Effect Of Boundary Conditions Research Articles

Stress localization investigation of additively manufactured GRCop-42 thin-wall structure

A full-field crystal plasticity (CP) framework is presented for the GRCop-42 alloy to study microscopic mechanical behavior and local stress heterogeneities. The microstructures of additively manufactured (AM) materials are often unique relative to conventionally processed materials, and the local thermal histories drive these differences during the build process. These thermal histories depend on the process parameters (laser power, scan speed, and scan strategy) and the part geometry. Prior research has shown that the mechanical properties of thin-walled structures can vary significantly with wall thickness due to changes in the thermal boundary conditions during manufacturing. It is, therefore, desirable to perform CP simulations based on the phenomenological constitutive model to predict the local mechanical responses induced by microstructural heterogeneities. This work generates representative microstructures based on experimentally collected grain information (i.e., texture) for grain scale stress analysis, and the material constitutive parameters are calibrated using the experimental mechanical testing data. We specifically investigated the effect of crystallographic texture and grain morphologies on the size-dependent mechanical properties of AM GRCop-42. The selection of appropriate material properties for implementing an effective free surface boundary condition and the influence of adjacent buffer layers are also discussed. Analysis of local field results reveals a strong correlation between stress localization and the initial grain orientation. However, no significant relationship between the misorientation of the individual adjacent grains and the average misorientation is observed.

Wave scattering and radiation by a surface-piercing vertical truncated metamaterial cylinder

In this paper, we study wave scattering and radiation by a surface-piercing vertical truncated metamaterial cylinder composed of a closely spaced array of thin vertical barriers, between which fluid can flow. A theoretical model is developed under full depth-dependent linearised water wave theory, where an effective medium equation and effective boundary conditions are employed, respectively, to describe the fluid motion inside the cylinder and match the flow between the fluid regions in and outside the metamaterial cylinder. A damping mechanism is introduced at the surface of the fluid occupied by the metamaterial cylinder to consider the wave power dissipation in narrow gaps between the thin vertical plates. The wave excitation forces acting on the cylinder and the hydrodynamic coefficients can be calculated straightforwardly in terms of the velocity potential inside the cylinder. An alternative way is by using the velocity potential outside the cylinder, the expression of which has the reduction of the integral and an infinite accumulation that are included in the straightforward expression. The results highlight the patterns of the radiated waves induced by the oscillation of the cylinder and the characteristics of the hydrodynamic coefficients. The metamaterial cylinder when fixed in place and with a damping mechanism included is found to capture more wave power than that of a traditional axisymmetric heaving wave energy converter over a wide range of wave frequencies.

The Robin boundary condition for modelling heat transfer

The heat exchange between a rigid body and a fluid is usually modelled by the Robin boundary condition saying that the heat flux through the interface is proportional to the difference between their temperatures. Such interface law describes only the unilateral heat exchange. The goal of this paper is to compare the Robin boundary condition starting with the transmission condition (the temperature and the flux continuity) using rigorous mathematical analysis. Our main results are the following. We first show that a generalized version of the Robin boundary condition can be justified. Second, we prove that replacing the generalized by the standard Robin condition can be justified for high convection velocity if the conductivity of the surrounding liquid is much lower than that of the body. On the other hand, if the fluid conducts much better than the body, then the effective boundary condition is shown not to be the Robin one, but it involves second-order derivatives. We strongly believe that those findings bring new insights to the physics of the heat exchange processes and, thus, could prove useful in engineering practice.

An approximate secular equation of Rayleigh-like waves in coated elastic half-space containing voids

Propagation of Rayleigh-like surface waves is studied in an isotropic elastic solid half-space coated with a thin isotropic elastic solid layer. The half-space and the thin coated layer are in welded contact with each other and contain a uniform distribution of small void pores. Effective boundary condition method is employed to derive an approximate secular equation of second-, third-, and fourth-orders in terms of dimensionless wavenumber. The corresponding secular equations are solved numerically to obtain the speed of propagating Rayleigh-like waves for a particular model. The computed results are presented graphically and compared with those obtained from exact secular equation. The fourth-order approximate secular equation is found to have high accuracy as it provides solutions that are in close vicinity of those obtained from the exact secular equation in the considered model. The presence of voids in the model is found to influence the speed of Rayleigh-like waves theoretically and verified numerically. By ignoring the presence of voids in the model, the secular equation is found to be in complete agreement to the earlier known results in the literature for the corresponding model.

