Engineering with Computers

, Volume 31, Issue 2, pp 193–213 | Cite as

Verification tests in solid mechanics

  • K. Kamojjala
  • R. BrannonEmail author
  • A. Sadeghirad
  • J. Guilkey
Original Article


Code verification against analytical solutions is a prerequisite to code validation against experimental data. Though solid-mechanics codes have established basic verification standards such as patch tests and convergence tests, few (if any) similar standards exist for testing solid-mechanics constitutive models under nontrivial massive deformations. Increasingly complicated verification tests for solid mechanics are presented, starting with simple patch tests of frame-indifference and traction boundary conditions under affine deformations, followed by two large-deformation problems that might serve as standardized verification tests suitable to quantify accuracy, robustness, and convergence of momentum solvers used in solid-mechanics codes. These problems use an accepted standard of verification testing, the method of manufactured solutions (MMS), which is rarely applied in solid mechanics. Body forces inducing a specified deformation are found analytically by treating the constitutive model abstractly, with a specific model introduced only at the last step in examples. One nonaffine MMS problem subjects the momentum solver and constitutive model to large shears comparable to those in penetration, while ensuring natural boundary conditions to accommodate codes lacking support for applied tractions. Two additional MMS problems, one affine and one nonaffine, include nontrivial traction boundary conditions.


Verification testing Solid mechanics Method of manufactured solutions MMS Generalized vortex Bending bar Frame indifference 



The support of Schlumberger Technology Corporation, SURVICE Engineering and Sandia National Laboratories is gratefully acknowledged. The authors would also like to acknowledge the constructive feedback given by the reviewers which helped to improve the manuscript.


