Fully coupled hydromechanical multiscale model with microdynamic effects

In this paper, a multiscale finite element framework is developed based on the first‐order homogenization method for fully coupled saturated porous media using an extension of the Hill‐Mandel theory in the presence of microdynamic effects. The multiscale method is employed for the consolidation problem of a 2‐dimensional saturated soil medium generated from the periodic arrangement of circular particles embedded in a square matrix, which is compared with the direct numerical simulation method. The effects of various issues, including the boundary conditions, size effects, particle arrangements, and the integral domain constraints for the microscale boundary value problem, are numerically investigated to illustrate the performance of a representative volume element in the proposed computational homogenization method of fully coupled saturated porous media. This study is aimed to clarify the effect of scale separation and size dependence, and to introduce characteristics of a proper representative volume element in multiscale modeling of saturated porous media.     

Size-dependent generalized thermoelasticity model for Timoshenko microbeams

A size-dependent, explicit formulation for coupled thermoelasticity addressing a Timoshenko microbeam is derived in this study. This novel model combines modified couple stresses and non-Fourier heat conduction to capture size effects in the microscale. To this purpose, a length-scale parameter as square root of the ratio of curvature modulus to shear modulus and a thermal relaxation time as the phase lag of heat flux vector are considered for predicting the thermomechanical behavior in a microscale device accurately. Governing equations and boundary conditions of motion are obtained simultaneously through variational formulation based on Hamilton’s principle. As for case study, the model is utilized for simply supported microbeams subjected to a constant impulsive force per unit length. A comparison of the results with those obtained by the classical elasticity and Fourier heat conduction theories is carried out. Findings indicate that simultaneous considering the length-scale parameter and thermal relaxation time has strong influence on the thermoelastic behavior of microbeams. In dynamic thermoelastic analysis of the microbeam, while the non-Fourier heat conduction model is employed, the modified couple stress theory predicts larger deflection compared with the classical theory.     

A two-scale homogenization framework for nonlinear effective thermal conductivity of laminated composites

In this study, we formulate the effective temperature-dependent thermal conductivity of laminated composites. The studied laminated composites consist of laminas (plies) made of unidirectional fiber-reinforced matrix with various fiber orientations. The effective thermal conductivity is obtained through a two-scale homogenization scheme. A simplified micromechanical model of a unidirectional fiber-reinforced lamina is formulated at the lower scale. Thermal conductivities of fiber and matrix constituents are allowed to change with temperature. The upper scale uses a sublaminate model to homogenize temperature-dependent thermal conductivities of only a representative lamina stacking sequence in laminated composites. The effective thermal conductivity of each lamina, in the sublaminate model, is obtained using the simplified micromechanical model. The thermal conductivities from the micromechanical and sublaminate models represent average nonlinear properties of fictitiously homogeneous composite media. Interface conditions between fiber and matrix constituents and within laminas are assumed to be perfect. Experimental data available in the literature are used to verify the proposed multi-scale framework. We then analyze transient heat conduction in the homogenized composites. Temperature profiles, during transient heat conduction, in the homogenized composites are compared to the ones in heterogeneous composites. The heterogeneous composites, having different fiber arrangements and sizes, are modeled using finite element (FE) method.     

Transverse heat transfer coefficients on a full size dual channel CICC ITER conductor

Dual channel Cable-In-Conduit Conductors (CICC) provide low hydraulic resistance and faster central channel circulation, limiting superconductors temperature rise. The Poloidal Field Insert Sample (PFIS) was tested in the SULTAN facility to evaluate the thermal coupling between the CICC channels upon an experimental heat transfer coefficient assessment. Simple assumptions on the flow – homogeneous central and annular temperatures, no jacket conduction, no steel inertia and diffusivity – lead to a one-dimensional thermal model fully solved in its transient response to a Heavyside temperature evolution at the inlet, using a Laplace transformation. Transient temperature step data fitted with the analytical resolution provide heat transfer coefficients as a function of mass flow rate, compared to crude predictions. The transient measurements provided consistent measurements on the full range of mass flow rate in both vertical flow directions, whereas steady state homogenization characteristic length measures pursuing the same goal suffer from annular isothermal assumption. Recommendations are made for the thermohydraulic instrumentation of future conductor samples.     

