Features 

Operates on all Macintosh and Windows based computers 

Supports color and math-coprocessor 

Graphical problem definition 

Zoom and drawing template import features 

Cartesian and cylindrical coordinates 

English and SI units 

Steady-state and transient problems 

Triangular finite elements 

Automatic mesh reduction 

Time-dependent boundary conditions 

Potential (e.g., temperature) dependent BC's 

Time, potential and position dependent properties 

Graphical output e.g., color contours, flux, flow 

Numerical output option 

Up to 12,000 nodes 

Extremely fast execution 

Presentation-quality output 

License holders of the Windows version can download the latest version of FEHT.  Install the update in your current FEHT folder.

Download the FEHT manual in Adobe Acrobat (.pdf) format.
 

FEHT is an acronym for Finite Element Heat Transfer. FEHT was originally designed to facilitate the numerical solution of steady-state and transient two-dimensional conduction heat transfer problems. However, the fundamental equations describing conduction heat transfer, bio-heat transfer, potential flow, steady electric currents, electrostatics, and scalar magnetostatics are similar. The current version of FEHT has been designed to solve problems in all of these disciplines. Versions of FEHT have been developed for the Apple Macintosh series computers and for PC compatible computers using the Microsoft Windows 95/98/NT operating systems. 

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. 

By default, FEHT is configured for steady-state heat transfer problems in Cartesian coordinates. These characteristics apply to this practice problem and do not need to be changed. 

It is usually best to set the unit system, scale, and grid spacing at the start of a problem although they can be changed at any time. Pull down the Setup menu and select the Scale and Size command which will bring up a dialog window in which the scale attributes can be entered. 

The small circles in the Scale and Size dialog window shown below are called radio buttons. Radio buttons control the unit system for heat transfer problems. To change the unit system, move the mouse to position the cursor on the appropriate button and click the mouse button. A reasonable scale for this problem is to have 1 cm on the screen represent 0.25 m. Any item within a box can be changed. The unit for length is, by default, centimeters but it can be changed from millimeters to kilometers by clicking in the units box to the right of the scale value. (In the English system, the length unit can be changed from inches to miles.) Click in the scale units box until it displays m for meters. Then enter 0.25 in the scale edit box. Note that double-clicking within any edit box causes the characters to be highlighted (shown in inverse). Typing any character will replace the highlighted field with the entered character. X0 and Y0 designate the location of the origin of the coordinate system on the screen in screen coordinates. The default values, X0=0.0, Y0=0.0 correspond to the origin being placed at the lower left of the screen. Gridlines make the drawing easier to prepare. Grid spacing is specified in the same coordinates as for the drawing. Set the grid spacing as shown below. Click the OK button or press the Enter key to set these scale attributes. 

The first step is to sketch material outlines. It is easier to prepare a scale drawing with a coordinate grid. Select Show Grid from the Display menu. Select Outline from the Draw menu. Since this problem is symmetrical, we only need to analyze one quarter of the furnace. Note that the X and Y coordinates of the cursor position for the selected unit system and scale are shown in the small window at the upper left, just below the menu bar. Move the mouse to locate the cursor at position X=0.50 m, Y=2.50 m. Click the mouse to fix a node at the corner. The first node is shown as a small closed circle. Now, position the cursor at X=3.00 m, Y=2.50 m and click the mouse. As a drawing aid for horizontal and vertical lines, the X or Y position of the cursor can be locked by holding down the Shift or Ctrl keys, respectively. Click on the remaining corners at X=3.00 m, Y=0.50 m; X=2.00 m, Y=0.50 m; X=2.00 m, Y=1.50 m; and X=0.50 m, Y=1.50 m. Now click on the first corner. The outline must begin and end on the same node without crossing any existing lines. At this point, the outline will flash, indicating that the outlining process is completed and the material within the flashing boundary is selected. The outline number and name are shown in the center information window below the menu bar. The area enclosed by the outline is shown in the right information window. The screen should now look like this. 

A material must be selected (flashing) in order to specify its properties. A material can be selected by clicking the mouse anywhere within it outline; it is automatically selected just after it has been drawn. Select Material Properties from the Specify menu. A property dialog box will appear with default property names listed on the left. Choose the material to be brick by clicking on Building Brick in the list on the left. You can choose the pattern which will be used to identify the material by holding the mouse button down while the cursor is in the pattern box. A pop-up palette will appear with the possible patterns. The color of the pattern can also be selected in the same manner by clicking in the Color box. The thermal properties of brick are displayed. These properties can be changed to other values. They may also be entered as a function of temperature and position. Leave the brick properties at their default values. The properties dialog box should now appear like this. 

Click the OK button and the screen will be redrawn with the pattern you have chosen identifying the material. 

Next we will set the boundary conditions. The vertical line at X=0.50 m and the horizontal line at Y=0.50 m are lines of symmetry and therefore there is zero heat flow across these lines. Move the cursor to a point near the center of one of these lines and click the button. The line should now be flashing. Move the cursor to the center of the other line and click. Both lines should now be flashing. Once a boundary is selected (flashing), the Boundary Conditions menu item in the Specify menu becomes accessible. Select this menu item to bring up the Boundary Conditions dialog window. Enter 0 (for adiabatic conditions) in the Heat Flux box and click the OK button. A check mark will automatically appear in before the word Heat Flux to confirm that you are setting a heat flux boundary condition. 

