Microscopic Boundary Conditions Research Articles

Two-Dimensional System of Moment Equations and Macroscopic Boundary Conditions Depending on the Velocity of Movement and the Surface Temperature of a Body Moving in Fluid

This article is dedicated to the derivation of a two-dimensional system of moment equations depending on the velocity of movement and the surface temperature of a body submerged in fluid, and macroscopic boundary conditions for the system of moment equations approximating the Maxwell microscopic boundary condition for the particle distribution function. The initial-boundary value problem for the Boltzmann equation with the Maxwell microscopic boundary condition is approximated by a corresponding problem for the system of moment equations with macroscopic boundary conditions. The number of moment equations and the number of macroscopic boundary conditions are interconnected and depend on the parity of the approximation of the system of moment equations. The setting of the initial-boundary value problem for a non-stationary, nonlinear two-dimensional system of moment equations in the first approximation with macroscopic boundary conditions is presented, and the solvability of the above-mentioned problem in the space of functions continuous in time and square-integrable in spatial variables is proven.

A multi-scale formulation for polycrystalline materials accounting for cohesive micro-cracks: Homogenisation of the traction-separation law

This contribution investigates the modelling of failure in polycrystalline materials due to the nucleation and propagation of cohesive cracks at the micro-scale. First, a two-scale formulation that enables the analysis of failure, even when the evolution and propagation of cohesive micro-cracks induce material instability, is proposed. The development of this macro-continuous/micro-discontinuous formulation is based on the Method of Multi-Scale Virtual Power, from which dynamic equilibrium and homogenisation equations are derived, ensuring a variationally consistent scale transition. The respective minimal kinematical constraint is retrieved and enforced with the Lagrange multiplier method. The formulation is also extended for periodic boundary conditions. The discretisation of the resulting equilibrium equations, with the finite element method and the generalised-α method, is described in detail.Second, a fracture-based computational homogenisation procedure is developed to obtain homogenised traction-separation laws and fracture properties from microstructural volume elements. It is derived from a crack-based Hill-Mandel principle and employs a novel energetic-based damage variable, within the Park-Paulino-Roesler cohesive model, proposed to define the crack domain and compute the crack homogenised quantities. A strategy for accurately computing the homogenised unit normal vector of the equivalent macroscopic crack is also suggested.Numerical examples demonstrate the formulation’s ability to analyse fracture problems, including a study on the impact of different microscopic boundary conditions. This work also reveals the convergence of the traction-separation law with increasing microstructure model size when using the fracture-based computational homogenisation. Furthermore, it investigates how advanced deformation mechanisms, like slip plasticity and martensitic transformation, affect intergranular crack propagation mechanisms.

A multiscale approach in modeling of chemically reactive porous media

Chemical reactions occur in geotechnical, biological and synthetic porous media. Swelling and shrinkage phenomena can be observed in clays, shales and gels, when they are in close contact with water. In petroleum engineering, wellbore-stability problems are caused by swelling or shrinking of the wellbore. Thus, it is significant to model the Chemo-Hydro-Mechanical (CHM) processes in porous media that include different physical and geometrical heterogeneities. In this paper, a multiscale computational homogenization approach is developed for the analysis of CHM problems. In this manner, the first-order homogenization technique is adopted for the two-scale formulation. The heterogeneities are assumed in the microscale level and a homogeneous domain is considered for the macroscale level, where the constitutive behavior is derived from the microscopic level. The governing equations and constitutive equations of the CHM processes are presented in the coupled manner. The primary variables are taken as the displacement, pore pressure, and solute concentration fields. Moreover, the transient terms are included in the microscale level to increase the accuracy of computations. Consequently, a generalized form of Hill-Mandel principle of macro-homogeneity is defined between the two scales. Appropriate microscopic boundary conditions, i.e. the linear and periodic boundary conditions, are employed to satisfy the averaging constraints. Finally, the accuracy and efficiency of the proposed computational multiscale method are illustrated through several numerical examples.

Investigation of Aerodynamic Characteristics of Aircrafts in a Rarefied Gas Flow Using the Moment Method

A one-dimensional nonstationary nonlinear moment system of equations and an approximation of Maxwell’s microscopic boundary condition will be introduced. The flight speed and surface temperature of the aircraft are included in the moment system of equations as coefficients. Macroscopic boundary conditions also depend on the surface temperature of the aircraft. The quantity of macroscopic boundary conditions for the moment system of equations depends on the parity of the approximation number of the moment system of equations. We state the initial and boundary value problem for the moment system of equations in the third approximation under macroscopic boundary conditions. This paper proves the existence and uniqueness of the solution of the abovementioned problem in the space of functions continuous in time and summable in the square by spatial variable. The theorem on the existence and uniqueness of a solution of the initial and boundary value problem for the moment system of equations in the third approximation is proved by the method of a priori estimation and using the Galerkin method and Tartar’s compensated compactness method.

