Planar multi-patch domain parameterization for isogeometric analysis based on evolution of fat skeleton

Abstract In this paper, we propose a new algorithm for computing planar multi-patch domain parameterizations which are suitable for consequent numerical computations based on isogeometric analysis. Firstly, a so-called fat skeleton of the given domain, represented by a multi-patch NURBS parameterization, is constructed, which determines the topology of a multi-patch NURBS parameterization suitable for the given domain. Then, we formulate an evolution process which successively transforms the initial shape represented by the fat skeleton to the target shape represented by the given domain. Moreover, a multi-patch optimization is applied in every time iteration to make the multi-patch NURBS parameterization as uniform and orthogonal as possible. The method is constructed such that it is fully automatic and it solves the biggest problems of the domain parameterization problem, i.e., it automatically determines a suitable topology of a multi-patch NURBS parameterization for the given domain and it also automatically solves the boundary correspondence problem. The functionality of the proposed method is demonstrated on several examples.

[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]  Bert Jüttler,et al.  Planar domain parameterization with THB-splines , 2015, Comput. Aided Geom. Des..

[3]  Bert Jüttler,et al.  Matrix Generation in Isogeometric Analysis by Low Rank Tensor Approximation , 2014, Curves and Surfaces.

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

[5]  Victor M. Calo,et al.  Gauss-Galerkin quadrature rules for quadratic and cubic spline spaces and their application to isogeometric analysis , 2016, Comput. Aided Des..

[6]  Giancarlo Sangalli,et al.  Fast formation and assembly of finite element matrices with application to isogeometric linear elasticity , 2019, Computer Methods in Applied Mechanics and Engineering.

[7]  Mark de Berg,et al.  Computational geometry: algorithms and applications , 1997 .

[8]  Sung-Kie Youn,et al.  Shape optimization and its extension to topological design based on isogeometric analysis , 2010 .

[9]  Bert Jüttler,et al.  Parameterization of Contractible Domains Using Sequences of Harmonic Maps , 2010, Curves and Surfaces.

[10]  Anders Logg,et al.  The FEniCS Project Version 1.5 , 2015 .

[11]  Bert Jüttler,et al.  Low rank tensor methods in Galerkin-based isogeometric analysis , 2017 .

[12]  Bohumír Bastl,et al.  PARAMETERIZATIONS OF GENERALIZED NURBS VOLUMES OF REVOLUTION , 2012 .

[13]  Régis Duvigneau,et al.  Constructing analysis-suitable parameterization of computational domain from CAD boundary by variational harmonic method , 2013, J. Comput. Phys..

[14]  Ann E. Jeffers,et al.  Isogeometric analysis of laminated composite and functionally graded sandwich plates based on a layerwise displacement theory , 2017 .

[15]  Bert Jüttler,et al.  Planar multi-patch domain parameterization via patch adjacency graphs , 2017, Comput. Aided Des..

[16]  Jiansong Deng,et al.  Two-dimensional domain decomposition based on skeleton computation for parameterization and isogeometric analysis , 2015 .

[17]  Victor M. Calo,et al.  The cost of continuity: A study of the performance of isogeometric finite elements using direct solvers , 2012 .

[18]  Thomas J. R. Hughes,et al.  Isogeometric shell analysis: The Reissner-Mindlin shell , 2010 .

[19]  Stefan Turek,et al.  Isogeometric Analysis of the Navier-Stokes equations with Taylor-Hood B-spline elements , 2015, Appl. Math. Comput..

[20]  Giancarlo Sangalli,et al.  Approximation estimates for isogeometric spaces in multipatch geometries , 2015 .

[21]  Alan E. Middleditch,et al.  Stable Computation of the 2D Medial Axis Transform , 1998, Int. J. Comput. Geom. Appl..

[22]  Martin Aigner,et al.  Swept Volume Parameterization for Isogeometric Analysis , 2009, IMA Conference on the Mathematics of Surfaces.

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

[24]  Seokchan Kim,et al.  A finite element method for computing accurate solutions for Poisson equations with corner singularities using the stress intensity factor , 2016, Comput. Math. Appl..

