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

black box decomposition approach for computational hemodynamics one dimensional models abstract increasing efforts exist in integrating different levels of detail in models of the cardiovascular system for instance one dimensional representations are employed to model the systemic circulation in this context effective and black box type decomposition strategies for one dimensional networks are needed so as to i employ domain decomposition strategies for large systemic models 1d 1d coupling and ii provide the conceptual basis for dimensionally heterogeneous representations 1d 3d coupling among various possibilities the strategy proposed in this article works for both of these two scenarios though the several applications shown to illustrate its performance focus on the 1d 1d coupling case a one dimensional network is decomposed in such a way that each coupling point connects two and not more of the sub networks at each of the m connection points two unknowns are defined the flow rate and pressure these 2 m unknowns are determined by 2 m equations since each sub network provides one non linear equation per coupling point it is shown how to build the 2 m 2 m non linear system with arbitrary and independent choice of boundary conditions for each of the sub networks the idea is then to solve this non linear system until convergence which guarantees strong coupling of the complete network in other words if the non linear solver converges at each time step the solution coincides with what would be obtained by monolithically modeling the whole network the decomposition thus imposes no stability restriction on the choice of the time step size effective iterative strategies for the non linear system that preserve the black box character of the decomposition are then explored several variants of matrix free broyden s and newton gmres algorithms are assessed as numerical solvers by comparing their performance on sub critical wave propagation problems which range from academic test cases to realistic cardiovascular applications a specific variant of broyden s algorithm is identified and recommended on the basis of its computer cost and reliability

singularly perturbed finite element methods this paper introduces a new piecewise linear finite element which is designed to handle singularly perturbed ordinary differential equations both pointwise and global estimates which are independent of the perturbation parameter are obtained

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

matrix analysis of statically and kinematically indeterminate frameworks abstract the paper is concerned with the structural mechanics of assemblies of bars and pinjoints particularly where they are simultaneously statically and kinematically indeterminate the physical significance of the four linear algebraic vector subspaces of the equilibrium matrix is examined and an algorithm is set up which determines the rank of the matrix and the bases for the four subspaces in particular this algorithm gives full details of any states of self stress and modes of inextensional deformation which an assembly may possess a scheme is devised for the segregation of inextensional modes into rigid body modes up to six of these may be allowed by the foundation constraints and internal mechanisms in some circumstances a state of self stress may impart first order stiffness to an inextensional mode a matrix method for detecting this effect is devised and it is shown that if there is no state of self stress which imparts first order stiffness to a given mode then that mode can undergo rather large distortion which involves either zero change in length of the bars or possibly changes in length of third or higher order in the displacements the significance of negative stiffness as indicated by the matrix method is discussed the paper contains simple examples which illustrate all of the main points of the work

iterative strong coupling of dimensionally heterogeneous models in this article we address decomposition strategies especially tailored to perform strong coupling of dimensionally heterogeneous models under the hypothesis that one wants to solve each submodel separately and implement the interaction between subdomains by boundary conditions alone the novel methodology takes full advantage of the small number of interface unknowns in this kind of problems existing algorithms can be viewed as variants of the natural staggered algorithm in which each domain transfers function values to the other and receives fluxes or forces and vice versa this natural algorithm is known as dirichlet to neumann in the domain decomposition literature essentially we propose a framework in which this algorithm is equivalent to applying gauss seidel iterations to a suitably defined linear or nonlinear system of equations it is then immediate to switch to other iterative solvers such as gmres or other krylov based method which we assess through numerical experiments showing the significant gain that can be achieved indeed the benefit is that an extremely flexible automatic coupling strategy can be developed which in addition leads to iterative procedures that are parameter free and rapidly converging further in linear problems they have the finite termination property copyright 2009 john wiley sons ltd

the construction of a null basis for a discrete divergence operator the divergence free finite element method dffem is a method to find an approximate solution of the navier stokes equations in a divergence free space that is the continuity equation is satisfied a priori dffem eliminates the pressure from the calculations and significantly reduces the dimension of the system to be solved at each time step for the standard 9 node velocity and 4 node pressure dffem a basis for the divergence free subspace is constructed such that each basis function has nonzero support on at most 4 contiguous elements given this basis discretely divergence free macro elements can be constructed and used in the implementation of the dffem

algorithms for strong coupling procedures abstract this paper considers algorithms for computing the response of a coupled problem with a partitioned approach one important example treated here is fluid structure interaction fsi often procedures or even software exists to solve each sub problem separately and one wants to couple both this setting seems to allow only the so called weak coupling which is not sufficient for some problems the so called strong coupling is often a totally implicit formulation where the system components are evaluated at the same time level this usually requires an iterative procedure the fsi problem is cast as an abstract differential algebraic equation dae for which the coupling procedures are developed with the partitioned approach one simple and frequently used computational procedure is similar to a block gauss seidel iteration it is shown why this approach may experience difficulties and how they may be circumvented with newton like methods still staying in the partitioned framework the functional and software engineering requirements for the simulation interface are described and analysed some simple coupled example problems demonstrate how the proposed procedures work

