Domain Boundary Conditions Research Articles

Convergence analysis of a hp‐finite element approximation of the time‐harmonic Maxwell equations with impedance boundary conditions in domains with an analytic boundary

AbstractWe consider a nonconforming hp‐finite element approximation of a variational formulation of the time‐harmonic Maxwell equations with impedance boundary conditions proposed by Costabel et al. The advantages of this formulation is that the variational space is embedded in H1 as soon as the boundary is smooth enough (in particular it holds for domains with an analytic boundary) and standard shift theorem can be applied since the associated boundary value problem is elliptic. Finally in order to perform a wavenumber explicit error analysis of our problem, a splitting lemma and an estimation of the adjoint approximation quantity are proved by adapting to our system the results from Melenk and Sauter obtained for the Helmholtz equation. Some numerical tests that illustrate our theoretical results are also presented. Analytic regularity results with bounds explicit in the wavenumber of the solution of a general elliptic system with lower order terms depending on the wavenumber are needed and hence proved.

Super-critical Neumann problems on unbounded domains**C O Alves was partially supported by CNPq/Brazil 304804/2017-7. A Moameni is pleased to acknowledge the support of the National Sciences and Engineering Research Council of Canada. A Moameni is also grateful to Craig Cowan for several fruitful discussions on this project.

In this paper, by making use of a new variational principle, we prove existence of nontrivial solutions for two different types of semilinear problems with Neumann boundary conditions in unbounded domains. Namely, we study elliptic equations and Hamiltonian systems on the unbounded domain where Br is a ball centred at the origin with radius r in . Our proofs consist of several new and novel ideas that can be used in broader contexts.

Principal eigenvalues of elliptic BVPs with glued Dirichlet-Robin mixed boundary conditions. Large potentials on the boundary conditions

In this paper we perform a complete study about the existence, uniqueness and simplicity of the principal eigenvalue of a class of elliptic eigenvalue problems with glued Dirichlet-Robin boundary conditions on a component of the boundary of the domain and Dirichlet boundary conditions on the other component of the boundary. Moreover, this principal eigenvalue and its normalized principal eigenfunction are approached in R×H1(Ω) by principal eigen-pairs of boundary eigenvalue problems with classical mixed boundary conditions. In some sense, the glued mixed boundary conditions analyzed in this work may be regarded as classical mixed boundary conditions with large potentials in the Robin boundary condition. The principal eigenvalues analyzed in this paper, play a crucial role in analyzing the existence, uniqueness and asymptotic behavior of the positive solutions of certain kinds of semilinear elliptic boundary value problems with nonlinear boundary conditions and spatial heterogeneities. The main technical tools used to carry out the mathematical analysis of this work are variational and monotonicity techniques. The results in [2], [3] and [4] play a crucial role to obtain some of the results of this work.

Numerical simulation of heat transfer in a parallel plate channel and promote dissipative particle dynamics method using different weight functions

In this paper heat transfer in a channel with parallel plates has been studied using Dissipative Particle Dynamics with energy conservation (eDPD). To simulate and present different thermal flows in finite geometries, a computer program was developed. The thermal boundary condition of the wall is assumed to be a constant temperature. Each particle in contact with the wall achieves the wall temperature. Also periodic boundary condition in the direction of the flow of the solution domain and no-slip boundary condition on the walls have been considered. The no-slip boundary condition is possible by freezing the layers of particles on the walls and applying bounce-back reflection. The results show dimensionless temperature and velocity profiles perpendicular to the direction of the flow along the channel. Finally, in this research it is tried to endeavored the fluctuations near the wall and promote the method by proposing different weight functions. Four different weight functions were used and the results on the velocity and temperature profiles were investigated. All of the results in the present work were compared and verified with the previous results.

A Predictor-Corrector Finite Element Method for Time-Harmonic Maxwell’s Equations in Polygonal Domains

The overall efficiency and accuracy of standard finite element methods may be severely reduced if the solution of the boundary value problem entails singularities. In the particular case of time-harmonic Maxwell’s equations in nonconvex polygonal domains Ω, H1-conforming nodal finite element methods may even fail to converge to the physical solution. In this paper, we present a new nodal finite element adaptation for solving time-harmonic Maxwell’s equations with perfectly conducting electric boundary condition in general polygonal domains. The originality of the present algorithm lies in the use of explicit extraction formulas for the coefficients of the singularities to define an iterative procedure for the improvement of the finite element solutions. A priori error estimates in the energy norm and in the L2 norm show that the new algorithm exhibits the same convergence properties as it is known for problems with regular solutions in the Sobolev space H2Ω2 in convex and nonconvex domains without the use of graded mesh refinements or any other modification of the bilinear form or the finite element spaces. Numerical experiments that validate the theoretical results are presented.