Properties of Hopf bifurcation to a reaction-diffusion population model with nonlocal delayed effect

This paper is concerned with a reaction-diffusion population model with nonlocal delayed effect and zero-Dirichlet boundary condition. Under the condition when the delayed feedback control is dominant, the normal form for spatially nonhomogeneous Hopf bifurcation from the sufficiently small positive equilibrium is computed by means of the normal form method and the center manifold theorem for partial functional differential equations. It is revealed that Hopf bifurcations appearing at the small positive equilibrium are forward and all bifurcating periodic solutions are locally orbitally asymptotically stable on the center manifold. To verify the validity of the obtained theoretical results, numerical simulations are also provided.

Pore-Scale Study of Gas Natural Convection in Confined Porous Media Based on Lattice Boltzmann Method

Abstract Gas natural convection is one common phenomenon in industrial applications, especially for the thermal management of electronic devices. In this study, a numerical model for gas natural convection in a confined porous cavity is constructed based on the lattice Boltzmann (LB) method, which predicts the density-difference-induced flow using a multiple relaxation time (MRT) collision operator. At the gas–solid interfaces, the microscale flow and heat transfer effects are formulated using an effective slip boundary condition. The established LB model is applied to investigate the Nusselt number for heated obstacles arranged in a staggered formation in the cavity. Based on the calculated data, the Nusselt number values obtained for a five-cylinder pore-scale (single pore, SP) domain are analyzed and compared to those for a 13-cylinder (multipore, MP) one. The Nusselt number shows a sharp decrease as soon as the microscale effect is considered at the obstacle walls. It was also observed that the Nusselt number for MP domain achieved lower values than that of SP one. The findings in this work can contribute to the design of thermal management device with confined porous media.

An Effective Model for the Simulation of Transpiration Cooling

AbstractTranspiration cooling is numerically investigated, where a cooling gas is injected through a carbon composite material into a hot gas channel. To account for microscale effects at the injection interface, an effective problem is derived. Here, effects induced by microscale structures on macroscale variables, e.g., cooling efficiency, are taken into account without resolving the microscale structures. For this purpose, effective boundary conditions at the interface between hot gas and porous medium flow are derived using an upscaling strategy. Numerical simulations in 2D with effective boundary conditions are compared to uniform and non-uniform injection. The computations confirm that the effective model provides a more efficient and accurate approximation of the cooling efficiency than the uniform injection.

Laminar flow in a channel bounded by porous/rough walls: Revisiting Beavers-Joseph-Saffman

The fully developed, steady, incompressible, laminar flow in a channel delimited by rough and/or permeable walls is considered. The influence of the micro-structured bounding surfaces on the channel flow behavior is mimicked by imposing high-order effective boundary conditions which stem from homogenization theory and do not contain empirical parameters. A closed-form solution of the Navier–Stokes equations is found for the flow in the channel, with conditions at each virtual boundary linking the slip velocities to shear stress and streamwise pressure gradient. The boundary condition for the longitudinal velocity coincides with a little-noticed extension of the so-called Beavers-Joseph condition, first derived by Saffman (1971); it applies to both permeable and rough surfaces, including the case of separated flow (Couette-Poiseuille motion with adverse pressure gradient). The analytical solution obtained for the velocity distribution in the channel is validated against full feature-resolving simulations of the flow for either highly ordered or random textures, highlighting the accuracy and the applicability range of the model. The Stokes-based model used to identify slip and interface permeability coefficients in the effective boundary conditions is found to be reliable and accurate up to ϵReτ≈10, with ϵ ratio of microscopic to macroscopic length scales and Reτ the shear-velocity Reynolds number. Above this threshold, the coefficients must account for advective effects: a new upscaling procedure, based on an Oseen’s approximation, is thus proposed and validated, extending considerably beyond the Stokes regime.

