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An automated approach for parallel adjoint-based error estimation and mesh adaptation

  • Brian N. GranzowEmail author
  • Assad A. Oberai
  • Mark S. Shephard
Original Article
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Abstract

In finite element simulations, not all of the data are of equal importance. In fact, the primary purpose of a numerical study is often to accurately assess only one or two engineering output quantities that can be expressed as functionals. Adjoint-based error estimation provides a means to approximate the discretization error in functional quantities and mesh adaptation provides the ability to control this discretization error by locally modifying the finite element mesh. In the past, adjoint-based error estimation has only been accessible to expert practitioners in the field of solid mechanics. In this work, we present an approach to automate the process of adjoint-based error estimation and mesh adaptation on parallel machines. This process is intended to lower the barrier of entry to adjoint-based error estimation and mesh adaptation for solid mechanics practitioners. We demonstrate that this approach is effective for example problems in Poisson’s equation, nonlinear elasticity, and thermomechanical elastoplasticity.

Keywords

Automated Adjoint A posteriori Error estimation Adaptation Finite element 

Notes

Acknowledgements

The authors acknowledge the support of IBM Corporation in the performance of this research. The computing resources of the Center for Computational Innovations at Rensselaer Polytechnic Institute are also acknowledged. The development of tools used in this work was partly supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, under award DE-SC00066117 (FASTMath SciDAC Institute). The authors would like to thank Li Dong for providing the discrete geometric model used for the microglial cell example and Max Bloomfield for the geometric model used for the solder ball example.

This work was performed at the time at Rensselaer Polytechnic Institute before the authors moved to different organizations. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.

Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

This work is supported by the U.S. Army grants W911NF1410301 and W911NF16C0117.