Effect of Domain Size, Boundary, and Loading Conditions on Mechanical Properties of Amorphous Silica: A Reactive Molecular Dynamics Study.

Mechanical properties are very important when choosing a material for a specific application. They help to determine the range of usefulness of a material, establish the service life, and classify and identify materials. The size effect on mechanical properties has been well established numerically and experimentally. However, the role of the size effect combined with boundary and loading conditions on mechanical properties remains unknown. In this paper, by using molecular dynamics (MD) simulations with the state-of-the-art ReaxFF force field, we study mechanical properties of amorphous silica (e.g., Young’s modulus, Poisson’s ratio) as a function of domain size, full-/semi-periodic boundary condition, and tensile/compressive loading. We found that the domain-size effect on Young’s modulus and Poisson’s ratio is much more significant in semi-periodic domains compared to full-periodic domains. The results, for the first time, revealed the bimodular and anisotropic nature of amorphous silica at the atomic level. We also defined a “safe zone” regarding the domain size, where the bulk properties of amorphous silica can be reproducible, while the computational cost and accuracy are in balance.

The dynamics of a zooplankton–fish system in aquatic habitats

Diel vertical migration is a common movement pattern of zooplankton in marine and freshwater habitats. In this paper, we use a temporally periodic reaction–diffusion–advection system to describe the dynamics of zooplankton and fish in aquatic habitats. Zooplankton live in both the surface water and the deep water, while fish only live in the surface water. Zooplankton undertake diel vertical migration to avoid predation by fish during the day and to consume sufficient food in the surface water during the night. We establish the persistence theory for both species as well as the existence of a time-periodic positive solution to investigate how zooplankton manage to maintain a balance with their predators via vertical migration. Numerical simulations discover the effects of migration strategy, advection rates, domain boundary conditions, as well as spatially varying growth rates, on persistence of the system.

Maximal regularity of the Stokes system with Navier boundary condition in general unbounded domains

Consider the instationary Stokes system in general unbounded domains $\Omega \subset \mathbb{R}^n$, $n \geq 2$, with boundary of uniform class $C^3$, and Navier slip or Robin boundary condition. The main result of this article is the maximal regularity of the Stokes operator in function spaces of the type $\tilde{L}^q$ defined as $L^q \cap L^2$ when $q \geq 2$, but as $L^q + L^2$ when $1 < q < 2$, adapted to the unboundedness of the domain.

Weighted gradient estimates for elliptic problems with Neumann boundary conditions in Lipschitz and (semi-)convex domains

Let n≥2 and Ω be a bounded Lipschitz domain in Rn. In this article, the authors investigate global (weighted) norm estimates for the gradient of solutions to Neumann boundary value problems of second order elliptic equations of divergence form with real-valued, bounded, measurable coefficients in Ω. More precisely, for any given p∈(2,∞), two necessary and sufficient conditions for W1,p estimates of solutions to Neumann boundary value problems, respectively, in terms of a weak reverse Hölder inequality with exponent p or weighted W1,q estimates of solutions with q∈[2,p] and some Muckenhoupt weights, are obtained. As applications, for any given p∈(1,∞) and ω∈Ap(Rn) (the class of Muckenhoupt weights), the authors establish weighted Wω1,p estimates for solutions to Neumann boundary value problems of second order elliptic equations of divergence form with small BMO coefficients on bounded (semi-)convex domains. As further applications, the global gradient estimates are obtained, respectively, in (weighted) Lorentz spaces, (Lorentz–)Morrey spaces, (weighted) Orlicz spaces, and variable Lebesgue spaces.

Regional climate model RCA4 simulations of temperature and precipitation over the Arabian Peninsula: sensitivity to CORDEX domain and lateral boundary conditions

Regional Climate Models (RCMs) are increasingly used to add small-scale processes at higher grid resolution that are not represented by their Lateral Boundary Conditions (LBCs). Rossby Centre regional atmospheric model, RCA4, has downscaled three Global Climate Models (GCMs), namely, CNRM-CM5, EC-EARTH and GFDL-ESM2M in the COordinated Regional climate Downscaling EXperiment (CORDEX) framework for Middle East North Africa (MENA) and South Asia (SA) domains. Arabian Peninsula is covered in both MENA and SA simulations, which gives a unique opportunity to study the effects of CORDEX domain and LBCs on the simulation of temperature and precipitation by RCM. It is examined by calculating the differences between MENA and SA simulations for different driving GCMs in the historical (1976–2005) and future (2071–2100) periods under RCP8.5 emission scenario, for both summer (dry) and winter (wet) seasons. RCA4 performs generally well when simulating the observed temperature and precipitation patterns, with some local wet biases over Asir Mountains and cold bias over the north eastern parts of Saudi Arabia. The simulations of temperature seem to be very sensitive to the simulation domain (i.e., MENA and SA) and less sensitive to different LBCs, whereas in case of precipitation LBCs seems to play a dominant role. The MENA simulations generally project about 2 °C warmer and drier climate compared to SA simulations by the end of this century, which is comparable to the differences arising due to different LBCs and climate change.

