Cartesian grid FEM (cgFEM): High performance h-adaptive FE analysis with efficient error control. Application to structural shape optimization

More and more challenging designs are required everyday in today¿s industries. The traditional trial and error procedure commonly used for mechanical parts design is not valid any more since it slows down the design process and yields suboptimal designs. For structural components, one alternative consists in using shape optimization processes which provide optimal solutions. However, these techniques require a high computational effort and require extremely efficient and robust Finite Element (FE) programs. FE software companies are aware that their current commercial products must improve in this sense and devote considerable resources to improve their codes. In this work we propose to use the Cartesian Grid Finite Element Method, cgFEM as a tool for efficient and robust numerical analysis. The cgFEM methodology developed in this thesis uses the synergy of a variety of techniques to achieve this purpose, but the two main ingredients are the use of Cartesian FE grids independent of the geometry of the component to be analyzed and an efficient hierarchical data structure. These two features provide to the cgFEM technology the necessary requirements to increase the efficiency of the cgFEM code with respect to commercial FE codes. As indicated in [1, 2], in order to guarantee the convergence of a structural shape optimization process we need to control the error of each geometry analyzed. In this sense the cgFEM code also incorporates the appropriate error estimators. These error estimators are specifically adapted to the cgFEM framework to further increase its efficiency. This work introduces a solution recovery technique, denoted as SPR-CD, that in combination with the Zienkiewicz and Zhu error estimator [3] provides very accurate error measures of the FE solution. Additionally, we have also developed error estimators and numerical bounds in Quantities of Interest based on the SPR-CD technique to allow for an efficient control of the quality of the numerical solution. Regarding error estimation, we also present three new upper error bounding techniques for the error in energy norm of the FE solution, based on recovery processes. Furthermore, this work also presents an error estimation procedure to control the quality of the recovered solution in stresses provided by the SPR-CD technique. Since the recovered stress field is commonly more accurate and has a higher convergence rate than the FE solution, we propose to substitute the raw FE solution by the recovered solution to decrease the computational cost of the numerical analysis. All these improvements are reflected by the numerical examples of structural shape optimization problems presented in this thesis. These numerical analysis clearly show the improved behavior of the cgFEM technology over the classical FE implementations commonly used in industry.Cada d'¿a dise¿nos m'as complejos son requeridos por las industrias actuales. Para el dise¿no de nuevos componentes, los procesos tradicionales de prueba y error usados com'unmente ya no son v'alidos ya que ralentizan el proceso y dan lugar a dise¿nos sub-'optimos. Para componentes estructurales, una alternativa consiste en usar procesos de optimizaci'on de forma estructural los cuales dan como resultado dise¿nos 'optimos. Sin embargo, estas t'ecnicas requieren un alto coste computacional y tambi'en programas de Elementos Finitos (EF) extremadamente eficientes y robustos. Las compa¿n'¿as de programas de EF son conocedoras de que sus programas comerciales necesitan ser mejorados en este sentido y destinan importantes cantidades de recursos para mejorar sus c'odigos. En este trabajo proponemos usar el M'etodo de Elementos Finitos basado en mallados Cartesianos (cgFEM) como una herramienta eficiente y robusta para el an'alisis num'erico. La metodolog'¿a cgFEM desarrollada en esta tesis usa la sinergia entre varias t'ecnicas para lograr este prop'osito, cuyos dos ingredientes principales son el uso de los mallados Cartesianos de EF independientes de la geometr'¿a del componente que va a ser analizado y una eficiente estructura jer'arquica de datos. Estas dos caracter'¿sticas confieren a la tecnolog'¿a cgFEM de los requisitos necesarios para aumentar la eficiencia del c'odigo cgFEM con respecto a c'odigos comerciales. Como se indica en [1, 2], para garantizar la convergencia del proceso de optimizaci'on de forma estructural se necesita controlar el error en cada geometr'¿a analizada. En este sentido el c'odigo cgFEM tambi'en incorpora los apropiados estimadores de error. Estos estimadores de error han sido espec'¿ficamente adaptados al entorno cgFEM para aumentar su eficiencia. En esta tesis se introduce un proceso de recuperaci'on de la soluci'on, llamado SPR-CD, que en combinaci'on con el estimador de error de Zienkiewicz y Zhu [3], da como resultado medidas muy precisas del error de la soluci'on de EF. Adicionalmente, tambi'en se han desarrollado estimadores de error y cotas num'ericas en Magnitudes de Inter'es basadas en la t'ecnica SPR-CD para permitir un eficiente control de la calidad de la soluci'on num'erica. Respecto a la estimaci'on de error, tambi'en se presenta un proceso de estimaci'on de error para controlar la calidad del campo de tensiones recuperado obtenido mediante la t'ecnica SPR-CD. Ya que el campo recuperado es por lo general m'as preciso y tiene un mayor orden de convergencia que la soluci'on de EF, se propone sustituir la soluci'on de EF por la soluci'on recuperada para disminuir as'¿ el coste computacional del an'alisis num'erico. Todas estas mejoras se han reflejado en esta tesis mediante ejemplos num'ericos de problemas de optimizaci'on de forma estructural. Los resultados num'ericos muestran claramente un mejor comportamiento de la tecnolog'¿a cgFEM con respecto a implementaciones cl'asicas de EF com'unmente usadas en la industria.Nadal Soriano, E. (2014). Cartesian grid FEM (cgFEM): High performance h-adaptive FE analysis with efficient error control. Application to structural shape optimization [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/35620TESI

