Numerical analysis of the dispersion of electroelastic waves in a cylinder whose physical properties have a longitudinal axis of symmetry

We develop a numerical method of analyzing the dispersion of electroelastic waves in a hollow circular cylinder. The method is based on the reduction of the wave problem to a spectral problem for a system of ordinary differential equations. This system is solved numerically. The roots of the dispersion equation are determined by the method of bisection. The practical convergence of the algorithm is shown using a specific example. One figure. Bibliography: 3 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 151–156.     

Numerical solution of the problem of the stress-strain state in hollow cylinders using spline approximations

The three-dimensional theory of elasticity is used for a study of the stress-strain state in a hollow cylinder with varying stiffness. The corresponding problem is solved by a method that is partly analytical and partly numerical in nature: Spline approximations and collocation are used to reduce the partial differential equations of elasticity to a boundary-value problem for a system of ordinary differential equations of higher order for the radial coordinate, which is then solved using the method of stable discrete orthogonalization. Results for an inhomogeneous cylinder for various types of stiffness are presented.     

Numerical analysis of the dispersion of electroelastic waves in a cylinder

The wave problem of electroelastic waves in a cylinder is reduced to a spectral problem for a system of eight ordinary differential equations. We give an algorithm for numerical solution based on reducing the original boundary-value problem to four Cauchy problems and determining the roots of the dispersion equation by the method of bisection. One figure. Bibliogrpahy: 4 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 149–153.     

The coupled problem of thermoelastic vibrations of a cylinder with cavities

Using a method based on the machinery of analytic functions of complex variables, we construct the solution of the problem of steady planar strain of an isotropic cylinder with curvilinear cavities caused by uneven heating of the surfaces of the cylinder and the cavities. It is an exact solution of the equations and an approximate solution of the boundary conditions. The results of numerical studies are given for a hollow cylinder undergoing thermoelastic radially symmetric vibrations. Two figures. Bibliography: 4 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 82–86.     

Solution of an axisymmetric problem of free vibrations of piezoceramic hollow cylinders of finite length by the spline collocation method

An axisymmetric problem of longitudinal free vibrations of piezoceramic hollow cylinders for some types of boundary conditions on the end faces is considered. The lateral surfaces of the cylinder are covered with thin short-circuited electrodes. The polarization of piezoceramics is directed along the axis of the cylinder. Using the method of separation of variables and method of spline collocation along the length of the cylinder, we reduce the problem to a system of ordinary differential equations, which is solved by the method of step-by-step search. We present results of calculations for a PZT 4 piezoceramic cylinder.     

Solution of two-dimensional boundary value problems corresponding to initial-boundary value problems of diffusion on a right cylinder

We consider boundary value problems for the differential equations Δ₂ u + B u = 0 with operator coefficients B corresponding to initial-boundary value problems for the diffusion equation Δ₃ u − pu = ∂ _t u (p > 0) on a right cylinder with inhomogeneous boundary conditions on the lateral surface of the cylinder with zero boundary conditions on the bases of the cylinder and with zero initial condition. For their solution, we derive specific boundary integral equations in which the space integration is performed only over the lateral surface of the cylinder and the kernels are expressed via the fundamental solution of the two-dimensional heat equation and the Green function of corresponding one-dimensional initial-boundary value problems of diffusion. We prove uniqueness theorems and obtain sufficient existence conditions for such solutions in the class of functions with continuous L ₂-norm.     

