Parallel three-dimensional simulations of quasi-static elastoplastic solids

Hypo-elastoplasticity is a flexible framework for modeling the mechanics of many hard materials under small elastic deformation and large plastic deformation. Under typical loading rates, most laboratory tests of these materials happen in the quasi-static limit, but there are few existing numerical methods tailor-made for this physical regime. In this work, we extend to three dimensions a recent projection method for simulating quasi-static hypo-elastoplastic materials. The method is based on a mathematical correspondence to the incompressible Navier-Stokes equations, where the projection method of Chorin (1968) is an established numerical technique. We develop and utilize a three-dimensional parallel geometric multigrid solver employed to solve a linear system for the quasi-static projection. Our method is tested through simulation of three-dimensional shear band nucleation and growth, a precursor to failure in many materials. As an example system, we employ a physical model of a bulk metallic glass based on the shear transformation zone theory, but the method can be applied to any elastoplasticity model. We consider several examples of three-dimensional shear banding, and examine shear band formation in physically realistic materials with heterogeneous initial conditions under both simple shear deformation and boundary conditions inspired by friction welding.Comment: Final version. Accepted for publication in Computer Physics Communication

Solving elliptic problems with discontinuities on irregular domains – the Voronoi Interface Method.

We introduce a simple method, dubbed the Voronoi Interface Method, to solve Elliptic problems with discontinuities across the interface of irregular domains. This method produces a linear system that is symmetric positive definite with only its right-hand-side affected by the jump conditions. The solution and the solution's gradients are second-order accurate and first-order accurate, respectively, in the L∞L∞ norm, even in the case of large ratios in the diffusion coefficient. This approach is also applicable to arbitrary meshes. Additional degrees of freedom are placed close to the interface and a Voronoi partition centered at each of these points is used to discretize the equations in a finite volume approach. Both the locations of the additional degrees of freedom and their Voronoi discretizations are straightforward in two and three spatial dimensions

An Eulerian projection method for quasi-static elastoplasticity

A well-established numerical approach to solve the Navier--Stokes equations for incompressible fluids is Chorin's projection method, whereby the fluid velocity is explicitly updated, and then an elliptic problem for the pressure is solved, which is used to orthogonally project the velocity field to maintain the incompressibility constraint. In this paper, we develop a mathematical correspondence between Newtonian fluids in the incompressible limit and hypo-elastoplastic solids in the slow, quasi-static limit. Using this correspondence, we formulate a new fixed-grid, Eulerian numerical method for simulating quasi-static hypo-elastoplastic solids, whereby the stress is explicitly updated, and then an elliptic problem for the velocity is solved, which is used to orthogonally project the stress to maintain the quasi-staticity constraint. We develop a finite-difference implementation of the method and apply it to an elasto-viscoplastic model of a bulk metallic glass based on the shear transformation zone theory. We show that in a two-dimensional plane strain simple shear simulation, the method is in quantitative agreement with an explicit method. Like the fluid projection method, it is efficient and numerically robust, making it practical for a wide variety of applications. We also demonstrate that the method can be extended to simulate objects with evolving boundaries. We highlight a number of correspondences between incompressible fluid mechanics and quasi-static elastoplasticity, creating possibilities for translating other numerical methods between the two classes of physical problems.Comment: 49 pages, 20 figure

An adaptive variational finite difference framework for efficient symmetric octree viscosity

While pressure forces are often the bottleneck in (near-)inviscid fluid simulations, viscosity can impose orders of magnitude greater computational costs at lower Reynolds numbers. We propose an implicit octree finite difference discretization that significantly accelerates the solution of the free surface viscosity equations using adaptive staggered grids, while supporting viscous buckling and rotation effects, variable viscosity, and interaction with scripted moving solids. In experimental comparisons against regular grids, our method reduced the number of active velocity degrees of freedom by as much as a factor of 7.7 and reduced linear system solution times by factors between 3.8 and 9.4. We achieve this by developing a novel adaptive variational finite difference methodology for octrees and applying it to the optimization form of the viscosity problem. This yields a linear system that is symmetric positive definite by construction, unlike naive finite difference/volume methods, and much sparser than a hypothetical finite element alternative. Grid refinement studies show spatial convergence at first order in L∞ and second order in L1, while the significantly smaller size of the octree linear systems allows for the solution of viscous forces at higher effective resolutions than with regular grids. We demonstrate the practical benefits of our adaptive scheme by replacing the regular grid viscosity step of a commercial liquid simulator (Houdini) to yield large speed-ups, and by incorporating it into an existing inviscid octree simulator to add support for viscous flows. Animations of viscous liquids pouring, bending, stirring, buckling, and melting illustrate that our octree method offers significant computational gains and excellent visual consistency with its regular grid counterpart.This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (RGPIN-04360-2014, CRDPJ-499952-2016) and the Rutgers University start-up grant

