lower bounds for the perron root of a nonnegative matrix abstract new lower bounds frequently better than previous bounds and readily computable for the perron root r a of a nonnegative matrix a are obtained the theoretical comparison of new bounds with the known ones is supplemented with two numerical examples

geometrically exact beam elements and smooth contact schemes for the modeling of fiber based materials and structures abstract this work focuses on finite element formulations for the accurate modeling and efficient simulation of the implicit dynamics of slender fiber or rod like components and their contact interaction when being embedded in complex systems of fiber based materials and structures recently the authors have proposed a novel all angle beam contact abc formulation that combines the advantages of existing point and line contact models in a variationally consistent manner however the abc formulation has so far only been applied in combination with a special torsion free beam model which yields a very simple and efficient finite element formulation but which is restricted to initially straight beams with isotropic cross sections in order to abstain from these restrictions the current work combines the abc formulation with a geometrically exact kirchhoff love beam element formulation that is capable of treating even the most general cases of slender beam problems in terms of initial geometry and external loads while the neglect of shear deformation that is inherent to this formulation has been shown to provide considerable numerical advantages in the range of high beam slenderness ratios alternative shear deformable beam models are required for examples with thick beams for that reason the current contribution additionally proposes a novel geometrically exact beam element based on the simo reissner theory similar to the torsion free and the kirchhoff love beam elements also this simo reissner element is based on a c1 continuous hermite interpolation of the beam centerline which will allow for smooth contact kinematics for this hermitian simo reissner element a consistent spatial convergence behavior as well as the successful avoidance of membrane and shear locking will be demonstrated numerically all in all the combination of the abc formulation with these different beam element variants i e the torsion free element the kirchhoff love element and the simo reissner element results in a very flexible and modular simulation framework that allows to choose the optimal element formulation for any given application in terms of accuracy efficiency and robustness based on several practically relevant examples the different variants are compared numerically and eventually a general recommendation concerning the optimal choice of beam elements is made

stability of elastic rods with self contact abstract in this paper we study the numerical stability of equilibrium configurations of an elastic rod with unilateral frictionless self contact we use a finite element method with penalty and augmented lagrangian approaches to account for the self penetration constraint the numerical solution is carried out using the arclength continuation method in the case where the energy expression satisfies some hypotheses we have proved that any isolated minimum of the continuous self contact problem is the strong limit of a sequence of local minima of the discrete problem our stability analysis is based on detecting the zeros of the determinant of a symmetrized version of the stiffness matrix and whether there is a sign change of the determinant or not

quadratically convergent direct calculation of critical points for 3d structures undergoing finite rotations in this work we present the implementation details of a quadratically converging newton method based algorithm for direct computation of instability points for 3d structures undergoing finite rotations the structural model chosen for illustration is the 3d geometrically exact beam the proposed algorithm makes use of an extended system where equilibrium equations are supplemented with the loss of stability condition which roughly doubles the total number of equations nonetheless the latter requires only an insignificant increase in computational cost due to judicious use of the bordering algorithm for computing the solution the main thrust of our work is directed towards a careful development of linearized forms of the governing equations employed by newton s method the corresponding results are presented both in material and spatial versions a set of numerical examples is used to illustrate a very satisfying performance of the proposed algorithm

a sequence of lower bounds for the spectral radius of nonnegatfve matrices abstract we present a sequence of progressively better lower bounds for the s pectral radius of a nonnegative matrix each element of the sequence is a function of the spectral radius of the geometric symmetrization of a power of the matrix as defined by schwenk each element provides a better approximation to the spectral radius than the elements in the sequences provided by other authors

