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Chapter 8 Conclusion and Outlook

This work presents major contributions to accelerate key computational steps of process TCAD simulations based on the level-set method. In particular, the focus is on the computational steps Velocity Extension and Re-Distancing for which three parallel algorithms, based on the FMM , were developed. Typically, process TCAD simulations require an adaptive spatial discretization to be computationally and practically feasible. The developed algorithms are, therefore, tailored towards hierarchical grids, are able to utilize parallel computational resources, and are thus able to reduce the turnaround time for a wide range of process TCAD simulations. In the following, the key contributions of this work are summarized:

For the computational step Velocity Extension an algorithm employing a relaxed computation order for the grid points reducing the computational complexity is derived, resulting in a reduction of the serial run-time (Chapter 5). Three different orders of computation based on different data structures are compared: The Queue data structure performs best on a representative set of test cases.

In addition, the changes that enable the reordering of the computation also provide the basis for parallelization of the algorithm on a Cartesian grid enabling a parallel speedup of 5.8 for eight threads. Further optimizations, i.e., tailoring the developed algorithm to hierarchical grids, increase the parallel speedup compared to a previous strategy centered around the so-called multi-block FMM. The increased parallel speedup of up to 7.1 for 10 threads is achieved by reducing global synchronization barriers in the developed algorithm and thus enabling a better load-balancing.

For the computational step Re-Distancing an algorithm utilizing block decomposition, which only splits large blocks of a given hierarchical grid, is developed to increase parallel scalability (Chapter 6). The decomposition enables a better implicit load-balancing, by creating a relatively high number of blocks compared to the used number of threads. Additionally, by optimizing the frequency of synchronization steps between the sub-blocks, the overall performance is increased. The so achieved parallel speedup is 17.4 for 24 threads. The performance increase is caused by reducing the influence of ghost points which are not sources for the final signed-distance field. The application of the decomposition algorithm prior to the setup of the FMM enables a straightforward extension of this approach to other methods for computing the signed-distance field, e.g., FIM and FSM. Another extension of the research could be considering cases in which the grid resolution along different axes differs strongly. In this case a dedicated block size along each axis might improve the spatial locality of the created blocks, thus potentially further increasing parallel performance.

For the computational step Re-Distancing an algorithm, which increases the accuracy of the computed signed-distance field via a bottom-up approach, is proposed (Chapter 7). The algorithm efficiently uses a previous top-down re-distancing algorithm to only re-compute the distance at grid points where an improvement of the accuracy is possible. The proposed bottom-up correction algorithm has a low run-time overhead (between 4 % and 10 %). The accuracy of the signed-distance field is significantly increased, e.g., around corners by a factor of up to 2.7. In cases where features smaller than the grid resolution are present on lower grid levels, e.g., a trench thinner than the grid resolution on Level 0, the proposed bottom-up correction algorithm enables correct representation of those features.

Finally, all developed algorithms are applied in combination to the simulation presented as the motivational example (cf. Chapter 1). Figure 8.1 shows a comparison of the original run-time for each time step (cf. Figure 1.7) to the run-time of the reference simulator using the new developed and parallelized algorithms. The overall run-time of the simulation in each time step is more than halved.

(image)

(a)

(image)

(b)
Figure 8.1: (a) Run-time of the level-set method utilizing the developed and presented algorithms for the thermal oxidation process shown for individual time steps, measured on the compute system ICS using 10 threads. The dashed line is the baseline (reference) run-time (combining the run-time of all computational steps for each time step as shown in Figure 1.7). (b) Highlighting the achievements for Velocity Extension and Re-Distancing.

Re-Distancing which has previously been the second biggest contributor to the run-time in a time step is reduced to second or third biggest contributor, depending on whether a Re-Gridding takes place in the specific time step. The overall speedup of 5.0 (cf. Figure 8.1b) for Re-Distancing is in the expected range. The blocks of the hierarchical grid are relatively small (about 45 grid points wide). For such small blocks the gained parallel performance is only slightly larger than the overhead from decomposition.

