Formulation Of Boundary Conditions Research Articles

О выборе методов решения уравнения Пуассона в общем случае распределения объемной плотности заряда и о постановке краевых условий в электрокинетических задачах (обзор)

О выборе методов решения уравнения Пуассона в общем случае распределения объемной плотности заряда и о постановке краевых условий в электрокинетических задачах (обзор)

Fluid flow and mass transfer over circular strands using the lattice Boltzmann method

Knowledge of the fluid flow and mass transfer over circular strands is fundamental to the cell culture of tissue scaffolds in bio-reactors. This paper presents a study on the simulation of fluid flow and mass transfer over the circular strands of a tissue scaffold by using the multiple relaxation time lattice Boltzmann method for the low Reynolds number regime, with Re D = 0.01 and 0.1, respectively. The mass transfer problem approximates the transport of a scalar nutrient from the bulk fluid to the strand surface, such as is encountered in the flow through tissue scaffolds placed in bio-reactors. The circular geometry of the scaffold strand is treated and implemented by means of the interpolated bounce-back boundary condition formulation. Our simulation illustrates that the flow accelerates around the strand, resulting in the maximum shear stress at the shoulder of the strand and that diffusion mass transfer plays the dominant role in the scalar transport. The local Sherwood number varies significantly over the surface of the strand, with a peak value located on the upstream surface. Increasing the Schmidt number of the scalar and decreasing the blockage ratio results in higher mass transfer rates on the surface of the stand. Overall, the simulation results provide one with the insight into the fluid flow and mass transfer over the circular strands of a tissue scaffold in a bio-reactor, which would be impractical to obtain by experiments.

Computational continua revisited

SummaryIn the recent paper, Fish and Kuznetsov introduced the so‐called computational continua (C2) approach, which is a variant of the higher order computational homogenization that does not require higher order continuity, introduces no new degrees of freedom, and is free of higher order boundary conditions. In a follow‐up paper on reduced order computational continua, the C2 formulation has been enhanced with a model reduction scheme based on construction of residual‐free fields to yield a computationally efficient framework coined as RC2. The original C2 formulations were limited to rectangular and box elements. The primary objectives of the present manuscript is to revisit the original formulation in three respects: (i) consistent formulation of boundary conditions for unit cells subjected to higher order coarse scale fields, (ii) effective solution of the unit cell problem for lower order approximation of eigenstrains, and (iii) development of nonlocal quadrature schemes for general two‐dimensional (quad and triangle) and three‐dimensional (hexahedral and tetrahedral) elements. Copyright © 2014 John Wiley & Sons, Ltd.

Modified gradient elasticity and its finite element method for shear boundary layer analysis

We attempt to present a modified gradient elasticity theory to analyze the shear boundary layer adjacent to the bi-material interface. By defining an internal length scale vector and assuming that the strain energy density depends on both the strain tensor and the strain gradient tensor, a modified gradient elasticity (MGE) theory is derived. When the internal characteristic length vector vanishes, the MGE can be simplified to classical elasticity theory. Based on the principle of virtual work, the corresponding variational principle, boundary condition and finite element formulation of the MGE are derived. The bi-material strip with shear boundary layers is numerically calculated by the MGE. It is found that the thickness of shear boundary layer strongly depends on the internal characteristic length, in addition, the mesh-dependence can be eliminated.

Hydrologic-Hydraulic Model for Simulating Dual Drainage and Flooding in Urban Areas: Application to a Catchment in the Metropolitan Area of Chicago

A one-dimensional (1D) hydrologic-hydraulic model for simulating dual drainage in urban areas is presented. It consists of four modules: (1) rainfall-runoff transformation, (2) 1D flow routing on a street network, (3) flow interception at street inlets, and (4) flow interaction between surface water on the streets and the underground storm-water system by interfacing with the EPA-SWMM5 engine (U.S. Environmental Protection Agency-Storm Water Management Model). The hydrologic model (first module) transforms rainfall to runoff using the kinematic wave approximation and simulates the infiltration process with the Green-Ampt method. The street network model (second module) is based on a finite-volume shock-capturing scheme that solves the fully conservative Saint-Venant equations and can be used to model both subcritical and supercritical flows. The inlet model (third module) computes the amount of water intercepted by inlets. The formulation of boundary conditions at the street crossings is generalized and can be used for any number of streets, any combination of inflowing and outflowing streets, and flow regime (e.g., subcritical and supercritical flows). Flow interaction between surface water on the streets and underground storm-water system is achieved by interfacing the proposed model with EPA-SWMM5. This interaction allows flow to enter from streets to the underground storm-water system and vice versa. The proposed model has several potential applications such as the identification of critical zones for flooding (e.g., zones with high water depths and flow velocities) in urban developments and can be used to take appropriate measures for drainage control (e.g., to increase number and/or size of inlets), to determine the consequences of different degrees of inlet clogging, and to assess flooding hazards through the application of suitable hazard criteria. A summary of criteria used for storm-water hazard assessment is presented. To demonstrate the dual-drainage model’s potential, an application is performed in a catchment of the metropolitan area of Chicago, Illinois. The results obtained are promising and show that the model can be a useful tool for storm-water management and flooding hazard assessment in urban areas.