Multiscale modeling of coupled thermo-hydro-mechanical analysis of heterogeneous porous media

This paper presents a numerical multiscale formulation for analysis of the transient heat and fluid flow in deformable heterogeneous porous media. Due to the heterogeneity of the media, the direct numerical simulation of the micro-structures leads to high computational costs. Hence, the multi-scale method can provide an efficient computational procedure. To this end, the first-order computational homogenization is adopted for two-scale simulation of THM problems. The governing equations of the problem contain a stress equilibrium equation, a mass continuity equation and an advection–diffusion equation in a fully coupled manner. Accordingly, the proper virtual power relations are defined as a bridging stage between the scales. The boundary conditions of the microscopic domain are specified in terms of the macroscopic information to fulfill the averaging constraints. In order to consider the transient effects, the micro-inertial terms are included in the micro-scale simulation, and then the microscopic boundary conditions are enhanced to assess the efficiency and accuracy. The macroscopic properties including the homogenized tangent operators and equivalent quantities are determined from the microscopic solution through the appropriate mathematical procedures. Moreover, an upwind finite element squared method is employed for highly advective heat transfer to achieve the accurate spatial results. Finally, several comprehensive examples are simulated to study the effectiveness and validity of the proposed computational model, particularly the boundary conditions; moreover, different case studies are discussed such as the highly advective flow, pure conduction heat transfer and a combined conduction–convection heat transfer.

The Solvability of Mixed Value Problem for the First and Second Approximations of One-Dimensional Nonlinear System of Moment Equations with Microscopic Boundary Conditions

The paper gives a derivation of a new one-dimensional non-stationary nonlinear system of moment equations, that depend on the flight velocity and the surface temperature of an aircraft. Maxwell microscopic condition is approximated for the distribution function on moving boundary, when one fraction of molecules reflected from the surface specular and another fraction diffusely with Maxwell distribution. Moreover, macroscopic boundary conditions for the moment system of equations depend on evenness or oddness of approximation {f}_{k}(t,x,c), where {f}_{k}(t,x,c) is partial expansion sum of the molecules distribution function over eigenfunctions of linearized collision operator around local Maxwell distribution. The formulation of initial and boundary value problem for the system of moment equations in the first and second approximations is described. Existence and uniqueness of the solution for the above-mentioned problem using macroscopic boundary conditions in the space of functions Cleft(left[0,Tright];{L}^{2}left[-a,aright]right) are proved.

Analytical Solutions of Model Problems for Large-Deformation Micromorphic Approach to Gradient Plasticity

The objective of this work is to present analytical solutions for several 2D model problems to demonstrate the unique plastic fields generated by the implementation of micromorphic approach for gradient plasticity. The approach is presented for finite deformations and several macroscopic and nonstandard microscopic boundary conditions are applied to a gliding plate to illustrate the capability to predict the size effects and inhomogeneous plastic fields promoted by the gradient terms. The constitutive behavior of the material undergoing plastic deformation is analyzed for softening, hardening and perfect plastic response and corresponding solutions are provided. The analytical solutions are also shown to match with the numerical results obtained by implementing a user element subroutine (UEL) to the commercial finite element software Abaqus/Standard.

Constraint and size effects in confined layer plasticity

Plane strain finite element analyses are used to model the experiments of Mu et al. (2014, 2016) for a thin metal layer confined between elastic solids. The thin metal layer undergoes elastic–plastic deformations, here modeled by strain gradient plasticity, while the rest of the solid deforms only elastically. Plane strain is assumed and finite strains are accounted for. The microscopic boundary condition at the band interface is initially taken to be zero plastic strain increment, but plastic straining at the interface is allowed when the plastic strain gradient at the interface has exceeded a critical value. When the narrow metal band is inclined 45° relative to the direction of compression of the solid, simple shear develops in the band and predictions are rather similar to those of a one-dimensional analysis for an infinite band. For a band at 90° the elastic parts give significant constraint on plastic flow, the stress fields are non-uniform along the band and the average nominal compressive stress shows strong sensitivity to the layer thickness and to the width of the specimen. Remarkably good qualitative, and in some cases quantitative, agreement is found with the experimental observations of Mu et al. (2014, 2016). Considering also other band angles, both the average nominal shear and compressive stress show variations with the angle.

