Two-scale Problem Research Articles

Evolution of parton distribution functions in the short-distance factorization scheme

Lattice QCD offers the possibility of computing parton distributions from first principles, although not in the usual MS¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{MS} $$\\end{document} factorization scheme. Calculations are therefore matched to MS¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{MS} $$\\end{document} using a perturbative procedure which is the source of significant uncertainty within the currently accessible kinematics. We present the possibility of computing the z2 evolution of non-singlet pseudo-parton distribution functions within the short factorization scheme in a numerically improvable way. The goal is to have tools to evolve a calculation to a scale where perturbative uncertainties are less pronounced. We compare a numerical extraction of the evolution operator from lattice data to the computation of z2 dependence in perturbation theory. Finally, we discuss how this numerical work may be extended to address the two-scale problem that arises when the Ioffe time range must be made large to extend the reach of the calculation of the pseudo-PDF to smaller values of the momentum fraction.

Multiscale simulation scheme by using a finite-difference derivative framework for lubrication surface

To exactly describe lubrication problems modelling micro-scale influence in rough lubricated contact is unavoidable. In this study a novel computational scheme for lubricated problem is presented based on two-scale homogenization method, which facilitates solving the two-scale problem. Homogenization model of mass fluxes is implemented at the micro-scale domain, and represents the influence of surface topography. A linearization method is developed to derive finite derivatives of mass fluxes at the micro-scale domain defined by scale separation. The key strength of the proposed procedure is the finite difference derivative coupling which represents the lubricated problem inclusive of the effect of surface topography. The effectiveness of the proposed multiscale scheme is validated by comparing with the deterministic method. The numerical results show that increasing topography amplitude can improve the performance of journal bearing in terms of load carrying capacity and friction coefficient.

A monolithic hyper ROM FE2 method with clustered training at finite deformations

The usage of numerical homogenization to obtain structure–property relations by applying the finite element method at both the micro- and macroscale has gained much interest in the research community. The computational cost of this so-called FE2 method, however, is typically so high that algorithmic modifications and reduction methods are essential. In the present contribution, a monolithic solution algorithm is combined with reduced order modeling (ROM) and the empirical cubature method (ECM) for hyper integration. It is further complemented by a clustered training strategy, which lowers the training effort and the number of necessary ROM modes immensely. The applied methods can be combined modularly as desired in finite element approaches. An implementation in terms of an extension to the previously established MonolithFE2 code is provided. Numerical examples show the efficiency and accuracy of the monolithic hyper ROM FE2 method and the advantages of the clustered training strategy. Even for two-scale problems with complex geometry and complex, inelastic material behaviors it was shown that speedup factors of almost 1000 (i.e., three orders of magnitude) regarding the online simulation time and of up to 30 regarding all necessary computing effort are obtainable in comparison to the conventional FE2 scheme. The training stage requires only around 3% of that time, meaning that the offline phase is relatively inexpensive, in contrast to many Neural Network approaches, whose employment, in terms of total computational efficiency, only pays off if a large number of online simulations is to be conducted, without requiring additional training.

MFSE-based two-scale concurrent topology optimization with connectable multiple micro materials

The concurrent design of different lattice material microstructures and their corresponding macro-scale distributions has great potential in achieving both lightweight and desired multiphysical performances. In such design problems, the lattice microstructures are usually separately optimized on the basis of the homogenization method, and the possibly poor connectivity between them is a key factor that severely hinders the fabrication and application of optimized two-scale structures. To handle the microstructure connectivity issue, this paper proposes a novel microstructure connectable strategy to bridge the gap between the two-scale and the full-scale model using the material-field series expansion (MFSE) method. Assuming different lattice material types in several pre-defined macro-scale regions, describe all types of microstructural topology with different portions of one material field function and update them simultaneously during the optimization process. Benefits from the material field definition with spatial correlation, the microstructures are well-connected without requiring additional constraints in the topology optimization model. The energy-based homogenization method is utilized for bridging the two-scale with different microstructures, while a decoupled sensitivities analysis for the microscale is employed to enhance the computation efficiency. Additionally, the proposed method significantly reduces the dimension of design variables, resulting in lower optimizer spending. The effectiveness and efficiency of the proposed method are demonstrated by several benchmark two-scale problems. Compared to density-based connectable methods, the proposed framework is easy to implement and reduces computational time by an order of magnitude in the 2D case.