contact between spheres and general surfaces abstract this work presents a mathematical model to establish the weak form and tangent operator contributions due to contact between spherical surfaces and general surfaces normal and friction components are included such as dissipative actions the main concern herein is the proper consideration of the kinematics of the spherical surface and its influence on contact forces including the appropriate description of the spherical motion by gap functions in normal and tangential directions this leads to the possibility of dealing with complex kinematics of rolling or alternating rolling sliding of the spherical surface mapping its motion on another general surface to achieve that the model is based on rotational and translational degrees of freedom used to map the motion of contacting surfaces one may employ such strategy for interactions between spheres and rigid or flexible bodies modeled using finite element meshes as examples parameterizations of rigid and flexible surfaces are proposed and used practical applications are shown with usage of beam and shell finite elements experiencing contact interactions

semismooth karush kuhn tucker equations and convergence analysis of newton and quasi newton methods for solving these equations there are several forms of systems of nonsmooth equations which are equivalent to the karush kuhn tucker kkt system of a nonlinearly constrained optimization problem nlp if the nlp is twice continuously differentiable and the hessian functions of its objective and constraint functions are locally lipschitzian then these kkt equations are strongly semismooth if furthermore the linear independence condition and the strong second order sufficiency condition are satisfied at a kkt point then the generalized jacobians of these kkt equations are nonsingular at that point and the sequence generated by the generalized newton method converges to this point q quadratically however direct application of quasi newton methods cannot guarantee q superlinear convergence we present a mixed quasi newton method which converges q superlinearly with common symmetrical updating rules under the above conditions for the generalized newton method superlinear convergence of the primal variables and global convergence are also discussed

nested bounds for the perron root of a nonnegative matrix abstract it is well known that for a nonnegative matrix a the smallest row sum r a and the largest row sum r a provide lower and upper bounds respectively for the perron root of a these bounds are generalized for a partitioned nonnegative matrix a the new bounds are better than r a and r a and they can be further improved by a refinement of the partition known monotonicity and convergence properties of r a and r a are generalized for the new bounds

elastic stability of dna configurations i general theory results are presented in the theory of the elastic rod model for dna among which are criteria enabling one to determine whether a calculated equilibrium configuration of a dna segment is stable in the sense that it gives a local minimum to the sum of the segment s elastic energy and the potential of forces acting on it the derived stability criteria are applicable to plasmids and to linear segments subject to strong anchoring end conditions their utility is illustrated with an example from the theory of configurations of the extranucleosomal loop of a dna miniplasmid in a mononucleosome with emphasis placed on the influence that nicking and ligation on one hand and changes in the ratio of elastic coefficients on the other have on the stability of equilibrium configurations in that example the configurations studied are calculated using an extension of the method of explicit solutions to cases in which the elastic rod modeling a dna segment is considered impenetrable and hence excluded volume effects and forces arising from self contact are taken into account

euler lagrange equations for nonlinearly elastic rods with self contact we derive the euler lagrange equations for nonlinearly elastic rods with self contact the excluded volume constraint is formulated in terms of an upper bound on the global curvature of the centre line this condition is shown to guarantee the global injectivity of the deformation of the elastic rod topological constraints such as a prescribed knot and link class to model knotting and supercoiling phenomena as observed e g in dna molecules are included by using the notion of isotopy and gaussian linking number the bound on the global curvature as a nonsmooth side condition requires the use of clarke s generalized gradients to obtain the explicit structure of the contact forces which appear naturally as lagrange multipliers in the euler lagrange equations transversality conditions are discussed and higher regularity for the strains moments the centre line and the directors is shown

a recursive quadratic programming algorithm that uses differentiable exact penalty functions in this paper a recursive quadratic programming algorithm for solving equality constrained optimization problems is proposed and studied the line search functions used are approximations to fletcher s differentiable exact penalty function global convergence and local superlinear convergence results are proved and some numerical results are given

level sets fields placement and velocity based formulations of contact impact problems by introducing unknown sign like fields of level sets type the signorini moreau dynamic contact conditions are set merely as boundary equations from this setting a continuous hybrid weak strong formulation for dynamic contact between deformable solids is derived and a new lagrangian formulation we call stabilized generalizing both the classical and augmented ones is obtained friction phenomena are treated similarly in the global problem the irregular sign like fields stand for the intrinsic contact unknown ones this problem is discretized by means of time space and collocation schemes some numerical experimentations are carried out showing the potential of our developments copyright 2006 john wiley sons ltd

