on improved predictions for structural equilibrium path evaluations the paper describes how several procedures for higher order predictions have been introduced in order to improve the convergence speed in a general finite element program for non linear structural

structural instability analyses based on generalised path following this paper describes how quasi static conservative instability problems can be completely described using generalised path following procedures for augmented equilibrium formulations in particul

canonical forms for linear differential algebraic equations with variable coefficients abstract we give a new set of local characterizing quantities for the treatment of linear differential algebraic equations with variable coefficients this leads to new global invariances under which we can find a generalization of the global or differentiation index and get a new existence and uniqueness theorem

entropy and boundary conditions in random rhombus tilings the tilings of rhombi in two dimensions and of rhomboedra in three dimensions are studied when they are constrained by fixed boundary conditions we establish a link between those conditions and free or periodic boundary ones the entropy is written as a functional integral which is treated via a saddle point method we can exhibit the dominant states of the statistical ensemble of tilings and show that they can display a strong structural inhomogeneity caused by the boundary this inhomogeneity is responsible for a difference of entropy between the studied fixed boundary tilings and free boundary ones this method uses a representation of tilings by membranes embedded in a higher dimensional hypercubic lattice it is illustrated in the case of 60 degree rhombus tilings

analysis of an isotropic finite wedge under antiplane deformation the antiplane deformation of an isotropic wedge with finite radius is studied in this paper depending upon the boundary data prescribed on the circular segment of the wedge traction or displacement two problems are analysed in each problem three different cases of boundary conditions on the radial edges are considered the radial boundary data are traction displacement displacement displacement and traction traction the solution of governing differential equations is accomplished by means of finite mellin transforms the closed form solutions are obtained for displacement and stress fields in the entire domain the geometric singularities of stress fields are identical to those cited in the literature however in displacement displacement case under certain representation of boundary condition another type of singularity has been observed

a geometric framework for the numerical study of singular points while bifurcation theory has developed rapidly in recent years there appears to be a need for a tighter framework for the numerical analysis of bifurcation problems this paper presents such a mathematical framework for the numerical study of the bifurcation phenomena associated with a parameter dependent equation f z lambda 0 the presentation draws from differential geometry and singularity theory and provides a basis for various numerical methods used to detect and compute certain types of bifurcation points

on the numerical solution of the euler lagrange equations the paper presents a new approach to the numerical solution of the euler lagrange equations based upon the reduction of the problem to a second order ordinary differential equation ode on the constraint manifold the algorithm guarantees that the constraints are automatically satisfied and requires a minimal number of evaluations of second order derivative terms in fact second order derivatives are involved only through the second fundamental tensor of the constraint manifold this tensor may be computed either explicitly when second derivatives are available or via an approximation procedure with excellent accuracy examples are given along with comparisons with state of the art software

on the computation of multi dimensional solution manifolds of parametrized equations a new algorithm is presented for computing vertices of a simplicial triangulation of thep dimensional solution manifold of a parametrized equationf x 0 wheref is a nonlinear mapping fromr n tor m p n m 1 an essential part of the method is a constructive algorithm for computing moving frames on the manifold that is of orthonormal bases of the tangent spaces that vary smoothly with their points of contact the triangulation algorithm uses these bases together with a chord form of the gauss newton process as corrector to compute the desired vertices the jacobian matrix of the mapping is not required at all the vertices but only at the centers of certain local triangulation patches several numerical examples show that the method is very efficient in computing triangulations even around singularities such as limit points and bifurcation points this opens up new possibilities for determining the form and special features of such solution manifolds

a geometrical method for the approximation of invariant tori we consider a numerical method based on the so called orthogonality condition for the approximation and continuation of invariant tori under flows the basic method was originally introduced by moore computation and parameterization of invariant curves and tori siam j numer anal 15 1991 245 263 but that work contained no stability or consistency results we show that the method is unconditionally stable and consistent in the special case of a periodic orbit however we also show that the method is unstable for two dimensional tori in three dimensional space when the discretization includes even numbers of points in both angular coordinates and we point out potential difficulties when approximating invariant tori possessing additional invariant sub manifolds e g periodic orbits we propose some remedies to these difficulties and give numerical results to highlight that the end method performs well for invariant tori of practical interest