A surface-piercing truncated cylindrical meta-structure operating as a wave energy converter

In this paper, we study the interaction of water waves with a surface-piercing truncated cylindrical meta-structure consisting of two overlapping arrays of closely spaced vertical thin plates. The fluid resonance promoted in the narrow vertical channels formed by the cylindrical meta-structure is exploited by a novel design concept of the wave power converter by covering the surface of the cylinder with an array of small cuboid buoys which float in the gaps between the intersecting plate arrays. Each buoy is attached to its own spring and power takeoff damping mechanism, and the vertical displacement of individual buoys is replaced by a continuous two-dimensional function of position which follows from homogenization of the plate/fluid structure of the cylinder. Effective medium equations and boundary conditions are derived under both full depth-dependent theory and shallow-water theory, allowing semi-analytical methods to be developed to investigate the wave scattering and wave energy absorption properties of this cylindrical meta-structure. Results illustrate that the internal resonance of the cylindrical meta-structure can lead to significant wave power capture across a broad range of frequencies.

Uniform approximations and effective boundary conditions for a high-contrast elastic interface

A difficulty in the theory of a thin elastic interface is that series expansions in its thickness become disordered in the high-contrast limit, i.e. when the interface is much softer or much stiffer than the media on either side. We provide a mathematical analysis of such series for an annular coating around a cylindrical fibre embedded in an elastic matrix subject to biaxial forcing. We determine the order of magnitude of successive terms in the series, and hence the terms that need to be retained in order to ensure that every neglected term is smaller in order of magnitude than at least one retained term. In this way, we obtain uniform approximations for quantities such as the jump in the displacement and stress across the coating, and explain some peculiarities that have been observed in numerical work. A key finding is that it is essential to distinguish three types of boundary-value problem, corresponding to ‘distant forcing’, ‘localized forcing’ and ‘the homogeneous problem’, since they give different patterns of disorder in the corresponding series expansions. This provides a meaningful correspondence between physical principles and our mathematical results.

Uncoupling system and environment simulation cells for fast-scaling modeling of complex continuum embeddings.

Continuum solvation models are becoming increasingly relevant in condensed matter simulations, allowing to characterize materials interfaces in the presence of wet electrified environments at a reduced computational cost with respect to all atomistic simulations. However, some challenges with the implementation of these models in plane-wave simulation packages still persists, especially when the goal is to simulate complex and heterogeneous environments. Among these challenges is the computational cost associated with large heterogeneous environments, which in plane-wave simulations has a direct effect on the basis-set size and, as a result, on the cost of the electronic structure calculation. Moreover, the use of periodic simulation cells is not well-suited for modeling systems embedded in semi-infinite media, which is often the case in continuum solvation models. To address these challenges, we present the implementation of a double-cell formalism, in which the simulation cell used for the continuum environment is uncoupled from the one used for the electronic-structure simulation of the quantum-mechanical system. This allows for a larger simulation cell to be used for the environment, without significantly increasing computational time. In this work, we show how the double-cell formalism can be used as an effective periodic boundary conditions correction scheme for nonperiodic and partially periodic systems. The accuracy of the double-cell formalism is tested using representative examples with different dimensionalities, both in vacuum and in a homogeneous continuum dielectric environment. Fast convergence and good speedups are observed for all the simulation setups, provided the quantum-mechanical simulation cell is chosen to completely fit the electronic density of the system.

Effective data sampling strategies and boundary condition constraints of physics-informed neural networks for identifying material properties in solid mechanics

Material identification is critical for understanding the relationship between mechanical properties and the associated mechanical functions. However, material identification is a challenging task, especially when the characteristic of the material is highly nonlinear in nature, as is common in biological tissue. In this work, we identify unknown material properties in continuum solid mechanics via physics-informed neural networks (PINNs). To improve the accuracy and efficiency of PINNs, we develop efficient strategies to nonuniformly sample observational data. We also investigate different approaches to enforce Dirichlet-type boundary conditions (BCs) as soft or hard constraints. Finally, we apply the proposed methods to a diverse set of time-dependent and time-independent solid mechanic examples that span linear elastic and hyperelastic material space. The estimated material parameters achieve relative errors of less than 1%. As such, this work is relevant to diverse applications, including optimizing structural integrity and developing novel materials.

