Isogeometric and immersogeometric analysis of incompressible flow problems

We study the application of the Isogeometric Finite Cell Method (IGAFCM) to mixed formulations in the context of the Stokes problem. We investigate the performance of the IGA-FCM when utilizing some isogeometric mixed finite elements, namely: Taylor-Hood, Sub-grid, Raviart-Thomas, and Nédélec elements. These element families have been demonstrated to perform well in the case of conforming meshes, but their applicability in the cut-cell context is still unclear. Dirichlet boundary conditions are imposed byNitsche’s method. Numerical test problems are performed, with a detailed study of the discrete inf-sup stability constants and of the convergence behavior under uniform mesh refinement. Reproduced from: T. Hoang, C.V. Verhoosel, F. Auricchio, E.H. van Brummelen, A. Reali, Mixed Isogeometric Finite Cell Methods for the Stokes problem, Computer Methods in Applied Mechanics and Engineering, Volume 316, 2017, Pages 400-423, DOI: 10.1016/j.cma.2016.07.027

[1]  Jindřich Nečas,et al.  Sur une méthode pour résoudre les équations aux dérivées partielles du type elliptique, voisine de la variationnelle , 1961 .

[2]  J. Nitsche.  Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind , 1971 .

[3]  I. Babuska.  The finite element method with Lagrangian multipliers , 1973 .

[4]  Marcel Crochet,et al.  On the stream function‐vorticity finite element solutions of Navier‐Stokes equations , 1978 .

[5]  D. Malkus.  Eigenproblems associated with the discrete LBB condition for incompressible finite elements , 1981 .

[6]  Ivo Babuška,et al.  The post-processing approach in the finite element method—part 1: Calculation of displacements, stresses and other higher derivatives of the displacements , 1984 .

[7]  F. Brezzi,et al.  On the Stabilization of Finite Element Approximations of the Stokes Equations , 1984 .

[8]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

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

[10]  Thomas J. R. Hughes,et al.  The Stokes problem with various well-posed boundary conditions - Symmetric formulations that converge for all velocity/pressure spaces , 1987 .

[11]  Tayfun E. Tezduyar,et al.  Petrov‐Galerkin methods on multiply connected domains for the vorticity‐stream function formulation of the incompressible Navier‐Stokes equations , 1988 .

[12]  Jim Douglas,et al.  An absolutely stabilized finite element method for the stokes problem , 1989 .

[13]  Tayfan E. Tezduyar,et al.  Stabilized Finite Element Formulations for Incompressible Flow Computations , 1991 .

[14]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[15]  C. Ross Ethier,et al.  Exact fully 3D Navier–Stokes solutions for benchmarking , 1994 .

[16]  G. Carey,et al.  High‐order compact scheme for the steady stream‐function vorticity equations , 1995 .

[17]  T. Hughes.  Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods , 1995 .

[18]  R. Rannacher,et al.  Benchmark Computations of Laminar Flow Around a Cylinder , 1996 .

[19]  I. Babuska,et al.  The partition of unity finite element method: Basic theory and applications , 1996 .

[20]  Les A. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communication.

[21]  Endre Süli,et al.  Adaptive error control for finite element approximations of the lift and drag coefficients in viscous flow , 1997 .

[22]  T. Hughes,et al.  The variational multiscale method—a paradigm for computational mechanics , 1998 .

[23]  W. Shyy,et al.  Regular Article: An Accurate Cartesian Grid Method for Viscous Incompressible Flows with Complex Immersed Boundaries , 1999 .

[24]  Ted Belytschko,et al.  A finite element method for crack growth without remeshing , 1999 .

[25]  Gerald Farin,et al.  NURBS: From Projective Geometry to Practical Use , 1999 .

[26]  Ramon Codina,et al.  Stabilized finite element method for the transient Navier–Stokes equations based on a pressure gradient projection , 2000 .

[27]  R. Codina.  Stabilization of incompressibility and convection through orthogonal sub-scales in finite element methods , 2000 .

[28]  Gershon Elber,et al.  Geometric modeling with splines - an introduction , 2001 .

[29]  R. Codina.  A stabilized finite element method for generalized stationary incompressible flows , 2001 .

[30]  R. Codina.  Pressure Stability in Fractional Step Finite Element Methods for Incompressible Flows , 2001 .

[31]  Thomas J. R. Hughes,et al.  Large eddy simulation of turbulent channel flows by the variational multiscale method , 2001 .

