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doi: 10.3934/naco.2021034
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A new hybrid method for shape optimization with application to semiconductor equations

1. 

Université Cadi Ayyad, laboratoire de Mathématiques Appliquées et Informatique

2. 

Faculté des Sciences et Techniques, Avenue Abdelkrim El khttabi B. P. 549, Marrakech, Maroc

* Corresponding author: Youness El Yazidi

Received  May 2021 Revised  July 2021 Early access August 2021

The aim of this work is to reconstruct the depletion region in pn junction. Starting with famous drift diffusion model, we establish the simplified equation for the considered semiconductor. There we call the shape optimization technique to formulate a minimization problem from the inverse problem at hand. The existence of an optimal solution of the optimization problem is proved. The proposed numerical algorithm is a combined Domain Decomposition method with an efficient hybrid conjugate gradient guided by differential evolution heuristic algorithm, the finite element method is used to discretize the state equation. At the end we establish several numerical examples, to prove the validity of theoretical results using the proposed algorithm, in addition we show some simulation of the depletion region approximation under two different functioning modes.

Citation: Youness El Yazidi, Abdellatif ELLABIB. A new hybrid method for shape optimization with application to semiconductor equations. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021034
References:
[1]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, SIAM, 2002. doi: 10.1137/1.9780898719208.  Google Scholar

[2]

M. Dashti Ardakani and M. Khodadad, Shape estimation of a cavity by inverse application of the 2D elastostatics problem, International Journal of Computational Methods, 10 (2013), 1350042. doi: 10.1142/S0219876213500424.  Google Scholar

[3]

Y. El Yazidi and A. Ellabib, Reconstruction of the depletion layer in MOSFET by genetic algorithms, Mathematical Modeling and Computing, 7 (2020), 96-103.  doi: 10.23939/mmc2020.01.096.  Google Scholar

[4]

Y. El Yazidi and A. Ellabib, An iterative method for optimal control of bilateral free boundaries problem, Mathematical Methodes in Applied Science, (2021), 1–20. doi: 10.1002/mma.7527.  Google Scholar

[5]

Y. El Yazidi and A. Ellabib, Augmented Lagrangian approach for a bilateral free boundary problem, Journal of Applied Mathematics and Computing, 2021. doi: 10.1007/s12190-020-01472-y.  Google Scholar

[6]

A. Ellabib and A. Nachaoui, On the numerical solution of a free boundary identification problem, Inverse Problems in Engineering, 9 (2001), 235-260.  doi: 10.1080/174159701088027764.  Google Scholar

[7]

J. Haslinger and R. A. E. Mäkinen, Introduction to Shape Optimization: Theory, Approximation, and Computation, SIAM, 2003. doi: 10.1137/1.9780898718690.  Google Scholar

[8]

M. Hinze and R. Pinnau, Second-order approach to optimal semiconductor design, Journal of Optimization Theory and Applications, 133 (2007), 179-199.  doi: 10.1007/s10957-007-9203-3.  Google Scholar

[9]

M. HinzeB. Kaltenbacher and T. N. T. Quyen, Identifying conductivity in electrical impedance tomography with total variation regularization, Numerische Mathematik, 138 (2018), 723-765.  doi: 10.1007/s00211-017-0920-8.  Google Scholar

[10]

C.-H. Huang and C.-C. Shih, A shape identification problem in estimating simultaneously two interfacial configurations in a multiple region domain, Applied Thermal Engineering, 26 (2006), 77-88.  doi: 10.1016/j.applthermaleng.2005.04.019.  Google Scholar

[11]

V. G. Korneev and U. Langer, Dirichlet-Dirichlet Domain Decomposition Methods for Elliptic Problems, World Scientific, 2013. doi: 10.1142/9035.  Google Scholar

[12]

P. A. Markowich, The Stationary Semiconductor Device Equations, Computational Microelectronics, Springer Vienna, 2013. doi: 10.1007/978-3-7091-3678-2.  Google Scholar

[13]

M. H. MozaffariM. Khodadad and M. Dashti Ardakani, Simultaneous identification of multi-irregular interfacial boundary configurations in non-homogeneous body using surface displacement measurements, Journal of Mechanical Engineering Science, 231 (2016), 1-12.  doi: 10.1177/0954406216636166.  Google Scholar

[14]

