A mixed boundary value problem for an infinite,piezoceramic bimorph

Summary An infinitely extended, piezoelectric bimorph is considered, consisting of two piezoceramic plates, glued to each other with in between an infinitesimally thin electrode. The bimorph is deformed by the application of a constant potential difference between shorted circular electrodes on the outer faces and the central electrode. By using Hankel transforms the linear, piezoelectric relations are reduced to dual integral equations for the charge on the outer electrodes. These equations are solved numerically. Also the charge on the central electrode and the potential on the faces are computed. The computed quantities are compared with corresponding values obtained from an approximated theory.

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Ein gemischtes Randwertproblem für einen unendlichen piezokeramischen Bimorph

Zusammenfassung Ein unendlicher piezoelektrischer Bimorph aus zwei piezoelektrischen Platten mit einer infinitesimal dünnen Zwischenelektrode wird untersucht. Ein Teil der Außenfläche ist mit kurzgeschlossenen kreisförmigen Elektroden versehen. Zwischen den Außenektroden und der Innenelektrode besteht eine konstante Potentialdifferenz. Mittels Hankel-Transformation werden die linearen Gleichungen reduziert auf duale Integralgleichungen bezüglich der Ladungsdichte der Außenelektroden. Eine numerische Lösung dieser Gleichungen wird angegeben. Berechnet werden auch die Ladungsdichte der Innenelektrode und das Potential der Außenfläche. Die Resultate werden mit den entsprechenden einer Näherungstheorie verglichen.

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With 5 Figures     

General solution of a certain mixed boundary value crack problem

An exact analytical solution is derived for a certain class of mixed boundary value crack problems which occur for example when analysing crack closure effects during fatigue crack growth. A general analysis is presented which is derived from the solution of a singular integral equation. The analysis is checked by comparing the general results to a known special solution (the triple crack problem). A technique is described for calculating the stress intensities for a central crack opened by two wedges of general shape which preserve symmetry about the crack plane and its perpendicular bisector. The stress distribution acting on the wedges is also calculated.     

Explicit solutions to a class of mixed boundary value problem

The potential problem in the semi-infinite region $\text{0?x?π,0?y?∞}$ is treated analytically for mixed Dirichlet and Neumann type boundary conditions on $\text{x = π}$. The existing techniques for such problems usually lead to integral equations requiring numerical inversion. A conformal mapping is developed to reduce the problem to sets of elementary Fourier series or dual series, from which the explicit solution may be found. Some closed form solutions are given as examples.     

The solution of a mixed boundary value problem in the theory of diffraction

Summary An exact solution is obtained for the problem of the diffraction of a cylindrical sound wave by an absorbent semi-infinite plane. The two faces of the half-plane have different impedance boundary conditions. The problem which is solved is a mathematical model for a noise barrier whose surface is treated with two different acoustically absorbent materials.The usual Wiener-Hopf method (which is the standard technique for solving half-plane problems) has to be modified to give a solution to the present mixed boundary value problem.     

A mixed boundary problem for a finite internally cracked plate

A displacements and resultant forces model (see eqns 2–5, 13 and 17) for a finite internally cracked plate is proposed. This model satisfies: (a) The equilibrium and compatibility condition in the region occupied by cracked plate; (b) Stress free condition on the surface of crack; (c) Single value condition of displacements around the crack. In this model, some undetermined coefficients are contained, these coefficients are derived from outer boundary condition.

It is proved that, this model is convenient not only for the displacements or resultant forces boundary problem, but also for the mixed boundary problem. Besides this, if the boundary problem is solved, to find the value of displacements of any points in cracked body is also convenient.

Two mixed boundary problems, one for the square cracked plate (see Fig. 3), and another for circular cracked plate (see Fig. 5), are solved. The numerical results obtained are shown in Tables 1 and 2 and Figs. 4 and 6 respectively. These results can explain how the constraint affects the values of the stress intensity factor on the crack tips.     

A mixed heat conduction boundary problem for a semi-infinite plate

An examination is made of a plane stationary heat conduction problem with mixed conditions assigned on the edge of a semi-infinite plate. There is a temperature field in the plate due to the action of a point source of heat located inside the plate.     

Solution of the mixed boundary value problem of heat conduction in a composite rectangle

The author offers a particular solution of the two-dimensional problem of steady heat conduction for a rectangular region composed of two different materials with mixed boundary conditions.     

Solution of one mixed problem of plate bending for a domain with partially unknown boundary

The paper addresses the problem of finding a full-strength contour in the problem of plate bending for a cycle-symmetric doubly connected domain. An isotropic elastic plate, bounded by a regular polygon, is weakened by a required full-strength hole whose symmetry axes are the regular polygon diagonals. Rigid bars are attached to each component of the broken line of the outer boundary of the plate. The plate bends under the action of concentrated moments applied to the middle points of the bars. An unknown part of the boundary is free from external forces. Using the methods of complex analysis, the analytical image of Kolosov?CMuskhelishvili??s complex potentials (characterising an elastic equilibrium of the body) and of an unknown full-strength contour are determined. A numerical analysis is performed and the corresponding plots are obtained by means of the Mathcad system.     