  1. 1.
    Appelo D, Anders PN (2009) A stable finite difference method for the elastic wave equation on complex geometries with free surfaces. Commun Comput Phys 5:84–107MathSciNetGoogle Scholar
  2. 2.
    Babuska I, Tinsley OJ (2004) Verification and validation in computational engineering and science: basic concepts. Comput Methods Appl Mech Eng 193:(10)4057–4066CrossRefzbMATHGoogle Scholar
  3. 3.
    Banerjee B (2006) Method of manufactured solutions.
  4. 4.
    Bardenhagen SG, Kober EM (2004) The generalized interpolation material point method. CMES Comput Model Eng Sci 5:477–495Google Scholar
  5. 5.
    Batra RC, Liang XQ (1997) Finite dynamic deformations of smart structures. Comput Mech 20:427–438CrossRefzbMATHGoogle Scholar
  6. 6.
    Benes M, Matous K (2010) Asynchronous multi-domain variational integrators for nonlinear hyperelastic solids. Comput Methods Appl Mech Eng 199:1992–2013CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Brannon RM, Leelavanichkul S (2010) A multi-stage return algorithm for solving the classical damage component of constitutive models for rocks, ceramics, and other rock-like media. Int J Fract 163:133–149CrossRefzbMATHGoogle Scholar
  8. 8.
    Brannon RM, Kamojjala K, Sadeghirad A (2011) Establishing credibility of particle methods through verification testing. In: II international conference on particle-based methods—fundamentals and applicationsGoogle Scholar
  9. 9.
    Brannon RM, Fossum AF, Strack OE (2009) Kayenta: theory and user’s guide. Technical Report SAND2009-2282, USDOEGoogle Scholar
  10. 10.
    Brunner TA (2006) Development of a grey nonlinear thermal radiation diffusion verification problem. Trans Am Nuclear Soc 95:876–878Google Scholar
  11. 11.
    Foster CD, Regueiro RA, Fossum AF, Borja RI (2005) Implicit numerical integration of a three-invariant, isotropic/kinematic hardening cap plasticity model for geomaterials. Comput Methods Appl Mech Eng 194:5109–5138CrossRefzbMATHGoogle Scholar
  12. 12.
    Guilkey J, Harman T, Luitjens J, Schmidt J, Thornock J, de St Germain JD, Shankar S, Peterson J, Brownlee C (2009) Uintah User guide. SCI Institute Technical ReportGoogle Scholar
  13. 13.
    Johnson GR, Cook WH (1983) A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures. In: Proceedings of the 7th international symposium on ballistics, The Netherlands, pp 541–547Google Scholar
  14. 14.
    Kamojjala K, Brannon RM (2011) Verification of frame indifference for complicated numerical constitutive models. In: ASME early career technical conferenceGoogle Scholar
  15. 15.
    Knupp P, Salari K (2003) Verification of computer codes in computational science and engineering. Chapman and Hall/CRC, LondonzbMATHGoogle Scholar
  16. 16.
    Kossa A, Szabó L (2009) Exact integration of the von Mises elastoplasticity model with combined linear isotropic-kinematic hardening. Int J Plasticity 25:1083–1106CrossRefzbMATHGoogle Scholar
  17. 17.
    Kozdon JE, Dunham EM, Nordstrom J (2013) Simulation of dynamic earthquake ruptures in complex geometries using high-order finite difference method. J Sci Comput 55:92–124CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Krieg RD, Krieg DB (1977) Accuracies of numerical solution methods for the elastic-perfectly plastic model. J Pressure Vessel Technol 99:510–515CrossRefGoogle Scholar
  19. 19.
    Kuo C-S, Hu H-T, Lin R-M, Huang K-Y, Lin P-C, Zhong Z-C, Hseih M-L (2010) Biomechanical analysis of the lumbar spine on facet joint force and intradiscal pressure—a finite element study. BMC Musculoskelet Disord 151:11Google Scholar
  20. 20.
    Love E, Sulsky DL (2006) An unconditionally stable, energy-momentum consistent implementation of the material-point method. Comput Methods Appl Mech Eng 195:33–36CrossRefMathSciNetGoogle Scholar
  21. 21.
    Malverin LE (1969) Introduction to the mechanics of a continuous medium. Prentice-Hall Inc., New JerseyGoogle Scholar
  22. 22.
    Monaghan JJ (1988) Introduction to sph. Comput Phys Commun 48:89–96CrossRefzbMATHGoogle Scholar
  23. 23.
    Oberkampf WL, Roy CJ (2010) Verification and validation in scientific computing. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  24. 24.
    Oberkampf WL, Trucano TG, Hirsh C (2002) Verification and validation of modeling and simulation in computer ccience and engineering applications. In: Foundations of verification and validation in 21st century workshopGoogle Scholar
  25. 25.
    Oberkampf WL, Trucano TG (2002) Verification and validation in computational fluid dynamics. Progress Aerospace Sci 38:209–272CrossRefGoogle Scholar
  26. 26.
    Pautz SD (2001) Verification of transport codes by the method of manufactured solutions: the attila experience. In: Proceedings of ANS international meeting on mathematical methods for nuclear applications, Salt Lake CityGoogle Scholar
  27. 27.
    Rashid MM (1993) Incremental kinematics for finite element applications. Int J Numer Methods Eng 36:(23)3937–3956CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Rieben R, White DA (2005) Verification of high-order mixed fem solution of transient magnetic diffusion problems. Technical report, Lawrence Livermore National LaboratoryGoogle Scholar
  29. 29.
    Roache Patrick J (1999) Fundamentals of Verification and Validation, volume 41. SIAM, Editor: Mark AnisworthGoogle Scholar
  30. 30.
    Roache Patrick J (2009) Fundamentals of verification and validation. Hermosa Publishers, SocorroGoogle Scholar
  31. 31.
    Roy CJ, Nelson CC, Smith TM, Ober CC (2004) Verification of euler/navier stokes codes using the method of manufactured solutions. Int J Numer Methods Fluids 44:599–620CrossRefzbMATHGoogle Scholar
  32. 32.
    Roy CJ, Oberkampf WL (2011) A comprehensive framework for verification, validation, and uncertainty quantification in scientific computing. Comput Methods Appl Mech Eng 200:2131–2144CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Sadeghirad A, Brannon RM, Burghardt J (2011) A convected particle domain interpolation technique to extend applicability of the material point method for problems involving massive deformations. Int J Numer Methods Eng 86(22):1435–1456CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Sadeghirad A, Brannon RM, Guilkey JE (2013) Second-order convected particle domain interpolation (CPDI2) with enrichment for weak discontinuities at material interfaces. Int J Numer Methods Eng 95:928–952CrossRefMathSciNetGoogle Scholar
  35. 35.
    Schwer LE (2006) Guide for verification and validation in computational solid mechanics. In: The American Society of Mechanical EngineersGoogle Scholar
  36. 36.
    Simo JC, Hughes TJR (1998) Computational inelasticity. Interdisciplinary applied mathematics: mechanics and materials. Springer, New YorkGoogle Scholar
  37. 37.
    Steffen M, Wallstedt PC, Guilkey JE, Kirby RM, Berzins M (2008) Examination and analysis of implementation choices within the material point method (mpm). Comput Model Eng Sci 2:107–127Google Scholar
  38. 38.
    Tremblay D, Etienne S, Pelletier D (2006) Code verification and the method of manufactured solutions for fluid-structure interaction problems. In: 36th AIAA fluid dynamics conference, vol 2, pp 882–892Google Scholar
  39. 39.
    Wallstedt PC, Guilkey JE (2008) An evaluation of explicit time integration schemes for use with the generalized interpolation material point method. J Comput Phys 227:9628–9642CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  • K. Kamojjala
    • 1
  • R. Brannon
    • 1
    Email author
  • A. Sadeghirad
    • 1
  • J. Guilkey
    • 1
  1. 1.Department of Mechanical EngineeringUniversity of UtahSalt Lake CityUSA

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