the multiscale robin coupled method for flows in porous media abstract a multiscale mixed method aiming at the accurate approximation of velocity and pressure fields in heterogeneous porous media is proposed the procedure is based on a new domain decomposition method in which the local problems are subject to robin boundary conditions the domain decomposition procedure is defined in terms of two independent spaces on the skeleton of the decomposition corresponding to interface pressures and fluxes that can be chosen with great flexibility to accommodate local features of the underlying permeability fields the well posedness of the new domain decomposition procedure is established and its connection with the method of douglas et al 1993 12 is identified also allowing us to reinterpret the known procedure as an optimized schwarz or two lagrange multiplier method the multiscale property of the new domain decomposition method is indicated and its relation with the multiscale mortar mixed finite element method mmmfem and the multiscale hybrid mixed mhm finite element method is discussed numerical simulations are presented aiming at illustrating several features of the new method initially we illustrate the possibility of switching from mmmfem to mhm by suitably varying the robin condition parameter in the new multiscale method then we turn our attention to realistic flows in high contrast channelized porous formations we show that for a range of values of the robin condition parameter our method provides better approximations for pressure and velocity than those computed with either the mmmfem and the mhm this is an indication that our method has the potential to produce more accurate velocity fields in the presence of rough realistic permeability fields of petroleum reservoirs

natural preconditioning and iterative methods for saddle point systems the solution of quadratic or locally quadratic extremum problems subject to linear ized constraints gives rise to linear systems in saddle point form this is true whether in the continuous or the discrete setting so saddle point systems arising from the discretization of partial differential equation problems such as those describing electromagnetic problems or incompressible flow lead to equations with this structure as do for example interior point methods and the sequential quadratic programming approach to nonlinear optimization this survey concerns iterative solution methods for these problems and in particular shows how the problem formulation leads to natural preconditioners which guarantee a fast rate of convergence of the relevant iterative methods these preconditioners are related to the original extremum problem and their effectiveness in terms of rapidity of convergence is established here via a proof of general bounds on the eigenvalues of the preconditioned saddle point matri

order reducing conjugate gradients versus block aor for constrained least squares problems abstract we compare the convergence properties of two iterative algorithms for solving equality constrained least squares problems the first algorithm due to barlow nichols and plemmons applies a variation of the conjugate gradient algorithm to a symmetric positive definite system which is smaller than the original problem the second block accelerated overrelaxation is a two parameter generalization of block sor barlow nichols and plemmons have proven that their order reducing conjugate gradient algorithm converges faster than block sor we extend their result to show that the algorithm is also superior to block aor numerical experiments on some structural engineering problems support the analysis

construction of null bases for the divergence operator associated with incompressible navier stokes equations the dual variable method dvm and the divergence free finite element method dffem both significantly reduce the dimension of the system to be solved at each time step of discrete transient navier stokes equations we establish an equivalence of these two methods and investigate various approaches to finding sparse bases for the null space of operators associated with each of the methods for the standard eight node velocity and four node pressure dffem a basis for the divergence free subspace is constructed such that each basis function has nonzero support on at most nine contiguous elements

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

sensitivity analysis in kinematically incompatible models abstract a natural question that arises when working with the so called kinematically incompatible models is about the positioning of the internal artificial boundary that splits the domain of analysis into the sub domains for which the different incompatible models are established although the experience in modeling might provide an ad hoc solution to such matter it may also draw misleading conclusions here the sensitivity analysis furnishes a systematic and reliable framework to study the impact of the positioning of the artificial boundary over the solution of the problem in this sense the present work is concerned with the sensitivity analysis of the variational formulation corresponding to the kinematically incompatible models when the boundary over which the incompatibility between the models takes place is changed assuming the presence of a discontinuity in the fields over a given interface between two incompatible models this analysis allows to measure the sensitivity of a given cost functional when the location of such interface is changed the application for which this analysis is envisaged is to assess the correctness or incorrectness in the definition of the positioning of such coupling interface between dimensionally heterogeneous domains also some numerical results are provided to give numerical evidence of the usefulness of the present tool