A cell-centred finite volume method for the Poisson problem on non-graded quadtrees with second order accurate gradients

The final publication is available at Elsevier via https://doi.org/10.1016/j.jcp.2016.11.035 © 2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/This paper introduces a two-dimensional cell-centred finite volume discretization of the Poisson problem on adaptive Cartesian quadtree grids which exhibits second order accuracy in both "the solution and its gradients, and requires no grading condition between adjacent cells. At T-junction configurations, which occur wherever resolution differs between neighboring cells, use of the standard centred difference gradient stencil requires that ghost values be constructed by interpolation. To properly recover second order accuracy in the resulting numerical gradients, prior work addressing block-structured grids and graded trees has shown that quadratic, rather than linear, interpolation is required; the gradients otherwise exhibit only first order convergence, which limits potential applications such as fluid flow. However, previous schemes fail or lose accuracy in the presence of the more complex T-junction geometries arising in the case of general non-graded quadtrees, which place no restrictions on the resolution of neighboring cells. We therefore propose novel quadratic interpolant constructions for this case that enable second order convergence by relying on stencils oriented diagonally and applied recursively as needed. The method handles complex tree topologies and large resolution jumps between neighboring cells, even along the domain boundary, and both Dirichlet and Neumann boundary conditions are supported. Numerical experiments confirm the overall second order accuracy of the method in the L-infinity norm. (C) 2016 Elsevier Inc. All rights reserved.This work was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada (Grant RGPIN-04360-2014)

Kartezyen Hesaplama Ağları Kullanılarak Üç Boyutlu Sıkıştırılabilir Akışlar için Navier-Stokes Çözücüsü Geliştirilmesi

TÜBİTAK MAG Proje01.06.2017Bu proje çerçevesinde Kartezyen hesaplama ağları için üç boyutlu bir Navier-Stokes çözücüsü geliştirilmiştir. Hesaplama ağının üretimi ile akış alanının çözümü aşamaları arasında gerekli olan kullanıcı müdahalesini ortadan kaldırmak üzere geliştirilecek yazılımın tam otomatik olarak gerçekleştirilmiştir. Akış alanındaki gövdenin geometrisi yapısal olmayan üçgen elemanlar kullanılarak üç boyutlu bir yüzey hesaplama ağı şeklinde verildiğinde, gövde uyumlu (body-fitted) üçgen prizma elemanlardan oluşan hesaplama ağı gövde geometrisinin şişirilmesiyle otomatik olarak oluşturulmaktadır. Daha sonra, gövde uyumlu hesaplama ağı, çözüm alanının sınırlarını tanımlayan bir kök hücrenin eşit hücrelere bölünmesi ile elde edilen Kartezyen hesaplama ağının içerisine yerleştirilmektedir. Her iki hesaplama ağının arasında kalan bölge dört yüzlü tetrahedral elemanlarla doldurulmaktadır. Navier-Stokes denklemlerinin sonlu hacim formülasyonu hücre merkezli yaklaşımla kullanılmaktadır. Hücre yüzlerindeki akılar akı fark ayrıştırması ve akı vektör ayrıştırması yöntemleriyle hesaplanmaktadır. Uzayda ikinci dereceden doğruluk elde edilebilmesi için basit değişkenlerin yeniden oluşturulmasında (reconstruction) yol tümleme (path integration) ve asgari kareler (least squares) yöntemleri kullanılmaktadır. Doğru ve sınırlı değerler elde edilebilmesi için yeniden oluşturma işlemi sırasında limitleyiciler kullanılmaktadır. Türbülans modeli olarak ise literatürde mevcut modellerden bir denklemli SpalartAllmaras türbülans modeli ile iki denklemli k-e ve k-w türbülans modellerinden yararlanılmaktadır. Yakınsamanın hızlandırılabilmesi için yerel zaman adımlarıyla birlikte çok kademeli (multistage) zaman adımlaması kullanılmaktadır. Çözüme bağlı hesaplama ağı adaptasyonu çözüm ile ağ arasındaki uyumun oluşmasını sağlayarak, akıştaki kritik bölgelerin daha iyi çözümlenmesine olanak sağlamaktadır. Çözüm adaptasyonu kayma tabakalarında hız dönümü, normal ve oblik şoklarda ise hız gradyanı kullanılarak gerçekleştirilmektedir. Geliştirilen yazılım NACA 0012 ve ONERA M6 kanadı etrafındaki üç boyutlu akış için test edilmiş ve elde edilen sayısal sonuçlar literatürde mevcut deneysel sonuçları ile karşılaştırılarak doğrulanmıştır

