an iterative substructuring algorithm for equilibrium equations the topic of iterative substructuring methods and more generally domain decomposition methods has been extensively studied over the past few years and the topic is well advanced with respect to first and second order elliptic problems however relatively little work has been done on more general constrained least squares problems or equivalent formulations involving equilibrium equations such as those arising for example in realistic structural analysis applications the potential is good for effective use of iterative algorithms on these problems but such methods are still far from being competitive with direct methods in industrial codes the purpose of this paper is to investigate an order reducing preconditioned conjugate gradient method proposed by barlow nichols and plemmons for solving problems of this type the relationships between this method and nullspace methods such as the force method for structures and the dual variable method for fluids are examined convergence properties are discussed in relation to recent optimality results for varga s theory ofp cyclic sor we suggest a mixed approach for solving equilibrium equations consisting of both direct reduction in the substructures and the conjugate gradient iterative algorithm to complete the computations

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

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

bi discontinuous time step integration algorithms part 2 second order equations abstract in this paper bi discontinuous time step integration algorithms for linear second order equations are investigated the second order equations are manipulated directly both the initial and final displacement and velocity are only weakly enforced at the beginning and at the end of the time interval as a result there may be discontinuous jumps for both the displacement and velocity at the two ends of the time interval if the displacement is assumed to be a polynomial of degree n 1 within the time interval there are n 4 unknowns in the formulation it is found that the order of accuracy would be at least n 1 for displacement and n for velocity in general it is shown that the relevant algorithmic parameters for the second order equations can be related to the algorithmic parameters presented in part 1 of this paper for the first order equations unconditionally stable higher order accurate time step integration algorithms with controllable numerical dissipation equivalent to the generalized pade approximations with an order of accuracy 2 n 3 can then be constructed systematically the weighting parameters and the corresponding weighting functions are given explicitly it is also shown that the accuracy of the particular solutions is compatible with the accuracy of the homogeneous solutions if the proposed weighting functions are employed

variants of the two machine flow shop problem connected with factorization of matrix functions in this paper we consider a number of variants of the two machine flow shop problem in these variants the makespan is given and the problem is to find a schedule that meets this makespan thereby minimizing the infeasibilities of the jobs in a prescribed sense in the max variant the maximum infeasibility of the jobs is to be minimized whereas in the sum variant the objective is to minimize the sum of the infeasibilities of the jobs for both variants observations about the structure of the optimal schedules are presented in particular it is proved that every instance of these problems has an optimal permutation schedule it is also shown that the max variant can be solved by johnson s rule for the sum variant this is not the case for solving this problem to optimality something quite different is necessary both variants are connected with factorization problems for certain rational matrix functions the factorizations involved are optimal in some sense and generalize the notion of complete factorization in this way a connection is established between job scheduling theory on one hand and mathematical systems theory on the other

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

numerically generated tangent stiffness matrices for nonlinear structural analysis accurate tangent stiffness matrices are essential for the solution of nonlinear structural problems via newton like methods the present paper presents an accurate numerical procedure for the generation of tangent stiffness matrices from the internal force generation modules the present procedure is attractive for problems that require complex constitutive relations and or kinematic nonlinearities whose models are difficult to linearize and for incorporating newton like solution strategies into vectorial based nonlinear structural analysis codes

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

minimum norm solutions to linear elastic analysis problems en analyse d elasticite lineaire un probleme consiste a trouver les vecteurs de contraintes et de deformations etant donnes un modele d elements finis d une structure et un systeme de charges externes on montre que ce probleme est un cas particulier du probleme de norme minimum pour des systemes indetermines d equations lineaires