Stokes flow around cylinders in a bounded two‐dimensional domain using multipole‐accelerated boundary element methods

The multipole technique has recently received attention in the field of boundary element analysis as a means of reducing the order of data storage and calculation time requirements from O(N²) (iterative solvers) or O(N³) (gaussian elimination) to O(N log N) or O(N), where N is the number of nodes in the discretized system. Such a reduction in the growth of the calculation time and data storage is crucial in applications where N is large, such as when modelling the macroscopic behaviour of suspensions of particles. In such cases, a minimum of 1000 particles is needed to obtain statistically meaningful results, leading to systems with N of the order of 10 000 for the smallest problems. When only boundary velocities are known, the indirect boundary element formulation for Stokes flow results in Fredholm equations of the second kind, which generally produce a well‐posed set of equations when discretized, a necessary requirement for iterative solution methods. The direct boundary element formulation, on the other hand, results in Fredholm equations of the first kind, which, upon discretization, produce ill‐conditioned systems of equations. The model system here is a two‐dimensional wide‐gap couette viscometer, where particles are suspended in the fluid between the cylinders. This is a typical system that is efficiently modelled using boundary element method simulations. The multipolar technique is applied to both direct and indirect formulations. It is found that the indirect approach is sufficiently well‐conditioned to allow the use of fast multipole methods. The direct approach results in severe ill‐conditioning, to a point where application of the multipole method leads to non‐convergence of the solution iteration. Copyright © 1999 John Wiley & Sons, Ltd.     

A multiwavelet Galerkin boundary element method for the stationary Stokes problem in 3D

In this paper, a multiwavelet Galerkin boundary element method is presented for the fast solution of the stationary Stokes problem in three dimensions. Piecewise linear discontinuous multiwavelet bases are constructed on each patch of piecewise smooth surface individually, which allow easy and efficient evaluation of the matrix entries. Because of the use of the multiwavelets, the system matrix can be compressed to O (N) (N denotes the number of unknowns) nonzero entries without compromising the order of convergence as for the conventional Galerkin boundary element method. Numerical results of two test samples are given to demonstrate the availability of the present method.     

A parallel multipolar indirect boundary element method for the Neumann interior Stokes flow problem

A multipolar expansion technique is applied to the indirect formulation of the boundary element method in order to solve the two‐dimensional internal Stokes flow second kind boundary value problems. The algorithm is based on a multipolar expansion for the far field and numerical evaluation for the near field. Due to the nature of the algorithm, it is necessary to resort to the use of an iterative solver for the resulting algebraic linear system of equations. A parallel implementation is designed to take advantage of the natural domain decomposition of fast multipolar techniques and bring further improvements. Copyright © 2000 John Wiley & Sons, Ltd.     

Two-dimensional mixed boundary value problem in heat and mass transfer for a characteristic equation with multiple roots

A system of two differential equations of parabolic type is examined. A boundary value problem is set up and solved. A system of integrodifferential equations is obtained for determining the unknown functions. A method of reducing this to a system of ordinary Volterra integral equations is given.     

Macro-elements in the mixed boundary value problems

 The symmetric Galerkin boundary element method (SGBEM), applied to elastostatic problems, is employed in defining a model with BE macro-elements. The model is governed by symmetric operators and it is characterized by a small number of independent variables upon the interface between the macro-elements. The kinematical and mechanical transmission conditions across the interface are imposed in global form, whereas the response of the boundary discretized elastic problem is provided in terms of forces on the interface boundary sides and of displacements at the interface nodes. A variational formulation is presented in which the boundary transmission conditions are derived by Polizzotto's boundary min-max variational principle. Simple numerical applications are shown. Received 8 December 99     

A Frobenius solution to the scaled boundary finite element equations in frequency domain for bounded media

The scaled boundary finite element method (FEM) is a recently developed semi‐analytical numerical approach combining advantages of the FEM and the boundary element method. Although for elastostatics, the governing homogeneous differential equations in the radial co‐ordinate can be solved analytically without much effort, an analytical solution to the non‐homogeneous differential equations in frequency domain for elastodynamics has so far only been obtained by a rather tedious series‐expansion procedure. This paper develops a much simpler procedure to obtain such an analytical solution by increasing the number of power series in the solution until the required accuracy is achieved. The procedure is applied to an extensive study of the steady‐state frequency response of a square plate subjected to harmonic excitation. Comparison of the results with those obtained using ABAQUS shows that the new method is as accurate as a detailed finite element model in calculating steady‐state responses for a wide range of frequencies using only a fraction of the degrees of freedom required in the latter. Copyright © 2006 John Wiley & Sons, Ltd.     

A mixed non-linear boundary value problem appearing in gas dynamics: generalized solutions and numerical results

Considering the class of mixed non-linear elliptic-hyperbolic problems, the paper deals with the significant case given by the model of Karman and Guderley. An entropy condition is also taken into account. The problem is ill-posed. This unfavourable situation leads first to construct an approach of the problem then to state numerical and functional methods. Involving suitable arguments, we show that the gradient method works to obtain a numerical solution in accordance with results of previous authors. Furthermore it is proved by a functional method that the solution of the problem reduces to the finite dimensional case.     

A nonlinear hyperbolic free boundary value problem

Summary The present paper is concerned with the application of a non-iterative transformation method to the numerical solution of a nonlinear hyperbolic free boundary value problem.Making use of the similarity analysis approach to the hyperbolic model describing time dependent velocity impact to nonlinear inhomogeneous thin rods we recover a free boundary value problem.Since exact solutions are known only in some particular cases, we consider application of numerical methods of integration.Usually iterative numerical methods of solution are known to be applicable to free boundary value problems. However, we can prove that the ordinary differential equation related to the model in point is invariant with respect to a stretching group of transformations. This is the hint to apply group properties and to develop an ad hoc non-iterative transformation method.     

Initial value problem formulation of time domain boundary element method for electromagnetic microwave simulations

To aim to obtain more stable solutions and wider area applications for the Time Domain Boundary Element Method (TDBEM), initial value problem formulation of the TDBEM is newly introduced for microwave simulations. The initial value problem formulation of the TDBEM allows us to solve transient microwave phenomena as interior region problems, which gives us well matrix property and interior resonance free solutions. This paper concentrates on applying the initial value problem formulation of the TDBEM to wake field phenomena in particle accelerator cavities.