[32]  Jungwoo Kim,et al.  An immersed-boundary finite-volume method for simulations of flow in complex geometries , 2001 .

[33]  R. Hiptmair.  Finite elements in computational electromagnetism , 2002, Acta Numerica.

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

[35]  F. Dubois,et al.  Vorticity–velocity-pressure and stream function-vorticity formulations for the Stokes problem , 2003 .

[36]  L. Heltai,et al.  A finite element approach to the immersed boundary method , 2003 .

[37]  Randall J. LeVeque,et al.  An Immersed Interface Method for Incompressible Navier-Stokes Equations , 2003, SIAM J. Sci. Comput..

[38]  C. Dohrmann,et al.  A stabilized finite element method for the Stokes problem based on polynomial pressure projections , 2004 .

[39]  Volker John,et al.  Reference values for drag and lift of a two‐dimensional time‐dependent flow around a cylinder , 2004 .

[40]  Antonio Huerta,et al.  Imposing essential boundary conditions in mesh-free methods , 2004 .

[41]  Gianluca Iaccarino,et al.  IMMERSED BOUNDARY METHODS , 2005 .

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

[43]  T. Hughes,et al.  ISOGEOMETRIC ANALYSIS: APPROXIMATION, STABILITY AND ERROR ESTIMATES FOR h-REFINED MESHES , 2006 .

[44]  T. Hughes,et al.  Isogeometric Fluid–structure Interaction Analysis with Applications to Arterial Blood Flow , 2006 .

[45]  P. Hansbo,et al.  Mathematical Modelling and Numerical Analysis Edge Stabilization for the Generalized Stokes Problem: a Continuous Interior Penalty Method , 2022 .

[46]  Alessandro Reali,et al.  Isogeometric Analysis of Structural Vibrations , 2006 .

[47]  D. Arnold,et al.  Finite element exterior calculus, homological techniques, and applications , 2006, Acta Numerica.

[48]  Miguel A. Fernández,et al.  Continuous Interior Penalty Finite Element Method for Oseen's Equations , 2006, SIAM J. Numer. Anal..

[49]  Erik Burman,et al.  Local Projection Stabilization for the Oseen Problem and its Interpretation as a Variational Multiscale Method , 2006, SIAM J. Numer. Anal..

[50]  Alessandro Reali,et al.  Studies of Refinement and Continuity in Isogeometric Structural Analysis (Preprint) , 2007 .

[51]  G. Sangalli,et al.  A fully ''locking-free'' isogeometric approach for plane linear elasticity problems: A stream function formulation , 2007 .

[52]  Miguel A. Fernández,et al.  Continuous interior penalty finite element method for the time-dependent Navier–Stokes equations: space discretization and convergence , 2007, Numerische Mathematik.

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

[54]  Ronald Fedkiw,et al.  The immersed interface method. Numerical solutions of PDEs involving interfaces and irregular domains , 2007, Math. Comput..

[55]  Rainald Löhner,et al.  Adaptive embedded and immersed unstructured grid techniques , 2008 .

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

[57]  Seung-Hyun Ha,et al.  Isogeometric shape design optimization: exact geometry and enhanced sensitivity , 2009 .

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

[59]  T. Hughes,et al.  B¯ and F¯ projection methods for nearly incompressible linear and non-linear elasticity and plasticity using higher-order NURBS elements , 2008 .

[60]  Elias Balaras,et al.  A strongly coupled, embedded-boundary method for fluid–structure interactions of elastically mounted rigid bodies , 2008 .

[61]  T. Hughes,et al.  Isogeometric fluid-structure interaction: theory, algorithms, and computations , 2008 .

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

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

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

[65]  Malte Braack,et al.  Finite elements with local projection stabilization for incompressible flow problems , 2009 .

[66]  Thomas J. R. Hughes,et al.  Patient-specific isogeometric fluid–structure interaction analysis of thoracic aortic blood flow due to implantation of the Jarvik 2000 left ventricular assist device , 2009 .

[67]  John A. Evans,et al.  Robustness of isogeometric structural discretizations under severe mesh distortion , 2010 .

[68]  Yuri Bazilevs,et al.  The bending strip method for isogeometric analysis of Kirchhoff–Love shell structures comprised of multiple patches , 2010 .

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

[70]  D. Arnold,et al.  Finite element exterior calculus: From hodge theory to numerical stability , 2009, 0906.4325.

[71]  Alfredo Pinelli,et al.  Immersed-boundary methods for general finite-difference and finite-volume Navier-Stokes solvers , 2010, J. Comput. Phys..

