52
4 Design Optimization of Micromixers
4.4 Surrogate Modeling
In optimization of thermo-fluid problems, a large number of numerical analyses are
required to evaluate objective function(s), but a single numerical simulation takes
long time to complete due to the non-linear governing differential equations. Surrogate modeling is introduced to alleviate this burden by constructing approximation
models, which mimic the behavior of the simulation model as closely as possible.
The major advantages of surrogate modeling are computational economy in evaluating the objective function(s), and accuracy in representing the characteristics of
the design space. The mathematical formulations for different surrogate models are
discussed below.
4.4.1 Response Surface Approximation
Response surface approximation (RSA) model is the simplest, yet widely used surrogate model for studying the underlying relationship in the data [35]. RSA models are
extension of linear regression models which contain additional features/predictors
generated using non-linear transformation of the input space to account for
non-linearity in the model.
In the linear regression model, a continuous response y is usually modeled as a
linear combination of the predictor, x =
x 1 x 2 . . . x p
T
plus a random error, ∈.
For N independent observations (i = 1…, N) of p design variables of the predictor x
and response y, the model can be represented as:
y i = β 0 + β 1 x 1,i + β 2 x 2,i + · · · + β p x p,i + ∈ i (0, σ
2
∈ )
(4.1)
where β j (j = 0…, p) are the coefficients of the model. Errors, ∈ i are assumed to be
uncorrelated and distributed with a mean of 0 and constant variance, σ
2
∈ . In compact
form, Eq. (4.1) can be written as:
Y = X β+ ∈
(4.2)
X =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 x 1,1 x 2,1 · · · x p,1
. . .
. . .
. . . · · ·
. . .
1 x 1,i x 2,i · · · x p,i
. . .
. . .
. . . · · ·
. . .
1 x 1,N x 1,N · · · x p,N
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
, Y =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
y 1
. . .
y i
. . .
y N
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
X is the design matrix of the system. For RSA model, Eq. (4.1) can be modified
as:
4 Design Optimization of Micromixers
4.4 Surrogate Modeling
In optimization of thermo-fluid problems, a large number of numerical analyses are
required to evaluate objective function(s), but a single numerical simulation takes
long time to complete due to the non-linear governing differential equations. Surrogate modeling is introduced to alleviate this burden by constructing approximation
models, which mimic the behavior of the simulation model as closely as possible.
The major advantages of surrogate modeling are computational economy in evaluating the objective function(s), and accuracy in representing the characteristics of
the design space. The mathematical formulations for different surrogate models are
discussed below.
4.4.1 Response Surface Approximation
Response surface approximation (RSA) model is the simplest, yet widely used surrogate model for studying the underlying relationship in the data [35]. RSA models are
extension of linear regression models which contain additional features/predictors
generated using non-linear transformation of the input space to account for
non-linearity in the model.
In the linear regression model, a continuous response y is usually modeled as a
linear combination of the predictor, x =
x 1 x 2 . . . x p
T
plus a random error, ∈.
For N independent observations (i = 1…, N) of p design variables of the predictor x
and response y, the model can be represented as:
y i = β 0 + β 1 x 1,i + β 2 x 2,i + · · · + β p x p,i + ∈ i (0, σ
2
∈ )
(4.1)
where β j (j = 0…, p) are the coefficients of the model. Errors, ∈ i are assumed to be
uncorrelated and distributed with a mean of 0 and constant variance, σ
2
∈ . In compact
form, Eq. (4.1) can be written as:
Y = X β+ ∈
(4.2)
X =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 x 1,1 x 2,1 · · · x p,1
. . .
. . .
. . . · · ·
. . .
1 x 1,i x 2,i · · · x p,i
. . .
. . .
. . . · · ·
. . .
1 x 1,N x 1,N · · · x p,N
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
, Y =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
y 1
. . .
y i
. . .
y N
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
X is the design matrix of the system. For RSA model, Eq. (4.1) can be modified
as: