Editorial Type:
Article Category: Research Article
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Online Publication Date: 16 Jul 2021

Isogeometric Analysis for Tire Simulations: From Mesh Generation to High Precision Results

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Page Range: 260 – 275
DOI: 10.2346/tire.21.20021
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ABSTRACT

Isogeometric analysis (IGA) has become an alternative to standard finite element analysis (FEA) in many areas of engineering. Its powerful tools for model generation and flexibility of basis functions make this relatively new approach attractive for tire analysis and its computational challenges.

This contribution summarizes the benefits of IGA for complex tire simulations starting from model generation and the subsequent transition to the environment of numerical analysis without losing accuracy at the parametrizing stage. It presents results of further development work on earlier pioneering examples of the application of IGA in pneumatic tire analysis. In addition to the analysis of vertical stiffness, for the first time, velocity and acceleration fields are addressed and compared with experimental results and standard FEA simulations, with a focus on benefits of the continuity of basis functions within the contact patch. The numerical issues that arise in IGA at the enforcement of contact and the application of inelastic materials with inclusions of reinforcing layers are studied. Moreover, the important advantages of the possibility to use higher order functions for simulations of tire maneuvers are addressed within the steady-state framework. Numerical examples are provided to illustrate the capabilities of IGA. Concluding remarks on the results close the publication.

Keywords: isogeometric analysis; NURBS; finite element method; ALE formulation; accelerations
FIG. 1 —
FIG. 1 —

Representations of a circular section by using NURBS and main parts of analytical discretization in IGA.

FIG. 2 —
FIG. 2 —

IGA concept for a single patch surface model defined by a quadratic basis.

FIG. 3 —
FIG. 3 —

Example of the knot insertion procedure in IGA: mesh for knot vectors Ξ1 = {0, 0, 0, 1, 1, 1}, Ξ2 = {0, 0, 0, 0.5, 1, 1, 1}, and Ξ3 = {0, 0, 0, 0.25, 0.5, 0.75, 1, 1, 1} is shown.

FIG. 4 —
FIG. 4 —

Example of the order elevation procedure in IGA: meshes of order 2, 3, and 5 are shown.

FIG. 5 —
FIG. 5 —

Example of k-refinement (combination of the knot insertion and order elevation procedure) in IGA: (a) knot insertion at 0.5 and then order elevation up to order 5; (b) order elevation up to order 5 and then knot insertion at 0.5.

FIG. 6 —
FIG. 6 —

Cross section of the tire model (size 205/55R16).

FIG. 7 —
FIG. 7 —

Discretization of the IGA and FEA tire models. Solid lines define C0-continuous elements' borders. Dashed lines define C1-continuity. Direction of the global axis are same for both models and shown only for FEA model.

FIG. 8 —
FIG. 8 —

Displacement fields in the compared models at free-rolling state.

FIG. 9 —
FIG. 9 —

IGA simulation results compared with FEA: displacements at sampling points plotted against the angle with respect to z-axis (defined in Fig. 7).

FIG. 10 —
FIG. 10 —

IGA simulation results compared with FEA: velocities at sampling points plotted against the angle with respect to z-axis (defined in Fig. 7).

FIG. 11 —
FIG. 11 —

IGA simulation results compared with FEA and experimental data: radial accelerations at sampling points plotted against the angle with respect to z-axis (defined in Fig. 7).

Contributor Notes

1 Institute for Structural Analysis, Technische Universität Dresden, D-01062 Dresden, Germany
2 Institute of Mathematics and Informatics, Tyumen State University, ul. Volodarskogo, 6, Tyumen 625003, Russia
3 Corresponding author. Email: michael.kaliske@tu-dresden.de
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