# FRF based material constant optimization

I need advice on how to optimize a finite element model of a transversely isotropic material. I have modeled the material with 5 nominal moduli in Abaqus and performed an eigenvalue extraction. Additionally I have done a modal analysis of the modeled part, have a full set of eigenvalues and have separated bending, torsional and in-plane modes. So far so good.

The initial moduli are just a starting point as the measurement is done on coupons and the material itself is sensitive to the measurement method. What I would like to do is use my initial eigenvalue extraction in conjunction with my FRF results to calibrate the material characteristics in the model input.

I have done some searching and it appears as though a DOE approach and creating a response surface may be an option. I am concerned about the time required(FEA jobs) for a full factorial DOE with 5 factors however. Additionally, I have investigated the Matlab optimization toolbox and read about using sequential quadratic programming for this correlation. I am comfortable with Matlab but have never used that toolbox and do not know how to create a transfer function between the inputs and outputs. Finally, I know about iSight but must find an alternative.

I have only done this sort of optimization using non-commercial software.

Rather than trying to fit "the FRF" (i.e. the complete shape of all your measured response functions) I would start by estimating the frequencies and mode shapes from the measured FRFs and fitting just those parameters. In fact a good first attempt would be just to fit the frequencies, and use the mode shapes extracted from the FRFs only to identify (manually) which modes in the measured data correspond to which modes in the FE model.

The efficient way to do this is to use an optimization algorithm that uses gradient information. If the model is "close enough" to the real structure, that should converge rapidly.

The best way to get the gradients, or sensitivity coefficients - i.e. $$\frac{\partial\, \text{frequency}_i}{\partial\, \text{property}_j}, \quad \text{for } i = 1,n, \ j = 1,m$$ is to use a FE package that will calculate them directly as part of the analysis, based on the sensitivity of the element stiffness and mass data to the material properties.

I don't use Abaqus so I don't know if it (or iSight) has that capability. Trying to estimate the gradients by doing several runs of the FE model with different material properties is likely to be expensive (and possibly inaccurate).

You can then use a gradient-based optimization algorithm to find the "best" values of the properties to fit the measured data. Re-run the FE model with those values, rinse and repeat.

• On the topic of gradients; I am going to experiment with a response surface design. My goal is actually to match the eigenvalues, so the actual shape is not important. Aug 31 '16 at 0:03