a theoretically supported scalable tfeti algorithm for the solution of multibody 3d contact problems with friction abstract a total feti tfeti based domain decomposition algorithm with preconditioning by a natural coarse grid defined by the rigid body motions of subdomains is adapted to the solution of multibody contact problems with friction in 3d and proved to be scalable for tresca friction the analysis admits floating bodies the algorithm finds an approximate solution at the cost asymptotically proportional to the number of variables provided the ratio of the decomposition and the discretization parameters is bounded the analysis is based on the classical results by farhat et al on scalability of feti for linear problems and on our development of optimal quadratic programming algorithms the algorithm preserves parallel scalability of the classical feti method both theoretical results and numerical experiments indicate that our algorithm is effective the effectiveness of the method in the inner loop of the fixed point iterations for the solution of the contact problems with coulomb s friction is illustrated on the analysis of a mine support

box constrained quadratic programming with proportioning and projections two new closely related concepts are introduced that depend on a positive constant gamma an iteration is proportional if the norm of violation of the kuhn tucker conditions at active variables does not excessively exceed the norm of the part of the gradient that corresponds to free variables while a progressive direction determines a descent direction that enables the released variables to move far enough from the boundary in a step called proportioning an algorithm that uses the conjugate gradient method to explore the face of the region defined by the current iterate until a disproportional iteration is generated is proposed it then changes the face by means of the progressive direction it is proved that for strictly convex problems the proportioning is a spacer iteration so that the algorithm converges to the solution if the solution is nondegenerate then the algorithm finds the solution in a finite number of steps moreover a simple lower bound on gamma is given to ensure finite termination even for problems with degenerate solutions the theory covers a class of algorithms allowing many constraints to be added or dropped at a time and accepting approximate solutions of auxiliary problems preliminary numerical results are promising

on the solution of convex qpqc problems with elliptic and other separable constraints with strong curvature the paper deals with an effective implementation of some algorithms for the solution of convex qpqc problems with elliptic and other separable constraints with strong curvature here we discuss robust quantitative refinement of the karush kuhn tucker conditions extend existing results on the decrease of the cost function along the projected gradient path to separable constraints with elliptic components and plug them into the existing algorithms for the solution of the qpqc problems with r linear rate of convergence in the bounds on the spectrum the results are then extended to the problems with separable inequality and linear equality constraints the performance of the algorithms is demonstrated on the solution of a problem of two cantilever beams in mutual contact with orthotropic tresca and coulomb friction discretized by up to one and half million nodal variables

augmented lagrangians with adaptive precision control for quadratic programming with simple bounds and equality constraints in this paper we discuss a specialization of the augmented lagrangian type algorithm of conn gould and toint to the solution of strictly convex quadratic programming problems with simple bounds and equality constraints the new feature of the presented algorithm is the adaptive precision control of the solution of auxiliary problems in the inner loop of the basic algorithm which yields a rate of convergence that does not have any term that accounts for inexact solution of auxiliary problems moreover boundedness of the penalty parameter is achieved for the precision control used numerical experiments illustrate the efficiency of the presented algorithm and encourage its usage

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

a numerically scalable domain decomposition method for the solution of frictionless contact problems we present a domain decomposition method with lagrange multipliers for solving iteratively frictionless contact problems this method which is based on the feti method and therefore is named here the feti c method incorporates a coarse contact system that guides the iterative prediction of the active zone of contact we demonstrate numerically that this method is numerically scalable with respect to both the problem size and the number of subdomains copyright 2001 john wiley sons ltd

quasistatic normal compliance contact problem of visco elastic bodies with coulomb friction implemented by qp and sgbem the quasistatic normal compliance contact problem of isotropic homogeneous linear visco elastic bodies with coulomb friction at small strains in kelvin voigt rheology is considered the discretization is made by a semi implicit formula in time and the symmetric galerkin boundary element method sgbem in space assuming that the ratio of the viscosity and elasticity moduli is a given relaxation time coefficient the obtained recursive minimization problem formulated only on the contact boundary has a nonsmooth cost function if the normal compliance responds linearly and the 2d problems are considered then the cost function is piecewise quadratic which after a certain transformation gets the quadratic programming qp structure however it would lead to second order cone programming in 3d problems finally several computational tests are presented and analysed with an additional discussion on numerical stability and convergence of the involved approximated poincare steklov operators