Scattering by a Perforated Sandwich Panel: Method of Riemann Surfaces

Summary The model problem of scattering of a sound wave by an infinite plane structure formed by a semi-infinite acoustically hard screen and a semi-infinite sandwich panel perforated from one side and covered by a membrane from the other is exactly solved. The model is governed by two Helmholtz equations for the velocity potentials in the upper and lower half-planes coupled by the Leppington effective boundary condition and the equation of vibration of a membrane in a fluid. Two methods of solution are proposed and discussed. Both methods reduce the problem to an order-2 vector Riemann–Hilbert problem. The matrix coefficients have different entries, have the Chebotarev–Khrapkov structure and share the same order-4 characteristic polynomial. Exact Wiener–Hopf matrix factorization requires solving a scalar Riemann–Hilbert on an elliptic surface and the associated genus-1 Jacobi inversion problem solved in terms of the associated Riemann θ-function. Numerical results for the absolute value of the total velocity potentials are reported and discussed.

Theoretical modelling of non-equilibrium reaction–diffusion of rarefied gas on a wall with microscale roughness

Heat and mass transports through a rough surface are among the most fundamental and important phenomena in either natural or engineering problems. In this paper, theoretical modelling and direct simulation Monte Carlo method are employed to study the heterogeneous reaction–diffusion features induced by microscale roughness which is comparable to the molecular mean free path of the ambient gas. A quasi-one-dimensional homogeneous model is proposed, and it consists of an external diffusion region outside the roughness elements and an internal reaction–diffusion region which could be equivalent to a smooth surface with an effective chemical property. The external macroscopic diffusion can be characterized by a non-equilibrium criterion – the Damköhler number. The internal diffusion in micro-cavities must be analysed by considering the rarefied gas effects on the diffusivity, and another non-equilibrium criterion, the Thiele number, is introduced to evaluate the effective boundary condition imposed on the external region. Analytical formulae based on these criteria are derived to predict the equivalent surface reaction–diffusion performance, and the predictions compare well with the numerical results of different types of surface reaction, even on the three-dimensional roughness. This reveals that the roughness could either enhance or weaken the apparent reaction rate depending on the non-equilibrium degree. This study could enrich our understanding of the gas–surface interactions on a rough wall, such as the oxidation, catalysis and energy accommodation, and also preliminarily provides a practical method for evaluation of the aerothermochemical performance of coating materials of hypersonic vehicles.

Surface properties of ferromagnetic films and spin wave modes

Ferromagnetic films are a fundamental platform in Magnonics, the area that studies spin wave propagation and its possible applications. These films have been studied for a long time experimentally and theoretically, either individually or as parts of multifilms structures. The present study delves into an issue of interest that remains to be studied in detail, which is the relation between the surface properties of these films and the spin wave propagation in them. The main idea of this work is that the frequencies of spin waves may be determined experimentally with precision and through their relation with surface properties one may theoretically validate and derive the main parameters of models that describe the behavior of the surfaces. Several parameters may be changed in the process of testing the relation between surface properties and spin waves, like an applied magnetic field orientation and strength, the angle of in-plane spin wave propagation, etc. This study uses a continuum micromagnetic model for the description of the magnetization dynamics, in this case, the surface properties enter through effective boundary conditions. Three models of surface properties are studied: uniaxial surface anisotropy, the latter combined with bulk uniaxial anisotropy, and surfaces with Dzyaloshinskii-Moriya interactions (DMI). In general, the boundary conditions affect the equilibrium magnetization in boundary layer regions close to the surfaces: an asymptotic analysis predicts their penetration lengths for reasonable surface anisotropies. For very thin films, this is quite relevant since the boundary layers are of the order of the film thickness, we propose this as a practical way to define what may be considered a very thin film. Effective physical parameters of thin films and very thin films clearly depend on the surface properties. The main effects of the surfaces on spin waves occur for angles of inclination of the applied magnetic field close to perpendicular to the plane, and they may be of significant magnitude. Several different numerical methods were used to solve for the spin waves, in the context of an orthogonality equation method.

Topological-insulator nanocylinders

Nanostructures, such a quantum dots or nanoparticles, made of three-dimensional topological insulators (3DTIs) have been recently attracting increasing interest, especially for their optical properties. In this paper we calculate the energy spectrum, the surface states and the dipole matrix elements for optical transitions with in-plane polarization of 3DTI nanocylinders of finite height LL and radius RR. We first derive an effective 2D Hamiltonian by exploiting the cylindrical symmetry of the problem. We develop two approaches: the first one is an exact numerical tight-binding model obtained by discretising the Hamiltonian. The second one, which allows us to obtain analytical results, is an approximated model based on a large-RR expansion and on an effective boundary condition to account for the finite height of the nanocylinder. We find that the agreement between the two models, as far as eigenenergies and eigenfunctions are concerned, is excellent for the lowest absolute value of the longitudinal component of the angular momentum. Finally, we derive analytical expressions for the dipole matrix elements by first considering the lateral surface alone and the bases alone, and then for the whole nanocylinder. In particular, we focus on the two limiting cases of tall and squat nanocylinders. The latter case is compared with the numerical results finding a good agreement.

