Higher order finite elements (FEs) based on the absolute nodal coordinate formulation (ANCF) may require the use of curvature vectors as nodal coordinates. The curvature vectors, however, can be difficult to define at the reference configuration, making such higher order ANCF FEs less attractive to use. It is the objective of this investigation to use the concept of the mixed-coordinate ANCF FEs to ensure that the gradient vectors are the highest spatial derivatives in the element nodal coordinate vector regardless of the order of the interpolating polynomials used. This concept is used to convert the curvature vectors to nodes, called position nodes, which have only position coordinates. These new position nodes can be defined at a preprocessing stage, leading to two different sets of nodes: one set of nodes has position and gradient coordinates, while the second set of nodes has position coordinates only. The new position nodes can be used to obtain better distribution of the forces, including contact forces. Higher degree of continuity, including curvature continuity, can still be achieved at the element interface by using, at a preprocessing stage, linear algebraic equations that can reduce significantly the model dimension and ensure higher degree of smoothness. The procedure proposed in this investigation also allows for the formulation of mechanical joints at arbitrary points and nodes using linear algebraic constraint equations. The difficulties that arise when formulating these joint constraints using B-spline and NURBS (Nonuniform Rational B-Splines) representations are discussed. In order to explain the concepts introduced in this paper, low and high order ANCF thin plate elements are used. For the high order thin plate element, the curvature vectors at the interface nodes are converted to internal nodes with position coordinates only, leading to a mixed-coordinate ANCF thin plate element. This element preserves the desirable ANCF features including a constant mass matrix and zero Coriolis and centrifugal forces. Kirchhoff plate theory is used to formulate the element elastic forces. The equations of motion of the structure are formulated in terms of an independent set of structure coordinates. The resulting mass matrix associated with the independent coordinates remains constant. Numerical examples are presented in order to demonstrate the use of the mixed-coordinate ANCF thin plate element when the continuity constraints are imposed.

References

References
1.
Dmitrochenko
,
O. N.
, and
Pogorelov
,
D. Y.
,
2003
, “
Generalization of Plate Finite Elements for Absolute Nodal Coordinate Formulation
,”
Multibody Syst. Dyn.
,
10
(1)
, pp.
17
43
.10.1023/A:1024553708730
2.
Mikkola
,
A.
, and
Shabana
,
A. A.
,
2003
, “
A Non-incremental Finite Element Procedure for the Analysis of Large Deformation of Plates and Shells in Mechanical System Applications
,”
Multibody Syst. Dyn.
,
9
(3)
, pp.
283
309
.10.1023/A:1022950912782
3.
Dufva
,
K.
, and
Shabana
,
A. A.
,
2005
, “
Analysis of Thin Plate Structures Using the Absolute Nodal Coordinate Formulation
,”
J. Multibody Dyn.
,
219
(4), pp.
345
355
.10.1243/146441905X50678
4.
Dmitrochenko
,
O. N.
, 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
.10.1115/1.2960479
5.
Mikkola
,
A.
,
Shabana
,
A. A.
,
Sanchez-Rebollo
,
C.
, and
Jimenez-Octavio
,
J. R.
,
2013
, “
Comparison Between ANCF and B-spline Surface
,”
Multibody Syst. Dyn.
,
30
(2)
, pp.
119
138
.10.1007/s11044-013-9353-z
6.
Shabana
,
A. A.
, and
Mikkola
,
A.
,
2002
, “
On the Use of the Degenerate Plate and the Absolute Nodal Coordinate Formulation in Multibody System Applications
,”
J. Sound Vib.
,
259
(
2
), pp.
481
489
.10.1006/jsvi.2002.5156
7.
Shabana
,
A. A.
, and
Christensen
,
A. P.
,
1997
, “
Three-Dimensional Absolute Nodal Coordinate Formulation: Plate Problem
,”
Int. J. Numer. Methods Eng.
,
40
(15), pp.
2775
2790
.10.1002/(SICI)1097-0207(19970815)40:15%3C2775::AID-NME189%3E3.0.CO;2-#
8.
Gerstmayr
,
J.
,
Sugiyama
,
H.
, and
Mikkola
,
A.
,
2013
, “
Review on the Absolute Nodal Coordinate Formulation for Large Deformation Analysis of Multibody Systems
,”
ASME J. Comput. Nonlinear Dyn.
,
8
(
3
), p.
031016
.10.1115/1.4023487
9.
García-Vallejo
,
D.
,
Mikkola
,
A.
, and
Escalona
,
J. L.
,
2007
, “
A New Locking-Free Shear Deformable Finite Element Based on Absolute Nodal Coordinates
,”
Nonlinear Dyn.
,
50
(1–2)
, pp.
249
264
.10.1007/s11071-006-9155-4
10.
Nachbagauer
,
K.
,
Gruber
,
P.
, and
Gerstmayr
,
J.
,
2013
, “
A 3D Shear Deformable Finite Element Based on the Absolute Nodal Coordinate Formulation
,”
Multibody Dynamics
, Vol. 28, J.-C. Samin and P. Fisette, eds., Springer Netherlands, Dordrecht, pp. 77–96.10.1007/978-94-007-5404-1
11.
Shabana
,
A. A.
,
Hamed
,
A. M.
,
Mohamed
,
A. A.
,
Jayakumar
,
P.
, and
Letherwood
,
M. D.
,
2012
, “
Use of B-Spline in The Finite Element Analysis: Comparison With ANCF Geometry
,”
ASME J. Comput. Nonlinear Dyn.
,
7
(1), p. 011008.10.1115/1.4004377
12.
Shabana
,
A. A.
,
1998
, “
Computer Implementation of the Absolute Nodal Coordinate Formulation for Flexible Multibody Dynamics
,”
Nonlinear Dyn.
,
16
(3)
, pp.
293
306
.10.1023/A:1008072517368
13.
Yakoub
,
R. Y.
, and
Shabana
,
A. A.
,
1999
, “
Use of Cholesky Coordinates and the Absolute Nodal Coordinate Formulation in the Computer Simulation of Flexible Multibody System
,”
Nonlinear Dyn.
,
20
(3)
, pp.
267
282
.10.1023/A:1008323106689
14.
Li
,
H.
, and
Schindler
,
C.
,
2013
, “
Analysis of Soil Compaction and Tire Mobility With Finite Element Method
,”
J. Multibody Dyn.
,
227
(3), pp.
275
291
.10.1177/1464419313486627
15.
Piegl
,
L.
, and
Tiller
,
W.
,
1997
,
The NURBS Book
,
2nd ed.
,
Springer
,
New York
.
16.
Lan
,
P.
, and
Shabana
,
A. A.
,
2010
, “
Integration of B-Spline Geometry and ANCF Finite Element Analysis
,”
Nonlinear Dyn.
,
61
(1–2)
, pp.
193
206
.10.1007/s11071-009-9641-6
17.
Timoshenko
,
S.
, and
Woinowsky-Krieger
,
S.
,
1959
,
Theory of Plates and Shells
,
2nd ed.
,
McGraw-Hill Book Co.
,
Singapore
.
18.
Shabana
,
A. A.
,
2008
,
Computational Continuum Mechanics
,
Cambridge University
, Cambridge.
19.
Bathe
,
K. J.
,
1996
,
Finite Element Procedures
,
Prentice-Hall
, Upper Saddle River,
NJ
.
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