Finite Elements in the Solution of Continuum Field Problems

Abstract

A finite element functional solution procedure was presented employing variational calculus. The Functionals of field continuum were developed on adoption of Euler minimum integral theorem and finite element procedures on Laplace model. The elements functionals minimization resulted to series of partial differential equations describing the variation of the function of interest at various discrete nodal points. The assembly of the partial differential equations gave a unifying algebraic system of equation was solved for the unique solutions of the function. To simulate the finite element model, boundary conditions of temperature field was assumed. The solution and post processing of FEM of this study showed that once the stiffness matrix of a continuum is established and the boundary conditions specified the continuum is solved uniquely. Regression method was used to establish the error associated with FEM results and to establish a simple prediction model for environmental temperatures. The procedure of this study presented the basis for insulation design for solid, hollow or shell pipes in fluid transport design in oil and gas transport system. The finite element method evaluated the temperature distribution of the region to serve as a guide in quantifying quantity of heat to the environment from the transit fluid. The error of FEM prediction was estimated at 0.006 and the coefficient of determination for goodness of regression fit is estimated as 0.99999. This study also presents an approximate procedure for processing polar systems as rectangular systems by using the circumference of the circular section as one dimensional independent variable and the difference between the inner and outer radius (thickness) as the second independent variable.