Two-scale Approach Research Articles

Two-scale optimal design of structures with thermal insulation materials

This paper introduces a two-scale topology optimization approach by integrating optimized structures with the design of their materials. The optimization aims to find a multifunctional structure composed of homogeneous porous material. Driven by the multi-objective functions, macrostructural stiffness and material thermal conductivity, stiff but lightweight structures composed of thermal insulation materials can be achieved through optimizing the topologies of the macrostructures and their material microstructure simultaneously. For such a two-scale optimization problem, the effective properties of materials derived from the homogenization method are applied to the analysis of macrostructure. Meanwhile, the displacement field of the macrostructure under given boundary conditions is used for the sensitivity analysis of the material microstructure. Then, the bi-directional evolutionary structural optimization (BESO) method is employed to iteratively update the macrostructures and material microstructures by ranking elemental sensitivity numbers at the both scales. Finally, some 2D and 3D numerical examples are presented to demonstrate the effectiveness of the proposed optimization algorithm. A variety of optimal macrostructures and their optimal material microstructures are obtained.

Experimental and numerical studies of initial cracking in CFRP cross-ply laminates

This work is concerned with the conditions for formation of the first (initial) cracks in composite laminates with cutouts or ply drop-offs subjected to in-plane loading. We study here the crack formation on the free edge of CFRP cross-ply laminates experimentally and by numerical stress and failure analysis. The free-edge surface strains are measured by the digital image correlation (DIC) technique. The numerical analysis consists of a two-scale approach, where the macro-level analysis is performed with a three-dimensional finite-element method (3D FEM) and the micro-level analysis uses a periodic unit-cell (PUC) in the transverse plies. The constitutive assumption made for the macro-level analysis is an orthotropic linear thermo-elastic solid for the unidirectional plies with a thin isotropic viscoplastic layer between the longitudinal and transverse plies. In the PUC, the fibers are assumed linear elastic, while the matrix is modeled as an elastic–viscoplastic solid. Crack formation is assumed to occur in the matrix by the dilatation induced brittle failure mechanism for which the dilatation energy density criterion is used.

A Multiscale Approach to Polycrystalline Materials Damage and Failure

A two-scale three-dimensional approach for degradation and failure in polycrystalline materials is presented. The method involves the component level and the grain scale. The damage-induced softening at the macroscale is modelled employing an initial stress boundary element approach. The microscopic degradation is explicitly modelled associating Representative Volume Elements (RVEs) to relevant points of the macro continuum and employing a cohesive-frictional 3D grain-boundary formulation to simulate intergranular degradation and failure in the Voronoi morphology. Macro-strains are downscaled as RVEs' periodic boundary conditions, while overall macro-stresses are obtained upscaling the micro-stress field via volume averages. The comparison between effective macro-stresses for the damaged and undamaged RVEs allows to define a macroscopic measure of local material degradation. Some attention is devoted to avoiding pathological damage localization at the macro-scale. The multiscale processing algorithm is described and some preliminary results are illustrated.

Modeling function-perfusion behavior in liver lobules including tissue, blood, glucose, lactate and glycogen by use of a coupled two-scale PDE-ODE approach.

This study focuses on a two-scale, continuum multicomponent model for the description of blood perfusion and cell metabolism in the liver. The model accounts for a spatial and time depending hydro-diffusion-advection-reaction description. We consider a solid-phase (tissue) containing glycogen and a fluid-phase (blood) containing glucose as well as lactate. The five-component model is enhanced by a two-scale approach including a macroscale (sinusoidal level) and a microscale (cell level). The perfusion on the macroscale within the lobules is described by a homogenized multiphasic approach based on the theory of porous media (mixture theory combined with the concept of volume fraction). On macro level, we recall the basic mixture model, the governing equations as well as the constitutive framework including the solid (tissue) stress, blood pressure and solutes chemical potential. In view of the transport phenomena, we discuss the blood flow including transverse isotropic permeability, as well as the transport of solute concentrations including diffusion and advection. The continuum multicomponent model on the macroscale finally leads to a coupled system of partial differential equations (PDE). In contrast, the hepatic metabolism on the microscale (cell level) was modeled via a coupled system of ordinary differential equations (ODE). Again, we recall the constitutive relations for cell metabolism level. A finite element implementation of this framework is used to provide an illustrative example, describing the spatial and time-depending perfusion-metabolism processes in liver lobules that integrates perfusion and metabolism of the liver.