Interaction of a hollow cylinder of finite length and a plate with a cylindrical cavity with a rigid insert

Two problems of the interaction of a hollow circular cylinder with load-free ends and an unbounded plate with a cylindrical cavity and a symmetrically imbedded rigid insert are considered. Homogeneous solutions are found and the generalized orthogonality of these solutions is used when the modified boundary conditions are satisfied. As a result, we have a system of two integral equations in functions of the displacements of the outer and inner surfaces of the hollow cylinder. These functions are sought in the form of sums of a trigonometric series and a power function with a root singularity. The ill-posed infinite systems of linear algebraic equations obtained are regularized by the introduction of small positive parameters. Since the elements of the matrices of the systems as well as the contact stresses are defined by poorly converging numerical and functional series, an efficient method for calculating of the remainders of the above-mentioned series is developed. Formulae are found for the contact pressure distribution function and the integral characteristic. Examples of the calculation of the interaction of the cylinder and the plate with an insert are given.The method of solving contact problems described here has been used earlier1, 2 and the generalized orthogonality of the solutions found for bodies of finite dimensions, that is, for a rectangle and cylinders of finite length, is its basis. Problems for hollow cylinders with a band ² and an insert reduce to a system of two integral equations, and the problem for a rectangle¹ reduces to one integral equation. Solving these integral equations, ill-posed systems of linear algebraic equations are obtained which are subject to regularization³.     

Solution of problems of free torsional vibrations of thick-walled orthotropic inhomogeneous cylinders

With the use of the 3D theory of elasticity, we investigate the problem of free torsional vibrations of an anisotropic hollow cylinder with different boundary conditions at its end faces. We have proposed a numerical-analytic approach for the solution of this problem. The original partial differential equations of the theory of elasticity with the use of spline approximation and collocation are reduced to an eigenvalue problem for a system of ordinary differential equations of high order in the radial coordinate. This system is solved by the stable numerical method of discrete orthogonalization together with the method of step-by-step search. We also present numerical results for the case of orthotropic and inhomogeneous material of the cylinder for some kinds of boundary conditions.     

Study of optimal control problems on the half-line by the averaging method

We justify the application of the averaging method to optimal control problems for systems of differential equations on the half-line. For optimal control problems for systems of differential equations linear in the control, we prove the existence of optimal controls for the exact and averaged problems. We show that an optimal control in the averaged problem is ɛ-optimal in the exact problem.     

Diffraction of a plane acoustic wave by a non-uniform thermoelastic cylindrical layer bounded by inviscid heat-conducting fluids

The diffraction of sound by a radially laminated isotropic thermoelastic cylindrical shell is considered. The system of equations for small perturbations of a hollow thermoelastic cylinder is reduced to a system of ordinary differential equations, the boundary-value problem for which is solved by the spline-collocation method. Expressions are obtained describing the wave field outside the cylindrical layer. Results of calculations of polar radiation patterns of the amplitude of the scattered acoustic wave in the far zone are presented.     

Summability of series in the root functions of boundary-value problems of Bitsadze-Samarskii type

We investigate the Abel summability of a system of eigenfunctions and associated functions of Bitsadze-Samarskii-type boundary-value problems for elliptic equations in a rectangle. These problems are reduced to a boundary-value problem for elliptic operator differential equations with an operator in boundary conditions in the corresponding spaces and are studied by the method of operator differential equations. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 4, pp. 443–452, April, 2008.     

On some axial Couette flows of non-Newtonian fluids

The velocity fields corresponding to some flows of second grade and Maxwell fluids, induced by a circular cylinder subject to a constantly accelerating translation along its symmetry axis, are presented as Fourier-Bessel series in terms of the eigenfunctions of some suitable boundary value problems. These solutions satisfy both the associate partial differential equations and all imposed initial and boundary conditions. For α or λ → 0, they are going to those for a Newtonian fluid. Finally, for comparison, some diagrams corresponding to the solutions for the flow through a circular cylinder are presented for different values of t and of the material constants.     