The leading contributor to the run-time in a time step is Advection, because significant parts of this computational step are not parallelized (and out of scope of this thesis) in the considered reference simulator.

It is especially important to point out that the contribution from Velocity Extension, which originally has been the third highest contributor to the run-time, is now the least contributor (using 10 threads). The serial and parallel speedup combined reduced the run-time by a factor of 18.5 (cf. Figure 8.1b). A further reduction of the run-time is possible, if the process model allows a combined extension of the scalar and vector velocity.

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  • [190] G. Diamantopoulos, J. Weinbub, A. Hössinger, and S. Selberherr. “Evaluation of the Shared-Memory Parallel Fast Marching Method for Re-Distancing Problems”. In: Proceedings of the International Conference on Computational Science and Its Applications (ICCSA). Trieste: IEEE, 2017, pp. 1–8. doi: 10.1109/ICCSA.2017.7999648.

  • [191] E. Becker, W. Ehrfeld, P. Hagmann, A. Maner, and D. Münchmeyer. “Fabrication of Microstructures with High Aspect Ratios and Great Structural Heights by Synchrotron Radiation Lithography, Galvanoforming, and Plastic Moulding (LIGA Process)”. In: Microelectronic Engineering 4.1 (1986), pp. 35–56. doi: 10.1016/0167-9317(86)90004-3.

  • [192] B. Radjenović, J. K. Lee, and M. Radmilović-Radjenović. “Sparse Field Level Set Method for Non-Convex Hamiltonians in 3D Plasma Etching Profile Simulations”. In: Computer Physics Communications 174.2 (2006), pp. 127–132. doi: 10.1016/j.cpc.2005.09.010.

  • [193] P. Liu, D. Zhang, J. Guo, W. Wang, and F. Yang. “Optimization of Photoresist Development and DRIE Processes to Fabricate High Aspect Ratio Si Structure in 5 nm Scale”. In: Journal of Micromechanics and Microengineering 29.3 (2019), p. 035006. doi: 10.1088/1361-6439/aaf940.

  • [194] A. Belyaev and P.-A. Fayolle. “An ADMM-Based Scheme for Distance Function Approximation”. In: Numerical Algorithms 84.3 (2020), pp. 983–996. doi: 10.1007/s11075-019-00789-5.

  • [195] N. Cornea, D. Silver, and P. Min. “Curve-Skeleton Applications”. In: Proceedings of IEEE Visualization (VIS). Minneapolis: IEEE, 2005, pp. 95–102. doi: 10.1109/VISUAL.2005.1532783.

Own Publications

Journal Articles

  • [1] M. Quell, V. Suvorov, A. Hössinger, and J. Weinbub. “Parallel Velocity Extension for Level-Set-Based Material Flow on Hierarchical Meshes in Process TCAD”. In: IEEE Transactions on Electron Devices 68.11 (2021), pp. 5430–5437. doi: 10.1109/TED.2021.3087451.

  • [2] M. Quell, G. Diamantopoulos, A. Hössinger, and J. Weinbub. “Shared-Memory Block-Based Fast Marching Method for Hierarchical Meshes”. In: Journal of Computational and Applied Mathematics 392 (2021), pp. 113488-1–113488-15. doi: 10.1016/j.cam.2021.113488.

  • [3] W. Auzinger, H. Hofstätter, O. Koch, and M. Quell. “Adaptive Time Propagation for Time-Dependent Schrödinger Equations”. In: International Journal of Applied and Computational Mathematics 7.1 (2021), pp. 6-1–6-14. doi: 10.1007/s40819-020-00937-9.

  • [4] A. Toifl, M. Quell, X. Klemenschits, P. Manstetten, A. Hössinger, S. Selberherr, and J. Weinbub. “The Level-Set Method for Multi-Material Wet Etching and Non-Planar Selective Epitaxy”. In: IEEE Access 8 (2020), pp. 115406–115422. doi: 10.1109/ACCESS.2020.3004136.

