Formulation Of Boundary Conditions Research Articles

A detailed numerical treatment of the boundary conditions imposed by the skull on a diffusion–reaction model of glioma tumor growth. Clinical validation aspects

The study of the diffusive behavior of glioma tumor growth is an active field of biomedical research with considerable therapeutic implications. An important aspect of the corresponding computational problem is the mathematical handling of boundary conditions. This paper aims at providing an explicit and thorough numerical formulation of the adiabatic Neumann boundary conditions imposed by the skull on the diffusive growth of gliomas and in particular on glioblastoma multiforme (GBM). Additionally, a detailed exposition of the numerical solution process for a homogeneous approximation of glioma invasion using the Crank–Nicolson technique in conjunction with the Conjugate Gradient system solver is provided. The entire mathematical and numerical treatment is also in principle applicable to mathematically similar physical, chemical and biological phenomena. A comparison of the numerical solution for the special case of pure diffusion in the absence of boundary conditions or equivalently in the presence of adiabatic boundaries placed in infinity with its analytical counterpart is presented. Numerical simulations for various adiabatic boundary geometries and non zero net tumor growth rate support the validity of the corresponding mathematical treatment. Through numerical experimentation on a set of real brain imaging data, a simulated tumor has shown to satisfy the expected macroscopic behavior of glioblastoma multiforme including the adiabatic behavior of the skull. The paper concludes with a number of remarks pertaining to both the biological problem addressed and the more generic diffusion–reaction context.

Bäcklund flux quantization in a model of electrodiffusion based on Painlevé II

A previously established model of steady one-dimensional two-ion electrodiffusion across a liquid junction is reconsidered. It involves three coupled first-order nonlinear ordinary differential equations and has the second-order Painlevé II equation at its core. Solutions are now grouped by Bäcklund transformations into infinite sequences, partially labelled by two Bäcklund invariants. Each sequence is characterized by evenly-spaced quantized fluxes of the two ionic species, and hence evenly-spaced quantization of the electric current density. Finite subsequences of exact solutions are identified, with positive ionic concentrations and quantized fluxes, starting from a solution with zero electric field found by Planck, and suggesting an interpretation as a ground state plus excited states of the system. Positivity of ionic concentrations is established whenever Planck’s charge-neutral boundary conditions apply. Exact solutions are obtained for the electric field and ionic concentrations in well-stirred reservoirs outside each face of the junction, enabling the formulation of more realistic boundary conditions. In an approximate form, these lead to radiation boundary conditions for Painlevé II. Illustrative numerical solutions are presented, and the problem of establishing compatibility of boundary conditions with the structure of flux-quantizing sequences is discussed.

Accelerate Iteration of Least-Squares Finite Element Method for Radiative Heat Transfer in Participating Media With Diffusely Reflecting Walls

A high reflectivity of walls often leads to prohibitive computation time in the numerical simulation of radiative heat transfer. Such problem becomes very serious in many practical applications, for example, metal processing in high-temperature environment. The present work proposes a modified diffusion synthetic acceleration model to improve the convergence of radiative transfer calculation in participating media with diffusely reflecting boundary. This model adopts the P1 diffusion approximation to rectify the scattering source term of radiative transfer equation and the reflection term of the boundary condition. The corrected formulation for boundary condition is deduced and the algorithm is realized by finite element method. The accuracy of present model is verified by comparing the results with those of Monte Carlo method and finite element method without any accelerative technique. The effects of emissivity of walls and optical thickness on the convergence are investigated. The results indicate that the accuracy of present model is reliable and its accelerative effect is more obvious for the optically thick and scattering dominated media with intensive diffusely reflecting walls.

The dynamics of hot-electron heating in direct-drive-implosion experiments caused by two-plasmon-decay instability

Two-plasmon-decay (TPD) instability is identified as a potential source of target preheat in direct-drive-implosion experiments on OMEGA. A physical model of electron heating is developed that relies on extended Zakharov simulations to predict the nonlinearly saturated Langmuir wave spectrum. Hot electron generation is estimated via a test-particle approach. It is noted that because of the relatively low areal density of the targets during the time of TPD instability, hot-electron recirculation and reheating are potentially important effects. This is modeled by a particular form of boundary conditions on the test particles. Such boundary conditions might prove useful in other kinetic simulations of particle heating where recirculation is a possibility.

