3.2 The method
3.2.2 Numerical optimization
The numerical optimization problem used to infer inter-particle forces from experi- mental measurements requires three sets of governing equations: momentum balance, stress-force relations, and constraint equations.
The momentum balance equations for a deformable body are derived as follows.
The balance of linear momentum arises from noting that the total force on a de- formable particle equals the time derivative of its linear momentum:
d dt
Z
Vp
ρvdv
!
= Z
Vp
ρbdv+ Z
∂Vp
tds (3.1)
whereρis the particle density,bis the body force,tare surface tractions,vis velocity, and Vp and ∂Vp represent deformed grain volumes and edges, respectively. Taking
x
˜ x
x1
x2
x3
Vp
xcmp
fj fi ti
xi
BOUNDARY OF REGION OF INTEREST
ei
fk
Figure 3.1: Illustration of various terms used in the governing equations.
the derivative and ignoring body forces, one obtains
Ncp
X
α=1
fα = Z
Vp
ρadv (3.2)
where fα is a contact point as shown in Fig. 3.1 and Ncp is the total number of contact points for particlep. We have made use of the fact that integrating tractions on disjoint sections of a surface is equivalent to summing the total force on each of those sections. We can further reduce Eq. (3.2) by making use of the definition of center-of-mass acceleration to obtain
Ncp
X
α=1
fα=mpacmp (3.3)
The balance of angular momentum arises from noting that the total torque on a deformable particle is equal to the time derivative of its angular momentum:
d dt
Z
Vp
x×ρvdv
!
= Z
Vp
x×ρbdv+ Z
∂Vp
x×tds (3.4)
wherex is a position vector and× represents a cross product of two vectors. Taking the derivative and ignoring body forces, one obtains
Ncp
X
c=1
xc×fc= Z
Vp
x×ρadv (3.5)
where we have made use of the same argument as above to turn the integral into a sum while approximating the contact point xc as the center of the contact area.
Now define x=xcmp + ˜x and a=acmp + ˜a, where xcmp and acmp are the position and acceleration of the center-of-mass, respectively. Plugging these into Eq. (3.5) yields
Ncp
X
c=1
xc×fc=mp(xcmp ×acmp ) + Z
Vp
ρ(˜x×a) dv˜ (3.6)
Note that the second term on the right hand side of this equation is negligibly small when accelerations within a grain remain small relative to its center-of-mass acceler- ation.
Linear and angular momentum balance, Eqs. (3.3) and (3.6), can be combined into a single matrix equation for a group of contacting particles, taking the form
i j
. .. 0 · · · 0 · · · p 0 Kim 0 Kjm 0
... 0 . .. 0 ... q 0 −Kim 0 0 0
... 0 ... 0 . ..
| {z }
K
m
... fi
... fj
...
| {z }
f
=
... bpm
... bqm
...
| {z }
b
m(3.7)
where p and q represent particles, i and j represent particle-particle and particle- boundary contacts as shown in Fig. 3.1, and Kim, fi, and bpm encompass Eqs. (3.3)
and (3.6) and in 2D take the form
Kim =
1 0
0 1
−xi2 xi1
; fi =
f1i f2i
(3.8)
bpm =
mpacm1 mpacm2
mp(xcm1 acm2 −xcm2 acm1 )
(3.9)
where we have omitted the integral term from Eq. (3.6) for brevity. In general, 3Np equations are contained in Eq. 5.1, where Np is the number of particles in the region of interest. Extension of Kim, fi, and bpm to 3D is straightforward.
The stress force relations are derived as follows. The volume-averaged stress in a deformable particle can be written as
σ¯p = 1 Vp
Z
Vp
σdv (3.10)
Left-multiplying the σ within the integral by the identity matrix xi,k and switching to index notation yields
¯ σij = 1
Vp Z
Vp
[(xiσkj),k−xiσkj,k] dv (3.11) Using the divergence theorem on the first term inside the integral of this equation and balance of linear momentum in the absence of body forces (σkj,k =ρaj) on the second term, we obtain
Ncp
X
c=1
xc⊗fc =Vpσ¯Vp+ Z
Vp
x⊗ρadv (3.12)
where ⊗ is a dyadic product (a ⊗b = aibj). Finally, defining x = xcmp + ˜x and
a=acmp + ˜a as above, we obtain
Ncp
X
c=1
xc⊗fc=Vpσ¯ +mp(xcm⊗acm) + Z
Vp
x˜⊗ρ˜adv (3.13)
Once again, we note that the third term on the right hand side of this equation is negligibly small when accelerations within a grain remain small relative to its center- of-mass acceleration.
The stress-force relation in Eq. (3.13) can also be written into a matrix equation for a group of contacting particles, taking the form
i j
. .. 0 · · · 0 · · · p 0 Kis 0 Kjs 0
... 0 . .. 0 ... q 0 −Kis 0 0 0
... 0 ... 0 . ..
| {z }
K
s
... fi
... fj
...
| {z }
f
=
... bps
... bqs
...
| {z }
b
s(3.14)
where fi is the same as in Eq. (3.8), and Kis and bps encompass Eq. (3.13) and in 2D take the form
Kis =
xi1 0
0 xi2 xi2 xi1
(3.15)
bps =
Vpσ¯11+mp(xcm1 acm1 ) Vpσ¯22+mp(xcm2 acm2 ) 2Vp¯σ12+mp(xcm1 acm2 +xcm2 acm1 )
(3.16)
where we have omitted the integral terms from Eq. (3.13) for brevity. In general, 3Np equations are also contained in Eq. 3.14. Extension of Kis and bps to 3D is
straightforward.