A. Novruzi and M. Pierre, Structure of shape derivatives, Journal of Evolution Equations, 2 (2002), 365-382.  doi: 10.1007/s00028-002-8093-y.  Google Scholar

[15]

O. Pironneau, Optimal Shape Design For Elliptic Systems, Springer Berlin Heidelberg, 1984. doi: 10.1007/978-3-642-87722-3.  Google Scholar

[16]

H. Prautzsch, W. Boehm and M. Paluszny, Bézier and B-Spline Techniques, Springer, Berlin, Heidelberg, 2002. doi: 10.1007/978-3-662-04919-8.  Google Scholar

[17]

K. V. Price, Differential evolution: a fast and simple numerical optimizer, in Proceedings of North American Fuzzy Information Processing, (1996), 524–527. doi: 10.1109/NAFIPS.1996.534790.  Google Scholar

[18]

W. Rudin, Functional Analysis, McGraw-Hill, 1991.  Google Scholar

[19]

S. Salhi, Heuristic Search The Emerging Science of Problem Solving, Palgrave Macmillan, 2017. doi: 10.1007/978-3-319-49355-8.  Google Scholar

[20]

R. Storn and K. Price, Differential evolution - A simple and efficient heuristic for global optimization over continuous spaces, Journal of Global Optimization, 11 (1997), 341-359.  doi: 10.1023/A:1008202821328.  Google Scholar

[21]

S. M. Sze, Semiconductor Devices: Physics and Technology, John Wiley & Sons Singapore Pte. Limited, 2012. Google Scholar

[22]

J. P. Zolesio and J. Sokolowski, Introduction to Shape Optimization Shape Sensitivity Analysis, Springer, Berlin, Heidelberg, 1992. doi: 10.1007/978-3-642-58106-9.  Google Scholar

show all references

References:
[1]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, SIAM, 2002. doi: 10.1137/1.9780898719208.  Google Scholar

[2]

M. Dashti Ardakani and M. Khodadad, Shape estimation of a cavity by inverse application of the 2D elastostatics problem, International Journal of Computational Methods, 10 (2013), 1350042. doi: 10.1142/S0219876213500424.  Google Scholar

[3]

Y. El Yazidi and A. Ellabib, Reconstruction of the depletion layer in MOSFET by genetic algorithms, Mathematical Modeling and Computing, 7 (2020), 96-103.  doi: 10.23939/mmc2020.01.096.  Google Scholar

[4]

Y. El Yazidi and A. Ellabib, An iterative method for optimal control of bilateral free boundaries problem, Mathematical Methodes in Applied Science, (2021), 1–20. doi: 10.1002/mma.7527.  Google Scholar

[5]

Y. El Yazidi and A. Ellabib, Augmented Lagrangian approach for a bilateral free boundary problem, Journal of Applied Mathematics and Computing, 2021. doi: 10.1007/s12190-020-01472-y.  Google Scholar

[6]

A. Ellabib and A. Nachaoui, On the numerical solution of a free boundary identification problem, Inverse Problems in Engineering, 9 (2001), 235-260.  doi: 10.1080/174159701088027764.  Google Scholar

[7]

J. Haslinger and R. A. E. Mäkinen, Introduction to Shape Optimization: Theory, Approximation, and Computation, SIAM, 2003. doi: 10.1137/1.9780898718690.  Google Scholar

[8]

M. Hinze and R. Pinnau, Second-order approach to optimal semiconductor design, Journal of Optimization Theory and Applications, 133 (2007), 179-199.  doi: 10.1007/s10957-007-9203-3.  Google Scholar

[9]

M. HinzeB. Kaltenbacher and T. N. T. Quyen, Identifying conductivity in electrical impedance tomography with total variation regularization, Numerische Mathematik, 138 (2018), 723-765.  doi: 10.1007/s00211-017-0920-8.  Google Scholar

[10]

C.-H. Huang and C.-C. Shih, A shape identification problem in estimating simultaneously two interfacial configurations in a multiple region domain, Applied Thermal Engineering, 26 (2006), 77-88.  doi: 10.1016/j.applthermaleng.2005.04.019.  Google Scholar

[11]

V. G. Korneev and U. Langer, Dirichlet-Dirichlet Domain Decomposition Methods for Elliptic Problems, World Scientific, 2013. doi: 10.1142/9035.  Google Scholar

[12]