Efficient Liquid Animation: New Discretizations for Spatially Adaptive Liquid Viscosity and Reduced-Model Two-Phase Bubbles and Inviscid Liquids

The work presented in this thesis focuses on improving the computational efficiency when simulating viscous liquids and air bubbles immersed in liquids by designing new discretizations to focus computational effort in regions that meaningfully contribute to creating realistic motion. For example, when simulating air bubbles rising through a liquid, the entire bubble volume is traditionally simulated despite the bubble’s interior being visually unimportant. We propose our constraint bubbles model to avoid simulating the interior of the bubble volume by reformulating the usual incompressibility constraint throughout a bubble volume as a constraint over only the bubble’s surface. Our constraint method achieves qualitatively similar results compared to a two-phase simulation ground-truth for bubbles with low densities (e.g., air bubbles in water). For bubbles with higher densities, we propose our novel affine regions to model the bubble’s entire velocity field with a single affine vector field. We demonstrate that affine regions can correctly achieve hydrostatic equilibrium for bubble densities that match the surrounding liquid and correctly sink for higher densities. Finally, we introduce a tiled approach to subdivide large-scale affine regions into smaller subregions. Using this strategy, we are able to accelerate single-phase free surface flow simulations, offering a novel approach to adaptively enforce incompressibility in free surface liquids without complex data structures. While pressure forces are often the bottleneck for inviscid fluid simulations, viscosity can impose orders of magnitude greater computational costs. We observed that viscous liquids require high simulation resolution at the surface to capture detailed viscous buckling and rotational motion but, because viscosity dampens relative motion, do not require the same resolution in the liquid’s interior. We therefore propose a novel adaptive method to solve free surface viscosity equations by discretizing the variational finite difference approach of Batty and Bridson (2008) on an octree grid. Our key insight is that the variational method guarantees a symmetric positive definite linear system by construction, allowing the use of fast numerical solvers like the Conjugate Gradients method. By coarsening simulation grid cells inside the liquid volume, we rapidly reduce the degrees-of-freedom in the viscosity linear system up to a factor of 7.7x and achieve performance improvements for the linear solve between 3.8x and 9.4x compared to a regular grid equivalent. The results of our adaptive method closely match an equivalent regular grid for common scenarios such as: rotation and bending, buckling and folding, and solid-liquid interactions

Numerical Simulations of the Two-phase flow and Fluid-Structure Interaction Problems with Adaptive Mesh Refinement