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 new implementation of the force method and the optimum design of large trusses an implementation of the force method is proposed in which the forces and the displacements are simultaneously obtained by the solution of a sparse symmetric indefinite system the matrix of coefficients is formed by just the concatenation of the element flexibility and equilibrium matrices no computational procedure is required to generate the compatibility conditions or the self stress matrix and no partitioning of the force vector is made into a basic set and a redundant set unlike the conventional force method a slightly modified sparse unsymmetric system can be written in which the stresses and the displacements are the unknowns this is used as constraints in the formulation of the minimum weight design problem for large structures under static loading conditions a sparse generalized reduced gradient package is used as the optimizer a class of test problems involving large truss structures is solved the results indicate that the present implementation of the force method is better than the displacement method for the optimum design of large structures

force method revisited computational mechanics is an important field in engineering with a strong foundation in continuum mechanics and finite element analysis the standard and integrated force methods are reviewed and a structural analysis technique using the finite element force method based on the complementary strain energy is proposed similar to the standard force and the integrated force methods the force method based on the complementary strain energy uses the same equilibrium equations however the compatibility conditions are satisfied through the complementary strain energy the gauss elimination technique has been employed successfully to generate automatically a basis determinate structure and redundant members extension of the force method to dynamics and benchmark problems with displacement stress and frequency constraints are presented

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

some current cfd issues relevant to the incompressible navier stokes equations abstract the goals of this paper are to carefully define a particular class of well set incompressible navier stokes problems in the continuum partial differential equation pde setting and to discuss some relevant and sometimes poorly understood issues related to these well posed pde problems both in the continuum world and in its computer counterpart emphasized are three common formulations used to generate computer codes the primitive variable velocity pressure formulation and two derived variable formulations pressure poisson equation and vorticity transport equation

discrete least squares finite element methods abstract a finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators 1 the differential operator defining the spatial boundary value problem and 2 a riesz map on the test space the resulting linear system is overdetermined two different approaches for solving the system are suggested although others are discussed 1 solving the associated normal equation with linear solvers for symmetric positive definite systems e g cholesky factorization and 2 solving the overdetermined system with orthogonalization algorithms e g qr factorization the finite element assembly algorithm for each of these approaches is described in detail the normal equation approach is usually faster for direct solvers and requires less storage the second approach reduces the condition number of the system by a power of two and is less sensitive to round off error the rectangular stiffness matrix of second approach is demonstrated to have condition number o h 1 for a variety of formulations of poisson s equation the stiffness matrix from the normal equation approach is demonstrated to be related to the monolithic stiffness matrices of least squares finite element methods and it is proved that the two are identical in some cases an example with poisson s equation indicates that the solutions of these two different linear systems can be nearly indistinguishable if round off error is not an issue and rapidly converge to each other the orthogonalization approach is suggested to be beneficial for problems which induce poorly conditioned linear systems experiments with poisson s equation in single precision arithmetic as well as the linear acoustics problem near resonance in double precision arithmetic verify this conclusion the methodology described here is an outgrowth of the discontinuous petrov galerkin dpg methodology of demkowicz and gopalakrishnan 2015 the strength of dpg is highlighted throughout however the majority of theory presented is more general extensions to constrained minimization principles are also considered throughout but are not analyzed in experiments

an effective fluid structure interaction formulation for vascular dynamics by generalized robin conditions in this work we focus on the modeling and numerical simulation of the fluid structure interaction mechanism in vascular dynamics we first propose a simple membrane model to describe the deformation of the arterial wall which is derived from the koiter shell equations and is applicable to an arbitrary geometry secondly we consider a reformulation of the fluid structure problem in which the newly derived membrane model thanks to its simplicity is embedded into the fluid equations and will appear as a generalized robin boundary condition the original problem is then reduced to the solution of subsequent fluid equations defined on a moving domain and may be achieved with a fluid solver only we also derive a stability estimate for the resulting numerical scheme finally we propose new outflow absorbing boundary conditions which are easy to implement and allow us to reduce significantly the spurious pressure wave reflections that typically appear in artificially truncated computational domains we present several numerical results showing the effectiveness of the proposed approaches

a new route to finding bounds on the generalized spectrum of many physical operators here we obtain bounds on the generalized spectrum of that operator whose inverse when it exists gives green s function we consider the wide range of physical problems that can be cast in a form where a constitutive equation j x l x e x h x with a source term h x holds for all x in some domain and relates fields e and j that satisfy appropriate differential constraints symbolized by e e 0 and j j where e 0 and j are orthogonal spaces that span the space h of square integrable fields in which h lies boundedness and coercivity conditions on the moduli l x ensure that there exists a unique e for any given h i e e g h which then establishes the existence of green s function g we show that the coercivity condition is guaranteed to hold if weaker conditions involving generalized quasiconvex functions are satisfied the advantage is that these weaker conditions are easier to verify and for multiphase materials they can be independent of the geometry of the phases for l x depending linearly on a vector of parameters z z1 z2 zn we obtain constraints on z that ensure that green s function exists and hence which provide bounds on the generalized spectrum here we obtain bounds on the generalized spectrum of that operator whose inverse when it exists gives green s function we consider the wide range of physical problems that can be cast in a form where a constitutive equation j x l x e x h x with a source term h x holds for all x in some domain and relates fields e and j that satisfy appropriate differential constraints symbolized by e e 0 and j j where e 0 and j are orthogonal spaces that span the space h of square integrable fields in which h lies boundedness and coercivity conditions on the moduli l x ensure that there exists a unique e for any given h i e e g h which then establishes the existence of green s function g we show that the coercivity condition is guaranteed to hold if weaker conditions involving generalized quasiconvex functions are satisfied the advantage is that these weaker conditions are easier to verify and for multiphase materials they can be independent of the geometry of the phases for l x depend

