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Contact between beams using a surface-to-surface formulation

Alfredo Gay Neto1

, Paulo de Mattos Pimenta1

, Peter Wriggers2

1Department of Structural and Geotechnical Engineering, Polytechnic School at

University of So Paulo, Brazil

E-mail: alfredo.gay@usp.br, ppimenta@usp.br

2Institute of Continuum Mechanics, Leibniz Universitt Hannover, Germany

E-mail: wriggers@ikm.uni-hannover.de

Keywords: surface-to-surface, beam, super-ellipse.

Beam structural elements in a finite element approach are often applied in engineering and

applied sciences. Structural models include the buildings, ropes and mooring lines, fabric threads

interaction, hair mechanics and many others. Some of these models involve contact interaction

between beams. Here, different approaches have already been proposed to model such contact.

Basic works, by our knowledge, are [1] and [2]. These papers proposed to model contact

pointwise. The basic idea is to find the location of minimum distance between two curves in space

that represent the beam axes. Then a gap function can be defined at that location which is basis for

the contact formulation. This leads in a finite element context to the residual and the tangent

operator for contact. These forces and matrices have to be included in the weak form and tangent

operator of the whole problem, in order to design a Newton-Raphson method to solve the general

nonlinear structural model. This contact approach was successfully applied in[3] for self-contactof beams, including comparisons with experiments. The examples included situations of large

twisting, but only circular cross sections were considered.

The present work derives a new technique to handle contact between beams. A surface-to-

surface approach is adopted. The two beams, candidate to contact, are parameterized, as shown in

equation (1). We assume that the parameterization is written using information from the two

extreme nodes of the beam, named A and B. Two convective coordinates (,) are selected,describing the length of the beam and the perimeter of a given cross section, respectively.

(, )= NA()[A+ A()]+ NB()[B+ B()] (1)

NA()and NB()are the linear Lagrange polynomials, shape functions of nodes A and B; Aand Bare the positions of nodes A and B, updated at each iteration of the Newton-

Raphson method with the accumulated displacements;

A and B are the rotation tensors of nodes A and B, updated at each iteration of theNewton-Raphson method with the rotations experienced by each node;

()is a parameterization of the beam cross section.We assume a pointwise contact. The contact location is determined by solving a minimum

distance problem. This is done based on an optimization criterion, using similar ideas as presented

in[4].For the cross section parameterization ()we chose a super-elliptic shape, see equation

(2).This provides flexibility in the model, since the super-ellipse can recover circular, elliptical

and rectangular cross sections by varying the parameters a, band the exponent n.

|xa|

+ |yb|

= 1 (2)

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An example of the external surface parameterization of a beam and the super-elliptic cross

section are depicted inFigure 1.

(a) (b)

Figure 1: An example of beam super-elliptical parameterization (a) The 3D visualization (b)

The cross section assumed

To test the new contact formulation we make use of the structural beam model presented in[5]. An example of multiple thread interactions is shown in Figure 2. All details regarding the

model will be addressed in future paper.

Figure 2: Multiple thread interactions example. A solid rendering is used for visualization of thebeams cross sections

The authors acknowledge FAPESP (Fundao de Amparo Pesquisa do Estado de So

Paulo) for the support under the grant 2014/17701-4 and CNPq (Conselho Nacional de

Desenvolvimento Cientfico e Tecnolgico) under the grant 303091/2013-4.

References:

[1] Wriggers, P. and Zavarise, G., On contact between three-dimensional beams undergoing

large deflections. Communications in Numerical Methods in Engineering, 13, 429-438,

(1997).

[2] Zavarise, G. and Wriggers, P., Contact with friction between beams in 3-D space. Int. J.

Numer. Meth. Engng., 49, 977-1006 (2000).

[3]

Gay Neto, A. Pimenta, P. M. & Wriggers, P.. Self-contact modeling on beams experiencingloop formation. Comp. Mechanics V. 55(1), 193-208, 2015.[4] Wellmann, C., Lillie, C. & Wriggers, P.. A contact detection algorithm for superellipsoids

based on the common-normal concept. Engineering Computations: International Journal for

Computer-Aided Engineering and Software V. 25 (5), 432-442, 2008.

[5] Gay Neto, A.; Martins, C. A. & Pimenta, P. M.. Static analysis of offshore risers with a

geometrically-exact 3D beam model subjected to unilateral contact. Comp. Mechanics V.53, 125-145, 2014.