Partial Refactorization in Sparse Matrix Solution: A New Possibility for Faster Nonlinear Finite Element Analysis

Algorithm 2

Sparse direct methods of refactorization.

Step 1.  Input original stiffness matrix 𝐊, upper triangular matrix 𝐔 and diagonal matrix 𝐃;

Step 2.  Create array CHANGE(𝑛) with the initial value zero, where 𝑛 is the number of

equations. CHANGE(𝑖)=0 indicates row 𝑖 is not changed, while CHANGE(𝑖)=1 indicates

row 𝑖 is changed;

Step 3.  Input the change of each element stiffness matrix and assemble them to original

total stiffness matrix. Then let CHANGE(𝑖) be 1 if row 𝑖 is changed;

Step 4.  According to Rule, spread the effect of changes over the whole matrix and mark

all the changed rows:

Step 5.  Assemble matrix to the original factor. If CHANGE(𝑖)=0, fill in row 𝑖 with

original factorization result; if CHANGE(𝑖)=1, fill in row 𝑖 with entries of new stiffness matrix;

Step 6.  Numerically factorize the matrix, only recalculating entries in changed rows

(CHANGE(𝑖)=1) according to (5). Only changed row is added to the elimination tree, in

implementation being represented by the linked list.

Step 7.  The solution procedure after factorization stays the same.