Poincaré and logarithmic Sobolev inequalities by decomposition of the energy landscape

We consider a diffusion on a potential landscape which is given by a smooth Hamiltonian $H:\mathbb {R}^n\to \mathbb {R}$ in the regime of low temperature $\varepsilon$. We proof the Eyring-Kramers formula for the optimal constant in the Poincar\'{e} (PI) and logarithmic Sobolev inequality (LSI) for the associated generator $L=\varepsilon \Delta -\nabla H\cdot\nabla$ of the diffusion. The proof is based on a refinement of the two-scale approach introduced by Grunewald et al. [Ann. Inst. Henri Poincar\'{e} Probab. Stat. 45 (2009) 302-351] and of the mean-difference estimate introduced by Chafa\"{i} and Malrieu [Ann. Inst. Henri Poincar\'{e} Probab. Stat. 46 (2010) 72-96]. The Eyring-Kramers formula follows as a simple corollary from two main ingredients: The first one shows that the PI and LSI constant of the diffusion restricted to metastable regions corresponding to the local minima scales well in $\varepsilon$. This mimics the fast convergence of the diffusion to metastable states. The second ingredient is the estimation of a mean-difference by a weighted transport distance. It contains the main contribution to the PI and LSI constant, resulting from exponentially long waiting times of jumps between metastable states of the diffusion.

Homogenisation for elastic photonic crystals and dynamic anisotropy

We develop a continuum model, valid at high frequencies, for wave propagation through elastic media that contain periodic, or nearly periodic, arrangements of traction free, or clamped, inclusions. The homogenisation methodology we create allows for wavelengths and periodic spacing to potentially be of similar scale and therefore is not limited to purely long-waves and low frequency.We treat in-plane elasticity, with coupled shear and compressional waves and therefore a full vector problem, demonstrating that a two-scale asymptotic approach using a macroscale and microscale results in effective scalar continuum equations posed entirely upon the macroscale; the vector nature of the problem being incorporated on the microscale. This rather surprising result is comprehensively verified by comparing the resultant asymptotics to full numerical simulations for the Bloch problem of perfectly periodic media. The dispersion diagrams for this Bloch problem are found both numerically and asymptotically. Periodic media exhibit dynamic anisotropy, e.g. strongly directional fields at specific frequencies, and both finite element computations and the asymptotic theory predict this. Periodic media in elasticity can be related to the emergent fields of metamaterials and photonic crystals in electromagnetics and relevant analogies are drawn. As an illustration we consider the highly anisotropic cases and show how their existence can be predicted naturally from the homogenisation theory.

Urban vegetation for reducing heat related mortality

The potential benefit of urban vegetation in reducing heat related mortality in the city of Melbourne, Australia is investigated using a two-scale modelling approach. A meso-scale urban climate model was used to quantify the effects of ten urban vegetation schemes on the current climate in 2009 and future climates in 2030 and 2050. The indoor thermal performance of five residential buildings was then simulated using a building simulation tool with the local meso-climates associated with various urban vegetation schemes. Simulation results suggest that average seasonal summer temperatures can be reduced in the range of around 0.5 and 2 °C if the city were replaced by vegetated suburbs and parklands, respectively. With the limited buildings and local meso-climates investigated in this study, around 5–28% and 37–99% reduction in heat related mortality rate have been estimated by doubling the city's vegetation coverage and transforming the city into parklands respectively.

Application of hp-Adaptive Finite Element Method to Two-Scale Computation

Application of a two-scale approach to modeling of inelastic deformations of metal matrix composites is presented in the paper. We use the computational homogenization based on RVE analysis with certain modifications that reduce the computational cost and some approaches to estimate the modeling error introduced by homogenization. Furthermore, the automatic \(hp\)-adaptive finite element method is used to deliver approximation error-controlled results in reasonable time. Numerical examples demonstrate the efficiency and accuracy of the developed methods and algorithms.