  • [5] W. Auzinger, I. Brezinova, H. Hofstätter, O. Koch, and M. Quell. “Practical Splitting Methods for the Adaptive Integration of Nonlinear Evolution Equations. Part II: Comparison of Local Error Estimation and Step-Selection Strategies for Nonlinear Schrödinger and Wave Equations”. In: Computer Physics Communications 234 (2019), pp. 55–71. doi: 10.1016/j.cpc.2018.08.003.

  • [6] W. Auzinger, H. Hofstätter, O. Koch, M. Quell, and M. Thalhammer. “A Posteriori Error Estimation for Magnus-Type Integrators”. In: ESAIM: Mathematical Modelling and Numerical Analysis 53.1 (2019), pp. 197–218. doi: 10.1051/m2an/2018050.

  • [7] W. Auzinger, O. Koch, and M. Quell. “Adaptive High-Order Splitting Methods for Systems of Nonlinear Evolution Equations with Periodic Boundary Conditions”. In: Numerical Algorithms 75.1 (2017), pp. 261–283. doi: 10.1007/s11075-016-0206-8.

Book Contributions

  • [8] M. Quell, G. Diamantopoulos, A. Hössinger, S. Selberherr, and J. Weinbub. “Parallel Correction for Hierarchical Re-Distancing Using the Fast Marching Method”. In: Advances in High Performance Computing, Studies in Computational Intelligence. Cham: Springer International Publishing, 2020, pp. 438–451. doi: 10.1007/978-3-030-55347-0_37.

  • [9] M. Quell, P. Manstetten, A. Hössinger, S. Selberherr, and J. Weinbub. “Parallelized Construction of Extension Velocities for the Level-Set Method”. In: Parallel Processing and Applied Mathematics, Lecture Notes in Computer Science. Cham: Springer International Publishing, 2020, pp. 348–358. doi: 10.1007/978-3-030-43229-4_30.

  • [10] W. Auzinger, H. Hofstätter, O. Koch, M. Quell, and M. Thalhammer. “A Posteriori Error Estimation for Magnus-Type Integrators”. In: ASC Report 1/2018. Wien: Vienna University of Technology, 2018, pp. 1–19.

  • [11] W. Auzinger, I. Brezinova, H. Hofstätter, O. Koch, and M. Quell. “Practical Splitting Methods for the Adaptive Integration of Nonlinear Evolution Equations. Part II: Comparison of Local Error Estimation and Step-Selection Strategies for Nonlinear Schrödinger and Wave Equations”. In: ASC Report 14/2017. Wien: Vienna University of Technology, 2017, pp. 1–40.

  • [12] W. Auzinger, O. Koch, and M. Quell. “Adaptive High-Order Splitting Methods for Systems of Nonlinear Evolution Equations with Periodic Boundary Conditions”. In: ASC Report 41/2015. Wien: Vienna University of Technology, 2015, pp. 1–29.

  • [13] W. Auzinger, O. Koch, and M. Quell. “Splittingverfahren für die Gray-Scott-Gleichung”. In: ASC Report 07/2015. Wien: Vienna University of Technology, 2015, pp. 1–11.

Conference Contributions

  • [14] C. Lenz, A. Scharinger, M. Quell, P. Manstetten, A. Hössinger, and J. Weinbub. “Evaluating Parallel Feature Detection Methods for Implicit Surfaces”. In: Proceedings of the Austrian-Slovenian HPC Meeting (ASHPC). Maribor: University of Ljubljana, 2021, p. 31. doi: 10.3359/2021hpc.

  • [15] M. Quell, G. Diamantopoulos, A. Hössinger, and J. Weinbub. “Shared-Memory Block-Based Fast Marching Method for Hierarchical Meshes”. In: Proceedings of the European Seminar on Computing (ESCO). Pilsen: University of West Bohemia, 2020.

  • [16] M. Quell, G. Diamantopoulos, A. Hössinger, S. Selberherr, and J. Weinbub. “Parallelized Bottom-Up Correction in Hierarchical Re-Distancing for Topography Simulation”. In: Procedings of the High Performance Computing Conference (HPC). Borovets: Bulgarian Academy of Sciences, 2019, p. 45.