References

  1. 1.
    Ainsworth M, Oden JT (2011) A posteriori error estimation in finite element analysis. Wiley, HobokenzbMATHGoogle Scholar
  2. 2.
    Alauzet F, Li X, Seol ES, Shephard MS (2006) Parallel anisotropic 3D mesh adaptation by mesh modification. Eng Comput 21(3):247–258CrossRefGoogle Scholar
  3. 3.
    Babuška I, Miller A (1984) The post-processing approach in the finite element method, Part 1: calculation of displacements, stresses and other higher derivatives of the displacements. Int J Numer Methods Eng 20(6):1085–1109zbMATHCrossRefGoogle Scholar
  4. 4.
    Babuška I, Miller A (1984) The post-processing approach in the finite element method, Part 2: the calculation of stress intensity factors. Int J Numer Methods Eng 20(6):1111–1129zbMATHCrossRefGoogle Scholar
  5. 5.
    Babuška I, Miller A (1984) The post-processing approach in the finite element method, Part 3: a posteriori error estimates and adaptive mesh selection. Int J Numer Methods Eng 20(12):2311–2324zbMATHCrossRefGoogle Scholar
  6. 6.
  7. 7.
    Bavier E, Hoemmen M, Rajamanickam S, Thornquist H (2012) Amesos2 and Belos: direct and iterative solvers for large sparse linear systems. Sci Progr 20(3):241–255Google Scholar
  8. 8.
    Becker R, Rannacher R (2001) An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica 10:1–102MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bloomfield MO, Li Z, Granzow B, Ibanez DA, Oberai AA, Hansen GA, Liu XH, Shephard MS (2017) Component-based workflows for parallel thermomechanical analysis of arrayed geometries. Eng Comput 33(3):509–517CrossRefGoogle Scholar
  10. 10.
    Boussetta R, Coupez T, Fourment L (2006) Adaptive remeshing based on a posteriori error estimation for forging simulation. Comput Methods Appl Mech Eng 195(48):6626–6645MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Burstedde C, Ghattas O, Stadler G, Tu T, Wilcox LC (2009) Parallel scalable adjoint-based adaptive solution of variable-viscosity stokes flow problems. Comput Methods Appl Mech Eng 198(21):1691–1700zbMATHCrossRefGoogle Scholar
  12. 12.
    Cyr EC, Shadid J, Wildey T (2014) Approaches for adjoint-based a posteriori analysis of stabilized finite element methods. SIAM J. Sci Comput 36(2):A766–A791MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Diamond G, Smith CW, Shephard MS (2017) Dynamic load balancing of massively parallel unstructured meshes. In: Proc. of the 8th Workshop on Latest Advances in Scalable Algorithms for Large-Scale Systems, Denver, CO, USA. Denver, CO, USAGoogle Scholar
  14. 14.
    Dong L, Oberai AA (2017) Recovery of cellular traction in three-dimensional nonlinear hyperelastic matrices. Comput Methods Appl Mech Eng 314:296–313MathSciNetCrossRefGoogle Scholar
  15. 15.
    Eriksson K, Estep D, Hansbo P, Johnshon C (1996) Computational differential equations, 2nd edn. Cambridge University Press, New YorkGoogle Scholar
  16. 16.
    Fidkowski KJ (2011) Output error estimation strategies for discontinuous galerkin discretizations of unsteady convection-dominated flows. Int J Numer Methods Eng 88(12):1297–1322MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Fidkowski KJ, Darmofal DL (2011) Review of output-based error estimation and mesh adaptation in computational fluid dynamics. AIAA J 49(4):673–694CrossRefGoogle Scholar
  18. 18.
    Gartland EC Jr (1984) Computable pointwise error bounds and the ritz method in one dimension. SIAM J Numer Anal 21(1):84–100MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Ghorashi SS, Rabczuk T (2017) Goal-oriented error estimation and mesh adaptivity in 3D elastoplasticity problems. Int J Fract 203:3–19CrossRefGoogle Scholar
  20. 20.
    Ghorashi SS, Amani J, Bagherzadeh AS, Rabczuk T (2014) Goal-oriented error estimation and mesh adaptivity in three-dimensional elasticity problems. In: WCCM XI-ECCM V-ECFD VI, Barcelona, Spain, Barcelona, SpainGoogle Scholar
  21. 21.
    Giles MB, Pierce NA (2016) Chapter 2—adjoint error correction for integral outputs. Springer, Berlin, pp 47–95zbMATHGoogle Scholar
  22. 22.
    González-Estrada OA, Nadal E, Ródenas JJ, Kerfriden P, Bordas SP-A, Fuenmayor FJ (2014) Mesh adaptivity driven by goal-oriented locally equilibrated superconvergent patch recovery. Comput Mech 53(5):957–976MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Granzow BN (2017) Goal GitHub repository. https://github.com/bgranzow/goal
  24. 24.
    Granzow BN, Oberai AA, Shephard MS (2018) Adjoint-based error estimation and mesh adaptation for stabilized finite deformation elasticity. Comput Methods Appl Mech Eng 337:263–280MathSciNetCrossRefGoogle Scholar
  25. 25.
    Granzow BN, Shephard MS, Oberai AA (2017) Output-based error estimation and mesh adaptation for variational multiscale methods. Comput Methods Appl Mech Eng 322:441–459MathSciNetCrossRefGoogle Scholar
  26. 26.
    Grätsch T, Bathe K-J (2005) A posteriori error estimation techniques in practical finite element analysis. Comput Struct 83(4–5):235–265MathSciNetCrossRefGoogle Scholar
  27. 27.
    Griewank A, Walther A (2008) Evaluating derivatives: principles and techniques of algorithmic differentiation. Soc Ind Appl Math, Philadelphia, PA, USA, 2 editionGoogle Scholar
  28. 28.
    Heroux MA, Bartlett RA, Howle VE, Hoekstra RJ, Hu JJ, Kolda TG et al (2005) An overview of the Trilinos project. ACM Trans Math Softw 31(3):397–423MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Heroux MA, Willenbring JM (2012) A new overview of the Trilinos project. Sci Program 20(2):83–88Google Scholar
  30. 30.
    Ibanez D, Shephard MS (2017) Modifiable array data structures for mesh topology. SIAM J Sci Comput 39(2):C144–C161MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Ibanez DA, Seol ES, Smith CW, Shephard MS (2016) PUMI: Parallel unstructured mesh infrastructure. ACM Trans Math Softw 42(3):17–45MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Larsson F, Hansbo P, Runesson K (2002) Strategies for computing goal-oriented a posteriori error measures in non-linear elasticity. Int J Numer Methods Eng 55(8):879–894MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Li X, Shephard MS, Beall MW (2005) 3D anisotropic mesh adaptation by mesh modification. Comput Methods Appl Mech Eng 194(48):4915–4950MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Li Z, Bloomfield MO, Oberai AA. Simulation of finite-strain inelastic phenomena governed by creep and plasticity. Comput. Mechanics (to be published)Google Scholar