Equations for the Missing Boundary Values in the Hamiltonian Formulation of Optimal Control Problems

Partial differential equations for the unknown final state and initial costate arising in the Hamiltonian formulation of regular optimal control problems with a quadratic final penalty are found. It is shown that the missing boundary conditions for Hamilton’s canonical ordinary differential equations satisfy a system of first-order quasilinear vector partial differential equations (PDEs), when the functional dependence of the H-optimal control in phase-space variables is explicitly known. Their solutions are computed in the context of nonlinear systems with ℝ ⁿ -valued states. No special restrictions are imposed on the form of the Lagrangian cost term. Having calculated the initial values of the costates, the optimal control can then be constructed from on-line integration of the corresponding 2n-dimensional Hamilton ordinary differential equations (ODEs). The off-line procedure requires finding two auxiliary n×n matrices that generalize those appearing in the solution of the differential Riccati equation (DRE) associated with the linear-quadratic regulator (LQR) problem. In all equations, the independent variables are the finite time-horizon duration T and the final-penalty matrix coefficient S, so their solutions give information on a whole two-parameter family of control problems, which can be used for design purposes. The mathematical treatment takes advantage from the symplectic structure of the Hamiltonian formalism, which allows one to reformulate Bellman’s conjectures concerning the “invariant-embedding” methodology for two-point boundary-value problems. Results for LQR problems are tested against solutions of the associated differential Riccati equation, and the attributes of the two approaches are illustrated and discussed. Also, nonlinear problems are numerically solved and compared against those obtained by using shooting techniques.     

On some axial Couette flows of non-Newtonian fluids

The velocity fields corresponding to some flows of second grade and Maxwell fluids, induced by a circular cylinder subject to a constantly accelerating translation along its symmetry axis, are presented as Fourier-Bessel series in terms of the eigenfunctions of some suitable boundary value problems. These solutions satisfy both the associate partial differential equations and all imposed initial and boundary conditions. For α or λ → 0, they are going to those for a Newtonian fluid. Finally, for comparison, some diagrams corresponding to the solutions for the flow through a circular cylinder are presented for different values of t and of the material constants. Received: March 18, 2004; revised: October 28, 2004     

The phase-temperature state of a hollow cylindrical casting

We use the finite-element method to study the problem of determining the temperature field and the evolution of the phase-state regions in a hollow cylindrical casting. Using weighted discrepancies we reduce the problem to solving systems of ordinary differential equations. We give a model solution of a specific problem and detailed analysis of the results. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 55–57.     

Structure of the frequency spectrum of nonaxisymmetric vibrations of a transversally isotropic hollow ball

We consider nonaxisymmetric vibrations of a hollow sphere made of a spherically Isotropic material. The original boundary-value problem is reduced to a system of ordinary differential equations by introducing new unknown junctions and expanding the sought solution in a series in spherical functions. The nonaxisymmetric problem of vibrations of a hollow ball decomposes into two independent problems that describe modes of first and second kind. The frequency spectrum for modes of first and second kind is analyzed.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 32–35, 1986.     

On the Existence and Stability of Solutions for a System of Elliptic Equations

We show, using a dual variational method developed in the paper, the existence of solutions for a system of Dirichlet problems for partial differential equations involving p– and q–Laplacian. The stability of solutions and the existence of positive solutions is considered in the superlinear case.      

Numerical solution of the propagation problem of longitudinal elastic waves in an orthotropic hollow cylinder

Wave processes in an orthotropic cylinder are analyzed. Harmonic oscillations are considered for which the original problem is reduced to a system of ordinary differential equations. This system is solved by the numerically stable discrete-orthogonalization method. Discrete curves for specific problems are given.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 56, pp. 75–78, 1985.     

On the solution of heat conduction problems for thermosensitive bodies heated by convective heat exchange

We propose a method of solving heat conduction problems for thermosensitive bodies, in particular for a hollow cylinder from whose surfaces a convective heat exchange with the external environment occurs.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 83–86.     

Robust multigrid method for the planar linear elasticity problems

We consider the solution of the system of equations that arise from the higher order conforming finite element (Scott–Vogelius element) discretizations of the boundary value problems associated with the differential operator −ρ ² Δ − κ ²∇div, where ρ and κ are nonzero parameters. Robust multigrid method is constructed, i.e., the convergence rate of multigrid method is optimal with respect to the mesh size, the number of levels, and weights on the two terms in the aforementioned differential operator.

---