Hierarchically refined isogeometric analysis of trimmed shells

This work focuses on the study of several computational challenges arising when trimmed surfaces are directly employed for the isogeometric analysis of Kirchhoff-Love shells. To cope with these issues and to resolve mechanical and/or geometrical features of interest, we exploit the local refinement capabilities of hierarchical B-Splines. In particular, we show numerically that local refinement is suited to effectively impose Dirichlet-type boundary conditions in a weak sense, where this easily allows to overcome the issue of over-constraining of trimmed elements. Moreover, we highlight how refinement can alleviate the spurious coupling stemming from disjoint supports of basis functions in the presence of “small” trimmed geometrical features such as thin holes. These phenomena are particularly pronounced in surface models defined by complex trimming patterns and with details at different scales, where we show through several numerical examples the benefits and computational efficiency of the proposed methodology.

[1]  Ernst Rank,et al.  Finite cell method , 2007 .

[2]  Alessandro Reali,et al.  An isogeometric collocation approach for Bernoulli–Euler beams and Kirchhoff plates , 2015 .

[3]  Klaus Höllig,et al.  Finite Element Analysis with B-Splines: Weighted and Isogeometric Methods , 2010, Curves and Surfaces.

[4]  Ernst Rank,et al.  The finite cell method for three-dimensional problems of solid mechanics , 2008 .

[5]  Thomas J. R. Hughes,et al.  Nonlinear Isogeometric Analysis , 2009 .

[6]  Thomas J. R. Hughes,et al.  Variationally consistent isogeometric analysis of trimmed thin shells at finite deformations, based on the STEP exchange format , 2018, Computer Methods in Applied Mechanics and Engineering.

[7]  John A. Evans,et al.  Isogeometric Analysis , 2010 .

[8]  Ernst Rank,et al.  Normal contact with high order finite elements and a fictitious contact material , 2015, Comput. Math. Appl..

[9]  Benjamin Marussig,et al.  A Review of Trimming in Isogeometric Analysis: Challenges, Data Exchange and Simulation Aspects , 2017, Archives of Computational Methods in Engineering.

[10]  Christoph Lehrenfeld,et al.  A note on the stability parameter in Nitsche's method for unfitted boundary value problems , 2017, Comput. Math. Appl..

[11]  Ernst Rank,et al.  The Finite Cell Method for linear thermoelasticity , 2012, Comput. Math. Appl..

[12]  David R. Forsey,et al.  Hierarchical B-spline refinement , 1988, SIGGRAPH.

[13]  T. Hughes,et al.  Isogeometric analysis of the Cahn–Hilliard phase-field model , 2008 .

[14]  Ernst Rank,et al.  The p-Version of the Finite Element and Finite Cell Methods , 2017 .

[15]  Alessandro Reali,et al.  A simplified Kirchhoff–Love large deformation model for elastic shells and its effective isogeometric formulation , 2019, Computer Methods in Applied Mechanics and Engineering.

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

[17]  Ernst Rank,et al.  Finite Cell Method: High-Order Structural Dynamics for Complex Geometries , 2015 .

[18]  Günther Greiner,et al.  Interpolating and approximating scattered 3D-data with hierarchical tensor product B-splines , 2010 .

[19]  Massimiliano Martinelli,et al.  Isogeometric Analysis on V-reps: First results , 2019, Computer Methods in Applied Mechanics and Engineering.

[20]  Alessandro Reali,et al.  Phase-field description of brittle fracture in plates and shells , 2016 .

[21]  Alessandro Reali,et al.  Suitably graded THB-spline refinement and coarsening: Towards an adaptive isogeometric analysis of additive manufacturing processes , 2018, Computer Methods in Applied Mechanics and Engineering.

[22]  Ernst Rank,et al.  Finite cell method compared to h-version finite element method for elasto-plastic problems , 2014 .

[23]  K. Höllig.  Finite element methods with B-splines , 1987 .

[24]  Roland Wüchner,et al.  Isogeometric shell analysis with Kirchhoff–Love elements , 2009 .

[25]  Ming-Chen Hsu,et al.  Penalty coupling of non-matching isogeometric Kirchhoff–Love shell patches with application to composite wind turbine blades , 2019, Computer Methods in Applied Mechanics and Engineering.

[26]  Graham F. Carey,et al.  Boundary penalty techniques , 1982 .

[27]  Ernst Rank,et al.  From geometric design to numerical analysis: A direct approach using the Finite Cell Method on Constructive Solid Geometry , 2017, Comput. Math. Appl..

[28]  Lei Liu,et al.  Weighted T-splines with application in reparameterizing trimmed NURBS surfaces , 2015 .

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

[30]  Giancarlo Sangalli,et al.  Mathematical analysis of variational isogeometric methods* , 2014, Acta Numerica.

[31]  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 .

[32]  I. Babuska.  The Finite Element Method with Penalty , 1973 .

[33]  Annalisa Buffa,et al.  Adaptive isogeometric analysis on two-dimensional trimmed domains based on a hierarchical approach , 2019, ArXiv.

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

[35]  Hector Gomez,et al.  Arbitrary-degree T-splines for isogeometric analysis of fully nonlinear Kirchhoff-Love shells , 2017, Comput. Aided Des..

[36]  Tom Lyche,et al.  Polynomial splines over locally refined box-partitions , 2013, Comput. Aided Geom. Des..

[37]  Ernst Rank,et al.  Efficient and accurate numerical quadrature for immersed boundary methods , 2015, Advanced Modeling and Simulation in Engineering Sciences.

[38]  Ernst Rank,et al.  Smart octrees: Accurately integrating discontinuous functions in 3D , 2016 .

[39]  Y. Bazilevs,et al.  Small and large deformation analysis with the p- and B-spline versions of the Finite Cell Method , 2012 .

[40]  Ernst Rank,et al.  Shell Finite Cell Method: A high order fictitious domain approach for thin-walled structures , 2011 .

[41]  Ernst Rank,et al.  The finite cell method for bone simulations: verification and validation , 2012, Biomechanics and modeling in mechanobiology.

[42]  Roland Wüchner,et al.  Isogeometric analysis of trimmed NURBS geometries , 2012 .

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

[44]  Bert Jüttler,et al.  THB-splines: An effective mathematical technology for adaptive refinement in geometric design and isogeometric analysis , 2016 .

[45]  Robert Schmidt,et al.  Realization of an integrated structural design process: analysis-suitable geometric modelling and isogeometric analysis , 2010, Comput. Vis. Sci..

[46]  Rafael Vázquez,et al.  Algorithms for the implementation of adaptive isogeometric methods using hierarchical splines , 2016 .

[47]  Yujie Guo,et al.  Weak Dirichlet boundary conditions for trimmed thin isogeometric shells , 2015, Comput. Math. Appl..

[48]  Roland Wüchner,et al.  Analysis in computer aided design: Nonlinear isogeometric B-Rep analysis of shell structures , 2015 .

[49]  Alessandro Reali,et al.  Multi-level Bézier extraction for hierarchical local refinement of Isogeometric Analysis , 2017 .

[50]  Michael C. H. Wu,et al.  Isogeometric Kirchhoff–Love shell formulations for general hyperelastic materials , 2015 .

[51]  Thomas J. R. Hughes,et al.  Hierarchically refined and coarsened splines for moving interface problems, with particular application to phase-field models of prostate tumor growth , 2017 .

[52]  Thomas J. R. Hughes,et al.  Volumetric T-spline construction using Boolean operations , 2014, Engineering with Computers.

[53]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[54]  Andreas Keller,et al.  Finite Element Approximation with Hierarchical B-Splines , 2014, Curves and Surfaces.