An Introduction to Modeling and Simulation of Particulate Flows Part 4 pot

05 book

2007/5/15

page 39

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Chapter 5

Inverse

problems/parameter

identification

An important aspect of any model is the identification of parameters that force the system

behavior to match a (desired) target response. For example, in the ideal case, one would

like to determine the type of near-field interaction that produces certain flow characteristics,

via numerical simulations, in order to guide or minimize time-consuming laboratory tests.

As a representative of a class of model problems, consider inverse problems, where the

parameters in the near-field interaction representation are sought, the α’s and β’s, that

deliver a target particulate flow behavior by minimizing a normalized cost function

 =

 T

0 |A − A∗| dt

 T

0 |A∗| dt , (5.1)

where the total simulation time is T , A is a computationally generated quantity of interest,

and A∗ is the target response. Typically, for the class of problems considered in this work,

formulations () such as in Equation (5.1) depend, in a nonconvex and nondifferentiable

manner, on the α’s and β’s. This is primarily due to the nonlinear character of the near￾field interaction, the physics of sudden interparticle impact, and the transient dynamics.

Clearly, we must have restrictions (for physical reasons) on the parameters in the near-field

interaction:

α−

1 or 2 ≤ α1 or 2 ≤ α+

1 or 2 (5.2)

and

β−

1 or 2 ≤ β1 or 2 ≤ β+

1 or 2, (5.3)

where α−

1 or 2, α+

1 or 2, β−

1 or 2, and β+

1 or 2 are the lower and upper limits on the coefficients

in the interaction forces.24 With respect to the minimization of Equation (5.1), classical

gradient-based deterministic optimization techniques are not robust, due to difficulties with

objective function nonconvexity and nondifferentiability. Classical gradient-based algo￾rithms are likely to converge only toward a local minimum of the objective function unless

a sufficiently close initial guess to the global minimum is not provided. Also, it is usually

24Additionally, we could also vary the other parameters in the system, such as the friction, particle densities,

and drag. However, we shall fix these parameters during the upcoming examples.

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