an optimal algorithm and superrelaxation for minimization of a quadratic function subject to separable convex constraints with applications we propose a modification of our mpgp algorithm for the solution of bound constrained quadratic programming problems so that it can be used for minimization of a strictly convex quadratic function subject to separable convex constraints our active set based algorithm explores the faces by conjugate gradients and changes the active sets and active variables by gradient projections possibly with the superrelaxation steplength the error estimate in terms of extreme eigenvalues guarantees that if a class of minimization problems has the spectrum of the hessian matrix in a given positive interval then the algorithm can find and recognize an approximate solution of any particular problem in a number of iterations that is uniformly bounded we also show how to use the algorithm for the solution of separable and equality constraints the power of our algorithm and its optimality are demonstrated on the solution of a problem of two cantilever beams in mutual contact with tresca friction discretized by almost twelve millions nodal variables

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

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

efficient contact solvers based on domain decomposition techniques abstract this paper deals with the construction of efficient algorithms for the solution of the finite dimensional constrained minimization problem arising from the finite element discretization of contact problems dualization techniques have been used to decrease the problem size from the large number of unknowns in the domain to the much smaller number of inequalities at the boundary the disadvantage of this direct schur complement approach is the need of the inversion of the stiffness matrix domain decomposition techniques meet very similar requirements a global boundary value problem is decoupled into local subproblems and one interface problem at the coupling boundary two complementary approaches are the dirichlet method and the neumann method the first one requires preconditioners for local dirichlet problems and for the interface problem in h 1 2 and extension operators from the boundary into the domain the second one needs preconditioners for local neumann problems and for the interface problem in h 1 2 efficient multilevel algorithms for all components are available in literature in this paper it is shown how to use exactly these components for the construction of solvers for contact problems new results for the analysis of convergence are presented at least at uniformly refined meshes we can prove optimal time complexity numerical results show high efficiency also on adaptively refined 3d meshes

inexact semimonotonic augmented lagrangians with optimal feasibility convergence for convex bound and equality constrained quadratic programming a variant of the augmented lagrangian type algorithm for strictly convex quadratic programming problems with bounds and equality constraints is considered the algorithm exploits the adaptive precision control in the solution of auxiliary bound constraint problems in the inner loop while the lagrange multipliers for the equality constraints are updated in the outer loop the update rule for the penalty parameter is introduced that depends on the increase of the augmented lagrangian global convergence is proved and an explicit bound on the penalty parameter is given a qualitatively new feature of our algorithm is a bound on the feasibility error that is independent of conditioning of the constraints when applied to the class of problems with the spectrum of the hessian matrix in a given interval the algorithm returns the solution in o 1 matrix vector multiplications the results are valid even for linearly dependent constraints theoretical results are illustrated by numerical experiments including the solution of an elliptic variational inequality

robust stabilization of uncertain descriptor fractional order systems this paper presents sufficient conditions for the robust asymptotical stabilization of uncertain descriptor fractional order systems with the fractional order a satisfying 0 a 2 the results are obtained in terms of linear matrix inequalities the parameter uncertainties are assumed to be time invariant and norm bounded appearing in the state matrix a necessary and sufficient condition for the normalization of uncertain descriptor fractional order systems is given via linear matrix inequality lmi formulation the state feedback control to robustly stabilize such uncertain descriptor fractional order systems with the fractional order a belonging to 0 a 2 is derived two numerical examples are given to demonstrate the applicability of the proposed approach

scalable tfeti algorithm for the solution of multibody contact problems of elasticity a total feti tfeti based domain decomposition algorithm with preconditioning by a natural coarse grid of the rigid body motions is adapted to the solution of multibody contact problems of elasticity in 2d and 3d and proved to be scalable the algorithm finds an approximate solution at the cost asymptotically proportional to the number of variables provided the ratio of the decomposition parameter and the discretization parameter is bounded the analysis is based on the classical results by farhat mandel and roux on scalability of feti with a natural coarse grid for linear problems and on our development of optimal quadratic programming algorithms for bound and equality constrained problems the algorithm preserves parallel scalability of the classical feti method both theoretical results and numerical experiments indicate a high efficiency of our algorithm in addition its performance is illustrated on a real world problem of analysis of the ball bearing copyright 2009 john wiley sons ltd