Two-scale blockage correction for an array of tidal turbines

It has been shown theoretically that tidal fences consisting of multiple turbines placed side-by-side can makeuse of constructive interference (local blockage) effects to raise the energy extraction efficiency of the fenceabove that of the Betz limit applicable to unblocked flow problems. For the two-scale problem of a longarray of turbines partially spanning the width of a much wider channel (vanishing global blockage) theefficiency of energy extraction, normalised on the undisturbed kinetic energy flux, rises from the Betz limitof 0.593 to the partial fence limit of 0.798 [1]. Experiments on pairs of side-by-side turbines at largelaboratory scale [2] have confirmed the important aspects of the underlying partial fence theory and thatsome of the performance benefits offered by constructive interference effects can be achieved in practice.
 Experimental validation in wind tunnels, towing tanks and other laboratory facilities are however prone toglobal blockage effects not seen in full-scale open flows due to the close proximity of flow boundaries to thebody. These global blockage effects modify the thrust and power performance of the turbines, such thatcorrections to experimental curves are necessary to either translate laboratory-scale experimental results tofull-scale conditions, or to calculate the expected loads and power on tidal turbines deployed in blocked-flowconditions [3][4]. The difficulty applying blockage corrections to turbine arrays is the non-linear interactionbetween local and global blockage. These two effects cannot be simply decoupled as for various turbine tip-to-tip spacings (affecting local blockage), changes in the global blockage have a different impact on turbineperformance.
 A number of blockage corrections have been developed for single turbines operating in blocked flowconditions. These corrections typically seek to describe an equivalent free-stream velocity which, in theabsence of global blockage, would result in the same thrust and velocity through the turbine as in the blockedcase. Thrust and power curves are then scaled non-linearly with the ratio of the experimental tank velocityand the equivalent free-stream velocity [5]. These single turbine blockage corrections can however onlyaccount for global blockage, and simplifications must currently be made based on the assumption that globaland local blockage effects can be linearly decoupled [2].
 This work therefore presents an analytical blockage correction for co-planar arrays of tidal turbines based ontwo-scale momentum theory [1]. This correction is then compared to other models, particularly for turbinearray experimental test data. Finally, RANS computations for a turbine array at various global blockageratios is compared to the analytical model, demonstrating its validity. A particularly useful aspect of thetheoretical model is to allow for experimental quantification of the local-blockage effect for finite lengthfences. For instance, doubling the fence length doubles the global blockage, but increases in fence thrust andpower cannot be attributed only to the change in global blockage due to non-linear coupling. This correctionallows for a decoupling of these two effects, such that the local blockage effect can be isolated andquantified.

Multiscale modeling of laminated thin-shell structures with Direct FE[formula omitted

Shell finite element is computationally efficient and thereby favored in large-scale structural analysis. With the application of advanced materials, understanding the relation between micro-heterogeneities and structural behavior is important. To this end, we propose a multiscale method in the current contribution for modeling thin-walled fiber reinforced composite laminates with Mindlin–Reissner shell theory and Direct FE2. By exploiting the length scales in layered structure together with a shell formulation degenerated from continuum solid element, both in-plane and through-thickness periodicity can be assumed. A single-fiber unit cell attached to each Gauss point is employed as a representative volume element (RVE) rather than a through-thickness microstructure. In this way, the scale transitions are naturally dealt with using a first-order homogenization framework, in which the downscaling of transverse shear strain, commonly troublesome for the through-thickness RVE, is straightforward. The other merit inherited from Direct FE2 is that the two-scale problem is formulated in a single finite element analysis wherein the macro and micro DOFs are coupled based on the downscaling kinematics. It avoids the upscaling of the macroscopic stresses and tangents in the conventional nested FE2 scheme, which is challenging in the case of complex microstructural response. The developed method is then verified through several benchmarking examples, including a flat plate and two curve shells. Isotropic materials and composite laminates with homogeneous and heterogeneous RVEs have been examined. Two examples with plasticity and cohesive cracks are further presented to showcase its potential in modeling damage propagation in composite structures.

