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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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Review of methods for local contact search
Kopačka, Ján ; Gabriel, Dušan ; Plešek, Jiří
The local contact search is a numerical procedure of a contact algorithm which is used for computing of a penetration vector between so-called master and slave segments of contact surfaces. There are several approaches. In this paper the minimum distance problem between the master segment and the integration point of the slave segment is used. For quadrilateral elements this approach leads to the system of two nonlinear equations. For its solution several methods was tested (Newton-Raphson method, least square projection, method of steepest descent, Broyden's method, BFGS method, DFP method and simplex method) and efficiency was compared.
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Modelling of nanoindentation of modern Nickel-based superalloys for turbofans
Hrubý, Zbyněk ; Plešek, Jiří ; Tin, S.
Stress and strain distribution underneath various types of indentors – spherical, conical, and Berkovich – can be provided by the finite element method. In the presented work, indentation of isotropic aluminium is introduced as a benchmark problem, in which plasticity and contact algorithms are tested. The knowledge obtained in this way passes on to the real-life indentation processes involving orthotropic materials such as FCC metals (Ni-based alloys) in the context of nonlinear continuum and finite strain elasto-plasticity, including homogenization approach on the material microscale.
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Stability Analysis of Plane Serendipity Finite Element for Explicit Linear Elastodynamics
Kolman, Radek ; Plešek, Jiří ; Gabriel, Dušan
The central difference method is widely used for the numerical solution of the transient elastodynamics problems by the finite element method. The effectiveness of this explicit conditional stable direct time integration methods is limited by using diagonal mass matrix, which entails significant computational savings and storage advantages. However, for the serendipity type element the construction of such diagonalized matrices is not uniquely defined and various class of lumped mass matrices can be assembled. In this paper the stability analysis for the plane square serendipity finite element is performed for various class of lumped mass matrices.
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