58
4 Design Optimization of Micromixers
problem is either dominated or non-dominated. The relative dominance can be established using the following conditions: A design, x 1 dominates a design, x 2 , if x 1 is
at least as good as x 2 for all objectives and x 1 is strictly better than x 2 for at least one
objective. All of the designs that are non-dominated to any other design, comprise a
Pareto-optimal set. The functional space representation of the Pareto-optimal solution set is the Pareto-optimal front. The Pareto-optimal solution represents the tradeoff among conflicting objectives, and can be used to analyze the trade-offs among
designs. Because each solution is a global Pareto-optimal solution, none of these
Pareto-optimal solutions is superior to the others for both objectives. Thus, the choice
by the designer is important when selecting a Pareto-optimal solution that meets a
given requirement.
To obtain Pareto-optimal solutions, a MATLAB built-in function, gamultiobj,
can be used to invoke MOGA [30–32]. The function uses a controlled elitist genetic
algorithm (a variant of NSGA-II [29]) which uses elite individuals differently than the
genetic algorithm. It sorts non-inferior individuals above inferior ones, thus the elite
individuals are automatically used. In non-elitist MOGAs, the genetic operator may
destroy some of the non-dominated solutions to explore the design space. Introducing
elitism in MOGAs alleviates this problem to some extent [45]. A hybrid function
was used for subsequent minimization after the genetic algorithm was terminated.
The hybrid solver starts at all points on the Pareto front returned by MOGA. The new
individuals returned by the hybrid solver are combined with the existing population,
and a new Pareto front is obtained. The algorithm terminates based on the convergence
criterion specified by the user. The inputs to the algorithm can affect the development
of Pareto-optimal front, and therefore, several cases need to be simulated to select
the correct parameters for the genetic algorithm.
The optimization procedure to obtain the Pareto-optimal solutions is shown in
Fig. 4.9. As mentioned earlier in Sect. 3.2, the mixing index is the primary objective
function related to the performance of a micromixer. Another candidate for objective
function is pressure loss. In the multi-objective optimizations of Afzal and Kim [30,
31], the aim was to simultaneously maximize mixing performance and minimize
pressure loss in a multi-objective optimization framework.
A Pareto-optimal front is established with optimized trade-offs among different
objective functions. The obtained Pareto-optimal front and the corresponding Paretooptimal designs (PODs) need to be analyzed to identify relations among different
objectives. Therefore, some of representative PODs, which cover both the design and
functional spaces, are selected by performing k-means clustering. It is an iterative
process for forming clusters [46]. Further, numerical simulations are conducted to
determine the objective function values at the representative PODs obtained from
clustering. These values are then compared with the corresponding objective function values obtained from the optimization. This exercise helps to determine the
relative accuracies of the CFD modeling, adopted function approximations, and
multi-objective optimization algorithm.
4 Design Optimization of Micromixers
problem is either dominated or non-dominated. The relative dominance can be established using the following conditions: A design, x 1 dominates a design, x 2 , if x 1 is
at least as good as x 2 for all objectives and x 1 is strictly better than x 2 for at least one
objective. All of the designs that are non-dominated to any other design, comprise a
Pareto-optimal set. The functional space representation of the Pareto-optimal solution set is the Pareto-optimal front. The Pareto-optimal solution represents the tradeoff among conflicting objectives, and can be used to analyze the trade-offs among
designs. Because each solution is a global Pareto-optimal solution, none of these
Pareto-optimal solutions is superior to the others for both objectives. Thus, the choice
by the designer is important when selecting a Pareto-optimal solution that meets a
given requirement.
To obtain Pareto-optimal solutions, a MATLAB built-in function, gamultiobj,
can be used to invoke MOGA [30–32]. The function uses a controlled elitist genetic
algorithm (a variant of NSGA-II [29]) which uses elite individuals differently than the
genetic algorithm. It sorts non-inferior individuals above inferior ones, thus the elite
individuals are automatically used. In non-elitist MOGAs, the genetic operator may
destroy some of the non-dominated solutions to explore the design space. Introducing
elitism in MOGAs alleviates this problem to some extent [45]. A hybrid function
was used for subsequent minimization after the genetic algorithm was terminated.
The hybrid solver starts at all points on the Pareto front returned by MOGA. The new
individuals returned by the hybrid solver are combined with the existing population,
and a new Pareto front is obtained. The algorithm terminates based on the convergence
criterion specified by the user. The inputs to the algorithm can affect the development
of Pareto-optimal front, and therefore, several cases need to be simulated to select
the correct parameters for the genetic algorithm.
The optimization procedure to obtain the Pareto-optimal solutions is shown in
Fig. 4.9. As mentioned earlier in Sect. 3.2, the mixing index is the primary objective
function related to the performance of a micromixer. Another candidate for objective
function is pressure loss. In the multi-objective optimizations of Afzal and Kim [30,
31], the aim was to simultaneously maximize mixing performance and minimize
pressure loss in a multi-objective optimization framework.
A Pareto-optimal front is established with optimized trade-offs among different
objective functions. The obtained Pareto-optimal front and the corresponding Paretooptimal designs (PODs) need to be analyzed to identify relations among different
objectives. Therefore, some of representative PODs, which cover both the design and
functional spaces, are selected by performing k-means clustering. It is an iterative
process for forming clusters [46]. Further, numerical simulations are conducted to
determine the objective function values at the representative PODs obtained from
clustering. These values are then compared with the corresponding objective function values obtained from the optimization. This exercise helps to determine the
relative accuracies of the CFD modeling, adopted function approximations, and
multi-objective optimization algorithm.