Применение уравнения Больцмана для определения аэродинамических характеристик летательных аппаратов

In this paper we present the derivation of new one-dimensional non-stationary nonlinear system of moment equations and an approximation of the microscopic boundary condition when part of molecules reflect from the surface specularly and part diffusive with Maxwell’s distribution. Macroscopic boundary conditions for the system of moment equations depend on the evenness and oddness of the approximation , where is the partial sum of expansion of the distribution function of molecules  into eigenfunctions of the linearized collision operator. The formulation of the initial and boundary value problem for the system of moment equations in the third approximation under the Maxwell-Auzhan macroscopic boundary conditions is given. To analyze aerodynamic characteristics of aircraft in transient regime was used complete integro-differential Boltzmann equation, which contains a term depending on the moving speed of aircraft, and under Maxwell's microscopic boundary conditions, depending on the surface temperature.

PHASE FIELD MODELING OF NORMAL AND STRESSED GRAIN GROWTH: THE EFFECT OF RVE SIZE AND MICROSCOPIC BOUNDARY CONDITIONS

PHASE FIELD MODELING OF NORMAL AND STRESSED GRAIN GROWTH: THE EFFECT OF RVE SIZE AND MICROSCOPIC BOUNDARY CONDITIONS

FE2 simulations of magnetorheological elastomers: influence of microscopic boundary conditions, microstructures and free space on the macroscopic responses of MREs

In the current work, the response of heterogeneous magnetorheological elastomers (MREs) which are loaded by external magnetic fields in the absence and also in the presence of free space is studied. A fully-coupled two-scale finite element computational homogenization procedure is used to derive the material response at the macro-scale from the averaged response of the underlying micro-scale problem. Different combinations of boundary conditions, that satisfy the Hill-Mandel condition and are based on the primary variables of the magneto-elastic enthalpy and energy functionals are applied to solve the micro-scale boundary value problem. Furthermore, the influences of various microstructures on the macroscopic response of the MREs are investigated. The results indicate that the choice of microscopic boundary conditions and microstructure types can significantly affect the macroscopic responses of MREs.

An assessment of multiscale asymptotic expansion method for linear static problems of periodic composite structures

The decoupled multiscale asymptotic expansion method (MsAEM) is applicable to the analysis of composite structures with periodic microstructures. When using the MsAEM to solve linear static problems, unit-cell influence functions must be solved from the microscale (or unit cell) problem with microscopic periodic boundary conditions, and homogenized displacements and their derivatives must be solved from the macroscale (or homogenized) problem with macrostructural boundary conditions. This paper compares the influence functions solved by using the normalization condition, the Dirichlet homogeneous boundary condition, and the oversampling technique or the super unit cell approach. It is concluded that the oversampling technique is a better way of dealing with periodic boundary conditions compared to the normalization condition. Another contribution of this work is to propose a combination method in which the MsAEM is used inside the structure and the multiscale eigenelement method (MEM) is used dealing with the boundary layer problem, and it is shown that this combination method can achieve accurate microscopic results for structures with any boundary conditions. In addition, the paper reveals the effects of different order expansion terms and gives some practical suggestions on how to determine the expansion order. Numerical comparisons validate the present combination method and all conclusions.

Microscopic Slip Boundary Conditions in Unsteady Fluid Flows.

An algebraic tail in the Green-Kubo integral for the solid-fluid friction coefficient hampers its use in the determination of the slip length. A simple theory for discrete nonlocal hydrodynamics near parallel solid walls with extended friction forces is given. We explain the origin of the algebraic tail and give a solution of the plateau problem in the Green-Kubo expressions. We derive the slip boundary condition with a microscopic expression for the slip length and the hydrodynamic wall position, and assess it through simulations of an unsteady plug flow.

Macroscopic Boundary Conditions on a Solid Surface in Rarefied Gas Flow for a One-Dimensional Nonlinear Nonstationary 12-Moment System of Boltzmann Equations

Boundary conditions for a one-dimensional nonlinear nonstationary system of Boltzmann equations are formulated in the fifth approximation. The Maxwell microscopic boundary conditions are approximated in the case of the one-dimensional Boltzmann equation when some of the molecules reflect specularly from the surface, while the others reflect diffusely with Maxwell’s distribution. An initial-boundary value problem for the 12-moment system of Boltzmann equations with Maxwell–Auzhani boundary conditions is stated. For the 12-moment system of Boltzmann equations, six boundary conditions are set at the left and right endpoints of the interval ( $$ - a$$ , $$a$$ ).

