Well-posed Boundary Conditions Research Articles

OBLC domain decomposition for artificial compressible equation

This paper is a continuation of the domain decomposition method according to the physics scale proposed in [1] and [2]. Starting from systems of ordinary differential equations, a solution is decomposed into an outer solution (0) and its boundary layer corrections (BLC) mainly on the fixed boundary. For efficient numerical solution, different equations, different numerical methods and different grids can be suitably chosen for the different scales. This paper also gives the characteristic nature and wellposed boundary condition about artificial compressible equations. Numerical experiments show that the computational method and the couple process presented in the paper are effective.

Numerical solution of two-carrier hydrodynamic semiconductor device equations employing a stabilized finite element method

A space-time Galerkin/least-squares finite element method was presented in [1] for numerical simulation of single-carrier hydrodynamic semiconductor device equations. The single-carrier hydrodynamic device equations were shown to resemble the ideal gas equations and Galerkin/least-squares finite element method, originally developed for computational fluid dynamics equations [16], was extended to solve semiconductor device applications. In this paper, the space-time Galerkin/least-squares finite element method is further extended and generalized to solve two-carrier hydrodynamic device equations. The proposed formulation is based on a time-discontinuous Galerkin method, in which physical entropy variables are employed. A standard Galerkin finite element method is applied to the Poisson equation. Numerical simulations are performed on the coupled Poisson and the two-carrier hydrodynamic equations employing a staggered approach. A mathematical analysis of the time-dependent multi-dimensional hydrodynamic model is performed to determine well-posed boundary conditions for electrical contacts. The number of boundary conditions that need to be specified for the hydrodynamic equations at inflow and outflow boundaries of the device are derived. Example boundary conditions that are based either on physical and/or mathematical basis are presented. Stability of the numerical algorithms is addressed. The space-time Galerkin/least-squares finite element method and the standard Galerkin finite element method for the hydrodynamic and Poisson equations, respectively, are shown to be stable. Specifically, a Clausius-Duhem inequality, a basic stability requirement, is derived for the hydrodynamic equations and the proposed numerical method automatically satisfies this stability requirement. Numerical simulations are performed on one- and two-dimensional two-carrier p-n diodes and the results demonstrate the effectiveness of the proposed numerical method.

High order finite differences methods on non-smooth domains

We demonstrate how to obtain stable high order approximations of parabolic and symmetric hyperbolic systems using finite difference methods. The idea behind the scheme is to use difference operators which satisfy a summation-by-parts rule, which in one space dimension reads ( u, Dv ) h = u Nv N — u 0 v 0 — ( Du, v ) h . This principle generalizes to domains with curved and non-smooth boundaries. Well-posed analytic boundary conditions are incorporated into the semi-discrete system by means of projection operators. The resulting system can be shown to be stable. In an appendix, we have included an explicit example of a stable difference operator, which is third order accurate at the boundary, and sixth order accurate in the interior.

Continuous potential Maxwell solutions on nodal-based finite elements

A nodal-based finite-element approach for computing electric fields in heterogeneous media is presented. The primary calculation is formulated in terms of continuous potentials, so that no special care is required on element assembly at dielectric interfaces. The resulting Galerkin weak-form matrices exhibit the special Helmholtz structure, which guarantees the absence of parasitic solutions in driven problems with physically well-posed boundary conditions. The enhanced sparsity of the Helmholtz form mitigates the extra coupling effort associated with introduction of a fourth degree of freedom relative to direct E solution. E can be extracted from the computed potentials as a postprocessing step either at nodal positions or element centroids. Solutions obtained with this approach for several benchmark and practical problems are shown to be parasite-free and essentially indistinguishable from previously reported direct E computations. >

Analysis of a Fluid-Mechanic Model of a Horizontal Tee-Junction

A physically based theoretical model of gas-liquid and steam-water two-phase flow in a horizontal tee-junction is developed. The model includes five independent equations: the mixture continuity equation, the vapor-phase continuity equation, the x direction momentum equation, the y direction momentum equation, and the energy equation. Any five unknown hydraulic parameters of the tee-junction can be solved in various ways, for virtually any choice of three well-posed boundary conditions. The uncertainty of the interfacial terms and the number of empirical constants that are used in the model are limited. The results of the calculations are compared with experimental data gathered from the literature. The comparison shows that the predictive ability of the model is reasonably good, except that the mass balance equation of the vapor phase is not suitable for some of the steam-water experimental data, and interfacial evaporation and condensation terms should be introduced into the model in the future.