P. A. Markowich, The Stationary Semiconductor Device Equations, Computational Microelectronics, Springer Vienna, 2013. doi: 10.1007/978-3-7091-3678-2.  Google Scholar

[13]

M. H. MozaffariM. Khodadad and M. Dashti Ardakani, Simultaneous identification of multi-irregular interfacial boundary configurations in non-homogeneous body using surface displacement measurements, Journal of Mechanical Engineering Science, 231 (2016), 1-12.  doi: 10.1177/0954406216636166.  Google Scholar

[14]

A. Novruzi and M. Pierre, Structure of shape derivatives, Journal of Evolution Equations, 2 (2002), 365-382.  doi: 10.1007/s00028-002-8093-y.  Google Scholar

[15]

O. Pironneau, Optimal Shape Design For Elliptic Systems, Springer Berlin Heidelberg, 1984. doi: 10.1007/978-3-642-87722-3.  Google Scholar

[16]

H. Prautzsch, W. Boehm and M. Paluszny, Bézier and B-Spline Techniques, Springer, Berlin, Heidelberg, 2002. doi: 10.1007/978-3-662-04919-8.  Google Scholar

[17]

K. V. Price, Differential evolution: a fast and simple numerical optimizer, in Proceedings of North American Fuzzy Information Processing, (1996), 524–527. doi: 10.1109/NAFIPS.1996.534790.  Google Scholar

[18]

W. Rudin, Functional Analysis, McGraw-Hill, 1991.  Google Scholar

[19]

S. Salhi, Heuristic Search The Emerging Science of Problem Solving, Palgrave Macmillan, 2017. doi: 10.1007/978-3-319-49355-8.  Google Scholar

[20]

R. Storn and K. Price, Differential evolution - A simple and efficient heuristic for global optimization over continuous spaces, Journal of Global Optimization, 11 (1997), 341-359.  doi: 10.1023/A:1008202821328.  Google Scholar

[21]

S. M. Sze, Semiconductor Devices: Physics and Technology, John Wiley & Sons Singapore Pte. Limited, 2012. Google Scholar

[22]

J. P. Zolesio and J. Sokolowski, Introduction to Shape Optimization Shape Sensitivity Analysis, Springer, Berlin, Heidelberg, 1992. doi: 10.1007/978-3-642-58106-9.  Google Scholar

Figure 1.  the geometry of the PN-Junction
Figure 2.  Example 1: the optimal interfaces versus the exact ones
Figure 3.  Example 2: the optimal interfaces versus the exact ones
Figure 4.  Example 1: the optimal interfaces versus the exact ones
Figure 5.  Example 2: the optimal interfaces versus the exact ones
Figure 6.  Example 1: the obtained results with noisy data
Figure 7.  Example 2: the obtained results with noisy data
Figure 8.  Forward mode
Figure 9.  Reverse mode
Table 1.  The used parameters in DE
Population size Max generations Mutation scale $ \varrho $ Crossover ratio $ \sigma $
25 10 0.1 0.75
Population size Max generations Mutation scale $ \varrho $ Crossover ratio $ \sigma $
25 10 0.1 0.75
Table 2.  Comparison of errors
Noise level 0% 1% 5% 10%
Example 1 0.0095 0.0184 0.0676 0.1779
Example 2 0.0213 0.0243 0.0458 0.0589
Noise level 0% 1% 5% 10%
Example 1 0.0095 0.0184 0.0676 0.1779
Example 2 0.0213 0.0243 0.0458 0.0589
Table 3.  Optimal cost for each functioning mode
Depletion mode Enhancement mode
$V^+$ $V^-$ cost $V^+$ $V^-$ cost
$+0.1V$ $-0.0V$ $0.018$ $-0.3V$ $+0.3V$ $0.029$
$+0.3V$ $-0.3V$ $0.047$ $-0.6V$ $+0.6V$ $0.063$
$+0.6V$ $-0.6V$ $0.079$ $-0.6V$ $+1.2V$ $0.085$
Depletion mode Enhancement mode
$V^+$ $V^-$ cost $V^+$ $V^-$ cost
$+0.1V$ $-0.0V$ $0.018$ $-0.3V$ $+0.3V$ $0.029$
$+0.3V$ $-0.3V$ $0.047$ $-0.6V$ $+0.6V$ $0.063$
$+0.6V$ $-0.6V$ $0.079$ $-0.6V$ $+1.2V$ $0.085$
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