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

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

Real time parameter identification and solution reconstruction from experimental data using the Proper Generalized Decomposition

Some industrial processes are modelled by parametric partial differential equations. Integrating computational modelling and data assimilation into the control process requires obtaining a solution of the numerical model at the characteristic frequency of the process (real-time). This paper introduces a computational strategy allowing to efficiently exploit measurements of those industrial processes, providing the solution of the model at the required frequency. This is particularly interesting in the framework of control algorithms that rely on a model involving a set of parameters. For instance, the curing process of a composite material is modelled as a thermo-mechanical problem whose corresponding parameters describe the thermal and mechanical behaviours. In this context, the information available (measurements) is used to update the parameters of the model and to produce new values of the control variables (data assimilation). The methodology presented here is devised to ensure the possibility of having a response in real-time of the problem and therefore the capability of integrating it in the control scheme. The Proper Generalized Decomposition is used to describe the solution in the multi-parametric space. The real-time data assimilation requires a further simplification of the solution representation that better fits the data (reconstructed solution) and it provides an implicit parameter identification. Moreover, the analysis of the assimilated data sensibility with respect to the points where the measurements are taken suggests a criterion to locate the sensors

Imposing Dirichlet boundary conditions in hierarchical Cartesian meshes by means of stabilized Lagrange multipliers

This is the pre-peer reviewed version of the following article: Tur, M., Albelda, J., Nadal, E. and Ródenas, J. J. (2014), Imposing Dirichlet boundary conditions in hierarchical Cartesian meshes by means of stabilized Lagrange multipliers. Int. J. Numer. Meth. Engng, 98: 399–417, which has been published in final form at http://dx.doi.org/10.1002/nme.4629 . This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.[EN] The use of Cartesian meshes independent of the geometry has some advantages over the traditional meshes used in the finite element method. The main advantage is that their use together with an appropriate hierarchical data structure reduces the computational cost of the finite element analysis. This improvement is based on the substitution of the traditional mesh generation process by an optimized procedure for intersecting the Cartesian mesh with the boundary of the domain and the use efficient solvers based on the hierarchical data structure. One major difficulty associated to the use of Cartesian grids is the fact that the mesh nodes do not, in general, lie over the boundary of the domain, increasing the difficulty to impose Dirichlet boundary conditions. In this paper, Dirichlet boundary conditions are imposed by means of the Lagrange multipliers technique. A new functional has been added to the initial formulation of the problem that has the effect of stabilizing the problem. The technique here presented allows for a simple definition of the Lagrange multipliers field that even allow us to directly condense the degrees of freedom of the Lagrange multipliers at element level.The authors acknowledge the financial support received from the research project DPI2010-20542 of the Ministerio de Economia y Competitividad. Also, we appreciated the financial support of the FPU program (AP2008-01086) of the Universitat Politecnica de Valencia and the Generalitat Valenciana (PROMETEO/2012/023). The authors are also grateful for the support of the Framework Program 7 Initial Training Network Funding under grant number 289361 'Integrating Numerical Simulation and Geometric Design Technology (INSIST)'.Tur Valiente, M.; Albelda Vitoria, J.; Nadal Soriano, E.; Ródenas García, JJ. (2014). Imposing Dirichlet boundary conditions in hierarchical Cartesian meshes by means of stabilized Lagrange multipliers. International Journal for Numerical Methods in Engineering. 98(6):399-417. https://doi.org/10.1002/nme.462939941798

Fast simulation of the pantograph-catenary dynamic interaction

Simulation of the pantograph-catenary dynamic interaction has now become a useful tool for designing and optimizing the system. In order to perform accurate simulations, including system non-linearities, the Finite Element Method is commonly employed combined with a time integration scheme, even though the computational time required may be longer than with the use of other simpler approaches. In this paper we propose a two-stage methodology (Offline/Online) which notably reduces the computational cost without any loss in accuracy and makes it possible to successfully carry out very efficient optimizations or even Hardware in the Loop simulations with real-time requirements.The authors would like to acknowledge the financial support received from the FPU program offered by the Ministerio de Educacion, Cultura y Deporte under grant number (FPU13/04191), and also funding from the Universitat Politecnica de Valencia and the Generalitat Valenciana (PROMETEO/2016/007).Gregori Verdú, S.; Tur Valiente, M.; Nadal Soriano, E.; Aguado, J.; Fuenmayor Fernández, FJ.; Chinesta, F. (2017). Fast simulation of the pantograph-catenary dynamic interaction. Finite Elements in Analysis and Design. 129:1-13. https://doi.org/10.1016/j.finel.2017.01.007S11312