Constraint equations can also be written for a group of cohesionless grains obeying a Coulomb-type friction law. These constraints in 2D require that (fi·ei) ≥ 0 and (µei±ti)·fi ≥0 whereei andtiare the normal and tangential to the contact surface of contact point i as shown in Fig. 3.1 and µ is the inter-particle friction coefficient.
This friction coefficient can be measured from a separate experiment performed on the particle material and may be either dynamic or static depending on the nature of the experiment. In 3D, the tangential direction must be specified by two vectors and the Coulomb friction constraint written above must be modified during implementation in the numerical optimization problem. We will continue with the 2D form and discuss the 3D extension briefly below.
The cohesionless force and Coulomb friction constraint equations can be expressed as a matrix equation for a group of contacting particles, taking the form Bf ≥ 0 where
B =
i i j j
i ei1 ei2 0 0 0
... 0 0 . .. 0 0
j 0 0 0 ej1 ej2
i µei1+ti1 µei2+ti2 0 0 0
... 0 0 . .. 0 0
j 0 0 0 µej1+tj1 µej2+tj2 i µei1−ti1 µei2−ti2 0 0 0
... 0 0 . .. 0 0
j 0 0 0 µej1−tj1 µej2−tj2
(3.17)
and f = (f1if2i. . . f1jf2j)>. In general, 3Ncp constraints are contained in Eq. (3.17), where Ncp is the total number of contacts.
The mechanics of a group of contacting grains is fully governed by the three matrix Eqs. (5.1), (3.14), and Bf ≥0. Every term in these equations except for the inter- particle forcesf must be measured during an experiment as described in section 3.2.1.
Once measured, finding the set of forces f that best fits the measured data involves solving a multi-objective optimization problem of the form:
f = arg min
f
(||Ksf −bs||2 +λ||Kmf −bm||2) (3.18a) subject to: Bf ≥0 (3.18b) Here, the objective functions ||Ksf −bs||2 and ||Kmf −bm||2 represent the desire to minimize the difference between inferred forces and experimental measurements of stress and momentum, respectively. The valueλis a weight that conveys the relative importance of the two objectives. No previous knowledge of boundary forces is needed as in past implementations of this inverse problem [13].
In order to solve Eq. (3.18), a suitable value ofλ must be selected. With no prior knowledge of the type or magnitude of error in stress or momentum measurements, the best strategy is to seek a solution which does not favor one set of measurements (e.g., momentum or stress) over another. To do this, we solve Eq. (3.18) with many values of λ and obtain a curve of ||Ksf −bs||2 versus ||Kmf −bm||2, as shown in Fig 3.2. All points on this curve for which neither objective can be decreased without increasing the other are referred to as Pareto optimal points, and the resulting curve is referred to as the optimal trade-off curve or the Pareto front [74].
||Kmf bm||2
||Ksf bs||2
KNEE POINT
1 1
OPTIMAL TRADE-OFF CURVE / PARETO FRONT
Figure 3.2: Pareto front.
In principle, any point on the optimal trade-off curve is a possible solution to the multi-objective problem. However, in order to use as much information in the two objective functions as possible, we wish to select a point where the slope of the trade-off curve is steep. At a steep point, a small decrease in one objective function is accompanied by a small increase in the other. This steep “knee” of the optimal trade- off curve therefore represents the solution where we satisfy each objective without making an unfavorably large sacrifice in the other.
Finally, we mention that when solving Eq. (3.18) for 2D experiments, special care must be taken because of the nonzero thickness of the disks or cylinders involved. In particular, the mass used in Eqs. (5.1) and (3.14) must be mass per unit thickness (i.e., actual mass divided by thickness of the particles) and the volume in Eq. (3.14) must be the area of the face of the particle. The result of solving Eq. (3.18) will be force per unit thickness, from which actual forces can be obtained by multiplying by the particle thickness.
In 3D, a contact surface between two particles must be specified by a normal vector and two orthogonal tangent vectors, ti1 and ti2, rather than one. Defining qi1 and q2i to be the magnitude of f projected onto each tangent vector, the multi-objective optimization problem takes the form
f = arg min (||Ksf −bs||2 +λ||Kmf −bm||2) (3.19a) subject to: ei·fi ≥0 (3.19b) q1i = (ti1·f) (3.19c) q2i = (ti2·f) (3.19d) µ(ei·fi)− ||qi|| ≥0 (3.19e)
where ||qi|| = ((q1i)2 + (q2i)2)1/2. Many of the constraints in Eq. (3.19) can be cast in matrix form as in Eq. (3.18). However, the last constraint is a second-order cone constraint which is not present in the 2D problem and cannot be written in matrix form, instead requiring a norm [74]. For more details on this type of constraint,
see [74].