on computational procedures for the force method it is known that the matrix force method has certain advantages over the displacement method for a class of structural problems it is also known that the force method when carried out by the conventional gauss jordan procedure tends to fill in the problem data making the method unattractive for large size sparse problems this poor fill in property however is not necessarily inherent to the method and the sparsity may be maintained if one uses what we call the turn back lu procedure the purpose of this paper is two fold first it is shown that there exist some close relationships between the force method and the least squares problem and that many existing algebraic procedures to perform the force method can be regarded as applications extensions of certain well known matrix factorization schemes for the least squares problem secondly it is demonstrated that these algebraic procedures for the force method can be unified form the matrix factorization viewpoint included in this unification is the turn back lu procedure which was originally proposed by topcu in his thesis 8 it is explained why this procedure tends to produce sparse and banded self stress and flexibility matrices with small band width some computational results are presented to demonstrate the superiority of the turn back lu procedure over the other schemes considered in this paper

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

a unified method for constructing the dynamic boundary stiffness and boundary flexibility for rod beam and circular membrane structures in this paper the dynamic boundary stiffness and boundary flexibility for rod beam and circular membrane structures are analytically derived using the dual integral formulation two approaches the real and the imaginary part kernels are employed to determine the dynamic boundary stiffness and boundary flexibility the continuous system for a circular membrane can be transformed into a discrete system with a circulant matrix based on the properties of circulant the analytical solution for dynamic boundary stiffness and dynamic boundary flexibility in the discrete system can be derived the exact formulae for the dynamic boundary stiffness and boundary flexibility matrices of a rod a beam and a circular membrane are obtained also the calculation for the static flexibility using the pseudo inverse technique is discussed

factorization and job scheduling a connection via companion based matrix functions a connection is made between two sets of problems the first set involves factorization problems of specific rational matrix functions the companion based matrix functions the second set is concerned with variants of the two machine flow shop problem 2mfsp from job scheduling theory in particular it is shown that with each companion based matrix function one can associate an instance of 2mfsp and vice versa the latter can be done in such a way that the factorization properties of the companion based matrix function correspond to the combinatorial properties of the instance of 2mfsp

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

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

shape functions for mixed p version finite elements in the time domain abstract the advantages of mixed formulations are well known in that they simplify the form of the governing equations and can lead to potentially significant savings in computational effort for given accuracy likewise the p version of the finite element method is known to offer savings in computational effort for some problems in this paper shape functions for a mixed p version finite element formulation in the time domain are developed these shape functions produce a high percentage of zeros in certain coefficient matrices the mixed formulation in time used here is based on a weak form of hamilton s extended principle the present formulation allows for accurate and efficient solution of time domain problems the accuracy and power of this formulation are shown for initial value problems such as time marching and for directly obtaining the periodic response of linear systems the method is shown to yield a family of accurate unconditionally stable algorithms for initial value problems based on a simple linear oscillator

numerical analysis of dissolution processes in cementitious materials using discontinuous and continuous galerkin time integration schemes the present paper is concerned with the numerical integration of non linear reaction diffusion problems by means of discontinuous and continuous galerkin methods the first order semidiscrete initial value problem of calcium leaching of cementitious materials based on a phenomenological dissolution model an electrolyte diffusion model and the spatial p finite element discretization is used as a highly non linear model problem a p finite element method is used for the spatial discretization in the context of discontinuous galerkin methods the semidiscrete mass balance and the continuity of the primary variables are weakly formulated within time steps and between time steps respectively continuous galerkin methods are obtained by the strong enforcement of the continuity condition as special cases the introduction of a natural time co ordinate allows for the application of standard higher order temporal shape functions of the p lagrange type and the well known gauss legendre quadrature of associated time integrals it is shown that arbitrary order accurate integration schemes can be developed within the framework of the proposed temporal p galerkin methods selected benchmark analyses of calcium dissolution demonstrate the robustness of these methods with respect to pronounced changes of the reaction term and non smooth changes of dirichlet boundary conditions copyright 2006 john wiley sons ltd

