4.4 Surrogate Modeling
55
The parameters of the Kriging model,
α, σ
2
f , θ
are found by using the
maximum likelihood estimation (MLE). More details on the Kriging model, and
its implementation can be found in the references [14, 37, 38].
4.5 Single-Objective Optimization
The standard form of a single-objective optimization problem is:
maximi ze
x
/ minimi ze
x
f (x)
subject to x min ≤ x ≤ x max
where f (x) is an objective function and x is a design variable vector. x min and x max
are vectors for the lower and upper bounds of the design variables, respectively.
In the case of micromixers, mixing index is a critical performance parameter
related to the performance of the device. In most cases [9, 10, 12, 14], the mixing index
evaluated using the method described in Sect. 3.2 was used as the objective function,
f (x). Parametric analysis is usually conducted to select the design variables and
their ranges for optimization. The detailed flowchart for the optimization procedure
is shown in Fig. 4.7.
The design space bounded by the lower and upper bounds of the chosen design
variables are discretized using DOE such as LHS to generate design points, which
are further used to construct the surrogate models to approximate the objective function. CFD simulations are carried out to determine the objective function values at
the design points. A surrogate model of the objective function, h(x) is approved
using a suitable error estimation technique, and is supplied as fitness function to the
optimization algorithm to determine the optimum point. Once the optimum point is
found by an algorithm, it is verified using CFD analysis. Some of the algorithms used
to find the optimum point on the surrogate model are: sequential quadratic programming (SQP) [39], particle swarm optimization (PSO) [40, 41], genetic algorithm
(GA) [42], simulated annealing (SA) [43], among others.
Afzal and Kim [4] performed a comparative evaluation of various global optimization algorithms for three practical design applications in the field of thermo-fluids
engineering. Figure 4.8 shows the results of this comparative analysis for optimization of a convergent-divergent micromixer coupled with pulsatile flow. Four different
algorithms were tested: GA, PSO, SA, and SQP, and their performances were evaluated comparatively. Among the tested algorithms, PSO showed the best overall
performance in the combined aspects of optimization result and computational time.
55
The parameters of the Kriging model,
α, σ
2
f , θ
are found by using the
maximum likelihood estimation (MLE). More details on the Kriging model, and
its implementation can be found in the references [14, 37, 38].
4.5 Single-Objective Optimization
The standard form of a single-objective optimization problem is:
maximi ze
x
/ minimi ze
x
f (x)
subject to x min ≤ x ≤ x max
where f (x) is an objective function and x is a design variable vector. x min and x max
are vectors for the lower and upper bounds of the design variables, respectively.
In the case of micromixers, mixing index is a critical performance parameter
related to the performance of the device. In most cases [9, 10, 12, 14], the mixing index
evaluated using the method described in Sect. 3.2 was used as the objective function,
f (x). Parametric analysis is usually conducted to select the design variables and
their ranges for optimization. The detailed flowchart for the optimization procedure
is shown in Fig. 4.7.
The design space bounded by the lower and upper bounds of the chosen design
variables are discretized using DOE such as LHS to generate design points, which
are further used to construct the surrogate models to approximate the objective function. CFD simulations are carried out to determine the objective function values at
the design points. A surrogate model of the objective function, h(x) is approved
using a suitable error estimation technique, and is supplied as fitness function to the
optimization algorithm to determine the optimum point. Once the optimum point is
found by an algorithm, it is verified using CFD analysis. Some of the algorithms used
to find the optimum point on the surrogate model are: sequential quadratic programming (SQP) [39], particle swarm optimization (PSO) [40, 41], genetic algorithm
(GA) [42], simulated annealing (SA) [43], among others.
Afzal and Kim [4] performed a comparative evaluation of various global optimization algorithms for three practical design applications in the field of thermo-fluids
engineering. Figure 4.8 shows the results of this comparative analysis for optimization of a convergent-divergent micromixer coupled with pulsatile flow. Four different
algorithms were tested: GA, PSO, SA, and SQP, and their performances were evaluated comparatively. Among the tested algorithms, PSO showed the best overall
performance in the combined aspects of optimization result and computational time.