partitioned feti method for the solution of structural engineering problems on parallel computers is presented the present algorithm consists of three attributes an explicit generation of the orthogonal null space matrix associated with the interface nodal forces the floating subdomain rigid body modes computed from the subdomain static equilibrium equation of the classical force method and the identification of redundant interface force constraint operator that emanates when the interface force computations are localized comparisons of the present method with the previously developed differentially partitioned feti method are offered in terms of the saddle point formulations at the end of the paper a companion paper reports implementation details and numerical performance of the proposed algorithm 1997 john wiley sons ltd

a variational principle for the formulation of partitioned structural systems a continuum based variational principle is presented for the formulation of the discrete governing equations of partitioned structural systems this application includes coupled substructures as well as subdomains obtained by mesh decomposition the present variational principle is derived by a series of modifications of a hybrid functional originally proposed by atluri for finite element development the interface is treated by a displacement frame and a localized version of the method of lagrange multipliers interior displacements are decomposed into rigid body and deformational components to handle floating subdomains both static and dynamic versions are considered an important application of the present principle is the treatment of nonmatching meshes that arise from various sources such as separate discretization of substructures independent mesh refinement and global local analysis the present principle is compared with that of a globalized version of the multiplier method copyright 2000 john wiley sons ltd

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

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 preconditioner for substructuring based on constrained energy minimization a preconditioner for substructuring based on constrained energy minimization concepts is presented the preconditioner is applicable to both structured and unstructured meshes and offers a straightforward approach for the iterative solution of second and fourth order structural mechanics problems the approach involves constraints associated with disjoint sets of nodes on substructure boundaries these constraints provide the means for preconditioning at both the substructure and global levels numerical examples are presented that demonstrate the good performance of the method in terms of iterations compute time and condition numbers of the preconditioned equations

comparison of different feti preconditioners for elastoplasticity this paper illustrates parallel solution of elastoplastic problems with hardening based on the tfeti domain decomposition method with several preconditioning strategies we consider von mises plasticity with isotropic hardening using the return mapping concept for one time step to treat nonlinearity and nonsmoothness we use the semismooth newton method in each newton iteration we solve a linear system of equations using the tfeti domain decomposition method with lumped dirichlet or no preconditioner our permon software is used for the numerical experiments the observed times and numbers of iterations are backed up by the regular condition number estimates both preconditioners accelerate the solution process in terms of the convergence rate for the worst conditioned benchmark the most expensive dirichlet preconditioner gives the lowest computational times this lets us suggest the dirichlet preconditioner as the default choice for engineering problems

efficient solvers for saddle point problems arising from domain decompositions with lagrange multipliers in this paper we are concerned with the domain decomposition method with lagrange multipliers for solving three dimensional elliptic problems with variable coefficients we shall first introduce a weighted saddle point problem resulting from this domain decomposition which can be solved by existing iterative methods then we will construct two simple preconditioners one for the system associated with the displacement variable and the other for the schur complement system associated with the multiplier variable that are applicable to various discretization schemes the new preconditioners possess such local properties that they can be implemented cheaply we will show that the condition number of the global preconditioned system grows only as the logarithm of the dimension of the local problem associated with an individual substructure and is independent of the large variations of the coefficients across the local interfaces

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

a stabilized explicit lagrange multiplier based domain decomposition method for parabolic problems a fully explicit stabilized domain decomposition method for solving moderately stiff parabolic partial differential equations pdes is presented writing the semi discretized equations as a differential algebraic equation dae system where the interface continuity constraints between subdomains are enforced by lagrange multipliers the method uses the runge kutta chebyshev projection scheme to integrate the dae explicitly and to enforce the constraints by a projection with mass lumping techniques and node to node matching grids the method is fully explicit without solving any linear system a stability analysis is presented to show the extended stability property of the method the method is straightforward to implement and to parallelize numerical results demonstrate that it has excellent performance

extraction of substructural flexibility from global frequencies and mode shapes a computational procedure for extracting substructure by substructure flexibility properties from global modal parameters is presented the present procedure consists of two key features an element based direct flexibility method which uniquely determines the global flexibility without resorting to case dependent redundancy selections and the projection of cinematically inadmissible modes that are contained in the iterated substructural matrices the direct flexibility method is used as the basis of an inverse problem whose goal is to determine substructural flexibilities given the global flexibility geometrically determined substructural rigid body modes and the local to global assembly operators the resulting procedure given accurate global flexibility extracts the exact element by element substructural flexibilities for determinate structures for indeterminate structures the accuracy depends on the iteration tolerance limits the procedure is illustrated using both simple and complex numerical examples and appears to be effective for structural applications such as damage localization and finite element model reconciliation

