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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 53 (1988), S. 165-181 
    ISSN: 0945-3245
    Keywords: AMS(MOS):65H10 ; CR: G1.5
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary 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.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Computing 36 (1986), S. 81-90 
    ISSN: 1436-5057
    Keywords: 65H10 ; Key words ; Systems of nonlinear equations ; Newton's method ; monotone convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Neue hinreichende Bedingungen für die monotone Konvergenz der Newton-Methode für die Lösung nichtlinearer Gleichung werden gegeben. Diese Bedingungen sind affin-invariant und allgemeiner als die Voraussetzungen des Satzes von Baluev.
    Notes: Abstract New sufficient conditions for the monotone convergence of Newton's method for solving nonlinear systems of equations are given. These conditions are affine-invariant and less restrictive than the hypothesis of Baluev's theorem.
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 25 (1988), S. 269-282 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The goal of mathematical modelling of sheet metal forming processes is to provide predictive tools for use in the design of stamping processes and the selection of sheet materials. Most current approaches to finite element modelling of large deformation, elastic-plastic sheet metal forming problems use a rate form of the virtual work (equilibrium) equations, and a single-field finite element representation of the displacement components. Called the incremental method, this approach does not produce approximations which satisfy the discrete equilibrium equations at all times, and consequently it demands small time steps to insure stability and numerical accuracy. This paper describes a variant of the mixed method in which displacements, stresses, effective strain and pressures are all given separate finite element representations. The equilibrium equations in non-rate form are discretized to produce a system of algebraic equations which are coupled with the constitutive equations and then integrated using state-of-the-art numerical software. When used to model rate sensitive sheet materials in hydrostatic bulging, plane strain punch stretching and hemispherical punch stretching, the new approach proved to be between 6 and 26 times as fast as the old incremental method.
    Additional Material: 9 Ill.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 23 (1986), S. 847-862 
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Many current approaches to finite element modelling of large deformation elastic - plastic forming problems use a rate form of the virtual work (equilibrium) equations, and a finite element representation of the displacement components. Called the incremental method, this approach produces a three-field formulation in which displacements, stresses and effective strain are dependent variables. Next, the formulation is converted to a one-field displacement formulation by an algebraic time discretization which uses a low order explicit time-stepping procedure to integrate the equations. This approach does not produce approximations which satisfy the discrete equilibrium equations at all times and, moreover, the advantage of the single-field algebraic formulation is realized at the expense of very small time steps needed to produce stability and accuracy in the numerical calculations.This paper describes a variant of the mixed method in which all three field variables (displacements, stresses and effective strain) are given finite element representations. The discrete equilibrium equations then generate a nonlinear system of algebraic equations whose solutions represent a manifold, while the constitutive equations form a system of ordinary differential equations. A commercially available, variable time step/variable order code is then used to integrate this differential/algebraic system. When applied to the problem of hydrostatic bulging of a membrane, the new approach requires far less computer time than the incremental method.
    Additional Material: 6 Ill.
    Type of Medium: Electronic Resource
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