Numerical simulations of two-phase flow and fluid structure interaction problems are of great interest in many environmental problems and engineering applications. To capture the complex physical processes involved in these problems, a high grid resolution is usually needed. However, one does not need or maybe cannot afford a fine grid of uniformly high resolution across the whole domain. The need to resolve local fine features can be addressed by the adaptive mesh refinement (AMR) method, which increases the grid resolution in regions of interest as needed during the simulation while leaving general estimates in other regions. In this work, we propose a block-structured adaptive mesh refinement (BSAMR) framework to simulate two-phase flows using the level set (LS) function with both the subcycling and non-subcycling methods on a collocated grid. To the best of our knowledge, this is the first framework that unifies the subcycling and non-subcycling methods to simulate two-phase flows. The use of the collocated grid is also the first among the two-phase BSAMR framework, which significantly simplifies the implementation of multi-level differential operators and interpolation schemes. We design the synchronization operations, including the averaging, refluxing, and synchronization projection, which ensures that the flow field is divergence-free on the multi-level grid. It is shown that the present multi-level scheme can accurately resolve the interfaces of the two-phase flows with gravitational and surface tension effects while having good momentum and energy conservation.Comment: 178 page

Recursive Rewarding Modified Adaptive Cell Decomposition (RR-MACD): A Dynamic Path Planning Algorithm for UAVs