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

the extended generalized finite element method an overview of the method and its applications an overview of the extended generalized finite element method gefm xfem with emphasis on methodological issues is presented this method enables the accurate approximation of solutions that involve jumps kinks singularities and other locally non smooth features within elements this is achieved by enriching the polynomial approximation space of the classical finite element method the gefm xfem has shown its potential in a variety of applications that involve non smooth solutions near interfaces among them are the simulation of cracks shear bands dislocations solidification and multi field problems copyright 2010 john wiley sons ltd

cohesive and non cohesive fracture by higher order enrichment of xfem summary a comprehensive study is performed on the use of higher order terms of the crack tip asymptotic fields as enriching functions for the extended fem xfem for both cohesive and traction free cracks for traction free cracks the williams asymptotic field is used to obtain highly accurate stress intensity factors sifs directly from the enriched degrees of freedom without any post processing the low accuracy of the results of the original research on this subject by liu et al int j numer meth engng 2004 59 1103 1118 is remedied here by appropriate modifications of the enrichment scheme the modifications are simple and can be easily included into an xfem computer code for cohesive cracks the relevant asymptotic field is used and two widely used criteria including the sifs criterion and the stress criterion are examined for the crack growth simulation both linear and nonlinear cohesive laws are used for the stress criterion averaging is avoided due to the highly accurate crack tip approximation because of the higher order enrichment then a modified stress criterion is proposed which is shown to be applicable to a wider class of problems several numerical examples including straight and curved cracks stationary and growing cracks single and multiple cracks and traction free and cohesive cracks are studied to investigate in detail the robustness and efficiency of the proposed enrichment scheme copyright 2012 john wiley sons ltd

a direct flexibility method we present a direct flexibility method dfm for the solution of finite element equations this method is based on a decomposition of the finite element model into substructures which may reduce to individual elements substructures are preprocessed by the direct stiffness method dsm to generate free free flexibility matrices for floating substructures the interface problem is solved for the interface forces and the solution recovered over substructure interiors the dfm shares with the dsm the advantages of being automatic maintaining locality and sparseness efficiently handling continuum elements and requiring only the availability of element stiffness libraries the new method appears to be advantageous for specific applications these include massively parallel processing inverse problems treatment of rigid members and inclusions and use of underintegrated elements without spurious mode stabilization

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

a proof of perkoʼs conjectures for the bogdanov takens system abstract the bogdanov takens system has at most one limit cycle and in the parameter space it exists between a hopf and a saddle loop bifurcation curves the aim of this paper is to prove the perko s conjectures about some analytic properties of the saddle loop bifurcation curve moreover we provide sharp piecewise algebraic upper and lower bounds for this curve

a hybrid multi time step framework for pore scale and continuum scale modeling of solute transport in porous media abstract in this paper we propose a computational framework which is based on a domain decomposition technique to employ both finite element method which is a popular continuum modeling approach and lattice boltzmann method which is a popular pore scale modeling approach in the same computational domain to bridge the gap across the disparate length and time scales we first propose a new method to enforce continuum scale boundary conditions i e dirichlet and neumann boundary conditions onto the numerical solution from the lattice boltzmann method this method is based on maximization of entropy and preserve the non negativity of discrete distributions under the lattice boltzmann method the proposed computational framework allows different grid sizes orders of interpolation and time steps in different subdomains this allows for different desired resolutions in the numerical solution in different subdomains through numerical experiments the effect of grid and time step refinement disparity of time steps in different subdomains domain partitioning and the number of iteration steps on the accuracy and rate of convergence of the proposed methodology are studied finally to showcase the performance of this framework in porous media applications we use it to simulate the dissolution of calcium carbonate in a porous structure

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

a stochastic deterministic coupling method for continuum mechanics in this paper we present a novel approach that allows to couple a deterministic continuum model with a stochastic continuum one the coupling strategy is performed in the arlequin framework which is based on a volume coupling and a partition of the energy a suitable functional space is chosen for the weak enforcement of the continuity between the two models the choice of this space ensures that the mean of the stochastic solution equals the deterministic solution point wise and enforces appropriate boundary conditions on the stochastic dimension the proof of the existence of the solution of the mixed problem is provided the numerical strategy is also reviewed in particular with a view at the monte carlo method finally examples show the interest of the method and possible strategies for use in adaptive modeling

a preconditioner for the feti dp formulation with mortar methods in two dimensions in this paper we consider a dual primal feti feti dp method for elliptic problems on nonmatching grids the feti dp method is a domain decomposition method that uses lagrange multipliers to match solutions continuously across subdomain boundaries in the sense of dual primal variables we use the mortar matching condition as the continuity constraints for the feti dp formulation we construct a preconditioner for the feti dp operator and show that the condition number of the preconditioned feti dp operator is bounded by c max i 1 ldots n left 1 log left h i h i right right 2 where h i and h i are sizes of domain and mesh for each subdomain respectively and c is a constant independent of h i s and h i s we allow jumps of coefficients of elliptic problems across subdomain boundaries numerical results are included