structural optimization with frequency constraints using the finite element force method a structural optimization algorithm is developed to minimize the weight of structures with truss and beam type members under single or multiple frequency constraints the cross sectional areas of the structural members are considered as the design variables the algorithm proposed combines the finite element technique based on the integrated force method with the mathematical programming technique the equilibrium matrix is generated automatically using the finite element analysis and the compatibility matrix is obtained directly using the displacement deformation relations and the single value decomposition technique when combining the equilibrium and the compatibility matrices with the force displacement relations the frequency eigenvalue equations are obtained with element forces as variables three structures composed of truss and frame type members are studied to illustrate the procedure and the results are compared with the literature it is shown that in structural problems with multiple frequency constraints the analysis procedure force or displacement method significantly affects the final optimum design the structural optimization based on the force method results in a lighter design the proposed structural optimization method is efficient to analyze and optimize both truss and beam type structures

perturbation bounds for constrained and weighted least squares problems we derive perturbation bounds for the constrained and weighted linear least squares ls problems both the full rank and rank deficient cases are considered the analysis generalizes some results of earlier works

a two dimensional finite element algorithm for the simultaneous solution of the semiconductor device equations with automatic convergence the proposed algorithm solves equations governing the behaviour of semiconductor devices using a finite element technique electrostatic potential and the hole and electron quasi fermi potentials are chosen as the solution variables the equation set is written in a steady state form using these three variables and this gives rise to a system of three nonlinear partial differential equations the equations which are intimately coupled are solved simultaneously using a weighted residual formulation convergence of the nonlinear solution procedure using any initial guess is guaranteed by employing incremental loading coupled to a test for divergence that is applied at each iterative step the triangular elements used in the program are automatically generated from a mesh of eight node isoparametric elements that is itself an automatically generated subdivision of a small number of eight node super elements a novel method of generating an initialisation state using the boundary element method is also described

dynamic boundary conditions in computational fluid dynamics abstract the number and type of boundary conditions to be used in the numerical modeling of fluid mechanics problems is normally chosen according to a simplified analysis of the characteristics and also from the experience of the modeler the problem is harder at inflow outflow boundaries which are in most cases artificial boundaries so that a bad decision about the boundary conditions to be imposed may affect the precision and stability of the whole computation for inviscid flows the analysis of the sense of propagation in the normal direction to the boundaries gives the number of conditions to be imposed and in addition the conditions that are absorbing for the waves impinging normally to the boundary in practice it amounts to counting the number of positive and negative eigenvalues of the advective flux jacobian projected onto the normal the problem is even harder when the number of incoming characteristics varies during the computation and the correct treatment of these cases poses both mathematical and practical problems one example considered here is a compressible flow where the flow regime at a certain part of an inlet outlet boundary can change from subsonic to supersonic and the flow can revert in this work the technique for dynamically imposing the correct number of boundary conditions along the computation using lagrange multipliers and penalization is discussed and several numerical examples are presented

a modular numerical method for implicit 0d 3d coupling in cardiovascular finite element simulations implementation of boundary conditions in cardiovascular simulations poses numerical challenges due to the complex dynamic behavior of the circulatory system the use of elaborate closed loop lumped parameter network lpn models of the heart and the circulatory system as boundary conditions for computational fluid dynamics cfd simulations can provide valuable global dynamic information particularly for patient specific simulations in this paper the necessary formulation for coupling an arbitrary lpn to a finite element navier stokes solver is presented a circuit analogy closed loop lpn is solved numerically and pressure and flow information is iteratively passed between the 0d and 3d domains at interface boundaries resulting in a time implicit scheme for neumann boundaries an implicit method regardless of the lpn is presented to achieve the desired stability and convergence properties numerical procedures for passing flow and pressure information between the 0d and 3d domains are described and implicit semi implicit and explicit quasi newton formulations are compared the issue of divergence in the presence of backflow is addressed via a stabilized boundary formulation the requirements for coupling dirichlet boundary conditions are also discussed and this approach is compared in detail to that of the neumann coupled boundaries having the option to select between dirichlet and neumann coupled boundary conditions increases the flexibility of current framework by allowing a wide range of components to be used at the 3d 0d interface

