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 Localization Introduction Conclusion Hypothesis Results Teaching Module Plan National Science Foundation Grant EEC- 0743068 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. http://www.matsceng.ohio-state.edu/faculty/flores/SB3_004300.jpg 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 The minimum eigenvalue method (unpublished) will detect localization even with symmetric strains.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.

Hypothesis Detection of Localization in Three Dimensional Models Andrew Mauer-Oats, RET Fellow 2009 Evanston Township and Curie Metro High School RET Mentor:

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Hypothesis

Detection of Localization in Three Dimensional Models

Andrew Mauer-Oats, RET Fellow 2009Evanston 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-0743068

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.

1Image courtesy of Dr. Katharine Flores. http://www.matsceng.ohio-state.edu/faculty/flores/SB3_004300.jpg

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.