Free bending vibrations of an isotropic plate with free holes

A solution of the problem of the free bending vibrations of a thin finite isotropic plate with stiffly clamped outer edge is obtained by using functions of a complex variable. Results of numerical investigations are presented for an elliptic plate with acircular hole.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 67–70, 1989.     

Numerical solution of some nonlinear problems of plate dynamics

We derive the equations of motion of plates for various cases of geometrical nonlinearity and shear strain. A finite-difference numerical solution is given.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 66, pp. 83–89, 1988.     

Numerical solution of minimum norm problems of distributed control of vibrations

This paper is concerned with the distributed control of a vibration process that can be described by a differential equation for a Hilbert-spacevalued functiony: [0, ) H. The control functions on the right-hand side of this equation are taken fromL ([0, ),H) equipped with the essential supremum norm. To be solved is the problem of time-minimal null-controllability by norm-bounded controls. This problem is essentially reduced to solving the problem of minimum norm control on a given time interval. This is solved via its dual problem which is approximately solved by truncation and discretization. Numerical results are presented for a vibrating string and a vibrating beam.     

Numerical solution of steady-state porous flow free boundary problems

Summary A new numerical method is used to solve stationary free boundary problems for fluid flow through porous media. The method also applies to inhomogeneous media, and to cases with a partial unsaturated flow.     

On the solution of certain integral equations of mixed problems of plate bending theory

The problem of the bending of a Kirchhoff-Love plate in the shape of a strip under the impression of a thin linear rigid inclusion fastened at one of the edges of the plate when the other edge of the plate is rigidly clamped is considered. The problem is reduced by a Fourier integral transform to the solution of a convolution-type integral equation of the first kind in a finite segment with a regular kernel. The exact inversion of the principal part of the corresponding integral operator is constructed in the class of functions with non-integrable singularities on the segment edges. An effective asymptotic solution is given for the integral equation under investigation in this class of functions in the whole range of variation of the characteristic parameter λ. The results obtained are verified numerically. Analogous integral equations were examined in /1, 2/. The mode of investigation is similar to that proposed in /3/.     

On bending vibrations of a periodically inhomogeneous orthotropic plate

The matrix technique which has been developed by the author in connection with solving the problem of elastic wave propagation in an unbounded medium with periodic inhomogeneities is applied. The formalism is extended to the case of bending vibrations of inhomogeneous and periodically inhomogeneous orthotropic plates.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 179–181, 1989.     

Elementary solution of some plate problems

Zusammenfassung Die Arbeit behandelt das Biegungsproblem einer am Rande fest eingespannten elliptischen Platte unter einer durch allgemeines Polynomn-ten Grades bestimmten Belastung. In alle Einzelheiten geht die Rechnung für den in der Literatur kaum zu findenden Spezialfalln=2.     

Solution of some plate bending problems using the boundary element method

Some fundamental aspects of the boundary element method of the Kirchhoff theory of thin plate flexure are given. The direct boundary integral equation method with higher conforming properties (using first-order Hermitian interpolation for plate displacement ω, and zero-order Hermitian interpolation for angle of rotation θ, the moment M andthe equivalent shear V) are used for several computational examples. They are: square plate with simply-supported or clamped edges, the same square plate with square central opening and the cantilevered triangular plates. The results of computation as compared with some known experimental and theoritical results showed that the numerical schemes seemed to be satisfactory for the practical applications.     

Numerical solution for exterior problems

Summary We solve the Helmholtz equation in an exterior domain in the plane. A perfect absorption condition is introduced on a circle which contains the obstacle. This boundary condition is given explicitly by Bessel functions. We use a finite element method in the bounded domain. An explicit formula is used to compute the solution out of the circle. We give an error estimate and we present relevant numerical results.     

Numerical solution of nonlinear diffraction problems

An adaptive finite element method is developed for solving Maxwell's equations in a nonlinear periodic structure. The medium or computational domain is truncated by a perfect matched layer (PML) technique. Error estimates are established. Numerical examples are provided, which illustrate the efficiency of the method.     

