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Alter to Compute Response Variance


Alter Name     cova.alt


Purpose

To compute the variance and standard deviation in response given the
user-specified variance in design variables.  Calculations are
performed by postprocessing design sensitivity results.  The following
is computed:

          COVR = S(t) * COVD * S

where     COVR  =  response variance
          S     =  sensitivity coefficient matrix
          COVD  =  design  variable variance


Solution Sequence

SOL 200.


Input:
 
Executive Control

Include 'cova.alt'
 
Case Control

None (other than standard design sensitivity requests).
 
Bulk Data

Three types of input are required:

1.  
Design sensitivity input (e.g., DESOBJ, DRESP1,
DSCREEN, DCONSTR, DESVAR, DVPREL1).

2.  
PARAM,OPTEXIT,4 (to exit after sensitivity).

3.  
DMI entries for the design variable variances.  This is a
square matrix of size ND by ND where ND is the number
of design variables.  The name of this matrix is COVD. 
The order of the terms must match the order of the
design variables (e.g., the 3,3 term in COVD is the third
design variable).  COVD can be diagonal (in which case
there is no correlation between design variables) or it can
be full (representing cross-correlation between design
variables).


Output

Five matrices are printed:

        1.  
DSCM2 - The design sensitivity coefficient matrix (size m by
n where m is the number of design variables and n is the
number of responses).

        2.  
COVD - The design variable variances (m by m, square).

        3.  
COVR - The response variances (n by n, square, full).

        4.  
VAR - The diagonal terms of COVR (n by 1).

        5.  
SDEV - The square root of each term in VAR giving the
standard deviations of responses (n by 1).

DSCM2 is computed in SOL 200;  the others are computed in
the alter.

Note:     
m is the number of independent design variables after
design variable linking (if any) and n is the number of
retained responses after constraint screening.


Limitations

This DMAP is applicable only for SOL 200.  The theory is valid for
small values of design variable variance.  With a large variance (for
example, 1000%), the theory breaks down because the sensitivity
matrix is not constant across such a wide range of design variable
values.


Sample Files

cov1.dat        10-bar truss with two load subcases (diagonal
                COVD).

cov2.dat        Same as cov1.dat, with full COVD.


Additional Documentation

K. Blakely, " Using Design Optimization for Statistical Response
Analysis," 1993 MSC World Users' Conference Proceedings.