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Visualizing Tensor Fields in Geomechanics

Visualizing Tensor Fields in Geomechanics. Alisa Neeman + Boris Jeremi Ć * Alex Pang +. + UC Santa Cruz, Computer Science Dept. * UC Davis, Dept. of Civil and Environmental Engineering. Motivation. Geomechanics uses tensors (stress, strain…)

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Visualizing Tensor Fields in Geomechanics

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  1. Visualizing Tensor Fields in Geomechanics Alisa Neeman +Boris JeremiĆ*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 node Gauss point • 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 y x z Finite Element (8 node brick)

  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 Limit plane by box edges • Convert plane from point-normal form to general form:Ax + By + Cz + D = 0 D = - Ax0- By0- Cz0 • A,B,C are respectively X,Y,Z components of normalized minor eigenvector • P0 is Gauss point location P0

  10. Intersection with Box Edge P1 Intersection occurs at P1 + t(P2-P1) Substitute into plane equation: A(x1 + t(x2-x1)) + B(y1 +t(y2-y1)) + C(z1 +t(z2-z1)) + D = 0and solve for tIterate through all 12 box edges. P0 P2 Source: http://astronomy.swin.edu.au/~pbourke/geometry/planeline

  11. There’s already a wayto 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 • Edge lies in the plane (infinite intersections) • 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 scalarsas filters to find stress features

  16. Seismic Moment Tensors Describes forces causing earthquake withvector dipoles: two equal and opposite vectors along an axis orthogonal to bothMxy: Y x Moment Tensor Decomposition (after diagonalizing): Mij = isotropy + anisotropy Isotropy = (λ1 + λ2 + λ3)/3

  17. Moment Tensor Anisotropy Anisotropy = Double Couple + Compensated Linear Vector Dipole mi* = λi – isotropy sort: |m3*| ≥ |m2*| ≥ |m1*| F = - m1* / m3* Double Couple: m3* (1 - 2F) CLVD: m3* F

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

  19. Eigen Difference • Measure for double degenerate tensors • K = 2λ2 − (λ1 + λ3) • 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 Linear Scale Isotropy Log Scale color and opacity Easily verify results through symmetry 157,861 0 -157,861 11.97 0 -11.97

  21. Boussinesq Dual Point Load • Double Couple problem: find high shear • Selects different regions than isotropy 1.0 0.0 All values Mid-range values(0.5-0.715) High values0.975-1.0

  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 Deck Pile • 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 Double Couple 0.0

  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 Log Scale Eigen Difference -15.03

  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 withcomputation cost) 14.5 Log Scale Isotropy 0.0 Shadow -14.5

  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 JeremiĆ*Alex Pang+ + UC Santa Cruz, Computer Science Dept. * UC Davis, Dept. of Civil and Environmental Engineering

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