Optimization of Nowcast Software WDSS-II for operational application over the Indian region

Nowcasting in the India Meteorological Department (IMD) is being provided for T + 0 to T + 2 h, using the Warning Decision Support System (WDSS-II) software. Prior to operational nowcasting over the Indian region, the parameters of the nowcast algorithm tool of the software were optimized, and accuracy was evaluated for various weather systems over Delhi. This optimization is demonstrated in this study with reference to three weather systems over Delhi, with each case representing one of three typical types of cloud systems over the region. These are—(a) convective lines associated with winter and early pre-monsoon weather systems, (b) deep convective cells that form in the pre-monsoon (April–June) and post-monsoon season (October–November) and (c) wide convective echoes that form during the monsoon season. The efficacy of the algorithm was assessed on a frame-by-frame basis as well as holistically for entire convective episodes. The important findings of the frame-by-frame study are (1) the inability of the inbuilt growth-decay algorithm to capture the evolution of storm cells, (2) setting of the threshold of detection of storms and tracking storms and (3) number of scales through which storms should be tracked. The holistic capabilities of the nowcast algorithm were tested for entire convective episodes using Model Evaluation Tools software. The results indicate that the advection algorithm tends to move the convective areas faster than observed at all time scales. Hence the multi-scale segmentation approach (over the two-scale approach) increases the smoothening of the output, at the cost of decreased nowcast skill. The inter-event comparison indicates that the low-intensity convective line zones, which are characteristic of winter and early pre-monsoon weather systems, have the most rapid temporal change in the overall area under convection. This leads to larger area errors during nowcasting of these systems. On the other hand, pre-monsoon systems comprised mostly isolated cells that reach great heights and move very fast, but do not have much horizontal area growth. The error in the nowcasting of these systems is mostly in respect of location error, as well as error in forecast of the intensity of the cells. The overall error in nowcasting is least for the monsoon systems over the Delhi region.

A Probabilistic Two-Scale Approach to Predict the High Cycle Fatigue Behaviour of Ti-Al6-V4 Parts Machined by High Pressure Water Jet Assisted Turning

The high cycle fatigue life of machined parts is affected by the so-called surface integrity induced by the machining process. To model the high cycle fatigue behaviour of turned parts a probabilistic two-scale continuum damage model is developed. While the macroscopic behaviour of the material is assumed to remain elastic during the fatigue loading, the fatigue prediction is based on the incremental evolution of micro-plasticity and damage. The non-standard initial mechanical state of the material in the sub-surface, viz. the plastic strains and residual stresses fields induced by the machining process are taken into account via an initial step prior to the fatigue loading. As far as the micro-geometry of the surface is concerned, an initial micro-crack distribution depending on the depth and shape of the micro-defects observed is introduced.

A two-scale approach for trabecular bone microstructure modeling based on computational homogenization procedure

In this paper the numerical implementation of two-scale modelling of bone microstructure is presented. The study is a part of long-term project on bone remodelling which drives bone microstructure change based directly on trabeculae surface energy. The proposed approach is based on a first-order computational homogenization technique. The coincidence of macro- and micro-model kinematics is done with the use of uniform displacement and traction boundary conditions. The computational homogenization procedure is driven by a self-prepared manager which is coded in Python. The computation on real bone structure (a piece of female Wistar rat bone) is performed as well.

Two-scale identification of composites’ material constants by means of computational intelligence methods

The paper deals with the two-scale approach to the identification of material constants in composite materials. Structures made of unidirectionally fibre-reinforced composites are examined. Composite constituents’ elastic constants in a micro scale are identified on the basis of measurements performed in a macro scale. Numerical homogenization methods using a representative volume element are employed. Static (displacements in sensor points) and dynamic (eigenfrequencies) data are considered as measurements. Ideal and disturbed measurements are taken into account. Computational intelligence methods in the form of evolutionary algorithms and artificial immune systems are used to perform the identification procedure. Finite element method is used to solve the boundary-value problem for composites in both scales. Numerical examples presenting the effectiveness of the proposed approach are attached. Statistical data are considered to compare the efficiency of the identification procedure for both algorithms and different measurement data.

The approach of Otto-Reznikoff revisited

In this article we consider a lattice system of unbounded continuous spins. Otto and Reznikoff used the two-scale approach to show that exponential decay of correlations yields a logarithmic Sobolev inequality (LSI) with uniform constant in the system size. We improve their statement by weakening the assumptions, for which a more detailed analysis based on two new ingredients is needed. The two new ingredients are a covariance estimate and a uniform moment estimate. We additionally provide a comparison principle for covariances showing that the correlations of a conditioned Gibbs measure are controlled by the correlations of the original Gibbs measure with ferromagnetic interaction. This comparison principle simplifies the verification of the hypotheses of the main result. As an application of the main result we show how sufficient algebraic decay of correlations yields the uniqueness of the infinite-volume Gibbs measure, generalizing a result of Yoshida from finite-range to infinite-range interaction.