An Angle-Dependent Impedance Boundary Condition for the Split-Step Parabolic Equation Method

A formulation of the impedance boundary condition (IBC) is derived, which can be used in split-step parabolic wave equation solvers to accurately represent an interface with arbitrary dielectric properties. The approach relies on ensuring that the plane-wave decomposition of the field satisfies the appropriate IBC for each spectral component. Numerical experiments illustrating the robustness of the approach for low-contrast interfaces and angles-of-incidence near the Brewster angle are presented.

On the boundary conditions for differential turbulence models

The distinctive features of the formulation of boundary conditions at the outer edge of a boundary layer and near a wall are considered. Approximate wall functions of turbulence parameters are proposed for three turbulence models.

Viscous incompressible flow simulation using penalty finite element method

Numerical analysis of Navier-Stokes equations in velocity- pressure variables with traction boundary conditions for isothermal incompressible flow is presented. Specific to this study is formulation of boundary conditions on synthetic boundary characterized by traction due to friction and surface tension. The traction and open boundary conditions have been investigated in detail. Navier-Stokes equations are discretized in time using Crank-Nicolson scheme and in space using Galerkin finite element method. Pressure being unknown and is decoupled from the computations. It is determined as post processing of the velocity field. The justification to simulate this class of flow problems is presented through benchmark tests - classical lid-driven cavity flow- widely used by numerous authors due to its simple geometry and complicated flow behavior and squeezed flow between two parallel plates amenable to analytical solution. Results are presented for very low to high Reynolds numbers and compared with the benchmark results.

Dynamic Modelling for a Trickle-Bed Reactor Using the Numerical Inverse Laplace Transform Technique

Abstract Experimental evaluation and dynamic modelling were presented for the transient behavior of a trickle bed reactor in which gas and liquid streams flow downwards through a bed of catalyst particles. One-dimensional dynamic mathematical model has been described to study the gas-liquid-solid process in which the gas phase (Ar) with the tracer (SO2) is treated as a continuum. The physical model has been analyzed, including the formulation of initial and boundary conditions and the description of the solution methodology. An experimental procedure to measure the concentrations of the tracer (SO2) has been performed. The concentration measurements for this tracer (SO2) were performed in a fixed be reactor on trickling flow of the gas phase for a range of operating conditions. The gas-liquid mass transfer (kgl) coefficient, liquid-solid mass transfer (kls) coefficient, gas holdup (hg) and partial wetting efficiency (fe) were chosen as the hydrodynamic parameters of the proposed mathematical model. Such parameters have been optimized with the experimental measurents of the tracer (SO2) at the exit of the trickle-bed reactor. The optimized parameters hg, kgl, kls and fe were calculated simultaneously using the equation (51) with minimization of the objective function. Results of the mathematical model was presented and compared to the two experimental cases. Each above parameters were correlated using empirical correlations.

Molecular Dynamics Study of Condensation/Evaporation and Velocity Distribution of N-Dodecane at Liquid-Vapour Phase Equilibria

Some recent results of molecular dynamics simulations of the condensation/evaporation and velocity distribution of n-dodecane (C12H26), the closest approximation to Diesel fuel, at a liquid-vapour interface in equilibrium state are briefly described. It is shown that molecules at the liquid surface need to gain relatively large translational energy to evaporate. Vapour molecules with large translational energy can easily penetrate deep into the transition layer and condense in the liquid phase. The evaporation/condensation coefficient is shown to be controlled mainly by the translational energy. The properties of the velocity distribution functions of molecules at the liquid, interface and vapour regions are summarised. It has been shown that the distribution functions of evaporated and reflected molecules for the velocity component normal to the surface deviate considerably from the Maxwellian, while the distribution function for all molecules leaving this surface (evaporated and reflected) is close to Maxwellian. The evaporation coefficient has been shown to increase with increasing molecular energy in the direction perpendicular to the surface. These properties have been recommended to be taken into account when formulating boundary conditions for kinetic modelling.