condition number estimates and weak scaling for 2 level 2 lagrange multiplier methods for general domains and cross points the 2 lagrange multiplier method is a domain decomposition method which can be used to parallelize the solution of linear problems arising from partial differential equations in order to scale to large numbers of subdomains and processors domain decomposition methods require a coarse grid correction to transport low frequency information more rapidly between subdomains that are far apart we introduce two new 2 level methods by adding a coarse grid correction to 2 lagrange multiplier methods we prove that if we shrink h the grid parameter while maintaining bounded the ratio frac h h where h is the size of the subdomains the condition number of the method remains bounded we confirm our analysis with experiments on the hector high end computing terascale resource supercomputer this proves that the new methods scale weakly opening the door to massively parallel implementations

a time staggered partitioned coupling algorithm for transient heat conduction we present a time staggered partitioned coupling algorithm for transient heat conduction finite element simulations this algorithm divides a large structural mesh into a number of smaller subdomains solves the individual subdomains separately and couples the solutions to obtain the response to the original problem the proposed algorithm is a mixed multi timestep algorithm and enables arbitrary time integration schemes and meshes to be coupled with different timesteps in the various subdomains in this procedure the solution of each partition is separately evaluated over a system timestep after which the interfacial conditions are enforced making this a staggered algorithm that facilitates parallel computation we present examples showing the feasibility of the coupling algorithm and discuss the merits in terms of convergence and stability

multi scale domain decomposition method for large scale structural analysis with a zooming technique application to plate assembly this article is concerned with a multi scale domain decomposition method ddm based on the feti dp solver for large scale structural elastic analysis and suited to problems that exhibit structural heterogeneities such as plate assemblies in the presence of structural details in this approach once a partition of the global fine mesh into subdomains has been performed all subdomains possess a fine mesh and to optimize the computational time the fine mesh is preserved only in the zones of interest with local phenomena due to discontinuity hole etc while the remaining subdomains are replaced by numerical homogenized coarse elements indeed the multi scale aspect is introduced by the description of subdomains with either a fine or a coarse scale mesh as a result an extension of the feti dp ddm is proposed in this article called herein feti dp micro macro that allows the simultaneous usage of different discretizations fine microscopic mesh for subdomains in zones of interest and coarse macroscopic or homogenized mesh for the complementary part of the structure using this strategy raises the problem of the determination of the stiffness of coarse subdomains and of the incompatible finite element connection between fine and coarse subdomains two approaches collocation and mortar are presented and compared the article ends with patch tests and some numerical examples in 2d and 3d the obtained numerical results exemplify the efficiency and capability of the feti dp micro macro approach and reveal that the mortar approach is more accurate at constant cost than the collocation approach copyright 2009 john wiley sons ltd

bifurcation of the separatrix skeleton in some 1 parameter families of planar vector fields this article deals with the bifurcation of polycycles and limit cycles within the 1 parameter families of planar vector fields x m k defined by y 3 x 2k 1 x my 4k 1 where m is a real parameter and k1 integer the bifurcation diagram for the separatrix skeleton of x m k in function of m is determined and the one for the global phase portraits of x 1 m mr is completed furthermore for arbitrary k1 some bifurcation and finiteness problems of periodic orbits are solved among others the number of periodic orbits of x m k is found to be uniformly bounded independent of mr and the hilbert number for x m k mr that thus is finite is found to be at least one

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

a comparison of deflation and the balancing preconditioner in this paper we compare various preconditioners for the numerical solution of partial differential equations we compare the well known balancing preconditioner used in domain decomposition methods with a so called deflation preconditioner we prove that the effective condition number of the deflated preconditioned system is always i e for all deflation vectors and all restrictions and prolongations below the condition number of the system preconditioned by the balancing preconditioner even more we establish that both preconditioners lead to almost the same spectra the zero eigenvalues of the deflation preconditioned system are replaced by eigenvalues which are one if the balancing preconditioner is used moreover we prove that the a norm of the errors of the iterates built by the deflation preconditioner is always below the a norm of the errors of the iterates built by the balancing preconditioner depending on the implementation of the balancing preconditioner the amount of work of one iteration of the deflation preconditioned system is less than or equal to the amount of work of one iteration of the balancing preconditioned system if the amount of work is equal both preconditioners are sensitive with respect to inexact computations finally we establish that the deflation preconditioner and the balancing preconditioner produce the same iterates if one uses certain starting vectors numerical results for porous media flows emphasize the theoretical results