[EN] A relevant task in unmanned aerial vehicles (UAV) flight is path planning in 3D environments. This task must be completed using the least possible computing time. The aim of this article is to combine methodologies to optimise the task in time and offer a complete 3D trajectory. The flight environment will be considered as a 3D adaptive discrete mesh, where grids are created with minimal refinement in the search for collision-free spaces. The proposed path planning algorithm for UAV saves computational time and memory resources compared with classical techniques. With the construction of the discrete meshing, a cost response methodology is applied as a discrete deterministic finite automaton (DDFA). A set of optimal partial responses, calculated recursively, indicates the collision-free spaces in the final path for the UAV flight.The authors would like to acknowledge the Spanish Ministry of Economy and Competitiveness for providing funding through the project DPI2015-71443-R and the local administration Generalitat Valenciana through the project GV/2017/029. Franklin Samaniego thanks IFTH (Instituto de Fomento al Talento Humano) Ecuador (2015-AR2Q9209), for its sponsorship of this work.Samaniego-Riera, FE.; Sanchís Saez, J.; Garcia-Nieto, S.; Simarro Fernández, R. (2019). Recursive Rewarding Modified Adaptive Cell Decomposition (RR-MACD): A Dynamic Path Planning Algorithm for UAVs. Electronics. 8(3):1-21. https://doi.org/10.3390/electronics8030306S12183Valavanis, K. P., & Vachtsevanos, G. J. (Eds.). (2015). Handbook of Unmanned Aerial Vehicles. doi:10.1007/978-90-481-9707-120 Great UAV Applications Areas for Droneshttp://air-vid.com/wp/20-great-uav-applications-areas-drones/Industry Experts—Microdroneshttps://www.microdrones.com/en/industry-experts/Li, J., & Han, Y. (2017). Optimal Resource Allocation for Packet Delay Minimization in Multi-Layer UAV Networks. IEEE Communications Letters, 21(3), 580-583. doi:10.1109/lcomm.2016.2626293Stuchlík, R., Stachoň, Z., Láska, K., & Kubíček, P. (2015). Unmanned Aerial Vehicle – Efficient mapping tool available for recent research in polar regions. Czech Polar Reports, 5(2), 210-221. doi:10.5817/cpr2015-2-18Pulver, A., & Wei, R. (2018). Optimizing the spatial location of medical drones. Applied Geography, 90, 9-16. doi:10.1016/j.apgeog.2017.11.009Claesson, A., Svensson, L., Nordberg, P., Ringh, M., Rosenqvist, M., Djarv, T., … Hollenberg, J. (2017). Drones may be used to save lives in out of hospital cardiac arrest due to drowning. Resuscitation, 114, 152-156. doi:10.1016/j.resuscitation.2017.01.003Reineman, B. D., Lenain, L., Statom, N. M., & Melville, W. K. (2013). Development and Testing of Instrumentation for UAV-Based Flux Measurements within Terrestrial and Marine Atmospheric Boundary Layers. Journal of Atmospheric and Oceanic Technology, 30(7), 1295-1319. doi:10.1175/jtech-d-12-00176.1LaValle, S. M. (2006). Planning Algorithms. doi:10.1017/cbo9780511546877Elbanhawi, M., & Simic, M. (2014). Sampling-Based Robot Motion Planning: A Review. IEEE Access, 2, 56-77. doi:10.1109/access.2014.2302442Hernandez, K., Bacca, B., & Posso, B. (2017). Multi-goal Path Planning Autonomous System for Picking up and Delivery Tasks in Mobile Robotics. IEEE Latin America Transactions, 15(2), 232-238. doi:10.1109/tla.2017.7854617Kohlbrecher, S., von Stryk, O., Meyer, J., & Klingauf, U. (2011). A flexible and scalable SLAM system with full 3D motion estimation. 2011 IEEE International Symposium on Safety, Security, and Rescue Robotics. doi:10.1109/ssrr.2011.6106777Aguilar, W., & Morales, S. (2016). 3D Environment Mapping Using the Kinect V2 and Path Planning Based on RRT Algorithms. Electronics, 5(4), 70. doi:10.3390/electronics5040070Aguilar, W. G., Morales, S., Ruiz, H., & Abad, V. (2017). RRT* GL Based Optimal Path Planning for Real-Time Navigation of UAVs. Lecture Notes in Computer Science, 585-595. doi:10.1007/978-3-319-59147-6_50Yao, P., Wang, H., & Su, Z. (2015). Hybrid UAV path planning based on interfered fluid dynamical system and improved RRT. IECON 2015 - 41st Annual Conference of the IEEE Industrial Electronics Society. doi:10.1109/iecon.2015.7392202Yan, F., Liu, Y.-S., & Xiao, J.-Z. (2013). Path Planning in Complex 3D Environments Using a Probabilistic Roadmap Method. International Journal of Automation and Computing, 10(6), 525-533. doi:10.1007/s11633-013-0750-9Yeh, H.-Y., Thomas, S., Eppstein, D., & Amato, N. M. (2012). UOBPRM: A uniformly distributed obstacle-based PRM. 