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Research Papers

# Use of B-Spline in the Finite Element Analysis: Comparison With ANCF Geometry

[+] Author and Article Information
Ahmed A. Shabana

Department of Mechanical and Industrial Engineering,  University of Illinois at Chicago, 842 West Taylor Street, Chicago, IL 60607shabana@uic.edu

Ashraf M. Hamed

Department of Mechanical and Industrial Engineering,  University of Illinois at Chicago, 842 West Taylor Street, Chicago, IL 60607aabdal5@uic.edu

Abdel - Nasser A. Mohamed

Department of Mechanical and Industrial Engineering,  University of Illinois at Chicago, 842 West Taylor Street, Chicago, IL 60607amoham25@uic.edu

Paramsothy Jayakumar

U.S. Army RDECOM-TARDEC, 6501 E. 11 Mile Road, Warren, MI 48397-5000paramsothy.jayakumar@us.army.mil

Michael D. Letherwood

U.S. Army RDECOM-TARDEC, 6501 E. 11 Mile Road, Warren, MI 48397-5000mike.letherwood@us.army.mil

J. Comput. Nonlinear Dynam 7(1), 011008 (Sep 06, 2011) (8 pages) doi:10.1115/1.4004377 History: Received January 31, 2011; Revised May 16, 2011; Published September 06, 2011; Online September 06, 2011

## Abstract

This paper examines the limitations of using B-spline representation as an analysis tool by comparing its geometry with the nonlinear finite element absolute nodal coordinate formulation (ANCF) geometry. It is shown that while both B-spline and ANCF geometries can be used to model nonstructural discontinuities using linear connectivity conditions, there are fundamental differences between B-spline and ANCF geometries. First, while B-spline geometry can always be converted to ANCF geometry, the converse is not true; that is, ANCF geometry cannot always be converted to B-spline geometry. Second, because of the rigid structure of the B-spline recurrence formula, there are restrictions on the order of the parameters and basis functions used in the polynomial interpolation; this in turn can lead to models that have significantly larger number of degrees of freedom as compared to those obtained using ANCF geometry. Third, in addition to the known fact that B-spline does not allow for straightforward modeling of T-junctions, B-spline representation cannot be used in a straightforward manner to model structural discontinuities. It is shown in this investigation that ANCF geometric description can be used to develop new spatial chain models governed by linear connectivity conditions which can be applied at a preprocessing stage allowing for an efficient elimination of the dependent variables. The modes of the deformations at the definition points of the joints that allow for rigid body rotations between ANCF finite elements are discussed. The use of the linear connectivity conditions with ANCF spatial finite elements leads to a constant inertia matrix and zero Coriolis and centrifugal forces. The fully parameterized structural ANCF finite elements used in this study allow for the deformation of the cross section and capture the coupling between this deformation and the stretch and bending. A new chain model that employs different degrees of continuity for different coordinates at the joint definition points is developed in this investigation. In the case of cubic polynomial approximation, $C1$ continuity conditions are used for the coordinate line along the joint axis; while $C0$ continuity conditions are used for the other coordinate lines. This allows for having arbitrary large rigid body rotation about the axis of the joint that connects two flexible links. Numerical examples are presented in order to demonstrate the use of the formulations developed in this paper.

## Figures

Figure 1

Structural and nonstructural discontinuities

Figure 2

Initial configuration of the belt drive mechanism for both C0/C1 and C1 models

Figure 3

Initial configuration of the chain model

Figure 4

Angular velocity of the driving and driven pulleys [

driving pulley,
C1 belt (driven pulley),
C0/C1 belt (driven pulley),
chain (driven pulley)]

Figure 5

Centerline of the C1 belt model at time 1 s

Figure 6

Centerline of the C0/C1 belt model at time 1 s

Figure 7

Centerline of the chain model at time 1 s

Figure 8

Area ratio along the belt centerline at time 1.9 s (

C1 belt,
C0/C1 belt,
chain)

Figure 9

Axial strain ɛxx along the belt centerline at time 1.9 s (

C1 belt,
C0/C1 belt,
chain)

Figure 10

Normal strain ɛyy along the belt centerline at time 1.9 s (

C1 belt,
C0/C1 belt,
chain)

Figure 11

Normal strain ɛzz along the belt centerline at time 1.9 s (

C1 belt,
C0/C1 belt,
chain)

Figure 12

Shear strain ɛxz along the belt centerline at time 1.9 s (

C1 belt,
C0/C1 belt,
chain)

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