on multi grid methods for variational inequalities we consider here a general class of algorithms for the numerical solution of variational inequalities a convergence proof is given and in particular a multi grid method is described numerical results are presented for the finite difference discretization of an obstacle problem for minimal surfaces

monotone multigrid methods for elliptic variational inequalities i we derive fast solvers for discrete elliptic variational inequalities of the first kind obstacle problems as resulting from the approximation of related continuous problems by piecewise linear finite elements using basic ideas of successive subspace correction we modify well known relaxation methods by extending the set of search directions extended underrelaxations are called monotone multigrid methods if they are quasioptimal in a certain sense by construction all monotone multigrid methods are globally convergent we take a closer look at two natural variants the standard monotone multigrid method and a truncated version for the considered model problems the asymptotic convergence rates resulting from the standard approach suffer from insufficient coarse grid transport while the truncated monotone multigrid method provides the same efficiency as in the unconstrained case

theoretically supported scalable feti for numerical solution of variational inequalities the feti method with a natural coarse grid is combined with recently proposed optimal algorithms for the solution of bound and or equality constrained quadratic programming problems in order to develop a scalable solver for elliptic boundary variational inequalities such as those describing equilibrium of a system of bodies in mutual contact a discretized model problem is first reduced by the duality theory of convex optimization to the quadratic programming problem with bound and equality constraints the latter is then modified by means of orthogonal projectors to the natural coarse grid introduced by farhat mandel and roux comput methods appl mech engrg 115 1994 pp 365 385 finally the classical results on linear scalability for linear problems are extended to boundary variational inequalities the results are validated by numerical experiments the experiments also confirm that the algorithm enjoys the same parallel scalability as its linear counterpart

multigrid algorithms for variational inequalities we present an iterative scheme for obstacle problems described by variational inequalities at each step of the iteration the method leads to a reduced linear resp nonlinear algebraic system whi

on the maximization of a concave quadratic function with box constraints a new method for maximizing a concave quadratic function with bounds on the variables is introduced the new algorithm combines conjugate gradients with gradient projection techniques as the algorithm of more and toraldo siam j optimization 1 1991 pp 93 113 and other well known methods do a new strategy for the decision of leaving the current face is introduced that makes it possible to obtain finite convergence even for a singular hessian and in the presence of dual degeneracy numerical experiments are presented

solution of contact problems by feti domain decomposition with natural coarse space projections abstract an efficient non overlapping domain decomposition algorithm of the neumann neumann type for solving both coercive and semicoercive contact problems is presented the discretized problem is first turned by the duality theory of convex programming to the quadratic programming problem with bound and equality constraints and the latter is further modified by means of orthogonal projectors to the natural coarse space introduced by farhat and roux in the framework of their feti method the resulting problem is then solved by an augmented lagrangian type algorithm with an outer loop for the lagrange multipliers for the equality constraints and an inner loop for the solution of the bound constrained quadratic programming problems the projectors are shown to guarantee fast convergence of iterative solution of auxiliary linear problems and to comply with efficient quadratic programming algorithms proposed earlier reported theoretical results and numerical experiments indicate high numerical scalability of the algorithm which preserves the parallelism of the feti methods

on r linear convergence of semi monotonic inexact augmented lagrangians for bound and equality constrained quadratic programming problems with application new convergence results for a variant of the inexact augmented lagrangian algorithm smalbe z dostal an optimal algorithm for bound and equality constrained quadratic programming problems with bounded spectrum computing 78 2006 311 328 for the solution of strictly convex bound and equality constrained quadratic programming problems are presented the algorithm smalbe m presented here uses a fixed regularization parameter and controls the precision of the solution of auxiliary bound constrained problems by a multiple of the norm of violation of the equality constraints and a constant which is updated in order to enforce the increase of lagrangian function a nice feature of smalbe m is its capability to find an approximate solution of important classes of problems in a number of iterations that is independent of the conditioning of the equality constraints here we prove the r linear rate of convergence of the outer loop of smalbe m for any positive regularization parameter after the strong active constraints of the solution are identified the theoretical results are illustrated by solving two benchmarks including the contact problem of elasticity discretized by two million of nodal variables the numerical experiments indicate that the inexact solution of auxiliary problems in the inner loop results in a very small increase of the number of outer iterations as compared with the exact algorithm the results do not assume independent equality constraints and remain valid when the solution is dual degenerate