analysis of prestressed mechanisms abstract a new theory is presented for the matrix analysis of prestressed structural mechanisms made from pin jointed bars the response of a prestressed mechanism to any external action is decomposed into two almost separate parts which correspond to extensional and inextensional modes a matrix algorithm which treats these two modes separately is developed and tested it is shown that the equilibrium requirements for the assembly in its initial configuration as well as in deformed configurations which are obtained through infinitesimal inextensional displacements can be fully described by a square equilibrium matrix it is also shown that any set of extensional nodal displacements has to satisfy some equilibrium conditions as well as standard compatibility equations and that the resulting system of linear equations defines a square kinematic matrix theoretical as well as experimental evidence supporting this approach is given in the paper two simple experiments which were of crucial importance in arriving at the equilibrium conditions on the extensional displacements are described the interaction between the two modes of action of a prestressed mechanism is discussed together with a rapidly converging iterative procedure to handle it a study of the non linear effect by which the self stress level in a statically indeterminate assembly rapidly increases if an inextensional mode is excited supported by further experimental results concludes the paper this work is relevant to the analysis of most cable systems pneumatic domes fabric roofs and tensegrity frameworks

the derivation of minimal support basis functions for the discrete divergence operator the divergence free finite element method dffem is a method to find an approximate solution of the navier stokes equations in a divergence free space that is the continuity equation is satisfied a priori dffem eliminates the pressure from the calculations and significantly reduces the dimension of the system to be solved at each time step for a low order and nonconforming element we construct a basis for the divergence free subspace such that basis functions have smallest nonzero support

a framework for equilibrium equations the equations of equilibrium in a wide class of applications share a common structure the discrete case leads to a positive definite symmetric matrix a t ca in which a is rectangular and c is often diagonal the continuous case linear or nonlinear repeats that pattern there is a minimum principle for the potentials and a dual principle for the flow variables we try to clarify these different expressions of the same law

on solving block structured indefinite linear systems we consider 2 2 block indefinite linear systems whose 2 2 block is zero such systems arise in many applications we discuss two techniques that are based on modifying the 1 1 block in a way that makes the system easier to solve the main part of the paper focuses on an augmented lagrangian approach a technique that modifies the 1 1 block without changing the system size the choice of the parameter involved the spectrum of the linear system and its condition number are discussed and some analytical observations are provided a technique of deflating the 1 1 block is then introduced finally numerical experiments that validate the analysis are presented

coupling vs decoupling approaches for pde ode systems modeling intercellular signaling we consider pde ode systems for the simulation of intercellular signaling in multicellular environments the intracellular processes for each cell described here by odes determine the long time dynamics but the pde part dominates the solving effort thus it is not clear if commonly used decoupling methods can outperform a coupling approach based on a sensitivity analysis we present a systematic comparison between coupling and decoupling approaches for this class of problems and show numerical results for biologically relevant configurations of the model our quantitative study shows that a coupling approach performs much better than a decoupling one

modeling dimensionally heterogeneous problems analysis approximation and applications in the present work a general theoretical framework for coupled dimensionally heterogeneous partial differential equations is developed this is done by recasting the variational formulation in terms of coupling interface variables in such a general setting we analyze existence and uniqueness of solutions for both the continuous problem and its finite dimensional approximation this approach also allows the development of different iterative substructuring solution methodologies involving dimensionally homogeneous subproblems numerical experiments are carried out to test our theoretical results

topology optimization of trusses using genetic algorithm force method and graph theory in this article size topology optimization of trusses is performed using a genetic algorithm ga the force method and some concepts of graph theory one of the main difficulties with optimization with a ga is that the parameters involved are not completely known and the number of operations needed is often quite high application of some concepts of the force method together with theory of graphs make the generation of a suitable initial population well matched with critical paths for the transformation of internal forces feasible in the process of optimization generated topologically unstable trusses are identified without any matrix manipulation and highly penalized identifying a suitable range for the cross section of each member for the ground structure in the list of profiles the length of the substrings representing the cross sectional design variables are reduced using a contraction algorithm the length of the strings is further reduced and a ga is performed in a smaller domain of design space the above process is accompanied by efficient methods for selection and by using a suitable penalty function in order to reduce the number of numerical operations and to increase the speed of the optimization toward a global optimum the efficiency of the present method is illustrated using some examples and compared to those of previous studies copyright 2003 john wiley sons ltd

an orthogonal self stress matrix for efficient analysis of cyclically symmetric space truss structures via force method abstract the analysis of structures is normally carried out through displacement method while the force method is considered as an alternative approach for this purpose and used on occasion the generation of compatibility conditions the transpose of self stress matrix is one of the major and complicated parts of any structural analysis using force method in this paper an efficient method is proposed for producing orthogonal self stress matrix related to space truss structures with cyclic symmetry this is actually performed by eigen decomposition of a special matrix having the same null basis as in equilibrium matrix then the advantages of the obtained compatibility conditions are demonstrated with respect to different formulations such as standard force method eigen force method and integrated force method finally the efficiency of the presented method is comprehensively compared with three well known numerical methods and tested on a set of practical examples the results indicate clearly the significant superiority of the proposed approach in terms of both computational time and the accuracy of the results

fluid structure algorithms based on steklov poincare operators in this paper we review some classical algorithms for fluid structure interaction problems and we propose an alternative viewpoint mutuated from the domain decomposition theory this approach yields preconditioned richardson iterations on the steklov poincare nonlinear equation at the fluid structure interface