computing the null space of finite element problems we present a method for computing the null space of finite element models including models with equality constraints the method is purely algebraic it requires access to the element matrices but not to the geometry or material properties of the model theoretical considerations show that under certain conditions both the amount of computation and the amount of memory required by our method scale linearly with model size memory scales linearly but computation scales quadratically with the dimension of the null space our experiments confirm this the method scales extremely well on 3 dimensional model problems in general large industrial models do not satisfy all the conditions that the theoretical results assume however experimentally the method performs well and outperforms an established method on industrial models including models with many equality constraints the accuracy of the computed null vectors is acceptable but the method is usually less accurate than a more naive and computationally much more expensive method

an algebraic multigrid method with interpolation reproducing rigid body modes for semi definite problems in two dimensional linear elasticity based on the geometric grid information as geometric coordinates an algebraic multigrid amg method with the interpolation reproducing the rigid body modes namely the kernel elements of semi definite operator arising from linear elasticity is constructed and such method is applied to the linear elasticity problems with a traction free boundary condition and crystal problems with free boundary conditions as well the results of various numerical experiments in two dimensions are presented it is shown from the numerical results that the constructed amg method is robust and efficient for such semi definite problems and the convergence is uniformly bounded away from one independent of the problem size furthermore the amg method proposed in this paper has better convergence rate than the commonly used amg methods simultaneously an amg method that can preserve the quotient space which means that if the exact solution of original problem belongs to the quotient space of discrete operator considered then the numerical solution of amg method is convergent in the same quotient space is obtained using the technique of orthogonal decomposition

weighting parameters for unconditionally stable higher order accurate time step integration algorithms part 2 second order equations in this paper unconditionally stable higher order accurate time step integration algorithms suitable for linear second order differential equations based on the weighted residual method are presented the second order equations are manipulated directly as in part 1 of this paper instead of specifying the weighting functions the weighting parameters are used to control the algorithm characteristics the algorithms are at least nth order accurate if the numerical solution for displacement is approximated by a polynomial of degree n 1 with n undetermined coefficients by choosing the weighting parameters carefully the order of accuracy can be improved the generalized pade approximations for the second order equations are considered the ultimate spectral radius is an algorithmic parameter by relating the approximate solutions to the equivalent formulations presented in part 1 of this paper the required weighting parameters are found explicitly any set of linearly independent functions can be used to construct the corresponding weighting functions from the weighting parameters the stabilizing weighting functions for the weighted residual method are found explicitly to ensure higher order accuracy in the general solution the accuracy of the particular solution due to excitation is also examined copyright 1999 john wiley sons ltd

an integrated geometric algebraic method for solving semi definite problems in structural mechanics a method for the general solution of structural semi definite problems in the presence of zero energy modes is described in this paper semi definite problems are usually associated in structural mechanics with floating structures namely totally unconstrained or partially constrained structures a general solution of these problems is obtained by the computation of a particular displacement field which ensures the equilibrium of the structure and of the zero energy modes of the structure the proposed method combines geometric and algebraic concepts and goes beyond the restrictions of existing methods in this field in particular it is robust cost effective and accounts for all rigid body and mechanism modes in either floating structures or semi definite subdomain problems encountered in domain decomposition methods furthermore it can be combined with any open or closed serial or parallel solver for symmetric positive definite spd problems at a very low cost

time modes and linear systems abstract a vector space approach for generating the response of linear multi degree of freedom time invariant and time varying dynamic systems using hamilton s law of varying action hlva is presented the boundary initial condition constraints on the temporal basis function expansions of the time dependent variables has been removed while preserving hlva in its original form this provides for the broadest choice of basis functions as a result of this new approach it has been demonstrated that the response of dynamic systems are composed of temporal modes herein denoted as fundamental time modes ftm using these fundamental time modes the general solution for the system response is obtained without reference to initial conditions or forcing functions the unique response of the system is subsequently generated by using the initial conditions and forcing functions to scale the ftm this new approach is demonstrated to provide the exact analytical response as well as provide accurate numerical response solutions to a forced damped spring mass system using admissible temporal basis functions tbf the numerical response solutions using gaussian radial basis functions are compared to those obtained by using power series and third order hermite polynomials the new methodology which will be referred to as the universal method in conjunction with gaussian tbf has permitted use of transition intervals ordinarily referred to as time steps of unprecedented length larger than the period of the motion while still maintaining an accurate response solution