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems. doi:10.1109/iros.2012.6385875Denny, J., & Amatoo, N. M. (2013). Toggle PRM: A Coordinated Mapping of C-Free and C-Obstacle in Arbitrary Dimension. Algorithmic Foundations of Robotics X, 297-312. doi:10.1007/978-3-642-36279-8_18Li, Q., Wei, C., Wu, J., & Zhu, X. (2014). Improved PRM method of low altitude penetration trajectory planning for UAVs. Proceedings of 2014 IEEE Chinese Guidance, Navigation and Control Conference. doi:10.1109/cgncc.2014.7007587Ortiz-Arroyo, D. (2015). A hybrid 3D path planning method for UAVs. 2015 Workshop on Research, Education and Development of Unmanned Aerial Systems (RED-UAS). doi:10.1109/red-uas.2015.7440999Thanou, M., & Tzes, A. (2014). Distributed visibility-based coverage using a swarm of UAVs in known 3D-terrains. 2014 6th International Symposium on Communications, Control and Signal Processing (ISCCSP). doi:10.1109/isccsp.2014.6877904Qu, Y., Zhang, Y., & Zhang, Y. (2014). Optimal flight path planning for UAVs in 3-D threat environment. 2014 International Conference on Unmanned Aircraft Systems (ICUAS). doi:10.1109/icuas.2014.6842250Fang, Z., Luan, C., & Sun, Z. (2017). A 2D Voronoi-Based Random Tree for Path Planning in Complicated 3D Environments. Advances in Intelligent Systems and Computing, 433-445. doi:10.1007/978-3-319-48036-7_31Khuswendi, T., Hindersah, H., & Adiprawita, W. (2011). UAV path planning using potential field and modified receding horizon A* 3D algorithm. Proceedings of the 2011 International Conference on Electrical Engineering and Informatics. doi:10.1109/iceei.2011.6021579Chen, X., & Zhang, J. (2013). The Three-Dimension Path Planning of UAV Based on Improved Artificial Potential Field in Dynamic Environment. 2013 5th International Conference on Intelligent Human-Machine Systems and Cybernetics. doi:10.1109/ihmsc.2013.181Rivera, D. M., Prieto, F. A., & Ramirez, R. (2012). Trajectory Planning for UAVs in 3D Environments Using a Moving Band in Potential Sigmoid Fields. 2012 Brazilian Robotics Symposium and Latin American Robotics Symposium. doi:10.1109/sbr-lars.2012.26Liu Lifen, Shi Ruoxin, Li Shuandao, & Wu Jiang. (2016). Path planning for UAVS based on improved artificial potential field method through changing the repulsive potential function. 2016 IEEE Chinese Guidance, Navigation and Control Conference (CGNCC). doi:10.1109/cgncc.2016.7829099Dijkstra, E. W. (1959). A note on two problems in connexion with graphs. Numerische Mathematik, 1(1), 269-271. doi:10.1007/bf01386390Verscheure, L., Peyrodie, L., Makni, N., Betrouni, N., Maouche, S., & Vermandel, M. (2010). Dijkstra’s algorithm applied to 3D skeletonization of the brain vascular tree: Evaluation and application to symbolic. 2010 Annual International Conference of the IEEE Engineering in Medicine and Biology. doi:10.1109/iembs.2010.5626112Hart, P., Nilsson, N., & Raphael, B. (1968). A Formal Basis for the Heuristic Determination of Minimum Cost Paths. IEEE Transactions on Systems Science and Cybernetics, 4(2), 100-107. doi:10.1109/tssc.1968.300136Ferguson, D., & Stentz, A. (s. f.). Field D*: An Interpolation-Based Path Planner and Replanner. Robotics Research, 239-253. doi:10.1007/978-3-540-48113-3_22De Filippis, L., Guglieri, G., & Quagliotti, F. (2011). Path Planning Strategies for UAVS in 3D Environments. Journal of Intelligent & Robotic Systems, 65(1-4), 247-264. doi:10.1007/s10846-011-9568-2Gautam, S. A., & Verma, N. (2014). Path planning for unmanned aerial vehicle based on genetic algorithm & artificial neural network in 3D. 2014 International Conference on Data Mining and Intelligent Computing (ICDMIC). doi:10.1109/icdmic.2014.6954257Maturana, D., & Scherer, S. (2015). 3D Convolutional Neural Networks for landing zone detection from LiDAR. 2015 IEEE International Conference on Robotics and Automation (ICRA). doi:10.1109/icra.2015.7139679Iswanto, I., Wahyunggoro, O., & Imam Cahyadi, A. (2016). Quadrotor Path Planning Based on Modified Fuzzy Cell Decomposition Algorithm. TELKOMNIKA (Telecommunication Computing Electronics and Control), 14(2), 655. doi:10.12928/telkomnika.v14i2.2989Duan, H., Yu, Y., Zhang, X., & Shao, S. (2010). Three-dimension path planning for UCAV using hybrid meta-heuristic ACO-DE algorithm. Simulation Modelling Practice and Theory, 18(8), 1104-1115. doi:10.1016/j.simpat.2009.10.006He, Y., Zeng, Q., Liu, J., Xu, G., & Deng, X. (2013). Path planning for indoor UAV based on Ant Colony Optimization. 