Spectral Analysis for Singularity Formation of the Two Dimensional Keller–Segel System

We analyse an operator arising in the description of singular solutions to the two-dimensional Keller-Segel problem. It corresponds to the linearised operator in parabolic self-similar variables, close to a concentrated stationary state. This is a two-scale problem, with a vanishing thin transition zone near the origin. Via rigorous matched asymptotic expansions, we describe the eigenvalues and eigenfunctions precisely. We also show a stability result with respect to suitable perturbations, as well as a coercivity estimate for the non-radial part. These results are used as key arguments in a new rigorous proof of the existence and refined description of singular solutions for the Keller–Segel problem by the authors [8]. The present paper extends the result by Dejak, Lushnikov, Yu, Ovchinnikov and Sigal [11]. Two major difficulties arise in the analysis: this is a singular limit problem, and a degeneracy causes corrections not being polynomial but logarithmic with respect to the main parameter.

A multiscale quasilinear system for colloids deposition in porous media: Weak solvability and numerical simulation of a near-clogging scenario

We study the weak solvability of a macroscopic, quasilinear reaction–diffusion system posed in a 2D porous medium which undergoes microstructural problems. The solid matrix of this porous medium is assumed to be made out of circles of not-necessarily uniform radius. The growth or shrinkage of these circles, which are governed by an ODE, has direct feedback to the macroscopic diffusivity via an additional elliptic cell problem.The reaction–diffusion system describes the macroscopic diffusion, aggregation, and deposition of populations of colloidal particles of various sizes inside a porous media made of prescribed arrangement of balls. The mathematical analysis of this two-scale problem relies on a suitable application of Schauder’s fixed point theorem which also provides a convergent algorithm for an iteration method to compute finite difference approximations of smooth solutions to our multiscale model. Numerical simulations illustrate the behavior of the local concentration of the colloidal populations close to clogging situations.

Two-scale analysis of transient diffusion problems through a homogenized enriched continuum

This paper addresses the two-scale problem underlying the enriched continuum for transient diffusion problems, which was previously developed and tested at the single scale level only (Waseem et al., Comp.Mech, 65, 2020). For a linear material model exhibiting a relaxed separation of scales, a model reduction was proposed at the micro-scale that replaces the micro-scale problem with a set of uncoupled ordinary differential equations (ODEs). At the macro-scale, the balance law, the ODEs and the macroscopic constitutive equations collectively represent an enriched continuum description. Examining different discretization techniques, distinct solution methods are presented for the macro-scale enriched continuum. Proof-of-principle examples are solved for a mass diffusion system in which species diffuse slower in the inclusion than in the matrix. The results from the enriched continuum formulation are compared with the computational transient homogenization (CTH) and direct numerical simulations (DNS). Without compromising the solution accuracy, significant computational gains are obtained through the enriched continuum approach.

Homogenization of a multiscale multi-continuum system

ABSTRACT We study homogenization of a locally periodic two-scale dual-continuum system where each continuum interacts with the other. Equations for each continuum are written separately with interaction terms added. The homogenization limit depends strongly on the scale of this continuum interaction term with respect to the microscopic scale. In J. S. R. Park and V. H. Hoang, Hierarchical multiscale finite element method for multi-continuum media, Journal of Computational and Applied Mathematics, we study in details the case where the interaction terms are scaled as where ε is the microscale of the problem. We establish rigorously homogenization limit for this case where we show that in the homogenization limit, the dual-continuum structure disappears. In this paper, we consider the case where this term is scaled as . This case is far more interesting and difficult as the homogenized problem is a dual-continuum system which contains features that are not in the original two-scale problem. In particular, the homogenized dual-continuum system contains extra convection terms and negative interaction coefficients, while the interaction coefficient between the continua in the original two-scale system obtains both positive and negative values. We prove rigorously the homogenization convergence and the convergence rate.

