A conservative coupling algorithm between a compressible flow and a rigid body using an Embedded Boundary method

This paper deals with a new solid-fluid coupling algorithm between a rigid body and an unsteady compressible fluid flow, using an Embedded Boundary method. The coupling with a rigid body is a first step towards the coupling with a Discrete Element method. The flow is computed using a Finite Volume approach on a Cartesian grid. The expression of numerical fluxes does not affect the general coupling algorithm and we use a one-step high-order scheme proposed by Daru and Tenaud [Daru V,Tenaud C., J. Comput. Phys. 2004]. The Embedded Boundary method is used to integrate the presence of a solid boundary in the fluid. The coupling algorithm is totally explicit and ensures exact mass conservation and a balance of momentum and energy between the fluid and the solid. It is shown that the scheme preserves uniform movement of both fluid and solid and introduces no numerical boundary roughness. The effciency of the method is demonstrated on challenging one- and two-dimensional benchmarks

A Correction Function Method for Poisson problems with interface jump conditions

In this paper we present a method to treat interface jump conditions for constant coefficients Poisson problems that allows the use of standard “black box” solvers, without compromising accuracy. The basic idea of the new approach is similar to the Ghost Fluid Method (GFM). The GFM relies on corrections applied on nodes located across the interface for discretization stencils that straddle the interface. If the corrections are solution-independent, they can be moved to the right-hand-side (RHS) of the equations, producing a problem with the same linear system as if there were no jumps, only with a different RHS. However, achieving high accuracy is very hard (if not impossible) with the “standard” approaches used to compute the GFM correction terms. In this paper we generalize the GFM correction terms to a correction function, defined on a band around the interface. This function is then shown to be characterized as the solution to a PDE, with appropriate boundary conditions. This PDE can, in principle, be solved to any desired order of accuracy. As an example, we apply this new method to devise a 4th order accurate scheme for the constant coefficients Poisson equation with discontinuities in 2D. This scheme is based on (i) the standard 9-point stencil discretization of the Poisson equation, (ii) a representation of the correction function in terms of bicubics, and (iii) a solution of the correction function PDE by a least squares minimization. Several applications of the method are presented to illustrate its robustness dealing with a variety of interface geometries, its capability to capture sharp discontinuities, and its high convergence rate.National Science Foundation (U.S.) (Grant DMS-0813648)Brazil. Coordenacao de Aperfeicoamento de Pessoal de Nivel SuperiorFulbright Program (Grant BEX 2784/06-8

A High-Order Numerical Method for the Nonlinear Helmholtz Equation in Multidimensional Layered Media

We present a novel computational methodology for solving the scalar nonlinear Helmholtz equation (NLH) that governs the propagation of laser light in Kerr dielectrics. The methodology addresses two well-known challenges in nonlinear optics: Singular behavior of solutions when the scattering in the medium is assumed predominantly forward (paraxial regime), and the presence of discontinuities in the % linear and nonlinear optical properties of the medium. Specifically, we consider a slab of nonlinear material which may be grated in the direction of propagation and which is immersed in a linear medium as a whole. The key components of the methodology are a semi-compact high-order finite-difference scheme that maintains accuracy across the discontinuities and enables sub-wavelength resolution on large domains at a tolerable cost, a nonlocal two-way artificial boundary condition (ABC) that simultaneously facilitates the reflectionless propagation of the outgoing waves and forward propagation of the given incoming waves, and a nonlinear solver based on Newton's method. The proposed methodology combines and substantially extends the capabilities of our previous techniques built for 1Dand for multi-D. It facilitates a direct numerical study of nonparaxial propagation and goes well beyond the approaches in the literature based on the "augmented" paraxial models. In particular, it provides the first ever evidence that the singularity of the solution indeed disappears in the scalar NLH model that includes the nonparaxial effects. It also enables simulation of the wavelength-width spatial solitons, as well as of the counter-propagating solitons.Comment: 40 pages, 10 figure

An Immersed Interface Method for Discrete Surfaces

Fluid-structure systems occur in a range of scientific and engineering applications. The immersed boundary(IB) method is a widely recognized and effective modeling paradigm for simulating fluid-structure interaction(FSI) in such systems, but a difficulty of the IB formulation is that the pressure and viscous stress are generally discontinuous at the interface. The conventional IB method regularizes these discontinuities, which typically yields low-order accuracy at these interfaces. The immersed interface method(IIM) is an IB-like approach to FSI that sharply imposes stress jump conditions, enabling higher-order accuracy, but prior applications of the IIM have been largely restricted to methods that rely on smooth representations of the interface geometry. This paper introduces an IIM that uses only a C0 representation of the interface,such as those provided by standard nodal Lagrangian FE methods. Verification examples for models with prescribed motion demonstrate that the method sharply resolves stress discontinuities along the IB while avoiding the need for analytic information of the interface geometry. We demonstrate that only the lowest-order jump conditions for the pressure and velocity gradient are required to realize global 2nd-order accuracy. Specifically,we show 2nd-order global convergence rate along with nearly 2nd-order local convergence in the Eulerian velocity, and between 1st-and 2nd-order global convergence rates along with 1st-order local convergence for the Eulerian pressure. We also show 2nd-order local convergence in the interfacial displacement and velocity along with 1st-order local convergence in the fluid traction. As a demonstration of the method's ability to tackle complex geometries,this approach is also used to simulate flow in an anatomical model of the inferior vena cava.Comment: - Added a non-axisymmetric example (flow within eccentric rotating cylinder in Sec. 4.3) - Added a more in-depth analysis and comparison with a body-fitted approach for the application in Sec. 4.