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

modeling holes and inclusions by level sets in the extended finite element method a methodology to model arbitrary holes and material interfaces inclusions without meshing the internal boundaries is proposed the numerical method couples the level set method s osher j a sethian j comput phys 79 1 1988 12 to the extended finite element method x fem n moes j dolbow t belytschko int j numer methods engrg 46 1 1999 131 in the x fem the finite element approximation is enriched by additional functions through the notion of partition of unity the level set method is used for representing the location of holes and material interfaces and in addition the level set function is used to develop the local enrichment for material interfaces numerical examples in two dimensional linear elastostatics are presented to demonstrate the accuracy and potential of the new technique

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

high order boundary and finite elements for 3d fracture propagation in brittle materials abstract the quasi static propagation of fracture in brittle materials was studied in several recent publications a variational formulation salvadori 2008 salvadori and carini 2011 salvadori and fantoni 2013 led to three dimensional crack tracking strategies salvadori and fantoni 2014 2 3 salvadori and fantoni 2016 one of the complexities of this new class of algorithms is the evaluation of high order terms of the expansion of the crack opening and sliding in this paper new types of finite and boundary elements are formulated that capture the near crack front asymptotical displacement behavior up to the order 3 2 the use of these elements with the quasi static crack propagation algorithms of the above references is demonstrated for a simple crack configuration

extended preconditioners for the feti method applied to constrained problems an efficient and simple approach for handling linear multipoint constraints in a class of substructure based solvers namely the finite element tearing and interconnecting feti method is proposed previously it was argued that multipoint constraints should be handled in feti by adding a second level iteration on a coarse grid such that the feti iterates satisfy the multipoint constraints exactly the procedure presented here does not require an additional coarse grid but instead takes account of the multipoint constraints in the preconditioning step the preconditioning strategy is shown to be mechanically consistent and to incur only a small additional computational cost this strategy is simpler and computationally less expensive than the two level feti procedure its numerical scalability even for highly heterogeneous problems is demonstrated in several test examples

complete bifurcation diagram and global phase portraits of lienard differential equations of degree four abstract li and llibre in j differential equations 252 2012 3142 3162 proved that a lienard system of degree four d x d t y a x b x 2 c x 3 x 4 d y d t x has at most one limit cycle moreover the limit cycle is stable and hyperbolic if it exists based on their works the aim of this paper is to give the complete bifurcation diagram and global phase portraits in the poincare disc of this system further first we analyze the equilibria at both finity and infinity then a necessary and sufficient condition for existence of separatrix loop is founded by the rotation property moreover a necessary and sufficient condition of the existence of limit cycles is also obtained finally we show that the complete bifurcation diagram includes one hopf bifurcation surface and one bifurcation surface of separatrix loop

an iterative substructuring method for coupled fluid solid acoustic problems a fast parallel iterative method is proposed for the solution of linear equations arising from finite element discretization of the time harmonic coupled fluid solid systems in fluid pressure and solid displacement formulation the fluid and the solid domains are decomposed into nonoverlapping subdomains continuity of the solution is enforced by lagrange multipliers the system is augmented by duplicating the degrees of freedom on the wet interface the original degrees of freedom are then eliminated and the resulting system is solved by iterations preconditioned by a coarse space correction in each iteration the method requires the solution of one independent local acoustic problem per subdomain and the solution of a global problem with several degrees of freedom per subdomain computational results show that the method is scalable with the problem size frequency and the number of subdomains the method generalizes the feti h method for the helmholtz equation to coupled fluid elastic scattering the number of iterations is about same as for the feti h method for the related helmholtz problem with neumann boundary condition instead of an elastic scatterer if enough coarse space functions are used convergence behavior is explained from the spectrum of the iteration operator and from numerical near decoupling of the equations in the fluid and in the solid regions

a global analysis of the bogdanov takens system this paper studies the bifurcations that occur in the bogdanov takens system dot x y qquad dot y x x 1 mu 1 y mu 2 xy both in the phase plane bf r 2 and at infinity i e on the poincare sphere bf s 2 for all parameter values mu 1 mu 2 in bf r 2 for all mu 2 ne 0 a unique limit cycle bifurcates from the origin at mu 1 0 and it expands monotonically until it intersects the saddle at 1 0 and forms a homoclinic loop at mu 1 h mu 2 the function h mu 2 describing the homoclinic loop bifurcation curve in bf r 2 is analytic for all mu 2 in bf r h 0 0 h 0 1 7 and h pm infty 0 the only other bifurcation for this system is a saddle saddle connection on bf s 2 which occurs when mu 1 1 mu 2 even for mu 1 mu 1 mu 2 1 mu 1 h mu 2 ne 0 this system is not structurally stable on bf s 2 due to the presence of nonhyperbolic critical points on the equ