A coarse-grid incremental pressure projection method for accelerating low Reynolds number incompressible flow simulations

Coarse grid projection (CGP) multigrid techniques are applicable to sets of equations that include at least one decoupled linear elliptic equation. In CGP, the linear elliptic equation is solved on a coarsened grid compared to the other equations, leading to savings in computations time and complexity. One of the most important applications of CGP is when a pressure correction scheme is used to obtain a numerical solution to the Navier-Stokes equations. In that case there is an elliptic pressure Poisson equation. Depending on the pressure correction scheme used, the CGP method and its performance in terms of acceleration rate and accuracy level vary. The CGP framework has been established for non-incremental pressure projection techniques. In this article, we apply CGP methodology for the first time to incremental pressure correction schemes. Both standard and rotational forms of the incremental algorithms are considered. The influence of velocity Dirichlet and natural homogenous boundary conditions in regular and irregular domains with structured and unstructured triangular finite element meshes is investigated. $L^2$ norms demonstrate that the level of accuracy of the velocity and the pressure fields is preserved for up to three levels of coarsening. For the test cases investigated, the speedup factors range from 1.248 to 102.715.

Consistent coupled optical and thermal analysis of volumetric solar receivers with honeycomb absorbers

In concentrating solar power plants with central towers, successful design of volumetric solar receivers requires proper understanding of the interaction between optical, heat transfer, and fluid flow phenomena occurring at the microscopic scale of receiver structure material and their effect on the overall solar-to-thermal efficiency. In the present article, coupled, 3D, optical, heat transfer and fluid flow numerical models have been developed for the analysis and design of honeycomb volumetric receiver modules. The optical model considers the absorptivity and micro dimensions of honeycomb absorber structure and employs a Monte Carlo ray tracing technique to calculate and analyze the absorbed solar heat flux distribution. This, in turn, is employed as a volumetric heat source term at the solid surface for consistent heat transfer and fluid flow modeling using a realistic solution domain and proper boundary conditions. The validated models have been employed to investigate the effects of different types of absorber materials, material absorptivity, and air flow rate on the performance of the solar receiver. It has been shown that positive volumetric effect and high solar-to-thermal efficiency can be obtained by controlling the absorbed radiation heat flux distribution within the honeycomb receiver using surface coating of the absorber material.

Optimal boundary control for the stationary Boussinesq equations with variable density

Certain classes of optimal boundary control problems for the Boussinesq equations with variable density are studied. Controls for the velocity vector and temperature are applied on parts of the boundary of the domain, while Dirichlet and Navier friction boundary conditions for the velocity and Dirichlet and Robin boundary conditions for the temperature are assumed on the remaining parts of the boundary. As a first step, we prove a result on the existence of weak solution of the dynamical equations; this is done by first expressing the fluid density in terms of the stream-function. Then, the boundary optimal control problems are analyzed, and the existence of optimal solutions are proved; their corresponding characterization in terms of the first-order optimality conditions are obtained. Such optimality conditions are rigorously derived by using a penalty argument since the weak solutions are not necessarily unique neither isolated, and so standard methods cannot be applied.

Hybrid-Trefftz finite elements for non-homogeneous parabolic problems using a novel dual reciprocity variant

A new hybrid-Trefftz finite element for the solution of transient, non-homogeneous parabolic problems is formulated. The governing equations are first discretized in time and reduced to a series of non-homogeneous elliptic problems in space. The complementary and particular solutions of each elliptic problem are approximated independently. The complementary solution is expanded in Trefftz bases, designed to satisfy exactly the homogeneous form of the problem. Trefftz bases are regular, and defined independently for each finite element, using arbitrary orders. A novel dual reciprocity method is used for the approximation of the particular solution, to avoid domain integration. The same, regular basis is used for the expansions of the source function and particular solution, avoiding the cumbersome expressions of the latter that typify conventional dual reciprocity techniques. Moreover, the bases of the complementary and particular solutions are defined by the same expressions, with different wave numbers. The finite element formulation is obtained by enforcing weakly the domain equations and boundary conditions. To enhance the reproducibility of this work, the formulation is implemented in the computational platform FreeHyTE, where it takes advantage of the pre-programmed numerical procedures and graphical user interfaces. The resulting software is open-source, user-friendly and freely distributed to the scientific community through the FreeHyTE web page.