a finite element approach for the line to line contact interaction of thin beams with arbitrary orientation abstract the objective of this work is the development of a novel finite element formulation describing the contact interaction of slender beams in complex 3d configurations involving arbitrary beam to beam orientations it is shown in a mathematically concise manner that standard beam contact models based on a point wise contact force fail to describe a considerable range of configurations which are however likely to occur in practical applications on the contrary the formulation proposed here models beam to beam contact by means of distributed line forces a procedure that is shown to be applicable for arbitrary geometrical configurations the proposed formulation is based on a gauss point to segment type contact discretization and a penalty regularization of the contact constraint by means of detailed theoretical and numerical investigations it is shown that this approach is more suitable for beam contact than possible alternatives based on mortar type contact discretizations or constraint enforcement by means of lagrange multipliers the proposed formulation is enhanced by a consistently linearized integration interval segmentation avoiding numerical integration across strong discontinuities in combination with a smoothed contact force law and the employed c 1 continuous beam elements this procedure drastically reduces the numerical integration error an essential prerequisite for optimal spatial convergence rates the resulting line to line contact algorithm is supplemented by contact contributions of the beam endpoints which represent boundary minima of the underlying minimal distance problem finally a series of numerical test cases is analyzed in order to investigate the accuracy and consistency of the proposed formulation regarding integration error spatial convergence behavior and resulting contact force distributions for one of these test cases an analytical solution based on the kirchhoff theory of thin rods is derived which can serve as valuable benchmark for the proposed model but also for future beam to beam contact formulations

approximation of general arch problems by straight beam elements in this paper we approximate the solution of a problem of a general arch by a nonconforming method using straight beam elements and taking into account numerical integration compatibility conditions which have to be satisfied at the mesh points are given these conditions ensure for this method the same order of convergence as usual conforming finite element methods

a new merit function and a successive quadratic programming algorithm for variational inequality problems recently various merit functions for variational inequality problems have been proposed and their properties have been studied unfortunately these functions may not be easy to evaluate unless the constraints of the problem have a relatively simple structure in this paper a new merit function for variational inequality problems with general convex constraints is proposed at each point the proposed function is defined as an optimal value of a quadratic programming problem whose constraints consist of a linear approximation of the given nonlinear constraints it is shown that the set of constrained minima of the proposed merit function coincides with the set of solutions to the original variational inequality problem it is also shown that this function is directionally differentiable in all directions and under suitable assumptions any stationary point of the function over the constraint set actually solves the original variational inequality problem finally a descent method for solving the variational inequality problem is proposed and its convergence is proved the method is closely related to a successive quadratic programming method for solving nonlinear programming problems

an isogeometric collocation method for frictionless contact of cosserat rods abstract a frictionless contact formulation for spatial rods is developed within the framework of isogeometric collocation the structural mechanics is described by the cosserat theory of geometrically nonlinear spatial rods the numerical discretization is based on an isogeometric collocation method where the geometry and solution fields are represented as nurbs curves and the strong forms of the equilibrium equations are collocated at greville points in this framework a frictionless rod to rod contact formulation is proposed contact points are detected by a coarse level and a refined search for close centerline points and reaction forces are computed by the actual penetration of rod surface points so that the enforcement of the contact constraints is performed with the penalty method an important aspect is the application of contact penalty forces as point loads within the collocation scheme and methods for this purpose are proposed and evaluated the overall contact algorithm is validated by and applied to several numerical examples

a lower bound for the perron root of a nonnegative matrix ii abstract this paper is a continuation of an earlier one by the author we obtain a new lower bound for the perron root r a of a nonnegative n n matrix a we show that for a with at least one principal submatrix of order two with different diagonal entries and with at least one positive off diagonal entry the bound is at least as good as the one from 7 and we determine a class of nonnegative matrices for which it is essentially better we also show that not every nonnegative square matrix is similar to a nonnegative square matrix with identical diagonal entries