COUPLING MULTIPHASE PORE-SCALE MODELS TO ACCOUNT FOR BOUNDARY CONDITIONS: APPLICATION TO 2D QUASI-STATIC PORE NETWORKS

Accurate predictions of macroscopic multiphase flow properties (relative permeability and capillary pressure) are necessary for modeling flow and transport in subsurface applications, such as hydrocarbon recovery, carbon sequestration and nuclear waste storage. These properties are usually measured experimentally, but pore-scale network modeling has become an efficient alternative for understanding fundamental flow behavior and predicting macroscopic properties. In many cases, network modeling gives excellent agreement with experiment by using models physically representative of real media. Void space within a rock sample can be extracted from high resolution images and converted to a topologically equivalent network of pores and throats. Multiphase fluid transport is then modeled in the network and macroscopic properties extracted from the model. Advancements continue to be made in making multiphase network models (both quasistatic and dynamic) predictive, but one limitation is that arbitrary (e.g., constant pressure) boundary conditions are usually assumed; they do not reflect the local saturations and pressure distributions that are affected by flow and transport in the surrounding media. In this work we demonstrate that transport behavior at the pore scale, and therefore, upscaled macroscopic properties are directly affected by the boundary conditions. Pore-scale drainage in 2D quasi-static networks is modeled by direct coupling to other pore-network models so that the boundary conditions reflect local variations of transport behavior in the surrounding media. Phase saturations are coupled at model boundaries to ensure continuity between adjacent models. Macroscopic petrophysical properties are shown to be largely dependent upon the surrounding media, which are manifested in the form of boundary conditions. The predictive ability of network simulations is thus improved using the novel network coupling scheme.

Dynamic evaluation for liquid tracer in a trickle bed reactor

A mathematical model is developed for a liquid flow on solid particles in a trickle bed reactor. A mathematical formulation is followed based on the liquid-solid model approach where the liquid phase with the (KCl) tracer is treated as a continuum. The physical modeling is discussed, including the formulation of initial and boundary conditions and the description of the solution methodology. Results of mathematical model are presented and validated. The model is validated through comparison using three experimental cases. The optimized values of the axial dispersion (Dax), liquid-solid mass transfer (kLS), and partial wetting efficiency (FM) coefficients are obtained simultaneously using the objective function. The behavior of Dax, kLS, and FM is analyzed by the empirical correlations.

On generalised penalty approaches for slip, free surface and related boundary conditions in viscous flow simulation

PurposeThe purpose of this paper is to extend the penalty concept to treat partial slip, free surface, contact and related boundary conditions in viscous flow simulation.Design/methodology/approachThe penalty partial‐slip formulation is analysed and related to the classical Navier slip condition. The same penalty scheme also allows partial penetration through a boundary, hence the implementation of porous wall boundaries. The finite element method is used for investigating and interpreting penalty approaches to boundary conditions.FindingsThe generalised penalty approach is verified by means of a novel variant of the circular‐Couette flow problem, having partial slip on one of the cylindrical boundaries, for which an analytic solution is derived. Further verificationis provided by consideration of viscous flow over a sphere with partial slip on the surface, and comparison of numerical and classical solutions. Numerical studies illustrate the versatility of the approach.Research limitations/implicationsThe penalty approach is applied to some different boundaries: partial slip and partial penetration with no/full slip/penetration as limiting cases; free surface; space‐ and time‐varying boundary conditions which allow progressive contact over time. Application is made to curved and inclined boundaries. Sensitivity of flow to penalty parameters is an avenue for continued research, as is application of the penalty approach for non‐Newtonian flows.Originality/valueThis is the first work to show the relation between penalty formulation of boundary conditions and physical boundary conditions. It provides a method that overcomes past difficulties in implementing partial slip on boundaries of general shape, and which handles progressive contact. It also provides useful benchmark problems for future studies.

Particle-Based Discrete Element Modeling: Geomechanics Perspective

This paper is a review of the use of particulate discrete element modeling (DEM) in geomechanics. The overall objective of the paper is to serve as an introduction to researchers and practitioners in geomechanics who are considering adopting DEM in their work or using the results of DEM simulations to guide other studies, for example, the development of constitutive models for continuum-based numerical analysis. It is hoped that prior converts to the use of DEM will also benefit from a relatively objective overview of current DEM use in geomechanics. The introductory sections present the background to the method and give an overview of the evolution of the use of particulate DEM in recent geotechnical research. The general principals of the algorithm are then presented, considering the types of particles typically used, the calculation of contact forces, and formulation of simulation boundary conditions. Some techniques available to interpret and postprocess of DEM results and provide the information to link the particle scale and overall response are outlined. Approaches used to validate and calibrate DEM models to verify that DEM simulation results representative of physical reality are discussed. An overview of the application of DEM modeling to field-scale problems is then presented. Finally the conclusions consider future developments in the area and emphasize the need to maintain quality in DEM simulations.

