Hypothesis Detection of Localization in Three Dimensional Models Andrew Mauer-Oats, RET Fellow 2009 Evanston Township and Curie Metro High School RET Mentor: Professor Craig Foster NSF-RET Program LocalizationIntroduction Conclusion Hypothesis Results Teaching Module Plan  National Science Foundation Grant EEC  Professor Andreas Linninger, Program Director  Dr. Gerardo Ruiz, RET Program Managing Director  Dr. Craig Foster, Research Mentor  University of Illinois at Chicago  IMSA students: Tasha A. and Irene C. Acknowledgements  Students will be able to explain the ideas of eigenvalues and eigenvectors.  Students will be able to explain the importance of linear algebra in computational modeling.  Students will be able to create hypotheses from data.  Students will be able to test and refine their hypotheses based on further experimental information. Computer Simulations  Computational mechanics studies the behavior of materials using computer simulations.  Benefits of simulations:  Cheaper than actual experiments.  Provide data that is not practical to measure in experiments. (Example: internal strain.)  Simulations must detect localization to be useful. cerebral artery Localization in steel reinforced concrete. 1 Image courtesy of Dr. Katharine Flores. Shear bands in glass. 1  Localization in a solid material is the formation of shear bands or cracks.  Localization is easy to see in real life, but in computer modeling it can be difficult to detect and handle correctly.  Localization usually leads to structural failure. Key Question  How can we detect localization when strains are equal in two different directions, like a cylindrical column supporting building? Determinants and Eigenvalues A matrix has two measures of how it changes vectors in the plane: determinant and eigenvalues.  One Determinant: Multiply a times b.  Fast to compute.  Tells total change in volume.  Two Eigenvalues: a, b.  Slower to compute.  Tell stretching along each axis.  The minimum determinant method of [Ortiz, 1985] fails to detect localization when there are symmetric strains because the symmetry causes the determinant to always be positive.  The minimum eigenvalue method (unpublished) will detect localization even with symmetric strains. Minimum eigenvalues change dramatically in a short period of time at the onset of localization.  The minimum eigenvalue method can be implemented with a reasonable computational overhead.  Visualization of minimum eigenvalues provides a way of seeing the approach of localization.  Graphical validation allows detection of anomalies in model. Norm of change vs. time shows evolution of state.  Minimum eigenvalue method is competitive with minimum determinant in von Mises J2 model.  Detection algorithm runs 40% faster after optimization.  Visual verification methods detected several problems in the underlying implementation of the model. Minimum eigenvalue vs. H parameter.