Proper generalized decomposition solutions within a domain decomposition strategy

Domain decomposition strategies and proper generalized decomposition are efficiently combined to obtain a fast evaluation of the solution approximation in parameterized elliptic problems with complex geometries. The classical difficulties associated to the combination of layered domains with arbitrarily oriented midsurfaces, which may require in‐plane–out‐of‐plane techniques, are now dismissed. More generally, solutions on large domains can now be confronted within a domain decomposition approach. This is done with a reduced cost in the offline phase because the proper generalized decomposition gives an explicit description of the solution in each subdomain in terms of the solution at the interface. Thus, the evaluation of the approximation in each subdomain is a simple function evaluation given the interface values (and the other problem parameters). The interface solution can be characterized by any a priori user‐defined approximation. Here, for illustration purposes, hierarchical polynomials are used. The repetitiveness of the subdomains is exploited to reduce drastically the offline computational effort. The online phase requires solving a nonlinear problem to determine all the interface solutions. However, this problem only has degrees of freedom on the interfaces and the Jacobian matrix is explicitly determined. Obviously, other parameters characterizing the solution (material constants, external loads, and geometry) can also be incorporated in the explicit description of the solution

Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference

The 6th ECCOMAS Young Investigators Conference YIC2021 will take place from July 7th through 9th, 2021 at Universitat Politècnica de València, Spain. The main objective is to bring together in a relaxed environment young students, researchers and professors from all areas related with computational science and engineering, as in the previous YIC conferences series organized under the auspices of the European Community on Computational Methods in Applied Sciences (ECCOMAS). Participation of senior scientists sharing their knowledge and experience is thus critical for this event.YIC 2021 is organized at Universitat Politécnica de València by the Sociedad Española de Métodos Numéricos en Ingeniería (SEMNI) and the Sociedad Española de Matemática Aplicada (SEMA). It is promoted by the ECCOMAS.The main goal of the YIC 2021 conference is to provide a forum for presenting and discussing the current state-of-the-art achievements on Computational Methods and Applied Sciences,including theoretical models, numerical methods, algorithmic strategies and challenging engineering applications.Nadal Soriano, E.; Rodrigo Cardiel, C.; Martínez Casas, J. (2022). Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. https://doi.org/10.4995/YIC2021.2021.15320EDITORIA

High-order discontinuous Galerkin method for time-domain electromagnetics on geometry-independent Cartesian meshes