  35. 35.
    Logg A, Mardal KA, Garth Wells (2012) Automated solution of differential equations by the finite element method: the FEniCS book. Springer, HeidelbergzbMATHCrossRefGoogle Scholar
  36. 36.
    Nemec M, Aftosmis MJ (2007) Adjoint error estimation and adaptive refinement for embedded-boundary Cartesian meshes. In: 18th AIAA Computational Fluid Dynamics Conf., Miami, FL, USA, Miami, FL, USAGoogle Scholar
  37. 37.
    Oden JT, Prudhomme S (2001) Goal-oriented error estimation and adaptivity for the finite element method. Comput Math Appl 41(5–6):735–756MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Pawlowski RP, Phipps ET, Salinger AG (2012) Automating embedded analysis capabilities and managing software complexity in multiphysics simulation, Part I: Template-based generic programming. Sci Program 20(2):197–219Google Scholar
  39. 39.
    Pawlowski RP, Phipps ET, Salinger AG, Owen SJ, Siefert CM, Staten ML (2012) Automating embedded analysis capabilities and managing software complexity in multiphysics simulation, Part II: application to partial differential equations. Sci Program 20(3):327–345Google Scholar
  40. 40.
    Phipps E, Pawlowski R (2012) Efficient expression templates for operator overloading-based automatic differentiation. In: Recent advances in algorithmic differentiation. Springer, Berlin, pp 309–319Google Scholar
  41. 41.
    Prokopenko A, Hu JJ, Wiesner TA, Siefert CM, Tuminaro RS (2014) MueLu user’s guide 1.0. Technical Report SAND2014-18874, Sandia Nat. Lab., Albuquerque, NM, USAGoogle Scholar
  42. 42.
    Prudhomme S, Oden JT (1999) On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors. Comput Methods Appl Mech Eng 176(1–4):313–331MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Rabizadeh E, Bagherzadeh AS, Rabczuk T (2015) Adaptive thermo-mechanical finite element formulation based on goal-oriented error estimation. Comput Mater Sci 102:27–44CrossRefGoogle Scholar
  44. 44.
    Ramesh B, Maniatty AM (2005) Stabilized finite element formulation for elastic–plastic finite deformations. Comput Methods Appl Mech Eng 194(6):775–800zbMATHCrossRefGoogle Scholar
  45. 45.
    Rannacher R, Suttmeier F-T (1997) A feed-back approach to error control in finite element methods: application to linear elasticity. Comput Mech 19(5):434–446MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Rannacher R, Suttmeier F-T (1998) A posteriori error control in finite element methods via duality techniques: application to perfect plasticity. Comput Mech 21(2):123–133MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Rannacher R, Suttmeier F-T (1999) A posteriori error estimation and mesh adaptation for finite element models in elasto-plasticity. Comput Methods Appl Mech Eng 176(1–4):333–361MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Richardson LF (1911) The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam. Philos Trans R Soc Lond 210:307–357zbMATHCrossRefGoogle Scholar
  49. 49.
    Richter T, Wick T (2015) Variational localizations of the dual weighted residual estimator. J Comput Appl Math 279:192–208MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Rognes ME, Logg A (2013) Automated goal-oriented error control I: stationary variational problems. SIAM J Sci Comput 35(3):C173–C193MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Salinger AG, Bartett RA, Chen Q, Gao X, Hansen G, Kalashnikova I, Mota A, Muller RP, Nielsen E, Ostien J et al (2013) Albany: a component-based partial differential equation code built on trilinos. Technical Report SAND2013-8430J, Sandia Nat. Lab., Albuquerque, NM, USAGoogle Scholar
  52. 52.
    Simo JC, Hughes TJR (2006) Computational inelasticity. Springer, New YorkzbMATHGoogle Scholar
  53. 53.
    Smith CW, Granzow B, Ibanez D, Sahni O, Jansen KE, Shephard MS (2016) In-memory integration of existing software components for parallel adaptive unstructured mesh workflows. In: Proc. of the XSEDE16 Conf. on Diversity, Big Data, and Science at Scale, Miami, FL, USA. Miami, FL, USAGoogle Scholar
  54. 54.
    Smith CW, Rasquin M, Ibanez D, Jansen KE, Shephard MS (2018) Improving unstructured mesh partitions for multiple criteria using mesh adjacencies. SIAM J Sci Comput 40:C47–C75MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Stein E, Rüter M, Ohnimus S (2007) Error-controlled adaptive goal-oriented modeling and finite element approximations in elasticity. Comput Methods Appl Mech Eng 196(37):3598–3613MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Taylor C, Hood P (1973) A numerical solution of the Navier-Stokes equations using the finite element technique. Comput Fluids 1(1):73–100MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Tezaur IK, Perego M, Salinger AG, Tuminaro RS, Price SF (2015) Albany/FELIX: a parallel, scalable and robust, finite element, first-order Stokes approximation ice sheet solver built for advanced analysis. Geosci Model Dev 8(4):1197–1220CrossRefGoogle Scholar
  58. 58.
    Venditti DA, Darmofal DL (2000) Adjoint error estimation and grid adaptation for functional outputs: application to quasi-one-dimensional flow. J Comput Phys 164(1):204–227MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Venditti DA, Darmofal DL (2002) Grid adaptation for functional outputs: application to two-dimensional inviscid flows. J Comput Phys 176(1):40–69MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Venditti DA, Darmofal DL (2003) Anisotropic grid adaptation for functional outputs: application to two-dimensional viscous flows. J Comput Phys 187(1):22–46zbMATHCrossRefGoogle Scholar
  61. 61.
    Verfürth R (1994) A posteriori error estimation and adaptive mesh-refinement techniques. J Comput Appl Math 50(1–3):67–83MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Whiteley JP, Tavener SJ (2014) Error estimation and adaptivity for incompressible hyperelasticity. Int Numer Methods Eng 99(5):313–332MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Sandia National LaboratoriesAlbuquerqueUSA
  2. 2.Scientific Computation Research CenterRensselaer Polytechnic InstituteTroyUSA
  3. 3.Aerospace and Mechanical EngineeringUniversity of Southern CaliforniaLos AngelesUSA

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