2013 25th Chinese Control and Decision Conference (CCDC). doi:10.1109/ccdc.2013.6561444Zhang, Y., Wu, L., & Wang, S. (2013). UCAV Path Planning by Fitness-Scaling Adaptive Chaotic Particle Swarm Optimization. Mathematical Problems in Engineering, 2013, 1-9. doi:10.1155/2013/705238Goel, U., Varshney, S., Jain, A., Maheshwari, S., & Shukla, A. (2018). Three Dimensional Path Planning for UAVs in Dynamic Environment using Glow-worm Swarm Optimization. Procedia Computer Science, 133, 230-239. doi:10.1016/j.procs.2018.07.028YongBo, C., YueSong, M., JianQiao, Y., XiaoLong, S., & Nuo, X. (2017). Three-dimensional unmanned aerial vehicle path planning using modified wolf pack search algorithm. Neurocomputing, 266, 445-457. doi:10.1016/j.neucom.2017.05.059Wang, G.-G., Chu, H. E., & Mirjalili, S. (2016). Three-dimensional path planning for UCAV using an improved bat algorithm. Aerospace Science and Technology, 49, 231-238. doi:10.1016/j.ast.2015.11.040Aghababa, M. P. (2012). 3D path planning for underwater vehicles using five evolutionary optimization algorithms avoiding static and energetic obstacles. Applied Ocean Research, 38, 48-62. doi:10.1016/j.apor.2012.06.002Mac, T. T., Copot, C., Tran, D. T., & De Keyser, R. (2016). Heuristic approaches in robot path planning: A survey. Robotics and Autonomous Systems, 86, 13-28. doi:10.1016/j.robot.2016.08.001Szirmay-Kalos, L., & Márton, G. (1998). Worst-case versus average case complexity of ray-shooting. Computing, 61(2), 103-131. doi:10.1007/bf02684409Berger, M. J., & Oliger, J. (1984). Adaptive mesh refinement for hyperbolic partial differential equations. Journal of Computational Physics, 53(3), 484-512. doi:10.1016/0021-9991(84)90073-1Min, C., & Gibou, F. (2006). A second order accurate projection method for the incompressible Navier–Stokes equations on non-graded adaptive grids. Journal of Computational Physics, 219(2), 912-929. doi:10.1016/j.jcp.2006.07.019Hasbestan, J. J., & Senocak, I. (2018). Binarized-octree generation for Cartesian adaptive mesh refinement around immersed geometries. Journal of Computational Physics, 368, 179-195. doi:10.1016/j.jcp.2018.04.039Pantano, C., Deiterding, R., Hill, D. J., & Pullin, D. I. (2007). A low numerical dissipation patch-based adaptive mesh refinement method for large-eddy simulation of compressible flows. Journal of Computational Physics, 221(1), 63-87. doi:10.1016/j.jcp.2006.06.011Ryde, J., & Hu, H. (2009). 3D mapping with multi-resolution occupied voxel lists. Autonomous Robots, 28(2), 169-185. doi:10.1007/s10514-009-9158-3Samet, H., & Kochut, A. (s. f.). Octree approximation an compression methods. Proceedings. First International Symposium on 3D Data Processing Visualization and Transmission. doi:10.1109/tdpvt.2002.1024101Samaniego, F., Sanchis, J., Garcia-Nieto, S., & Simarro, R. (2017). UAV motion planning and obstacle avoidance based on adaptive 3D cell decomposition: Continuous space vs discrete space. 2017 IEEE Second Ecuador Technical Chapters Meeting (ETCM). doi:10.1109/etcm.2017.8247533Skoldstam, M., Akesson, K., & Fabian, M. (2007). Modeling of discrete event systems using finite automata with variables. 2007 46th IEEE Conference on Decision and Control. doi:10.1109/cdc.2007.4434894Yang, Y.-H. E., & Prasanna, V. K. (2011). Space-time tradeoff in regular expression matching with semi-deterministic finite automata. 2011 Proceedings IEEE INFOCOM. doi:10.1109/infcom.2011.5934986Normativa Sobre Drones en España [2019]—Aerial Insightshttp://www.aerial-insights.co/blog/normativa-drones-espana/Disposición 15721 del BOE núm. 316 de 2017 - BOE.eshttps://www.boe.es/boe/dias/2017/12/29/pdfs/BOE-A-2017-15721.pdfVelasco-Carrau, J., García-Nieto, S., Salcedo, J. V., & Bishop, R. H. (2016). Multi-Objective Optimization for Wind Estimation and Aircraft Model Identification. Journal of Guidance, Control, and Dynamics, 39(2), 372-389. doi:10.2514/1.g001294Vanegas, G., Samaniego, F., Girbes, V., Armesto, L., & Garcia-Nieto, S. (2018). Smooth 3D path planning for non-holonomic UAVs. 2018 7th International Conference on Systems and Control (ICSC). doi:10.1109/icosc.2018.8587835Samaniego, F., Sanchis, J., Garcia-Nieto, S., & Simarro, R. (2018). Comparative Study of 3-Dimensional Path Planning Methods Constrained by the Maneuverability of Unmanned Aerial Vehicles. 2018 7th International Conference on Systems and Control (ICSC). doi:10.1109/icosc.2018.858781