This is the peer reviewed version of the following article: Navarro¿García, H., Sevilla, R., Nadal, E., & Ródenas, J. J. (2021). High¿order discontinuous Galerkin method for time¿domain electromagnetics on geometry¿independent Cartesian meshes. International Journal for Numerical Methods in Engineering, 122(24), 7632-7663, which has been published in final form at https://doi.org/10.1002/nme.6846. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.[ES] En este trabajo presentamos el método de los elementos finitos con mallados cartesianos y formulación Galerkin discontinua (cgDG), una técnica novedosa que permite la obtención de soluciones numéricas para problemas dominados por términos convectivos. Esta técnica combina la alta precisión y eficiencia de la discretización discontinua de alto orden característica de la formulación Galerkin discontinua con la simplicidad y estructura jerárquica de los mallados cartesianos independientes de la geometría. El correcto tratamiento de los elementos localizados sobre la frontera del dominio de cálculo es crucial a fin de asegurar un buen desempeño del algoritmo. El método tiene en cuenta la definición exacta de la geometría, evitando la aparición de artefactos derivados de una pobre representación de las fronteras. Por otra parte, se ha definido un procedimiento de estabilización que elimina la restricción que impone sobre el paso temporal del integrador explícito la presencia de elementos intersecados con patrones de corte extremos. La estrategia de estabilización elimina los grados de libertad inestables y reasigna los dominios de soporte de sus funciones de forma asociadas a elementos vecinos. En esta publicación presentamos u algoritmo de emparejamiento de subdominios y una estrategia de enriquecimiento a posteriori. La discretización espacial resultante de combinar estas estrategias preserva la estabilidad y precisión de la aproximación con discretizaciones conformes con la geometría. El método se valida a través de un conjunto de ejemplos numéricos de prueba y se aplica de forma satisfactoria a la resolución de problemas de interés en el ámbito de la reflexión de ondas electromagnéticas.[EN] In this work we present the Cartesian grid discontinuous Galerkin (cgDG)finite element method, a novel numerical technique that combines the high accuracy and efficiency of a high-order discontinuous Galerkin discretization with the simplicity and hierarchical structure of a geometry-independent Cartesian mesh. The elements that intersect the boundary of the physical domain require special treatment in order to minimize their effect on the performance of the algorithm. We considered the exact representation of the geometry for the boundary of the domain avoiding any nonphysical artefacts. We also define a stabilization procedure that eliminates the step size restriction of the time marching scheme due to extreme cut patterns. The unstable degrees of freedom are eliminated and the supporting regions of their shape functions are reassigned to neighbouring elements. A subdomain matching algorithm and a posterior enrichment strategy are presented. Combining these techniques we obtain a final spatial discretization that preserves stability and accuracy of the standard body-fitted discretization. The method is validated through a series of numerical tests and it is successfully applied to the solution of problems of interest in the context of electromagnetic scattering with increasing complexity.Engineering and Physical Sciences Research Council, Grant/Award Number: EP/T009071/1; Ministerio de Ciencia, Innovacion y Universidades, Grant/Award Number: FPU17/03993; Ministerio de Economia y Competitividad, Grant/Award Number: DPI2017-89816-RNavarro-García, H.; Sevilla, R.; Nadal, E.; Ródenas, JJ. (2021). High-order discontinuous Galerkin method for time-domain electromagnetics on geometry-independent Cartesian meshes. International Journal for Numerical Methods in Engineering. 122(24):7632-7663. https://doi.org/10.1002/nme.6846763276631222

A recovery-explicit error estimator in energy norm for linear elasticity

Significant research effort has been devoted to produce one-sided error estimates for Finite Element Analyses, in particular to provide upper bounds of the actual error. Typically, this has been achieved using residual-type estimates. One of the most popular and simpler (in terms of implementation) techniques used in commercial codes is the recovery-based error estimator. This technique produces accurate estimations of the exact error but is not designed to naturally produce upper bounds of the error in energy norm. Some attempts to remedy this situation provide bounds depending on unknown constants. Here, a new step towards obtaining error bounds from the recovery-based estimates is proposed. The idea is (1) to use a locally equilibrated recovery technique to obtain an accurate estimation of the exact error, (2) to add an explicit-type error bound of the lack of equilibrium of the recovered stresses in order to guarantee a bound of the actual error and (3) to efficiently and accurately evaluate the constants appearing in the bounding expressions, thus providing asymptotic bounds. The numerical tests with h-adaptive refinement process show that the bounding property holds even for coarse meshes, providing upper bounds in practical applications