The influence of open boundary conditions on the convergence to steady state for the Navier-Stokes equations

The question of open boundary conditions of inflow and outflow type for the Navier-Stokes equations and its influence on the convergence to steady state is addressed. Both the continuous and semi-discrete problem are analysed using the energy method and the Laplace transform technique. The energy method is used to derive well-posed boundary conditions for the continuous problem. For the semi-discrete problem we use the energy method to prove that by using the well-posed boundary conditions for the continuous problem and adding a suitable numerical boundary condition well-posedness is preserved. By employing the Laplace transform technique the spectra for different types of boundary conditions are obtained. The spectra are analysed and it is shown how the choice of boundary conditions strongly affects the convergence to steady state. One-dimensional Navier-Stokes calculations are performed and the resulting convergence rates agree well with the theoretical analysis. Finally, the spectra obtained using inflow and outflow types of boundary conditions are compared with spectra obtained using periodic boundary conditions and the choice of a time-integration method for the Navier-Stokes equations is discussed.

On minimalization of time varying linear systems with well-posed boundary conditions

The notion of minimal realization of a pair of finite rank operators between Hilbert spaces is introduced. Constructions of such minimal realizations are made. A minimal realization of an integral operator with semi-separable kernel is constructed for the case that one of the realizations has analytic coefficients. A consequence for the characterization of minimal stability of an analytic minimal system is given.

Complementary solutions of electrostatic field problems

A general error form the corresponds to mismatch in the constitutive relationship is used for systematic derivation of complementary solution formulations, and the associated bounds, in electrostatic field problems. A close kinship between the error-based treatment and a general theorem on well-posed boundary conditions is made use of to highlight the role of boundary specifications in determining the expression for the complementary functionals. A combination of complementary computational results, obtained using two-dimensional finite elements is shown to provide valuable insight into numerical behavior.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Implicit Normal Mode Initialization

Abstract It is shown that nonlinear normal mode initialization (NMI) can be implemented without knowing the normal modes of a model. The implicit form of nonlinear NMI is particularly useful for models whose normal modes cannot readily be found; for example, if the underlying linear equations are nonseparable. An implicit nonlinear NMI scheme is formulated for the shallow-water equations on a polar stereographic projection. The linear equations which define the implicit normal modes include most of the beta terms as well as variable Coriolis parameter and map scale factor. Even in this nonseparable case, the equivalence between implicit and conventional nonlinear NMI is shown to be exact. The scheme is implemented in a regional model on a quasi-hemispheric domain, which uses a finite-element discretization on a nonuniform grid. The well-posed lateral boundary conditions of this model lead to consistent boundary conditions for the initialization. Results are presented not only for the implicit form of Mach...

Lithospheric flexure and deformation-induced gravity changes: effect of elastic compressibility and gravitation on a multilayered, thick-plate model

SUMMARY An approach is formulated by which the theoretical admittance can be evaluated for the flexure of a multi-layered elastic plate overlying an inviscid fluid. The multi-layered elastic and compressible plate is subjected to well-posed boundary conditions. Gravitational body force, elastic compressibility and elastic coupling at the interfaces with elastic modulus contrast are fully accounted for. The elastic field solution was obtained by the propagator matrix technique. The contribution to gravity change from local density perturbation within the compressible plate is also included. Our calculation shows that the density perturbation component is smaller than the Bouguer component by at least an order of magnitude over the compensated waveband. The thin-plate approximation can be used with an effective flexural rigidity evaluated from the thickness-averaged values of the elastic moduli. If there is significant elastic coupling between the crust and the mantle, the discrepancy between our results and the incompressible, thick-plate model can be quite appreciable due to the different isostatic compensation mechanisms operative within the crust. Preliminary calculations show that similar conclusions would apply to the response functions for the sub-surface loading case. Our technique can also be used to compute the stress distribution in a multilayered plate. The numerical results show that the stresses can be appreciably different from that in a homogeneous plate.

