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Application of Methods for Unconstrained Optimization in Computation of Normal Contact Vector
Kopačka, Ján ; Gabriel, Dušan ; Plešek, Jiří ; Ulbin, M.
The stability of the contact algorithm using the penalty method is significantly affected by choosing of the penalty function. The penalty function is defined like a magnitude of the penetration vector multiplied by the users-defined constant - the penalty parameter. The penetration vector is obtained by solution of the minimum distance problem between the node/Gaussian integration point and the segment of the element. For a general quadrilateral contact segment this task leads to the system of two nonlinear equations. It is shown that the popular Newton-Raphson method is inadvisable for this problem. In this paper, alternative methods like quasi-Newton methods, gradient methods and the simplex method are presented. Especial attention is put on the line-search method that is crucial for a general success of quasi-Newton methods as well as gradient methods. All mentioned methods are tested by means of numerical example, which involves bending of two rectangular plates over a cylinder.
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Analysis of classical and spectral finite element spatial discretization in one-dimensional elastic wave propagation
Kolman, Radek ; Plešek, Jiří ; Okrouhlík, Miloslav ; Gabriel, Dušan
The spatial discretization of continuum by finite element method introduces the dispersion error to numerical solutions of stress wave propagation. For higher order finite elements there are the optical modes in the spectrum resulting in spurious oscillations of stress and velocity distributions near the sharp wavefront. Spectral finite elements are of h-type finite element, where nodes have special positions along the elements corresponding to the numerical quadrature schemes, but the displacements along element are approximated by Lagrangian interpolation polynomials. In this paper, the classical and Legendre and Chebyshev spectral finite elements are tested in the one-dimensional wave propagation in an elastic bar.
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Přehled metod pro lokální vyhledávání kontaktu
Kopačka, Ján ; Gabriel, Dušan ; Plešek, Jiří
Lokální vyhledávání kontaktu je procedura kontaktního algoritmu sloužící k výpočtu vektoru penetrace mezi tzv. master a slave segmenty kontaktních ploch. Existuje několik technik jak toho docílit. V této práci je zmíněný problém řešen minimalizací funkce vzdálenosti mezi master segmentem a integračním bodem slave segmentu. Pro kvadratické isoparametrické prvky tato formulace vede na soustavu dvou nelineárních rovnic. K jejich řešení bylo použito několik metod (Newtonova-Raphsonova metoda, metoda projekce nejmenších čtverců, metoda největšího spádu, Broydenova metoda, BFGS metoda, DFP metoda a simplexová metoda), jejichž efektivita je v závěru porovnána.
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