partitioned component mode synthesis via a flexibility approach a new flexibility based component mode synthesis method is presented which is derived by approximating the partitioned equations of motion that employ a localized method of lagrange multipliers the use of the localized lagrange multipliers leads to unlike the classical lagrange multipliers a linearly independent set of interface forces without any redundancies at multiply connected interface nodes hence the resulting interface flexibility matrices are uniquely determined and well suited for the present method development an attractive feature of the present method is its substructural mode selection criterion that is independent of loading conditions numerical experiments show that the present modal selection criterion is reliable and that the proposed flexibility based component mode synthesis method performs well relative to the classical craig bampton method in terms of accuracy and of its ability to include dominant substructural modes for simple plates and a relatively complex solid ring

a generalized finite element method for the simulation of three dimensional dynamic crack propagation this paper is aimed at presenting a partition of unity method for the simulation of three dimensional dynamic crack propagation the method is a variation of the partition of unity finite element method and hp cloud method in the context of crack simulation this method allows for modeling of arbitrary dynamic crack propagation without any remeshing of the domain in the proposed method the approximation spaces are constructed using a partition of unity pu and local enrichment functions the pu is provided by a combination of shepard and finite element partitions of unity this combination of pus allows the inclusion of arbitrary crack geometry in a model without any modification of the initial discretization it also avoids the problems associated with the integration of moving least squares or conventional shepard partitions of unity used in several meshless methods the local enrichment functions can be polynomials or customized functions these functions can efficiently approximate the singular fields around crack fronts the crack propagation is modeled by modifying the partition of unity along the crack surface and does not require continuous remeshings or mappings of solutions between consecutive meshes as the crack propagates in contrast with the boundary element method the proposed method can be applied to any class of problems solvable by the classical finite element method in addition the proposed method can be implemented into most finite element data bases several numerical examples demonstrating the main features and computational efficiency of the proposed method for dynamic crack propagation are presented

a simple explicit implicit finite element tearing and interconnecting transient analysis algorithm summary a simple explicit implicit finite element tearing and interconnecting feti algorithm afeti ei algorithm is presented for partitioned transient analysis of linear structural systems the present algorithm employs two decompositions first the total system is partitioned via spatial or domain decomposition to obtain the governing equations of motions for each partitioned domain second for each partitioned subsystem the governing equations are modally decomposed into the rigid body and deformational equations the resulting rigid body equations are integrated by an explicit integrator for its stability is not affected by step size restriction on account of zero frequency contents 0 the modally decomposed partitioned deformation equations of motion are integrated by an unconditionally stable implicit integration algorithm it is shown that the present afeti ei algorithm exhibits unconditional stability and that the resulting interface problem possesses the same solution matrix profile as the basic feti static problems the present simple dynamic algorithm as expected falls short of the performance of the feti dp but offers a similar performance of implicit two level feti d algorithm with a much cheaper coarse solver hence its simplicity may offer relatively easy means for conducting parallel analysis of both static and dynamic problems by employing the same basic scalable feti solver especially for research mode numerical experiments copyright 2011 john wiley sons ltd

a nitsche type formulation and comparison of the most common domain decomposition methods in isogeometric analysis this paper provides a detailed elaboration and assessment of the most common domain decomposition methods for their application in isogeometric analysis the methods comprise a penalty approach lagrange multiplier methods and a nitsche type method for the nitsche method a new stabilized formulation is developed in the context of isogeometric analysis to guarantee coercivity all these methods are investigated on problems of linear elasticity and eigenfrequency analysis in 2d in particular focus is put on non uniform rational b spline patches which join nonconformingly along their common interface thus the application of isogeometric analysis is extended to multi patches which can have an arbitrary parametrization on the adjacent edges moreover it has been shown that the unique properties provided by isogeometric analysis that is high order functions and smoothness across the element boundaries carry over for the analysis of multiple domains copyright 2013 john wiley sons ltd

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

a domain decomposition method with lagrange multipliers and inexact solvers for linear elasticity a new domain decomposition method with lagrange multipliers for elliptic problems is introduced it is based on a reformulation of the well known finite element tearing and interconnecting feti method as a saddle point problem with both primal and dual variables as unknowns the resulting linear system is solved with block structured preconditioners combined with a suitable krylov subspace method this approach allows the use of inexact subdomain solvers for the positive definite subproblems it is shown that the condition number of the preconditioned saddle point problem is bounded independently of the number of subregions and depends only polylogarithmically on the number of degrees of freedom of individual local subproblems numerical results are presented for a plane stress cantilever membrane problem

