COMSOL 6.4 - Optimization of Chemical Etching

This model is based on the model Chemical Etching in the Chemical Engineering folder of the COMSOL Multiphysics Application Library, which models wet etching under laminar flow using the Deformed Geometry interface.

In the model at hand, the symmetry of the etching is optimized by allowing the concentration and convection to change over time, while constraining the average etching depth.

Model Definition

The objective is to minimize the squared deviation between the two sides:

where yspatial,mirror is constructed using a operator. The minimization is achieved by allowing the concentration at the boundaries and the wall movement to be controlled by two separate features. The geometry is symmetric before etching, so a design without etching exists as a trivial solution to this optimization. This solution can be made infeasible by imposing a minimum value on the average etching depth

Note that the constraint is scaled so that a bound equal to 1 can be used. Similarly, the objective is scaled based on the initial value. A solution to the optimization problem is found with the MMA method using move limits equal to 0.1. The optimization is limited to 25 iterations to avoid a long computational time.

Results and Discussion

Figure 1 displays the optimized control functions. The convection causes the asymmetry, so it makes sense that this decreases when the concentration is increased. However, such changes tend to violate the constraint. Therefore, those trends are reversed toward the end of the simulation, where the convection introduces less anisotropy, because the cavity is larger than early on.

Figure 1: The plot shows that the concentration is doubled throughout the first half of the simulation and then drops sharply to half the initial value toward the end. In contrast, the wall movement behaves in the opposite way, dropping to the minimum value throughout the first half and then increasing to the maximum value toward the end.

The final shape of the cavity before and after optimization is shown in Figure 2. Some of the asymmetry remains, but most of it has been removed by the optimization.

Figure 2: The plot shows the concentration and velocity field at the end of the process before (top) and after optimization (bottom). The optimization increases the symmetry significantly without reducing the etching depth.

Notes About the COMSOL Implementation

The time-dependent part of the problem is initialized with the result of a stationary problem. However, there is no support for gradient-based optimization over the combination of a solver followed by a solver. Therefore, the features are set up with a Dirichlet boundary condition for the initial time, so that the optimization can be restricted to the study step.

Optimization_Module/Optimal_Control/chemical_etching_optimization

Modeling Instructions

This example starts from an existing model from the COMSOL Multiphysics Application Library.

Application Libraries

From the menu, choose .

In the window, select > > in the tree.

Click

Study 1: Initial

In the window, click .

In the window for , type Study 1: Initial in the text field.

Results

Evaluation Group 1

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Line Average 1

Right-click and choose > .

In the window for , locate the section.

From the list, choose .

Locate the section. In the table, enter the following settings:


Expression	Unit	Description
y	m	y-coordinate

In the toolbar, click

Add the initial etching depth as a parameter together with the parameter for the time to allow the use of features with this argument in stationary solvers. Note that the value of the parameter will be overwritten by time dependent solvers.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
yavg	-319[um]	-3.19E-4 m	Initial etching depth
t	0[s]	0 s	Time

Component 1 (comp1)

The initial profile is symmetric, so it is easy to achieve a symmetric profile by avoiding etching. This trivial solution can be avoided by imposing a constraint on the etching depth using a .

Definitions

Boundary Probe 1 (bnd1)

In the window, expand the node.

Right-click > and choose > .

In the window for , type constr in the text field.

Locate the section. Click

From the list, choose .

Locate the section. In the text field, type y/yavg.

Control Function 1 (cfunc1)

In the toolbar, click

and choose .

In the window for , locate the section.

In the xend text field, type tmax.

Locate the section. In the fmin text field, type 0.1.

In the fmax text field, type 1.5.

From the list, choose .

In the f(xstart) text field, type 1 to avoid setting unphysical initial conditions for the solver.

In the c0 text field, type 1.

Locate the section. From the list, choose .

In the text field, type 3 to increase the design freedom a bit.

Control Function 2 (cfunc2)

Right-click and choose .

Use the control function to scale the concentration and wall movement.

Transport of Diluted Species (tds)

In the window, expand the > node.

Transport of Diluted Species (tds)

Concentration 1

In the window, expand the > node, then click > > .

In the window for , locate the section.

In the c0,cCuCl2 text field, type cCuCl2_bulk*cfunc1(t).

Laminar Flow (spf)

Wall 2

In the window, under > click .

In the window for , click to expand the section.

Specify the utr vector as


1[mm/s]*cfunc2(t)	x

Add a operator so that this can be used to construct an objective function that quantifies the asymmetry.

Definitions (comp1)

General Extrusion 1 (genext1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the text field, type -x.

In the text field, type Yg.

Locate the section. Select the checkbox.

In the i text field, type Yg.

Boundary Probe 2 (bnd2)

In the toolbar, click

and choose .

In the window for , type obj in the text field.

Locate the section. Click

From the list, choose .

Locate the section. In the text field, type if(isnan(genext1(y)),0, (y-genext1(y))^2) to avoid contributions where the asymmetry prevents retrieval of a y coordinate for the comparison.

Add Study

In the toolbar, click