Research Papers: Gas Turbines: Structures and Dynamics

Harmonic Convergence Estimation Through Strain Energy Superconvergence

[+] Author and Article Information
Alexander A. Kaszynski

Universal Technology Corporation,
Dayton, OH 45434
e-mail: akascap@gmail.com

Joseph A. Beck, Jeffrey M. Brown

Turbine Engine Division,
U.S. Air Force Research Laboratory,
Wright-Patterson AFB, OH 45431

Contributed by the Structures and Dynamics Committee of ASME for publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received January 15, 2016; final manuscript received February 22, 2016; published online April 12, 2016. Editor: David Wisler.

J. Eng. Gas Turbines Power 138(10), 102501 (Apr 12, 2016) (9 pages) Paper No: GTP-16-1017; doi: 10.1115/1.4033059 History: Received January 15, 2016; Revised February 22, 2016

Grid convergence in finite element analysis (FEA), despite a wide variety of tools available to date, remains an elusive and challenging task. Due to the complex and time-consuming process of remeshing and solving the finite element model (FEM), convergence studies can be a part of the most arduous portion of the modeling process and can even be impossible with FEMs unassociated with CAD. Existing a posteriori methods, such as relative error in the energy norm, provide a near arbitrary indication of the model convergence for eigenfrequencies. This paper proposes a new approach to evaluate the harmonic convergence of an existing model without conducting a convergence study. Strain energy superconvergence (SES) takes advantage of superconvergence points within a FEM and accurately recovers the strain energy within the model using polyharmonic splines, thus providing a more accurate estimate of the system's eigenfrequencies without modification of the FEM. Accurate eigenfrequencies are critical for designing for airfoil resonance avoidance and mistuned rotor response prediction. Traditional error estimation strategies fail to capture harmonic convergence as effectively as SES, potentially leading to a less accurate airfoil resonance and rotor mistuning prediction.

Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.


