A family of higher order mixed finite element methods for plane elasticity

Summary The Dirichler problem for the equations of plane elasticity is approximated by a mixed finite element method using a new family of composite finite elements having properties analogous to those possessed by the Raviart-Thomas mixed finite elements for a scalar, second-order elliptic equation. Estimates of optimal order and minimal regularity are derived for the errors in the displacement vector and the stress tensor inL ²(), and optimal order negative norm estimates are obtained inH ^s () for a range ofs depending on the index of the finite element space. An optimal order estimate inL () for the displacement error is given. Also, a quasioptimal estimate is derived in an appropriate space. All estimates are valid uniformly with respect to the compressibility and apply in the incompressible case. The formulation of the elements is presented in detail.This work was performed while Professor Arnold was a NATO Postdoctoral Fellow     

Maintaining the positive definiteness of the matrices in reduced secant methods for equality constrained optimization

We propose an algorithm for minimizing a functionf on ⁿ in the presence ofm equality constraintsc that locally is a reduced secant method. The local method is globalized using a nondifferentiable augmented Lagrangian whose decrease is obtained by both a longitudinal search that decreases mainlyf and a transversal search that decreases mainly c. Our main objective is to show that the longitudinal path can be designed to maintain the positive definiteness of the reduced matrices by means of the positivity of _k ^T _k , where _k is the change in the reduced gradient and _k is the reduced longitudinal displacement.Work supported by the FNRS (Fonds National de la Recherche Scientifique) of Belgium.     

Optimal L∞-error estimates for nonconforming and mixed finite element methods of lowest order

Summary For second order linear elliptic problems, it is proved that theP ₁-nonconforming finite element method has the sameL -asymptotic accuracy as theP ₁-conforming one. This result is applied to derive optimalL -error estimates for both the displacement and the stress fields of the lowest order Raviart-Thomas mixed finite element method, and a superconvergence result at the barycenter of each element.Performed in the research program of Istituto di Analisi Numerica of C.N.R. of PaviaPartially supported by MPI, GNIM of CNR, ItalySupported by Consejo Nacional de Investigaciones Cientificas y Técnicas, Argentina     

A Levenberg–Marquardt Scheme for Nonlinear Image Registration

This paper presents a Levenberg—Marquardt scheme to obtain a displacement vector field u(x)=(u ₁(x),u ₂(x)) ^t , which matches two images recorded with the same imaging machinery. The displacement vector should transform the image location x=(x ₁,x ₂) ^t of an image T, such that the grey level are equal to another image R. The so-called mono-modal image registration problem leads to minimize the nonlinear least squares functional D(u(x))=R(x)–T(x–u(x))².To apply the Levenberg—Marquardt method, we replace the nonlinear functional D by its linearization around a current approximation. The resulting quadratic minimization problem is ill-posed, due to the fact that determining the unknown components of the displacements merely from the images is an underdetermined problem. We use an auxiliary Lagrange term borrowed from linear elasticity theory, which incorporates smoothness constraints to the displacement field. Finally, numerical experiments demonstrate the robustness and effectiveness of the proposed approach.This revised version was published online in October 2005 with corrections to the Cover Date.     

Large deflections of eccentrically loaded arches

Zusammenfassung Auf Grund der Hypothesen von Ebenbleiben und Normalität der Querschnitte werden die Differentialgleichungen der nichtlinearen Theorie der Bogenträger abgeleitet und im Falle des schlanken, durch Einzellasten belasteten Kreisbogenträgers mit undehnbarer Mittellinie auf die Form der Pendelgleichung gebracht. Diese Gleichung wird dann benutzt, um die grossen Durchbiegungen und die Spannungsresultierenden eines Zweigelenkkreisbogens, der durch eine lotrechte exzentrische Einzellast belastet wird, zu berechnen. In der Nähe der kritischen Last bewirken kleine Exzentrizitäten bedeutende Grössenänderungen der Spannungsresultierenden und der Durchbiegungen.

