Linear Independence of T-Spline Blending Functions of Degree One for Isogeometric Analysis

Aizeng Wang,Ling Li,Zhanchuan Cai,Wei Wang,Gang Zhao,Feng Xiao,Xiaochen Du

doi:10.3390/math9121346

Aizeng Wang, Ling Li + Show 5 more

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https://doi.org/10.3390/math9121346

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Abstract

Linear independence of the blending functions is a necessary requirement for T-spline in isogeometric analysis. The main work in this paper focuses on the analysis about T-splines of degree one, we demonstrate that all the blending functions of such T-spline of degree one are linearly independent. The advantage owned by one degree T-spline is that it can avoid the problem of judging whether the model is analysis-suitable or not, especially for occasions that need a quick response from the analysis results. This may provide a new way of using T-spline for a CAD and CAE integrating scenario, since one degree T-spline still guarantees the topology flexibility and is compatible with the spline-based modeling system. In addition, we compare the numerical approximations of isogeometric analysis and finite element analysis, and the experiment indicates that isogeometric analysis using T-spline of degree one can reach a comparable result with classical method.

Highlights

Linear independence of the blending functions is a necessary requirement for T-spline in isogeometric analysis
Finite element analysis (FEA), which has a wide range of applications in industry, is a classic numerical simulation, but during the analysis progress the spline model generated by computer-aided design (CAD) needs to discretize into a computer-aided engineering (CAE) mesh model
This result provides a mathematical foundation of one degree T-splines for isogeometric analysis, because it can avoid the problem of judging whether it is analysis-suitable or not. This may provide a new way of using T-spline for a CAD and CAE integrating scenario, since one degree

Summary

Introduction

Finite element analysis (FEA), which has a wide range of applications in industry, is a classic numerical simulation, but during the analysis progress the spline model generated by computer-aided design (CAD) needs to discretize into a computer-aided engineering (CAE) mesh model. The most essential difference is that the linear functions in the literature [21] are used for finite element analysis, and our T-spline of degree one is applied to IGA The main results of this paper can be summarized as follows: Firstly, we prove that the blending functions for any T-spline of degree one are linearly independent This result provides a mathematical foundation of one degree T-splines for isogeometric analysis, because it can avoid the problem of judging whether it is analysis-suitable or not. At the end of this paper, we give a brief conclusion and future work

B-Spline Functions of Degree One

Blending Functions of Degree One

Examples of One Degree T-Splines

Isogeometric Analysis Example

Conclusions and Future Work