is physically admissible since noise may bias the result towards a force distribution with unrealistically high tangential to normal force ratios or attractive contact forces.
The constraint (2.13c) can also be modified to eliminate the dependence on Coulomb friction when a Coulomb friction law is unjustified or when the friction coefficient cannot be estimated. When Coulomb friction is abandoned, the restriction will only require that normal forces be repulsive.
Conditions for the existence and uniqueness of a solution to equation (2.13) are easy to understand. It must be emphasized, however, that existence and uniqueness of a solution to equation (2.13) does not imply that the solution corresponds to the true inter-particle forces but rather to a solution that minimizes the error between experimental observations and calculations made with the governing equations of the problem in an L2-norm sense.
Let S = {f|Keqf = 0,Bf ≥ 0}, the set of force vectors that satisfy the con- straints (2.13b) and (2.13c). A solution to (2.13) exists when S is nonempty since the cost function in (2.13a) is bounded below by 0, e.g., when Kst is positive semi- definite. Furthermore, S is nonempty when Keq has more columns than rows, and when Bf ≥ 0 is solvable, which can be ensured in practice by choosingµ conserva- tively. The solution is unique if there is no w 6=0 such thatw ∈ N(Kst)∩ N(Keq) (see proof and other conditions in Theorem 1 of [66]); such a w could be added to any existing solution without changing the value of the cost function in (2.13a) or violating the equality (2.13b).
From a physical perspective, such a w is unlikely to exist: nonzero forces in N(Kst) satisfy Kstf = 0 and must cause rigid body particle motion while forces in N(Keq) satisfy Keqf = 0 and must result in equilibrium. When Keq and Kst
are rank deficient, such a w may exist and additional criteria must be satisfied to ensure uniqueness; namely, the w ∈ N(Kst)∩ N(Keq) must violate the restrictions of constraint (2.13c) to ensure that the solution to (2.13) remains unique. It is merely stated here that the authors have never found such a w to exist in both numerical simulations and experiments, ensuring the uniqueness of solution to (2.13).
2.3.2 Measurement noise and alternative formulation
Experimental imaging techniques contain error, or measurement noise. In addition, algorithms used to extract strain fields, contact locations, contact planes, and con- stitutive law parameters introduce noise. This noise manifests itself in the matrices Keq, Kst, and B and in the vector bst. While high-fidelity imaging can typically ensure negligibly small error in the point-wise quantities used to populateKeq,Kst, and B, the vectorbst requires accurate estimation of the particle constitutive model and associated parameters, and involves a sum over all point-wise measured stresses in a particle, potentially introducing significant error.
To account for the possibility of significant measurement error in bst a simple alternative to the inverse problem (2.13) is proposed which incorporates knowledge of boundary forces, quantities that are typically found by using load cells in experi- ments. The alternative method is motivated by the experience that when the solution to (2.13) is affected by measurement error in the constitutive law, the relative sizes of forces remains relatively unchanged (i.e., all forces are generally over- or underes- timated). The alternative inverse problem is given by:
f = arg min
f kKstf −bstk2+λ2kfk2 (2.14a) subject to: Keqf =0 (2.14b)
Bf ≥0 (2.14c)
where λ is a regularization parameter, to be discussed.
Problem (2.14) employs Tikhonov regularization, a common technique in the so- lution of ill-posed or rank-deficient inverse problems (e.g., [67, 68]). Tikhonov regu- larization can be interpreted as a method for incorporating prior knowledge of the magnitude of f and as a technique for selecting a “smoother” solution. This can be seen intuitively: as λ is increased, the solution will decrease until each value of f approaches 0. Furthermore, the solution will typically decrease with some uniformity in that each value of f will approach 0 at a rate proportional to its size. It is im-
portant to note, however, that the solution to problem (2.14) will still satisfy particle equilibrium.
The primary challenge in using the alternative form (2.14) is selecting λ to find a tradeoff between noise reduction and loss of information [68]. When the structure of noise in bst is well known, many methods exist to select λ, although all methods have known issues and limitations [69]. The technique used in this chapter to select λ is simple and has been proven effective by experience: gradually increase λ until a solutionf to (2.14) minimizes the difference between known and calculated boundary forces. Example of implementing this procedure are presented in the next section and the experimental example in section 2.1. In general, the mathematical framework presented in this section may be extended to incorporate penalty functions other than Tikhonov regularization, but it is beyond the scope of this chapter to address all of these possibilities. It is important to note that solution of equation (2.14), as with the solution of equation (2.13), will produce forces that satisfy particle equilibrium.
2.3.3 Implementation
Many solvers and optimization packages exist to solve the minimization problem (2.13) or (2.14). In particular, implementations of SeDuMi [70] such as CVX [71] and Yalmip [72] provide efficient ways to solve the problem in Matlab. In addition, any numerical optimization package capable of solving quadratic programming problems can be used to solve (2.13) and (2.14) when the cost functions in these problems are expanded to quadratic form (e.g.: min{fTKTstKstf − 2bTKstf}). Since the matrices Keq and Kst may not be full rank in practice, solvers capable of handling rank deficient matrices may be preferred.