Strain resolving method of composite plane plates

The paper deals with the extension of isotropic plates problem to the case of composite plates. In order to perform it, the Kirchhoff-Love hypotheses were softened by some additional ones. Considering the constitutive laws for composite materials the stress functions were eliminated by using Cauchy equations. As a result a partial derivative equation in displacements was obtained. Finally the boundary condition formulation was extended for the case of complex composite plates.

Involving the Navier–Stokes equations in the derivation of boundary conditions for the lattice Boltzmann method

By means of the continuity equation of the incompressible Navier-Stokes equations, additional physical arguments for the derivation of a formulation of the no-slip boundary condition for the lattice Boltzmann method for straight walls at rest are obtained. This leads to a boundary condition that is second-order accurate with respect to the grid spacing and conserves mass. In addition, the boundary condition is stable for relaxation frequencies close to two.

Adapting Generally Filled Cylindrical Sheet Impedances for Electromagnetic Compatibility [Measurements Corner

The ability to control the bistatic scattering characteristics from cylindrical objects using a tapered cylindrical impedance sheet is presented. It is shown that with the capability to readily control both the real and imaginary components of a percolated lossy film's effective impedance, specific impedance distributions can be obtained to effectively introduce and direct the overall nulls in the bistatic scattering pattern. The bistatic scattering width calculations are based on a modal analysis and an impedance-sheet boundary condition formulation. Potential uses of tapered cylindrical impedance sheets in bistatic EMC/EMI applications are discussed.

An exact linear dispersion relation for CRM instability

An exact self-consistent linear dispersion relation of a large orbit electron beam including two principles of cyclotron emission with oscillation frequencies above and below the relativistic electron frequency is derived and analyzed numerically for the first time in the literature. The two principles are cyclotron resonance maser (CRM) instability and Cherenkov instability in the azimuthal direction. Self-consistency in the formulation and inclusion of proper boundary conditions have removed the unphysical instability existing for infinitely large kz observed in conventional dispersion relations of CRM instability.

Molecular dynamics study on evaporation and condensation of n-dodecane at liquid–vapor phase equilibria

Molecular dynamics simulations are performed to study the evaporation and condensation of n-dodecane (C(12)H(26)) at temperatures in the range 400-600 K. A modified optimized potential for liquid simulation model is applied to take into account the Lennard-Jones, bond bending and torsion potentials with the bond length constrained. The equilibrium liquid-vapor n-dodecane interface thickness is predicted to be ~1.2-2.0 nm. It is shown that the molecular chains lie preferentially parallel to the interface in the liquid-vapor transition region. The predicted evaporation/condensation coefficient decreased from 0.9 to 0.3 when temperature increased from 400 to 600 K. These values can be used for the formulation of boundary conditions in the kinetic modeling of droplet heating and evaporation processes; they are noticeably different from those predicted by the transition state theory. We also present the typical molecular behaviors in the evaporation and condensation processes. The molecular exchange in condensation, typical for simple molecules, has never been observed for n-dodecane molecular chains.

Boundary operator from matrix field formulation of boundary conditions for Friedrichs systems

Following the recent progress in understanding the abstract setting for Friedrichs symmetric positive systems by Ern, Guermond and Caplain (2007) [8], as well as Antonić and Burazin (2010) [3], an attempt is made to relate these results to the classical Friedrichs theory. A comparison of two approaches, via the trace operator and the boundary operator, has been made, favouring the latter. Finally, a particular set of sufficient conditions for a boundary matrix field to define a boundary operator in that case is given, and the applicability of this procedure in realistic situations is shown by examples.

On the stability of the high-order Higdon Absorbing Boundary Conditions

The Higdon sequence of Absorbing Boundary Conditions (ABCs) for the linear wave equation is considered. Building on a previous work of Ha-Duong and Joly, which related to other forms of boundary conditions, the Higdon ABCs are proved to be energy-stable (on the continuous level) up to any order. This type of stability is stronger than the more standard notion of stability of initial boundary value problems in the sense of Kreiss; in particular it leads to stability estimates which are uniform in time. In consequence to the theorem proved here, energy-stability is immediately implied for the high-order Givoli–Neta and Hagstrom–Warburton ABCs, which are reformulations of the Higdon ABCs using auxiliary variables. A weakness of this theory is that it requires sufficiently smooth data, and that the required smoothness level increases with the order of the ABC. This issue and its implications are discussed.