Mathematical Problems in Engineering

Volume 2016, Article ID 3457649, 9 pages

http://dx.doi.org/10.1155/2016/3457649

## Application of the Least Squares Method in Axisymmetric Biharmonic Problems

Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of National Academy of Sciences of Ukraine, 3-b Naukova Street, Lviv 79060, Ukraine

Received 29 March 2016; Revised 29 May 2016; Accepted 2 June 2016

Academic Editor: Masoud Hajarian

Copyright © 2016 Vasyl Chekurin and Lesya Postolaki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An approach for solving of the axisymmetric biharmonic boundary value problems for semi-infinite cylindrical domain was developed in the paper. On the lateral surface of the domain homogeneous Neumann boundary conditions are prescribed. On the remaining part of the domain’s boundary four different biharmonic boundary pieces of data are considered. To solve the formulated biharmonic problems the method of least squares on the boundary combined with the method of homogeneous solutions was used. That enabled reducing the problems to infinite systems of linear algebraic equations which can be solved with the use of reduction method. Convergence of the solution obtained with developed approach was studied numerically on some characteristic examples. The developed approach can be used particularly to solve axisymmetric elasticity problems for cylindrical bodies, the heights of which are equal to or exceed their diameters, when on their lateral surface normal and tangential tractions are prescribed and on the cylinder’s end faces various types of boundary conditions in stresses in displacements or mixed ones are given.

#### 1. Introduction

Many practically important problems bring the biharmonic equation considering in 2D domains with Lipschitz-continuous boundary .

Here is Laplace differential operator: , where stands for the gradient operator in and the dot denotes scalar product. Function is considered as four-time differentiable one in .

The classical formulations of biharmonic problems distinguish the Dirichlet and Neumann boundary value problems. Two kinds of Dirichlet problems are usually considered for biharmonic equation (1). In the problem of the first and second kinds the biharmonic functions should be subordinated to boundary conditions (2) and (3) correspondingly [1]:Here is the operator of normal derivative on , is the outward unit normal vector to , and and are given functions.

Various kinds of Neumann problems for (1) can also be considered [2, 3]. In simplest cases these are the problems with boundary conditions (4), (5), or (6):

One can distinguish two kinds of biharmonic mixed problems [4, 5]. In the mixed problem of the first kind (1) is considered subject to boundary conditions which are weighted combinations of Dirichlet boundary conditions and Neumann boundary conditions (so-called Robin boundary condition). In another case Dirichlet data are prescribed on one part of the boundary and Neumann data are prescribed on the remainder.

Various methods are used for solving of the biharmonic problems. Among them are iterative methods [6], boundary integral method [7], method of finite differences [8], finite element method [9], and so forth.

Significant interest in biharmonic problems in rectangle arises in 2D theory of elasticity [10]. In this connection we should refer to the so-called method of homogeneous solutions [11–14] used for these problems’ solving. An idea of the method consists in representing the solution as a series expansion in some complete system of biharmonic functions being solutions of a homogeneous biharmonic problem on infinite strip [11, 14]. These functions satisfy homogeneous Neumann-type boundary conditions on the strip’s sides. As such representation automatically satisfies (1) and homogeneous boundary conditions on two opposite sides of the rectangle, to find the solution it is necessary to determine the expansion coefficients by subordinating the solution to the boundary data prescribed on the other two opposite rectangle’s sides. To do that the least squares method was applied in [14]. In such way the problem was reduced to a problem of nonconstrained optimization. The approach was applied to solve Neumann problem and some mixed biharmonic problems on rectangle.

The method of least squares on the boundary was also used in [15] to solve the first Dirichlet problem on rectangle. Here the solution of the biharmonic equation was presented as a linear combination of finite system of biharmonic polynomials.

In [16] an axisymmetric biharmonic problem for a finite cylindrical domain was considered. The solution was presented there as the Fourier-Bessel expansion. In [17] the method of fundamental solution was applied to solve axisymmetric Dirichlet biharmonic problem (1) and (2).

In this paper we consider the method of least squares on the boundary combined with the method of homogeneous solutions in application to axisymmetric biharmonic problems for a semi-infinite cylindrical domain.

#### 2. Problem Formulation

Problems of elastic equilibrium in axisymmetric case can be reduced to axisymmetric biharmonic equation (1), where is axisymmetric Laplace operator in cylindrical coordinate ( and stand for radial and axial coordinates).

Biharmonic function in this case has the sense of Love stress function, through which displacement components and stress components can be expressed as [10]where is Poisson ratio and stands for shear modulus.

We will consider four biharmonic problems for semi-infinite cylindrical domain with prescribed stresses and on its lateral surface :Here and are integrable functions which decay when tends to infinity.

Problems I to IV are distinguished by boundary conditions prescribed on the plane circular area .

*Problem I*. Consider the following:

*Problem II*. Consider the following:*Problem III*. Consider the following:*Problem IV*. Consider the following:where , , , and are given function.

With the use of relations (7) and (8) we can express the boundary conditions (9)–(13) in terms of function . We can see that problem I is of Neumann type. It is solvable only if the functions and satisfy the condition Problems III and IV should be classified as mixed ones.

To use the method of homogeneous solutions we reduce the problems to corresponding problems with homogeneous boundary conditions on . To do that we consider an auxiliary biharmonic problem for infinite cylindrical domain with boundary conditionswhere and are defined on the line integrable functions that satisfy conditions , , for , and both decay when . For instance, we can choose and asor as

Problem (1) and (15) was solved with the use of the Fourier integral transform. One can find the solution in [18].

Let be the solution of problem (1) and (15) and , and , , , the functions, calculated due to formulas (7) and (8) correspondingly for solution . We introduce the functions

Now the solutions of the biharmonic problems I to IV can be presented as , where is the solution of the biharmonic problems for domain on the lateral surface of which Neumann’s homogeneous conditions are prescribed:

On the surface the function obeys one pair of conditions (20)–(23): Here

So, biharmonic problems I to IV reduced to biharmonic problems (problems I′ to IV′) with homogeneous conditions (19) on the cylinder lateral surface and corresponding nonhomogeneous conditions (20)–(23) on the circular area . We will solve these problems using the method of homogeneous solution.

#### 3. Systems of Homogeneous Solutions in Cylindrical Coordinates

We look for a solution of biharmonic equation (1) in the form

Substituting (25) into (1) brings the next ordinary differential equation for the radial function :

Due to relations (19) and (25) the radial function obeys at the boundary conditions:The function and its derivative should be finite at .

The general solution of (26) is Here , , , and stand for arbitrary constants, , and , are Bessel and Hankel functions of orders zero and one correspondingly.

To provide finiteness of solution (28) at the point we put , . Then, with accounting of the property of Bessel functions the radial function takes the form

Substitution of (30) into boundary conditions (27) brings the linear homogeneous system regarding the constants and :

Nontrivial solutions of system (31) exist under the condition

Transcendental equation (32) does not have any real roots except the doubly degenerate root . It does not have imaginary roots too. Hence complex roots should be considered. The set of roots of (32) contains four infinite sequences of complex roots [10]:

The values of first 15 roots of (32) are presented in Table 1:; . The data were obtained by numerical solving of (32) at .