We present control strategies of a diffusion process for chemical vapor deposition for metallic bipolar plates. In the models, we discuss the application of different models to simulate the plasma-transport of chemical reactants in the gas-chamber. The contribution are an optimal control problem based on a PID control to obtain a homogeneous layering. We have taken into account one- and two-dimensional problems that are given with constraints and control functions. A finite-element formulation with adaptive feedback control for time-step selection has been developed for the diffusion process. The optimization is presented with efficient algorithms. Numerical experiments are discussed with respect to the diffusion processes of the macroscopic model.

We motivate our studying on simulating a low-temperature
low-pressure plasma that can be found in chemical vapor deposition (CVD)
processes. In the last years the research and optimization in producing
high-temperature films by depositing low-pressure processes have increased by
using simulation tools, see [

We discuss a model for low-temperature and
low-pressure plasma that can be used to implant or deposit thin layers of
important materials, see [

To solve such optimization problems, we present a
PID-controller (proportional, integral, differential) to control our deposition
process, see [

Numerical methods are described in the context of time- and spatial-discretization methods for the mesoscopic-scale model. We discussed different experiments and their convergence rates.

For the simulations we apply analytical and also numerical methods to obtain results to control the grow of thin layers.

The paper is outlined as follows. In Section

In the following, the models are discussed in two directions of far-field and near-field problems:

reaction-diffusion equations, see [

Boltzmann-Lattice equations, see [

Dynamic processes with modifications in time and space will be reshaped by partial differential equations. There is (i) the PDE-formula itself which describes the physical laws of nature that influence the process and (ii) initial and boundary conditions in which specific characteristics of the process, like boundary behavior, can be coded.

There are two types of boundary conditions, namely,
Dirichlet and Neumann boundary. With the Dirichlet type the exact value of the
boundary is known, however, with Neumann boundaries the spatial derivation of
the boundary values in normal direction is known, see an example of the
boundary conditions in Figure

Dirichlet and Neumann boundary conditions.

Spatial-discretization.

We will concentrate us on a continuum model of mass
transportation and assume that the energy and momentum is conserved, see
[

We modify our model equation (

We assume an optimal concentration at the
layer

Additionally, we have to solve the minimization
problem:

We choose the

In a first part, we only solve the transport equation
with UG software-tool (unstructed grid software, see [

In a second part, we consider the optimal control problem and solve also the backward problem.

In what follows we discuss the approximation methods and errors for the simulation of the CVD processes.

For the numerical solutions we need to apply
approximation methods, for example, finite-difference methods and iterative
solver methods for the nonlinear differential equations, see
[

The finite-element discretization is based on

This leads to the following linear semidiscretized
system of ordinary differential equations:

Here we have taken into account the
Courant-Friedrichs-Levy- (CFL-) condition, which is given as

For the explicit time-discretization, we apply explicit Euler or Runge-Kutta methods.

We use the explicit lower-order Runge-Kutta
methods:

Furthermore we use the following Heun method
(third-order):

The implicit time-discretization is done with implicit Euler or Runge-Kutta methods.

Here, we use the implicit trapezoidal
rule:

Furthermore we use the following Gauss Runge-Kutta
method:

We apply implicit
time-discretization methods for the pure diffusion part, where we apply
explicit time-discretization methods for the pure convection part. Here we have
to respect the CLF-condition, see [

For studying the errors and the convergence-rates in our test example, we have to define the following norm in two space-dimensions:

discrete

discrete

discrete

The numerical convergence rate are given as follows.

For the spatial error, we define

For the time error, we define

We often use

Here we discuss the control of a diffusion equation with a feedback based on a PID-controller.

The first controller we discuss is the simple
P-controller, see [

In Figure

P-controller
for the solution

Our control problem is given with the control of the
error to the optimal concentration of the layer and correct the
source-flux:

We assume an optimal concentration at the layer with
the concentration

Our constraints are bounded as

Taken into account the hysteresis of
the deposition process, we apply a linear increase of

Linear
constraint

The PID-controller is used to control temperature,
motion, and flow. The controller is available in analog and digital forms, see
[

We have three elements in the PID-control, where

These terms describe three basic mathematical
functions applied to the error signal,

The errors represented the difference between constraint (optimal set) and computed results in the simulation.

