Research Papers

Three-Dimensional Solid Brick Element Using Slopes in the Absolute Nodal Coordinate Formulation

[+] Author and Article Information
Alexander Olshevskiy

School of Mechanical Engineering,
Konkuk University,
1 Hwanyang-Dong, Gwangjin-Gu,
Seoul 143-701, South Korea
Applied Mechanics,
Bryansk State Technical University,
Bulvar 50 let Oktyabrya 7,
Bryansk 241035, Russia

Oleg Dmitrochenko

Department of Mechanical Engineering,
Lappeenranta University of Technology,
Skinnarilankatu 34,
Lappeenranta 53850, Finland
Applied Mechanics,
Bryansk State Technical University,
Bulvar 50 let Oktyabrya 7,
Bryansk 241035, Russia

Chang-Wan Kim

School of Mechanical Engineering,
Konkuk University,
1 Hwanyang-Dong, Gwangjin-Gu,
Seoul 143-701, South Korea
e-mail: goodant@konkuk.ac.kr

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received April 22, 2012; final manuscript received April 6, 2013; published online September 12, 2013. Assoc. Editor: Arend L. Schwab.

J. Comput. Nonlinear Dynam 9(2), 021001 (Sep 12, 2013) (10 pages) Paper No: CND-12-1062; doi: 10.1115/1.4024910 History: Received April 22, 2012; Revised April 06, 2013

The present paper contributes to the field of flexible multibody systems dynamics. Two new solid finite elements employing the absolute nodal coordinate formulation are presented. In this formulation, the equations of motion contain a constant mass matrix and a vector of generalized gravity forces, but the vector of elastic forces is highly nonlinear. The proposed solid eight node brick element with 96 degrees of freedom uses translations of nodes and finite slopes as sets of nodal coordinates. The displacement field is interpolated using incomplete cubic polynomials providing the absence of shear locking effect. The use of finite slopes describes the deformed shape of the finite element more exactly and, therefore, minimizes the number of finite elements required for accurate simulations. Accuracy and convergence of the finite element is demonstrated in nonlinear test problems of statics and dynamics.

Copyright © 2014 by ASME
Your Session has timed out. Please sign back in to continue.