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

a new approach to numerical solution of defective boundary value problems in incompressible fluid dynamics we consider the incompressible navier stokes equations where on a part of the boundary flow rate and mean pressure boundary conditions are prescribed there are basically two strategies for solving these defective boundary problems the variational approach see j heywood r rannacher and s turek internat j numer methods fluids 22 1996 pp 325 352 and the augmented formulation see l formaggia j f gerbeau f nobile and a quarteroni siam j numer anal 40 2002 pp 376 401 and a veneziani and c vergara internat j numer methods fluids 47 2005 pp 803 816 however both of these approaches present some drawbacks for the flow rate problem the former resorts to nonstandard functional spaces which are quite difficult to discretize on the other hand for the mean pressure problem it yields exact solutions only in very special cases the latter is applicable only to the flow rate problem since for the mean pressure problem it provides unfeasible boundary conditions in this paper we propose a new strategy based on a reformulation of the problems at hand in terms of the minimization of an appropriate functional this approach allows us to treat the two kinds of problems flow rate and mean pressure successfully within the same framework which can be useful in view of a mixed problem where the two conditions are simultaneously prescribed on different artificial boundaries moreover it is more versatile being prone to be extended to other kinds of defective conditions we analyze the problems obtained with this approach and propose some algorithms for their numerical solution several numerical results are presented supporting the effectiveness of our approach

a unified variational approach for coupling 3d 1d models and its blood flow applications in this work we propose an extended variational formulation in order to handle the problem of the flow of an incompressible fluid in compliant vessels with discontinuous fields this approach makes possible to tackle in a consistent variational manner problems like the coupling between models of different dimensionality the main ideas of the work are presented for a simple heat transfer problem and then are used in the context of modelling blood flow in large vessels a computational implementation of the proposed formulation is also shown and discussed by presenting examples of practical applications

convergence of a direct iterative method for large scale least squares problems abstract in 1975 chen and gentleman suggested a 3 block sor method for solving least squares problems based on a partitioning scheme for the observation matrix a into a a 1 a 2 where a1 is square and nonsingular in many cases a1 is obvious from the nature of the problem this combined direct iterative method was discussed further and applied to angle adjustment problems in geodesy where a1 is easily formed and is large and sparse by plemmons in 1979 recently niethammer de pillis and varga have rekindled interest in this method by correcting and extending the sor convergence interval the purpose of our paper is to discuss an alternative formulation of the problem leading to a 2 block sor method for this formulation it is shown that the resulting direct iterative method always converges for sufficiently small sor parameter in contrast to the 3 block formulation formulas for the optimum sor parameter and the resulting asymptotic convergence factor based upon a2a 11 2 are given furthermore it is shown that this 2 cyclic block sor method always gives better convergence results than the 3 cyclic one for the same amount of work per iteration the direct part of the algorithm requires only a sparse matrix factorization of a1 our purpose here is to establish theoretical convergence results in line with the purpose of the recent paper by niethammer de pillis and varga practical considerations of choosing a1 in certain applications and of estimating the resulting a2a 11 2 will be addressed elsewhere

numerical treatment of defective boundary conditions for the navier stokes equations we present a formulation for accommodating defective boundary conditions for the incompressible navier stokes equations where only averaged values are prescribed on measurable portions of the boundary in particular we consider the case where the flow rate is imposed on several domain sections this methodology has an interesting application in the numerical simulation of flow in blood vessels when only a reduced set of boundary data are generally available for the upstream and downstream sections

efficient finite element analysis using graph theoretical force method rectangular plane stress and plane strain lagrange family elements formation of a suitable null basis for equilibrium matrix is the main part of finite elements analysis via force method for an optimal analysis the selected null basis matrices should be sparse and banded corresponding to sparse banded and well conditioned flexibility matrices in this paper an efficient method is developed for the formation of null bases of finite element models fems consisting of rectangular plane stress and plane strain lagrange family elements corresponding to highly sparse and banded flexibility matrices this is achieved by associating special graphs with the fem and selecting appropriate subgraphs and forming the self equilibrating systems sess on these subgraphs the efficiency of the present method is illustrated through three examples