Numerical solution of Hopf bifurcation problems

We present two numerical methods for the solution of Hopf bifurcation problems involving ordinary differential equations. The first one consists in a discretization of the continuous problem by means of shooting or multiple shooting methods. Thus a finite-dimensional bifurcation problem of special structure is obtained. It may be treated by appropriate iterative algorithms. The second approach transforms the Hopf bifurcation problem into a regular nonlinear boundary value problem of higher dimension which depends on a perturbation parameter ?. It has isolated solutions in the ?-domain of interest, so that conventional discretization methods can be applied. We also consider a concrete Hopf bifurcation problem, a biological feedback inhibition control system. Both methods are applied to it successfully.     

Numerical solution of bilinear programming problems

A bilinear programming problem with uncoupled variables is considered. First, a special technique for generating test bilinear problems is considered. Approximate algorithms for local and global search are proposed. Asymptotic convergence of these algorithms is analyzed, and stopping rules are proposed. In conclusion, numerical results for randomly generated bilinear problems are presented and analyzed.     

On the convergence of a mixed finite-element method for plate bending problems

Summary In this paper we justify a finite element method for biharmonic boundary value problems. The method is based on a stationary variational principle (the Reissner principle), and was introduced by Hellan, Herrmann and Visser. We prove error estimates and the existence of a finite element solution.     

Numerical solution of subsurface moisture transport problems

An algorithm is proposed for solving difference problems of subsurface hydromechanics by the alternating triangular method (ATM). For second, third, and mixed boundary-valued problems of subsurface moisture transport in the elastic regime, we use an improved ATM generalization scheme, which has been tested by the authors in application to the first boundary-value problem. The proposed algorithm can be applied to various problems of subsurface mass transport.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 56, pp. 83–91, 1985.     

Numerical solution of some two-parameter eigenvalue problems

We consider an iteration algorithm for determining the eigenvalues of an algebraic two-parameter spectral problem with the use of Newton’s method and a new efficient numerical procedure for calculation of the derivative of a determinant. Numerical examples are given.     

Numerical solution of the optimal boundary control of transverse vibrations of a beam

Boundary control is an effective means for suppressing excessive structural vibrations. By introducing a quadratic index of performance in terms of displacement and velocity, as well as the control force, and an adjoint problem, it is possible to determine the optimal control. This optimal control is expressed in terms of the adjoint variable by utilizing a maximum principle. With the optimal control applied, the determination of the corresponding displacement and velocity is reduced to solving a set of partial differential equations involving the state variable, as well as the adjoint variable, subject to boundary, initial, and terminal conditions. The set of equations may not be separable and analytical solutions may only be found in special cases. Furthermore, the computational effort to determine an analytic solution may also be excessive. Herein a numerical algorithm is presented, which easily solves the optimal boundary control problem in the space‐time domain. An example of a continuous system is analyzed. This is the case of the vibrating cantilever beam. Using a finite element recurrence scheme, numerical solutions are obtained, which compare the behavior of the controlled and uncontrolled systems. Also, the analytic solution to the problem is compared with the results obtained using the numerical scheme presented. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 558–568, 1999     

Numerical solution of stochastic differential problems in the biosciences

Stochastic differential equations (SDEs) models play a prominent role in many application areas including biology, epidemiology and population dynamics, mostly because they can offer a more sophisticated insight through physical phenomena than their deterministic counterparts do. So, suitable numerical methods must be introduced to simulate the solutions of the resulting stochastic differential systems. In this work we take into account both Euler–Taylor expansion and Runge–Kutta-type methods for stochastic ordinary differential equations (SODEs) and the Euler–Maruyama method for stochastic delay differential equations (SDDEs), focusing on the most relevant implementation issues. The corresponding Matlab codes for both SODEs and SDDEs problems are tested on mathematical models arising in the biosciences.     

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.