Shabana, A. A., 2005, Dynamics of Multibody Systems, 3rd ed., Cambridge University, New York.
Rankin, C. C., and Brogan, F. A., 1984, “An Element-Independent Corotational Procedure for the Treatment of Large Rotations,” ASME J. Pressure Vessel Technol., 108(2), pp. 165–174. [CrossRef]
Kane, T. R., Ryan, R. R., and Banerjee, A. K., 1987, “Dynamics of a Cantilever Beam Attached to a Moving Base,” J. Guid. Control, 10, pp. 139–151. [CrossRef]
Belytschko, T., and Hsieh, B. J., 1973, “Nonlinear Transient Finite Element Analysis With Convected Coordinates,” Int. J. Numer. Methods Eng., 7, pp. 255–271. [CrossRef]
Simo, J. C., 1985, “A Finite Strain Beam Formulation. The Three-Dimensional Dynamic Problem, Part I,” Comput. Methods Appl. Mech. Eng., 49, pp. 55–70. [CrossRef]
Simo, J. C., and Vu-Quoc, L., 1986, “A Three-Dimensional Finite Strain Rod Model, Part II: Computational Aspects,” Comput. Meth. Appl. Mech. Eng., 58, pp. 79–116. [CrossRef]
Shabana, A. A., 1997, “Flexible Multibody Dynamics: Review of Past and Recent Developments,” Multibody Syst. Dyn., 1, pp. 189–222. [CrossRef]
Wu, G., He, X., and Pai, F., 2011, “Geometrically Exact 3D Beam Element for Arbitrary Large Rigid-Elastic Deformation Analysis of Aerospace Structures,” Finite Elem. Anal. Design, 47(4), pp. 402–412. [CrossRef]
Pai, P. F., 2007, Highly Flexible Structures: Modeling, Computation, and Experimentation, AIAA, Reston, VA.
Shabana, A. A., 1997, “Definition of the Slopes and the Finite Element Absolute Nodal Coordinate Formulation,” Multibody Syst. Dyn., 1(3), pp. 339–348. [CrossRef]
Escalona, J. L., Hussien, H. A., and Shabana, A. A., 1998, “Application of the Absolute Nodal Coordinate Formulation to Multibody System Dynamics,” J. Sound Vib., 5, pp. 833–851. [CrossRef]
von Dombrowski, S., 2002, “Analysis of Large Flexible Body Deformation in Multibody Systems Using Absolute Coordinates,” Multibody Syst. Dyn., 8, pp. 409–432. [CrossRef]
Dmitrochenko, O., Yoo, W. S., and Pogorelov, D., 2006, “Helicoseir as Shape of a Rotating Chain (II): 3D Theory and Simulation Using ANCF,” Multibody Syst. Dyn., 15(2), pp. 181–200. [CrossRef]
Yoo, W.-S., Dmitrochenko, O., Park, S.-J., and Lim, O.-K., 2005, “A New Thin Spatial Beam Element Using the Absolute Nodal Coordinates: Application to a Rotating Strip,” Mech. Based Des. Struct. Mach., 33(3–4), pp. 399–422. [CrossRef]
Nachbagauer, K., Gruber, P., Vetyukov, Yu., and Gerstmayr, J., 2011, “A Spatial Thin Beam Finite Element Based on the Absolute Nodal Coordinate Formulation Without Singularities,” Proceedings of the ASME 2011 International Design Engineering Technical Conference & Computers and Information in Engineering Conference, Washington, DC, Aug. 28–31, IDETC/CIE 201.
Dmitrochenko, O., and Pogorelov, D., 2003, “Generalization of Plate Finite Elements for Absolute Nodal Coordinate Formulation,” Multibody Syst. Dyn., 10(1), pp. 17–43. [CrossRef]
Dmitrochenko, O., and Mikkola, A., 2008, “Two Simple Triangular Plate Elements Based on the Absolute Nodal Coordinate Formulation,” ASME J. Comput. Nonlinear Dyn., 3(4), p. 041012. [CrossRef]
Dmitrochenko, O., and Mikkola, A., 2008, “Shear Correction of a Thin Plate Element in Absolute Nodal Coordinates,” Proceedings of 8th World Congress On Computational Mechanics (WCCMS) and 5th European Congress on Computational Methods in Applied Science and Engineering (ECCOMAS 2008), Venice, Italy, June 30–July 4.
Dmitrochenko, O., and Mikkola, A., 2009, “Shear Correction for Thin Plate Finite Elements Based on the Absolute Nodal Coordinate Formulation,” Proceedings of the ASME 2009 IDETC/CIE, San Diego, CA, Aug. 30–Sept. 2, pp. 1–9.
Sereshk, M., and Salimi, M., 2011, “Comparison of Finite Element Method Based on Nodal Displacement and Absolute Nodal Coordinate Formulation (ANCF) in Thin Shell Analysis,” Int. J. Numer. Methods Biomed. Eng., 27(8), pp. 1185–1198. [CrossRef]
Gerstmayr, J., and Schöberl, J., 2006, “A 3D Finite Element Method for Flexible Multibody Systems,” Multibody Syst. Dyn., 15, pp. 309–324. [CrossRef]
Kübler, L., Eberhard, P., and Geisler, J., 2003, “Flexible Multibody Systems With Large Deformations and Nonlinear Structural Damping Using Absolute Nodal Coordinates,” Nonlinear Dyn., 34, pp. 31–52. [CrossRef]
Dmitrochenko, O., and Mikkola, A., 2011, “Digital Nomenclature Code for Topology and Kinematics of Finite Elements Based on the Absolute Nodal Coordinate Formulation,” Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics March 1, 2011, Vol. 225, No. 1, 34–51.
Dmitrochenko, O., and Mikkola, A., 2011, “Extended Digital Nomenclature Code for Description of Complex Finite Elements and Generation of New Elements,” Mech. Based Des. Struct. Mach., 39(2), pp. 229–252. [CrossRef]
Friedman, Z., and Kosmatka, J. B., 1993, “An Improved Two-Node Timoshenko Beam Finite Element,” Comput. Struct., 47(3), pp. 473–481. [CrossRef]
Olshevskiy, A., Dmitrochenko, O., and Kim, C. W., 2013, “Three- and Four-Noded Planar Elements Using Absolute Nodal Coordinate Formulation,” Multibody Syst. Dyn., 29, pp. 255–269. [CrossRef]
Zienkiewicz, O. C., and Taylor, R. L., 1991, The Finite Element Method: Fourth Edition. Vol 2: Solid and Fluid Mechanics, McGraw-Hill, New York.
FelippaC., 2004, “A Compendium of FEM Integration Formulas for Symbolic Work,” Eng. Comput., 21(8), pp. 867–890. [CrossRef]
Gere, J. M., and Timoshenko, S. P., 1997, Mechanics of Materials, PWS Publishing, Boston, MA.
Pogorelov, D., 1997, “Some Developments in Computational Techniques in Modeling Advanced Mechanical Systems,” Proceedings of the IUTAM Symposium on Interaction Between Dynamics and Control in Advanced Mechanical Systems, pp. 313–320.
Bazeley, G. P., Cheung, Y. K., Irons, B. M., and Zienkiewicz, O. C., 1966, “Triangular Elements in Bending—Conforming and Non-Conforming Solutions,” Proceedings of the First Conference on Matrix Methods in Structural Mechanics, Air Force Flight Dynamics Laboratory, Wright Patterson Air Force Base, OH, Paper No. AFFDL-TR-66-90, pp. 547–576.