A new finite element formulation for computational fluid dynamics: VII. The stokes problem with various well-posed boundary conditions: Symmetric formulations that converge for all velocity/pressure spaces

Symmetric finite element formulations are proposed for the primitive-variables form of the Stokes equations and shown to be convergent for any combination of pressure and velocity interpolations. Various boundary conditions, such as pressure, are accommodated.

Minimality and irreducibility of time-invariant linear boundary-value systems

Abstract Minimality and irreducibility in the class of time-invariant linear systems with well-posed boundary conditions are introduced and characterized. Necessary and sufficient conditions for similarity of irreducible (or minimal) time-invariant systems with well-posed boundary conditions are given. The problem of stable minimality is analysed. Some of the results are specified for stationary and displacement systems.

Similarity and Reduction for Time Varying Linear Systems with Well-Posed Boundary Conditions

The classical theory of similarity and reduction of causal and anti-causal systems is generalized and extended to time varying linear systems with well-posed boundary conditions. Two main problems are solved. The first is to determine to what extent a system with boundary conditions can be simplified by similarity and reduction. The second problem is to find the invariants of such a simplification. Special attention is given to the time invariant case.

On minimality and stable minimality of time-varying linear systems with well-posed boundary conditions

For time-varying linear systems with well-posed boundary conditions the concept of stable minimality is introduced. A complete characterization is given of stably minimal systems with analytic coefficients.

SUNION—an algorithm for one-dimensional laser-plasma interaction

SUNION solves the combined problem of ion expansion and of resonance absorption of p-polarized electromagnetic radiation. Well-posed initial and boundary conditions are derived within the ponderomotive approximation. Efficiency and accuracy of the code are checked by reproducing known results found in literature for various simplifications. A new type of numerical instability is exposed within the Lagrangian description of the expanding ions, pointing out a fundamental difficulty in treating plasma expansion into a vacuum. It originates from charge separation and is correlated with a singular behavior of the plasma flow. The solution method is considerably simplified by taking into account the first time-derivative in the complex second-order Schrödinger-type wave equation, and, by solving the latter by a Crank-Nicholson scheme, without reference to shooting methods. The absorption coefficient reaching values up to 60 per cent is found to be rather insensitive to the global density structure, and reflects more or less the local scale length at the critical density. Profile steepening caused by radiation pressure is accomplished within the first twenty ion plasma periods.

Incompletely Parabolic Problems in Fluid Dynamics

Some partial differential equations encountered in physical applications are of incompletely parabolic type; the Navier–Stokes equations in fluid dynamics are a typical example. In this paper we analyze such systems; in particular we treat the mixed initial-boundary value problem. In many applications there is a small parameter $\varepsilon $ multiplying the coefficient for the highest derivative. The energy method is used to derive well-posed boundary conditions such that, when $\varepsilon $ tends to zero, the reduced problem is also well posed.

Liquid distribution and redistribution in packed columns-I Theoretical

When liquid is uniformly distributed at the top of a packed column it is found that there is a preferential flow to the wall, and early theoretical work has suggested that the observed radial velocity is proportional to the radial gradient of axial velocity. A set of consistent boundary conditions has not been found. In this paper the experimental observation of preferential liquid flow is interpreted as a difference in permeability between the wall and bulk region of packing, and the existence of a potential for liquid redistribution is inferred from an examination of experimental work on two-phase flow in porous media, and of the internal consistency of the early relation between radial velocity and the radial gradient of axial velocity. The existence of a potential for liquid redistribution, and a difference in permeability between the wall and bulk regions are shown to lead to a differential equation describing liquid distribution in the packing, defined by consistent and well-posed boundary conditions that are determined from the physical analysis.The solution to the partial differential equation describing redistribution from an initial axisymmetric distribution is given.