[72]  B. Simeon,et al.  Adaptive isogeometric analysis by local h-refinement with T-splines , 2010 .

[73]  I. Akkerman,et al.  Large eddy simulation of turbulent Taylor-Couette flow using isogeometric analysis and the residual-based variational multiscale method , 2010, J. Comput. Phys..

[74]  T. Hughes,et al.  ISOGEOMETRIC COLLOCATION METHODS , 2010 .

[75]  J. Dolbow,et al.  Imposing Dirichlet boundary conditions with Nitsche's method and spline‐based finite elements , 2010 .

[76]  A. Peirce.  Computer Methods in Applied Mechanics and Engineering , 2010 .

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

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

[79]  T. Belytschko,et al.  The extended/generalized finite element method: An overview of the method and its applications , 2010 .

[80]  Peter Wriggers,et al.  A large deformation frictional contact formulation using NURBS‐based isogeometric analysis , 2011 .

[81]  Giancarlo Sangalli,et al.  IsoGeometric Analysis: Stable elements for the 2D Stokes equation , 2011 .

[82]  Thomas J. R. Hughes,et al.  A large deformation, rotation-free, isogeometric shell , 2011 .

[83]  Kang Li,et al.  Isogeometric analysis and shape optimization via boundary integral , 2011, Comput. Aided Des..

[84]  T. Belytschko,et al.  X‐FEM in isogeometric analysis for linear fracture mechanics , 2011 .

[85]  Thomas J. R. Hughes,et al.  An isogeometric approach to cohesive zone modeling , 2011 .

[86]  Peter Wriggers,et al.  Contact treatment in isogeometric analysis with NURBS , 2011 .

[87]  Yuri Bazilevs,et al.  Rotation free isogeometric thin shell analysis using PHT-splines , 2011 .

[88]  Giancarlo Sangalli,et al.  Isogeometric Discrete Differential Forms in Three Dimensions , 2011, SIAM J. Numer. Anal..

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

[90]  John A. Evans,et al.  Isogeometric finite element data structures based on Bézier extraction of NURBS , 2011 .

[91]  Brummelen van Eh,et al.  Flux evaluation in primal and dual boundary-coupled problems , 2011 .

[92]  D. Schillinger,et al.  An unfitted hp-adaptive finite element method based on hierarchical B-splines for interface problems of complex geometry , 2011 .

[93]  Thomas J. R. Hughes,et al.  An isogeometric analysis approach to gradient damage models , 2011 .

[94]  C. V. Verhoosel,et al.  Isogeometric analysis-based goal-oriented error estimation for free-boundary problems , 2011 .

[95]  D. F. Rogers,et al.  An Introduction to NURBS: With Historical Perspective , 2011 .

[96]  Jia Lu,et al.  Isogeometric contact analysis: Geometric basis and formulation for frictionless contact , 2011 .

[97]  Giancarlo Sangalli,et al.  Some estimates for h–p–k-refinement in Isogeometric Analysis , 2011, Numerische Mathematik.

[98]  Fehmi Cirak,et al.  Subdivision-stabilised immersed b-spline finite elements for moving boundary flows , 2012 .

[99]  G. Sangalli,et al.  IsoGeometric Analysis using T-splines , 2012 .

[100]  T. Hughes,et al.  Isogeometric collocation for elastostatics and explicit dynamics , 2012 .

[101]  Peter Wriggers,et al.  A mortar formulation for 3D large deformation contact using NURBS-based isogeometric analysis and the augmented Lagrangian method , 2012 .

[102]  P. Hansbo,et al.  Fictitious domain finite element methods using cut elements , 2012 .

[103]  Ernst Rank,et al.  Geometric modeling, isogeometric analysis and the finite cell method , 2012 .

[104]  T. Hughes,et al.  A Simple Algorithm for Obtaining Nearly Optimal Quadrature Rules for NURBS-based Isogeometric Analysis , 2012 .

[105]  Yuri Bazilevs,et al.  Isogeometric fluid–structure interaction analysis with emphasis on non-matching discretizations, and with application to wind turbines , 2012 .

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

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

[108]  I. Babuska,et al.  Stable Generalized Finite Element Method (SGFEM) , 2011, 1104.0960.

[109]  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 .

[110]  Ahmed Ratnani,et al.  Isogeometric analysis in reduced magnetohydrodynamics , 2012 .

