Original Boundary Conditions Research Articles

An Augmented Method for 4th Order PDEs with Discontinuous Coefficients

A fast finite difference method is developed for solving 4th order partial differential equations with discontinuous coefficients across an arbitrary interface. The method is based on an augmented approach by introducing an intermediate (augmented variable) boundary condition $$\varDelta u|_{\partial \varOmega }$$ along the boundary so that the problem can be treated as two separated Poisson equations with jumps in the source terms along the interface. Thus a fast Poisson solver can be utilized, which makes the proposed method fast. The augmented variable should be chosen such that the original boundary condition $$\frac{\partial u}{\partial n}|_{\partial \varOmega }$$ is satisfied. In discretization, the augmented variable is solved first using the Schur complement associated with the method. The proposed method is probably the first finite difference method for such an interface problem although the paper is partially motivated by the immersed finite element method (Lin et al in J Comput Appl Math 235(13):3953–3964, 2011) for such a problem. Numerical experiments against analytic solutions show that the computed solution using the proposed method has second order accuracy (convergence) in the maximum norm. Numerical results and analysis are also presented for various jump ratios and arbitrary interfaces.

Linear electron stability for a bi-kinetic sheath model

We establish the linear stability of an electron equilibrium for an electrostatic and collisionless plasma in interaction with a wall. The equilibrium we focus on is called in plasma physics a Debye sheath. Specifically, we consider a two species (ions and electrons) Vlasov–Poisson–Ampère system in a bounded and one dimensional geometry. The interaction between the plasma and the wall is modeled by original boundary conditions: On the one hand, ions are absorbed by the wall while electrons are partially re-emitted. On the other hand, the electric field at the wall is induced by the accumulation of charged particles at the wall. These boundary conditions ensure the compatibility with the Maxwell–Ampère equation. A global existence, uniqueness and stability result for the linearized system is proven. The main difficulty lies in the fact that (due to the absorbing boundary conditions) the equilibrium is a discontinuous function of the particle energy, which results in a linearized system that contains a degenerate transport equation at the border.

Variational formulation of dynamic problems for a nonlinear Cosserat medium

Abstract A variational formulation of dynamic problems for a geometrically and physically nonlinear elastic Cosserat medium is obtained in the form of the problem of finding the stationary point of Hamilton's functional. Variations of the strain and rotation tensors and the linear and angular velocity vectors are calculated. The equivalence of the Euler equations with natural boundary conditions to the equations of motion with the original boundary conditions in the case of potential force and torque loads is proved. A nontrivial potentiality condition of the torque (bulk and surface) loads is obtained.

Analysis of High-Efficiency Cross-Polarized Converter at Oblique Incidence

In this letter, we present a theoretical study on a cross-polarized converter based on the reflective metasurface that is composed of slanted metallic cut-lines on a perfect electrical conductor backed substrate. According to the model, although reflection and refraction laws are still applicable in the interface between the air and the substrate, the metallic cut-lines break the original boundary condition. In order to describe the broken boundary condition, the generalized Fresnel's formulas are established based on the traditional Fresnel's formulas and the induced radiation of the metallic cut-lines. Finally, multiple reflections and refractions are resulted from the back metallic sheet in the interface of metallic cut-lines, and the total reflected wave is obtained by summing all waves except the incident wave in air. The theoretical results agree well with the simulated and measured results at the oblique incidence. When incidence angle varies from 0° to 25°, the metasurface can convert more than 99% incident wave into its cross-polarized reflected wave within a relative bandwidth of 22%.

A highly accurate and efficient algorithm for electrostatic interactions of charged particles confined by parallel metallic plates.

We present an accurate and efficient algorithm to calculate the electrostatic interaction of charged point particles with partially periodic boundary conditions that are confined along the non-periodic direction by two parallel metallic plates. The method preserves the original boundary conditions, leading to an exact solution of the problem. In addition, the scaling complexity is quasilinear O(Nln(N)), where N is the number of particles in the simulation box. Based on the superposition principle in electrostatics, the problem is split into two electrostatic problems where each can be calculated by the appropriate Poisson solver. The method is applied to NaCl ultra-thin films where its dielectric response with respect to an external bias voltage is investigated. Furthermore, the total charge induced on the metallic boundaries can be calculated to an arbitrary precision.