two scale dirichlet neumann preconditioners for elastic problems with boundary refinements abstract the present work deals with the efficient resolution of elastostatics problems on domains with boundary refinements the proposed approach separates the boundary refinements from the interior of the domain by the mortar method and uses dirichlet neumann preconditioners to solve the corresponding algebraic system we prove that the simplest dirichlet neumann algorithm achieves independence of the condition number of the preconditioned system with respect to the number and the size of the small details nevertheless the situation no longer prevails when the refined boundary is clamped an enhanced preconditioner is then designed by the introduction of a coarse space to mitigate the aforementioned sensitivity some numerical tests are performed to confirm the analysis and the tools are extended by the proposition of a quasi newton method in the case of nonlinear elasticity this paper is an extended version of a work presented at the dd16 conference with proofs and complete numerical results

partitioned formulation of internal and gravity waves interacting with flexible structures abstract this paper presents a partitioned modeling of internal and gravity fluid waves that interact with flexible structures the governing interaction model consists of three completely partitioned entities fluid model structural model and interface model that acts as an internal constraint on the fluid structure interface boundary thus the proposed partitioned multi physics modeling can employ two completely modular fluid and structure software modules plus an interface solver hence amenable to partitioned solution algorithms the interface discretization can exploit the nonmatching interface algorithm previously developed via the method of localized lagrange multipliers also noted is that the present fluid model can make use of widely available finite element software for standard poisson type problems

the mosaic of high performance domain decomposition methods for structural mechanics part ii formulation enhancements multiple right hand sides and implicit dynamics domain decomposition methods ddm constitute a family of high performance solvers for large scale problems of computational mechanics and of structural mechanics in particular this paper compares popular ddm for structural analysis in both theory and computational performance while it investigates the impact of enhanced mapping operators in their convergence behavior multiple right hand side problems are considered while standard and new ddm solution techniques for implicit dynamics are tested the performed benchmark tests lead to useful conclusions regarding the comparative performance of the methods

on a mixed and multiscale domain decomposition method this paper presents a reexamination of a multiscale computational strategy with homogenization in space and time for the resolution of highly heterogeneous structural problems focusing on its suitability for parallel computing spatially this strategy can be viewed as a mixed multilevel domain decomposition method or more accurately as a structure decomposition method regarding time a parallel property is also described we also draw bridges between this and other current approaches

hybrid analytical and extended finite element method hax fem a new enrichment procedure for cracked solids a new numerical method is proposed to estimate directly stress intensity factors at the tip of a crack in an elastic body in the same spirit as for the extended finite element method the approximation of the displacement field is enriched in the vicinity of the crack tip yet the method proposed herein differs by the way the enrichment is introduced instead of using partition of unity concepts a two description formulation is implemented the first one uses standard finite elements while the second one in the vicinity of the crack tip resorts to a few purely analytical expressions these two descriptions are then coupled by partitioning the energy with an overlapping zone the performance of this new enrichment technique is illustrated and compared with existing techniques by means of two examples

non planar 3d crack growth by the extended finite element and level sets part i mechanical model a methodology for solving three dimensional crack problems with geometries that are independent of the mesh is described the method is based on the extended finite element method in which the crack discontinuity is introduced as a heaviside step function via a partition of unity in addition branch functions are introduced for all elements containing the crack front the branch functions include asymptotic near tip fields that improve the accuracy of the method the crack geometry is described by two signed distance functions which in turn can be defined by nodal values consequently no explicit representation of the crack is needed examples for three dimensional elastostatic problems are given and compared to analytic and benchmark solutions the method is readily extendable to inelastic fracture problems copyright 2002 john wiley sons ltd

a tfeti domain decomposition solver for elastoplastic problems we propose an algorithm for the efficient parallel implementation of elastoplastic problems with hardening based on the so called tfeti total finite element tearing and interconnecting domain decomposition method we consider an associated elastoplastic model with the von mises plastic criterion and the linear isotropic hardening law such a model is discretized by the implicit euler method in time and the consequent one time step elastoplastic problem by the finite element method in space the latter results in a system of nonlinear equations with a strongly semismooth and strongly monotone operator the semismooth newton method is applied to solve this nonlinear system corresponding linearized problems arising in the newton iterations are solved in parallel by the above mentioned tfeti domain decomposition method the proposed tfeti based algorithm was implemented in matlab parallel environment and its performance was illustrated on a 3d elastoplastic benchmark numerical results for different time discretizations and mesh levels are presented and discussed and a local quadratic convergence of the semismooth newton method is observed

