Curved Domain Boundaries Research Articles

Curving of a Domain Boundary in the Presence of an Inclined Symmetrical Magnetic Inhomogeneity in Film Materials with High Anisotropy

The theory of magnetic domain ordering in magnetic film materials when there is an inclined and symmetrically located magnetic inhomogeneity in the strip domain is developed, allowing for variation in both the magnetostatic energy and the energy of anisotropy of the curved domain boundary.

Features of a Domain Structure with Asymmetric Magnetic Heterogeneity in Film Materials with High Anisotropy

The theory of magnetic domain ordering in magnetic film materials with asymmetric magnetic heterogeneity is considered, with allowance for changes in the magnetostatic energy and that of the anisotropy of a curved domain boundary. The effect the parameters that determine changes in the energy of the magnetostatic interaction of the domain structure and the energy of the anisotropy of the curved domain boundary have on the shape and value of domain boundary bending is analyzed.

A staggered space–time discontinuous Galerkin method for the incompressible Navier–Stokes equations on two-dimensional triangular meshes

In this paper we propose a novel arbitrary high order accurate semi-implicit space–time discontinuous Galerkin method for the solution of the two dimensional incompressible Navier–Stokes equations on staggered unstructured triangular meshes. Isoparametric finite elements are used to take into account curved domain boundaries. The discrete pressure is defined on the primal triangular grid and the discrete velocity field is defined on an edge-based staggered dual grid. While staggered meshes are state of the art in classical finite difference approximations of the incompressible Navier–Stokes equations, their use in the context of high order DG schemes is novel and still quite rare. Formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse four-point block system for the scalar pressure, which is conveniently solved with a matrix-free GMRES algorithm. Note that the same space–time DG scheme on a collocated grid would lead to ten non-zero blocks per element, since substituting the discrete velocity into the discrete continuity equation on a collocated mesh would involve also neighbors of neighbors. From numerical experiments we find that our linear system is well-behaved and that the GMRES method converges quickly even without the use of any preconditioner, which is a unique feature in the context of high order implicit DG schemes. A very simple and efficient Picard iteration is then used in order to derive a space–time pressure correction algorithm that achieves also high order of accuracy in time, which is in general a non-trivial task in the context of high order discretizations for the incompressible Navier–Stokes equations. The flexibility and accuracy of high order space–time DG methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both, space and time. The use of a staggered grid allows to avoid the use of Riemann solvers in several terms of the discrete equations and significantly reduces the total stencil size of the linear system that needs to be solved for the pressure. The proposed method is verified for approximation polynomials of degree up to p=4 in space and time by solving a series of typical numerical test problems and by comparing the obtained numerical results with available exact analytical solutions or other numerical reference data.To the knowledge of the authors, this is the first time that a space–time discontinuous Galerkin finite element method is presented for the incompressible Navier–Stokes equations on staggered unstructured grids.

Analysis using higher-order XFEM: implicit representation of geometrical features from a given parametric representation

We present a promising approach to reduce the difficulties associated with meshing complex curved domain boundaries for higher-order finite elements. In this work, higher-order XFEM analyses for strong discontinuity in the case of linear elasticity problems are presented. Curved implicit boundaries are approximated inside an unstructured coarse mesh by using parametric information extracted from the parametric representation (the most common in Computer Aided Design CAD). This approximation provides local graded sub-mesh (GSM) inside boundary elements (i.e. an element split by the curved boundary) which will be used for integration purpose. Sample geometries and numerical experiments illustrate the accuracy and robustness of the proposed approach.

High-order central ENO finite-volume scheme for ideal MHD

A high-order accurate finite-volume scheme for the compressible ideal magnetohydrodynamics (MHD) equations is proposed. The high-order MHD scheme is based on a central essentially non-oscillatory (CENO) method combined with the generalized Lagrange multiplier divergence cleaning method for MHD. The CENO method uses k-exact multidimensional reconstruction together with a monotonicity procedure that switches from a high-order reconstruction to a limited low-order reconstruction in regions of discontinuous or under-resolved solution content. Both reconstructions are performed on central stencils, and the switching procedure is based on a smoothness indicator. The proposed high-order accurate MHD scheme can be used on general polygonal grids. A highly sophisticated parallel implementation of the scheme is described that is fourth-order accurate on two-dimensional dynamically-adaptive body-fitted structured grids. The hierarchical multi-block body-fitted grid permits grid lines to conform to curved boundaries. High-order accuracy is maintained at curved domain boundaries by employing high-order spline representations and constraints at the Gauss quadrature points for flux integration. Detailed numerical results demonstrate high-order convergence for smooth flows and robustness against oscillations for problems with shocks. A new MHD extension of the well-known Shu–Osher test problem is proposed to test the ability of the high-order MHD scheme to resolve small-scale flow features in the presence of shocks. The dynamic mesh adaptation capabilities of the approach are demonstrated using adaptive time-dependent simulations of the Orszag–Tang vortex problem with high-order accuracy and unprecedented effective resolution.

