A new method for T-spline parameterization of complex 2D geometries

We present a new strategy, based on the idea of the meccano method and a novel T-mesh optimization procedure, to construct a T-spline parameterization of 2D geometries for the application of isogeometric analysis. The proposed method only demands a boundary representation of the geometry as input data. The algorithm obtains, as a result, high quality parametric transformation between 2D objects and the parametric domain, the unit square. First, we define a parametric mapping between the input boundary of the object and the boundary of the parametric domain. Then, we build a T-mesh adapted to the geometric singularities of the domain to preserve the features of the object boundary with a desired tolerance. The key of the method lies in defining an isomorphic transformation between the parametric and physical T-mesh finding the optimal position of the interior nodes by applying a new T-mesh untangling and smoothing procedure. Bivariate T-spline representation is calculated by imposing the interpolation conditions on points sited both in the interior and on the boundary of the geometry. The efficacy of the proposed technique is shown in several examples. Also we present some results of the application of isogeometric analysis in a geometry parameterized with this technique.

[1]  Régis Duvigneau,et al.  Variational Harmonic Method for Parameterization of Computational Domain in 2D Isogeometric Analysis , 2011, 2011 12th International Conference on Computer-Aided Design and Computer Graphics.

[2]  B. Simeon,et al.  A hierarchical approach to adaptive local refinement in isogeometric analysis , 2011 .

[3]  John A. Evans,et al.  Isogeometric analysis using T-splines , 2010 .

[4]  Hong Qin,et al.  Generalized PolyCube Trivariate Splines , 2010, 2010 Shape Modeling International Conference.

[5]  Thomas J. R. Hughes,et al.  Isogeometric Analysis: Toward Integration of CAD and FEA , 2009 .

[6]  L. Freitag,et al.  Tetrahedral mesh improvement via optimization of the element condition number , 2002 .

[7]  J. M. Cascón,et al.  A new approach to solid modeling with trivariate T-splines based on mesh optimization , 2011 .

[8]  Rafael Montenegro,et al.  Simultaneous aligning and smoothing of surface triangulations , 2010, Engineering with Computers.

[9]  Patrick M. Knupp,et al.  Algebraic Mesh Quality Metrics , 2001, SIAM J. Sci. Comput..

[10]  T. Hughes,et al.  Solid T-spline construction from boundary representations for genus-zero geometry , 2012 .

[11]  Régis Duvigneau,et al.  Parameterization of computational domain in isogeometric analysis: Methods and comparison , 2011 .

[12]  Elaine Cohen,et al.  Volumetric parameterization and trivariate B-spline fitting using harmonic functions , 2009, Comput. Aided Geom. Des..

[13]  David L. Marcum,et al.  Proceedings of the 16th International Meshing Roundtable , 2008 .

[14]  F. Cirak,et al.  A subdivision-based implementation of the hierarchical b-spline finite element method , 2013 .

[15]  Rafael Montenegro,et al.  The Meccano Method for Automatic Tetrahedral Mesh Generation of Complex Genus-Zero Solids , 2009, IMR.

[16]  Gerald E. Farin,et al.  Discrete Coons patches , 1999, Comput. Aided Geom. Des..

[17]  Springer-Verlag London,et al.  The meccano method for isogeometric solid modeling and applications , 2014 .

[18]  J. M. González-Yuste,et al.  Simultaneous untangling and smoothing of tetrahedral meshes , 2003 .

[19]  Thomas J. R. Hughes,et al.  Trivariate solid T-spline construction from boundary triangulations with arbitrary genus topology , 2012, Comput. Aided Des..

[20]  L. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communications.

[21]  José María Escobar,et al.  The Meccano Method for Automatic Three-Dimensional Triangulation and Volume Parametrization of Complex Solids , 2010 .

[22]  T. Hughes,et al.  Local refinement of analysis-suitable T-splines , 2012 .

[23]  Victor M. Calo,et al.  Isogeometric Analysis: Toward Unification of Computer Aided Design and Finite Element Analysis , 2008 .

[24]  John A. Evans,et al.  An Isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces , 2012 .

[25]  L. Freitag,et al.  Local optimization-based simplicial mesh untangling and improvement. , 2000 .

[26]  Hanan Samet,et al.  Foundations of multidimensional and metric data structures , 2006, Morgan Kaufmann series in data management systems.

[27]  J. M. Cascón,et al.  An automatic strategy for adaptive tetrahedral mesh generation , 2009 .

[28]  Rafael Montenegro,et al.  A New MeccanoTechnique for Adaptive 3-D Triangulations , 2007, IMR.

[29]  Rüdiger Verfürth,et al.  A posteriori error estimation and adaptive mesh-refinement techniques , 1994 .

[30]  Hong Qin,et al.  Harmonic volumetric mapping for solid modeling applications , 2007, Symposium on Solid and Physical Modeling.

[31]  Patrick M. Knupp.  A method for hexahedral mesh shape optimization , 2003 .

[32]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[33]  Rafael Montenegro,et al.  Smoothing and local refinement techniques for improving tetrahedral mesh quality , 2005 .

[34]  Rafael Montenegro,et al.  The meccano method for isogeometric solid modeling and applications , 2014, Engineering with Computers.