Variationally consistent computational homogenization of transient heat flow

A framework for variationally consistent homogenization, combined with a generalized macro‐homogeneity condition, is exploited for the analysis of non‐linear transient heat conduction. Within this framework the classical approach of (model‐based) first‐order homogenization for stationary problems is extended to transient problems. Homogenization is then carried out in the spatial domain on representative volume elements (RVE), which are (in practice) introduced in quadrature points in standard fashion. Along with the classical averages, a higher order conservation quantity is obtained. An iterative FE²‐algorithm is devised for the case of non‐linear diffusion and storage coefficients, and it is applied to transient heat conduction in a strongly heterogeneous particle composite. Parametric studies are carried out, in particular with respect to the influence of the ‘internal length’ associated with the second‐order conservation quantity. Copyright © 2009 John Wiley & Sons, Ltd.     

Prediction of transport properties of wood below the fiber saturation point - A multiscale homogenization approach and its experimental validation. Part II: Steady state moisture diffusion coefficient

In this publication a multiscale homogenization model for moisture transport in wood is developed and validated. The model aims at prediction of macroscopic transport properties of clear wood samples from their microstructure and the physical properties of a few microscale constituents. In the first part of this two-part paper, the theoretical background and fundamentals of the model were presented, and its specification for the estimation of macroscopic thermal conductivities was shown. In this second part the model is applied to steady state moisture diffusion below the fiber saturation point. The model starts on a scale of about 50 μm, where the wood cells form a honeycomb-like structure. In a first homogenization step the effective moisture transport behavior of the cell structure is determined from moisture diffusion properties of the cell walls and the (moist) air in lumens, respectively. Further homogenization steps account for the larger vessels that exist in hardwood species, the annual rings which are a succession of layers with different densities, and finally wood rays, that form pathways in the radial direction throughout the stem. The model validation rests on experiments as in the case of heat conduction: The macroscopic diffusion coefficients predicted by the multiscale homogenization model for tissue-specific composition data (input data set II) are compared to corresponding experimentally determined tissue-specific diffusion coefficients under steady state conditions (experimental data set). As for thermal conductivity, the good agreement of model predictions and test data underlines the suitability of the presented multiscale model.     

Formulation and implementation of stress‐driven and/or strain‐driven computational homogenization for finite strain

In this paper, we present a homogenization approach that can be used in the geometrically nonlinear regime for stress‐driven and strain‐driven homogenization and even a combination of both. Special attention is paid to the straightforward implementation in combination with the finite‐element method. The formulation follows directly from the principle of virtual work, the periodic boundary conditions, and the Hill–Mandel principle of macro‐homogeneity. The periodic boundary conditions are implemented using the Lagrange multiplier method to link macroscopic strain to the boundary displacements of the computational model of a representative volume element. We include the macroscopic strain as a set of additional degrees of freedom in the formulation. Via the Lagrange multipliers, the macroscopic stress naturally arises as the associated ‘forces’ that are conjugate to the macroscopic strain ‘displacements’. In contrast to most homogenization schemes, the second Piola–Kirchhoff stress and Green–Lagrange strain have been chosen for the macroscopic stress and strain measures in this formulation. The usage of other stress and strain measures such as the first Piola–Kirchhoff stress and the deformation gradient is discussed in the Appendix. Copyright © 2015 John Wiley & Sons, Ltd.     

A meshless analysis of three-dimensional transient heat conduction problems

In this paper, we consider a numerical modeling of a three-dimensional transient heat conduction problem. The modeling is carried out using a meshless reproducing kernel particle (RKPM) method. In the mathematical formulation, a variational method is employed to derive the discrete equations. The essential boundary conditions of the formulated problems are enforced by the penalty method. Compared with numerical methods based on meshes, the RKPM needs only scattered nodes, rather than having to mesh the domain of the problem. An error analysis of the RKPM for three-dimensional transient heat conduction problem is also presented in this paper. In order to demonstrate the applicability of the proposed solution procedures, numerical experiments are carried out for a few selected three-dimensional transient heat conduction problems.     

Coupled heat conduction and thermal stress analyses in particulate composites

This study introduces two micromechanical modeling approaches to analyze spatial variations of temperatures, stresses and displacements in particulate composites during transient heat conduction. In the first approach, a simple micromechanical model based on a first order homogenization scheme is adopted to obtain effective mechanical and thermal properties, i.e., coefficient of linear thermal expansion, thermal conductivity, and elastic constants, of a particulate composite. These effective properties are evaluated at each material (integration) point in three dimensional (3D) finite element (FE) models that represent homogenized composite media. The second approach treats a heterogeneous composite explicitly. Heterogeneous composites that consist of solid spherical particles randomly distributed in homogeneous matrix are generated using 3D continuum elements in an FE framework. For each volume fraction (VF) of particles, the FE models of heterogeneous composites with different particle sizes and arrangements are generated such that these models represent realistic volume elements “cut out” from a particulate composite. An extended definition of a RVE for heterogeneous composite is introduced, i.e., the number of heterogeneities in a fixed volume that yield the same expected effective response for the quantity of interest when subjected to similar loading and boundary conditions. Thermal and mechanical properties of both particle and matrix constituents are temperature dependent. The effects of particle distributions and sizes on the variations of temperature, stress and displacement fields are examined. The predictions of field variables from the homogenized micromechanical model are compared with those of the heterogeneous composites. Both displacement and temperature fields are found to be in good agreement. The micromechanical model that provides homogenized responses gives average values of the field variables. Thus, it cannot capture the discontinuities of the thermal stresses at the particle-matrix interface regions and local variations of the field variables within particle and matrix regions.     

