Isogeometric Analysis for Tire Simulation at Steady-State Rolling

Mario A. Garcia; Michael Kaliske

doi:10.2346/tire.19.170164

Article Contents

ABSTRACT
Introduction
Steady-State Rolling and the ALE Formulation
Isogeometric Analysis
Implementation into FEA
Numerical Examples
Conclusions and Outlook
References

Editorial Type:

Article Category: Research Article

Online Publication Date: 01 Jul 2019

Isogeometric Analysis for Tire Simulation at Steady-State Rolling

and

Page Range: 174 – 195

DOI: 10.2346/tire.19.170164

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ABSTRACT

The use of isogeometric analysis (IGA) in industrial applications has increased in the past years. One of the main advantages is the combination of finite element analysis (FEA) with the capability of representing the exact geometry by means of non-uniform rational B-splines (NURBS). This framework has proven to be an efficient alternative to standard FEA in solid mechanics and fluid dynamics, in cases in which sensitivity to geometry is found. The numerical simulation of rolling tires requires a proper discretization for the curved boundaries and complex cross sections, which often leads to the use of higher-order or cylindrical elements. As remeshing operations are numerically costly in tire models, IGA stands as an attractive alternative for the modeling of rolling tires.

In this contribution, an arbitrary Lagrangian Eulerian formulation is implemented into IGA to provide the basic tools for the numerical analysis of rolling bodies at steady-state conditions. The solid basis of the formulation allows the employment of standard material models, but tire constructive elements, such as reinforcing layers, require special attention. Streamlines are constructed based on the locations of the integration points, and therefore, linear and nonlinear viscoelastic models can be implemented. Numerical examples highlight the advantage of the new approach of requiring fewer degrees of freedom for an accurate description of the geometry.

Keywords: finite element; ALE formulation; steady-state rolling; isogeometric analysis

FIG. 1

Motion decomposition for ALE kinematics.

FIG. 2

Basis functions with equally spaced nodes for p = 2 in (a) standard finite element and (b) B-spline.

FIG. 3

Circular arc from quadratic NURBS.

FIG. 4

Refinement by knot insertion: (a) original knot vector, (b) new knot vector.

FIG. 5

Refinement by elevation of the basis functions: (a) original knot vector p = 2, (b) elevated knot vector p = 3.

FIG. 6

Dirichlet boundary conditions in a circular beam.

FIG. 7

Solid disc: (a) dimensions, (b) disc modeled for IGA in four sections (control points are visible), and (c) disc discretized using eight-node solid elements.

FIG. 8

Displacements in x-direction for a solid disc and IGA, angular velocity Ω = 50 rad/s, discretized in circumferential direction into (a) four elements, (b) eight elements, and (c) 12 elements.

FIG. 9

Displacements in the x-direction for solid disc and FEA, angular velocity Ω = 50 rad/s, discretized in the circumferential direction into (a) eight elements with eight nodes, (b) 64 elements with eight nodes, and (c) 32 elements with 27 nodes.

FIG. 10

Comparison of degrees of freedom for converged solution between IGA and FEA.

FIG. 11

Comparison of computational time vs degrees of freedom between IGA and FEA.

FIG. 12

Cross section of tire model and details of bead and shoulder; control points are visible for the NURBS model.

FIG. 13

Discretization of the tire in circumferential direction: (a) initial NURBS model with control points for the external surface, (b) NURBS refinement in contact region, and (c) discretization for FEA.