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

consistent mesh tying methods for topologically distinct discretized surfaces in non linear solid mechanics this work presents a new technique for joining dissimilar meshes in non linear finite element analysis with the novel result that the adjoining surfaces do not need to be spatially coincident for patch tests to be passed the work builds in some sense on results previously presented by dohrmann et al international journal for numerical methods in engineering 1991 40 1205 but considerably generalizes upon the results presented there most importantly it is shown that consistency of the tied finite element formulation may be achieved such that implementation of element formulations is not affected through the use of a three field variational principle applied over the contacting region a key conclusion of the work is that although the approach advocated bears some resemblance to mortar element schemes for contact analysis see for example mcdevitt and laursen international journal for numerical methods in engineering 2000 48 1525 a straightforward application of a mortaring scheme is not sufficient to guarantee patch test passage when dissimilar meshing of curved surfaces leads to local open volumes or overlaps between tied surfaces in their reference configurations the proposed algorithm is demonstrated through a number of numerical examples performed within a matrix free non linear equation solving framework published in 2003 by john wiley sons ltd

the construction of free free flexibility matrices for multilevel structural analysis abstract this article considers a generalization of the classical structural flexibility matrix it expands on previous papers by taking a deeper look at computational considerations at the substructure level direct or indirect computation of flexibilities as influence coefficients has traditionally required pre removal of rigid body modes by imposing appropriate support conditions mimicking experimental arrangements with the method presented here the flexibility of an individual element or substructure is directly obtained as a particular generalized inverse of the free free stiffness matrix this generalized inverse preserves the stiffness spectrum the definition is element independent and only involves access to the stiffness generated by a standard finite element program and the separate construction of an orthonormal rigid body mode basis the free free flexibility has proven useful in special application areas of finite element structural analysis notably massively parallel processing model reduction and damage localization it can be computed by solving sets of linear equations and does not require processing an eigenproblem or performing a singular value decomposition if substructures contain thousands of d o f exploitation of the stiffness sparseness is important for that case this paper presents a computation procedure based on an exact penalty method and a projected rank regularized inverse stiffness with diagonal entries inserted by the sparse factorization process these entries can be physically interpreted as penalty springs this procedure takes advantage of the stiffness sparseness while forming the full free free flexibility or a boundary subset and is backed by an in depth null space analysis for robustness

high fidelity flexibility based component mode synthesis method with interface degrees of freedom reduction the partitioned structural equations of motion via the method of localized lagrange multipliers palm consist of internal and interface variables the model reduction achieved based on palm is called flexibility based component mode synthesis f cms and it thus far has concentrated on reducing the internal degrees of freedom dofs or reducing the internal and interface dofs at once in this work well defined three level reduction formulations of the f cms method are proposed for better precision internal dofs the localized lagrange multipliers and the interface boundary displacement the proposed formulations include three different reduction techniques for the interface boundary displacement for better computational efficiency numerical experiments indicate that the proposed formulations can substantially reduce the overall reduced order models further without compromising the accuracy thus achieving a substantial additional reduction of computational cost compared with the previous works in a

automatic spectral coarse spaces for robust finite element tearing and interconnecting and balanced domain decomposition algorithms we introduce spectral coarse spaces for the balanced domain decomposition and the finite element tearing and interconnecting methods these coarse spaces are specifically designed for the two level methods to be scalable and robust with respect to the coefficients in the equation and the choice of the decomposition we achieve this by solving generalized eigenvalue problems on the interfaces between subdomains to identify the modes that slow down convergence theoretical bounds for the condition numbers of the preconditioned operators which depend only on a chosen threshold and the maximal number of neighbors of a subdomain are presented and proved for the finite element tearing and interconnecting method there are two versions of the two level method one based on the full dirichlet preconditioner and the other on the cheaper lumped preconditioner some numerical tests confirm these results copyright 2013 john wiley sons ltd

partitioned analysis of flexible multibody systems using filtered linear finite element deformational modes summary this work presents a partitioned finite element formulation for flexible multibody systems based on the floating frame ff approach under the assumption of small deformations but arbitrarily large rotations of the bodies in classical ff of reference methods deformational modes are normally computed by modal analysis in this approach free floating modes are eliminated from the linear model using projection techniques and substituted by a complete set of non linear finite rotations in this way all deformational modes are retained and no modal selection is needed the main difference between this work and a classical ff of reference formulation is an algebraic separation of pure deformational modes from rigid body motions the proposed methodology presents the following advantages first the position and orientation of the ff has no restriction and can be freely located in the body with identical results second the formulation uses only the linear finite element matrices of a classical vibration problem hence they can be easily obtained from linear fem packages third no selection of modes is needed all deformational modes are retained through the filtering process and finally thanks to the use of localized lagrangian multipliers llm a partitioned system is obtained that can be solved iteratively and in a distributed manner by available scalable solvers copyright 2014 john wiley sons ltd

