Mathcad

Worksheet for Rotation of a Rigid Ellipsoid in Slow Flows (by D. Jiang, [email protected] last updated Jan. 2014)

Please refer to Jiang (2007, Journal of Structural Geology 29, 189-200) and Jiang (2012, Computers & Geosciences 38, 52-61) for theoretical background. 5 a  3   2

The state of the ellipsoid is given by a shape vector, like:

and an orientation specified by 3 spherical angles, written as a vector: T

x0  ( 123deg 26deg 35deg )

The flow field is defined by a velocity gradient tensor like: 0   0.3 1  L  0 0.2 0     0 0.0 0.1 

Select a step length of computation, δt like:

δt  0.01

See Jiang (2012) for the choice of δt to ensure that every step of calculation corresponds to an infinitesimal deformation. Slect the total number of computation:

STEPS  3000

and the number of calculations between output states:

mm  10

This means one output state every 10 steps of calculation. There will be a total of 300 output states (STEPS/mm =300)

Orientation tensor Q & Auxiliary Functions  sin x2  cos x1  a( x )   sin x 2  sin x 1     cos x 2    c( x )  a( x )  b ( x ) T

Q( x )  augment( a( x ) b ( x ) c( x ) )

 sin x1 sin x3  π b ( x )   cos x 1 sin x 3  if x = 2   2  cos x 3     cos atan tan x 2  cos x 1  x3    cos x3   cos atan tan x   cos x  x     sin x   otherwise 2 1 3 3    sin atan tan x 2  cos x 1  x3       u3    u 

phi( u )  acos

I  identity( 3 )

theta( u ) 

orien( u )  augment( theta( u ) phi ( u ) )



 

 u 2   u1   

z  atan

z if u  0 1

 

( π  z) otherwise

 

1 2 3 orienMatrix( x )  augment orien x orien x orien x Exp( A) 

ω

norme( A)

ω1 

2

Rodrigues rotation

A ω

I  ω1 sin( ω)  ( 1  cos( ω) )  ω1

rou1( φ) 

for i  1 

2

STEPS

rou2( φ) 

mm

for i  1 

 φi   if φi  π 2 2

2  sin

a  i

a  i

NaN otherwise

STEPS mm

 φi   if φi  π 2 2

2  cos

NaN otherwise

a

a

Upper-hemisphere points



 

Lower-hemisphere points

 

 

 

 

 

1 2 2 3 4 4 5 6 6 EqArea( z)  augment z rou1 z rou2 z z rou1 z rou2 z z rou1 z rou2 z

Angular velocity of ellipsoid and incremental rotation Θ( x ) 

D

1 2

L  LT

WL D q  Q( x ) T

d  q  D q

T

w  q  W q wd 

for i  1  3 for j  1  3 m

i j



aj 2  ai2 aj 

2

 

 ai

2

d

i j

m w  wd INCR( x ) 

a  Q( x ) q1  Θ( x ) K1  Exp( q1 δt)  a

 TT

x1  orienMatrix K1 q2  Θ( x1)

K2  Exp( q2 0.5 δt)  K1

 TT

x2  orienMatrix K2 q3  Θ( x2)

K3  Exp( q3 0.5 δt)  K2

 TT

x3  orienMatrix K3 q4  Θ( x3) K4  Exp( q4 δt)  K3 bb 





1 T T T T q1  a 2 K2  q2 K2  K3  q3 K3  K4  q4 K4  a  6

b  Exp( bb δt)  a

 T

orienMatrix b

T

PP ( x ) 

y  INCR( x ) 1

a  Θ( x ) for i  2  mm

 i1T

y  INCR y i

y AA 

T mm T

x  PP ( x0) 1

mx

T 1

for i  2 

STEPS mm

 i1T

M  EqArea( AA)

x  PP x i

m  stack m x



T i 

In the equal-area plots below, x-axis = east; y-axis = north, and z-axis = up. Dark points are in the upper hemisphere and light ones the lower hemisphere

2 M 3 M

1 M

a1-axis path

5 M 6 M

4 M

a2-axis path

8 M 9 M

7 M

a3-axis path