the best possible lower bound for the perron root using traces abstract let a be an n n matrix with real eigenvalues wolkowicz and styan presented bounds for the eigenvalues using only n tr a and tr a 2 we show that their lower bound for the largest eigenvalue works also as a lower bound for the perron root of a if a is nonnegative and its eigenvalues are not necessarily real we also show that this bound is optimal under certain conditions finally we solve completely the problem to find the optimal lower bound for the perron root using only n tr a and tr a 2

on finite element implementation of geometrically nonlinear reissner s beam theory three dimensional curved beam elements abstract finite element implementation of the three dimensional finite strain beam theory of reissner is considered in this work in contrast with some earlier works on the subject discussed here are the beam elements whose reference axes are arbitrary space curved lines we have shown that an improved representation of curved reference geometry significantly increases the accuracy of the results however it also makes the choice of non locking finite element interpolations somewhat more delicate a hierarchical displacement interpolation proposed here is proved to be capable of eliminating both shear and membrane locking phenomena a very satisfying non locking performance is demonstrated for a set of problems in nonlinear elasto statics

an augmented lagrangian function with improved exactness properties in this paper we introduce a new exact augmented lagrangian function for the solution of general nonlinear programming problems for this lagrangian function a complete equivalence between its unconstrained minimization on an open set and the solution of the original constrained problem can be established under mild assumptions and without requiring the boundedness of the feasible set of the constrained problem moreover we describe an unconstrained algorithmic model which is globally convergent toward kkt pairs of the original constrained problem the algorithmic model can be endowed with a superlinear rate of convergence by a proper choice of the search direction in the unconstrained minimization without requiring strict complementarity

a continuum based finite element formulation for the implicit solution of multibody large deformation frictional contact problems in this paper a formulation is presented for the finite element treatment of multibody large deformation frictional contact problems the term multibody is used to mean that when two bodies mechanically contact both may be deformable a novel aspect of the approach advocated is that the equations governing contact are developed in the continuum setting first before deriving the corresponding finite element equations this feature distinguishes the current work from many earlier treatments of contact problems and renders it considerably more general in particular the approach yields a characterization of the frictional constraint assuming a coulomb law suitable for arbitrary discretizations in either two or three dimensions a geometric framework is constructed within which both frictionless and frictional response are naturally described making subsequent finite element discretization a straightforward substitution of finite dimensional solution spaces for their continuum counterparts to our knowledge this general formulation and implementation of the frictional contact problem in a finite element setting has not been reported previously in the literature the development includes exact linearization of the statement of virtual work which enables optimal convergence properties for newton raphson solution strategies and which appears to be highly desirable if not essential for the general robustness of implicit finite element techniques since the theory and subsequent linearization require no limitations on the amount of deformation or relative sliding that can occur the resulting treatment of frictional contact is suitable for a wide range of examples displaying significant non linear behaviour this assertion is substantiated through presentation of a variety of examples in both two and three dimensions

modeling and numerical treatment of elastic rods with frictionless self contact in this paper we present a hyperelastic rod model that takes into account self contact forces the model is based on cosserat rod theory that incorporates shear elongation flexure and twist deformations the problem of avoiding self penetration of parts of the rod is handled by the introduction of a contact distance function and the incorporation of associated contact forces we present a penalty method for the treatment of the multi valued and non differentiable contact law we also describe an augmented lagrangian formulation of this problem we then give the details of the finite element discretization of the elastic rod self contact problem as well as some numerical examples

