Visualizing Tensor Fields in Geomechanics
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Transcript of Visualizing Tensor Fields in Geomechanics
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Visualizing Tensor Visualizing Tensor Fields in Fields in GeomechanicsGeomechanics
Alisa Neeman Alisa Neeman ++
Boris JeremiBoris JeremiĆĆ**
Alex PangAlex Pang ++
++ UC Santa Cruz, Computer Science Dept. UC Santa Cruz, Computer Science Dept. * UC Davis, Dept. of Civil and * UC Davis, Dept. of Civil and Environmental EngineeringEnvironmental Engineering
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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)
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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
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Limitations of current techniques Hedgehog glyphs inadequate to easily
understand tensor fields Hyperstreamlines/surfaces require
separate visuals for each principal stress
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Visualization ContributionsNew 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
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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
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Element Mesh and Element Mesh and Gauss PointsGauss Points
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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
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How to make a plane in a box
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
Limit plane by box edges
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Intersection with Box Edge
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 = 0
and solve for tIterate through all 12 box edges.
P0
P2
P1
Source: http://astronomy.swin.edu.au/~pbourke/geometry/planeline
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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
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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?
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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
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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
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Seismic Moment TensorsIdea: apply moment tensor
decomposition to stress tensors, use scalars as filters to find stress features
GeomechanicsStress Tensor
Seismic/Acoustic Moment Tensor
Symmetric 3 x 3 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
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Seismic Moment Tensors
Moment Tensor Decomposition (after diagonalizing):
Mij = isotropy + anisotropy
Isotropy = (λ1 + λ2 + λ3)/3
Describes forces causing earthquake withvector dipoles: two equal and opposite vectors
along an axis orthogonal to both
Mxy:x
Y
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Moment Tensor AnisotropyAnisotropy = Double Couple +
Compensated Linear Vector Dipolemi
* = λi – isotropy
sort: |m3*| ≥ |m2
*| ≥ |m1*|
F = - m1* / m3
*
Double Couple: m3* (1 - 2F)
CLVD: m3* F
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Seismic Failure Measures1. 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 motionEigenvalues 1, 0, -1
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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
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Boussinesq Dual Point LoadEasily verify results through symmetry
Linear Scale Isotropy Log Scale color and opacity
0157,861 -157,861 011.97 -11.97
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Boussinesq Dual Point Load•Double Couple problem: find high shear
•Selects different regions than isotropy
All values High values0.975-1.0
Mid-range values(0.5-0.715)
1.0
0.0
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Bridges and EarthquakesSeries of bents support bridge
Frequency and amplitude vary with soil/rock foundation. Worst case, high amplitude (soft soils)
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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
Dou
ble
Cou
ple
DeckPile
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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
Sca
le E
igen
Diff
eren
ce
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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
-14.5
0.0
Log
Sca
le Is
otro
py
Shadow
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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
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Acknowledgements
Sponsors: NSF and GAANN Thanks to the reviewers for feedback Thanks to Dr. Xiaoqiang Zheng for
discussions
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Visualizing Tensor Visualizing Tensor Fields in Fields in GeomechanicsGeomechanics
Alisa Neeman Alisa Neeman ++
Boris JeremiBoris JeremiĆĆ**
Alex PangAlex Pang ++
++ UC Santa Cruz, Computer Science Dept. UC Santa Cruz, Computer Science Dept. * UC Davis, Dept. of Civil and * UC Davis, Dept. of Civil and Environmental EngineeringEnvironmental Engineering