a direct analytical method to extract mixed mode components of strain energy release rates from irwin s integral using extended finite element method summary a new analytical approach within the extended finite element framework is proposed to compute mixed mode components of strain energy release rates directly from irwin s integral crack tip enrichment functions in extended fem allow for evaluation of integral quantities in closed form for some crack configurations studied and therefore resulting in a simple and accurate method several benchmark examples on pure and mixed mode problems are studied in particular we analyze the effects of high order enrichments mesh refinement and the integration limits of irwin s integral the results indicate that high order enrichment functions have significant effect on the convergence in particular when the integral limits are finite when the integral limits tend to zero simpler strain energy release rate expressions are obtained and high order terms vanish nonetheless these terms contribute indirectly via coefficients of first order terms the numerical results show that high accuracy can be achieved with high order enrichment terms and mesh refinement however the effect of the integral limits remains an open question with finite integration intervals chosen as h 2 tending to give more accurate results copyright 2013 john wiley sons ltd

the d alembert lagrange principal equations and applications to floating flexible systems this paper addresses the dynamics and quasi statics of floating flexible structures as well as extensions to unconstrained substructures and partitions of coupled mechanical systems the principal solution is defined as the state of self equilibrated forces obtained as the particular solution of the rigid motion and interface equilibrium equations this solution is independent of the stress strain constitutive properties as well as of the compatibility equations for statically determinate systems the principal solution is the final force solution for statically indeterminate systems the correction due to flexibility and compatibility is orthogonal to the principal solution the formulation is done in the context of d alembert s principle which supplies the d alembert lagrange principal equations for floating bodies these are obtained by summation of virtually working forces and moments acting on the floating systems applications of this approach are demonstrated on a set of dynamic and quasi static example problems of increasing generality linkage to variational principles with an interface potential is eventually discussed as providing the theoretical foundation for handling interacting semi discrete subsystems linked by node collocated lagrange multipliers copyright 2008 john wiley sons ltd

on micro to macro connections in domain decomposition multiscale methods abstract micro to macro connection techniques constitute a key ingredient in the formulation of multiscale strategies in this contribution several connection methods are explored for the concurrent multiscale analysis of brittle heterogeneous materials particularly these techniques are investigated in a domain decomposition strong coupling multiscale framework the structural component under analysis is partitioned into a number of non overlapping domains and a fine scale resolution is assigned therein in an adaptive manner exploiting a zoom in technique mesh refinement is employed in the domains where crack coalescence and growth take place the original contribution presented in this manuscript consists in the study of different strong and weak locality constraints that connect coarse and fine resolution domains standard collocation and average compatibility are considered and serve as a basis for the development of two new interscale links the influence of different locality constraints is studied in terms of the mechanical response and the error distribution is compared to a full fine scale analysis

a feti domain decomposition method for edge element approximations in two dimensions with discontinuous coefficients a class of finite element tearing and interconnecting feti methods for the edge element approximation of vector field problems in two dimensions is introduced and analyzed first an abstract framework is presented for the analysis of a class of feti methods where a natural coarse problem associated with the substructures is lacking then a family of feti methods for edge element approximations is proposed it is shown that the condition number of the corresponding method is independent of the number of substructures and grows only polylogarithmically with the number of unknowns associated with individual substructures the estimate is also independent of the jumps of both of the coefficients of the original problem numerical results validating our theoretical bounds are given the method and its analysis can be easily generalized to raviart thomas element approximations in two and three dimensions

generalized methodology for assembly and reduction of component models for dynamic substructuring a classic issue in component mode synthesis methods is the choice for fixed or free boundary conditions at the interface degrees of freedom and the associated vibration modes in the component s reduction basis closely related is the aspect of substructure assembly in this paper a general approach to mixed boundary component mode synthesis problems is proposed by considering component reduction and assembly separately first a general framework for substructure assembly is presented enabling assembly of all types of reduced component models regardless of their interface conditions thereafter a general mixed boundary component mode synthesis method called the mixed craig bampton method is introduced the method is derived by dividing the substructure degrees of freedom into a set of internal degrees of freedom free interface degrees of freedom and fixed interface degrees of freedom using a simple but effective selection scheme based on this selection a reduction basis is computed consisting of vibration modes constraint modes and residual flexibility modes which leads to relatively sparse reduced matrices as such the mixed craig bampton method forms a natural generalization of the classic craig bampton and more recent dual craig bampton methods finally the method is applied to a test problem revealing that the proposed method has comparable or better accuracy and superior versatility with respect to existing methods

a simple algorithm for localized construction of non matching structural interfaces a simple and effective algorithm for the modular construction of non matched interfaces is presented for the partitioned solution of large scale structural problems the formulation is based on a recently developed four field variational principle which introduces a connection frame between the interfaced partitions a key result of the present study is a frame nodal placement criterion that uniquely determines the frame discretization into piecewise linear elements so that the interface patch test condition is satisfied a priori the method is demonstrated with several 2d and 3d example problems