Studying Stiffness Equation Assembly and Solution

  

Basic usage of the function

This function is only to simulate the assembly and solution process of the stiffness equations. There fore, the actual data, analysis results and setting of analysis options are not affected by this function. The usage of this function is described below.

Starting the function

In order to initiate this function, first create the model whose global stiffness matrix is to be displayed, and assign the element properties and the boundary conditions.

Select "System Equations" item of "FEM Simulation" submenu of menu Then, "System Equations" window opens. The window opens only when a model is created and assigned with element properties. The process of assembling and solving the system equations is simulated and visualized on this window.

Ending the function
The function is terminated by closing "Stiffness Matrix" window. Clicking the close box of the window will close the window. This function may also be terminated by starting any other menu function.

Contents of the "System Equations" window
The window contains the stiffness matrix, the force vector, a popup menu, control buttons, and texts as shown in the figure below.

 

Stiffness matrix: The stiffness matrix is initially displayed in square matrix form, but can also be displayed in a few other forms. Each entry is represented by graphics or by text depending on setting of the display mode.
  An entry is represented by one of the following 4 different appearances:
  - Empty Entry : These are the entries with no contribution from element stiffness matrices. They are not associated with any d.o.f. in the model. They have 0 values in actual stiffness matrix. Their space are left blank.
  - Zero Entry : These are the entries associated with one or more d.o.f. in the model, but their numerical value is zero. These entries are represented by hollow square boxes, , or numerical value 0.
  - Non-boundary entries : These entries are not associated with d.o.f. constrained by boundary conditions. They are represented by black square boxes, , or by black text like .
  - Boundary entries : These entries are associated with constrained d.o.f. They are represented by gray square boxes, , or by gray text like .
 

Force vector: The force vector is displayed to the right of the stiffness matrix. The appearance and the control of the force vector entries are the same as the stiffness matrix.
  Popup menu: The popup menu has the items for selection of assembly methods. The stiffness matrix is displayed by the selected method.


<Displaying the global Stiffness matrix>

 

Control buttons: There are 9 buttons to control the display of the stiffness matrix.
  Texts: There are texts indicating the half band width of the matrix, the required computer memory to store the assembled matrix, and the numerical value of the selected entry.

Selecting the assembly method

There are numerous methods of assembling and solving the system equations in finite element method. A number of typical methods are included in the function, and are shown as items in the popup menu at the top left corner of the window.

 

¡°Square Matrix¡± : The stiffness matrix is displayed in full square matrix form. It is the initial setting.
  ¡°Symmetric Sqr. Matrix¡± : The stiffness matrix is re p resented by the upper triangular part based on its symmetry.
  ¡°Band matrix¡± : The stiffness matrix is displayed in band matrix form.
  ¡°Skyline¡± : The stiffness matrix is displayed in skyline assembly form.
 

¡°Symmetric + Skyline¡± : The stiffness matrix is displayed in both upper triangular and skyline assembly so that each entry of the two assembly forms can be compared.
  ¡°Frontal solver¡± : The stiffness matrix is displayed in frontal solution form, and its elimination process is animated.

Texts related to the matrix as well as the image of the matrix is renewed immediately after the menu item is selected.

Using control buttons


There are 7 control buttons at the top of ¡°Stiffness Assembly¡± window. They are used in controlling the display of the global stiffness matrix or animation of its assembly and elimination procedure.

 

Stop button : This button is used to stop or pause the animation of assembly or elimination process.
 

Backward button : This button is used to move one step backward in the display of the assembly or elimination process.
 

Forward button : This button is used to proceed one step forward in the display of the assembly or elimination process.
 

Play button : This button is used to start or resume the animation of the assembly or animation process.
 

Text mode button : This button is used to switch the mode of displaying the stiffness matrix. Each entry of the matrix is displayed in text when the button is pressed, and represented by a small rectangle otherwise.
 

Manual assembly mode button : This button is used to turn on or off the manual assembly mode of the stiffness matrix. While this button is pressed, the element stiffness matrix can be assembled into the global stiffness matrix in any order as manipulated by the user.
  Matrix partitioning button : This button is used to partition or unpartition the stiffness matrix. The matrix is partitioned when the button is pressed, and unpartitioned otherwise.
  Matrix decomposition button : This button is used to decompose the stiffness matrix.
  Back substitution button : This button is used to simulate the back substitution phase in solving the system equations.

Getting the information on an entry of the equations


An entry of the stiffness matrix and the force vector can be selected by clicking the displayed text or image of the entry. The identity of a selected entry and its related information can be obtained from the following responses.

  The selected entry is highlighted in red.
  If the assembly method is set as ¡°Symmetric + Skyline¡±, the corresponding entries in both matrices are highlighted.
  The numerical value of the entry is displayed with label ¡°Selected value¡± in ¡°System Equations¡± window. If the stiffness matrix assembly is not
completed at the time of selection, a partially assembled value may be displayed.
  Elements and nodes associated with the selected entry are highlighted in the main window.

The selection can be switched from one entry to another without interruption by moving the cursor with the mouse button pressed.

Finding entries associated with an element

Simply click a desired element, in order to find which entries get contribution from the stiffness matrix of the element. The element is highlighted in red, and at the same time, all the entries associated with the element are also marked in red. If you select multiple elements using shift click or rubber band rectangle, the selected elements are highlighted in blue, and all the entries associated with the selected elements are highlighted in blue. In this case, the last selected element is highlighted in red, and its associated entries are also highlighted in red.

Displaying the system equations in text mode


There are 2 modes of displaying the system equations: graphic mode and text mode. The numerical values of the stiffness matrix and the force vector are represented by text strings in text mode. If button is shown as pressed like , the system equations are displayed in text mode , and in graphic mode otherwise. Click button to toggle the display mode between graphic mode and text mode. In text mode, only part of the equations are usually displayed, as the display takes larger space. Scrolling and resizing the window are necessary to display the
hidden part of the system equations.

Scrolling, resizing and zooming the window


The contents of ¡°Stiffness Matrix¡± window can be scrolled using horizontal or vertical scroll bars. Only the stiffness matrix part is scrolled. The window may be resized or zoomed if necessary, by dragging the resize box or clicking the zoom box of the window.

< Displaying the system equations in graphic mode and in text mode >