# Overview

Most practical conduction heat transfer problems can not be solved analytically. As a result, numerical solutions provide the only feasible way in which these problems can be solved. There are two common approaches: finite-difference and finite-element methods. In both approaches, the governing partial differential conduction equation subject to specified boundary (and for transient problems, initial) conditions is transformed into a system of ordinary differential equations (for transient problems) or algebraic equations (for steady-state problems) which are solved to yield an approximate solution for the temperature distribution. In the finite difference method, spatial discretization of the problem using a set of nodal points followed by application of energy balances and rate equations for each of the discrete segments directly results in a system of equations which are solved to obtain the temperature at each nodal point.

In the finite-element method, the partial differential equation is transformed into an integral form. Numerical approximation of the integral results in the system of algebraic or ordinary differential equations. Just as in the finite-difference approach, the accuracy of the finite-element approach is improved as the number of nodes used to discretize the region is increased. Though less intuitive, the finite-element method has been chosen over the finite-difference method primarily because its use of triangular elements greatly simplifies the discrete approximation of non-rectangular geometries.

FEHT provides three essential functions: Problem Definition, Calculations, and Output. The Problem Definition commands provide a drawing environment in which the mouse is used to draw the outlines of the materials with straight lines. A variety of drawing aids are provided. Triangular elements of arbitrary size needed in the finite-element analysis are formed simply by clicking the mouse button on the endpoints of the lines. The program monitors the discretization process to ensure that lines do not cross. The Problem Definition is completed by specifying the boundary and (for transient problems) the initial conditions. These specifications are made by tagging the line, node, or material with a mouse click (causing it to flash) and then selecting the desired specification from a pull-down menu. The accuracy of a solution is improved as the number of elements increases. An automatic mesh command can be used to reduce the mesh size.

Calculations are initiated from a pull-down menu. The program first checks to see that all materials are properly discretized and the properties, boundary, and initial conditions are specified. Any error detected during the checking is marked and described in a separate window at the top of the screen. For transient problems, the computational method (Euler or Crank-Nicolson) and start, stop and time step are selected from a dialog box; if no errors are detected, the calculations are initiated.

A variety of output capabilities are provided. For steady-state problems, the potentials (temperature,. voltages, magnetic potential, streamlines, or pressures) within the material may be shown at the nodal positions or in one of several types of contour plots. The potential at the cursor position is displayed when the mouse button is depressed. The potential gradients (temperature gradient, current density, electrical or magnetic flux density) within the materials can be displayed by arrows pointing in the direction of the gradient with the shaft length proportional to the gradient magnitude. The flow of heat, charge, current, or magnetic flux across any element line may be determined by simply clicking the mouse button while the cursor is on the line. For transient heat transfer problems, the temperatures of selected nodes may be displayed in a temperature versus time plot. Heat flow can be plotted as a function of time. The contours and/or temperature gradients for each time step may be shown in sequence providing a 'movie' depicting the changes with time.

The motivation behind the development of FEHT was for instruction. Although undergraduate engineers are exposed to numerical solution methods, they are typically not in a position to solve the more interesting practical problems after completing the course due to a lack of experience. The problems encountered by students in an undergraduate course are typically one-dimensional or two-dimensional with very simple geometry that are used to illustrate the basic concepts. As a result, students do not have the opportunity to learn how to choose appropriate nodal networks as needed for complex geometries and/or boundary conditions, to apply variable nodal spacing, or to ensure smooth transitions at the interface between different materials. These subjects do not receive more attention because such analyses currently involve a prohibitive amount of programming effort and student time.

FEHT offers the advantages of a simple set of intuitive commands with which a novice can quickly learn to use for solving complex two-dimensional problems. FEHT is ideally suited for instruction in electrical engineering fields and mechanical engineering heat transfer and bio-engineering courses and for the practicing engineer faced with the need for solving practical problems.

FEHT received a Distinguished Software award in the 1990 EDUCOM/NCRIPTAL Higher Education Software Awards Competition.