[111]  Cv Clemens Verhoosel,et al.  A phase-field description of dynamic brittle fracture , 2012 .

[112]  Tomas Bengtsson,et al.  Fictitious domain methods using cut elements : III . A stabilized Nitsche method for Stokes ’ problem , 2012 .

[113]  Erik Burman,et al.  A Penalty-Free Nonsymmetric Nitsche-Type Method for the Weak Imposition of Boundary Conditions , 2011, SIAM J. Numer. Anal..

[114]  T. Rabczuk,et al.  A two-dimensional Isogeometric Boundary Element Method for elastostatic analysis , 2012 .

[115]  John A. Evans,et al.  Isogeometric boundary element analysis using unstructured T-splines , 2013 .

[116]  ANDRÉ MASSING,et al.  Efficient Implementation of Finite Element Methods on Nonmatching and Overlapping Meshes in Three Dimensions , 2013, SIAM J. Sci. Comput..

[117]  Thomas J. R. Hughes,et al.  Isogeometric divergence-conforming B-splines for the unsteady Navier-Stokes equations , 2013, J. Comput. Phys..

[118]  Thomas J. R. Hughes,et al.  Explicit trace inequalities for isogeometric analysis and parametric hexahedral finite elements , 2013, Numerische Mathematik.

[119]  M. Fortin,et al.  Mixed Finite Element Methods and Applications , 2013 .

[120]  Alessandro Reali,et al.  Isogeometric Collocation: Cost Comparison with Galerkin Methods and Extension to Adaptive Hierarchical NURBS Discretizations , 2013 .

[121]  Y. Bazilevs,et al.  Weakly enforced essential boundary conditions for NURBS‐embedded and trimmed NURBS geometries on the basis of the finite cell method , 2013 .

[122]  Luca F. Pavarino,et al.  Isogeometric Schwarz preconditioners for linear elasticity systems , 2013 .

[123]  J. Kraus,et al.  Algebraic multilevel preconditioning in isogeometric analysis: Construction and numerical studies , 2013, 1304.0403.

[124]  John A. Evans,et al.  Isogeometric divergence-conforming b-splines for the darcy-stokes-brinkman equations , 2013 .

[125]  John A. Evans,et al.  ISOGEOMETRIC DIVERGENCE-CONFORMING B-SPLINES FOR THE STEADY NAVIER–STOKES EQUATIONS , 2013 .

[126]  Giancarlo Sangalli,et al.  Isogeometric discretizations of the Stokes problem: stability analysis by the macroelement technique , 2013 .

[127]  Victor M. Calo,et al.  The Cost of Continuity: Performance of Iterative Solvers on Isogeometric Finite Elements , 2012, SIAM J. Sci. Comput..

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

[129]  Vinh Phu Nguyen,et al.  Nitsche’s method for two and three dimensional NURBS patch coupling , 2013, 1308.0802.

[130]  Benedikt Schott,et al.  A new face-oriented stabilized XFEM approach for 2D and 3D incompressible Navier–Stokes equations , 2014 .

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

[132]  Hung Nguyen-Xuan,et al.  An isogeometric analysis for elliptic homogenization problems , 2013, Comput. Math. Appl..

[133]  Ernst Rank,et al.  Weak coupling for isogeometric analysis of non-matching and trimmed multi-patch geometries , 2014 .

[134]  R. A. Silverman,et al.  The Mathematical Theory of Viscous Incompressible Flow , 2014 .

[135]  Giancarlo Sangalli,et al.  Isogeometric methods for computational electromagnetics: B-spline and T-spline discretizations , 2012, J. Comput. Phys..

[136]  R. Schmidt,et al.  Isogeometric shape optimization of shells using semi-analytical sensitivity analysis and sensitivity weighting , 2014 .

[137]  Thomas J. R. Hughes,et al.  A higher-order phase-field model for brittle fracture: Formulation and analysis within the isogeometric analysis framework , 2014 .

[138]  I. Temizer,et al.  Multiscale thermomechanical contact: Computational homogenization with isogeometric analysis , 2014 .

[139]  Mats G. Larson,et al.  $L^2$-error estimates for finite element approximations of boundary fluxes , 2014 .

[140]  D. Arnold,et al.  Periodic Table of the Finite Elements , 2014 .

[141]  Roland Wüchner,et al.  A Nitsche‐type formulation and comparison of the most common domain decomposition methods in isogeometric analysis , 2014 .

