# Optimization Algorithm : Prony Series

1 Definition of a Prony Series

Let GR(t) be the shear stress relaxation modulus. Define G1 and G0 by the following limits:

...

From the shear relaxation modulus we can define a dimensionless relaxation modulus from:

....

The normalized shear stress relaxation modulus is often represented by a series expansion in exponential

terms:

....

where gi and i are material parameters. This series expansion is called the Prony series.

2 Dynamic Frequency Experiments

The viscoelastic behavior of a material is often determined from dynamic vibrational experiments.

In these experiments the material is exposed to small strain vibrations and the resulting storage

modulus (G0) and loss modulus (G00) are determined as a function of the applied frequency: G0(!),

G00(!).

3 Conversion from Dynamic Frequency Data to Prony Series

Data

In most finite element packages the linear viscoelasticity model is specified by a Prony series (Eq.

4). If only dynamic frequency data is available for a material then the Prony series can be determined from the following implicit equations [see the ABAQUS theory manual for a derivation]:

.....

These equations determine the storage modulus and the loss modulus for a given Prony series. In

this case we are interested in the reverse relationship, that is, if we know G0(!) and G00(!) then

what is the Prony series (G0, N, gi, i)?

To solve this problem we can use the following approach.

1. Pick a value for N.

2. Guess values for G0, gi, i.

3. Calculate G0(!) and G00(!).

4. Calculate the residual between the calculated dynamic data and the experimental dynamic

data.

5. Optimize the parameters G0, gi, and i using any minimization algorithm (e.g. the Simplex

method). Goto 3.

.....

("A Stochastic BFGS.pdf" file has codes in the end but there is a calling function missing,i think THIS IS EXACTLY WHAT I NEED TO GET LIKE THE "Result_1_Xugang.pdf" SOLUTION FILE)

NOTE: IGNORE THE FILE "A Stochastic BFGS Based Algorithm for Estimating the Angles of Arriving Signals Given Signal Measure"

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#### Solution Summary

Prony series are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.