Quasi-harmonic Bézier approximation of minimal surfaces for finding forms of structural membranes

Gang Xu, Timon Rabczuk, Erhan Güler, Qing Wu, Kin Chuen Hui, Guozhao Wang

Research output: Contribution to journalArticlepeer-review

25 Citations (Scopus)


Numerical approximation of minimal surface is an important problem in form-finding of structural membranes. In this paper, we present a novel approach to construct minimal surface from a given boundary by quasi-harmonic Bézier approximation. A new energy functional called quasi-harmonic energy functional is proposed as the objective function to obtain the quasi-harmonic Bézier surface from given boundaries. The quasi-harmonic mask is also proposed to generate approximate minimal surfaces by solving a sparse linear system. We propose a framework to construct multi-patch quasi-harmonic Bézier approximation from N-sided boundary curves. The efficiency of the proposed methods is illustrated by several modeling examples.

Original languageEnglish
Pages (from-to)55-63
Number of pages9
JournalComputers and Structures
Publication statusPublished - 2015 Dec 1
Externally publishedYes

Bibliographical note

Funding Information:
Dr. Gang Xu is partially supported by the National Nature Science Foundation of China (Nos. 61472111 and 61272300 ), the Zhejiang Provincial Natural Science Foundation of China under Grant No. LR16F020003 , and the Open Project Program of the State Key Lab of CAD & CG(A1406) . Prof. Timon Rabczuk is supported by the European Union Initial Training Network (ITN) INSIST from the Framework Programme 7 (No. 289361 ) “Integrating Numerical Simulation and Geometric Design Technology.” Prof. K.C Hui is supported by the Direct Grant from the Chinese University of Hong Kong (Project No. 2050492 ).


  • Bézier approximation
  • Form finding
  • Minimal surfaces
  • Multi-patch structure
  • Plateau-Bézier problem
  • Quasi-harmonic method

ASJC Scopus subject areas

  • Civil and Structural Engineering
  • Modelling and Simulation
  • General Materials Science
  • Mechanical Engineering
  • Computer Science Applications


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