An atomic-continuum multiscale modeling approach for interfacial thermal behavior between materials

The aim of this paper is to provide a systematic method to perform interfacial thermal behavior between materials. A multiscale modeling method is proposed to investigate the interfacial thermal properties about copper nano interface structure. The interface stress element (ISE) method is set as a coupling button to a span-scale model combined with molecular dynamics (MD) and finite element (FE) methods. The handshake regions can simulate the structure transfer properties between the transition with MD and ISE, ISE and FE. The multiscale model is used to calculate the interfacial thermal characters under different temperatures. Some examples about numerical experiments with copper materials demonstrate the performance of MD–ISE–FE multiscale model is more successful compared with the approach applying MD–FE model. The results indicate that the accuracy of the MD–ISE–FE model is higher than that of MD–FE mode. This investigation implies a potential possibility of multiscale analysis from atomic to continuum scales.     

Selective Damping for Weakly Coupled FE-MD models

When dynamic problems are of interest, the high-frequency waves reflected from the coupling boundary will pollute the solution in the MD domain quickly and render the results from the multiscale model meaningless. In this contribution, we show that by applying artificial damping only to the high-frequency part of the displacement field in the coupling domain in the weak coupling method can significantly reduce the reflections in the MD domain while not affecting the low-frequency part that can be transmitted into the FE domain. The frequency analysis shows the errors in the magnitudes and phases of wave components are orders of magnitudes smaller than that in the bridging domain model or undamped weak coupling model. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A stochastic multiscale model for electricity generation capacity expansion

Long-term planning for electric power systems, or capacity expansion, has traditionally been modeled using simplified models or heuristics to approximate the short-term dynamics. However, current trends such as increasing penetration of intermittent renewable generation and increased demand response requires a coupling of both the long and short term dynamics. We present an efficient method for coupling multiple temporal scales using the framework of singular perturbation theory for the control of Markov processes in continuous time. We show that the uncertainties that exist in many energy planning problems, in particular load demand uncertainty and uncertainties in generation availability, can be captured with a multiscale model. We then use a dimensionality reduction technique, which is valid if the scale separation present in the model is large enough, to derive a computationally tractable model. We show that both wind data and electricity demand data do exhibit sufficient scale separation. A numerical example using real data and a finite difference approximation of the Hamilton–Jacobi–Bellman equation is used to illustrate the proposed method. We compare the results of our approximate model with those of the exact model. We also show that the proposed approximation outperforms a commonly used heuristic used in capacity expansion models.     

Multiscale Coupling through Locally Enriched Finite Elements

Multiscale modeling helps us to focus our limited computational power into those special places where traditional models based on continuum mechanics will fail while not losing the big picture of the macro scale behavior. An hourglass shaped development can be observed in today's simulation technologies. Simulation tools in the macroscale category and that for the micro phenomenons are both relatively well developed. Many algorithms and methods have been proposed in recent years to fill the gap between them. However, rather than trying to bridging different techniques, many tend to replace them completely and become independent simulation tools. Since many single–scale models have already been widely adopted by both the industry and the academy, it would be more beneficial to concentrate just on coupling techniques which can be applied without significant modifications of the original simulation framework. In this work, we present a multiscale idea of coupling the fine–scale model with the coarse–scale model through local enrichment within the elements at the coupling boundary. Higher order shape functions have been used to ‘enrich’ the coarse–scale model, allowing softer transition of the displacement field from the fine–scale model to the coarse–scale model. A least–square process has been used to fit the displacement gradients of different models at the coupling region. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Multiscale Approach of Turbulent Premixed Flames treated as a Gasdynamic Discontinuity

A multiscale approach describing a turbulent flame as a gasdynamic discontinuity [1] on each scale is proposed. The methodology becomes attractive if simulations on individual scales are implemented in a very efficient way. Simulations in a 2D rectangular box with an artificial turbulence field exploiting a robust phase-field/level-set method fit this purpose. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Spatial Geometric Nonlinearity Spline Beam Element With Nodal Parameters Containing Strains北大核心CSCD