A Path Planning Algorithm for a Dynamic Environment Based on Proper Generalized Decomposition

[EN] A necessity in the design of a path planning algorithm is to account for the environment. If the movement of the mobile robot is through a dynamic environment, the algorithm needs to include the main constraint: real-time collision avoidance. This kind of problem has been studied by different researchers suggesting different techniques to solve the problem of how to design a trajectory of a mobile robot avoiding collisions with dynamic obstacles. One of these algorithms is the artificial potential field (APF), proposed by O. Khatib in 1986, where a set of an artificial potential field is generated to attract the mobile robot to the goal and to repel the obstacles. This is one of the best options to obtain the trajectory of a mobile robot in real-time (RT). However, the main disadvantage is the presence of deadlocks. The mobile robot can be trapped in one of the local minima. In 1988, J.F. Canny suggested an alternative solution using harmonic functions satisfying the Laplace partial differential equation. When this article appeared, it was nearly impossible to apply this algorithm to RT applications. Years later a novel technique called proper generalized decomposition (PGD) appeared to solve partial differential equations, including parameters, the main appeal being that the solution is obtained once in life, including all the possible parameters. Our previous work, published in 2018, was the first approach to study the possibility of applying the PGD to designing a path planning alternative to the algorithms that nowadays exist. The target of this work is to improve our first approach while including dynamic obstacles as extra parameters.This research was funded by the GVA/2019/124 grant from Generalitat Valenciana and by the RTI2018-093521-B-C32 grant from the Ministerio de Ciencia, Innovacion y Universidades.Falcó, A.; Hilario, L.; Montés, N.; Mora, MC.; Nadal, E. (2020). A Path Planning Algorithm for a Dynamic Environment Based on Proper Generalized Decomposition. Mathematics. 8(12):1-11. https://doi.org/10.3390/math8122245S111812Gonzalez, D., Perez, J., Milanes, V., & Nashashibi, F. (2016). A Review of Motion Planning Techniques for Automated Vehicles. IEEE Transactions on Intelligent Transportation Systems, 17(4), 1135-1145. doi:10.1109/tits.2015.2498841Rimon, E., & Koditschek, D. E. (1992). Exact robot navigation using artificial potential functions. IEEE Transactions on Robotics and Automation, 8(5), 501-518. doi:10.1109/70.163777Khatib, O. (1986). Real-Time Obstacle Avoidance for Manipulators and Mobile Robots. The International Journal of Robotics Research, 5(1), 90-98. doi:10.1177/027836498600500106Kim, J.-O., & Khosla, P. K. (1992). Real-time obstacle avoidance using harmonic potential functions. IEEE Transactions on Robotics and Automation, 8(3), 338-349. doi:10.1109/70.143352Connolly, C. I., & Grupen, R. A. (1993). The applications of harmonic functions to robotics. Journal of Robotic Systems, 10(7), 931-946. doi:10.1002/rob.4620100704Garrido, S., Moreno, L., Blanco, D., & Martín Monar, F. (2009). Robotic Motion Using Harmonic Functions and Finite Elements. Journal of Intelligent and Robotic Systems, 59(1), 57-73. doi:10.1007/s10846-009-9381-3Bai, X., Yan, W., Cao, M., & Xue, D. (2019). Distributed multi‐vehicle task assignment in a time‐invariant drift field with obstacles. IET Control Theory & Applications, 13(17), 2886-2893. doi:10.1049/iet-cta.2018.6125Bai, X., Yan, W., Ge, S. S., & Cao, M. (2018). An integrated multi-population genetic algorithm for multi-vehicle task assignment in a drift field. Information Sciences, 453, 227-238. doi:10.1016/j.ins.2018.04.044Falcó, A., & Nouy, A. (2011). Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces. Numerische Mathematik, 121(3), 503-530. doi:10.1007/s00211-011-0437-5Chinesta, F., Leygue, A., Bordeu, F., Aguado, J. V., Cueto, E., Gonzalez, D., … Huerta, A. (2013). PGD-Based Computational Vademecum for Efficient Design, Optimization and Control. Archives of Computational Methods in Engineering, 20(1), 31-59. doi:10.1007/s11831-013-9080-xFalcó, A., Montés, N., Chinesta, F., Hilario, L., & Mora, M. C. (2018). On the Existence of a Progressive Variational Vademecum based on the Proper Generalized Decomposition for a Class of Elliptic Parameterized Problems. Journal of Computational and Applied Mathematics, 330, 1093-1107. doi:10.1016/j.cam.2017.08.007Domenech, L., Falcó, A., García, V., & Sánchez, F. (2016). Towards a 2.5D geometric model in mold filling simulation. Journal of Computational and Applied Mathematics, 291, 183-196. doi:10.1016/j.cam.2015.02.043Falcó, A., & Nouy, A. (2011). A Proper Generalized Decomposition for the solution of elliptic problems in abstract form by using a functional Eckart–Young approach. Journal of Mathematical Analysis and Applications, 376(2), 469-480. doi:10.1016/j.jmaa.2010.12.003Falcó, A., & Hackbusch, W. (2012). On Minimal Subspaces in Tensor Representations. Foundations of Computational Mathematics, 12(6), 765-803. doi:10.1007/s10208-012-9136-6Canuto, C., & Urban, K. (2005). Adaptive Optimization of Convex Functionals in Banach Spaces. SIAM Journal on Numerical Analysis, 42(5), 2043-2075. doi:10.1137/s0036142903429730Ammar, A., Chinesta, F., & Falcó, A. (2010). On the Convergence of a Greedy Rank-One Update Algorithm for a Class of Linear Systems. Archives of Computational Methods in Engineering, 17(4), 473-486. doi:10.1007/s11831-010-9048-

