# Solving the Laplace-Beltrami Equation on Curved Two-dimensional Manifolds using the Finite Element Method with Catmull-Clark Subdivision Surfaces

, , and (2019) Solving the Laplace-Beltrami Equation on Curved Two-dimensional Manifolds using the Finite Element Method with Catmull-Clark Subdivision Surfaces. VII International Conference on Isogeometric Analysis, Munich, 18-20 September 2019.

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## Abstract

The early works on isogeometric analysis focused on geometries modelled through Non-Uniform Rational B-Splines (NURBS). However, a NURBS sur- face is a tensor product surface generated by two NURBS curves. This has limitations in modelling geometries with complex topologies. Catmull-Clark subdivision surfaces are derived as tensor-products of Catmull-Clark subdivi- sion curves, but allowing ”extraordinary vertices”. This ensure the Catmull- Clark subdivision surfaces can be used for modelling complex geometries with arbitrary topologies. The present work proposes and analyses an isogeometric approach for solving the Laplace-Beltrami equation on a two-dimensional manifold embed- ded in three-dimensional space using the finite element method with Catmull- Clark subdivision surfaces. The Catmull-Clark subdivision bases are used to discretise both the geometry and physical fields. A fitting method is also proposed to generate control meshes to approximate any given geometries with Catmull-Clark subdivision surfaces. Sample points are chosen from the given surface and least square fitting is used to obtain an optimal control mesh for the surface. The Catmull-Clark subdivision method is also com- pared to the conventional finite element method. Subdivision surfaces with no extraordinary vertices have the optimal p + 1 convergence rate. However the extraordinary vertices introduce relatively large errors in the analysis which decreases the convergence rate. A comparative study shows the ef- fects of the number and valences of the extraordinary vertices. An adaptive quadrature scheme and other remedies are also implemented to reduce the error.

Item Type: Conference or Workshop Item Published No McBride, Dr Andrew and Saxena, Dr Prashant and Steinmann, Professor Paul and Liu, Dr Zhaowei Q Science > QA MathematicsT Technology > TA Engineering (General). Civil engineering (General) College of Science and Engineering > School of Engineering > Infrastructure and Environment

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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
3001290Strategic Support Package: Engineering of Active Materials by Multiscale/Multiphysics Computational MechanicsChristopher PearceEngineering and Physical Sciences Research Council (EPSRC)EP/R008531/1ENG - Infrastructure & Environment