Application of Differential Equations to Calculations of Large Deformations of Structural Members

The article discusses the application of differential equations to solve large deformations of the designed slender structural elements. The structural element under design has a variable circular cross-section in longitudinal direction. It is made of a composite material based on carbon fiber and is capable of large deformations. In the solution it was necessary to consider the nonlinear deformation theory, which led to the solution of a nonlinear differential equation of second order. The creation of the differential equation describing the line of elastic deformation of a significantly curved structural element required the formulation of input equations and boundary conditions for its solution. The results obtained by means of a differential equation were compared with the numerical solution applying the finite element method. The solution presented and methodology can be used for practical application in the engineering design of fishing rods.

Finite element formulation of unilateral boundary conditions for unsaturated flow in porous continua

AbstractThis paper presents the numerical resolution of unilateral boundary conditions able to effectively model several problems of unsaturated flow, as those involving rainfall infiltration and seepage faces. Besides the penalty technique, we also consider the novel regularization of these conditions by means of the more effective augmented Lagrangian method. The performance of the so‐obtained finite element method is carefully investigated in terms of accuracy and ill‐conditioning effects, including comparisons with analytical solutions and a complete identification of the analogies with the problem of frictionless contact. In this way, we provide a priori estimates of optimal and admissible ranges for the penalty coefficient as functions of permeability and spatial discretization. The proposed method and the estimated coefficient ranges are validated in further numerical examples, involving the propagation of a wetting front due to rainfall and the partial saturation of an aged concrete dam. These applications show that the proposed regularizations do not induce any detrimental effect on solution accuracy and on convergence rate of the employed Newton‐Raphson method. Hence, the present approach should be preferred to the commonly used iterative switching between the imposed‐flow and the imposed‐pressure conditions, which often leads to spurious oscillations and convergence failures.

Modeling of Water Flows around a Circular Cylinder with the SPH Method

Abstract The paper describes the SPH modeling of a plane problem of fluid flow around a rigid circular cylinder. In the model considered, the cylinder is placed in a rectangular fluid domain at a certain distance from a horizontal plane boundary, and it is subjected to fluid flow forces. The fluid motion is induced by a piston type generator. The generator - fluid system starts to move from rest at a certain moment of time. The work aims at a discrete description of the fluid flow around the cylinder and, at the same time, calculation of the pressure distribution along the circumference of the cylinder and the resultant of the pressure on the cylinder. In order to solve the initial value problem considered, a new SPH formulation of boundary conditions on the cylinder surface is proposed which match the physical condition for the fluid velocity at this boundary. For a viscous fluid, an approximate description of the stress tensor is formulated which allows to reduce the differentiation of field functions to the first order in calculating the shear forces in the SPH approach.

Determining Temperature Dependencies of Sand Mould Thermal Properties

The presented work is aimed at determining thermal diffusivity, thermal conductivity and heat capacity coefficients of silica quartz bentonite foundry sand. The values of the above thermo-physical properties were determined for temperature range of about 30 - 450 °C using theCasting measuring method[1-.The results obtained during the examinations presented in the paper can be useful when formulating boundary conditions in numerical models of heat and mass transfer in the system: casting mould ambient. TheCasting methodallows preserving real conditions during the experiment, i.e. contact of the mass with liquid metal and solidifying casting, and the obtained results are in a good agreement with the mean values available in literature. From the obtained results it follows that examinations should be also focused onthermo-physical properties vs. mass densitydependency.

Accounting for depth-wise linear change of stresses in the intact rock mass in geomechanical problems

The authors discuss formulation of boundary conditions in problems on stress-strain state of rocks around an underground excavation. The classical formulations neglect the presupposition of linearity of stress field in an intact rock mass with depth and disregard the discrepancy of the calculated and actual behavior of rocks. The presented study is based on the linearity of initial stresses and the computational domain boundary conditions that assume stress linearity and unaltered boundary conditions during drivage. The problem on stress-strain state is divided into two problems due to option of formulating accurately boundary conditions for the horizontal boundary and the absence of such option for the vertical boundary.

