High-order non-reflecting boundary conditions for dispersive waves

High-order local non-reflecting boundary conditions: a review

Local artificial boundary conditions for Schrödinger and heat equations by using high-order azimuth derivatives on circular artificial boundary

Non-reflecting boundary conditions (NRBCs) play an important role in computational fluid dynamics (CFD). A novel NRBC based on the method of characteristics using timeline interpolations is proposed for fluid dynamics solved by smoothed particle hydrodynamics (SPH). It is performed by four layers of particles whose pressures and velocities are obtained through the Lagrange interpolation in the time domain which is derived from the propagation of characteristic waves between particles. The proposed NRBC can allow outward travelling pressure and velocity messages to pass through the boundary without obvious reflection. That is, with the implementation of the NRBC, the solution in a finite computational domain of interest is close to that in an infinite domain. Several numerical tests show that this NRBC is robust and applicable for a broad variety of hydrodynamics ranging from low to high speed.

Scaled Boundary Perfectly Matched Layer (SBPML): A novel 3D time-domain artificial boundary method for wave problem in general-shaped and heterogeneous infinite domain

Subject index of volume 101

An analysis of wave-motion problems in unbounded or infinite domains is presented. Analytical solutions that express outgoing waves from an artificial boundary such as a circle or a sphere can be obtained easily. These solutions may be transformed into discretized relations between values of velocity potential and volumetric flux on the artificial boundary. These relations are in the same forms as the FEM equations. They can be joined directly to finite elements within the artificial boundary in order to construct an equation system for the whole domain. Some problems of radiation and scattering are calculated numerically by this method. The so-called fictitious interior eigenvalue failure does not arise in this method.

Padded continued fraction absorbing boundary conditions for dispersive waves

Nonlinear characteristic boundary conditions based on nonlinear multidimensional characteristics are proposed for 2‐ and 3‐D compressible Navier–Stokes equations with/without scalar transport equations. This approach is consistent with the flow physics and transport properties. Based on the theory of characteristics, which is a rigorous mathematical technique, multidimensional flows can be decomposed into acoustic, entropy, and vorticity waves. Nonreflecting boundary conditions are derived by setting corresponding characteristic variables of incoming waves to zero and by partially damping the source terms of the incoming acoustic waves. In order to obtain the resulting optimal damping coefficient, analysis is performed for problems of pure acoustic plane wave propagation and arbitrary flows. The proposed boundary conditions are tested on two benchmark problems: cylindrical acoustic wave propagation and the wake flow behind a cylinder with strong periodic vortex convected out of the computational domain. This new approach substantially minimizes the spurious wave reflections of pressure, density, temperature, and velocity as well as vorticity from the artificial boundaries, where strong multidimensional flow effects exist. The numerical simulations yield accurate results, confirm the optimal damping coefficient obtained from analysis, and verify that the method substantially improves the 1‐D characteristics‐based nonreflecting boundary conditions for complex multidimensional flows. Copyright © 2009 John Wiley & Sons, Ltd.

Exact boundary condition for time-dependent wave equation based on boundary integral

Solving partial differential equations (PDEs) on infinite domains has been a challenging task in physical simulations and geometry processing. We introduce a general technique to transform a PDE problem on an unbounded domain to a PDE problem on a bounded domain. Our method uses the Kelvin Transform, which essentially inverts the distance from the origin. However, naive application of this coordinate mapping can still result in a singularity at the origin in the transformed domain. We show that by factoring the desired solution into the product of an analytically known (asymptotic) component and another function to solve for, the problem can be made continuous and compact, with solutions significantly more efficient and well-conditioned than traditional finite element and Monte Carlo numerical PDE methods on stretched coordinates. Specifically, we show that every Poisson or Laplace equation on an infinite domain is transformed to another Poisson (Laplace) equation on a compact region. In other words, any existing Poisson solver on a bounded domain is readily an infinite domain Poisson solver after being wrapped by our transformation. We demonstrate the integration of our method with finite difference and Monte Carlo PDE solvers, with applications in the fluid pressure solve and simulating electromagnetism, including visualizations of the solar magnetic field. Our transformation technique also applies to the Helmholtz equation whose solutions oscillate out to infinity. After the transformation, the Helmholtz equation becomes a tractable equation on a bounded domain without infinite oscillation. To our knowledge, this is the first time that the Helmholtz equation on an infinite domain is solved on a bounded grid without requiring an artificial absorbing boundary condition.

For exterior scattering problems one of the chief difficulties arises from the unbounded nature of the problem domain. Inhomogeneous obstacles may require a volumetric discretization, such as the Finite Element Method (FEM), and for this approach to be feasible the exterior domain must be truncated and an appropriate condition enforced at the far, artificial, boundary. An exact, non-reflecting boundary condition can be stated using the classical method if the Artificial Boundary's shape is quite specific: circular or elliptical. Recently, this approach has been generalized to permit quite general Artificial Boundaries which are shaped as perturbations of a circle resulting in the Enhanced DtN-FE method. In this paper we extend this method to a two-dimensional FEM featuring high-order polynomials in order to realize a high rate of convergence. This is more involved than simply specifying high-order test and trial functions as now the scatterer shape and Artificial Boundary must be faithfully represented. This entails boundary elements which conform (to high order) to the true boundary shapes. As we show, this can be accomplished and we realize an arbitrary order FEM without spurious reflections.

The propagation of small perturbations in complex geometries can involve hydrodynamic-acoustic interactions, coupling acoustic waves and vortical modes. A propagation model, based on the linearized Navier–Stokes equations, is proposed. It includes the mechanism responsible for the generation of vorticity associated with the hydrodynamic modes. The linearized Navier–Stokes equations are discretized in space using a discontinuous Galerkin formulation for unstructured grids. Explicit time integration and non-reflecting boundary conditions are described. The linearized Navier–Stokes (LNS) model is applied to two test cases. The first one is the time-harmonic source line in an incompressible inviscid two-dimensional mean shear flow in an infinite domain. It is shown that the proposed model is able to capture the trailing vorticity field developing behind the mass source and to represent the redistribution of the vorticity. The second test case deals with the analysis of the acoustic propagation of an incoming perturbation inside a circular duct with a sudden area expansion in the presence of a mean flow and the evaluation of its scattering matrix. The computed coefficients of the scattering matrix are compared to experimental data for three different Mach numbers of the mean flow, M0 = 0.08, 0.19 and 0.29. The good agreement with the experimental data shows that the proposed method is suitable for characterizing the acoustic behavior of this kind of network.

Nonreflecting boundary condition for the Helmholtz equation

A simple impedance-infinite element for the finite element solution of the three-dimensional wave equation in unbounded domains

Finite element method for time dependent scattering: nonreflecting boundary condition, adaptivity, and energy decay