a unified approach for beam to beam contact abstract existing beam contact formulations can be categorized in point to point contact models that consider a discrete contact force at the closest point of the beams and line to line contact models that assume distributed contact forces in this work it will be shown that line contact formulations applied to slender beams provide accurate and robust mechanical models in the range of small contact angles whereas the computational efficiency considerably decreases with increasing contact angles on the other hand point contact formulations serve as sufficiently accurate and very efficient models in the regime of large contact angles while they are not applicable for small contact angles as a consequence of non unique closest point projections in order to combine the advantages of these basic formulations a novel all angle beam contact abc formulation is developed that applies a point contact formulation in the range of large contact angles and a recently developed line contact formulation in the range of small contact angles the two being smoothly connected by means of a variationally consistent model transition based on a stringent analysis two different transition laws are investigated optimal algorithmic parameters are suggested and conservation of linear momentum angular momentum and total energy is shown all configuration dependent quantities within the point the line and the transition contact regime are consistently linearized thus allowing for their application within implicit time integration schemes furthermore a step size control of the nonlinear solution scheme is proposed that allows for displacement increments per time step that exceed the order of magnitude of the beam cross section radius for many standard beam to beam contact algorithms this is the typical limitation concerning possible time step sizes especially when considering high beam slenderness ratios finally an efficient two stage contact search based on dynamically adapted search segments is proposed this algorithm yields in a tight set of potential contact pairs and enables a subdivision into potential point and potential line contact pairs which is essential in order to fully exploit the efficiency potential of the proposed contact formulation a series of numerical test cases is analyzed in order to verify the accuracy and consistency of the proposed contact model transition regarding contact force distributions and conservation properties but also for quantifying the efficiency gains as compared to standard beam contact formulations

finite rotation analysis and consistent linearization using projectors abstract a systematic procedure is presented that may be used to extend the capabilities of an existing linear finite element to accomodate finite rotation analysis this procedure is a generalization of the work presented in computers and structures volume 30 pp 257 267 the basis of our approach is the element independent co rotational algorithm where the element rigid body motion translations and rotations is separated from the deformational part of its total motion the variation of this co rotational relation results in a projector matrix with the property that a consistent internal force vector is invariant under its action the consistent tangent stiffness matrix is shown to depend on this invariance condition through the derivative of the projector this results in an unsymmetric tangent matrix whose anti symmetric part depends on the out of balance force vector in this paper we prove that using the symmetric part of the tangent matrix the newton iteration retains its quadratic rate of convergence this approach has been used to solve a number of large rotation test example problems the results demonstrate that it is possible to analyze structures undergoing large rotations within a general co rotational framework using simple and economical finite elements the resulting improvements in the performance of these simple elements are brought about through the use of convenient software utilities as pre and post processors to the element routines

serial and parallel computation of karush kuhn tucker points via nonsmooth equations in this paper a robust iterative algorithm is presented for computing karush kuhn tucker kkt points of nonlinear programs this algorithm is a variation of the ne sqp method for solving the nonlinear complementarily problem it makes use of a merit function that combines the original objective function of the nonlinear program with a residual function of the kkt conditions formulated as a system of nonsmooth equations global convergence and a q quadratic rate of convergence of the algorithm are established under some standard constraint qualifications in nonlinear programming theory but without the strict complementarily condition parallel implementations of the algorithm are discussed

global convergence on an active set sqp for inequality constrained optimization sequential quadratic programming sqp has been one of the most important methods for solving nonlinearly constrained optimization problems in this paper we present and study an active set sqp algorithm for inequality constrained optimization the active set technique is introduced which results in the size reduction of quadratic programming qp subproblems the algorithm is proved to be globally convergent thus the results show that the global convergence of sqp is still guaranteed by deleting some redundant constraints

on the choice of finite rotation parameters in this work we discuss some aspects of the three dimensional finite rotations pertinent to the formulation and computational treatment of the geometrically exact structural theories among various possibilities to parameterize the finite rotations special attention is dedicated to a choice featuring an incremental rotation vector some computational aspects pertinent to the implementation of the newton iterative scheme and the newmark time stepping algorithm applied to solving these problems are examined representative numerical simulations are presented in order to illustrate the performance of the proposed formulation