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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

## Abstract

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.

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## Figures

Fig. 1

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

Fig. 2

Solid element using positions of nodes and slopes as nodal coordinates

Fig. 3

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

Fig. 4

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

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

Fig. 11

Positions of the flexible pendulum (1 element 3843)

Fig. 12

Z-coordinate of the pendulum free end

Fig. 13

Deformed shape of the pendulum at t = 0.35 s

Fig. 10

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

Fig. 6

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

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

Fig. 8

Error in cantilever beam deflection (see Table 2)

Fig. 7

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

Fig. 14

Coordinate of the middle point of the ellipsograph during motion

Fig. 15

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

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