Bathe, K.-J. , 1996, Finite Element Procedures, Prentice-Hall, Upper Saddle River, NJ.
Heath, S. , Slater, T. , Mansfield, L. , and Loftus, P. , 1997, “ Turbomachinery Blade Tip Measurement Techniques,” 90th Symposium on Advanced Non-Intrusive Instrumentation for Propulsion Engines, Propulsion and Energetics Panel (PEP), Brussels, Belgium, Oct. 20–24, AGARD 598, p. 32.
Chan, Y. J. , 2009, “ Variability of Blade Vibration in Mistuned Bladed Disks,” Ph.D. thesis, University of London, London.
Roy, C. J. , 2010, “ Review of Discretization Error Estimators in Scientific Computing,” AIAA Paper No. 2010-126.
Walz, J. E. , Fulton, R. E. , and Cyrus, N. J. , 1968, “ Accuracy and Convergence of Finite Element Approximations,” 2nd Conference on Matrix Methods in Structural Mechanics, Air Force Flight Dynamics Laboratory, Wright-Patterson AFB, OH, Oct. 15–17, Report No. ADA447390.
Schmidt, H. , Alber, T. , Wehner, T. , Blakytny, R. , and Wilke, H. , 2009, “ Discretization Error When Using Finite Element Models: Analysis and Evaluation of an Underestimated Problem,” J. Biomech., 42(12), pp. 1926–1934. [CrossRef] [PubMed]
Baušys, R. , 1995, “ Accuracy Estimates in Free Vibration Analysis,” Statyba, 1(4), pp. 5–19. [CrossRef]
Ainsworth, M. , and Oden, J. T. , 1997, “ A Posteriori Error Estimation in Finite Element Analysis,” Comput. Methods Appl. Mech. Eng., 142(12), pp. 1–88. [CrossRef]
Oh, H.-S. , and Batra, R. , 1999, “ Application of Zienkiewiczzhu's Error Estimate With Superconvergent Patch Recovery to Hierarchical p-Refinement,” Finite Elem. Anal. Des., 31(4), pp. 273–280. [CrossRef]
Craig, A. , Ainsworth, M. , Zhu, J. , and Zienkiewicz, O. , 1989, “ h and h-p Version Error Estimation and Adaptive Procedures From Theory to Practice,” Eng. Comput., 5(3–4), pp. 221–234. [CrossRef]
Barlow, J. , 1976, “ Optimal Stress Locations in Finite Element Models,” Int. J. Numer. Methods Eng., 10(2), pp. 243–251. [CrossRef]
Zienkiewicz, O. , Gago, J. D. S. , and Kelly, D. , 1983, “ The Hierarchical Concept in Finite Element Analysis,” Comput. Struct., 16(14), pp. 53–65. [CrossRef]
Voltera, E. , and Zachmanoglou, E. C. , 1965, Dynamics of Vibrations, Charles E. Merrill Books, Columbus, OH.
Yue, Z. , and Robbins, D. , 2003, “ Rank Deficiency in Superconvergent Patch Recovery Techniques With 4-Node Quadrilateral Elements,” Commun. Numer. Methods Eng., 23(1), pp. 1–10. [CrossRef]
Li, B. , and Zhang, Z. , 1999, “ Analysis of a Class of Superconvergence Patch Recovery Techniques for Linear and Bilinear Finite Elements,” Numer. Methods Partial Differ. Equations, 15(2), pp. 151–167. [CrossRef]
Rdenas, J. J. , Tur, M. , Fuenmayor, F. J. , and Vercher, A. , 2007, “ Improvement of the Superconvergent Patch Recovery Technique by the Use of Constraint Equations: The SPR-C Technique,” Int. J. Numer. Methods Eng., 70(6), pp. 705–727. [CrossRef]
Gu, H. , Zong, Z. , and Hung, K. C. , 2004, “ A Modified Superconvergent Patch Recovery Method and Its Application to Large Deformation Problems,” Finite Elem. Anal. Des., 40(5–6), pp. 665–687. [CrossRef]
Beatson, R. , Powell, M. J. D. , and Tan, A. M. , 2006, “ Fast Evaluation of Polyharmonic Splines in Three Dimensions,” IMA J. Numer. Anal., 27(3), pp. 427–450. [CrossRef]
MacQueen, J. , 1967, “Some Methods for Classification and Analysis of Multivariate Observations,” Fifth Berkeley Symposium on Mathematical Statistics and Probability, Berekely, CA, June 21–July 18, Vol. 1, pp. 281–297.
Castanier, M. P. , and Pierre, C. , 2002, “ Using Intentional Mistuning in the Design of Turbomachinery Rotors,” AIAA J., 40(10), pp. 2077–2086. [CrossRef]
Petrov, E. P. , and Ewins, D. J. , 2003, “ Analysis of the Worst Mistuning Patterns in Bladed Disk Assemblies,” ASME J. Turbomach., 125(4), pp. 623–631. [CrossRef]


Grahic Jump Location
Fig. 4

SERR and frequency error for twisted cantilever beam

Grahic Jump Location
Fig. 3

Twisted cantilever beam strain energy in 3B

Grahic Jump Location
Fig. 2

SERR and frequency error with respect to nodal density

Grahic Jump Location
Fig. 1

Cantilever beam SERR (top) and strain energy (bottom)

Grahic Jump Location
Fig. 5

Improved eigenfrequency estimation comparison

Grahic Jump Location
Fig. 9

Tetrahedral FEM convergence study results

Grahic Jump Location
Fig. 7

Annulus recovered strain energy variation

Grahic Jump Location
Fig. 8

Annulus eigenfrequency estimation comparison

Grahic Jump Location
Fig. 10

Strain energy in original beam (top) and polyharmonic interpolated beam (bottom)

Grahic Jump Location
Fig. 11

Interpolation accuracy versus PSIM cloud size

Grahic Jump Location
Fig. 12

Interpolation accuracy versus distance to cluster center

Grahic Jump Location
Fig. 6

Recovered strain energy variation

Grahic Jump Location
Fig. 13

Production IBR SES convergence study

Grahic Jump Location
Fig. 14

Recovered strain energy variation in first bend for a production IBR sector

Grahic Jump Location
Fig. 15

Nonaxisymmetric incomplete inflated torus

Grahic Jump Location
Fig. 16

Convergence study comparing original and interpolation based eigenvalues



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In