[25]  Franz Aurenhammer,et al.  Medial axis computation for planar free-form shapes , 2009, Comput. Aided Des..

[26]  Falai Chen,et al.  Low-rank parameterization of planar domains for isogeometric analysis , 2018, Comput. Aided Geom. Des..

[27]  Bert Jüttler,et al.  Bounding the influence of domain parameterization and knot spacing on numerical stability in Isogeometric Analysis , 2014 .

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

[29]  Bohumír Bastl,et al.  Automatic generators of multi-patch B-spline meshes of blade cascades and their comparison , 2020, Math. Comput. Simul..

[30]  Alfio Quarteroni,et al.  Isogeometric Analysis and error estimates for high order partial differential equations in Fluid Dynamics , 2014 .

[31]  Gabriella Sanniti di Baja,et al.  Euclidean skeleton via centre-of-maximal-disc extraction , 1993, Image Vis. Comput..

[32]  Hendrik Speleers,et al.  THB-splines: The truncated basis for hierarchical splines , 2012, Comput. Aided Geom. Des..

[33]  Bert Jüttler,et al.  H2 regularity properties of singular parameterizations in isogeometric analysis , 2012, Graph. Model..

[34]  Xiaoping Qian,et al.  Full analytical sensitivities in NURBS based isogeometric shape optimization , 2010 .

[35]  Falai Chen,et al.  Computing IGA-suitable planar parameterizations by PolySquare-enhanced domain partition , 2018, Comput. Aided Geom. Des..

[36]  D. T. Lee,et al.  Medial Axis Transformation of a Planar Shape , 1982, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[37]  Falai Chen,et al.  Planar domain parameterization for isogeometric analysis based on Teichmüller mapping , 2016 .

[38]  Hwan Pyo Moon,et al.  MATHEMATICAL THEORY OF MEDIAL AXIS TRANSFORM , 1997 .

[39]  Falai Chen,et al.  Boundary correspondence of planar domains for isogeometric analysis based on optimal mass transport , 2019, Comput. Aided Des..

[40]  T. Hughes,et al.  Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows , 2007 .

[41]  Bert Jüttler,et al.  Existence of stiffness matrix integrals for singularly parameterized domains in isogeometric analysis , 2011 .

[42]  Bert Jüttler,et al.  Geometry + Simulation Modules: Implementing Isogeometric Analysis , 2014 .

[43]  Alfio Quarteroni,et al.  Multipatch Isogeometric Analysis for electrophysiology: Simulation in a human heart , 2021 .

[44]  Paul G. Tucker,et al.  Differential equation-based wall distance computation for DES and RANS , 2003 .

[45]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[46]  Jens Gravesen,et al.  Isogeometric Shape Optimization of Vibrating Membranes , 2011 .

[47]  B. Gurumoorthy,et al.  Constructing medial axis transform of planar domains with curved boundaries , 2003, Comput. Aided Des..

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

[49]  Paul A. Beata,et al.  A mixed isogeometric analysis and control volume approach for heat transfer analysis of nonuniformly heated plates , 2019, Numerical Heat Transfer, Part B: Fundamentals.

[50]  Alessandro Reali,et al.  Duality and unified analysis of discrete approximations in structural dynamics and wave propagation : Comparison of p-method finite elements with k-method NURBS , 2008 .

[51]  Marek Brandner,et al.  Isogeometric analysis for turbulent flow , 2016, Math. Comput. Simul..

[52]  Marek Brandner,et al.  IgA-Based Solver for turbulence modelling on multipatch geometries , 2017, Adv. Eng. Softw..

[53]  Jens Gravesen,et al.  Planar Parametrization in Isogeometric Analysis , 2012, MMCS.

[54]  Bert Jüttler,et al.  Low rank interpolation of boundary spline curves , 2017, Comput. Aided Geom. Des..

[55]  Tom Lyche,et al.  Analysis-aware modeling: Understanding quality considerations in modeling for isogeometric analysis , 2010 .