multidimensional modelling for the carotid artery blood flow abstract in this work a multidimensional 3d 1d fem model of the whole arterial tree is implemented it comprises a 3d compliant model of the carotid bifurcation coupled with a 1d model for the remaining part of the arterial tree with this approach difficulties arising from the treatment of boundary conditions for the 3d model are naturally handled the navier stokes equations are used as the governing equations for the blood flow while an elastic compliant model is implemented for the arterial wall also the a l e formulation is considered within the 3d blood regions taking into account the domain deformations produced by the moving artery wall this 3d model is complemented with a 1d model of the entire arterial tree in order to appropriately set inflow and outflow boundary conditions for the former the reduced 1d model solves the momentum and continuity equations in compliant tubes so as to reproduce the propagation of the pressure pulse in the arterial network also a volumetric flow rate is imposed at the inlet to model the systolic work of the heart the peripheral arteriole beds are simulated with the well known lumped windkessel model a standard geometry of the carotid bifurcation is discretized with p1 bubble p1 tetrahedral elements the obtained results properly reproduce the general flow patterns reported in the literature very good agreement between the outcomes of a pure 1d model and those of the combined multidimensional model was obtained it is worth noting that this kind of model may provide useful information to gain understanding in the genesis and development of arterial diseases

an iteration by subdomain overlapping dirichlet robin domain decomposition method for advection diffusion problems we present a new overlapping dirichlet robin domain decomposition method the method uses dirichlet and robin transmission conditions on the interfaces of an overlapping partitioning of the computational domain we derive interface equations to study the convergence of the method and show its properties through four numerical examples the mathematical framework is general and can be applied to derive overlapping versions of all the classical nonoverlapping methods

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

inverse problems and derivatives of determinants this paper is about a well posed and well conditioned inverse problem instead of measurements at the boundary we go right inside a network all nodes and edges are treated equally by probing the network we find its response to m different source terms from that information about the system the goal is to find the conductances on the m edges the equations are highly nonlinear what is attractive is the appearance of a convex potential function whose gradient is zero at the solution the analysis of the problem becomes an analysis of this potential function

structured backward errors for kkt systems karush kuhn tucker kkt systems are linear systems with coefficient matrices of the form graphics where h is symmetric a normwise structured backward error for kkt systems is defined and a com

time dependent viscous incompressible navier stokes equations the finite difference galerkin formulation and streamfunction algorithms abstract the finite difference galerkin fdg method is extended to time dependent incompressible navier stokes equations two algorithm development examples are given that use a staggered grid and centered differencing scheme for the primitive variables mass balance is used to solve the essential problems associated with applying the fdg method the use of the fdg method with this underlying discretization is shows to be the discrete analog of the continuum manipulations that lead to the fourth order streamfunction equation asymptotic and time evolution results obtained with a crank nicolson adams basforth algorithm are compared with published computations for re 400 1000 and 3200

an approximate method for solving incompressible navier stokes problems with flow rate conditions we consider the incompressible navier stokes problem with flow rate boundary conditions this problem has been investigated in j heywood r rannacher s turek artificial boundaries and flux and pressure conditions for the incompressible navier stokes equations int j numer methods fluids 22 1996 325 352 l formaggia j f gerbeau f nobile a quarteroni numerical treatment of defective boundary conditions for the navier stokes equation siam j numer anal 40 1 2002 376 401 a veneziani c vergara flow rate defective boundary conditions in haemodynamics simulations int j numer methods fluids 47 2005 803 816 in the latter two papers a feasible numerical formulation of this problem based on a lagrange multiplier approach has been investigated this approach has the drawback of high computational costs in this paper we propose an approximate formulation of this problem yielding a strong reduction of the computational costs the error analysis shows that the error introduced by this approximate formulation is confined in a small region of the boundary this is confirmed by the numerical simulations

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

reduction of equilibrium compatibility and flexibility matrices in the force method this paper develops analytical expressions and numerical for reducing the structural matrices which are used in the force method of structural analysis two simple examples illustrate the calculations for pin jointed trusses two advanced applications a beam cross consisting of two coplanar beams which are linked by a shear connector and an equilibrium linear stress finite element are also presented

perturbation analysis of generalized saddle point systems we discuss the perturbation analysis of generalized saddle point systems in this paper we give the nonlinear perturbation bounds then derive the condition numbers and analyze the sensitivity of the computed solutions

the construction of an optimal weakly divergence free macroelement the divergence free finite element method dffem is a method to find an approximate solution of the navier stokes equations in a divergence free space that is the continuity equation is satisfied a priori dffem eliminates the pressure from the calculations and reduces significantly the dimension of the system to be solved at each time step for the standard 8 node velocity and 4 node pressure dffem an optimal basis for the weakly divergence free subspace is constructed such that each basis function has non zero support on at most nine contiguous elements given this basis weakly divergence free macroelements are constructed

structural computations with the singular value decomposition of the equilibrium matrix abstract the singular value decomposition of the equilibrium matrix makes it possible to answer any question of a static kinematic or static kinematic nature for any structural assembly within a unified computational framework the paper introduces this decomposition relates it to the physical properties of a given assembly and derives formulae for stresses displacements etc a simple example illustrates the calculations

