Efficient recovery-based error estimation for the smoothed finite element method for smooth and singular linear elasticity

[EN] An error control technique aimed to assess the quality of smoothed finite element approximations is presented in this paper. Finite element techniques based on strain smoothing appeared in 2007 were shown to provide significant advantages compared to conventional finite element approximations. In particular, a widely cited strength of such methods is improved accuracy for the same computational cost. Yet, few attempts have been made to directly assess the quality of the results obtained during the simulation by evaluating an estimate of the discretization error. Here we propose a recovery type error estimator based on an enhanced recovery technique. The salient features of the recovery are: enforcement of local equilibrium and, for singular problems a ¿smooth + singular¿ decomposition of the recovered stress. We evaluate the proposed estimator on a number of test cases from linear elastic structural mechanics and obtain efficient error estimations whose effectivities, both at local and global levels, are improved compared to recovery procedures not implementing these features.Stephane Bordas would like to thank the partial financial support of the Royal Academy of Engineering and of the Leverhulme Trust for his Senior Research Fellowship Towards the next generation surgical simulators as well as the financial support for Octavio A. Gonzalez-Estrada and Stephane Bordas from the UK Engineering Physical Science Research Council (EPSRC) under grant EP/G042705/1 Increased Reliability for Industrially Relevant Automatic Crack Growth Simulation with the eXtended Finite Element Method. Stephane Bordas also thanks partial financial support of the European Research Council Starting Independent Research Grant (ERC Stg grant agreement No. 279578) and the FP7 Initial Training Network Funding under grant number 289361 "Integrating Numerical Simulation and Geometric Design Technology, INSIST". This work has been carried out within the framework of the research project DPI2010-20542 of the Ministerio de Ciencia e Innovacion (Spain). The financial support from Universitat Politecnica de Valencia, PROMETEO/2012/023 and Generalitat Valenciana are also acknowledged.González Estrada, OA.; Natarajan, S.; J.J. Ródenas; Nguyen-Xuan, H.; Bordas, S. (2013). Efficient recovery-based error estimation for the smoothed finite element method for smooth and singular linear elasticity. Computational Mechanics. 52(1):37-52. https://doi.org/10.1007/s00466-012-0795-6S3752521Liu GR, Dai KY, Nguyen TT (2006) A smoothed finite element method for mechanics problems. 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Evaluación del daño en la vértebra lumbar L5 mediante análisis por elementos finitos

International audienceBone metastasis to the spine, pelvis, or hip in patients suffering from prostate cancer occurs in about 80% of cases. A spinal metastasis may cause pain, instability and neurological injuries. Thus, it is relevant to assess when critical conditions have been reached and bone structural integrity is compromised. Data-driven numerical methods based on the postprocessing of medical imaging provide a tool to model the material complexity of biological tissues. Computed axial tomography (CAT) together with segmentation tools allow the reconstruction of 3D models of the bone that include mechanical properties representing the anisotropic condition of the bone structures. In this work, we present the model of the L5 lumbar vertebra of a patient affected by metastasis, and evaluate biomarkers to indicate the level of damage, compared with the reference case of a healthy bone in an initial stage.La metástasis ósea a la columna vertebral, la pelvis o la cadera en pacientes con cáncer de próstata es una patología que ocurre en aproximadamente el 80% de los casos. La metástasis en la columna vertebral puede causar dolor, inestabilidad y lesiones neurológicas. Por lo tanto, es relevante evaluar cuándo se han alcanzado condiciones críticas y la integridad estructural del hueso está comprometida. Los métodos numéricos basados en datos del paciente, obtenidos mediante el postprocesamiento de imágenes médicas, proporcionan una herramienta para modelar la complejidad del material de los tejidos biológicos. La tomografía axial computarizada (TAC) junto con las herramientas de segmentación permiten la reconstrucción de modelos 3D del hueso que incluyen propiedades mecánicas, y que representan la condición anisotrópica de las estructuras óseas. En este trabajo, presentamos el modelo de la vértebra lumbar L5 de un paciente afectado por metástasis y evaluamos biomarcadores para indicar el nivel de daño, comparando con el caso de referencia del hueso sano en una etapa inicial. Palabras clave: daño óseo; segmentación ósea; vértebras; metástasis; FEA basado en imagen