derivatives of tangential stiffness matrices for equilibrium path descriptions the paper describes how several procedures based on expressions from analytical elastic stability theory are introduced as numerical tools in a general finite element program for geometrically non linear structural analysis especially is discussed how derivatives of the tangential stiffness matrix can be utilized in several contexts in the solution algorithm these include improved predictions for the step wise solution of equilibrium states identification of critical points and accurate descriptions of initial post bifurcation behaviour for two plane beam and bar elements formulations have been developed giving analytical expressions for these derivatives the corresponding numerical approximations needed in other element types are also discussed the paper discusses the relative efficiency of higher order predictions in relation to these different element types and different solution strategies some numerical examples showing different types of behaviour are analysed and discussed

derivative and proportional state feedback for linear descriptor systems with variable coefficients abstract we study linear descriptor systems with rectangular variable coefficient matrices using local and global equivalence transformations we introduce normal and condensed forms and get sets of characteristic quantities these quantities allow us to decide whether a linear descriptor system with variable coefficients is regularizable by derivative and or proportional state feedback or not regularizable by feedback means for us that there exists a feedback which makes the closed loop system uniquely solvable for every consistent initial vector

on the computation of impasse points of quasi linear differential algebraic equations abstract we present computational algorithms for the calculation of impasse points and higher order singularities in quasilinear differential algebraic equations our method combines a reduction step transforming the dae into a singular ode with an augmentation procedure inspired by numerical bifurcation theory singularities are characterized by the vanishing of a scalar quantity that may be monitored along any trajectory two numerical examples with physical relevance are given

on a computational method for the second fundamental tensor and its application to bifurcation problems an algorithm is presented for the computation of the second fundamental tensorv of a riemannian submanifoldm ofr n fromv the riemann curvature tensor ofm is easily obtained moreover v has a close relation to the second derivative of certain functionals onm which in turn provides a powerful new tool for the computational determination of multiple bifurcation directions frequently in applications thed dimensional manifoldm is defined implicitly as the zero set of a submersionf onr n in this case the principal cost of the algorithm for computingv p at a given pointp m involves only the decomposition of the jacobiandf p off atp and the projection ofd d 1 neighboring points ontom by means of a local iterative process usingdf p several numerical examples are given which show the efficiency and dependability of the method

estimating the asymptotics of solid partitions we study the asymptotic behavior of solid partitions using transition matrix monte carlo simulations if p 3 n denotes the number of solid partitions of an integer n we show that lim n rightarrow infty n 3 4 log p 3 n sim 1 822 pm 0 001 this shows clear deviation from the value 1 7898 attained by macmahon numbers m 3 n that was conjectured to hold for solid partitions as well in addition we find estimates for other sub leading terms in log p 3 n in a pattern deviating from the asymptotics of line and plane partitions we need to add an oscillatory term in addition to the obvious sub leading terms the period of the oscillatory term is proportional to n 1 4 the natural scale in the problem this new oscillatory term might shed some insight into why partitions in dimensions greater than two do not admit a simple generating function

on the sign characteristics of hermitian matrix polynomials the sign characteristics of hermitian matrix polynomials are discussed and in particular an appropriate definition of the sign characteristics associated with the eigenvalue infinity the concept of sign characteristic arises in different forms in many scientific fields and is essential for the stability analysis in hamiltonian systems or the perturbation behavior of eigenvalues under structured perturbations we extend classical results by gohberg lancaster and rodman to the case of infinite eigenvalues we derive a systematic approach studying how sign characteristics behave after an analytic change of variables including the important special case of mobius transformations and we prove a signature constraint theorem we also show that the sign characteristic at infinity stays invariant in a neighborhood under perturbations for even degree hermitian matrix polynomials while it may change for odd degree matrix polynomials we argue that the non uniformity can be resolved by introducing an extra zero leading matrix coefficient

solution of the dimer problem on a hexagonal lattice with boundary the dimer problem is solved exactly for a hexagonal lattice with general boundary using a known generating function from the theory of partitions it is shown that the leading term in the entropy depends on the shape of the boundary

binomial determinants paths and hook length formulae abstract we give a combinatorial interpretation for any minor or binomial determinant of the matrix of binomial coefficients this interpretation involves configurations of nonintersecting paths and is related to young tableaux and hook length formulae