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

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

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

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

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

weighting parameters for time step integration algorithms with predetermined coefficients in this paper the effect of using the predetermined coefficients in constructing time step integration algorithms is investigated both first and second order equations are considered the approximate solution is assumed to be in a form of polynomial in the time domain it can be related to the truncated taylor s series expansion of the exact solution therefore some of the coefficients can be predetermined from the known initial conditions if there are m predetermined coefficients and r unknown coefficients in the approximate solution the unknowns can be solved by the weighted residual method the weighting parameter method is used to investigate the resultant algorithm characteristics it is shown that the formulation is consistent with a minimum order of accuracy m r the maximum order of accuracy achievable is m 2r unconditionally stable algorithms exist if m r for first order equations and m 1 r for second order equations hence the dahlquist s theorem is generalized algorithms equivalent to the pade approximations and unconditionally stable algorithms equivalent to the generalized pade approximations are constructed the corresponding weighting parameters and weighting functions for the pade and generalized pade approximations are given explicitly copyright 2000 john wiley sons ltd

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

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

an algorithm to compute a sparse basis of the null space leta be a realm n matrix with full row rankm in many algorithms in engineering and science such as the force method in structural analysis the dual variable method for the navier stokes equations or more generally null space methods in quadratic programming it is necessary to compute a basis matrixb for the null space ofa hereb isn r r n m of rankr withab 0 in many instancesa is large and sparse and often banded the purpose of this paper is to describe and test a variation of a method originally suggested by topcu and called the turnback algorithm for computing a banded basis matrixb two implementations of the algorithm are given one using gaussian elimination and the other using orthogonal factorization by givens rotations the fortran software was executed on an ibm 3081 computer with an fps 164 attached array processor at the triangle universities computing center and on a cyber 205 vector computer test results on a variety of structural analysis problems including two and three dimensional frames plane stress plate bending and mixed finite element problems are discussed these results indicate that both implementations of the algorithm yielded a well conditioned banded basis matrixb whena is well conditioned however the orthogonal implementation yielded a better conditionedb for large illconditioned problems

quasicomplete factorization and the two machine flow shop problem a connection is made between the two machine flow shop problem 2mfsp from job scheduling theory and the issue of quasicomplete factorization of rational matrix functions a quasicomplete factorization is a factorization into elementary i e degree one factors such that the number of factors is minimal for a companion based matrix function w the number of factors in a quasicomplete factorization of w is related in a simple way to the minimum makespan of an instance j of 2mfsp which can be associated to w as a consequence of this result variants of the 2mfsp and other types of factorization can be related too

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

multi time step explicit implicit method for non linear structural dynamics we present a method with domain decomposition to solve time dependent non linear problems this method enables arbitrary numeric schemes of the newmark family to be coupled with different time steps in each subdomain this coupling is achieved by prescribing continuity of velocities at the interface we are more specifically interested in the coupling of implicit explicit numeric schemes taking into account material and geometric non linearities the interfaces are modelled using a dual schur formulation where the lagrange multipliers represent the interfacial forces unlike the continuous formulation the discretized formulation of the dynamic problem is unable to verify simultaneously the continuity of displacements velocities and accelerations at the interfaces we show that within the framework of the newmark family of numeric schemes continuity of velocities at the interfaces enables the definition of an algorithm which is stable for all cases envisaged to prove this stability we use an energy method i e a global method over the whole time interval in order to verify the algorithms properties then we propose to extend this to non linear situations in the following cases implicit linear explicit non linear explicit non linear explicit non linear and implicit non linear explicit non linear finally we present some examples showing the feasibility of the method copyright 2001 john wiley sons ltd

