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Numerical Investigations of the Errors of the Bubnov–Galerkin Method
Awrejcewicz J., Krysko V.A.
In Chap.
3
, many prior error estimates for the Bubnov–Galerkin method applied to coupled thermoelastic problems of shallow shells and plates were derived. The estimates related to the general case of a shell with a variable thickness subjected to time-varying mechanical and thermal loads possess an important theoretical meaning (they guarantee, for a wide class of problems, strong convergence of successive approximations to the exact solution with a velocity larger than the estimated velocity). However, in applications, when specific real problems have to be considered the estimates possess a more generalized meaning. This question is addressed, for instance, in the case of the estimates obtained in Sect.
3.3
for vibrations of a simply supported plate with constant thickness, subjected to mechanical and thermal loads that are constant in time. In this chapter, numerical results are given and the efficiency of the estimate used is verified.
Coupled Thermoelasticity and Transonic Gas Flow
Awrejcewicz J., Krysko V.A.
This chapter includes considerations of the coupled linear thermoelasticity of shallow shells and of a cylindrical panel within a transonic gas flow. First, the fundamental assumptions related to the stress–strain relation of the Timoshenko kinematic model are formulated, and then the differential equations are derived. Both the Timoshenko and the Kirchhoff–Love models are taken into account. Then the boundary and initial conditions are formulated. Next, an abstract Cauchy problem for a coupled system of two differential equations in a Hilbert space is considered. This includes the thermoelastic problems of shallow shells modelled by the Kirchhoff–Love and Timoshenko theories defined earlier. In Sect. 2.1.5, theorems related to the existence and uniqueness of a general, “classical” solution to the coupled abstract program are given, and then the corresponding theorems for coupled thermoelastic problems of shallow shells are formulated.
Chaotic Vibrations of Flexible Shallow Axially Symmetric Shells vs. Different Boundary Conditions
Awrejcewicz J., Krysko V.A.
In the present chapter, novel results of the study of chaotic vibration of flexible circular axially symmetric shallow shells subjected to sinusoidal transverse load are presented for four different boundary conditions are presented. The study is conducted with the use of the finite difference method (FDM), which differentiates the study from the majority of research conducted with the finite element method (FEM).
Chaotic Vibrations of Two Euler-Bernoulli Beams With a Small Clearance
Awrejcewicz J., Krysko V.A.
In this chapter a methodology to detect true/reliable chaos (in terms of non-linear dynamics) is developed on an example of a structure composed of two beams with a small clearance. The Euler-Bernoulli hypothesis is employed, and the contact interaction between beams follows the Kantor model. The complex non-linearity results from the von Kármán geometric non-linearity as well as the non-linearity implied by the contact interaction. The governing PDEs are reduced to ODEs by the second-order Finite Difference Method (FDM). The obtained system of equations is solved by Runge-Kutta methods of different accuracy. To purify the signal from errors introduced by numerical methods, the principal component analysis is employed and the sign of the first Lyapunov exponent is estimated by the Kantz, Wolf and Rosenstein methods and the method of neural networks. In the latter case a spectrum of the Lyapunov exponents is estimated. It has been illustrated how the number of nodes in the employed FDM influences the numerical results regarding chaotic vibrations. We have also shown how increase of the beams distance implies stronger action of the geometric nonlinearity, and hence influence of convergence of the used numerical algorithm for FDM has been demonstrated. Effect of essential dependence of the initial conditions choice on the numerical results of the studied contact problem is presented and discussed.
Mathematical Models of Chaotic Vibrations of Closed Cylindrical Shells with Circle Cross Section
Awrejcewicz J., Krysko V.A.
This chapter presents investigation of convergence of the Bubnov-Galerkin method for the cylindrical shells under action of the transverse harmonic local load on an example of a shell with λ = 2; 6 and under the width of the external pressure band measured by the central angle φ
0 = 343.