Homogenization results for a class of parabolic equations with a non-local interface condition via time-periodic unfolding

We study the thermal properties of a composite material in which a periodic array of finely mixed perfect thermal conductors is inserted. The suitable model describing the behaviour of such physical materials leads to the so-called equivalued surface boundary value problem. To analyze the overall conductivity of the composite medium (when the size of the inclusions tends to zero), we make use of the homogenization theory, employing the unfolding technique. The peculiarity of the problem under investigation asks for a particular care in developing the unfolding procedure, giving rise to a non-standard two-scale problem.

On using mesh-based and mesh-free methods in problems defined by Eringen’s non-local integral model: issues and remedies

With the recent success of nonlocal theories in modeling of engineering problems involving small intrinsic length scales, such as modeling of crack propagation, this paper addresses issues pertaining to cost-ineffectiveness of Eringen’s integral model. The cost effectiveness of the computation may be considered as a twofold issue; one pertaining to the non-local model and another pertaining to the numerical tool. First of all, we shall show that during the solution of problems with Eringen’s non-local integral model, there is no need to consider the integral model for the whole computational domain. In fact, the problems may be solved by just using the integral model close to the boundaries, i.e. a boundary layer effect, or around the points with singularities. In this paper we propose a partitioning strategy to remarkably reduce the computational cost. This may be considered as a gateway for solving some types of two-scale problems, e.g. those with macro/micro and nano scales, in which the small scale effects are localized just at parts of the domain. To demonstrate the efficiency of the numerical tools, we examine the performance of the finite element method (FEM), the element free Galerkin method (EFG) and the finite point method (FPM). This paves the way for using mesh-free methods in the solution of problems with non-local integral models. Examples with smooth and non-smooth solutions are considered for examining the efficiency of the methods. It will be shown that, by considering the boundary layer effect, the FEM and FPM will be efficient enough for being used in problems defined by Eringen’s non-local integral model.

Analyticity, Regularity, and Generalized Polynomial Chaos Approximation of Stochastic, Parametric Parabolic Two-Scale Partial Differential Equations

We study two-scale parabolic partial differential equations whose coefficient is stochastic and depends linearly on a sequence of pairwise independent random variables which are uniformly distributed in a compact interval. We cast the problem into a deterministic two-scale parabolic problem which depends on a sequence of real parameters in a compact interval. Passing to the limit when the microscale tends to zero, using two-scale homogenization, we obtain the parametric two-scale homogenized problem. This problem contains the solution to the homogenized equation which describes the solution to the original two-scale parabolic problem macroscopically, and the corrector which encodes the microscopic information. The solution of this two-scale homogenized equation is represented as a generalized polynomial chaos (gpc) expansion according to a polynomial basis of the L2 space of the parameters. We use a semidiscrete Galerkin approximation which projects the solution into a space of parametric functions which contain a finite number of pre-chosen gpc modes. Analyticity of the solution of the two-scale homogenized equation with respect to the parameters is established. Under mild assumptions, we show the summability of the coefficients of the solution’s gpc expansion. From this, an explicit error estimate for the semidiscrete Galerkin approximation in terms of the number of the chosen gpc modes is derived when these gpc modes are chosen as the N best ones according to the norms of the gpc coefficients. Regularity of the gpc coefficients and summability of their norms in the regularity spaces are also established. Using the solution of the best N term semidiscrete Galerkin approximation, we derive an approximation for the solution of the original two-scale problem, with an explicit convergence rate in terms of the microscopic scale and the number of gpc modes in the Galerkin approximation.

