Isogeometric analysis for accurate modeling of rolling tires
Introduction
The growing interest on Isogeometric Analysis (IGA) has been extending to a large variety of research fields [1], [2], [3], [4]. The advantage of modeling exactly the geometry by using higher order basis functions, which at the same time provide continuity across element boundaries, is presented in this contribution within the context of tire analysis.
Traditionally, automotive and tire industry make use of models for describing the behavior of a tire, especially depending on the type of desired response. Useful approaches can be found from empirical models widely used like the Magic Formula [5] approach, physical models [6], [7] to complex three-dimensional numerical models. In the latter category, Finite Element Analysis (FEA) has grown in importance due to its capabilities in the context of analyzing the response of the tire based on the geometry and material properties of the rubber compounds. A major challenge is the adequate discretization of complex tire models using elements with linear shape functions, where a large amount of nodes is often required to obtain an accurate description. IGA stands as an excellent alternative for the modeling of pneumatic tires, especially by linking the use of computer-aided design (CAD) formulations with the large capabilities of FEA.
One of the first attempts to apply IGA in pneumatic tire simulation is done by Garcia and Kaliske [8]. They proved the efficiency of the method by comparing simulation results to those obtained by FEA for the same geometry, by using simplified tire models. Later, their framework was extended to include the reinforcing elements and to locally refine the mesh [9]. Furthermore, Kuraishi et al. [10] analyzed aerodynamics of the tire using a combination of a so-called time–space method and rational B-spline basis functions.
Due to the dynamic nature of the rolling process, transient simulations are often required for tire analysis. However, in this case, a large amount of computational resources is required. The use of an Arbitrary Lagrangian–Eulerian (ALE) formulation provides an efficient scheme for the analysis of rolling bodies in steady state, which is implemented straightforwardly into FEA [11] and also into IGA [8].
An important maneuver to be considered in tire analysis is braking, which is a complex physical phenomenon and, thus, a challenging task for numerical modeling where significant frictional effects are involved leading to large deformations and braking forces.
Nevertheless, the analysis of the relation between acceleration and forces in the tire enables a more efficient and optimized design. Moreover, the adequate numerical framework could be used for in situ parameter estimation for driving optimization and could become a contribution to the development of accelerometer-based intelligent tire systems.
Braking is, strictly speaking, hardly a stationary process as velocities are not constant and tire torsional oscillations occur. However, it is possible to simulate this maneuver in a steady-state manner as a set of incremental snapshots reaching equilibrium within each increment. Hence, the assessment of braking is often performed either as stationary fully locked sliding, or in relation to a specific slip rate in the contact area. These approaches show good agreement to experimental results and are implemented in common commercial codes under consideration of different frictional formulations [12], [13].
The main aim of the contribution at hand is the introduction of overall continuous models to tire analysis based on unclamped B-splines. As both, tire and rim are represented by single patches, no discontinuities are observed in the variable fields. Basic concepts of B-splines as basis functions are provided in Section 3. Important details on the implementation into an in-house finite element code are given in Section 4, where material models, rebar and contact formulation are discussed. Important advantages of using IGA for rolling bodies are presented in Section 5 by means of numerical examples including braking simulations, where hyperelastic material models at finite strains are used. In these examples, the presented novel approach is compared to traditional tire analysis in FEA, and validated using experimental results from bench tests, highlighting the capabilities and limitations in each case. The presented analysis implies the ALE-formulation of steady-state rolling problem, its approximate description within the Bubnov-Galerkin approach followed by the subsequent solution of the resulting system of nonlinear algebraic equations by the Newton–Raphson iterative method. It is shown, that the combination of a continuous formulation and the efficient computation of higher order derivatives of B-splines leads to an accurate evaluation of accelerations, which is further validated by experiments in physical tires, taking advantage of intelligent tire sensors. Concluding remarks and an outlook on suggested future steps are given in Section 6.
The versatility of this formulation can be used in combination with different analytical models for the estimation of forces in real time, especially in order to identify mechanical behavior which may not be evident in experiments, leading to a powerful tool for further calibration of said models as well as the potential for a combination with machine learning.
Section snippets
ALE-formulation for rolling bodies
The Arbitrary Lagrangian-Eulerian (ALE) framework has gained importance in industry due to the efficient description of the rolling phenomenon at steady state. This formulation preserves advantages and overcomes the drawbacks of the combined approaches [14], [15] and it has been proved to be much more efficient for rolling bodies in comparison to traditional transient analysis. More details on the concept of ALE-formulation for steady-state rolling can be found in [11].
The fundamental idea of
Isogeometric Analysis
The isogeometric analysis is built on the isoparametric concept, which employs the unique set of functions for the exact geometry description and the approximation procedures of the numerical solution. Thus, instead of approximating the geometry with functions, which are used for treating the sought variables, within IGA, the set of basis functions, which can reproduce the exact geometry, is taken for the approximation of the variable field. The term “isogeometric” itself is first mentioned in
FEA procedures
The spline-based model is analyzed using NURBS-based Galerkin FEA, as shown in [17]. The process is similar to standard FEA, only the set of basis functions used is different. Thus, the parametrization of the variable field takes the common formwhere denotes the degree of freedom evaluations of the variable field, in this case, the displacements. However, in IGA the set of basis-functions can be associated with a patch rather than with a single element. Note that,
Numerical examples
In this section, a set of numerical examples is presented to illustrate the capabilities of IGA for tire analysis. The simulations are performed by an in-house finite element code based on Fortran programming language. Results are validated by bench tests carried out for the physical tire at study. All experimental measurements are taken using micro-electro-mechanical system (MEMS) accelerometers due to their lightweight, robust, and cost-effective design [46].
It should be noted that the
Conclusions
A full framework for the simulation of rolling tires is presented, where closed unclamped knot vectors of higher order are introduced for the continuous description of the circumference of the tire. Basic concepts of the method are summarized with special focus at the kinematics for steady-state rolling within an ALE formulation.
The results obtained in numerical simulations show good agreement with experimental results and standard FEA simulations, with the advantage of retrieving smooth and
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors would like to acknowledge the financial support provided by the German Research Foundation (KA 1163/42), by the Scholarship Program for the Promotion of Early-Career Female Scientists of TU Dresden, and by the National Natural Science Foundation of China (No. 51761135124 and No. 11672148) throughout this research, as well as the support from tire manufacturer CEAT Limited, Mumbai, and ANSYS, Inc., Canonsburg, USA.
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