Nonlinear eigenvalue problem in the integral transforms solution of convection-diffusion with nonlinear boundary conditions

Purpose – The purpose of this paper is to propose the generalized integral transform technique (GITT) to the solution of convection-diffusion problems with nonlinear boundary conditions by employing the corresponding nonlinear eigenvalue problem in the construction of the expansion basis. Design/methodology/approach – The original nonlinear boundary condition coefficients in the problem formulation are all incorporated into the adopted eigenvalue problem, which may be itself integral transformed through a representative linear auxiliary problem, yielding a nonlinear algebraic eigenvalue problem for the associated eigenvalues and eigenvectors, to be solved along with the transformed ordinary differential system. The nonlinear eigenvalues computation may also be accomplished by rewriting the corresponding transcendental equation as an ordinary differential system for the eigenvalues, which is then simultaneously solved with the transformed potentials. Findings – An application on one-dimensional transient diffusion with nonlinear boundary condition coefficients is selected for illustrating some important computational aspects and the convergence behavior of the proposed eigenfunction expansions. For comparison purposes, an alternative solution with a linear eigenvalue problem basis is also presented and implemented. Originality/value – This novel approach can be further extended to various classes of nonlinear convection-diffusion problems, either already solved by the GITT with a linear coefficients basis, or new challenging applications with more involved nonlinearities.

Selection of optimal artificial boundary condition (ABC) frequencies for structural damage identification

In this paper, the sensitivities of artificial boundary condition (ABC) frequencies to the damages are investigated, and the optimal sensors are selected to provide the reliable structural damage identification. The sensitivity expressions for one-pin and two-pin ABC frequencies, which are the natural frequencies from structures with one and two additional constraints to its original boundary condition, respectively, are proposed. Based on the expressions, the contributions of the underlying mode shapes in the ABC frequencies can be calculated and used to select more sensitive ABC frequencies. Selection criteria are then defined for different conditions, and their performance in structural damage identification is examined with numerical studies. From the findings, conclusions are given.

Modeling of Hydrodynamics of a Highly Concentrated Granular Medium on the Basis of a Power-Law

The paper deals with the movement of the granular medium at a high concentration on the basis of the “power” of the liquid. Based on the original partial slip boundary conditions on the walls of protection obtained with experimental and numerical data to flow in the channel at a flow obstacle.

A comparison of the continuous and discrete adjoint approach extended based on the standard lattice Boltzmann method in flow field inverse optimization problems

The objective of this research is the numerical implementation and comparison between the performance of the continuous and discrete adjoint Lattice Boltzmann (LB) methods in optimization problems of unsteady flow fields. For this purpose, a periodic two-dimensional incompressible channel flow affected by the constant and uniform body forces is considered as the base flow field. The standard LB method and D2Q9 model are employed to solve the flow field. Moreover, the inverse optimization of the selected flow field is defined by considering the body forces as the design variables and the sum of squared errors of flow field variables on the whole field as the cost function. In this regard, the continuous and discrete adjoint approaches extended based on the LB method are used to achieve the gradients of the cost function with respect to the design variables. Finally, the numerical results obtained from the continuous adjoint LB method are compared with the discrete one, and the accuracy and efficiency of them are discussed. In addition, the validity of the obtained cost function gradients is investigated by comparing with the results of the standard forward finite difference and complex step methods. The numerical results show that regardless of the implementation cost of the two approaches, the computational cost to evaluate the gradients in each optimization cycle for the discrete adjoint LB approach is slightly more than the other one but has a little higher convergence rate and needs a smaller number of cycles to converge. Besides, the gradients obtained from the discrete version have a better agreement with those of the complex step method. Eventually, based on the structural similarities of the continuous LB equation and its corresponding adjoint one and using the simple periodic and complete bounce-back boundary conditions for the LB equation, the improved boundary conditions for the continuous adjoint LB equation are presented. The numerical results show that the use of these boundary conditions instead of the original adjoint boundary conditions significantly improves the relative accuracy and also the convergence rate of the continuous adjoint LB method.

Spectral element modeling and analysis of the transverse vibration of a laminated composite plate

This paper presents a frequency-domain spectral element model for the symmetric laminated composite plates which have finite dimensions in two orthogonal directions, i.e., in the x- and y-directions. The spectral element model is developed by using two methods in combination: the splitting of original boundary conditions and the so-called super spectral element method in which both the Kantorovich method-based finite strip element method and the frequency-domain waveguide method are utilized. The present spectral element model has nodes (or degrees of freedom (DOFs)) only on four edges of a finite element, i.e., no nodes inside the finite element. Accordingly the total number of DOFs used in the dynamic analysis can be drastically decreased to lead to a significant decrease of the computation cost, when compared with the standard finite element method (FEM). The high accuracy of the present spectral element model is verified in due course by the comparison with the results by two solution methods: the exact theory available in the literature and the standard finite element model developed in this study.