An approach to geometric optimisation of railway catenaries

[EN] The quality of current collection becomes a limiting factor when the aim is to increase the speed of the present railway systems. In this work an attempt is made to improve current collection quality optimising catenary geometry by means of a genetic algorithm (GA). As contact wire height and dropper spacing are thought to be highly influential parameters, they are chosen as the optimisation variables. The results obtained show that a GA can be used to optimise catenary geometry to improve current collection quality measured in terms of the standard deviation of the contact force. Furthermore, it is highlighted that apart from the usual pre-sag, other geometric parameters should also be taken into account when designing railway catenaries.The authors would like to acknowledge the financial support received from the FPU program offered by the Ministerio de Educación, Cultura y Deporte (MECD), under grant number [FPU13/04191], and also the funding provided by the Generalitat Valenciana [PROMETEO/2016/007].Gregori Verdú, S.; Tur Valiente, M.; Nadal, E.; Fuenmayor Fernández, F. (2017). An approach to geometric optimisation of railway catenaries. Vehicle System Dynamics. 1-25. https://doi.org/10.1080/00423114.2017.1407434S125Nåvik, P., Rønnquist, A., & Stichel, S. (2015). The use of dynamic response to evaluate and improve the optimization of existing soft railway catenary systems for higher speeds. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, 230(4), 1388-1396. doi:10.1177/0954409715605140Harèll, P., Drugge, L., & Reijm, M. (2005). Study of Critical Sections in Catenary Systems During Multiple Pantograph Operation. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, 219(4), 203-211. doi:10.1243/095440905x8934Bruni, S., Ambrosio, J., Carnicero, A., Cho, Y. H., Finner, L., Ikeda, M., … Zhang, W. (2014). The results of the pantograph–catenary interaction benchmark. Vehicle System Dynamics, 53(3), 412-435. doi:10.1080/00423114.2014.953183Shabana, A. A. (1998). Nonlinear Dynamics, 16(3), 293-306. doi:10.1023/a:1008072517368Zhou, N., & Zhang, W. (2011). Investigation on dynamic performance and parameter optimization design of pantograph and catenary system. Finite Elements in Analysis and Design, 47(3), 288-295. doi:10.1016/j.finel.2010.10.008Kim, J.-W., & Yu, S.-N. (2013). Design variable optimization for pantograph system of high-speed train using robust design technique. International Journal of Precision Engineering and Manufacturing, 14(2), 267-273. doi:10.1007/s12541-013-0037-7Ambrósio, J., Pombo, J., & Pereira, M. (2013). Optimization of high-speed railway pantographs for improving pantograph-catenary contact. Theoretical and Applied Mechanics Letters, 3(1), 013006. doi:10.1063/2.1301306Lee, J.-H., Kim, Y.-G., Paik, J.-S., & Park, T.