piecewise linear methods for nonlinear equations and optimization piecewise linear methods had their beginning in the mid 1960s with lemke s algorithm for calculating solutions to linear complementarity problems in the 1970s and 1980s activity moved on to computing fixed points of rather general maps and economic equilibria more recently they have been used to approximate implicitly defined manifolds with applications being made to computer graphics and approximations of integral over implicitly defined manifolds in this paper we present the basic ideas of piecewise linear algorithms and a selection of applications further references to the literature on piecewise linear algorithms are indicated

an algorithm for piecewise linear approximation of an implicitly defined manifold a simplicial method is used to approximate the solution manifold to a system of nonlinear equations h x theta where h mathbb r n k to mathbb r n the method begins at a point x 0 in the solution set where the derivative dh x 0 is of full rank given any varepsilon 0 a piecewise linear manifold is constructed along which h x infty varepsilon an algorithm is presented to carry out this construction in an efficient fashion

manpak a set of algorithms for computations on implicitly defined manifolds abstract mathematical models often involve differentiable manifolds that are implicitly defined as the solution sets of systems of nonlinear equations the resulting computational tasks differ considerably from those arising for manifolds defined in parametric form here a collection of algorithms is presented for performing a range of essential tasks on general implicitly specified submanifolds of a finite dimensional space this includes algorithms for determining local parametrizations and their derivatives and for evaluating quantities related to the curvature with sensitivity measures the methods have been implemented as a fortran 77 package called manpak

continuation methods in semiconductor device simulation predictor corrector continuation methods for characterizing the voltage current v i behavior of semiconductor devices are presented numerical simulations of some complex cmos structures demonstrate the efficacy of continuation methods in particular it is possible to accurately determine the limit points of certain v i curves corresponding to latchup triggering and holding points continuation techniques coupled with grid adaption provide substantial improvement in computational efficiency over previous approaches and are well suited to deal with multivalued current responses

a note on the computation of an orthonormal basis for the null space of a matrix a highly regarded method to obtain an orthonormal basis z for the null space of a matrix a t is the qr decomposition of a where q is the product of householder matrices in several optimization contexts a x varies continuously with x and it is desirable the z x vary continuously also in this note we demonstrate that the standard implementation of the qr decomposition does not yield an orthonormal basis z x whose elements vary continuously with x we suggest three possible remedies

a new algorithm for two dimensional numerical continuation abstract consider a smooth mapping f from r n 2 into r n such a mapping can be interpreted as a system of n functions in n 2 unknowns the zero set of the mapping is typically a smooth two dimensional manifold surface embedded in the ambient space of dimension n 2 our paper describes a numerical algorithm for generating a representation of such a surface which can be displayed graphically the defining equations are given in implicit form not only is such a formulation convenient but it allows surfaces with folds in which there is not a functional relationship between the n state space variables and the two distinguished continuation parameters the algorithm is built as a layer on top of an existing one dimensional continuation solver and has been successfully applied to problems with a high dimensional state space using efficient sparse matrix techniques applications in electronic circuit simulation are described

equilibrium subsets for multi parametric structural analysis the paper discusses multi parametric formulations for structural equilibrium problems by seeking subsets of equilibrium states which also fulfil extra conditions more detailed information on structural behaviour can be obtained these conditions can be related to analyses of e g critical behaviour imperfections or optimisation the paper shows a general problem setting and some interesting augmentations a solution method for the special case of one dimensional equilibrium subsets is described the algorithm is a generalisation of a path following algorithm where new methods are developed for the evaluation of the path tangent and for the isolation of special solution points examples are used to show the properties and the possibilities of the ideas

local bisection refinement for n simplicial grids generated by reflection a simple local bisection refinement algorithm for the adaptive refinement of n simplicial grids is presented the algorithm requires that the vertices of each simplex be ordered in a special way re

smooth factorizations of matrix valued functions and their derivatives in this paper we study the numerical factorization of matrix valued functions in order to apply them in the numerical solution of differential algebraic equations with time varying coefficients the main difficulty is to obtain smoothness of the factors and a numerically accessible form of their derivatives we show how this can be achieved without numerical differentiation if the derivative of the given matrix valued function is known these results are then applied in the numerical solution of differential algebraic riccati equations for this a numerical algorithm is given and its properties are demonstrated by a numerical example

numerical continuation methods a perspective in this historical perspective the principal numerical approaches to continuation methods are outlined in the framework of the mathematical sources that contributed to their development notably homotopy and degree theory simplicial complexes and mappings submanifolds defined by submersions and singularity and foldpoint theory

