An Isogeometric Analysis approach for the study of the gyrokinetic quasi-neutrality equation

In this work, a new discretization scheme for the gyrokinetic quasi-neutrality equation is proposed. It is based on Isogeometric Analysis; the IGA which relies on NURBS functions, accommodates arbitrary coordinates and the use of complicated computation domains. Moreover, arbitrary high order degree of basis functions can be used and their regularity enables the use of a low number of elements. Here, this approach is successfully tested on elliptic problems like the quasi-neutrality equation arising in gyrokinetic models. In this last application, when polar coordinates are considered, a fast solver can be used and the non locality is dealt with a suitable decomposition which reduces the resolution of the gyrokinetic quasi-neutrality equation to a sequence of local 2D elliptic problems.

[1]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[2]  Olivier Czarny,et al.  Bézier surfaces and finite elements for MHD simulations , 2008, J. Comput. Phys..

[3]  J. Krommes,et al.  Nonlinear gyrokinetic equations , 1983 .

[4]  T. Hughes,et al.  Efficient quadrature for NURBS-based isogeometric analysis , 2010 .

[5]  J. L. V. Lewandowski,et al.  A finite element Poisson solver for gyrokinetic particle simulations in a global field aligned mesh , 2006, J. Comput. Phys..

[6]  R. A. Kolesnikov,et al.  On higher order corrections to gyrokinetic Vlasov―Poisson equations in the long wavelength limit , 2009 .

[7]  Tom Lyche,et al.  T-spline Simplication and Local Renement , 2004 .

[8]  Hartmut Prautzsch,et al.  A fast algorithm to raise the degree of spline curves , 1991, Comput. Aided Geom. Des..

[9]  W. W. Lee,et al.  Gyrokinetic approach in particle simulation , 1981 .

[10]  J. Kruth,et al.  NURBS curve and surface fitting for reverse engineering , 1998 .

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

[12]  T. S. Hahm,et al.  Nonlinear gyrokinetic equations for tokamak microturbulence , 1988 .

[13]  Virginie Grandgirard,et al.  Some parallel algorithms for the Quasineutrality solver of GYSELA , 2011 .

[14]  R. Bartels,et al.  Fitting Uncertain Data with NURBS , 1997 .

[15]  Charlson C. Kim,et al.  Comparisons and physics basis of tokamak transport models and turbulence simulations , 2000 .

[16]  Guillaume Latu,et al.  Parallel bottleneck in the Quasineutrality solver embedded in GYSELA , 2011 .

[17]  John R. Rice,et al.  Direct solution of partial difference equations by tensor product methods , 1964 .

[18]  李幼升,et al.  Ph , 1989 .

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

[20]  C. D. Boor,et al.  Collocation at Gaussian Points , 1973 .

[21]  Tom Lyche,et al.  T-spline simplification and local refinement , 2004, ACM Trans. Graph..

[22]  Laurent Villard,et al.  A drift-kinetic Semi-Lagrangian 4D code for ion turbulence simulation , 2006, J. Comput. Phys..

[23]  Ahmed Ratnani,et al.  An Arbitrary High-Order Spline Finite Element Solver for the Time Domain Maxwell Equations , 2012, J. Sci. Comput..

[24]  C. Loan The ubiquitous Kronecker product , 2000 .

[25]  Perkins,et al.  Fluid moment models for Landau damping with application to the ion-temperature-gradient instability. , 1990, Physical review letters.

[26]  Guillaume Latu,et al.  Scalable Quasineutral Solver for Gyrokinetic Simulation , 2011, PPAM.

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

[28]  Ralph R. Martin,et al.  Fast degree elevation and knot insertion for B-spline curves , 2005, Comput. Aided Geom. Des..

[29]  R. Hatzky,et al.  Energy conservation in a nonlinear gyrokinetic particle-in-cell code for ion-temperature-gradient-driven modes in θ-pinch geometry , 2002 .

[30]  Thomas J. R. Hughes,et al.  n-Widths, sup–infs, and optimality ratios for the k-version of the isogeometric finite element method , 2009 .

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

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