Well‐Posed Treatment of Space‐Charge Layers in the Electroneutral Limit of Electrodiffusion

AbstractThe electroneutral model describes cellular electrical activity, accounting for ionic concentration dynamics without resolution of the fine spatial scales of the space‐charge layer. This is done by asserting that the ionic solution is electrically neutral at each point in space. However, electroneutrality is inconsistent with the original boundary conditions at cell membranes. We consider three separate methods of resolving this inconsistency that result in well‐posed models that are accurate approximations to a detailed model in which the space‐charge layer is fully resolved. A particular electrodiffusion problem is utilized to make the discussion specific.© 2015 Wiley Periodicals, Inc.

New Bondi-type outgoing boundary condition for the Einstein equations with cosmological constant

In the middle of last century, Bondi and his coworkers proposed an outgoing boundary condition for the Einstein equations. Recently, more and more observations imply that the Einstein equations should include a nonzero cosmological constant. A spacetime with a positive cosmological constant approaches to a de Sitter space asymptotically. Bondi's original boundary condition is not valid for these asymptotically de Sitter spacetimes. But the traditional conformally flat boundary condition excludes the gravitational radiation for the asymptotically de Sitter spacetimes. In this work, a new Bondi-type outgoing boundary condition based on Bondi–Sachs coordinates is considered. With this new boundary condition, the gravitational wave behavior for the asymptotically de Sitter spacetime is similar to the one for the asymptotically Minkowski spacetime. The traditional conformally flat boundary condition falls into a special subclass of the new boundary condition.

Viscosity measurement techniques in Dissipative Particle Dynamics

In this study two main groups of viscosity measurement techniques are used to measure the viscosity of a simple fluid using Dissipative Particle Dynamics, DPD. In the first method, a microscopic definition of the pressure tensor is used in equilibrium and out of equilibrium to measure the zero-shear viscosity and shear viscosity, respectively. In the second method, a periodic Poiseuille flow and start-up transient shear flow is used and the shear viscosity is obtained from the velocity profiles by a numerical fitting procedure. Using the standard Lees–Edward boundary condition for DPD will result in incorrect velocity profiles at high values of the dissipative parameter. Although this issue was partially addressed in Chatterjee (2007), in this work we present further modifications (Lagrangian approach) to the original LE boundary condition (Eulerian approach) that will fix the deviation from the desired shear rate at high values of the dissipative parameter and decrease the noise to signal ratios in stress measurement while increases the accessible low shear rate window. Also, the thermostat effect of the dissipative and random forces is coupled to the dynamic response of the system and affects the transport properties like the viscosity and diffusion coefficient. We investigated thoroughly the dependency of viscosity measured by both Eulerian and Lagrangian methodologies, as well as numerical fitting procedures and found that all the methods are in quantitative agreement.

Velocity dependent Maxwell boundary conditions in DSMC

Recently Struchtrup (2013) proposed an extension to the original Maxwell boundary conditions for the Boltzmann equation which introduces velocity dependent accommodation coefficients. These boundary conditions are implemented into the direct simulation Monte Carlo (DSMC) method. The effect of the velocity dependent Maxwell (VDM) boundary conditions on thermal transpiration phenomena is studied for two-dimensional micro-cavities. Variation of the three microscopic parameters provided by the VDM boundary condition yields changes in slip velocity, temperature jump and the thermal transpiration effect. The results indicate that the strength of thermal transpiration can change and, depending on the values of the coefficients, the rarefied flow can be driven from warmer toward colder regions.

Comparison of Anisotropy Reduction Strategies for Transformation Optics Designs

In this paper, we present new strategies to reduce anisotropy in transformation optics designs and compare them to other techniques. Perturbation functions are used to modify the original transformation to achieve a quasi-conformal map, resulting in a medium with isotropic properties. All strategies investigated have no effect on the original boundary conditions of the transformation, such that neither the initial design is affected nor are reflections at the boundaries introduced. The results show that there exists a clear compromise between the residual anisotropy and the required refractive index contrast in the optimized transformation, but the former can be made as small as desired when the degree of freedom provided by perturbation functions is increased, although it is never exactly zero.

A finite difference method for fractional diffusion equations with Neumann boundary conditions

Abstract A finite difference numerical method is investigated for fractional order diffusion problems in one space dimension. The basis of the mathematical model and the numerical approximation is an appropriate extension of the initial values, which incorporates homogeneous Dirichlet or Neumann type boundary conditions. The wellposedness of the obtained initial value problem is proved and it is pointed out that each extension is compatible with the original boundary conditions. Accordingly, a finite difference scheme is constructed for the Neumann problem using the shifted Grünwald–Letnikov approximation of the fractional order derivatives, which is based on infinite many basis points. The corresponding matrix is expressed in a closed form and the convergence of an appropriate implicit Euler scheme is proved.