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Notation A cross-sectional area of curved beam - a radius of centroidal circle - E modulus of elasticity - e eccentricity of the load (Fig. 2) - F an arbitrary function - H horizontal component of the internal forceR acting on a cross section of the arch rib (Fig. 2) - h _P horizontal displacement of the loadP (Fig. 2) - I moment of inertia of the cross-sectional area - k ² =4p ²/(1+4p ² sin²₀) - L span (distance between supports),L=2a sin - M internal bending couple (Figs. 1 and 2) - N internal normal tensile force (Figs. 1 and 2) - n distributed tangential load (Fig. 1) - P downward point load (Fig. 2) - p ² –R a ² /E I - Q internal shearing force (Figs. 1 and 2) - q distributed normal load (Fig. 1) - R internal resultant force (Fig. 2);R ²=H ²+V ²=N ²+Q ² - radius of curvature of the undeformed centroidal curve - s length along the unextended centroidal curve measured from the left support - length along the unextended centroidal curve measured from the right support - u tangential displacement component of the centroidal curve (Fig. 1) - V vertical component ofR (Fig. 2) - v _P vertical displacement of the loadP (Fig. 2) - w normal displacement component (Fig. 1) - x, y rectangular coordinates of the deformed left portion of the centroidal curve (Fig. 2) - Z - z normal distance (positive inward) from centroidal curve (Fig. 1) - half subtending angle of the arch (Fig. 2) - angle of rotation of the centroidal curve (Fig. 1) - extensional strain of the centroidal curve - ^z extensional strain of the linez=constant - y cos–x sin - angle between the tangent to the formed left portion of the centroidal curve and the horizontal (Fig. 2) - (u–w)/r, whereu=du/dø - angle betweenH andR - x cos+y sin - normal stress along the centroidal curve - ^z normal stress along the linez=constant - angle measured from the radius at the left support of the undeformed arch - (–)/2 (Fig. 2) - (+u)/r, where =d/dø A bar over a letter indicates that the entity pertains to the right portion of the arch. Asterisk indicates the deformed configuration. Primes indicate derivatives with respect to ø.     

Finite element method for incompressible miscible displacement in porous media

Under the assumptions of nonlinear finite element and t =o(h), Ewing and Wheeler discussed a Galerkin method for the single phase incompressible miscible displacement of one fluid by another in porous media. In the present paper we give a finite element scheme which weakens the t =o(h)-restriction to t =o(H ), 0 < 1/2. Furthermore, this scheme is suitable for both linear element and nonlinear element. We also derive the optimal approximation estimates for concentrationc, its gradient c and the gradient p of the pressurep.     

Computation of an Infinite Integral Using Romberg'S Method

A numerical computation in crystallography involves the integral g(a)=₀ ⁺[(exp ^x +exp^–x ) ^a –exp ^ax –exp^–ax ]dx, 0<a<2. A first approximation value for g(5/3)=4.45 has been given. This result has been obtained by a classical method of numerical integration. It has been followed in an other paper by a second one 4.6262911 obtained from a theoretical formula which seems to lead to a more reliable result. The difficulty when one wants to use a numerical method is the choice of parameters on which the method depends, in this case, the size of the integration interval for instance and the number of steps in Romberg's method. We present a new approach of numerical integration which dynamically allows to take into account both the round-off error and the truncation error and leads to reliable results for every value of a.     

Fast Preconditioned Iterative Methods for Convolution-Type Integral Equations

We consider solving the Fredholm integral equation of the second kind with the piecewise smooth displacement kernel x(t) + _j=1 ^m µ_j x(t – t _j) + ₀ k(t – s)x(s) ds = g(t), 0 t , where t _j (–, ), for 1 j m. The direct application of the quadrature rule to the above integral equation leads to a non-Toeplitz and an underdetermined matrix system. The aim of this paper is to propose a numerical scheme to approximate the integral equation such that the discretization matrix system is the sum of a Toeplitz matrix and a matrix of rank two. We apply the preconditioned conjugate gradient method with Toeplitz-like matrices as preconditioners to solve the resulting discretization system. Numerical examples are given to illustrate the fast convergence of the PCG method and the accuracy of the computed solutions.     