Thermo‐mechanical analysis of periodic multiphase materials by a multiscale asymptotic homogenization approach

A spatial and temporal multiscale asymptotic homogenization method used to simulate thermo‐dynamic wave propagation in periodic multiphase materials is systematically studied. A general field governing equation of thermo‐dynamic wave propagation is expressed in a unified form with both inertia and velocity terms. Amplified spatial and reduced temporal scales are, respectively, introduced to account for spatial and temporal fluctuations and non‐local effects in the homogenized solution due to material heterogeneity and diverse time scales. The model is derived from the higher‐order homogenization theory with multiple spatial and temporal scales. It is also shown that the modified higher‐order terms bring in a non‐local dispersion effect of the microstructure of multiphase materials. One‐dimensional non‐Fourier heat conduction and dynamic problems under a thermal shock are computed to demonstrate the efficiency and validity of the developed procedure. The results indicate the disadvantages of classical spatial homogenization. Copyright © 2006 John Wiley & Sons, Ltd.     

A generalized theory of thermoelasticity based on thermomass and its uniqueness theorem

In this paper, a new theory of generalized thermoelasticity has been proposed by taking into account the general heat conduction law, which depends on the motion of the thermomass defined as the equivalent mass of phonon gas in dielectrics according to Einstein’s mass–energy relation and involves the inertia effect on the time and space of the heat flux and temperature. The formulations are derived and given for anisotropic heterogeneous and isotropic homogenous materials. The uniqueness theorem of equations for the isotropic homogenous materials is proved. By comparison with the other theories of generalized thermoelasticity, the theory based on the motion of thermomass is more reasonable to predict the propagation of thermal and elastic waves in the microscale heat conduction conditions.     

Boundary element method analyses of transient heat conduction in an unbounded solid layer containing inclusions

The boundary element method (BEM) is used to compute the three-dimensional transient heat conduction through an unbounded solid layer that may contain heterogeneities, when a pointwise heat source placed at some point in the media is excited. Analytical solutions for the steady-state response of this solid layer when subjected to a spatially sinusoidal harmonic heat line source are presented when the solid layer has no inclusions. These solutions are incorporated into a BEM formulation as Greens functions to avoid the discretization of flat media interfaces. The solution is obtained in the frequency domain, and time responses are computed by applying inverse (Fast) Fourier Transforms. Complex frequencies are used to prevent the aliasing phenomena. The results provided by the proposed Greens functions and BEM formulation are implemented and compared with those computed by a BEM code that uses the Greens functions for an unbounded media which requires the discretization of all solid interfaces with boundary elements. The proposed BEM model is then used to evaluate the temperature field evolution through an unbounded solid layer that contains cylindrical inclusions with different thermal properties, when illuminated by a plane heat source. In this model zero initial conditions are assumed. Different simulation analyses using this model are then performed to evaluate the importance of the thermal properties of the inclusions on transient heat conduction through the solid layer.     

Multiscale design optimization of lithium ion batteries using adjoint sensitivity analysis

The capacity of lithium ion batteries can be improved through the use of functionally graded electrodes. Here, we present a computational framework for optimizing the layout of electrodes using a multiscale lithium ion battery cell model. The model accounts for nonlinear transient transport processes and mechanical deformations at multiple scales. A key component of the optimization methodology is the formulation of the adjoint sensitivity equations of the multiscale battery model. The efficient solution of the adjoint equations relies on the decomposition of the multiscale problem into multiple, computationally small problems associated with the individual realizations of the microscale model. This decomposition method is shown to significantly reduce the computational time needed for sensitivity analysis versus numerical finite differencing. The potential of the proposed optimization framework is illustrated with numerical problems involving both macroscale and microscale performance criteria and design variables. The usable capacity of a lithium ion battery cell is maximized while limiting the stress level in the electrode particles through manipulation of the local porosities and particle radii. The optimization results suggest that optimal functionally graded electrodes improve the performance of a battery cell over using uniform porosity and particle radius distributions. Copyright © 2012 John Wiley & Sons, Ltd.     

