1. Visualizing Tensor Fields in Geomechanics Alisa Neeman + Boris Jeremi Ć * Alex Pang + Alisa Neeman + Boris Jeremi Ć * Alex Pang + + UC Santa Cruz, Computer Science Dept. * UC Davis, Dept. of Civil and Environmental Engineering

2. Motivation Geomechanics uses tensors (stress, strain…) –to understand the behavior of soil and their relation to foundations, structures –analysis of failure of bridges, dams, buildings, etc. Understand accumulated stress and strain in geological subduction zones (which trigger earthquakes and tsunamis)

3. Features of Interest Positive stress in piles: cracking concrete large shear stresses: shear deformation or shear failure zones of sign changes: tensile failure These usually occur at soil-pile boundary but can happen anywhere Bonus Features Capture global stress field Verification and Validation: assess accuracy of simulation

4. Limitations of current techniques Hedgehog glyphs inadequate to easily understand tensor fields Hyperstreamlines/surfaces require separate visuals for each principal stress

5. Visualization Contributions New stress glyph, plane-in-a-box  Cheap and interactive  Shows general trends in volume  Glyph placement issues addressed through size, thresholding, opacity Test of four scalar measures to detect critical features

6. Geomechanics Data Symmetric 3 x 3 stress tensors (diagonalizable) Materials with memory –Single time step OR single loading iteration Gauss points –As nodes move, stress induced at Gauss points –Gauss rule provides most accurate integration –Irregular layout on X, Y, and Z Finite Element (8 node brick) node Gauss point x z y

7. Element Mesh and Gauss Points

8. Plane-in-a-Box Plane created from 2 major eigenvectors Normal implies minor eigenvector Box size limited by half-distance to neighbors (reduce occlusion) Given connectivity, grid need not be regular to establish box

9. How to make a plane in a box Convert plane from point-normal form to general form: Ax + By + Cz + D = 0 D = - Ax 0 - By 0 - Cz 0 A,B,C are respectively X,Y,Z components of normalized minor eigenvector P 0 is Gauss point location P0P0 Limit plane by box edges

10. Intersection with Box Edge Intersection occurs at P 1 + t(P 2 -P 1 ) Substitute into plane equation: A(x 1 + t(x 2 -x 1 )) + B(y 1 +t(y 2 -y 1 )) + C(z 1 +t(z 2 -z 1 )) + D = 0 and solve for t Iterate through all 12 box edges. P0P0 P2P2 P1P1 Source: http://astronomy.swin.edu.au/~pbourke/geometry/planeline

11. There’s already a way to draw planes in boxes… Marching Cubes designed for isosurfaces in regular grids –Above-below index –Interpolation points Loop around interpolation points to draw triangles Some ambiguous cases

12. Drawing with Marching Cubes! Edge index: sum of box edges the plane intersects (labels 1,2,4,8,..) Map from edge intersection index to Marching Cubes index Intersection points act as interpolation points No ambiguous marching cubes cases –We build a continuous surface so no holes occur Ambiguous edge index cases, though Why?

13. Ambiguous Edge Index Cases 1. Edge lies in the plane (infinite intersections) 2. Plane coincides with a box corner (three edges claim intersection) Workaround: shift box along an axis slightly - proper marching cubes case forms Shift box back BEFORE forming triangles - get correct plane

14. Filtering With Physical Parameters Scalar Features –Color, opacity show feature magnitude Threshold, Inverted Threshold Filtering Isosurface and isovolume-like selections (without smooth surface) Opacity Filtering Goal: find zones of positive stress, sign changes, and high shear

15. Seismic Moment Tensors Idea: apply moment tensor decomposition to stress tensors, use scalars as filters to find stress features Geomechanics Stress Tensor Seismic/Acoustic Moment Tensor Symmetric 3 x 3 tensor Elastic or elastic-plastic material Elastic material Describes force on external surface Force across internal surface (causing movement along fault) Values throughout volume from simulation A few point sources from measured acoustic emission

16. Seismic Moment Tensors Moment Tensor Decomposition (after diagonalizing): M ij = isotropy + anisotropy Isotropy = (λ 1 + λ 2 + λ 3 )/3 Describes forces causing earthquake with vector dipoles: two equal and opposite vectors along an axis orthogonal to both M xy : x Y

17. Moment Tensor Anisotropy Anisotropy = Double Couple + Compensated Linear Vector Dipole m i * = λ i – isotropy sort: |m 3 * | ≥ |m 2 * | ≥ |m 1 * | F = - m 1 * / m 3 * Double Couple: m 3 * (1 - 2F) CLVD: m 3 * F

18. Seismic Failure Measures 1.Pure isotropy: explosion or implosion 2.CLVD: change in volume compensated by particle movement along plane of largest stress. Eigenvalues  2, -1, -1 3.Double couple: two linear vector dipoles of equal magnitude, opposite sign, resolving shear motion Eigenvalues  1, 0, -1

19. Eigen Difference Measure for double degenerate tensors K = 2λ 2 − (λ 1 + λ 3 ) K > 0 planar (identical major and medium eigenvectors) K < 0 –linear (identical medium and minor eigenvectors). Applied universally across volume

20. Boussinesq Dual Point Load Easily verify results through symmetry Linear Scale IsotropyLog Scale color and opacity 0157,861-157,861011.97-11.97

21. Boussinesq Dual Point Load Double Couple problem: find high shear Selects different regions than isotropy All valuesHigh values 0.975-1.0 Mid-range values (0.5-0.715) 1.0 0.0

22. Bridges and Earthquakes Series of bents support bridge Frequency and amplitude vary with soil/rock foundation. Worst case, high amplitude (soft soils)

23. Two Pile Bridge Bent Piles penetrate halfway down into soil Circular appearance in planes’ orientation –boundary effects in simulation –model needs to be expanded to more realistically simulate half- space No pure double couple –Discrete nature of field 1.0 0.0 Double Couple Deck Pile

24. Bridge Bent Pushover Force applied at bridge deck (top of columns) Simulation: pushover followed by shaking Eigen difference shows sudden flip between linear and planar in piles +15.03 -15.03 Log Scale Eigen Difference

25. Isotropy: Inverted threshold Log scale isotropy: lowest 25% and top 25% Zones switching sign highlighted Shadowing effect: right hand pile ‘in shadow’ Border effects (tradeoff with computation cost) 14.5 -14.5 0.0 Log Scale Isotropy Shadow

26. Conclusions Isotropy most useful scalar feature Thresholding/inverted thresholding highlights behavior under stress Plane-in-a-box provides global perspective of stress orientation Algorithm cheap and interactive Assists with simulation verification and validation

27. Acknowledgements Sponsors: NSF and GAANN Thanks to the reviewers for feedback Thanks to Dr. Xiaoqiang Zheng for discussions

28. Visualizing Tensor Fields in Geomechanics Alisa Neeman + Boris Jeremi Ć * Alex Pang + Alisa Neeman + Boris Jeremi Ć * Alex Pang + + UC Santa Cruz, Computer Science Dept. * UC Davis, Dept. of Civil and Environmental Engineering