Eero-Matti Salonen

Coping with Curvilinear Coordinates in Fluid Mechanics

The method of local Cartesian coordinates introduced in the first part of a series of three articles on curvilinear coordinates was applied in the second article to some solid mechanics problems. In this final, third article of the series, the method is applied to fluid mechanics and for intrinsic coordinates. The use of the principle of virtual power is advocated to generate the approriate equations of motion.

Coping with curvilinear coordinates

A systematic method to generate expressions appearing in physics and engineering and valid in curvilinear orthogonal coordinates is presented. The method, which is called ‘the method of local cartesian coordinates’ achieves the same as ‘the method of moving axes’ presented in Love [1] but with a smaller effort and with more familiar mathematical tools. The main idea is: if we have an expression valid in rectangular cartesian coordinates, a corresponding expression for curvilinear orthogonal coordinates can be formed with simple steps. Some general expressions are generated, but the method can be used equally well to produce the desired formulas directly in specific cases. For this purpose, polar coordinates are employed extensively as a demonstration example.

Regenerator Analysis with Finite Elements

The classic one-dimensional regenerator problem has been solved numerically in a new way. Instead of marching in time, the hot and cold periods are treated as boundary value problems in rectangular domains. To solve the boundary value problems, a space-time finite element method using the commercial package COMSOL 4.3 with MATLAB with a Galerkin formulation has been applied. This approach minimizes the coding effort. The code solves a problem automatically with three consecutive meshes of increasing densities. We present results of three example cases in some detail.

Strain and Stress Analysis of a Curved Tapered Beam Model

Abstract The strain and stress resultant expressions for a tapered curved beam model are derived. The model is a one-dimensional version of the finite element based shell theory model of Irons and it may be considered as a generalization of the well-known Timoshenko beam model. To gain insight into the properties of the model, the expressions needed are developed analytically without use of finite elements. An effort is made to clarify a statement given previously, which is needed in the application of the theory and apt to lead to some confusion in its interpretation. Small displacement theory is applied. The stress resultant expressions are derived using the principle of virtual work and in addition directly from the stresses acting on the beam cross-section. The stresses obtained by the model in the isotropic elastic case are compared in two simple example problems with those by the Timoshenko model and with the exact values.