Sparse tensor product finite element method for nonlinear multiscale variational inequalities of monotone type

Abstract We study an essentially optimal finite element (FE) method for locally periodic nonlinear multiscale variational inequalities of monotone type in a domain $D\subset{\mathbb{R}}^d$ that depend on a macroscopic and $n$ microscopic scales. The scales are separable. Using multiscale convergence we deduce a multiscale homogenized variational inequality in a tensorized domain in the high-dimensional space ${\mathbb R}^{(n+1)d}$. Given sufficient regularity on the solution the sparse tensor product FE method is developed for this problem, which attains an essentially equal (i.e., it differs by only a logarithmic factor) level of accuracy to that of the full tensor product FE method, but requires an essentially optimal number of degrees of freedom which is equal to that for solving a problem in ${{\mathbb{R}}}^d$ apart from a logarithmic factor. For two-scale problems we deduce a new homogenization error for the nonlinear monotone variational inequality. A numerical corrector is then constructed with an explicit error in terms of the homogenization and the FE errors. For general multiscale problems we deduce a numerical corrector from the FE solution of the multiscale homogenized problem, but without an explicit error as such a homogenization error is not available.

Optimization of the porous material described by the Biot model

The paper is devoted to the shape optimization of microstructures generating porous locally periodic materials saturated by viscous fluids. At the macroscopic level, the porous material is described by the Biot model defined in terms of the effective medium coefficients, involving the drained skeleton elasticity, the Biot stress coupling, the Biot compressibility coefficients, and by the hydraulic permeability of the Darcy flow model. By virtue of the homogenization, these coefficients are computed using characteristic responses of the representative unit cell consisting of an elastic solid skeleton and a viscous pore fluid. For the purpose of optimization, the sensitivity analysis on the continuous level of the problem is derived. We provide sensitivities of objective functions constituted by the Biot model coefficients with respect to the underlying pore shape described by a B-spline box which embeds the whole representative cell. We consider material design problems in the framework of which the layout of a single representative cell is optimized. Then we propose a sequential linearization approach to the two-scale problem in which local microstructures are optimized with respect to macroscopic design criteria. Numerical experiments are reported which include stiffness maximization with constraints guaranteeing minimum required permeability, and vice versa. Issues of the design and anisotropy and the spline box parametrization are discussed. In order to avoid remeshing, a geometric regularization technique based on injectivity constraints is applied.

Computational homogenization of fresh concrete flow around reinforcing bars

Motivated by casting of fresh concrete in reinforced concrete structures, we introduce a numerical model of a steady-state non-Newtonian fluid flow through a porous domain. Our approach combines homogenization techniques to represent the reinforced domain by the Darcy law with an interfacial coupling of the Stokes and Darcy flows through the Beavers-Joseph-Saffman conditions. The ensuing two-scale problem is solved by the Finite Element Method with consistent linearization and the results obtained from the homogenization approach are verified against fully resolved direct numerical simulations.

Multiscale model for moisture transport with adsorption phenomenon in concrete materials

We consider a new two-scale problem which is given as a mathematical model for moisture transport arising in a concrete carbonation process. In research for moisture transport, it is a crucial step how to describe the relationship between the relative humidity and the degree of saturation, mathematically. Here, we have proposed the two-scale model consisting of the diffusion equation for the relative humidity in a macro domain and the free boundary problems describing the relationship in infinitely micro domains. Accordingly, the structures of the micro domains are unknown in our model. This is a significant feature of our new model to emphasize. The aim of this paper is to establish local existence in time and uniqueness of a solution to the model.