Sequential gradient-restoration algorithm for the minimization of constrained functions—Ordinary and conjugate gradient versions

The problem of minimizing a functionf(x) subject to the constraint (x)=0 is considered. Here,f is a scalar,x ann-vector, and aq-vector. Asequential algorithm is presented, composed of the alternate succession of gradient phases and restoration phases.In thegradient phase, a nominal pointx satisfying the constraint is assumed; a displacement x leading from pointx to a varied pointy is determined such that the value of the function is reduced. The determination of the displacement x incorporates information at only pointx for theordinary gradient version of the method (Part 1) and information at both pointsx and for theconjugate gradient version of the method (Part 2).In therestoration phase, a nominal pointy not satisfying the constraint is assumed; a displacement y leading from pointy to a varied point is determined such that the constraint is restored to a prescribed degree of accuracy. The restoration is done by requiring the least-square change of the coordinates.If the stepsize of the gradient phase is ofO(), then x=O() and y=O(²). For sufficiently small, the restoration phase preserves the descent property of the gradient phase: the functionf decreases between any two successive restoration phases.This research, supported by the NASA Manned Spacecraft Center, Grant No. NGR-44-006-089, and by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-828-67, is a condensation of the investigations reported in Refs. 1 and 2.     

A Translation and Rotation Invariant Gauss–Newton Like Scheme for Image Registration

A Gauss–Newton like method is considered to obtain a d–dimensional displacement vector field , which minimizes a suitable distance measure D between two images. The key to find a minimizer is to substitute the Hessian of D with the Sobolev-H²(Ω)^d norm for . Since the kernel of the associated semi-norm consists only of the affine linear functions we can show in this way, that the solution of each Newton step is a linear combination of an affine linear transformation and an affine-free nonlinear deformation. Our approach is based on the solution of a sequence of quadratic subproblems with linear constraints. We show that the resulting Karush–Kuhn–Tucker system, with a 3×3 block structure, can be solved uniquely and the Gauss–Newton like scheme can be separated into two separated iterations. Finally, we report on synthetic as well as on real-life data test runs. AMS subject classification (2000) 65F20, 68U10     

Conjugate-gradient method for computing the Moore-Penrose inverse and rank of a matrix

A conjugate-gradient method is developed for computing the Moore-Penrose generalized inverseA of a matrix and the associated projectors, by using the least-square characteristics of both the method and the inverseA . Two dual algorithms are introduced for computing the least-square and the minimum-norm generalized inverses, as well asA . It is shown that (i) these algorithms converge for any starting approximation; (ii) if they are started from the zero matrix, they converge toA ; and (iii) the trace of a sequence of approximations multiplied byA is a monotone increasing function converging to the rank ofA. A practical way of compensating the self-correcting feature in the computation ofA is devised by using the duality of the algorithms. Comparison with Ben-Israel's method is made through numerical examples. The conjugate-gradient method has an advantage over Ben-Israel's method.After having completed the present paper, the author received from Professor M. R. Hestenes his paper entitledPseudo Inverses and Conjugate Gradients. This paper treated the same subject and appeared in Communications of the ACM, Vol. 18, pp. 40–43, 1975.     

Der Quotienten-Differenzen-Algorithmus

Summary The quotient-difference (=QD) algorithm developed by the author may be considered as an extension ofBernoulli's method for solving algebraic equations. WhereasBernoulli's method gives the dominant root as the limit of a sequence of quotientsq ₁ ^(v) =s ₁ ^(v+1) /s ₁ ^(v) formed from a certain numerical sequences ₁ ^(v) , the QD-algorithm gives (under certain conditions) all the roots as the limits of similiar quotient sequencesq ^(v) =s ^(v+1) /s ^(v) . Close relationship exists between this method and the theory of continued fractions. In fact the QD-algorithm permits developing a function given in the form of a power series into a continued fraction in a remarkably simple manner.In this paper only the theoretical aspects of the method are discussed. Practical applications will be discussed later.     

Upper and lower bounds for the frequencies of rectangular free plates

Zusammenfassung Eine neuentwickelte Methode für untere Schranken und das Rayleigh-Ritzverfahren für obere Schranken werden von den Verfassern dazu angewandt, die Eigenfrequenzen der Schwingungen von dünnen gleichförmigen, rechteckigen Platten mit freien Rändern abzuschätzen. Die Anwendbarkeit des Verfahrens für untere Schranken wird hervorgehoben, und es werden Berechnungen für eine Symmetrieklasse von rechteckigen sowie für eine Unterklasse von quadratischen Platten angegeben. Die dadurch entstehenden Schranken führen zu einer Verbesserung bisher veröffentlichter Resultate und weisen auf die Brauchbarkeit der Methode für untere Schranken hin.

