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

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Abstract

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.

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Notes

  1. The use of enrichment to improve recovery based error estimates for enriched approximations was discussed in detail in some of the first papers discussing derivative recovery techniques for enriched finite element approximations, see References [11–14, 18, 20]. A very detailed and clear discussion of a wide variety of error estimators and adaptive procedures for discontinuous failure is provided in [34].

  2. The explanations are restricted to linear QoI. In the developments, affine quantities of the displacement will also be considered, but we will show that this particular case can be recast into the linear case.

  3. The interested reader is referred to the recent paper by Moumnassi and colleagues [44] which discusses recent advances in “ambient space finite elements” and proposes hybrid level set/FEMs able to handle sharp corners and edges.

  4. The interested reader may want to refer to [50–53] regarding preconditioning techniques for systems sharing similar features.

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Acknowledgments

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”. Stéphane 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 Economía y Competitividad (Spain). The financial support of the FPU program (AP2008-01086), the funding from Universitat Politècnica de València 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.”

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Authors and Affiliations

  1. Institute of Mechanics and Advanced Materials (IMAM), Cardiff School of Engineering, Cardiff University, Queen’s Buildings, The Parade, Cardiff, Wales, CF24 3AA, UK

    O. A. González-Estrada, P. Kerfriden & S. P. A. Bordas

  2. Centro de Investigación de Tecnología de Vehículos (CITV), Universitat Politècnica de València, 46022, Valencia, Spain

    E. Nadal, J. J. Ródenas & F. J. Fuenmayor

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  1. O. A. González-Estrada
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  2. E. Nadal
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  3. J. J. Ródenas
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  4. P. Kerfriden
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  5. S. P. A. Bordas
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  6. F. J. Fuenmayor
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Correspondence to O. A. González-Estrada.

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González-Estrada, O.A., Nadal, E., Ródenas, J.J. et al. Mesh adaptivity driven by goal-oriented locally equilibrated superconvergent patch recovery. Comput Mech 53, 957–976 (2014). https://doi.org/10.1007/s00466-013-0942-8

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