# American Institute of Mathematical Sciences

doi: 10.3934/naco.2021034
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## 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 and 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. [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. [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. [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. [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. [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. [7] J. Haslinger and R. A. E. Mäkinen, Introduction to Shape Optimization: Theory, Approximation, and Computation, SIAM, 2003. doi: 10.1137/1.9780898718690. [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. [9] M. Hinze, B. 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. [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. [11] V. G. Korneev and U. Langer, Dirichlet-Dirichlet Domain Decomposition Methods for Elliptic Problems, World Scientific, 2013. doi: 10.1142/9035. [12] P. A. Markowich, The Stationary Semiconductor Device Equations, Computational Microelectronics, Springer Vienna, 2013. doi: 10.1007/978-3-7091-3678-2. [13] M. H. Mozaffari, M. 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. [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. [15] O. Pironneau, Optimal Shape Design For Elliptic Systems, Springer Berlin Heidelberg, 1984. doi: 10.1007/978-3-642-87722-3. [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. [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. [18] W. Rudin, Functional Analysis, McGraw-Hill, 1991. [19] S. Salhi, Heuristic Search The Emerging Science of Problem Solving, Palgrave Macmillan, 2017. doi: 10.1007/978-3-319-49355-8. [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. [21] S. M. Sze, Semiconductor Devices: Physics and Technology, John Wiley & Sons Singapore Pte. Limited, 2012. [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.

show all references

##### References:
 [1] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, SIAM, 2002. doi: 10.1137/1.9780898719208. [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. [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. [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. [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. [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. [7] J. Haslinger and R. A. E. Mäkinen, Introduction to Shape Optimization: Theory, Approximation, and Computation, SIAM, 2003. doi: 10.1137/1.9780898718690. [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. [9] M. Hinze, B. 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. [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. [11] V. G. Korneev and U. Langer, Dirichlet-Dirichlet Domain Decomposition Methods for Elliptic Problems, World Scientific, 2013. doi: 10.1142/9035. [12] P. A. Markowich, The Stationary Semiconductor Device Equations, Computational Microelectronics, Springer Vienna, 2013. doi: 10.1007/978-3-7091-3678-2. [13] M. H. Mozaffari, M. 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. [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. [15] O. Pironneau, Optimal Shape Design For Elliptic Systems, Springer Berlin Heidelberg, 1984. doi: 10.1007/978-3-642-87722-3. [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. [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. [18] W. Rudin, Functional Analysis, McGraw-Hill, 1991. [19] S. Salhi, Heuristic Search The Emerging Science of Problem Solving, Palgrave Macmillan, 2017. doi: 10.1007/978-3-319-49355-8. [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. [21] S. M. Sze, Semiconductor Devices: Physics and Technology, John Wiley & Sons Singapore Pte. Limited, 2012. [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.
the geometry of the PN-Junction
Example 1: the optimal interfaces versus the exact ones
Example 2: the optimal interfaces versus the exact ones
Example 1: the optimal interfaces versus the exact ones
Example 2: the optimal interfaces versus the exact ones
Example 1: the obtained results with noisy data
Example 2: the obtained results with noisy data
Forward mode
Reverse mode
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
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
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 $+0.1V$ $-0.0V$ $0.018$ $-0.3V$ $+0.3V$ $0.029$ $+0.3V$ $-0.3V$ $0.047$ $-0.6V$ $+0.6V$ $0.063$ $+0.6V$ $-0.6V$ $0.079$ $-0.6V$ $+1.2V$ $0.085$
 [1] Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095 [2] Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 1-21. doi: 10.3934/dcdss.2021006 [3] Jaroslav Haslinger, Raino A. E. Mäkinen, Jan Stebel. Shape optimization for Stokes problem with threshold slip boundary conditions. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1281-1301. doi: 10.3934/dcdss.2017069 [4] Gianmarco Manzini, Annamaria Mazzia. A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem. Journal of Computational Dynamics, 2022, 9 (2) : 207-238. doi: 10.3934/jcd.2021020 [5] Barbara Kaltenbacher, Gunther Peichl. The shape derivative for an optimization problem in lithotripsy. Evolution Equations and Control Theory, 2016, 5 (3) : 399-430. doi: 10.3934/eect.2016011 [6] Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768 [7] Toyohiko Aiki. On the existence of a weak solution to a free boundary problem for a model of a shape memory alloy spring. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 1-13. doi: 10.3934/dcdss.2012.5.1 [8] Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic and Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639 [9] Daniele Boffi, Lucia Gastaldi. Discrete models for fluid-structure interactions: The finite element Immersed Boundary Method. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 89-107. doi: 10.3934/dcdss.2016.9.89 [10] Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 61-79. doi: 10.3934/dcdsb.2020351 [11] Javier A. Almonacid, Gabriel N. Gatica, Ricardo Oyarzúa, Ricardo Ruiz-Baier. A new mixed finite element method for the n-dimensional Boussinesq problem with temperature-dependent viscosity. Networks and Heterogeneous Media, 2020, 15 (2) : 215-245. doi: 10.3934/nhm.2020010 [12] Yinnian He, Yanping Lin, Weiwei Sun. Stabilized finite element method for the non-stationary Navier-Stokes problem. Discrete and Continuous Dynamical Systems - B, 2006, 6 (1) : 41-68. doi: 10.3934/dcdsb.2006.6.41 [13] Cornel M. Murea, H. G. E. Hentschel. A finite element method for growth in biological development. Mathematical Biosciences & Engineering, 2007, 4 (2) : 339-353. doi: 10.3934/mbe.2007.4.339 [14] Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic and Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59 [15] Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems and Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025 [16] Yu-Feng Sun, Zheng Zeng, Jie Song. Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation. Numerical Algebra, Control and Optimization, 2020, 10 (2) : 157-164. doi: 10.3934/naco.2019045 [17] Zhongwen Chen, Songqiang Qiu, Yujie Jiao. A penalty-free method for equality constrained optimization. Journal of Industrial and Management Optimization, 2013, 9 (2) : 391-409. doi: 10.3934/jimo.2013.9.391 [18] Bun Theang Ong, Masao Fukushima. Global optimization via differential evolution with automatic termination. Numerical Algebra, Control and Optimization, 2012, 2 (1) : 57-67. doi: 10.3934/naco.2012.2.57 [19] Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10 [20] Yang Zhang. A free boundary problem of the cancer invasion. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1323-1343. doi: 10.3934/dcdsb.2021092

Impact Factor:

## Tools

Article outline

Figures and Tables