EM Optimization using Coarse and Fine Mesh Space Mapping

Chao Zhang

IntroductionSpace mapping is a recognized engineering optimization methodology in the microwave area. The coarse models : empirical functions , equivalent circuits computationally efficient low accuracyThe fine models: provided by an electromagnetic (EM) simulator accurate computationally intensiveSpace mapping combines the computational efficiency of coarse models with the accuracy of fine models.

Algorithm

The fine model: fine mesh EM simulation high accuracy achieved by the mesh convergence process

The coarse model: coarse mesh EM local meshing without the mesh convergence process.

Models

Algorithm

the neural networkweight parameters in the neural networkthe optimal parameters of the neural network

The fine model

The coarse model

The surrogate model

The response

Input variables

The optimal solution

( )f fR x

fx

*fx

( )c cR x

cx

( )s sR x

sx

*cx *

sx

NNPw

*w

Notions

AlgorithmFunctions

The original EM optimization:

(1)Build an input space mapping neural network:

(2)Establish a surrogate model:

(3)Unit mapping: train (2) to learn .Finding the optimal solution:

(4)Error function:

(5)Termination condition: (6)

* = arg min ( ( ))f

f f fUx

x R x

= ( , )c NN s x P x w

( ) ( ) ( ( , ))s s c c c NN s R x R x R P x w=c s x x

* = arg min ( ( ))s

s s sUx

x R x

= ( ( )) d f fE U R x

dE

AlgorithmFunctions

Optimize our surrogate model to match the fine model data. The error function for this optimization is defined as

(7)The solution of the optimization:

(8)Next we perform the neural network training,

(9)Perform the surrogate design optimization. Then perform fine model evaluation and repeat (3)-(9). When termination condition is satisfied, the algorithm stops.

2( ) ( ) ( )c f f c ce x R x R x

* = arg min ( )c

c cex

x x

2* * *= arg min ( , )NN s c w

w P x w x

AlgorithmFlowchart

Start

Stopyes

dE no

DESIGN OPTIMIZATION WITH MAPPED COARSE MODEL:We perform the surrogate design optimization in EM simulator to

find , by solving .

COARSE MODEL OPTIMIZATON: Initialize the surrogate model with unit

mapping and find , by solving .

*sx

FINE MODEL EVALUATION:Evaluate the at ,

and calculate the design error .

*=f s x xUPDATE MAPPING

PARAMETERS BY COARSE MODEL OPTIMIZATION:

Optimize the design variables , by solving .

Then find the optimal , by neural network training

.

*cx

*w

*sx

( )f fR xdE

* = arg min ( ( ))s

s s sUx

x R x * = arg min ( ( ))s

s s sUx

x R x

* = arg min ( )c

c cex

x x

2* * *= arg min ( , )NN s c w

w P x w x

ExamplesA. Bandpass HTS Microstrip Filter

The bandpass microstrip filter example: the coarse mesh and fine mesh EM simulations of this structure are used as the coarse and fine models, respectively.

Space mapping optimization on bandpass filter. (a) The comparison of the coarse and fine mesh

models at initial point. (b) Responses of the fine model from each iteration of

the space mapping optimization.

3.8 3.9 4.0 4.1 4.20

0.2

0.4

0.6

0.8

1

Frequency(GHz)

|S21

|

Coarse MeshFine Mesh

3.8 3.9 4.0 4.1 4.20

0.2

0.4

0.6

0.8

1

Frequency(GHz)

|S21

| InitialIteration1

(a) (b)

ExamplesA. Bandpass HTS Microstrip Filter

COMPARISON OF DIRECT FINE MODEL OPTIMIZATION AND PROPOSED SPACE MAPPING OPTIMIZATION

Optimization algorithm

Number of fine model evaluations

Optimization time Final design error

Direct fine model optimization 71 23h34min 0.0783

Proposed space mapping 2 1h46min 0

ExamplesB. Two-Section Lowpass Elliptic Microstrip Filter

(a) (b)The lowpass elliptic microstrip filter example: the coarse mesh and fine mesh EM simulations of this structure are used as the coarse and fine models, respectively.

Space mapping optimization on lowpass filter. (a) The comparison of the coarse and fine mesh

models at initial point. (b) Responses of the fine model from each iteration

of the space mapping optimization.

1 2 3 40

0.2

0.4

0.6

0.8

1

Frequency(GHz)

|S21

|

Coarse MeshFine Mesh

1 2 3 40

0.2

0.4

0.6

0.8

1

Frequency(GHz)

|S21

|

Initial Iteration1

ExamplesB. Two-Section Lowpass Elliptic Microstrip Filter

COMPARISON OF DIRECT FINE MODEL OPTIMIZATION AND PROPOSED SPACE MAPPING OPTIMIZATION

Optimization algorithm

Number of fine model evaluations

Optimization time Final design error

Direct fine model optimization 69 25h13min 0.0001

Proposed space mapping 2 3h44min 0

Conclusion

1. The method uses coarse and fine mesh EM evaluations.2. The coarse model: coarse mesh EM simulation.3. Useful when equivalent coarse model is not available.4. Much more efficient than direct fine mesh EM optimization.