# Adaptive local surface refinement based on LR NURBS and its application to contact

## TL;DR

This paper introduces an adaptive local surface refinement method using LR NURBS, enabling precise modeling and efficient contact analysis of complex 3D geometries with reduced computational effort.

## Contribution

It develops a new LR NURBS-based refinement technique with an automatic remeshing algorithm for improved contact simulations in 3D solids and membranes.

## Key findings

- LR NURBS achieve high accuracy with less computation than uniform refinement.
- The method effectively models complex geometries in contact problems.
- Numerical examples demonstrate the efficiency and accuracy of the approach.

## Abstract

A novel adaptive local surface refinement technique based on Locally Refined Non-Uniform Rational B-Splines (LR NURBS) is presented. LR NURBS can model complex geometries exactly and are the rational extension of LR B-splines. The local representation of the parameter space overcomes the drawback of non-existent local refinement in standard NURBS-based isogeometric analysis. For a convenient embedding into general finite element code, the B\'ezier extraction operator for LR NURBS is formulated. An automatic remeshing technique is presented that allows adaptive local refinement and coarsening of LR NURBS. In this work, LR NURBS are applied to contact computations of 3D solids and membranes. For solids, LR NURBS-enriched finite elements are used to discretize the contact surfaces with LR NURBS finite elements, while the rest of the body is discretized by linear Lagrange finite elements. For membranes, the entire surface is discretized by LR NURBS. Various numerical examples are shown, and they demonstrate the benefit of using LR NURBS: Compared to uniform refinement, LR NURBS can achieve high accuracy at lower computational cost.