Contact Interaction of Two Rectangular Plates Made from Different Materials with an Account of Physical Non-Linearity
Awrejcewicz J., Krysko V.A.
In this chapter an approach convenient for analysis of stress-strain state (SSS) of multi-layer plates and beams with welded layers and with the occurrence of separated contacting zones due to the technological and exploitation reasons is proposed. The method offers a possibility of studying the constructions with contacts between layers, when the contact condition may depend on coordinates and all non-homogenity properties of the studied contact.
Introduction
Awrejcewicz J., Krysko V.A.
In this chapter, we give a brief discussion of the literature on the nonlinear theory of structural members, paying attention particularly to Eastern references, where many interesting results have been obtained and which are (unfortunately) still not well distributed among the worldwide scientific community.
Dynamics of Thin Elasto-Plastic Shells
Awrejcewicz J., Krysko V.A.
In Chap.
6
, a mathematical model governing the oscillations of a flexible shell with physical nonlinearity and coupling between the thermal and deformation fields has been presented. However, the loading and unloading processes overlap in the σ
i(e
i) diagram, and the remaining elasto-plastic deformation has not been taken into account. In a theoretical treatment of complicated shell oscillations, in order to analyse the stress–strain state properly, the elasto-plastic deformation should be considered, as well as the fatigue behaviour of the material. Only in this case can the mathematical model be close to the real behaviour of a structure. The aim of this chapter is to describe the development of complex mathematical models of structures, including elasto-plastic deformation and cyclic loading.
Coupled Nonlinear Thermoelastic Problems
Awrejcewicz J., Krysko V.A.
In this chapter, we formulate fundamental assumptions and relations similar to those presented in Chap.
2
for coupled linear thermoelasticity problems of shallow shells. A Timoshenko-type model including the inertial effect of rotation of shell elements is used. Both the generalized heat transfer equation and the equations governing vibration of a shell are formulated in Sect. 5.2, and then some special cases of these equations are analysed. In the next section, boundary and initial conditions are attached to the differential equations. In Sect. 5.4, the existence and uniqueness of a solution as well as the convergence of the Bubnov–Galerkin method, are rigorously discussed.
Estimation of the Errors of the Bubnov–Galerkin Method
Awrejcewicz J., Krysko V.A.
In Sect. 3.1, an abstract coupled problem is considered and a few theorems related to the estimation of the accuracy of the Bubnov–Galerkin method are formulated and proved. The error estimates hold for a system of differential equations of a rather general form with homogeneous boundary conditions, which corresponds to coupled thermoelastic problems for plates and shallow shells with variable thickness. In addition, a particular case of this problem (with nonhomogeneous initial conditions), where a prior estimate of the errors of the Bubnov–Galerkin method is most effective, is illustrated and discussed. Finally, a prior estimate for the Bubnov–Galerkin method to a problem generalizing a class of dynamical problems of elasticity (without a heat transfer equation) for both three-dimensional and thin-walled elements of structures is given.
Mathematical Model of Cylindrical/Spherical Shell Vibrations
Awrejcewicz J., Krysko V.A.
In this chapter a mathematical model of flexible shallow shells with an account of geometric non-linearity is derived. The results reliability is discussed. The stiff stability loss versus the cylindrical panels curvature k
y and versus the spherical shell with rectangular planform curvatures k
x, k
y under static load and an impulse of infinite length is analysed (conservative systems).
Chaotic Dynamics of Flexible Closed Cylindrical Nano-Shells Under Local Load
Awrejcewicz J., Krysko V.A.
In this chapter the mathematical model of a geometrically nonlinear closed cylindrical nano-shell is constructed. The solution convergence depending on the used series number terms in the Bubnov-Galerkin method is investigated. In particular, scenarios of transition into chaos of the nano-shell versus a number of the mentioned series terms N and the size dependent parameter l are analyzed. We have also detected, illustrated and discussed the scenarios of the Feigenbaum, Ruelle-Takens-Newhouse and their combination for differential values of N and l.