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Nomenclature The following symbols are not defined in the text ² Laplacian differential operator (= ²/x ² + ²/y ²) - ⁴ Biharmonic differential operator (= ⁴/x ⁴ + 2 ⁴/x ² y ² + ⁴/y ⁴) - u Generic name for displacement functions - Generic name for eigenvalues - a,b Plate side lengths - Circular frequency of free vibration - Mass density of plate material - h Plate thickness - D Plate flexural rigidity - First variation - d Element of surface area - _ij Kronecker's delta - {} Matrix or vector of the included elements The research reported in this article has been sponsored by the Department of the Navy under Contract NOw-62-0604-c with the Bureau of Naval Weapons.     

Eine einfache Verallgemeinerung der Rayleigh-Grenzschicht

Summary A simple generalization of the theory of the compressible boundary layer near an infinite flat plate to the case with suction or blowing out is given if at the timet=0 the plate is set into motion in its own plane with velocityu _w t ⁿ. The normal velocity at the wall shall vary with time according tov _wt ^–1/2. In that case one gets similar boundary layer profiles for allt>0, which can be reduced to the profiles without suction or blowing (v _w=0) by a simple parallel displacement and stretching of the coordinates. As an example the Rayleigh boundary layer (n=0,u _w=const) is discussed.     

Global optimization and stochastic differential equations

Let ⁿ be then-dimensional real Euclidean space,x=(x ₁,x ₂, ...,x _n)^{T n} , and letf: ⁿ R be a real-valued function. We consider the problem of finding the global minimizers off. A new method to compute numerically the global minimizers by following the paths of a system of stochastic differential equations is proposed. This method is motivated by quantum mechanics. Some numerical experience on a set of test problems is presented. The method compares favorably with other existing methods for global optimization.This research has been supported by the European Research Office of the US Army under Contract No. DAJA-37-81-C-0740.The third author gratefully acknowledges Prof. A. Rinnooy Kan for bringing to his attention Ref. 4.     

Quadrature method for singular integral equations on closed curves

Summary A method which combines quadrature with trigonometric interpolation is proposed for singular integral equations on closed curves. For the case of the circle, the present method is shown to be equivalent to the trigonometric -collocation method together with numerical quadrature for the compact term, and is shown to be stable inL ² provided the operatorA is invertible inL ². The results are extended to arbitraryC curves, to give a complete error analysis in the scale of Sobolev spacesH ^s . In the final section the case of a non-invertible operatorA is considered.     

A general descent framework for the monotone variational inequality problem

We present a framework for descent algorithms that solve the monotone variational inequality problem VIP _v which consists in finding a solutionv ^* _v satisfyings(v ^*)^T(v–v ^*)0, for allv _v. This unified framework includes, as special cases, some well known iterative methods and equivalent optimization formulations. A descent method is developed for an equivalent general optimization formulation and a proof of its convergence is given. Based on this unified logarithmic framework, we show that a variant of the descent method where each subproblem is only solved approximately is globally convergent under certain conditions.This research was supported in part by individual operating grants from NSERC.     

Total flux estimates for a finite element approximation of the Dirichlet problem using the boundary penalty method

Summary This paper considers a fully practical piecewise linear finite element approximation of the Dirichlet problem for a second order self-adjoint elliptic equation,Au=f, in a smooth region< ⁿ (n=2 or 3) by the boundary penalty method. Using an unfitted mesh; that is ^h , an approximation of with dist (, ^h )Ch ² is not in general a union of elements; and assuminguH ⁴ () we show that one can recover the total flux across a segment of the boundary of with an error ofO(h ²⁾. We use these results to study a fully practical piecewise linear finite element approximation of an elliptic equation by the boundary penalty method when the prescribed data on part of the boundary is the total flux.Supported by a SERC research studentship     

Random walks on generalized lattices

A generalized lattice is a graph on which the groupZ ^d acts almost transitively. The relations among various features of random walks on generalized lattices are studied. In particular we relate the mean displacement, the drift-freeness of the random walk and the existence of linear harmonic functions. Applications to recurrence criteria are given.     

On Birational Maps and Jacobian Matrices

One is concerned with Cremona-like transformations, i.e., rational maps from ⁿ to ^m that are birational onto the image Y ^m and, moreover, the inverse map from Y to ⁿ lifts to ^m . We establish a handy criterion of birationality in terms of certain syzygies and ranks of appropriate matrices and, moreover, give an effective method to explicitly obtaining the inverse map. A handful of classes of Cremona and Cremona-like transformations follow as applications.