Computational dynamic homogenization for the analysis of dispersive waves in layered rock masses with periodic fractures

The analysis of the wave propagation in layered rocks masses with periodic fractures is tackled via a two-scale approach in order to consider shape and size of the rock inhomogeneities. To match the displacement fields at the two scales, an approximation of the micro-displacement field is assumed that depends on the first and second gradients of the macro-displacement through micro-fluctuation displacement functions obtained by the finite element solution of cell problems derived by the classical asymptotic homogenization. The resulting equations of motion of the equivalent continuum at the macro-scale result to be not local in space, thus a dispersive wave propagation is obtained from the model. The simplifying hypotheses assumed in the multi-scale kinematics limit the validity of the model to the first dispersive branch in the frequency spectrum corresponding to the lowest modes.Although the homogenization procedure is developed to study the macro-scale wave propagation in rock masses with bounded domain, the reliability of the proposed method has been evaluated in the examples by considering unbounded rock masses and by comparing the dispersion curves provided by the rigorous process of Floquet–Bloch with those obtained by the method presented. The accuracy of the method is analyzed for compressional and shear waves propagating in the intact-layered rocks along the orthotropic axes. Therefore, the influence of crack density in the layered rock mass has been analyzed. Vertical cracks have been considered, periodically located in the stiffer layer, and two different crack densities have been analyzed, which are differentiated in the crack spacing. A good agreement is obtained in case of compressional waves travelling along the layering direction and in case of both shear and compressional waves normal to the layering. The comparison between two crack systems with different spacing has shown this aspect to have a remarkable effect on waves travelling along the direction of layering, and limited in the case of waves propagating normal to the layers.The equivalent continuous model obtained through the dynamic homogenization technique here presented may be applied to the computational analysis of non-stationary wave propagation in rock masses of finite size, also consisting of sub-domains with different macro-mechanical characteristics. This avoids the use of computational models represented at the scale of the heterogeneities, which may be too burdensome or even unfeasible.

Can pinewoods provide habitat for a deciduous forest specialist? A two-scale approach to the habitat selection of Bechstein's bat

Populations of Myotis bechsteinii in Sierra de Cazorla, Segura y Las Villas Natural Park (southwestern Spain) survive in a large, old-growth coniferous woodland, which is in contrast with known ecological preferences of the species. We tracked ten lactating females and studied patterns of habitat selection by Bechstein's bat (Myotis bechsteinii) at two spatial scales: macrohabitat and microhabitat, regarding forest essence (coniferous vs. deciduous). The tracked individuals always foraged within forested areas and did not use areas devoid of trees. At the macrohabitat level, no positive selection of deciduous stands was apparent, suggesting selection studies of coarse resolution may not be able to capture subtle selection patterns. At the microhabitat level Myotis bechsteinii selected deciduous patches within the coniferous matrix, therefore, our results corroborate the perception of this species as dependent of services provided by deciduous woodlands. Larger foraging home ranges and commuting distances as compared with other Mediterranean localities suggest that our studied population inhabits a marginal or suboptimal environment in terms of habitat quality. We argue that this population of Bechstein's bat has survived as a relict one probably as a consequence of fragmentation and transformation of deciduous forest ecosystems in the Mediterranean range.

Micromechanical damage modelling using a two-scale method for laminated composite structures

Failure mechanisms of fibre reinforced composite structures are closely related to processes within the heterogeneous material. In order to include these processes in numerical simulations a micromechanical model needs to be coupled to the finite element solution. This two-scale framework has in the current work been achieved using the reformulated High Fidelity Generalized Method of Cells (HFGMC) micromechanical model and the finite element code Abaqus/Explicit. The two-scale approach enables calculation of the stress field within the unit cell, based on the constitutive behaviour of each subcell and the unit cell morphology. As the stress distribution is determined for the representative unit cells, calculation of failure criteria and constitutive response of the composite are performed at the micro-level. Damage effects are also being modelled on the micromechanical level. Failure initiation has been predicted using three micromechanical failure criteria found in the literature – the 3D Tsai–Hill model, the MultiContinuum Theory model and the 3D Hashin type strain based failure criteria. Degradation of mechanical properties of the composite material has been introduced to the model using the damage model described in the work of Bednarcyk et al. [19]. The damage model relies on the 3D Hashin type strain based failure criteria and shows good agreement with experimental results.