Scenarios of Transition from Periodic to Chaotic Shell Vibrations
Awrejcewicz J., Krysko V.A.
In this chapter we carry out numerical analysis of the scenarios of transition from periodic to chaotic vibrations exhibited by shells. Namely, we have detected and studied the so called modified Ruelle-Takens-Newhouse (RTN) scenario the modified Ruelle-Takens-Feigenbaum (RTF) scenario as well as the modified Pomeau-Manneville scenario found in our analysed mechanical continuous systems. We have numerically investigated the Sharkovsky’s orders of periodicity for PDEs governing dynamics of shallow shells. In the case of analysis of vibrations of the closed cylindrical shell with k
y = 112.5 and λ = 3 under transversal uniformly distributed harmonic load
$$q={{q}_{0}}\sin ({{\omega }_{p}}t)$$
, localized in the shell zone characterized by the angle φ
0 = 6, we have detected windows of periodicity which coincide with the order predicted by Sharkovsky’s theorem, i.e. 3; 5; 22 ⋅ 5; 9. We have also investigated a cylindrical panel under action of the longitudinal two-parametric load
$${{p}_{x}}={{p}_{0}}\sin ({{\omega }_{p}}t)$$
and p
y = α ⋅ p
x for k
y = 24 and α = 3 where we have found also the periodicity windows following the Sharkovsky’s order 3; 2 ⋅ 3; 5; 2 ⋅ 5; 7; 2 ⋅ 7; 11; 13. The carried out analysis of the phase and model portraits constructed for both periodic and chaotic shell vibrations revealed that the space and temporal (timing) chaos appear simultaneously, i.e. in the case of shells it refers to occurrence of the space-temporal chaos.
Comprehensive Method
Liseikin V.D.
Many physical phenomena involve the rapid formation, propagation, and disintegration of small-scale structures. Examples include shock waves in compressible flows; shear layers in laminar and turbulent flows; phase boundaries in nonequilibrium and boundary and interior layers; tearing layers and magnetic reconnection regions in magnetically confined plasmas; and combustion and detonation fronts.
Unstructured Methods
Liseikin V.D.
Unstructured mesh techniques occupy an important niche in grid generation. The major feature of unstructured grids consists, in contrast to structured grids, of a nearly absolute absence of any restrictions on grid cells, grid organization, or grid structure.
General Considerations
Liseikin V.D.
An indispensable tool of the numerical solution of partial differential equations by finite-element or finite-difference methods on general regions is a grid which represents the physical domain in a discrete form. In fact, the grid is a preprocessing tool or a foundation on which physical, continuous quantities are described by discrete functions and on which the differential equations are approximated by algebraic relations for discrete values that are then numerically analyzed by the application of computational codes.
Numerical Implementations of Comprehensive Grid Generators
Liseikin V.D.
The systems of inverted Beltrami or diffusion equations in control metrics, and their modifications described in Chap.
9
, allow one to generate grids in domains and/or on surfaces in a unified manner, regardless of their dimension. In particular, these systems can be applied to produce grids in spatial blocks by means of the successive generation of grids on their curvilinear edges, faces, and volumes, using the solution at a step
$$i
Grid Quality Measures
Liseikin V.D.
It is very important to develop grid generation techniques which sense grid quality features and possess means to eliminate the deficiencies of the grids. These requirements give rise to the problem of selecting and adequately formulating the necessary grid quality measures and finding out how they affect the solution error and the solution efficiency, in order to control the performance of the numerical analysis of physical problems with grids. Commonly, these quality measuresMeasure
of quality
Grid
quality encompass grid skewness,
Skewness
stretching,
Stretching
torsion,
Torsion cell aspect ratio, cell volume, departure from conformality, cell deformation and various related constructions (centroids, circumcenters, circumcircles, incircles, etc.).
Cell
deformation
