# Towards real time analysis of coupled parametric multiphysic models: application to parametric CVD

Jose Aguado, Domenico Borzacchiello, Francisco Chinesta

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Jose Aguado, Domenico Borzacchiello, Francisco Chinesta. Towards real time analysis of coupled parametric multiphysic models: application to parametric CVD. 12e Colloque national en calcul des structures, CSMA, May 2015, Giens, France. hal-01517309

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# Towards real time analysis of coupled parametric multiphysic models: application to parametric CVD

J.V. Aguado<fnref n="1" />, D. Borzacchiello<fnref n="1" />, F. Chinesta<fnref n="1" />

**Abstract** — This paper is concerned with model order reduction of multiphysical simulations, with application in Numerical Prototyping. In our approach, the physics involved are solved independently and then coupled in a final stage to produce the quantities of interest. This approach allows combining different model reduction methods. Unlike classic multiphysical simulations, the reduced model allows a very cheap exploration of the design parametric space. This is of great interest at the conception stage, when rather than accuracy the main concern is evaluating different design alternatives.

**Keywords** — Model Order Reduction, Multiphysics, Coupling, Preconception, Numerical Prototyping.

## 1 Introduction

At the conception phase of industrial equipments, engineers explore different alternatives to find a compromise between performance, cost and other criteria. The main concern is to evaluate different alternatives as quickly as possible. Numerical Prototyping uses numerical simulations as a tool to evaluate the performances of the equipment before it is manufactured. In many cases, multiphysical simulations[1] combining different physical phenomena have to be solved. However they are generally complex and time-consuming, restricting the chance of estimating a nearly optimal design.

In consequence, there is a need of reducing the computational complexity of multiphysical simulations with the aim of rendering the exploration of the parametric space fast. In this context, Model Order Reduction (MOR) methods find an interesting and innovative field of application. Regardless the MOR method considered, two computational stages apply. First, the *offline* stage in which the *reduced model* is built upon the *full model*. Then, in the *online stage* the quantities of interest can be computed for a given parameter choice by performing almost inexpensive computations.

We consider a *distributed approach*, that is, each physical model is solved separately, rather than a *monolithic approach*, in which the different physical models are solved all together. Observe that the distributed approach is more flexible as it facilitates selecting appropriate solvers as well as appropriate MOR methods for each physics. On the other hand, it provides the challenge of being able to couple reduced models. In this work, we combine Reduced Basis method (RBM)[2] and Proper Generalized Decomposition (PGD)[3, 4], and we demonstrate how different reduced models can be coupled. Once the subsystems are coupled, it is possible to perform an optimization of the design parameters with respect to an objective function to find a nearly optimal design point.

To illustrate the methodology, we consider a multiphysical problem of practical interest, which is the design of a hot-wall Chemical Vapor Deposition (CVD) reactor [5]. CVD reactors are used to produce synthetic diamonds, carbon nanofibers (CNF) as well as in semiconductor industry to grow high quality films. Without going into details, the chemistry depends mainly on achieving a proper temperature distribution on the substrate where the layer is grown, and so the quality of the film. The substrate is placed onto a hollow cylindrical graphite susceptor, which is inductively heated by a copper coil. The susceptor is coated by insulation, and the whole is enclosed in a quartz cylinder. The copper coil is placed outside the quartz cylinder. The precursor gases that react with the substrate are diluted in the carrier gas which, in turn, removes the by-products. The carrier gas also has a cooling effect that must be accounted.

<footnote n="1">¹ ESI Corporate International Chair. GeM Institute, Ecole Centrale Nantes, (jose.aguado-lopez, domenico.borzacchiello, francisco.chinesta)@ec-nantes.fr</footnote>
<figure><img src="image_2.png" /><figcaption>Figure 1: Axisymmetric cross section of the CVD reactor</figcaption></figure>

## 2 Governing equations and computational domain

In this section the multiphysical model is described. To this end, we recall very synthetically the main equations involved in each physics as well as their coupling interactions. Since the goal is to perform simulations with application into a preliminary design stage, some simplification hypothesis concerning geometry and the coupling conditions will also be described. Thanks to the cylindrical shape of the reactor, a 2D axisymmetric model is can be considered. Figure 1 shows a schematic representation of the CVD reactor.