Grahic Jump Location
Fig. 1

A variety of locally d-dimensional n-node elements with c nodal m-vectors per node

Grahic Jump Location
Fig. 2

Solid element using positions of nodes and slopes as nodal coordinates

Grahic Jump Location
Fig. 3

Deformed shape of the pendulum at t = 0.1 s for the elements 2412 (straight edges) and 2432 (curved edges)

Grahic Jump Location
Fig. 4

Construction of interpolation polynomials: beam 122, plate 243, brick 384

Grahic Jump Location
Fig. 5

Calculating scheme (a), front (b), side (c), top (d), and general (e) views of the deformed shape of a single element with dncm 3843

Grahic Jump Location
Fig. 6

Vertical deflection of the beam's free end cross-section and relative error

Grahic Jump Location
Fig. 7

Deflection of the cantilever beam subjected to large bending (see Table 2)

Grahic Jump Location
Fig. 8

Error in cantilever beam deflection (see Table 2)

Grahic Jump Location
Fig. 9

A bar with rectangular cross-section in torsion: (a) calculating scheme; (b) 1 × 3843: φ = 82.78 deg; (c) 2 × 3843: φ = 83.87 deg; (d) 4 × 3843: φ = 84.48 deg; (e) 8 × 3843: φ = 84.81 deg

Grahic Jump Location
Fig. 10

Vertical deflection of the centroid of the pendulum free end cross-section

Grahic Jump Location
Fig. 11

Positions of the flexible pendulum (1 element 3843)

Grahic Jump Location
Fig. 12

Z-coordinate of the pendulum free end

Grahic Jump Location
Fig. 13

Deformed shape of the pendulum at t = 0.35 s

Grahic Jump Location
Fig. 14

Coordinate of the middle point of the ellipsograph during motion

Grahic Jump Location
Fig. 15

Nodal coordinates (a) and deformed shape (b) of the element with dncm 3443




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