  • [17] M. Quell, P. Manstetten, A. Hössinger, S. Selberherr, and J. Weinbub. “Parallelized Construction of Extension Velocities for the Level-Set Method”. In: Proceedings of the International Conference on Parallel Processing and Applied Mathematics (PPAM). Bialystok: Czestochowa University of Technology, 2019, p. 42.

  • [18] M. Quell, A. Toifl, A. Hössinger, S. Selberherr, and J. Weinbub. “Parallelized Level-Set Velocity Extension Algorithm for Nanopatterning Applications”. In: Proceedings of the International Conference on Simulation of Semiconductor Processes and Devices (SISPAD). Udine: IEEE, 2019, pp. 335–338. doi: 10.1109/SISPAD.2019.8870482.

  • [19] A. Toifl, M. Quell, A. Hössinger, A. Babayan, S. Selberherr, and J. Weinbub. “Novel Numerical Dissipation Scheme for Level-Set Based Anisotropic Etching Simulations”. In: Proceedings of the International Conference on Simulation of Semiconductor Processes and Devices (SISPAD). Udine: IEEE, 2019, pp. 327–330. doi: 10.1109/SISPAD.2019.8870443.

  • [20] G. Diamantopoulos, P. Manstetten, L. Gnam, V. Simonka, L. F. Aguinsky, M. Quell, A. Toifl, A. Hössinger, and J. Weinbub. “Recent Advances in High Performance Process TCAD”. In: Proceedings of the SIAM Conference on Computational Science and Engineering (CSE). Spokane: Society for Industrial and Applied Mathematics, 2019, p. 335.

  • [21] L. Gnam, P. Manstetten, M. Quell, K. Rupp, S. Selberherr, and J. Weinbub. “A Flexible Shared-Memory Parallel Mesh Adaptation Framework”. In: Proceedings of the International Conference on Computational Science and Its Applications (ICCSA). Saint Petersburg: IEEE, 2019, pp. 158–165. doi: 10.1109/ICCSA.2019.00016.

  • [22] A. Hössinger, P. Manstetten, G. Diamantopoulos, M. Quell, and J. Weinbub. “High Performance Computing Aspects in Semiconductor Process Simulation”. In: Proceedings of the Workshop on High Performance TCAD (WHPTCAD). Chicago: Institute for Microelectronics, TU Wien, 2019, pp. 3–4.

  • [23] P. Manstetten, G. Diamantopoulos, L. Gnam, L. F. Aguinsky, M. Quell, A. Toifl, A. Scharinger, A. Hössinger, M. Ballicchia, M. Nedjalkov, and J. Weinbub. “High Performance TCAD: From Simulating Fabrication Processes to Wigner Quantum Transport”. In: Proceedings of the Workshop on High Performance TCAD (WHPTCAD). Chicago: Institute for Microelectronics, TU Wien, 2019, p. 13.

Curriculum Vitae

Personal Information
Name Michael Julian Augustus Quell
Nationality Austrian
Place of Birth Vienna, Austria
Education
06/2018 - present

Doctoral Program, Electrical Engineering,
Institute for Microelectronics,
Technische Universität (TU) Wien

04/2016 - 04/2018

Graduate Studies (MSc), Technical Mathematics,
TU Wien, Faculty of Mathematics and Geoinformation

10/2012 - 04/2016

Graduate Studies (BSc), Technical Mathematics,
TU Wien, Faculty of Mathematics and Geoinformation

09/2011 - 09/2012

Active Reserve Officer Training, Signaling,
Fernmeldetruppenschule, Wien

09/2003 - 06/2011

Matura, Majors: Mathematics, German, Applied Computer Aided Geometry, English(FL)
BRG 8 Albertgasse, Wien

Research Positions
06/2019 - present

University Assistant,
Institute for Microelectronics, TU Wien

06/2018 - present

Project Assistant, Christian Doppler Laboratory for High Performance TCAD,
Institute for Microelectronics, TU Wien

10/2014 - 10/2016

Study Assistant, Institute for Analysis and Scientific Computing, TU Wien