Isogeometric Analysis of the Steady-State Incompressible MHD Equations

Abstract

\bfA \bfb \bfs \bft \bfr \bfa \bfc \bft. This paper presents an isogeometric (IGA) solver for steady-state incompressible magnetohydrodynamics (MHD). MHD is the study of the behavior of electrically conducting fluids and can be viewed mathematically as a coupled system: the Navier-Stokes equations (for the fluid) and a reduced form of Maxwell's equations (for the electromagnetic field). A key feature of MHD flow is the potential development of very strong shear, usually in proximity to walls. This results in two correlated demands on numerical simulation: the need to represent the geometry and the near-wall shear accurately. In addition, for both the Navier-Stokes and the Maxwell's equations, appropriate discretizations are required for the problem to be well-posed. IGA analysis is a variant of the conventional finite element (FE) method, but utilizing the underlying approximations commonly used in computer-aided design (CAD) to represent geometry, basis functions, and test functions. As a result, IGA can represent curved shapes such as circles and conic sections exactly using Bsplines and nonuniform rational B-splines (NURBS). Furthermore, IGA can obtain much improved accuracy in the computed solution, for a given number of degrees of freedom, due to its inherent smoothness and higher continuity of basis functions when compared with standard FE and finite volume (FV) methods. To address the issue of well-posedness, a set of stable IGA discretizations for the Navier-Stokes and Maxwell's equations is developed and incorporated into the IGA solver. A detailed convergence study is carried out to verify the convergence, accuracy, and performance of the IGA scheme for certain benchmark cases. \bfK \bfe \bfy \bfw \bfo \bfr \bfd \bfs. isogeometric, incompressible, steady-state, MHD \bfA \bfM \bfS \bfs \bfu \bfb \bfj \bfe \bfc \bft \bfc \bfl \bfa \bfs \bfs \bfi fi\bfc \bfa \bft \bfi \bfo \bfn \bfs. 76W05, 65N30

Isogeometric analysis of the time-dependent incompressible MHD equations

“…In this case, suitable B-spline or NURBS spaces (see Section 2) need to be employed. These discretisations for the steady MHD problems are studied in [5,22]. For completeness, the inf-sup stable spline spaces considered in this paper are given in following sections.…”