To accelerate a PID-controller means to adjust the
three multipliers

Figure

PID-control. Effect on control system: the main
influence in a control loop,

The algorithm of the initialization of the PID-control (i.e.,
search

(1)
We initialize the P-controller:

(2) The amplifying factor

(3) We obtain for

(4) The period-length of the permanent oscillation is given
as

(5) We obtain the parameters from Table

Heuristic derivation of the control parameters (Nichols-Ziegler).

Controller | |||
---|---|---|---|

P | |||

PI | |||

PID |

Further we compute the rest parameters as

Often the heuristic assumptions of the PID-parameters are too coarse.

One can improve the method by applying an adaptive step-size control.

We discuss the step-size control with respect to our underlying error, that is, given by the computed and optimal output of our differential equation.

Based on the adaptive control, we can benefit to accelerate the control problem.

According to Hairer and Wanner [

The automatically step-size is given as (see [

We can control the step-size with respect to our
heuristically computed

(1) Define Tolerance, Min and Max of the concentration.

(2) Apply the parameters:

(3) Optimize the computations with a first feedback.

In the forward problem, we computed a PDE to
simulate the CVD process in the underlying domain.
For our control problem, we have the behavior of two points in the underlying
domain,

(1)
We determine the model of the control path, for example, PT1 (Proportional time 1), PT2
(Proportional time 2), see [

(2) We determine the parameters of our model: Kp, T1, T2.

(3) Our goal is to control the system with a
controller. Also we have to determine the control-parameter [

With the step response

Because

For such control path which
has PT2 behavior, there
exists a model as ordinary differential equation. Here a generalization can be
obtained with four parameters and be divided by

So that we have the analytical solution for the normed
step response:

Firstly, we have

Step response and inflection tangent.

Empirical determination of the model of the control
path. The three main cases: (1)

The parameter occurrence also in the transfer function

Derivation of the control parameters (Chien-Hrones-Reswick and Takahashi).

PID | Takahashi | ||
---|---|---|---|

MATLAB-toolbox pdetool.

MATLAB function | Description | Enhancement |
---|---|---|

Parabolic | Solve parabolic PDE (heat equation) | In the future we plan to use
academical code to compute the PDE-solutions (e.g., |

Pdetool | MATLAB toolbox to create the geometry of the FEM-structure and the boundary conditions | We have also implemented an alternative mesh, with orthogonal triangles. MATLAB always uses equilateral triangles |

Refine | This function refines the geometry of the mesh. All triangles, were replaced with four new triangles | For convergence rate calculations we must guarantee that the geometry near the points, which are changed in the backward step (source), are similar |

Guide | MATLAB toolbox to create graphical user interfaces (gui) |

We use the MATLAB-toolbox pdetool for the time- and
spatial-discretizations, where we have a finite-element method with

The PID-controller is also programmed in MATLAB. Our combined code is given in the DEPOSIT-PID toolbox and described in what follows. The DEPOSIT-PID toolbox is manipulated by a graphical user interface, by which simulation- and control-models are chosen and corresponding parameters can be manually adjusted.

The objective is to simulate the diffusion and deposition of the vapor in the apparatus and to obtain the optimal vapor concentration at the measuring point.

We can divide the process into three phases, namely, forward step, control step, backward step, which have a cyclic repetition. In the forward step, the diffusion takes place, which can be simulated by a time-step of the heat equation.

After that, in the control step, the actual
concentration at the under boundary

The control step is followed by the backward step:
from the error in control step and the control model, the optimal alteration of
the source can be computed. The vapor flows through the source point

Backward step.

Layout of the DEPOSIT-PID Gui.

Convergence diagram tool.

In Section

This Gui contains the following:

short-time plot (2D) of computed, optimal, and SourceOutput;

long-time plot (2D);

listbox with names parameters:

textbox with actual value of the parameter chosen in [

textbox
to change the value of the parameter chosen in [

listbox with names
parameters:

textbox with actual value of the parameter chosen in [

textbox to change the value of the parameter chosen in [

3D plot of distribution;

3D grid plot of distribution;

listbox with names
of parameters:

checkbox with actual value of the parameter chosen in [

push button: save;

push button: reset;

radio button: start.