A method of fundamental solutions for transient heat conduction in layered materials

In this paper we investigate an application of the method of fundamental solutions (MFS) to transient heat conduction in layered materials, where the thermal diffusivity is piecewise constant. Recently, in Johansson and Lesnic [A method of fundamental solutions for transient heat conduction. Eng Anal Boundary Elem 2008;32:697–703], a MFS was proposed with the sources placed outside the space domain of interest, and we extend that technique to numerically approximate the heat flow in layered materials. Theoretical properties of the method, as well as numerical investigations are included.     

A method of two-scale thermo-mechanical analysis for porous solids with micro-scale heat transfer

A two-scale thermo-mechanical model for porous solids is derived and is implemented into a multi-scale multi-physics analysis method. The model is derived based on the mathematical homogenization method and can account for the scale effect of unit cells, which is our particular interest in this paper, on macroscopic thermal behavior and, by extension, on macroscopic deformation due to thermal expansion/contraction. The scale effect is thought to be the result of microscopic heat transfer, the amount of which depends on the micro-scale pore size of porous solids. We first formulate a two-scale model by applying the method of asymptotic expansions for homogenization and, by using a simple numerical model, verify the validity and relevancy of the proposed two-scale model by comparing it with a corresponding single-scale direct analysis with detailed numerical models.     

Wavelet-based homogenization of unidirectional multiscale composites

The main aim is to present a homogenization algorithm for the multiscale heterogeneous (composite) materials, which is based on the wavelet representation of material properties and the relevant multiscale reduction. It is shown that classical homogenization method used before for two-scale composites (with micro and macro scales) is a special case of general multiresolutional strategy, where a single scale parameter tends to 0. The approach presented is applied to unidirectional wavelet-based homogenization of linear elasticity heterogeneous problem and to wave propagation, which may be applied in conjunction with various discrete numerical methods for efficient modeling of heterogeneous solids, fluids and multiphase media.     

瞬态热传导区间反演分析

基于区间有限元和矩阵摄动理论, 引入同伦技术, 建立了瞬态热传导不确定性区间参数反演识别的数值求解模式。利用测量信息和计算信息的区间残差构造同伦函数, 将反演识别问题转化为一个优化问题进行求解。时间域上, 引入时域精细算法进行离散, 空间上, 采用八节点等参元技术进行离散, 并结合区间有限元法, 建立了便于敏度分析的不确定性正反演数值模型。该模型不仅考虑了非均质和参数分布的影响, 而且也便于正演和反演问题的敏度分析, 可对导热系数和热边界条件等宗量的区间范围进行有效的单一和组合识别, 并给出了相关的数值算例。数值结果表明了所建数值模型的有效性和可行性, 并具有较高的计算精度。     

多维非经典热传导问题的时间2空间多尺度高阶均匀化分析

采用一种时间-空间多尺度高阶渐近均匀化分析方法,模拟了热冲击载荷条件下多维微尺度多相周期性结构中的非经典热传导问题。通过引入放大空间尺度和缩小时间尺度,在不同时间尺度上获得由空间非均匀性引起的波动效应和非局部效应。根据高阶均匀化理论在空间和时间上进行均匀化,消去缩小时间尺度,确定各阶等效均匀化热传导系数的关系并对该系数进行数值求解,获得了多维非傅里叶热传导高阶非局部温度场控制方程。进而对二维周期性多相材料中的非傅里叶热传导问题进行分析,结果证明了本文中所提出的多维非傅里叶热传导高阶非局部模型的正确性与有效性。     

Thermoelastic analysis of a partially insulated crack in a strip under thermal impact loading using the hyperbolic heat conduction theory

In this paper, the transient temperature and thermal stresses around a partially insulated crack in a thermoelastic strip under a temperature impact are obtained using the hyperbolic heat conduction theory. Fourier and Laplace transforms are applied and the thermal and mechanical problems are reduced to solving singular integral equations. Numerical results show that the hyperbolic heat conduction parameters, the thermal conductivity of crack faces, and the geometric size of the strip have significant influence on the dynamic temperature and stress field. The results based on hyperbolic heat conduction show much higher temperature and much more dynamic thermal stress concentrations in the very early stage of impact loading comparing to the Fourier heat conduction model. It is suggested that to design materials and structures against fracture under transient thermal loading, the hyperbolic model is more appropriate than the Fourier heat conduction model.