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

on the sensitivity of solutions of parameterized equations the sensitivity of a solution of a parameterized equation f z lambda 0 with respect to the parameter vector lambda is usually defined as the change of the state z in dependence of lambda in other words for any solution expressible in the form z lambda lambda with some smooth function z z lambda the sensitivity is the derivative dz lambda typically the solutions form a manifold m in the product of 1the state space and the parameter space and this sensitivity is available only at those points of m where the parameters can be used to define a local coordinate system this paper introduces a general sensitivity concept which applies at all solutions on m and which includes the earlier definition some general geometric interpretations of the new measure are presented and it is shown that the sensitivity analysis can be easily integrated into the solution process the theory also suggests the introduction of a readily computable second order sensitivity measure reflectin

nonintersecting paths pfaffians and plane partitions gessel and viennot have developed a powerful technique for enumerating various classes of plane partitions the purpose of this article is to show by similar means that one may use pfaffiants to enumerate configurations of nonintersecting paths in which the initial and or terminal vertices of the paths are allowed to vary over specified regions of the digraph this leads to the possibility of enumerating classes of plane partitions in which the shape is allowed to vary whereas the previous applications of gessel and viennot were largely confined to plane partitions of a given shape

a formula for the number of tilings of an octagon by rhombi we propose the first algebraic determinantal formula to enumerate tilings of a centrosymmetric octagon of any size by rhombi this result uses the gessel viennot technique and generalizes to any octagon a formula given by elnitsky in a special case

on the theory and error estimation of the reduced basis method for multi parameter problems abstract in an earlier paper zamm 63 1983 21 j p fink and w c rheinboldt developed a priori local error estimates for the scalar parameter case of the reduced basis method by considering the method in a differential geometric setting here it is shown that an analogous setting can be used for the analysis of the method applied to problems with a multidimensional parameter vector and that this leads to a corresponding local error theory also in this general case

unitary integrators and applications to continuous orthonormalization techniques in this paper the issue of integrating matrix differential systems whose solutions are unitary matrices is addressed such systems have skew hermitian coefficient matrices in the linear case and a

characterization and computation of generalized turning points a minimal set of augmenting equations is given so that general parameter dependent problems together with these equations will be nonsingular at points of simple rank deficiency applications to de

numerical solution of differential algebraic riccati equations abstract we consider riccati matrix differential algebraic equations arising from singular or descriptor control problems we discuss the solvability of such equations under different conditions in order to apply numerical methods for differential algebraic systems one has to transform the equation unfortunately these equations then have a linear part which is described by a singular pencil and thus the usual integration methods do not apply under some conditions which we discuss these singularities can be removed by a preprocessing algorithm and the equation can then be solved by well known methods for differential algebraic systems like dassl of l petzold or limex of deuflhard hairer and zugck we discuss the numerical procedures and give some numerical examples

analysis of perfectly bonded wedges and bonded wedges with an interfacial crack under antiplane shear loading antiplane shear deformation of perfectly bonded wedges as well as bonded wedges with an interface crack are studied in this paper the solution of governing differential equations is accomplished by means of the mellin transform for two edge bonded isotropic wedges with perfect bonding along the common edge closed form solutions are obtained for stress fields and analytical relations are given for the strength of singularity at the apex however for bonded wedges with an interfacial crack first it is necessary to express the traction free condition of the crack faces in the form of a singular integral equation which is done in this paper by describing an exact analytical method the resultant singular integral equations are then solved analytically and the obtained results including the stress intensity factors at the crack tips are plotted a comparison of the results in the special cases shows a complete agreement with those cited in the literature however when the crack tip coincides with the wedge apex a strength of singularity of unity has been observed

solution manifolds and submanifolds of parametrized equations and their discretization errors the paper concerns solution manifolds of nonlinear parameterdependent equations 1 f u y0 involving a fredholm operatorf between infinite dimensional banach spacesx z andy and a finitedimensional parameter space differntial geometric ideas are used to discuss the connection between augmented equations and certain onedimensional submanifolds produced by numerical path tracing procedures then for arbitrary finite dimension of estimates of the error between the solution manifold of 1 and its discretizations are developed these estimates are shown to be applicable to rather general nonlinear boundaryvalue problems for partial differential equations

equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings the two main techniques for the generation of quasiperiodic tilings de bruijn s grid method 1981 and the projection formalism are generalised a very broad class of quasiperiodic tilings is obtained in this way the two generalised methods are shown to be equivalent the standard calculation of fourier spectra is extended to the whole general class of tilings various applications are discussed

numerical computation of an analytic singular value decomposition of a matrix valued function this paper extends the singular value decomposition to a path of matricese t an analytic singular value decomposition of a path of matricese t is an analytic path of factorizationse t x t s t y t t wherex t andy t are orthogonal ands t is diagonal to maintain differentiability the diagonal entries ofs t are allowed to be either positive or negative and to appear in any order this paper investigates existence and uniqueness of analytic svd s and develops an algorithm for computing them we show that a real analytic pathe t always admits a real analytic svd a full rank smooth pathe t with distinct singular values admits a smooth svd we derive a differential equation for the left factor develop euler like and extrapolated euler like numerical methods for approximating an analytic svd and prove that the euler like method converges

continuous bordering of matrices and continuous matrix decompositions let cal m n be the set of all real n times n matrices of rank ge n 1 we prove that for n ge 2 there are no continuous vector fields l r cal m n rightarrow bbb r n such that the bordered matrix is regular for all a in cal m n this result has some relevance for the numerical analysis of steady state bifurcation as a by product we show that there is no nonvanishing continuous vector field u cal m n n 1 rightarrow bbb r n with au a 0 for all a in cal m n n 1 where cal m n n 1 subset cal m n is the set of all matrices of rank deficiency one this implies that there is no singular value decomposition of a depending continuously on a in any matrix set which contains cal m n n 1 as another application we prove that in general there is no global analytic singular value decomposition for analytic matrix valued functions of more than one real variable

inverse eigenvalue problems a collection of inverse eigenvalue problems are identified and classified according to their characteristics current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper this exposition also reveals many open questions that deserve further study an extensive bibliography of pertinent literature is attached

fixed boundary octagonal random tilings a combinatorial approach some combinatorial properties of fixed boundary rhombus random tilings with octagonal symmetry are studied a geometrical analysis of their configuration space is given as well as a description in terms of discrete dynamical systems thus generalizing previous results on the more restricted class of codimension one tilings in particular this method gives access to counting formulas which are directly related to questions of entropy in these statistical systems methods and tools from the field of enumerative combinatorics are used

enumeration of octagonal tilings random tilings are interesting as idealizations of atomistic models of quasicrystals and for their connection to problems in combinatorics and algorithms of particular interest is the tiling entropy density which measures the relation of the number of distinct tilings to the number of constituent tiles tilings by squares and 45 rhombi receive special attention as presumably the simplest model that has not yet been solved exactly in the thermodynamic limit however an exact enumeration formula can be evaluated for tilings in finite regions with fixed boundaries we implement this algorithm in an efficient manner enabling the investigation of larger regions of parameter space than previously were possible our new results appear to yield monotone increasing and decreasing lower and upper bounds on the fixed boundary entropy density that converge toward s 0 36021 3

the computation of simplicial approximations of implicitly defined p dimensional manifolds abstract a method is presented for the computation of a simplicial approximation covering a subset m 0 of a p dimensional manifold m in r n defined implicitly as the solution set of a nonlinear system f x 0 of n p equations in n unknowns n p 2 the subset m 0 is either the intersection of m with some polyhedral domain in r n and is assumed to be bounded and nonempty or it is an entire manifold component which has no boundary such as a sphere the method represents a generalization of a simplicial approximation process for the case p 2 developed earlier by the author and w rheinboldt

on the computation of multiple bifurcations with multiple parameters and symmetry this paper concerns numerical methods for the determination of bifurcation points of certain steady state multiparameter problems in the presence of symmetries the approach is based on the fact that under general conditions the solution set forms a manifold in the space of all state and parameter variables the reduced manifold with respect to some subsymmetry is introduced methods are presented for the local computation of the submanifold of bifurcation points of the same symmetry