[142]  André Massing,et al.  A Stabilized Nitsche Fictitious Domain Method for the Stokes Problem , 2012, J. Sci. Comput..

[143]  Yuri Bazilevs,et al.  An immersogeometric variational framework for fluid-structure interaction: application to bioprosthetic heart valves. , 2015, Computer methods in applied mechanics and engineering.

[144]  Sven Klinkel,et al.  The weak substitution method – an application of the mortar method for patch coupling in NURBS‐based isogeometric analysis , 2015 .

[145]  Mats G. Larson,et al.  High order cut finite element methods for the Stokes problem , 2015, Advanced Modeling and Simulation in Engineering Sciences.

[146]  Yuri Bazilevs,et al.  Dynamic and fluid–structure interaction simulations of bioprosthetic heart valves using parametric design with T-splines and Fung-type material models , 2015, Computational mechanics.

[147]  Ernst Rank,et al.  Theoretical and Numerical Investigation of the Finite Cell Method , 2015, Journal of Scientific Computing.

[148]  Peter Hansbo,et al.  CutFEM: Discretizing geometry and partial differential equations , 2015 .

[149]  Barbara Wohlmuth,et al.  Isogeometric mortar methods , 2014, 1407.8313.

[150]  E. H. Brummelen,et al.  Elasto-Capillarity Simulations Based on the Navier–Stokes–Cahn–Hilliard Equations , 2015, 1510.02441.

[151]  Jörg Peters,et al.  Matched Gk-constructions always yield Ck-continuous isogeometric elements , 2015, Comput. Aided Geom. Des..

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

[153]  Cv Clemens Verhoosel,et al.  Image-based goal-oriented adaptive isogeometric analysis with application to the micro-mechanical modeling of trabecular bone , 2015 .

[154]  C. Bona-Casas,et al.  A NURBS-based immersed methodology for fluid–structure interaction , 2015 .

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

[156]  Dominik Schillinger,et al.  The Finite Cell Method: A Review in the Context of Higher-Order Structural Analysis of CAD and Image-Based Geometric Models , 2015 .

[157]  Benedikt Schott,et al.  A stabilized Nitsche cut finite element method for the Oseen problem , 2016, 1611.02895.

[158]  S. Hassanizadeh,et al.  Pore‐scale network modeling of microbially induced calcium carbonate precipitation: Insight into scale dependence of biogeochemical reaction rates , 2016 .

[159]  Giancarlo Sangalli,et al.  Isogeometric Preconditioners Based on Fast Solvers for the Sylvester Equation , 2016, SIAM J. Sci. Comput..

[160]  Ming-Chen Hsu,et al.  The tetrahedral finite cell method: Higher‐order immersogeometric analysis on adaptive non‐boundary‐fitted meshes , 2016 .

[161]  Christina Kluge,et al.  Fluid Structure Interaction , 2016 .

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

[163]  Ming-Chen Hsu,et al.  The tetrahedral finite cell method for fluids: Immersogeometric analysis of turbulent flow around complex geometries , 2016 .

[164]  D.W. Fellner,et al.  Isogeometric shell analysis with NURBS compatible subdivision surfaces , 2016, Appl. Math. Comput..

[165]  F. de Prenter,et al.  Condition number analysis and preconditioning of the finite cell method , 2016, 1601.05129.

[166]  Yuri Bazilevs,et al.  Isogeometric divergence-conforming variational multiscale formulation of incompressible turbulent flows , 2017 .

[167]  Alessandro Reali,et al.  Mixed isogeometric finite cell methods for the stokes problem , 2017 .

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

[169]  John A. Evans,et al.  Immersogeometric cardiovascular fluid-structure interaction analysis with divergence-conforming B-splines. , 2017, Computer methods in applied mechanics and engineering.

[170]  T. Hughes,et al.  Smooth cubic spline spaces on unstructured quadrilateral meshes with particular emphasis on extraordinary points: Geometric design and isogeometric analysis considerations , 2017 .

[171]  Giancarlo Sangalli,et al.  Fast formation of isogeometric Galerkin matrices by weighted quadrature , 2016, 1605.01238.

[172]  G. Sangalli,et al.  An isogeometric method for linear nearly-incompressible elasticity with local stress projection , 2017 .

[173]  Yuri Bazilevs,et al.  Three-dimensional dynamic simulation of elastocapillarity , 2018 .

[174]  I. Babuska,et al.  THE PLATE PARADOX FOR HARD AND SOFT SIMPLE SUPPORT by , 2022 .