טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentZarrouk David
SubjectSolution Convergence of the Functional Perturbation Method
for Problems of Nonhomogeneous Structures
DepartmentDepartment of Mechanical Engineering
Supervisor Professor Emeritus Eli Altus
Full Thesis textFull thesis text - English Version


Abstract

Micro-sized beams, routinely used in sensors and actuators, present non-homogeneities in dimensions and properties. Morphology simulations are usually based on pre-determined Two Point Correlation (C2) functions, while higher correlations are neglected. There are several commonly used C2-based methods for modeling such fields, for comparing analytical methods with Monte-Carlo Simulations (MCS). In this study, different “linear” methods which satisfy S’=Lη are studied. S’ is the deviation of the property, L is a matrix and η is the uncorrelated random variables. It has been found that each method leads to different high order statistical correlation values and that they do not ensure high order statistical homogeneity. These higher order terms become important in predicting strength reliability especially in problems with high heterogeneity. A new method (HVM) that ensures homogeneity in all orders is presented herein.

The present study proposes a more effective utilization of the one and two point morphological data by finding their contributions to the higher order terms and adding them to the solution.

    The research includes:

  • An examination of the effects of using different modeling methods on the three and four point morphological terms.
  • An evaluation of the expanded contribution of the one and two point correlations to the higher order functional expansion using the Functional Perturbation Method (FPM).
  • A comparison of the average and variance of reactions and buckling forces of heterogeneous beams, with the MCS using the new methodology.

    Conclusions and results:
  • The different stochastic fields constructed using different methods may visually look pretty different even though they share the same C2. Quantitatively, the difference is exhibited in the higher order correlations.
  • MCS made on fields built using different methods yielded to different solutions. In many cases, the difference between the MCS performed on different fields is around 10%.
  • A new homemade method (HVM) which ensures higher order homogeneity is introduced in this study. The HVM is the only method that can produce large vectors of correlated variables on a regular PC (the order of 106.
  • An exact value of the C4 as a function of the matrix L and the density function of η is found.
  • The 4th order FPM solution is calculated for the first time. The comparison between the 2nd and 4th FPM solutions to the MCS shows that 4th order FPM solution reduced the error by at least 50% and sometimes up to 90% .