a variational approach for coupling kinematically incompatible structural models abstract in this work an extended variational framework aimed at properly addressing the coupling of kinematically incompatible structural models is presented the main goal is to variationally state the theoretical bases to deal with the coupling of structural models with different dimensionality in this approach the coupling conditions are naturally derived from the governing variational principle formulated at the continuous level furthermore by means of a real parameter we manage to build different continuous mechanical models that have different mechanical and mathematical features in particular the coupling of 3d solid models and 2d shell models under naghdi hypothesis is treated by introducing the corresponding kinematical assumptions into the proposed extended variational principle also the coupling between 3d solid and 1d beam models under bernoulli hypothesis is presented moreover for the continuous 3d 2d coupled problem a numerical approximation is addressed via the finite element method and some numerical results are given comparing the responses of the system when the discrete model varies by changing the value of the parameter finally a discussion comprising the main conclusions of the work is given

block pivoting and shortcut strategies for detecting copositivity abstract in recently proposed quadratic optimization algorithms copositivity detection procedures are frequently employed which deliver a feasible direction yielding a negative value of the considered quadratic form if the answer is negative to improve the computational performance of this routine here 1 recursive characterizations of copositivity are presented which enable efficient reduction of the dimension of the problem using block pivoting techniques and 2 shortcut strategies are described which are connected with diagonalization

on the automation of the force method in the optimal plastic design of frames it is well known that linear programming provides computationally the only solution to the optimal plastic design of plane frames it is also known that the formulation of the problem by the force method using kinematic variables requires for its solution by the simplex method much less computation than the corresponding program which uses for its formulation the displacement method however the force method is not easily automated this problem is circumvented in the paper by presenting a new algorithm which automatically selects a subminimal cycle base in any planar graph based on a shortest path technique between two points of a connected graph the algorithm forms cycles by finding the shortest paths between the end nodes of members of the graph the check for independence is done by fictitiously increasing the lengths of the members that form the cycle this leads to an easy computer implementation of the algorithm the algorithm is then used to provide a statical basis for any plane frame equilibrium with applied loads is satisfied by the use of cantilevers that follow the shortest load path of each load to the ground the whole process automates the data generation of the force method and makes possible to use in a computer program the same input data as in the displacement method a few examples of applications are given

sparse self stress matrices for the finite element force method a basic problem in the finite element force method is that of obtaining a sparse and banded self stress matrix and a sparse and banded structure flexibility matrix traditionally the self stress matrix is obtained through the application of algebraic procedures to the equilibrium matrix the self stress matrix for an indeterminate structure is not unique and it is possible to obtain another self stress matrix from an existing one through algebraic operations and grouping of redundants the purpose of this paper is to describe and test an algorithm called reduc which combines the vectors of the self stress matrix obtained from the lu procedure of the force method the rows of the transpose of this matrix are combined by using a special form of the gaussian elimination technique a plane frame example is presented to demonstrate the algorithm at work the algorithm reduc is applied to a plane truss and physical interpretation of the resulting self stress matrix highlights the grouping of redundants improved sparsity and bandwidth improvements in the conditioning and bandwidth of the structure flexibility matrix are also observed the algorithm yields results similar to those of the turn back lu procedure but requires less computation time and programming effort copyright 2001 john wiley sons ltd

on convergent schemes for diffuse interface models for two phase flow of incompressible fluids with general mass densities we are concerned with convergence results for fully discrete finite element schemes suggested in grun and klingbeil j comput phys in press doi 10 1016 j jcp 2013 10 028 they were developed for the diffuse interface model in h abels h garcke and g grun math models methods appl sci 22 2012 1150013 which describes two phase flow of immiscible incompressible viscous fluids we formulate general conditions on discretization spaces and projection operators which allow us to prove compactness of discrete solutions with respect to both time and space and which hence permit us to establish convergence of the scheme to a generalized solution we identify a simple quantitative and physical criterion to decide whether this generalized solution is in fact a weak solution in this case our analysis provides another pathway to establish existence of weak solutions to the aforementioned model in two and in three space dimensions our argument is particularly based on higher regularity results

sizing optimization of truss structures by method of centers and force formulation this paper describes a new approach to optimum weight design of truss structures the force method is incorporated in an optimization algorithm based on the method of center points design variables are the member cross sectional areas and the redundant forces evaluated for each independent loading condition acting on the structure the optimization method utilizes the largest hyperspheres inscribed within the feasible space the method of hyperspheres has been enhanced here to handle the compatibility equality constraints as well by including the analysis step in the optimization cycle there is no longer the need to perform separate structural analyses thus saving computation time the viability and efficiency of the proposed method are demonstrated for truss structures subject to multiple loading conditions and constraints on member stresses nodal displacement and minimum gage numerical results are compared with those reported in the literature

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