On the large-scale structure and spectral dynamics of two-dimensional turbulence in a periodic channel

This paper reports on a numerical study of forced two-dimensional turbulence in a periodic channel with flat no-slip walls. Since corners or curved domain boundaries, which are met in the standard rectangular, square, or circular geometries, are absent in this geometry, the (statistical) analysis of the flow is substantially simplified. Moreover, the use of a standard Fourier–Chebyshev pseudospectral algorithm enables high integral-scale Reynolds number simulations. The paper focuses on (i) the influence of the aspect ratio of the channel and (ii) the integral-scale Reynolds number on the large-scale self-organization of the flow. It is shown that for small aspect ratios, a unidirectional flow spontaneously emerges, notably in the absence of a pressure gradient in the longitudinal direction. For larger aspect ratios, the flow tends to organize into an array of counter-rotating vortical structures. The computed energy and enstrophy spectra provide further evidence that the injection of small-scale vorticity at the no-slip walls modify the inertial-range scaling. Additionally, the quasistationary final state of decaying turbulence is interpreted in terms of the Stokes modes of a viscous channel flow. Finally, the transport of a passive tracer material is studied with emphasis on the role of the large-scale flow on the dispersion and the spectral properties of the tracer variance in the presence of no-slip boundaries.

Powell–Sabin splines with boundary conditions for polygonal and non-polygonal domains

Powell–Sabin splines are piecewise quadratic polynomials with a global C 1 -continuity, defined on conforming triangulations. Imposing boundary conditions on such a spline leads to a set of constraints on the spline coefficients. First, we discuss boundary conditions defined on a polygonal domain, before we treat boundary conditions on a general curved domain boundary. We consider Dirichlet and Neumann conditions, and we show that a particular choice of the PS-triangles at the boundary can greatly simplify the corresponding constraints. Finally, we consider an application where the techniques developed in this paper are used: the numerical solution of a partial differential equation by the Galerkin and collocation method.

Dynamics of curved domain boundaries in convection patterns.

Curved domain boundaries (DB's) between locally stable convection patterns are studied near the onset of convection, within the framework of the Newell-Whitehead-Segel theory [J. Fluid Mech. 38, 279 (1969); 38, 203 (1969)]. We consider the case where there exists a Lyapunov functional. By means of asymptotic methods, the equations of motion for DB's are derived, and their solutions are obtained. It is shown that the behavior of a DB depends strongly on the difference between Lyapunov functional's densities of the coexisting patterns. In the case of a nonzero difference, the normal velocity depends on the orientation of the DB, and caustics can be produced in a finite time. In the case of zero difference, the normal velocity depends on both orientation and distortion of the DB, and the DB tends typically to straighten after a long time.

Domain boundary energies in finite regions at 2D criticality via conformal field theory

By use of conformal field theory, the authors calculate universal results for the energies of curved domain boundaries at two-dimensional critical points in circular and rectangular regions. This work extends the theory of finite-size effects at criticality, complementing previous exact results for the energies of domain boundaries in infinite strips.

Domain boundary energies and interactions at 2D criticality via conformal field theory.

By use of conformal field theory, we calculate the energies of one or more curved domain boundaries at two-dimensional critical points. We find the interaction of two separated boundaries of the same type to be weakly attractive, and proportional to the square of an operator-product expansion coefficient. In a strip of width L, the interaction vanishes exponentially with separation, over a distance set by L. It is also small for nearby domains when one of them is much smaller than L. Many-body interaction energies are weak as well.

Domain Structure of Perminvar Having a Rectangular Hysteresis Loop

An investigation has been made of the magnetic domain structure of Perminvar (43 percent Ni, 34 percent Fe, 23 percent Co) ring specimens having rectangular hysteresis loops after heat-treatment in a magnetic field. Domain patterns obtained with colloidal magnetite showed curved domain boundaries extending completely around the rings, forming circles concentric with them. Changes in magnetization occur when an applied field causes the circular boundaries either to expand or contract so that there is a change in the relative values of clockwise and counter-clockwise flux. A nucleus of reversed magnetization was formed by making a small notch in a specimen, and this decreased the coercive force and hysteresis loss by a factor of two. It was found that in a 180° domain boundary it was possible to make the change in spin orientations, which occurs in going from one side of the boundary to the other, have either a right- or left-hand screw relation, by the application of a field of appropriate sign perpendicular to the surface. The effect of super-posing an applied alternating field was also investigated, and an effective permeability of 4,000,000 was obtained.