-W. (2012). Performance evaluation and design optimization using differential evolutionary algorithm of the pantograph for the high-speed train. Journal of Mechanical Science and Technology, 26(10), 3253-3260. doi:10.1007/s12206-012-0833-5Massat, J.-P., Laurent, C., Bianchi, J.-P., & Balmès, E. (2014). Pantograph catenary dynamic optimisation based on advanced multibody and finite element co-simulation tools. Vehicle System Dynamics, 52(sup1), 338-354. doi:10.1080/00423114.2014.898780Cho, Y. H., Lee, K., Park, Y., Kang, B., & Kim, K. (2010). Influence of contact wire pre-sag on the dynamics of pantograph–railway catenary. International Journal of Mechanical Sciences, 52(11), 1471-1490. doi:10.1016/j.ijmecsci.2010.04.002Zhang, W., Mei, G., & Zeng, J. (2002). A Study of Pantograph/Catenary System Dynamics with Influence of Presag and Irregularity of Contact Wire. Vehicle System Dynamics, 37(sup1), 593-604. doi:10.1080/00423114.2002.11666265Koziel, S., & Yang, X.-S. (Eds.). (2011). Computational Optimization, Methods and Algorithms. Studies in Computational Intelligence. doi:10.1007/978-3-642-20859-1Hare, W., Nutini, J., & Tesfamariam, S. (2013). A survey of non-gradient optimization methods in structural engineering. Advances in Engineering Software, 59, 19-28. doi:10.1016/j.advengsoft.2013.03.001Tur, M., Baeza, L., Fuenmayor, F. J., & García, E. (2014). PACDIN statement of methods. Vehicle System Dynamics, 53(3), 402-411. doi:10.1080/00423114.2014.963126Tur, M., García, E., Baeza, L., & Fuenmayor, F. J. (2014). A 3D absolute nodal coordinate finite element model to compute the initial configuration of a railway catenary. Engineering Structures, 71, 234-243. doi:10.1016/j.engstruct.2014.04.015Gregori, S., Tur, M., Nadal, E., Aguado, J. V., Fuenmayor, F. J., & Chinesta, F. (2017). Fast simulation of the pantograph–catenary dynamic interaction. Finite Elements in Analysis and Design, 129, 1-13. doi:10.1016/j.finel.2017.01.007Gerstmayr, J., & Shabana, A. A. (2006). Analysis of Thin Beams and Cables Using the Absolute Nodal Co-ordinate Formulation. Nonlinear Dynamics, 45(1-2), 109-130. doi:10.1007/s11071-006-1856-1Collina, A., & Bruni, S. (2002). Numerical Simulation of Pantograph-Overhead Equipment Interaction. Vehicle System Dynamics, 38(4), 261-291. doi:10.1076/vesd.38.4.261.8286Ambrósio, J., Pombo, J., Antunes, P., & Pereira, M. (2014). PantoCat statement of method. Vehicle System Dynamics, 53(3), 314-328. doi:10.1080/00423114.2014.969283Nåvik, P., Rønnquist, A., & Stichel, S. (2017). Variation in predicting pantograph–catenary interaction contact forces, numerical simulations and field measurements. Vehicle System Dynamics, 55(9), 1265-1282. doi:10.1080/00423114.2017.130852

José de Lugo y Molina : cónsul y agente general de España (1754-1835)

La singladura personal del orotavense José de Lugo y Molina, así como la más conocida de su hermano Estanislao, pueden encuadrarse en las coordenadas temporales de la transición de la Ilustración al Liberalismo, o si se prefiere, de la descomposición del Antiguo Régimen en España