Stress recovery for the polygonal finite element method

El método de los elementos finitos es una de las herramientas numéricas más utilizadas para el diseño en ingeniería. En los últimos años se han desarrollado nuevas aproximaciones numéricas para extender el uso del método de elementos finitos a mallas poligonales. Dichas aproximaciones mejoran la precisión de la solución y aumentan la flexibilidad en el mallado. Sin embargo, como toda aproximación, es necesario cuantificar el valor del error inducido para poder validar los resultados obtenidos. En este trabajo se presenta el uso de una técnica de estimación del error de discretización para mallas de elementos finitos poligonales. La técnica está basada en la reconstrucción de la solución en tensiones mediante un procedimiento de mínimos cuadrados ponderados que considera la influencia de las ecuaciones de equilibrio. Se ha utilizado un problema con solución exacta para evaluar la efectividad del estimador, obteniendo buenos resultados a nivel local y global. The finite element method is one of the most used numerical tools for engineering design. In recent years, novel numerical approximations have been proposed to extend the finite element method to meshes using arbitrary polygons. Such approximations are aimed to improve accuracy and increase the flexibility during the meshing process. However, as any approximation, they exhibit an error that requires to be quantified in order to validate the numerical results. In this paper, we present a technique to estimate the discretization error in energy norm for arbitrary polygonal finite element meshes. This recovery-based technique uses a moving least squares approach that considers constraints to represent the equilibrium equations.  We use two benchmark problems to evaluate the effectivity of the error estimator, with good results both locally and globally.&nbsp

Stress recovery for the polygonal finite element method

El método de los elementos finitos es una de las herramientas numéricas más utilizadas para el diseño en ingeniería. En los últimos años se han desarrollado nuevas aproximaciones numéricas para extender el uso del método de elementos finitos a mallas poligonales. Dichas aproximaciones mejoran la precisión de la solución y aumentan la flexibilidad en el mallado. Sin embargo, como toda aproximación, es necesario cuantificar el valor del error inducido para poder validar los resultados obtenidos. En este trabajo se presenta el uso de una técnica de estimación del error de discretización para mallas de elementos finitos poligonales. La técnica está basada en la reconstrucción de la solución en tensiones mediante un procedimiento de mínimos cuadrados ponderados que considera la influencia de las ecuaciones de equilibrio. Se ha utilizado un problema con solución exacta para evaluar la efectividad del estimador, obteniendo buenos resultados a nivel local y global. The finite element method is one of the most used numerical tools for engineering design. In recent years, novel numerical approximations have been proposed to extend the finite element method to meshes using arbitrary polygons. Such approximations are aimed to improve accuracy and increase the flexibility during the meshing process. However, as any approximation, they exhibit an error that requires to be quantified in order to validate the numerical results. In this paper, we present a technique to estimate the discretization error in energy norm for arbitrary polygonal finite element meshes. This recovery-based technique uses a moving least squares approach that considers constraints to represent the equilibrium equations.  We use two benchmark problems to evaluate the effectivity of the error estimator, with good results both locally and globally.&nbsp

Finite element computations over quadtree meshes : Strain smoothing and semi-analytical formulation

In this paper, we discuss two alternate techniques to treat hanging nodes in a quadtree mesh. Both the techniques share similarities, in that, they require only boundary information. Moreover, they do not require an explicit form of the shape functions, unlike the conventional approaches, for example, as in the work of Gupta (Int J Numer Methods Eng 12:35, 1978) or Tabarraei and Sukumar (Finite Elem Anal Des 41:686, 2005). Hence, no special numerical integration technique is required. One of the techniques relies on the strain projection procedure, whilst the other is based on the scaled boundary finite element method. Numerical examples are presented to demonstrate the accuracy and the convergence properties of the two techniques

Efficient recovery-based error estimation for the smoothed finite element method for smooth and singular linear elasticity

An error control technique aimed to assess the quality of smoothed finite element approximations is presented in this paper. Finite element techniques based on strain smoothing appeared in 2007 were shown to provide significant advantages compared to conventional finite element approximations. In particular, a widely cited strength of such methods is improved accuracy for the same computational cost. Yet, few attempts have been made to directly assess the quality of the results obtained during the simulation by evaluating an estimate of the discretization error. Here we propose a recovery type error estimator based on an enhanced recovery technique. The salient features of the recovery are: enforcement of local equilibrium and, for singular problems a “smooth+singular” decomposition of the recovered stress. We evaluate the proposed estimator on a number of test cases from linear elastic structural mechanics and obtain efficient error estimations whose effectivities, both at local and global levels, are improved compared to recovery procedures not implementing these features

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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