Beyond the Continuum: How Molecular Solvent Structure Affects Electrostatics and Hydrodynamics at Solid–Electrolyte Interfaces

Standard continuum theory fails to predict several key experimental results of electrostatic and electrokinetic measurements at aqueous electrolyte interfaces. In order to extend the continuum theory to include the effects of molecular solvent structure, we generalize the equations for electrokinetic transport to incorporate a space dependent dielectric profile, viscosity profile, and non-electrostatic interaction potential. All necessary profiles are extracted from atomistic molecular dynamics (MD) simulations. We show that the MD results for the ion-specific distribution of counterions at charged hydrophilic and hydrophobic interfaces are accurately reproduced using the dielectric profile of pure water and a non-electrostatic repulsion in an extended Poisson-Boltzmann equation. The distributions of Na(+) at both surface types and Cl(-) at hydrophilic surfaces can be modeled using linear dielectric response theory, whereas for Cl(-) at hydrophobic surfaces it is necessary to apply nonlinear response theory. The extended Poisson-Boltzmann equation reproduces the experimental values of the double-layer capacitance for many different carbon-based surfaces. In conjunction with a generalized hydrodynamic theory that accounts for a space dependent viscosity, the model captures the experimentally observed saturation of the electrokinetic mobility as a function of the bare surface charge density and the so-called anomalous double-layer conductivity. The two-scale approach employed here-MD simulations and continuum theory-constitutes a successful modeling scheme, providing basic insight into the molecular origins of the static and kinetic properties of charged surfaces, and allowing quantitative modeling at low computational cost.

Erratum to “A two-scale Weibull approach to the failure of porous ceramic structures made by robocasting: Possibilities and limits” [J. Eur. Ceram. Soc. 33 (2013) 679–688

Erratum to “A two-scale Weibull approach to the failure of porous ceramic structures made by robocasting: Possibilities and limits” [J. Eur. Ceram. Soc. 33 (2013) 679–688

Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential

We consider a noninteracting unbounded spin system with conservation of the mean spin. We derive a uniform logarithmic Sobolev inequality (LSI) provided the single-site potential is a bounded perturbation of a strictly convex function. The scaling of the LSI constant is optimal in the system size. The argument adapts the two-scale approach of Grunewald, Villani, Westdickenberg and the second author from the quadratic to the general case. Using an asymmetric Brascamp-Lieb-type inequality for covariances, we reduce the task of deriving a uniform LSI to the convexification of the coarse-grained Hamiltonian, which follows from a general local Cram\'{e}r theorem.

Two-scale approach to predict multi-site cracking potential in 3-D structures using the generalized finite element method

This research utilized the novel computational framework of the generalized finite element method (GFEM) to predict the potential for crack propagation in concrete slabs. A two-scale approach, using the global–local concept within the GFEM framework (GFEMg–l), is applied to multi-site cracking problems (MSC), where different crack geometries are placed simultaneously at different positions in a three-dimensional airfield slab loaded by new generation aircraft gears. The GFEMg–l approach efficiently simulated multiple cracks without discretization in the global mesh, but only in the local domain. The GFEMg–l enrichment functions allow the displacements of the local problem to be represented in the global domain through enrichment functions from the local problems rather than explicitly modeling each crack discretely in the global domain. The main contribution of this work was extension of the GFEMg–l approach to a class of three-dimensional MSC problems involving realistic boundaries conditions and existence of multiple cracks spanning different orders of magnitude in size (scales) within the domain. For the linear elastic structure, bottom- and surface-initiated cracks with small dimensions were considered in conjunction with a larger macro-crack. Unlike traditional numerical methods, the proposed GFEMg–l made it possible to tackle this class of problems by avoiding refined crack front meshes in the global domain as well as numerical round-off errors. Furthermore, the two-scale approach significantly reduces the computational cost for large-scale 3-D MSC problems.