On the weakly degenerate Allen-Cahn equation

AbstractIn this paper we consider a one-dimensional Allen-Cahn equation with degeneracy in the interior of the domain and Neumann boundary conditions. We allow the diffusivity coefficient vanish at some point of the space domain and we are addressed on the existence of stable non-constant solution.

Global in Space Regularity Results for the Heat Equation with Robin-Neumann Type Boundary Conditions in Time-varying Domains

This article deals with the heat equation @tu @2x u = f in D; D = {(t; x) 2 R2 : a &lt; t &lt; b; (t) &lt; x &lt; +1} with the function satisfying some conditions and the problem is supplemented with boundary conditions of Robin-Neumann type. We study the global regularity problem in a suitable parabolic Sobolev space. We prove in particular that for f 2 L2(D) there exists a unique solution u such that u; @tu; @jx u 2 L2 (D) ; j = 1; 2: The proof is based on the domain decomposition method. This work complements the results obtained in [10].

Reverse Thrust Aerodynamics of Variable Pitch Fans

Abstract Variable pitch fans are of interest for future low-pressure ratio fan systems since they provide improved operability relative to fixed pitch fans. If they can also be re-pitched such that they generate sufficient reverse thrust they could eliminate the engine drag and weight penalty associated with bypass duct thrust reversers. This paper sets out to understand the details of the 3D fan stage flow field in reverse thrust operation. This study uses the Advanced Ducted Propulsor variable pitch fan test case, which has a design fan pressure ratio of 1.29. Comparison with spanwise probe measurements show that the computational approach is valid for examining the variation of loss and work in the rotor in forward thrust. The method is then extended to a reverse thrust configuration using an extended domain and appropriate boundary conditions. Computations, run at two rotor stagger settings, show that the spanwise variation in relative flow angle onto the rotor aligns poorly to the rotor inlet metal angle. This leads to two dominant rotor loss sources: one at the tip associated with positive incidence and the second caused by negative incidence at lower span fractions. The second loss is reduced by opening the rotor stagger setting, and the first increases with rotor suction surface Mach number. The higher mass flow at more open rotor settings provide higher gross thrust, up to 49% of the forward take-off value, but is limited by the increased loss at high speed.

Analytic laws for direct calibration of discrete element modeling of brittle elastic media using cohesive beam model

In this study, a new methodology for the calibration of microscopic parameters for the cohesive beam model of discrete element method (DEM) applied to elastic brittle material is presented. This method enables the entry of material mechanical values directly into DEM simulations without any calibration steps. Several DEM simulations of tensile tests with different microscopic parameter values were carried out to generate a database of macroscopic parameter responses. This database was analyzed in order to deduce analytic laws by using nonlinear least square method. To validate the proposed calibration method, DEM simulations, which use the results of the calibrated microscopic parameters, were carried out. The macroscopic responses were compared to theoretical or experimental values. These validation tests were performed separately for two typical brittle elastic materials, i.e., soda-lime glass and alumina, with different shapes/sizes of discrete domain and various boundary conditions. Results between numerical and experimental values are in good accordance regarding the variability induced by this stochastic approach.

Asymptotic analysis of unsteady neutron transport equation

Consider the unsteady neutron transport equation with diffusive boundary condition in 2D convex domains. We establish the diffusive limit with both initial layer and boundary layer corrections. The major difficulty is the lack of regularity in the boundary layer with geometric correction. Our contribution relies on a detailed analysis of asymptotic expansions inspired by the compatibility condition and an intricate L2m − L∞ framework, which yields stronger remainder estimates.

Bilinear Optimal Control Problem for the Stationary Navier–Stokes Equations with Variable Density and Slip Boundary Condition

An optimal control problem for the stationary Navier–Stokes equations with variable density is studied. A bilinear control is applied on the flow domain, while Dirichlet and Navier boundary conditions for the velocity are assumed on the boundary. As a first step, we enunciate a result on the existence of weak solutions of the dynamical equation; this is done by firstly expressing the fluid density in terms of the stream-function. Then, the bilinear optimal control problem is analyzed, and the existence of optimal solutions are proved; their corresponding characterization regarding the first-order optimality conditions are obtained. Such optimality conditions are rigorously derived by using a penalty argument since the weak solutions are not necessarily unique neither isolated, and so standard methods cannot be applied.