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

companion based matrix functions description and minimal factorization companion based matrix functions are rational matrix functions admitting a minimal realization involving state space matrices that are first companions necessary and sufficient conditions are given for a rational matrix function to be companion based minimal factorization of such functions is discussed in detail it is shown that the property of being companion based is hereditary with respect to minimal factorization also the issue of minimal factorization is reduced to a division problem for pairs of monic polynomials of the same degree in this context a connection with the euclidean algorithm is made the results apply to canonical wiener hopf factorization as well as to complete factorization the analysis of the latter leads to a combinatorial problem involving the eigenvalues of the state space matrices the algorithmic aspects of this problem are intimately related to the two machine flow shop problem and johnson s rule from job scheduling theory

helicopter trim analysis by shooting and finite element methods with optimally damped newton iterations helicopter trim settings of periodic initial state and control inputs are investigated for convergence of newton iteration in computing the settings sequentially and in parallel the trim analysis uses a shooting method and a weak version of two temporal finite element methods with displacement formulation and with mixed formulation of displacements and momenta these three methods broadly represent two main approaches of trim analysis adaptation of initial value and finite element boundary value codes to periodic boundary conditions particularly for unstable and marginally stable systems in each method both the sequential and in parallel schemes are used and the resulting nonlinear algebraic equations are solved by damped newton iteration with an optimally selected damping parameter the impact of damped newton iteration including earlier observed divergence problems in trim analysis is demonstrated by the maximum condition number of the jacobian matrices of the iterative scheme and by virtual elimination of divergence the advantages of the in parallel scheme over the conventional sequential scheme are also demonstrated

helicopter rotor dynamics by finite element time approximation abstract starting from a variational formulation based on hamilton s principle the paper exploits the finite element technique in the time domain in order to generate by an automated procedure a numerical approximation of the nonlinear periodic equations peculiar to the dynamics of helicopter rotors with this method a unified approach is developed which can be used to solve both the response and stability problem two simple examples are presented that show how the method can be used to study the trimmed response of a fully articulated rigid blade and the mechanical instability analysis of a four bladed rotor

free vibration analysis of rotating beams by a variable order finite element method the free vibration of rotating beams is analyzed by means of a finite element method of variable order this method entails displacement functions that are a complete power series of a variable number of terms the terms are arranged so that the generalized coordinates are composed of displacements and slopes at the element extremities and additionally displacements at certain points within the element the displacement is assumed to be analytic within an element and thus can be approximated to any degree of accuracy desired by a complete power series numerical results are presented for uniform beams with zero and nonzero hub radii tapered beams and a nonuniform beam with discontinuities since the present method reduces to a conventional beam finite element method for a cubic displacement function the results are compared and found to be superior to the conventional results in terms of accuracy for a given number of degrees of freedom indeed essentially exact eigenvalues and eigenvectors are obtained with this technique which is far more rapidly convergent than other approaches in the literature

the null space and non conventional basis functions in the mixed finite element method the two matrix equations obtained from the mixed finite element method are uncoupled through the use of the qr decomposition the procedure obtains the basis for the null space which is composed of solutions to the homogeneous flux equilibrium equation the result is that even for statically indeterminate problems stresses can be obtained without solving for the displacement field furthermore the stress field can be decomposed directly into contributions from applied forces and from prescribed displacements the appearance of kinetic modes can easily be monitored the possibility of utilizing non conventional basis functions for use in conjunction with the qr decomposition is explored briefly simple examples are given to illustrate the theoretical results

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 variational energy formulation for the integrated force method the integrated force method ifm is one of the five formulations of mechanics the others being the flexibility stiffness mixed and total methods to date all but the ifm have been associated with variational functionals a variational functional vf has been developed for the ifm the stationary condition of the vf for the ifm yields the equilibrium and compatibility equations as well as the force and displacement boundary conditions the stationary condition also yields a new set of boundary equations identified as the boundary compatibility conditions this paper presents the theory of the variational functional for the ifm it is illustrated by examples from discrete structures the plane stress problem of elasticity and kirchhoff s plate bending problem the properties of the vf and its relationship to the potential and complementary energy functionals are shown for discrete analysis