High-dimensional finite elements for multiscale Maxwell-type equations

We consider multiscale Maxwell-type equations in a domain |$D\subset\mathbb{R}^d$| (⁠|$d=2,3$|⁠), which depend on |$n$| microscopic scales. Using multiscale convergence, we derive the multiscale homogenized problem, which is posed in |$\mathbb{R}^{(n+1)d}$|⁠. Solving it, we get all the necessary macroscopic and microscopic information. Sparse tensor product finite elements (FEs) are employed, using edge FEs. The method achieves a required level of accuracy with essentially an optimal number of degrees of freedom, which, apart from a multiplying logarithmic term, is equal to that for solving a problem in |$\mathbb{R}^d$|⁠. Numerical correctors are constructed from the FE solutions. In the two-scale case, an explicit homogenization error is deduced. To get this error, the standard procedure in the homogenization literature requires the solution |$u_0$| of the homogenized problem to belong to |$H^1({\rm curl\,},D)$|⁠. However, in polygonal domains, |$u_0$| belongs only to a weaker regularity space |$H^s({\rm curl\,},D)$| for |$0<s<1$|⁠. We derive a homogenization error estimate for this case. Though we prove the result for two-scale Maxwell-type equations, the approach works verbatim for elliptic and elasticity problems when the solution to the homogenized equation belongs to |$H^{1+s}(D)$| (standard procedure requires |$H^2(D)$| regularity). This homogenization error estimate is new in the literature. Thus, for two-scale problems, an explicit error for the numerical corrector is obtained; it is of the order of the sum of the homogenization error and the FE error. For the case of more than two scales, we construct a numerical corrector, albeit without a rate of convergence, as such a homogenization error is not available. Numerical experiments confirm the theoretical results.

Localized Orthogonal Decomposition for two-scale Helmholtz-typeproblems

In this paper, we present a Localized Orthogonal Decomposition (LOD) in Petrov-Galerkinformulation for a two-scale Helmholtz-type problem. The two-scale problem is, for instance, motivatedfrom the homogenization of the Helmholtz equation with high contrast, studied together with acorresponding multiscale method in a previous paper of the authors. There, an unavoidable resolutioncondition on the mesh sizes in terms of the wave number has been observed, which is known as “pollutione ect” in the finite element literature. Following ideas of Gallistl and Peterseim, we use standardfinite element functions for the trial space, whereas the test functions are enriched by solutions of subscsaleproblems (solved on a finer grid) on local patches. Provided that the oversampling parameter m , which indicates the size of the patches, is coupled logarithmically to the wave number, we obtain aquasi-optimal method under a reasonable resolution of a few degrees of freedom per wave length, thusovercoming the pollution effect. In the two-scale setting, the main challenges for the LOD lie in thecoupling of the function spaces and in the periodic boundary conditions.

Numerical simulation of rolling contact fatigue crack growth in rails with the rail bending and the frictional contact

Due to the repeated passages of the wheels, the rail is subjected to time-dependent, multiaxial, non-proportional, mixed mode cyclic loading. Under these conditions of loading, the fatigue cracks can appear in the surface or subsurface of the rails. This paper attempts to investigate the multiaxial fatigue crack growth in rails using 3D numerical simulations taking into account the frictional contact between crack faces and the rail bending. Firstly, a quick description is given of the two-scale strategy (global structure and local interface) for the cracked body problem with frictional contact between the crack faces. In this strategy, a stabilized three-field weak formulation is proposed to avoid possible oscillations in the local solution linked to the LBB condition. This formulation is discretized within the framework of the eXtended Finite Element Method (XFEM). Then, the LATIN non-linear solver is used to solve the two-scale problem. Secondly, we focus on the 3D railway track model with an initial crack on the rail surface. The sleepers and the rail fastenings are modelled by introducing constraints on the vertical displacement of some nodes on the rail foot. This 3D cracked rail model is used to simulate the propagation of different types of crack (in curve, in straight line) under the multiaxial wheel-rail contact loading. Finally, a parametric study is investigated in order to clarify the influence of the friction coefficient between crack faces, the position of the crack in relation to the sleepers and the stiffness of the track on the crack growth behavior.