Asymptotic Propagation Speeds of the Fisher-KPP Equation with an Effective Boundary Condition on a Road

Asymptotic Propagation Speeds of the Fisher-KPP Equation with an Effective Boundary Condition on a Road

Effective Boundary Conditions and Stochastic Crack Distribution for Modelling Guided Waves Scattering by a Partially Closed Interfacial Delamination in a Laminate

Cohesive and adhesive bindings degrade during operation and maintenance even if contacting materials in a manufactured laminated structure are perfectly matched at the interfaces. Two modelling approaches for describing partially closed delaminations or imperfect contact zones, which often occurs at the interfaces, are examined and considered. To confirm the adequateness of the applicability of the effective spring boundary conditions for guided wave scattering by a finite length delamination, guided wave propagation through a damaged zone with a distribution of micro-cracks is compared with an equivalent cohesive zone model, where the spring stiffnesses for the effective boundary conditions are calculated using the properties of the considered crack distribution. Two kinds of local interfacial decohesion zones with an imperfect contact at the interfaces are considered: uniform partially closed delaminations and bridged cracks. The possibility of the employment of the effective spring boundary conditions to substitute a distribution of micro-cracks is analysed and discussed. Two algorithms of generation of a distribution of open micro-cracks providing characteristics equivalent to the effective boundary conditions are presented and examined. The influence of the characteristics of a delamination on wave characteristics (eigenfrequencies, eigenforms, transmission coefficient) is investigated for several kinds of partially closed delaminations.

Nanoscale flow-induced nonlinear vibration of multilayer graphene based-resonators: Slip and transition flow regimes

This paper presents a nanoscale fluid-structure interaction (FSI) model for flow-induced nonlinear vibration of multilayer graphene based-resonators in the slip and transition regimes. Based on the multibeam interlayer shear model with von Kármán geometric nonlinearity and nonlocal modified couple stress theory, the size-dependent equations of motion are obtained for beam resonators laminated of NG-graphene layer and (NG−1) interlayer crosslinks. The small-scale effect on effective viscosity and slip boundary conditions of nanoscale flow is formulated by introducing the average velocity correction factor. For unbounded and bounded (by a rigid wall) nanoflows, the closed-form expressions of velocity correction factor in terms of Knudsen number, as a dimensionless discriminant parameter, are developed to correct the pressure distribution predicted by the potential theory. The present FSI model considers the size effects through the length scale parameter for microstructure local rotation, the nonlocal parameter for long-range interatomic interactions, and Knudsen number for slip condition of nanoscale flow. The Galerkin's method in conjunction with the multiple-scale perturbation technique is used to obtain closed-form solutions of frequency ratio and time-history response. The natural frequencies of cantilever multilayer graphene nanostrips are validated with the ones existing in the literature. Parametric studies are carried out to investigate the influences of small-scale, number of layers, interlayer shear modulus, and fluid characteristics on the linear and nonlinear natural frequencies. Findings show that the critical upstream speed for dynamic instability of the coupled system can change considerably by taking into account the size effects of nanoflow and nanostructures. Therefore, the classical FSI model is inadequate for analyzing nanoscale flow-induced vibration of nanoresonators. Also, it is seen that the effect of geometric nonlinearity becomes very significant for multilayer graphene beams with small interlayer shear modulus.

Rigorous derivation of the Darcy boundary condition on the porous wall

An effective boundary condition on a porous wall is derived, starting from basic principles of mechanics. Stokes system, governing the viscous flow through a reservoir with an array of small pores on the boundary, was studied, and the corresponding macroscopic model via rigorous asymptotic analysis is found. Under the assumption of periodicity of the pores, the effective boundary condition of the Darcy type is derived, using homogenization and boundary layer techniques. Further asymptotic analysis with respect to the porosity yields a recursive sequence of boundary value problems showing that the large pressure jump occurs on the boundary.