Transient response of general one-dimensional distributed systems through eigenfunction expansion with an implicit filter scheme

The current work is concerned with eigenfunction expansion of the transient response of general one-dimensional non-self-adjoint distributed dynamic systems. The main goal of the current work is to propose an alternative eigenfunction expansion approach for transient analysis of general one-dimensional non-self-adjoint systems which avoids complex-valued eigenvalue problems. With the implicit filter scheme proposed here one splits the solution into two additive components, namely, a filtered solution for which all boundary conditions are homogeneous and a carefully selected filter. Moreover, the eigenvalue problem associated with the differential equation for the filtered solution is self-adjoint and, therefore, all eigenvalues are real-valued and the eigenfunctions satisfy the classical orthogonality relations. The unknown time-dependent coefficients of the filter are then solved simultaneously with the modal co-ordinates of the filtered solution in order to satisfy the original boundary conditions. The proposed approach is demonstrated on a distributed system composed of a cantilever beam with end mass, viscous damper and linear spring.

Analytical solutions for stream‐aquifer flow exchange under varying head asymmetry and river penetration: Comparison to numerical solutions and use in regional groundwater models

AbstractAn analytical approach is presented to characterize the local flow exchange conductance and boundary condition between a stream and a hydraulically connected aquifer. Due to the curvilinear nature of the flow pattern in the vicinity of the stream, with numerical procedures, very fine grids would have to be used in order to secure accurate results. This is a waste of numerical efforts when the solution can be obtained analytically. Here, we show that the local analytical procedure for the determination of the stream‐aquifer flow exchange (SAFE) coefficient can be integrated within a regional numerical groundwater model through an original boundary condition. This is a very practical result. With this new approach, it is quite simple to calculate accurately the discharges on the two sides of the river when the heads are different. This is an important capability for example when wells are present near a river. The results show also that the extent of penetration of the stream is a factor in the estimation of the discharges. At the very least, the results can be used to choose a grid size that will yield correct estimates within an a priori selected acceptable level of accuracy.

Crop yield response to soil fertility and N, P, K inputs in different environments: Testing and improving the QUEFTS model

Global food production strongly depends on availability of nutrients. Assessment of future global phosphorus (P) fertilizer demand in interaction with nitrogen (N) and potassium (K) fertilizers under different levels of food demand requires a model-based approach. In this paper we tested use of the QUEFTS model (Quantitative Evaluation of Fertility of Tropical Soils) for assessing crop yields in response to N, P and K application in different environments. QUEFTS was initially developed to simulate interactions between N, P and K for tropical soils under maize crop. We performed an extensive model analysis of crop yields in relation to soil and fertilizer nutrients for six field data sets with maize, rice, and wheat crops grown in tropical and temperate regions. The model equations had to be adapted to broaden the model applicability beyond the original boundary conditions of pH, rain-fed cropping systems, optimum harvest index and temperature. Recalibration and modification resulted in a good agreement between model predicted and observed yields. Our results indicate that the adjustments increased the applicability of the model. However, for application in global studies QUEFTS is data demanding and, also, further testing (and probably improvement) is needed, since various processes (e.g. inputs of other nutrients than N, P and K, sub-soil properties and water supply) are ignored in the model, but may differ dramatically across the globe.

A semi-analytical method to generate g-functions for geothermal bore fields

This paper introduces a new methodology for the generation of thermal response factors of geothermal bore fields using the concept of g-functions introduced by Eskilson. Boreholes are divided into segments to consider the variation of the heat extraction rates along the length of the boreholes and the analytical finite line source (FLS) solution is used to calculate the temperature variations at the wall of each borehole segment along the axial direction. The proposed methodology accounts for the time variation of the heat extraction rates among boreholes and along the length of individual boreholes to obtain a uniform borehole wall temperature equal for all boreholes in accordance with the original boundary conditions proposed by Eskilson. In addition, the methodology is generalized to account for boreholes of different lengths and buried depths. g-Functions calculated with the proposed methodology are compared to the numerical technique used by Eskilson to derive the g-functions for fields of 1 to 12×12 boreholes. The difference between the two models is within 5% for all studied bore fields, except for fields of boreholes located on a single row. The variation of the heat extraction rates of individual boreholes along their length as well as in time also showed good agreement with the numerical model. It is shown that using 12 borehole segments is adequate to calculate the g-functions in most practical cases. For instance, the error on the g-function of a 10×10 bore field calculated using 12 borehole segments is 2.2% after 20years and 4.7% at steady-state.