# Background: computational model updating

Computational model updating is an interesting ICS service to validate FE models. By updating multiple selected parameters of the model simultaneously a minimization of the test/analysis differences can be achieved.

The model is validated in two steps:

- Updating of stiffness and mass properties (physical parameters)
- Updating of modal damping (modal parameters)

In a first step only stiffness and mass properties (physical parameters) are updated by minimizing the test/analysis differences e. g. of eigenvalues and mode shapes. ICS.sysval, a special MATLAB® program, is utilized that takes advantage of the MSC.Nastran™ analysis capabilities (in particular the sensitivity module). Thus large scale FE models can be processed. The parameter changes are directly applied to the "Bulk Data" section (the section in which the FE model is defined) of the MSC.Nastran™ input deck. Proceeding this way allows for an update of any physical parameter that can be defined in MSC.Nastran™ (e. g. shell thicknesses, beam section properties, Young's moduli, mass densities).

In order to handle complex elastomechanical systems a decomposition into components is usually necessary (reduction of uncertain parameters). The components are individually updated and the quality of the (modified) assembly is assessed subsequently. If the model quality is not yet sufficient further updating may be performed considering only the interface parameters (e. g. stiffnesses of connection elements).

After successful updating of stiffness and mass properties (physical parameters) modal damping (modal parameters) may be updated in a second step. Here the differences between measured and analytic frequency responses are minimized in the vicinity of the resonance peaks. This is accomplished utilizing another special MATLAB® program. The overall goal is to achieve high quality FE analysis predictions (at least in the covered frequency range) and thus a validated model.

In the following the updating theory will shortly be summarized.

*Updating of physical parameters:*

The basis for computational model updating of physical stiffness and mass parameters is the parameterization of the model matrices according to (1) (see [1], [2]). This parameterization allows for local updating of uncertain model areas.

(1a) K = K_{A} + ∑ α_{i} K_{i}, i = 1...n_{α}

(1b) M = M_{A} + ∑ β_{j} M_{j}, j = 1...n_{β}

with: | |

K_{A}, M_{A} |
initial analytical stiffness/mass matrices |

p = [α_{i},β_{j}] |
vector of unknown design parameters |

K_{i}, M_{i} |
given substructure matrices defining the location and type of model uncertainties |

Using equations (1) and appropriate residuals (containing different test/analysis differences, e. g. eigenvalue and mode shape differences, frequency response differences) the following objective function can be derived:

(2) J(p) = Δz^{T} W Δz + p^{T} W_{p} p → min

with: | |

Δz | residual vector |

W, W_{p} |
weighting matrices |

The minimization of equation (2) yields the desired design parameters p. The second term in equation (2) is used to constrain the parameter variation. The weighting matrix W_{p} as to be selected with care since for W_{p} >> 0 no design parameter changes will occur.

The residuals Δz = z_{T} - z(p) (z_{T}: test data vector, z(p): orresponding analytical data vector) are usually nonlinear functions of the design parameters. Thus the minimization problem is also nonlinear and must be solved iteratively. One way is to utilize the classical sensitivity approach (see reference [2]) where the analytical data vector is linearized at point 0 by a Taylor series expansion truncated after the first term. Proceeding this way leads to:

(3) Δz = Δz_{0} - G_{0} Δp

with: | |

Δp = p - p_{0} |
design parameter change |

Δz_{0} = z_{T} - z(p_{0}) |
test/analysis difference at linearization point 0 |

G_{0} = ∂z/∂p|p = p_{0} |
sensitivity matrix at linearization point 0 (the determination for different types of residuals can be found in [1], [2]) |

p_{0} |
design parameter vector at linearization point 0 |

If the design parameters are not bounded the minimization problem (2) leads to the linear problem (4) which has to be solved in each iteration step for the actual linearization point.

(4) (G_{0}^{T} W G_{0} + W_{p}) Δp = G_{0}^{T} W Δz_{0}

For W_{p} = 0 equation (4) represents a standard weighted least squares problem. Of course any other mathematical minimization technique may be applied as well to solve equation (2).

*Updating of modal parameters:*

In order to update modal damping parameters the classic sensitivity approach can also be used. The residual in this case is the difference between measured and analytic frequency responses (see e. g. [3]).

## References::

[1] Link, M.:

"Updating of Analytical Models - Review of Numerical Procedures and Application Aspects"

Proceedings of the Structural Dynamics Forum SD2000, Los Alamos, New Mexico, USA, April 1999

[2] Natke, H. G.:

"Einführung in die Theorie und Praxis der Zeitreihen- und Modalanalyse"

3., überarb. Aufl., Vieweg Verlag, Braunschweig, Wiesbaden, 1992

[3] Schedlinski C.:

""Computational Model Updating of Large Scale Finite Element Models" "

Proceedings of the 18th International Modal Analysis Conference, IMAC, San Antonio, Texas, USA, 2000

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