Weakly enforced boundary conditions for the NURBS-based Finite Cell Method

In this paper, we present a variationally consistent formulation for the weak en- forcement of essential boundary conditions as an extension to the finite cell method, a fictitious domain method of higher order. The absence of boundary fitted elements in fictitious domain or immersed boundary methods significantly restricts a strong enforcement of essential boundary conditions to models where the boundary of the solution domain coincides with the embedding analysis domain. Penalty methods and Lagrange multiplier methods are adequate means to overcome this limitation but often suffer from various drawbacks with severe consequences for a stable and accurate solution of the governing system of equations. In this contribution, we follow the idea of NITSCHE (29) who developed a stable scheme for the solution of the Laplace problem taking weak boundary conditions into account. An extension to problems from linear elasticity shows an appropriate behavior with regard to numerical stability, accuracy and an adequate convergence behavior. NURBS are chosen as a high-order approximation basis to benefit from their smoothness and flexibility in the process of uniform model refinement.

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

[2]  Rainald Löhner,et al.  Adaptive Embedded/Immersed Unstructured Grid Techniques , 2007 .

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

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

[5]  Ernst Rank,et al.  An efficient integration technique for the voxel‐based finite cell method , 2012 .

[6]  C. Peskin.  The immersed boundary method , 2002, Acta Numerica.

[7]  K. Bathe.  Finite Element Procedures , 1995 .

[8]  Ivo Babuška,et al.  The p-Version of the Finite Element Method for Parabolic Equations. Part 1 , 1981 .

[9]  Victor M. Calo,et al.  The role of continuity in residual-based variational multiscale modeling of turbulence , 2007 .

[10]  Ernst Rank,et al.  The p‐version of the finite element method for domains with corners and for infinite domains , 1990 .

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

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

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

[14]  D. Tiba,et al.  An Embedding of Domains Approach in Free Boundary Problems andOptimal Design , 1995 .

[15]  Ernst Rank,et al.  The hp‐d‐adaptive finite cell method for geometrically nonlinear problems of solid mechanics , 2012 .

[16]  Michael Griebel,et al.  A Particle-Partition of Unity Method-Part III: A Multilevel Solver , 2002, SIAM J. Sci. Comput..

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

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

[19]  T. Hughes,et al.  Isogeometric variational multiscale modeling of wall-bounded turbulent flows with weakly enforced boundary conditions on unstretched meshes , 2010 .

[20]  W. Wall,et al.  An eXtended Finite Element Method/Lagrange multiplier based approach for fluid-structure interaction , 2008 .

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

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

[23]  Victor M. Calo,et al.  Weak Dirichlet Boundary Conditions for Wall-Bounded Turbulent Flows , 2007 .

[24]  Satya N. Atluri,et al.  A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method , 1998 .

[25]  Michael E. Mortenson,et al.  Geometric Modeling , 2008, Encyclopedia of GIS.

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

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

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

[29]  Peter Hansbo,et al.  A Lagrange multiplier method for the finite element solution of elliptic interface problems using non-matching meshes , 2005, Numerische Mathematik.

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

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

[32]  Yuri Bazilevs,et al.  3D simulation of wind turbine rotors at full scale. Part II: Fluid–structure interaction modeling with composite blades , 2011 .

[33]  Michael Griebel,et al.  A Particle-Partition of Unity Method Part V: Boundary Conditions , 2003 .

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

[35]  P. Hansbo,et al.  A finite element method for the simulation of strong and weak discontinuities in solid mechanics , 2004 .

[36]  Peter Hansbo,et al.  Nitsche's method for interface problems in computa‐tional mechanics , 2005 .

[37]  E. Rank,et al.  Organizing a p-Version Finite Element Computation by an Octree-Based Hierarchy , 2005 .

[38]  Ted Belytschko,et al.  An extended finite element method for modeling crack growth with frictional contact , 2001 .

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