Design and verification methodology of boundary conditions for finite volume schemes

Despite the critical importance of resolving flow at or near domain boundaries, boundary condition formulations are often implemented in an ad hoc fashion. The purpose of this work is to provide a general framework for the implementation and, importantly, the verification of boundary conditions for node- and cell-centered finite volume schemes. These conditions may include any combination of Dirichlet conditions, Neumann conditions, extrapolation, and the conservation equations themselves. Specific conditions for inviscid walls, inflow, and outflow are systematically tested using manufactured and exact solutions to assess well-posedness and stability. The procedures in this work provide a method for the verification of complex boundary conditions as well as shed physical insight for different problem configurations.

Direct numerical simulation of laminar flame–wall interaction for a novel H2-selective membrane/injector configuration

Direct numerical simulations are performed to investigate the transient processes of laminar flame–wall interaction and quenching near a porous, permeable wall and compared against a reference case of a non-porous impermeable wall. A boundary condition formulation that models species (hydrogen in this case) transport through a permeable wall, driven by the fuel species partial pressure difference between the feed and the permeate side of a selective membrane, has been implemented in a high-order finite difference direct numerical simulation code for reactive flows (S3D) by Chen et al. (2009) [1]. The present results are obtained for lean, stoichiometric and rich initial mixture conditions on the permeate side of the permeable wall and indicate that the characteristic parameters of the flame–wall interaction (wall heat flux, quenching distance) are affected to a large extent by the presence of the membrane hydrogen flux. Concurrently, the hydrogen flux through the membrane is also strongly affected by the presence of the flame during the transient flame–wall interaction process, finally resulting in a strong feedback mechanism between the membrane hydrogen flux and the flame that greatly increases boundary layer flashback speeds at fuel lean conditions.

An immersed boundary method for fluids using the XFEM and the hydrodynamic Boltzmann transport equation

This paper presents a stabilized finite element formulation of the hydrodynamic Boltzmann transport equation (HBTE) to predict nearly incompressible fluid flow. The HBTE is discretized with Hermite polynomials in the velocity variable, and a streamline upwind Petrov–Galerkin formulation is used to discretize the spatial variable. A nonlinear stabilization scheme is presented, from which a simple linear stabilization scheme is constructed. In contrast to the Navier–Stokes (NS) equations, the HBTE is a first order equation and allows for conveniently enforcing Dirichlet conditions along immersed boundaries. A simple and efficient formulation for enforcing Dirichlet boundary conditions is presented and its accuracy is studied for immersed boundaries captured by the extended finite element method (XFEM). Numerical experiments indicate that both the linear and non-linear stabilization methods are sufficiently accurate and stable, but the linear formulation reduces the computational cost significantly. The accuracy of enforcing boundary conditions is satisfactory and shows second order convergence as the mesh is refined. Augmenting the boundary condition formulation with a penalty term increases the accuracy of enforcing the boundary condition constraints, but may degrade the accuracy of the global solution. Comparisons with results of a single relaxation time lattice Boltzmann method show that the proposed finite element method features greater robustness and lesser dependence of the computational costs on the level of mesh refinement.

Formulation of wall boundary conditions in turbulent flow computations on unstructured meshes

Features of the formulation and numerical implementation of wall boundary conditions in turbulent flow computations on unstructured meshes are discussed. A method is proposed for implementing weak wall boundary conditions for a finite-volume discretization of the Reynolds-averaged Navier-Stokes equations on unstructured meshes. The capabilities of the approach are demonstrated in several gasdynamic simulations in comparison with the method of near-wall functions. The influence of the near-wall resolution on the accuracy of the computations is analyzed, and the grid dependence of the solution is compared in the case of the near-wall function method and weak boundary conditions.

Elliptic theory of differential edge operators II: boundary value problems

This is a continuation of the first author's development of the theory of elliptic differential operators with edge degeneracies. That first paper treated basic mapping theory, focusing on semi-Fredholm properties on weighted Sobolev and H\older spaces and regularity in the form of asymptotic expansions of solutions. The present paper builds on this through the formulation of boundary conditions and the construction of parametrices for the associated boundary problems. As before, the emphasis is on the geometric microlocal structure of the Schwartz kernels of parametrices and generalized inverses.

