
Analysis of vibration of rotors with a thick rigid disc on the overhanging end supported by hydrodynamic bearings and loaded by uncertain unbalance effects
Zapoměl, Jaroslav ; Ferfecki, P. ; Molčan, M. ; Kozánek, Jan
Rotors of some rotating machines consist of a shaft and a rigid disc attached to its overhanging end. Because of manufacturing and assembling inaccuracies, the disc is unbalanced. In general case, the disc principal axis of inertia deviates from the shaft axis (both axes are skew lines) and the disc center of gravity is not situated on the shaft center line. Then the disc inertia effects are the source inducing the rotor lateral vibration. As magnitude of parameters specifying the disc unbalance is uncertain, application of an adequate method is required to perform the analysis. In addition, if the rotor is supported by hydrodynamic bearings, their nonlinear properties lead to rising complexity of the system vibration and to reducing its predictability. In the presented paper, the motion equation of a rigid disc has been derived. The shaft was represented by a beam body that was discretized into finite elements. The stiffness and damping parameters of the hydrodynamic bearings were linearized in vicinity of the rotor equilibrium position. Then vibration of the rotor is governed by a set of linearized motion equations. The fuzzy numbers were applied to consider uncertainty of the disc unbalance parameters. The presented procedure provides the approach to computational analysis of rotating machines with the rotor having a disc attached to its overhanging end and the geometrical, mechanical, or technological parameters of which have uncertain values.


Mathematical modelling and computational methods in applied sciences and engineering  Modelling 2019
Blaheta, Radim ; Starý, Jiří ; Sysala, Stanislav
Modelling 2019 is an international conference on Mathematical Modelling and Computational Methods in Applied Sciences and Engineering held in\nOlomouc, Czech Republic, in September 16  20, 2019. It aims to be a forum for an exchange of ideas, insights and experiences in different areas\nof mathematical modelling. It includes the fundamental formulation and analysis of mathematical models, the development of numerical methods and exploitation of capabilities of the contemporary highperformance computers or applications of mathematical modelling.\nThis conference belongs to a series of conferences previously held in Roznov in 2014 and 2009, in Pilsen in 2005 and 2001, and in Prague in 1998 and 1994. During the period of 25 years, the focus of the conference has been substantially enlarged. Besides the topics aiming at the development\nof numerical methods and analysis of mathematical models described by the partial differential equations, the conference relates to the inverse\nproblems, quantification of uncertainties in the input data, machine learning and exploitation of highperformance computing systems of petaflops and preexaflops performance. Increased attention is devoted to challenging industrial problems and collaboration with industry.


Reliable computation and local mesh adaptivity in limit analysis
Sysala, Stanislav ; Haslinger, Jaroslav ; Repin, S.
The contribution is devoted to computations of the limit load for a perfectly plastic model with the von Mises yield criterion. The limit factor of a prescribed load is defined by a specific variational problem, the socalled limit analysis problem. This problem is solved in terms of deformation fields by a penalization, the finite element and the semismooth Newton methods. From the numerical solution, we derive a guaranteed upper bound of the limit factor. To achieve more accurate results, a local mesh adaptivity is used.


Algebraic preconditioning for BiotBarenblatt poroelastic systems
Blaheta, Radim ; Luber, Tomáš
Poroelastic systems describe fluid flow through porous medium coupled with deformation of the porous matrix. In this paper, the deformation is described by linear elasticity, the fluid flow is modelled as Darcy flow. The main focus is on the BiotBarenblatt model with double porosity/double permeability flow, which distinguishes flow in two regions considered as continua. The main goal is in proposing block diagonal preconditionings to systems arising from the discretization of the BiotBarenblatt model by a mixed finite element method in space and implicit Euler method in time and estimating the condition number for such preconditioning. The investigation of preconditioning includes its dependence on material coefficients and parameters of discretization.


Implicit constitutive solution scheme for MohrCoulomb plasticity
Sysala, Stanislav ; Čermák, M.
This contribution summarizes an implicit constitutive solution\nscheme of the elastoplastic problem containing the MohrCoulomb yield cri\nterion, a nonassociative \now rule, and a nonlinear isotropic hardening. The\npresented scheme builds upon the subdifferential formulation of the \now rule\nleading to several improvements. Mainly, it is possible to detect a position\nof the unknown stress tensor on the MohrCoulomb pyramid without blind\nguesswork. Further, a simplifed construction of the consistent tangent opera\ntor is introduced. The presented results are important for an efficient solution\nof incremental boundary value elastoplastic problems.


Overlapping domain decomposition preconditioners for elliptic and parabolic problems in primal and mixed form
Blaheta, Radim
In this lecture, we concern the numerical solution of PDE problems and describe overlapping domain decomposition, which provides a tool for the construction of parallelizable Schwarz type iterative solvers and preconditioners. The idea of using overlapping domain decomposition goes back to Schwarz alternating method from 1870, see [1]. The analysis of this alternating iterative method was evolved by great mathematicians, see e.g. S.L. Sobolev (1936), R. Courant and D. Hilbert (1937), S.G. Michlin (1951), M. Práger (1958), I. Babuška (1958), F.E. Browder (1958). The use of overlapping domain decomposition for parallel computations started in the late eighties in the work of M. Dryja and O. Widlund [4], P.L. Lions [5, 6], S. Nepomnyaschikh [7] and others and continue up to the present days. The origin of the alternating Schwarz method is nicely described in [8].


SNA´17  Seminar on Numerical Analysis
Blaheta, Radim ; Starý, Jiří ; Sysalová, Dagmar
Seminar on Numerical Analysis 2017 (SNA'17) is a continuation in a series of SNA events held in different places in the Czech Republic and organized alternatively by Ostrava and Prague institutions. TheSNA'17isorganizedbytheInstituteofGeonicsoftheCASincollaborationwith VŠB  Technical University of Ostrava and IT4Innovations National Supercomputing Centre. Conference location is one of the lecture rooms in New Aula of the VŠBTU Ostrava. Let us note that SNA 2016 was reshaped to EMS School in Applied Mathematics (ESSAM) devoted to mathematical modelling, numerical analysis and scientific computing. The SNA'17 is turning back to more traditional winter event. It provides opportunity for meeting and mutual information of the community working in computational mathematics and computer science, but an important part of SNA is devoted to the Winter School with tutorial lectures focused on selected important topics within the scope of numerical methods and modelling.


Bifurcations in contact problems with Coulomb friction
Ligurský, Tomáš ; Renard, Y.
To explore the bifurcation in this contact problem, we have taken uniform meshes with 4096, 16384, 65536 and 262144 triangles. We shall show that the bifurcation behaviour is more complex here. Branches 1 and 4 approach one another for finer meshes, and they disappear both for the finest mesh. Nevertheless, regarding the branching of the corresponding contact problem with forces h = (h1,h2) over the plane h1h2, one can find it stable and convergent, again. \n