Another software-tool, KONTOOL, is programmed to compute the numerical
convergence rates of the applications. The software-tool has implemented the errors
and convergence rates defined in Section

The software-tool can be modified and applied to arbitrary spatial- and time-discretization methods. The interface of KONTOOL needs at least the parameters of the spatial- and time-grid and the starting parameters of the underlying methods.

(1) We compute reference solutions: (a) numerically: fine time and spatial steps or (b) analytically (if there exists an analytical solution).

(2) We apply one spatial discretization
of step

(3) We continue the next fine
spatial steps, for example,

(4) We compute the convergence tableau with time and space.

In the next section, we discuss the numerical experiments.

In this section, we present our numerical experiments for the CVD processes in a plasma reactor.

Here we simulate a diffusion equation with Neumann
boundary conditions and right-hand side

We have the following
equation:

We have the following analytical
solution:

2D experiment of the diffusion equation at the end-time

2D experiment of the diffusion equation and control of a single point.

We see that for
large

Offset convergence (

offset(analy.) | max(num.) | min(num.) | |||
---|---|---|---|---|---|

0.90 | 0.90 | 0.90 | |||

0.90 |
0.90 | 0.90 | |||

0.90 | 0.90 | 0.90 |

Numerical results for the P-controller for
different spatial-steps with

Convergence rate | |||
---|---|---|---|

0.1 | 0.1 | 0.077007 | 3.2531 |

0.1 | 0.05 | 0.016153 | 4.0077 |

0.1 | 0.025 | 0.0020085 | 3.9447 |

0.1 | 0.0125 | 0.00026087 | 1.8276 |

0.1 | 0.00625 | 0.00014699 | 0 |

0.05 | 0.1 | 0.27873 | 2.8591 |

0.05 | 0.05 | 0.076833 | 3.7481 |

0.05 | 0.025 | 0.011437 | 4.052 |

0.05 | 0.0125 | 0.001379 | 3.7776 |

0.05 | 0.00625 | 0.00020111 | 0 |

0.025 | 0.1 | 0.6564 | 2.4552 |

0.025 | 0.05 | 0.23939 | 3.2449 |

0.025 | 0.025 | 0.050505 | 4.008 |

0.025 | 0.0125 | 0.0062781 | 3.9999 |

0.025 | 0.00625 | 0.00078482 | 0 |

0.0125 | 0.1 | 1.04 | 2.1424 |

0.0125 | 0.05 | 0.4711 | 2.7173 |

0.0125 | 0.025 | 0.14327 | 3.7251 |

0.0125 | 0.0125 | 0.021668 | 4.0516 |

0.0125 | 0.00625 | 0.0026133 | 0 |

0.00625 | 0.1 | 1.2092 | 1.586 |

0.00625 | 0.05 | 0.80554 | 2.4179 |

0.00625 | 0.025 | 0.30148 | 3.2179 |

0.00625 | 0.0125 | 0.064803 | 4.005 |

0.00625 | 0.00625 | 0.0080726 | 0 |

The associated

.10000 | .65000 | 0.62 |

.05000 | .19000 | 2.4 |

.02500 | 0.04 | 4.64 |

.01250 | .01000 | 6.64 |

The associated

.10000 | 0.0700 | 3.84 |

.05000 | 0.0170 | 5.88 |

.02500 | 0.0040 | 7.97 |

.01250 | 0.0010 | 9.97 |

Convergence of

Time-step | |||
---|---|---|---|

790.000 | 0.031267612875889 | 0.031262218711872 | |

592.500 | 0.017928791447504 | 0.018012608268341 | 3 |

493.750 | 0.009755520357640 | 0.009663115651239 | 4 |

444.375 | 0.005011636633412 | 0.005007035757358 | 5 |

419.687 | 0.002464209438841 | 0.002504970973550 | 6 |

407.343 | 0.001131628414131 | 0.001182533773415 | 7 |

401.171 | 0.000466907455284 | 0.000465794576191 | 8 |

398.085 | 0.000109260068267 | 0.000189637255604 | 9 |

397.314 | 0.000080312399314 | 0.000043201893613 | 11 |

396.928 | 0.000041628241361 | 0.000012118492674 | 12 |

396.735 | 0.000002923384293 | 0.000006047318776 | 13 |

396.723 | 17 | ||

396.711 | 16 | ||

396.687 | 15 | ||

396.639 | 14 | ||

396.542 | 10 | ||

395.000 |

We test for the pure diffusion equation our underlying discretization methods and apply finite elements for the spatial-discretization and implicit Runge-Kutta methods for the time-discretization. In the results, we obtain decreasing errors for the different time- and spatial-steps.

Here we simulate a first example of a diffusion equation and control the concentrations in the deposition process.

We have the following
equation:

P-control with constant optimal constraint,

PID-control with constant optimal constraint,

PID-control with Linear optimal constraint.

The results for the
control methods are given in Figure

To determine the function

The function of the numerical convergence rate is