Three-dimensional vortex structures on heated micro-lattice in a gas

This paper addresses the microscale heat transfer problem from heated lattice to the gas. A micro-device for enhanced heat transfer is presented and numerically investigated. Thermal creep induces 3-D vortex structures in the vicinity of the lattice. The gas flow is in the slip flow regime (Knudsen number Kn⩽0.1). The simulations are performed using slip flow Navier–Stokes equations with boundary condition formulations proposed by Maxwell and Smoluchowski. In this study the wire thicknesses and distances of the heated lattice are varied. The surface geometrical properties alter significantly heat flux through the surface.

A stable approach for coupling multidimensional cardiovascular and pulmonary networks based on a novel pressure‐flow rate or pressure‐only Neumann boundary condition formulation

In many biomedical flow problems, reversed flows along with standard treatment of Neumann boundary conditions can cause instabilities. We have developed a method that resolves these instabilities in a consistent way while maintaining correct pressure and flow rate values. We also are able to remove the necessary prescription of both pressure and velocities/flow rates to problems where only pressure is known. In addition, the method is extended to coupled 3D/reduced-D fluid and fluid-structure interaction models. Numerical examples mainly focus on using Neumann boundary condition in cardiovascular and pulmonary systems, particularly, coupled with 3D-1D and 3D-0D models. Inflow pressure, traction, and impedance boundary conditions are first tested on idealized tubes for various Womersley numbers. Both pressure and flow rate are shown to match the analytical solutions for these examples. Our method is then tested on a coupled 1D-3D-1D artery example, demonstrating the power and simplicity of extending this method toward fluid-structure interaction. Finally, the proposed method is investigated for a coupled 3D-0D patient-specific full lung model during spontaneous breathing. All coupled 3D/reduced-D results show a perfect matching of pressure and flow rate between 3D and corresponding reduced-D boundaries. The methods are straight-forward to implement in contrast to using Lagrange multipliers as previously proposed in other studies.

A Perturbation Solution of the Thickness Natural Vibrations of a Piezoelectric Plate

This paper discusses a perturbation approach to the calculation of the natural frequencies and mode shapes for both the displacement and the electrostatic potential through-thickness vibration of an infinite piezoelectric plate. The problem is formulated within the coupled theory of linear piezoelectricity. It is described by a set of two coupled differential equations with unknown thickness displacement, the electrostatic potential and a general form of boundary conditions. A consistent perturbation solution to the natural vibration problem is described. An important element not present in the classical eigenvalue perturbation solution is that the small parameter appears in the boundary conditions; a way to handle this complication has been discussed. The natural frequencies and mode shapes obtained using the perturbation approach are compared to exact solutions, demonstrating the effectiveness of the proposed method. The advantage of the perturbation method derives from the fact that coupled piezoelectric results can be obtained from the elastic solution during the postprocessing stage.

The cosmological constant as an eigenvalue of a Sturm-Liouville problem

It is observed that one of Einstein-Friedmann’s equations has formally the aspect of a Sturm-Liouville problem, and that the cosmological constant, Λ, plays thereby the role of spectral parameter (what hints to its connection with the Casimir effect). The subsequent formulation of appropriate boundary conditions leads to a set of admissible values for Λ, considered as eigenvalues of the corresponding linear operator. Simplest boundary conditions are assumed, namely that the eigenfunctions belong to L 2 space, with the result that, when all energy conditions are satisfied, they yield a discrete spectrum for Λ>0 and a continuous one for Λ<0. A very interesting situation is seen to occur when the discrete spectrum contains only one point: then, there is the possibility to obtain appropriate cosmological conditions without invoking the anthropic principle. This possibility is shown to be realized in cyclic cosmological models, provided the potential of the matter field is similar to the potential of the scalar field. The dynamics of the universe in this case contains a sudden future singularity.

Boundary conditions for axisymmetric piezoelectric cylinder

For axisymmetric piezoelectric cylinder, the reciprocal theorem and the axisymmetric general solution of piezoelasticity are applied in a novel way to obtain the appropriate stress and mixed boundary conditions accurate to all orders for the cylinder of general edge geometry and loadings. A decay analysis technique developed by Gregory and Wan is converted into necessary conditions on the end-data of axisymmetric piezoelectric circular cylinder, and the rapidly decaying solution is established. The prescribed end-data of the circle cylinder must satisfy these conditions in order that they could generate a decaying state within the cylinder. When stress and mixed conditions are imposed on the end of cylinder, these decaying state conditions for the case of axisymmetric deformation of piezoelectric cylinder are derived explicitly. They are then used for the correct formulation of boundary conditions for the theory solution (or the interior solution) of axisymmetric piezoelectric cylinder. The results of the present paper enable us to establish a set of correct boundary conditions, most of which are obtained for the first time.