feti based algorithms for contact problems scalability large displacements and 3d coulomb friction theoretical and experimental results concerning feti based algorithms for contact problems of elasticity are reviewed a discretized model problem is first reduced by the duality theory of convex optimization to the quadratic programming problem with bound and equality constraints the latter is then optionally modified by means of orthogonal projectors to the natural coarse space introduced by farhat and roux in the framework of their feti method the resulting problem is then solved either by special algorithms for bound constrained quadratic programming problems combined with penalty that imposes the equality constraints or by an augmented lagrangian type algorithm with the inner loop for the solution of bound constrained quadratic programming problems recent theoretical results are reported that guarantee certain optimality and scalability of both algorithms the results are confirmed by numerical experiments the performance of the algorithm in solution of more realistic engineering problems by basic algorithm is demonstrated on the solution of 3d problems with large displacements or coulomb friction

a numerically scalable dual primal substructuring method for the solution of contact problems part i the frictionless case abstract we present a dual primal non linear domain decomposition method with lagrange multipliers for solving iteratively frictionless contact problems this method which is based on the feti dp substructuring algorithm features a non linear krylov type acceleration scheme it addresses both cases of restrained and unrestrained bodies when the bodies in contact behave linearly it does not perform any newton like iteration to solve the non linear contact problem we present performance results for several numerical simulations which suggest that the proposed method is numerically scalable with respect to both the problem size and the number of subdomains we also illustrate the parallel scalability of this method on a cluster of silicon graphics systems for a three million degree of freedom static contact problem

monotone multigrid methods on nonmatching grids for nonlinear multibody contact problems nonconforming domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations we use a generalized mortar method based on dual lagrange multipliers for the discretization of a nonlinear contact problem between linear elastic bodies in the case of unilateral contact problems pointwise constraints occur and monotone multigrid methods yield efficient iterative solvers here we generalize these techniques to nonmatching triangulations where the constraints are realized in terms of weak integral conditions the basic new idea is the construction of a nested sequence of nonconforming constrained spaces we use suitable basis transformations and a multiplicative correction in contrast to other approaches no outer iteration scheme is required the resulting monotone method is of optimal complexity and can be implemented as a multigrid method numerical results illustrate the performance of our approach in two and three dimensions

toward infinite horizon optimality in nonlinear model predictive control a new model predictive control scheme with guaranteed stability is presented for constrained discrete time nonlinear systems the open loop optimization problem does not involve explicit terminal constraints and employs a nonstandard terminal cost function the closed loop system is shown to be infinite horizon optimal provided that the terminal cost exactly captures the infinite horizon optimal value in a neighborhood of the origin stability and optimality are proven for a set of initial states which is invariant and approaches the set of all controllable states as the prediction horizon increases

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

the feti family of domain decomposition methods for inequality constrained quadratic programming application to contact problems with conforming and nonconforming interfaces two domain decomposition methods with lagrange multipliers for solving iteratively quadratic programming problems with inequality constraints are presented these methods are based on the feti and feti dp substructuring algorithms in the case of linear constraints they do not perform any newton like iteration instead they solve a constrained problem by an active set strategy and a generalized conjugate gradient based descent method equipped with controls to guarantee convergence monotonicity both methods possess the desirable feature of minimizing numerical oscillations during the iterative solution process performance results and comparisons are reported for several numerical simulations that suggest that both methods are numerically scalable with respect to both the problem size and the number of subdomains their parallel scalability is also illustrated on a linux cluster for a complex 1 4 million degree of freedom multibody problem with frictionless contact and nonconforming discrete interfaces

an optimal algorithm for minimization of quadratic functions with bounded spectrum subject to separable convex inequality and linear equality constraints an in a sense optimal algorithm for minimization of quadratic functions subject to separable convex inequality and linear equality constraints is presented its unique feature is an error bound in terms of bounds on the spectrum of the hessian of the cost function if applied to a class of problems with the spectrum of the hessians in a given positive interval the algorithm can find approximate solutions in a uniformly bounded number of simple iterations such as matrix vector multiplications moreover if the class of problems admits a sparse representation of the hessian it simply follows that the cost of the solution is proportional to the number of unknowns theoretical results are illustrated by numerical experiments