Approximation of the boundary value problem for 3-dimensional Boltzmann’s equation with Maxwell’s microscopic conditions

The aim of the research is to formulate the boundary conditions for the moment system equations. We put boundary conditions for arbitrary approximation of the three-dimensional nonlinear nonstationary Boltzmann’s moment system equations. We approximate microscopic Maxwell boundary condition for three-dimensional Boltzmann’s equation. We formulate initial and boundary value problem for twenty-moment system equations with Maxwell-Auzhan boundary conditions.

Microscopic determination of macroscopic boundary conditions in Newtonian liquids.

We study boundary conditions applied to the macroscopic dynamics of Newtonian liquids from the view of microscopic particle systems. We assume the existence of microscopic boundary conditions that are uniquely determined from a microscopic description of the fluid and the wall. By using molecular dynamical simulations, we examine a possible form of the microscopic boundary conditions. In the macroscopic limit, we may introduce a scaled velocity field by ignoring the higher-order terms in the velocity field that is calculated from the microscopic boundary condition and standard fluid mechanics. We define macroscopic boundary conditions as the boundary conditions that are imposed on the scaled velocity field. The macroscopic boundary conditions contain a few phenomenological parameters for an amount of slip, which are related to a functional form of the given microscopic boundary condition. By considering two macroscopic limits of the nonequilibrium steady state, we propose two different frameworks for determining macroscopic boundary conditions.

Characterizing the strange term in critical size homogenization: Quasilinear equations with a general microscopic boundary condition

Abstract The aim of this paper is to consider the asymptotic behavior of boundary value problems in n-dimensional domains with periodically placed particles, with a general microscopic boundary condition on the particles and a p-Laplace diffusion operator on the interior, in the case in which the particles are of critical size. We consider the cases in which {1<p<n} , {n\geq 3} . In fact, in contrast to previous results in the literature, we formulate the microscopic boundary condition in terms of a Robin type condition, involving a general maximal monotone graph, which also includes the case of microscopic Dirichlet boundary conditions. In this way we unify the treatment of apparently different formulations, which before were considered separately. We characterize the so called “strange term” in the homogenized problem for the case in which the particles are balls of critical size. Moreover, by studying an application in Chemical Engineering, we show that the critically sized particles lead to a more effective homogeneous reaction than noncritically sized particles.

Interfacial microscopic boundary conditions associated with backstress-based higher-order gradient crystal plasticity theory

Interfacial microscopic boundary conditions associated with backstress-based higher-order gradient crystal plasticity theory

Unified treatment of microscopic boundary conditions and efficient algorithms for estimating tangent operators of the homogenized behavior in the computational homogenization method

This work provides a unified treatment of arbitrary kinds of microscopic boundary conditions usually considered in the multi-scale computational homogenization method for nonlinear multi-physics problems. An efficient procedure is developed to enforce the multi-point linear constraints arising from the microscopic boundary condition either by the direct constraint elimination or by the Lagrange multiplier elimination methods. The macroscopic tangent operators are computed in an efficient way from a multiple right hand sides linear system whose left hand side matrix is the stiffness matrix of the microscopic linearized system at the converged solution. The number of vectors at the right hand side is equal to the number of the macroscopic kinematic variables used to formulate the microscopic boundary condition. As the resolution of the microscopic linearized system often follows a direct factorization procedure, the computation of the macroscopic tangent operators is then performed using this factorized matrix at a reduced computational time.

Experimental investigation of higher-order homogenization schemes under large strain

Multiscale methods and homogenization techniques are usually compared to full field simulations. The accuracy of the full field simulations is well controlled in the linear elastic regime and for small strain. With non-linear behavior, large strain or multi-axial strain states the comprehension is hampered by the lack of experimental results. In this study, the way to enforce the microscopic boundary conditions from the macroscopic scale is investigated by comparison to experimental results. A series of tests has been carried out on a model multiscale structure over which the displacement field is captured at microscopic and macroscopic scales. These experimental results allow to analyze the results of different homogenization methods such as classical or higher order ones. The study focuses and exhibits different conclusions about the unit cells with periodic or non-periodic kinematics.