The AC current flowing in the copper coil creates an alternating magnetic field which induces eddy currents in the electrically conductive parts of the reactor. By Joule's effect, conductive parts are heated up. Besides, the carrier gas flow affects to the temperature distribution, so that there are three physics that must be taken into account: electrodynamics, fluid mechanics and heat transfer.

The equation for the magnetic vector potential $\underline{A}$ is deduced from Maxwell equations for constant magnetic permeability, see [5] for details:

$$ \frac{1}{\mu_0 \mu_r} \nabla^2 \underline{A} = i \omega \sigma \underline{A} - \underline{J}, \qquad (1) $$

where $\mu_0$ is the magnetic permeability in vacuum, $\mu_r$ is the relative magnetic permeability of the material, $\omega = 2\pi f$ is the pulsation associated to the AC current frequency $f$, $\sigma$ is the electrical conductivity and $\underline{J}$ is the current density in the coil turns. Its amplitude is given by the following expression:

$$ |\underline{J}| = \frac{1}{A_{\text{coil}}} \sqrt{\frac{P}{R}}, \quad \text{with} \quad R = \frac{2\pi r_{\text{coil}} N \rho}{A_{\text{eff}}}, \qquad (2) $$

being $r_{\text{coil}}$ the radius of each coil turn, $N$ the number of turns, $\rho$ the electrical resistivity and $A_{\text{eff}}$ the effective cross section area through the current flows due to skin effect. Once $\underline{A}$ is known, the induced heating source (i.e. the right-hand side of the heat equation) is computed from:

$$ Q = \frac{1}{2} \sigma \omega^2 |\underline{A}|^2. \qquad (3) $$

Regarding the fluid model, the incompressible Navier-Stokes equation is assumed. It writes as follows:

$$ \left\{ \begin{array}{l} \text{Re} \underline{u} \cdot \underline{\nabla} \underline{u} - \Delta \underline{u} + \underline{\nabla} p = \underline{0} \\ \underline{\nabla} \cdot \underline{u} = \underline{0}, \end{array} \right. \qquad (4) $$

where $\underline{u}$ is the velocity field and $p$ is the pressure field. By “Re” we denote the Reynolds number.

Finally from the heat source $Q$ and the velocity $\underline{u}$, computed from Eq. (3) and (4) respectively, the temperature can be obtained by solving the convection-diffusion heat equation:

$$ \underline{u} \cdot \underline{\nabla} T - \lambda \underline{\nabla}^2 T = Q, \qquad (5) $$

being $\lambda$ the thermal conductivity. Materials are assumed homogeneous and linear so that the electrical resistivity (coil), the electrical conductivity (induced parts), the mass density of the gas and the thermal conductivity ($\rho$, $\sigma$, $\rho_g$ and $\lambda$, respectively) are constant. From these assumptions, it follows that neither the magnetic field $\underline{A}$ nor the velocity field $\underline{u}$ are affected by the temperature change. Electrodynamics and fluid flow simulations are therefore weakly coupled with the heat transfer simulation.

Figure 2 shows the computational domain used to solve the problem, which is in fact partitioned in three subdomains for the sake of clarity. Since electrodynamics, fluid mechanics and heat transfer are
<figure><img src="image_3.png" /><figcaption>Figure 2: Computational domain of the multiphysical problem</figcaption></figure>

solved independently, we can associate a computational domain to each physics. The electrodynamic model is solved in the union of the three subdomains, that is, the whole computational domain is considered. This is to be able to impose appropriate far-field conditions in the upper part of the computational domain; see Figure 2a and Figure 2b. The fluid model is solved only in the subdomain 2; Figure 2c shows the finite element mesh used. Finally, the computational domain of the heat transfer model is the union of subdomains 2 and 3, excluding the coil turns; see Figure 2c and Figure 2d.

In our approach, the design parameters are included in the reduced models as parameters. Here, the design parameters are the frequency of the input current (equivalently, $\omega$), the input power and the gas flow rate (equivalently, Re). It is estimated that they can vary in the following ranges: $\omega \in I_{\omega} = [20, 70]$ kHz, $P \in I_{P} = [2.3, 2.9]$ kW and $Re \in I_{Re} = [0, 100]$.

### 3 Coupling reduced models

In this section, we apply MOR methods to compute parametric solutions of the models presented in Section 2. Rather than using a brute-force approach in which each time a parameter changes a full simulation must be carried out, we prefer to invest some computational effort to compute a reduced model in an offline stage because this allows a fast exploration of the parametric space in the online stage.

