Reanalysis Method:Direct Method

Ashvini Kumar

Structural Optimization – September 8,2014

What are reanalysis methods?

Allow you to reanalysis a structure if design variables are changed without worry about analysing whole structure again.

Pre formulated algorithms for efficient and fast reanalysis.

Direct Method

One of many methods Most common implementation of the finite element

method (i.e. [R] = [K] [r]) Most efficient if the number of modified elements in the stiffness

matrix is limited

Finite Element Method

[K] [r] = {R} [K]: Global Stiffness Matrix [r]: Vector of unknown displacement [R]: Vector of externally applied loads

Why? Suited for computer-automated analysis of complex structure including the

statically indeterminate type. Accurate representation of complex geometry

Let’s try it!

We’re going to reanalysis a very basic structure

Formulation of Reanalysis

Consider a design problem where the design variables are only the cross-sectional dimension and the loading are fixed.

The elements of [K] are functions of the design variables [X], and the element of [R] are assumed to be fixed.

If [X] changes by [dX], then [K] = [K*] + [∆K] [r] = [r*] + [∆r]

( [K*]+[∆K] ) ( [r*] + [∆r] ) = [R] [∆K] can be compressed by eliminating zero columns and rows to size

equal to the number of changed columns(or rows) in matrix [∆K] = [b]’ [∆Kr] [b]

Formulation of Reanalysis

Before Modification [K*] = [9 -6 0; -6 8 -2; 0 -2 2] [r*] = [36; 54; 96] [R] = [0; 24; 84]

After Modification [∆K] = [0 0 0; 0 5 -5; 0 -5 5] [∆Kr] = [5 -5; -5 5] [b] = [0 1 0; 0 0 1]

Direct method to reanalyse

Modified Inverse of the stiffness Matrix Modified Displacement Matrix

Modified Inverse of the Stiffness Matrix

Computing [K]-1 based on Sherman-Morrison identity

[K]-1 = [K*]-1 - [K*]-1 [b]’ ( [I] +[∆Kr] [b] [K*]-1 [b]’)-1 [∆Kr][b][K*]-1 [∆r] = [K]-1 [R] – [r*]

Upside More efficient in certain cases, compared with the Gauss elimination

procedure. Downside

Not well adapted to modification of large structures because its extremely expensive to calculate for large band matrices.

Modified Inverse of the Stiffness Matrix

Modified Displacement Matrix

Instead of solving [K]-1 , it is more efficient to solve for modified displacement using algorithm by Argyris and Roy.

[∆r] = - [K*]-1 [b]’ ( [I] +[∆Kr] [b] [K*]-1 [b]’)-1 [∆Kr][b][r*]

Upside More efficient than modified Inverse of the stiffness matrix.

Downside Not well adapted to modification of large structures because its extremely

expensive to calculate for large band matrices.

Modified Displacement Matrix

[∆r] = - [K*]-1 [b]’ ( [I] +[∆Kr] [b] [K*]-1 [b]’)-1 [∆Kr][b][r*] if [∆Kr] is singular then also ( [I] +[∆Kr] [b] [K*]-1 [b]’) can be shown

non singular Choleski decomposition of [K*] -> [K*] = [U*]’ [U] [Z] = ( [U*]’ )-1 [b]’ [Q] = [b] [K*]-1 [b]’ = [b] [U*]-1 ([U*]’)-1 [b]’ = [Z]’ [Z]

[∆r] = -[U*]-1 [Z] [Q]-1 ( [Q]-1 + [∆Kr])-1 [∆Kr][b][r*]

Evaluation of [Z] and [Q] are usually most expensive step in analysis.

Modified Displacement Matrix

Thanks!