Mesh adaptivity driven by goal-oriented locally equilibrated superconvergent patch recovery

[EN] Goal-oriented error estimates (GOEE) have become popular tools to quantify and control the local error in quantities of interest (QoI), which are often more pertinent than local errors in energy for design purposes (e.g. the mean stress or mean displacement in a particular area, the stress intensity factor for fracture problems). These GOEE are one of the key unsolved problems of advanced engineering applications in, for example, the aerospace industry. This work presents a simple recovery-based error estimation technique for QoIs whose main characteristic is the use of an enhanced version of the Superconvergent Patch Recovery (SPR) technique previously used for error estimation in the energy norm. This enhanced SPR technique is used to recover both the primal and dual solutions. It provides a nearly statically admissible stress field that results in accurate estimations of the local contributions to the discretisation error in the QoI and, therefore, in an accurate estimation of this magnitude. This approach leads to a technique with a reasonable computational cost that could easily be implemented into already available finite element codes, or as an independent postprocessing tool.This work was supported by the EPSRC Grant EP/G042705/1 "Increased Reliability for Industrially Relevant Automatic Crack Growth Simulation with the eXtended Finite Element Method". Stephane Bordas also thanks partial funding for his time provided by the European Research Council Starting Independent Research Grant (ERC Stg Grant Agreement No. 279578) "RealTCut Towards real time multiscale simulation of cutting in non-linear materials with applications to surgical simulation and computer guided surgery". This work has received partial support from the research project DPI2010-20542 of the Ministerio de Economia y Competitividad (Spain). The financial support of the FPU program (AP2008-01086), the funding from Universitat Politecnica de Valencia and Generalitat Valenciana (PROMETEO/2012/023) are also acknowledged. All authors also thank the partial support of the Framework Programme 7 Initial Training Network Funding under Grant No. 289361 "Integrating Numerical Simulation and Geometric Design Technology."González Estrada, OA.; Nadal Soriano, E.; Ródenas, J.; Kerfriden, P.; Bordas, S.; Fuenmayor Fernández, FJ. (2014). Mesh adaptivity driven by goal-oriented locally equilibrated superconvergent patch recovery. Computational Mechanics. 53(5):957-976. https://doi.org/10.1007/s00466-013-0942-8S957976535Ainsworth M, Oden JT (2000) A posteriori error estimation in finite element analysis. 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Classifying and communicating risks in prediabetes according to fasting glucose and/or glycated hemoglobin : PREDAPS cohort study

Information about prognostic outcomes can be of great help for people with prediabetes and for physicians in the face of scientific controversy about the cutoff point for defining prediabetes. We aimed to estimate different prognostic outcomes in people with prediabetes. Prospective cohort of subjects with prediabetes according to American Diabetes Association guidelines. The probabilities of diabetes onset versus non-onset, the odds against diabetes onset, and the probability of reverting to normoglycemia according to different prediabetes categories were calculated. The odds against diabetes onset ranged from 29:1 in individuals with isolated FPG of 100-109 mg/dL to 1:1 in individuals with FPG 110-125 mg/dL plus HbA1c 6.0-6.4%. The probability of reversion to normoglycemia was 31.2% (95% CI 24.0-39.6) in those with isolated FPG 100-109 mg/dL and 6.2% (95% CI 1.4-10.0) in those with FPG 110-125 mg/dL plus HbA1c 6.0-6.4%. Of every 100 participants in the first group, 97 did not develop diabetes and 31 reverted to normoglycemia, while in the second group those figures were 52 and 6. Using odds of probabilities and absolute numbers might be useful for people with prediabetes and physicians to share decisions on potential interventions