The reduced electrodynamics model is first explained. The objective is to compute the heat source for every possible value of the pulsation and the input power: $Q(x, \omega, P)$. This requires computing the magnetic potential in advance. A PGD approach is used in this case, and therefore pulsation and power are regarded as coordinates of the problem: $A(x, \omega, P)$. However, since Eq. (1) is linear, $A$ is proportional to $J$ and therefore its dependence on the input power $P$ is trivial. In consequence, only a parametric dependence on the pulsation is kept and we solve for $A(x, \omega)$. A $N_A$-rank separated representation is computed using the classic PGD procedure, which is not described here for the sake of brevity:

$$ \underline{A}(x, \omega) = \sum_{i=1}^{N_A} \underline{X}_A^i(x) \circ \underline{W}_A^i(\omega), \quad \text{then} \quad Q(x, \omega) = \sum_{i=1}^{N_Q} X_Q^i(x) W_Q^i(\omega), \qquad (6) $$

where “$\circ$” stands for the component-wise product. In order to compute the velocity field for every possible Reynolds number, we use an approach mainly inspired from RBM. Although it is quite different to PGD, a separated representation is also obtained:

$$ \underline{u}(x, \text{Re}) = \sum_{i=1}^{N_u} X_{\underline{u}}^i(x) \circ \underline{R}_{\underline{u}}^i(\text{Re}). \qquad (7) $$

Once $Q$ and $\underline{u}$ are available, they can be coupled by solving the heat equation using a PGD approach.
<figure><img src="image_4.png" /><figcaption>Figure 3: First four terms in temperature separated representation</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 4: Temperature field for some parameter combinations</figcaption></figure>

The temperature is sought as an $N_T$-rank separated representation:

$$T(\underline{x}, \omega, \text{Re}) = \sum_{i=1}^{N_T} X_T^i(\underline{x}) W_T^i(\omega) R_T^i(\text{Re}). \quad (8)$$

One term of the expansion is added at a time. Using an alternating directions fixed-point algorithm, each separated function of the new term can be successively updated until convergence.

## 4 Results

The separated representations defined in Section 3 had $N_A = 11$ ($\epsilon_A = 10^{-8}$), $N_Q = 5$ ($\epsilon_Q = 10^{-8}$), $N_u = 64$ ($\epsilon_u = 10^{-8}$) and $N_T = 18$ ($\epsilon_T = 10^{-5}$) terms, being $\epsilon_{A,Q,T}$ the error measured as the norm of the residual, and $\epsilon_u$ an error estimate provided by RBM. Figure 3 shows some of the computed modes. Modes associated with Reynolds number have been converted to flow rate through a trivial transformation.

Figure 4 shows the temperature field obtained inside the reactor. Although the computational domain considers only half the reactor thanks to symmetry, see Fig. 1, the temperature is depicted on the physical domain for the sake of clarity. It is worth to emphasize that the exploration of the parametric space does not involve any additional computational effort and it can be performed practically in real time.

## References

[1] Q. Zhang, S. Cen. *Multiphysics Modeling: Numerical Methods and Engineering Applications*, Tsinghua University, 2015.

[2] C. Prud'homme, D.V. Rovas, K. Veroy, L. Machiels, Y. Maday, A.T. Patera, G. Turinici. *Reliable Real-Time Solution of Parametrized Partial Differential Equations: Reduced-Basis Output Bound Methods*, Journal of Fluids Engineering, ASME, 70-80, 2001.

[3] F. Chinesta, A. Leygue, F. Bordeu, J.V. Aguado, E. Cueto, D. Gonzalez, I. Alfaro, A. Ammar, A. Huerta. *PGD-based computational vademecum for efficient design, optimization and control*, Archives of Computational Methods in Engineering. State of the Art Reviews, Springer, 31-59, 2013.

[4] M.S. Aghighi, A. Ammar, C. Metivier, M. Normandin, F. Chinesta. *Non-incremental transient solution of the Rayleigh-Bénard convection model by using the PGD*, Journal of Non-Newtonian Fluid Mechanics, Elsevier, 65-78, 2013.

[5] Ö. Danielsson, U. Forsberg, A. Henry, E. Janzén. *Investigation of the temperature profile in a hot-wall SiC chemical vapor deposition reactor*, Journal of Crystal Growth, Elsevier, 352-364, 2002.