The two boundary lines are now shown with bold lines to indicate that the boundary conditions have been specified. 

The inside and outside walls of the brick are convective boundaries. Click the mouse at a point near the center of each outside line causing them to flash. Again, select Boundary Conditions from the Specify menu Enter a convection coefficient of 5 W/m²-K and a fluid temperature of 30 C. The dialog window will appear as shown. Click OK. 

The convection boundary information for the inside furnace walls is entered in the same manner. Select both boundaries and again issue the Boundary Conditions command. Enter the convection coefficient of 10 W/m²-K and a fluid temperature of 560 C. Click the OK button. 

At this point, you may first wish to exactly locate the node positions if you were unable to do so while drawing with the mouse. Click on the node at the upper left of the drawing and then select the Boundary Conditions command in the Specify menu. (Note that, as a short cut, you can accomplish the same result by double-clicking the mouse on the node.) The Specify Node Temperature dialog window will appear with edit boxes for the node temperature and for the X and Y coordinates of the node. In this case, we wish to alter the coordinates, not the node temperature. Enter the coordinates X=0.50 and Y=2.50 as shown below. FEHT will not let the node move to a position which causes existing lines to cross. Note the FEHT displays the coordinates in exponential notation. The E means 'time ten to the power of'. You may wish to repeat this process for all of the other nodes so that they are exactly positioned at the proper locations. 

To complete the problem definition, it is necessary to discretize the brick material into triangular elements. The triangular elements are formed by placing element lines within the material. The positions of these lines and the number of triangular elements is determined by the user. One purpose of FEHT is, in fact, to allow the effect of differing discretizations to be quickly determined. 

It is easier to construct element lines when the brick pattern is hidden; the pattern can be removed by selecting Hide Patterns from the Display menu. The grid lines are no longer needed; select Hide Grid from the Display menu. Select Element Lines from the Draw menu. Move the cursor to the upper left node at position X=0.50 m, Y=2.50 m and click the button. Now, click at point at X=1 m, Y=2.00 m. (Note that is it not important to exactly locate the nodes. Anywhere near this point is fine.) A node will be created at this point and an element line will be drawn between the two nodes. Continue this process of constructing element lines until the screen appears as shown below. 

The following rules apply to manual element line construction. 

1. The first end of the line must be on an existing line or node. A new node will form at this point if one is not already there. 
2. Element lines can not cross existing lines. 
3. Clicking in the area surrounding the drawing or pressing the Esc key cancels the Element Lines command. 

The problem definition is complete. Select the Calculate command from the Run menu to initiate the calculations. FEHT will first check the problem definition to ensure that the distributed materials are properly discretized and all properties and boundary conditions are specified. Any errors detected will be listed in the information window at the upper right of the screen, just below the menu bar. This example problem is assumed to be steady-state. Had this been a transient problem (by selecting Transient from the Setup menu), a dialog box would have appeared in which the start, stop and step times would be entered. If no errors are found, a dialog window will appear indicating that the calculations are in progress. When the calculations have been completed, the dialog box will display the elapsed time and other information. 

Click the Continue button. A number of the output display windows in the View menu will now be accessible. 

The temperatures within the wall can be displayed at each node by selecting Temperatures from the View menu. 

Temperature can also be displayed as a contour plot by selecting Temperature Contours which will bring up the following dialog window. 

Three types of contour plots are available: continuous colors from red to blud, a banded plot showing gradations of hot to cold (as shown below) and a plot of lines of constant temperatures. The minimum and maximum values in the contour plot can be entered manually or FEHT will automatically find the limits if you click in the User/Auto box at the upper left. Click the OK button or press the enter key to show the contour plot. Note that the contour plot show appears crude because very few nodes were employed.  FEHT allows up to 8000 nodes but only 16 were used in this analysis. 

In either the temperature or contour plot output, the temperature at the cursor position will be displayed at the upper left of the screen below the menu bar when the mouse button is depressed. 

One objective of this problem was to determine the total heat flow through the brick wall. Select Heat Flows from the View menu. (It is also possible to determine the heat flow using the Nodal Balances command, as described in Chapter 2 of the FEHT manual.) The screen will be redrawn with the nodes hidden. Click on any line segment of the inside wall. An arrow will appear indicating the direction of heat flow. The magnitude of the heat flow is shown in the information window at the top right of the screen below the menu bar. Clicking on adjacent lines forming the inside boundary in a clockwise or counterclockwise manner will allow the heat flows to be summed. For one quarter of the problem, the heat flow through the furnace wall is 961.9 W. The total heat flow through all sides of the wall is then 3847.6 W. 

Select Input from the View menu to return to the drawing window. At this point, you may wish to explore. Try using smaller triangular elements. FEHT will automatically reduce the mesh size for you if you select the Reduce Mesh Size command in the Draw menu. Will the smaller mesh significantly change the heat transfer rate?