34 research outputs found
High-Order Numerical Integration on Domains Bounded by Intersecting Level Sets
We present a high-order method that provides numerical integration on
volumes, surfaces, and lines defined implicitly by two smooth intersecting
level sets. To approximate the integrals, the method maps quadrature rules
defined on hypercubes to the curved domains of the integrals. This enables the
numerical integration of a wide range of integrands since integration on
hypercubes is a well known problem. The mappings are constructed by treating
the isocontours of the level sets as graphs of height functions. Numerical
experiments with smooth integrands indicate a high-order of convergence for
transformed Gauss quadrature rules on domains defined by polynomial, rational,
and trigonometric level sets. We show that the approach we have used can be
combined readily with adaptive quadrature methods. Moreover, we apply the
approach to numerically integrate on difficult geometries without requiring a
low-order fallback method
Generalized Gilat-Raubenheimer method for density-of-states calculation in photonic crystals
Efficient numeric algorithm is the key for accurate evaluation of density of
states (DOS) in band theory. Gilat-Raubenheimer (GR) method proposed in 1966 is
an efficient linear extrapolation method which was limited in specific
lattices. Here, using an affine transformation, we provide a new generalization
of the original GR method to any Bravais lattices and show that it is superior
to the tetrahedron method and the adaptive Gaussian broadening method. Finally,
we apply our generalized GR (GGR) method to compute DOS of various gyroid
photonic crystals of topological degeneracies.Comment: 7 pages, 2 figures; typos added, appendix B added. Programs are
available at: https://github.com/boyuanliuoptics/DOS-calculatio
The diffuse Nitsche method: Dirichlet constraints on phase-field boundaries
We explore diffuse formulations of Nitsche's method for consistently imposing Dirichlet boundary conditions on phase-field approximations of sharp domains. Leveraging the properties of the phase-field gradient, we derive the variational formulation of the diffuse Nitsche method by transferring all integrals associated with the Dirichlet boundary from a geometrically sharp surface format in the standard Nitsche method to a geometrically diffuse volumetric format. We also derive conditions for the stability of the discrete system and formulate a diffuse local eigenvalue problem, from which the stabilization parameter can be estimated automatically in each element. We advertise metastable phase-field solutions of the Allen-Cahn problem for transferring complex imaging data into diffuse geometric models. In particular, we discuss the use of mixed meshes, that is, an adaptively refined mesh for the phase-field in the diffuse boundary region and a uniform mesh for the representation of the physics-based solution fields. We illustrate accuracy and convergence properties of the diffuse Nitsche method and demonstrate its advantages over diffuse penalty-type methods. In the context of imaging based analysis, we show that the diffuse Nitsche method achieves the same accuracy as the standard Nitsche method with sharp surfaces, if the inherent length scales, i.e., the interface width of the phase-field, the voxel spacing and the mesh size, are properly related. We demonstrate the flexibility of the new method by analyzing stresses in a human vertebral body
Efficient operator-coarsening multigrid schemes for local discontinuous Galerkin methods
An efficient -multigrid scheme is presented for local discontinuous
Galerkin (LDG) discretizations of elliptic problems, formulated around the idea
of separately coarsening the underlying discrete gradient and divergence
operators. We show that traditional multigrid coarsening of the primal
formulation leads to poor and suboptimal multigrid performance, whereas
coarsening of the flux formulation leads to optimal convergence and is
equivalent to a purely geometric multigrid method. The resulting
operator-coarsening schemes do not require the entire mesh hierarchy to be
explicitly built, thereby obviating the need to compute quadrature rules,
lifting operators, and other mesh-related quantities on coarse meshes. We show
that good multigrid convergence rates are achieved in a variety of numerical
tests on 2D and 3D uniform and adaptive Cartesian grids, as well as for curved
domains using implicitly defined meshes and for multi-phase elliptic interface
problems with complex geometry. Extension to non-LDG discretizations is briefly
discussed