Visualizing Tensor Visualizing Tensor Fields in Fields in GeomechanicsGeomechanics

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

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)

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

Limitations of current techniques Hedgehog glyphs inadequate to easily

understand tensor fields Hyperstreamlines/surfaces require

separate visuals for each principal stress

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

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

Element Mesh and Element Mesh and Gauss PointsGauss Points

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

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

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

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

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?

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

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

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

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

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

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 motionEigenvalues 1, 0, -1

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

Boussinesq Dual Point LoadEasily verify results through symmetry

Linear Scale Isotropy Log Scale color and opacity

0157,861 -157,861 011.97 -11.97

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

Bridges and Earthquakes

Series of bents support bridge

Frequency and amplitude vary with soil/rock foundation. Worst case, high amplitude (soft soils)

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

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

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 I

sotr

opy

Shadow

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

Acknowledgements

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

discussions

Visualizing Tensor Visualizing Tensor Fields in Fields in GeomechanicsGeomechanics

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