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

unconditionally stable time step integration algorithms based on hamilton s principle unconditionally stable time step integration algorithms derived from hamilton s principle are presented the corresponding constrained variations are assumed to be in the form of k k 1 to ensure vanishing variations at the ends of a time interval the initial conditions are strongly enforced in the formulation the order of accuracy can be 2n 1 or 2n where n is the number of unknown variables the ultimate spectral radii of the algorithms can be controlled directly the approximate solutions are in fact equivalent to the generalized pade approximations for linear second order differential equations hence hamilton s principle can be used to construct unconditionally stable higher order accurate time step integration algorithms directly as well provided the appropriate constrained variations are used

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

the p version of the finite element method in the p version of the finite element method the triangulation is fixed and the degree p of the piecewise polynomial approximation is progressively increased until some desired level of precision is reached in this paper we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation these properties lead to an a priori estimate of the asymptotic rate of convergence of the p version the estimate shows that the p version gives results which are not worse than those obtained by the conventional finite element method called the h version in which h represents the maximum diameter of the elements when quasi uniform triangulations are employed and the basis for comparison is the number of degrees of freedom furthermore in the case of a singularity problem we show under conditions which are usually satisfied in practice that the rate of convergence of the p version is twice that of the h version with quasi uniform mesh inve

time domain finite element analysis of dynamic systems in this study dynamic analyses of systems are conducted using the time domain finite element analysis when initial and final conditions are specified dynamics of systems can be analyzed using the time marching procedure based on hamilton s weak principle a new method is developed to describe the motion of a system and is extended to accommodate the initial displacement as a state variable system matrices are constructed to evaluate time histories of the system from initial time to a specified time without the information on the velocity and momentum at each time step galerkin s weak principle is used to construct element matrices in the time domain and assemble the whole matrices the neumann boundary condition is modified to reflect the effect of initial displacement and velocity that are inherent to dynamic problems through the momentum conservation of the system the matrices obtained thereafter conserve the symmetricity and enable us to reduce the model order by the systematic approach spatial propagation equation is built up and two point boundary conditions are used to estimate the unknown initial conditions at one end of the beam modal domain analysis is introduced to reduce the size of matrices it can be seen that the matrices constructed using the time domain finite element analysis are applicable to the model reduction that is analogous to the space domain model reduction methodology several numerical examples show that a consequence of the suggested method can be applied to describe the motion of various dynamic systems successfully and to validate the effect of the model reduction in the time domain

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 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

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

complementary triangular forms of pairs of matrices realizations with prescribed main matrices and complete factorization of rational matrix functions abstract the problem considered is the following given two square matrices a and z when does there exist an invertible matrix s such that s 1 as is upper triangular and s 1 zs is lower triangular several sufficient conditions for such an s to exist are presented in some cases the conditions are also necessary attention is paid to the order in which the eigenvalues of a and z may appear on the diagonals of s 1 as and s 1 zs respectively there is an intimate connection with the notions of realization and minimal factorization from systems theory

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 comprehensive unified set of single step algorithms with controllable dissipation for dynamics part i formulation a new class of time integration algorithms for dynamics is presented the algorithms are based on finite element in time the equations of motion are expressed in first order form and the variables are interpolated via time polynomials the unknown nodal variables are solved by the generalized galerkin method algorithms of any desired order of time accuracy can be achieved by proper choice of the finite element interpolation in time the salient feature of the algorithms is that it can be either asymptotic annihilating or non dissipative besides any desired algorithmic damping between these two extreme cases could be obtained the algorithms are analysed in detail the spectral characteristics equivalent to the upper diagonal diagonal of the pade approximations and other rational approximations are given