discrete in the spatial-discretization variable

In the temporal discretization variable

We consider the
convergence rate

We observe that, for instance, in the case of
spatial-discretization

In
Figure

Convergence diagram (KONTOOL). We can see in the loglog-plot

Convergence diagram (KONTOOL). The loglog-plot

The experiment shows the linear
convergence rate of the P-controller with different

In the second example, we simulate the diffusion
equation and control the temperature with and adaptive control based on a
PID-controller, see [

We have the following
equation:

We have the following constraint:

The
automatically step-size is given as (see [

The errors are given as

The parameters are given as

2D experiment with and without the adaptive time-step control.

Furthermore, we change the parameter

Adaptive PID with modified parameters.

In Figures

A modified automatically step-size control, which minimize the oscillations is given in the following algorithm.

(1) We compute the reference control parameters

(2) We apply the automatically step-size control for
the global control parameters with

(3) We stop the computation till we reach the optimal
solution and mark remember the time

(4) We compute the local control parameters

(5) We restart the computation with the local control
parameters and smaller step-size parameters

(6) We stop the computation if we reach

The modified automatically step-size control had taken into account
the local behavior of the control problem. We could adapt the control
parameters

In what follows, we present the adaptive control based
on Algorithm

To be more precise the computation of global control
parameters

We improve the automatically step-size, which is given
in Section

With approximations to each subintervals of

The improved globalized control parameters are computed in the following algorithm.

Computing of the control parameters

(1) Compute the critical control function with the
minima and maxima. Based on the optimal oscillated function, we can derive the

(2) Redo the step (1) for the intervals

(3) The optimal control parameters are used in
Algorithm

Further ideas to determine the time-dependent parameters

To initialize the PID-controller, we have to derive
the

An idea is derived with
the step function response, see [

This idea has also be included into the time-step control of the PID-controller.

In Figure

Error of the time-step control.

Exponential
regression of min and max. Behavior of

So we use the exponential regression

Let

The adaptive step-size control of
the PID-controller had taken into account the
optimization in each local time-interval. We consider a larger time-interval
and derive the optimal control parameters

To automatize the adaptive time-step algorithm, we
have to compute the step response and derive the

A further algorithm to derive the control parameters
is done by Chien et al., see [

The results are given in Figure

The

In these experiments, we concentrate on the
Chien-Hrones-Reswick method, see [

To automatize the adaptive time-step algorithm, we
have to compute the step response and derive the

Parameters chosen by Chien-Hrones-Reswick: Summands of the minimization problem.

In Figure

Parameters chosen by Chien-Hrones-Reswick: Actual value.

In Figure

Parameters chosen by Chien-Hrones-Reswick: Source.

The benefit of the time-dependent
control parameters

We present a continuous or kinetic model, due to the fare-field or near-field effect of our deposition process. We discuss the PID-controller to automatize our deposition process. Due to heuristic methods of deriving the PID parameters, we discuss aposteriori error estimates to automatize the time-stepping methods. A modified automatically step-size control is discussed and the best approximations are obtained with the time-dependent control method based on the Chien-Hrones-Reswick algorithm. For the mesoscopic-scale model, we discussed different experiments and their convergence rates. In future, we will analyze the validity of the models with physical experiments.