Three-dimensional varying-order NURBS discretization method for enhanced IGA of large deformation frictional contact problems

TL;DR

The paper introduces a three-dimensional varying-order NURBS discretization for isogeometric analysis of large-deformation frictional contact problems, enabling higher-order NURBS on the contact surface while preserving minimum-order NURBS in the bulk. The method integrates with a Gauss-point-to-surface (GPTS) contact formulation and penalty regularization, requiring only minor changes to standard IGA implementations. Across patch tests, Hertzian contact, frictional ironing, and twisting benchmarks, VO discretization delivers higher accuracy at fixed meshes and substantial reductions in degrees of freedom (DOFs) compared to uniform $N_2$ discretizations, e.g., reductions by factors of 2–4 with comparable or improved accuracy. These results demonstrate the significant practical impact of piecewise higher-order refinement localized to the contact surface, with potential extensions to self-contact and dynamic frictional problems.

Abstract

We introduce a varying-order (VO) NURBS discretization method to enhance the performance of the IGA technique for three-dimensional large deformation frictional contact problems. Based on the promising results obtained with the previous work on the 2D isogeometric contact analysis, the present work extends the capability of the method for tri-variate NURBS discretization. The proposed method enables independent employment of the user-defined higher-order NURBS for the discretization of the contact surface and the minimum order of NURBS for the remaining solid volume. Such a method provides the possibility to refine a NURBS solid with the controllable order elevation-based approach while preserving its volume parametrization at a fixed mesh. The advantages of the method are twofold. First, the higher-order NURBS for the evaluation of contact integral enhances the accuracy of the contact responses at a fixed mesh, hence fully exploiting the advantage of higher-order NURBS specifically for contact computations. Second, the minimum order of NURBS for the computations in the remaining volume considerably reduces the computational cost associated with the uniform order NURBS-based isogeometric contact analyses. The capabilities of the proposed method are demonstrated using various contact problems with or without considering friction between deformable solids. The results with the standard uniform order of NURBS-based discretization are also included to provide a comparative assessment. We show that to attain similar accuracy results, the VO NURBS discretization uses a much coarser mesh resolution than the standard NURBS-based discretization, leading to a major gain in computational efficiency for isogeometric contact analysis. The convergence study demonstrates the consistent performance of the method for efficient IGA of three-dimensional (3D) frictional contact problems.

Figures (20)

• Figure 1: Schematic illustration of the two-body contact in 3D.
• Figure 2: A schematic illustration of the three-dimensional frictional contact quantities for a given slave point $\bm{x}^{\mathrm{s}}$. The master contact surface $\Gamma^{m}_{\mathrm{c}}$ is parametrized by the coordinates $\xi^i$. The quantities computed at the projection point $\bar{\bm{\xi}}_{\mathrm{m}}$ are denoted with bar.
• Figure 3: Schematic illustration of the standard and VO-based NURBS discretization of an example tri-variate geometry. (a) Standard NURBS discretization with minimum $p_1 = p_2 = 2$ and $p_3 = 1$ order NURBS along the $\xi^1$, $\xi^2$, and $\xi^3$ parametric directions at a given mesh, where $\Xi^1 = \Xi^2 = [0,0,0,1,2,3,4,5,5,5]$, and $\Xi^3 = [0,0, 1, 2,2]$. (b) VO NURBS discretization, where the higher-order NURBS, i.e. N$_{2 \cdot 1}$ are used for the description of the contact surface in both the $\xi^1$ and $\xi^2$ parametric direction, and the minimum N$_2$ NURBS in its remaining volume.
• Figure 4: Patch test: setup including the loading and coarsest mesh used for the cubes.
• Figure 5: Patch test: Error in the vertical stress for GPTS contact algorithm over different meshes but at a fix number of quadrature points (column-wise), and with varying the number of quadrature points at a fixed mesh (row-wise).