On the physical interpretation of Gaussian elimination

1.   Introduction

 Gaussian elimination is widely used in finite element analysis software’s for the solution of the equilibrium equation: {F} = [k] {u}. The physical interpretation of Gaussian elimination is an interesting fact which often goes un-noticed by most of the Engineers and mechanists!  One might conclude after reading this write-up that the resultant stiffness matrix obtained during each step in Gaussian elimination process corresponds to the stiffness matrix obtained by “condensing out” or “releasing” a degree of freedom in each successive step.

 

This write-up attempts to explain the physical interpretation of Gaussian elimination. Section 2 of this write up revisits the mathematical operations in Gaussian elimination process. Section 3 describes the physical process during the Gaussian elimination procedure.

     

Let us consider a structural system as shown in the Figure 1 below for which the equilibrium equation [k] {u} = {F} is denoted as below;

Equation Set 1

EI = 1 unit and L = 5 units

 

Figure 1: Structure with 4 translational degrees of freedom

 

If one were to generate the above stiffness matrix [k] from the first principles, one would apply unit translations at degrees of freedom 1,2,3,4 successively and each time measures the forces at all the degrees of freedom. The resultant matrix obtained will be as in the above equation. Next, if one were to solve the above system of equation for the kinematic degrees of freedom U1, U2, U3, U4 the mathematical operations as illustrated in section 2 will have to be carried out.

2 .   Mathematical operations involving Gaussian elimination

 

The mathematical operations involved in solving the above set of simultaneous equations can be outlined as below;

 

Step 1: Subtract a multiple of the first equation from the second and the third equations to obtain zero elements in the first column of the stiffness matrix [K]. This means that -4/5 times the first row is subtracted from the second row and 1/5 times the first row is subtracted from the third row. The resulting equations are;

Equation Set 2

Step 2: Subtract a multiple of the second equation from the third and the fourth equations to obtain zero elements in the second column of the stiffness matrix [K]. This means that we need to subtract -16/4 times the second equation from the third equation and 5/14 times the second equation from the fourth equation to get zero elements in the second column of the above stiffness matrix. The resulting equations are;

Equation Set 3

Step 3: Subtract a multiple of the third equation from the fourth equation to obtain zero elements in the third column of the stiffness matrix [K]. This means that we need to subtract -20/15 times the third equation from the fourth equation. The resulting equations are;

Equation Set 4

Thus, the procedure in the solution is therefore: to subtract in step number i in succession multiples of equation i from equations i+1, i+2, …., n -1. In this way the coefficient matrix K is reduced to upper triangular form, i.e. a form in which all elements below the diagonal elements are zero. Starting from the last equation, it is then possible to solve all for all unknowns in order Un, Un-1,..U1.

3.   Physical interpretation of the mathematical operations

Considering the mathematical operations carried out during the Gaussian elimination process, it may be noted that the operations are independent of load vector on the right hand side of the equation. For, ease of explanation; let us consider the case wherein no loads have been applied on the structure, so that,

Equation Set 5

Let us now, statically condense out u1 from the above,

Considering the first equation from the above system, we can write;

Equation Set 6

Substituting for U1 in the equations 2, 3, and 4 of the above matrix system will result in the following;

Writing the above in matrix form,                                                  Equation Set 7

Equation Set 8

The above Equation Set 8 is in fact the same as Equation Set obtained during the Gaussian elimination process- the step 1 wherein zero elements are obtained in the first column of the stiffness matrix.

 

This means that the resulting coefficient stiffness matrix obtained during the first step of the Gaussian elimination process corresponds to the stiffness matrix that is obtained by “statically condensing” out or “releasing” the first degree of freedom of the structure as below.

 Figure 2: Structure with the first degree of freedom condensed out

 

 

Step 1 (of Gaussian elimination process): Subtract a multiple of the first equation from the second and the third equations to obtain zero elements in the first column of the stiffness matrix [K]. This means that -4/5 times the first row is subtracted from the second row and 1/5 times the first row is subtracted from the third row. The resulting equations are;

 

 

 Equation Set 2

Note: The resulting coefficient stiffness matrix obtained in the above step of Gaussian elimination is the same as the matrix obtained by “statically condensing” / releasing the first degree of freedom. The corresponding structure is shown in Figure 2

 

With the same reasoning, it is possible to show that the resulting coefficient stiffness matrix obtained during the second step of the Gaussian elimination process corresponds to the stiffness matrix that is obtained by “statically condensing” out or “releasing” the first and the second degrees of freedom of the structure as below.

 

Figure 3: Structure with the first and second degrees of freedom condensed out

Step 2 (of Gaussian elimination process): Subtract a multiple of the second equation from the third and the fourth equations to obtain zero elements in the second column of the stiffness matrix [K]. This means that we need to subtract -16/4 times the second equation from the third equation and 5/14 times the second equation from the fourth equation to get zero elements in the second column of the above stiffness matrix. The resulting equations are;

 

 

 Equation Set 3

 

Note: The resulting coefficient stiffness matrix obtained in the above step of Gaussian elimination is the same as the matrix obtained by “statically condensing” / releasing the first and the second degrees of freedom. The corresponding structure is shown in Figure 3.

 

Again, the resulting coefficient stiffness matrix obtained during the third step of the Gaussian elimination process corresponds to the stiffness matrix that is obtained by “statically condensing” out or “releasing” the first, second and third degrees of freedom of the structure as below.

Figure 4: Structure with the first, second and the third degrees of freedom condensed out

Step 3: Subtract a multiple of the third equation from the fourth equation to obtain zero elements in the third column of the stiffness matrix [K]. This means that we need to subtract -20/15 times the third equation from the fourth equation. The resulting equations are;

Equation Set 4

Note: The resulting coefficient stiffness matrix obtained in the above step of Gaussian elimination is the same as the matrix obtained by “statically condensing” / releasing the first second and third degrees of freedom. The corresponding structure is shown in Figure 4.