Application of the Least Squares Method in Axisymmetric Biharmonic Problems

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.

Residual stresses in a finite cylinder. Direct and inverse problems and their solving using the variational method of homogeneous solutions

“…As B k are complex constants, it makes possible to satisfy both conditions (14), prescribed on the bases of the cylinder. We use for that the variational approach [8,11].…”

“…The convergence of the solution, obtaining with the use of reduction method, can be evaluated by investigation of how the value of the functional is changing with increasing of the number N . Results of such studies, conducted earlier, are presented in the paper [8,11].…”

Axially symmetric elasticity problems for the hollow cylinder with the stress-free ends. Analytical solving via a variational method of homogeneous solutions

“…If function ϕ(ζ) in presentation (12) is taken in form (21), then function χ(ξ, ζ) defines symmetrical with respect to the plane ζ = 0 strain-stressed state. In the other case, when ϕ(ζ) is taken in form (22), χ(ξ, ζ) determine the antisymmetric state of the cylinder.…”

“…Taking even axial function ϕ(ζ) in presentation (12), we come to other linear homogeneous system for the unknown coefficients in function (22):…”

“…In [20][21][22], the variational method of homogeneous solutions was extended to axially symmetric elasticity problems for the solid cylinder of semi-infinite and finite length, with the traction-free lateral boundary and loaded end faces. In [23], this method was used to solve an inverse problem for determining axially symmetric residual stresses in a finite solid cylinder with the use of the data obtained by the photo-elasticity method.…”