A Cut Finite Element Method for the Bernoulli Free Boundary Value Problem

We develop a cut finite element method for the Bernoulli free boundary problem. The free boundary, represented by an approximate signed distance function on a fixed background mesh, is allowed to i ...

[1]  Helmut Harbrecht,et al.  Analytical and numerical methods in shape optimization , 2008 .

[2]  C. Dapogny,et al.  Computation of the signed distance function to a discrete contour on adapted triangulation , 2012, Calcolo.

[3]  G. Allaire,et al.  Shape optimization with a level set based mesh evolution method , 2014 .

[4]  Erik Burman,et al.  Finite element methods with symmetric stabilization for the transient convection―diffusion-reaction equation , 2009 .

[5]  Timo Tiihonen,et al.  Shape optimization and trial methods for free boundary problems , 1997 .

[6]  Peter Hansbo,et al.  A cut finite element method for coupled bulk-surface problems on time-dependent domains , 2015, 1502.07142.

[7]  R. Hiptmair,et al.  Comparison of approximate shape gradients , 2014, BIT Numerical Mathematics.

[8]  Thomas Y. Hou,et al.  Numerical Solutions to Free Boundary Problems , 1995, Acta Numerica.

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

[10]  Frédéric de Gournay,et al.  Velocity Extension for the Level-set Method and Multiple Eigenvalues in Shape Optimization , 2006, SIAM J. Control. Optim..

[11]  P. Hansbo,et al.  Minimal surface computation using a finite element method on an embedded surface , 2014, 1403.3535.

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

[13]  J. Toland,et al.  Bernoulli Free-boundary Problems , 2009 .

[14]  K. Sturm,et al.  Distributed shape derivative via averaged adjoint method and applications , 2015, 1509.01816.

[15]  Mario S. Mommer,et al.  A new fictitious domain method in shape optimization , 2008, Comput. Optim. Appl..

[16]  Peter Hansbo,et al.  Cut finite element methods for coupled bulk–surface problems , 2014, Numerische Mathematik.

[17]  Jan Sokolowski,et al.  Introduction to shape optimization , 1992 .

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

[19]  J. Zou,et al.  Finite element methods and their convergence for elliptic and parabolic interface problems , 1998 .

[20]  Grégoire Allaire,et al.  Structural Optimization by the Level-Set Method , 2003 .

[21]  L. R. Scott,et al.  Finite element interpolation of nonsmooth functions satisfying boundary conditions , 1990 .

[22]  R. Correa,et al.  Directional derivates in minimax problems , 1985 .

[23]  G. Allaire,et al.  Structural optimization using sensitivity analysis and a level-set method , 2004 .

[24]  Helmut Harbrecht,et al.  A Newton method for Bernoulli’s free boundary problem in three dimensions , 2008, Computing.

[25]  Ralf Hiptmair,et al.  Shape Optimization by Pursuing Diffeomorphisms , 2015, Comput. Methods Appl. Math..

[26]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[27]  M. Flucher,et al.  Bernoulli's free-boundary problem, qualitative theory and numerical approximation. , 1997 .

[28]  Rachid Touzani,et al.  Fast Numerical Methods for Bernoulli Free Boundary Problems , 2007, SIAM J. Sci. Comput..