Many slender rods in engineering can be modeled as Euler-Bernoulli beams. For the analysis of their dynamic behaviors, it is necessary to establish the dynamic models for the flexible multi-body systems. Geometric nonlinear elements with absolute nodal coordinates help solve a large number of dynamic problems of flexible beams, but they still face such problems as shear locking, nodal stress discontinuity and low computation efficiency. Based on the theory of large deformation beams’ virtual power equations, the functional formulas between displacements and rotation angles at the nodes were established, which can satisfy the deformation coupling relationships. The generalized strains to describe geometric nonlinear effects in this case were derived. Some parameters of boundary nodes were replaced by axial strains and sectional curvatures to obtain a more accurate and concise constraint method for applying external forces. To improve the numerical efficiency and stability of the system’s motion equations, a model-smoothing method was used to filter high frequencies out of the model. The numerical examples verify the rationality and effectiveness of the proposed element. © 2022 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

平面弹性裂纹分析的一种有效边界元方法

提出了一种简单而有效的平面弹性裂纹应力强度因子的边界元计算方法．该方法由Crouch与Starfield建立的常位移不连续单元和闫相桥最近提出的裂尖位移不连续单元构成A·D2在该边界元方法的实施过程中,左、右裂尖位移不连续单元分别置于裂纹的左、右裂尖处,而常位移不连续单元则分布于除了裂尖位移不连续单元占据的位置之外的整个裂纹面及其它边界．算例(如单向拉伸无限大板中心裂纹、单向拉伸无限大板中圆孔与裂纹的作用)说明平面弹性裂纹应力强度因子的边界元计算方法是非常有效的．此外,还对双轴载荷作用下有限大板中方孔分支裂纹进行了分析．这一数值结果说明平面弹性裂纹应力强度因子的边界元计算方法对有限体中复杂裂纹的有效性,可以揭示双轴载荷及裂纹体几何对应力强度因子的影响．     

QTT-finite-element approximation for multiscale problems I: model problems in one dimension

Tensor-compressed numerical solution of elliptic multiscale-diffusion and high frequency scattering problems is considered. For either problem class, solutions exhibit multiple length scales governed by the corresponding scale parameter: the scale of oscillations of the diffusion coefficient or smallest wavelength, respectively. As is well-known, this imposes a scale-resolution requirement on the number of degrees of freedom required to accurately represent the solutions in standard finite-element (FE) discretizations. Low-order FE methods are by now generally perceived unsuitable for high-frequency coefficients in diffusion problems and high wavenumbers in scattering problems. Accordingly, special techniques have been proposed instead (such as numerical homogenization, heterogeneous multiscale method, oversampling, etc.) which require, in some form, a-priori information on the microstructure of the solution. We analyze the approximation properties of tensor-formatted, conforming first-order FE methods for scale resolution in multiscale problems without a-priori information. The FE methods are based on the dynamic extraction of principal components from stiffness matrices, load and solution vectors by the quantized tensor train (QTT) decomposition. For prototypical model problems, we prove that this approach, by means of the QTT reparametrization of the FE space, allows to identify effective degrees of freedom to replace the degrees of freedom of a uniform “virtual” (i.e. never directly accessed) mesh, whose number may be prohibitively large to realize computationally. Precisely, solutions of model elliptic homogenization and high-frequency acoustic scattering problems are proved to admit QTT-structured approximations whose number of effective degrees of freedom required to reach a prescribed approximation error scales polylogarithmically with respect to the reciprocal of the target Sobolev-norm accuracy ε with only a mild dependence on the scale parameter. No a-priori information on the nature of the problems and intrinsic length scales of the solution is required in the numerical realization of the presently proposed QTT-structured approach. Although only univariate model multiscale problems are analyzed in the present paper, QTT structured algorithms are applicable also in several variables. Detailed numerical experiments confirm the theoretical bounds. As a corollary of our analysis, we prove that for the mentioned model problems, the Kolmogorov n-widths of solution sets are exponentially small for analytic data, independently of the problems’ scale parameters. That implies, in particular, the exponential convergence of reduced basis techniques which is scale-robust, i.e., independent of the scale parameter in the problem.     

Isogeometric analysis of 3D straight beam-type structures by Carrera Unified Formulation

This paper applies the isogeometric analysis (IGA) based on unified one-dimensional (1D) models to study static, free vibration and dynamic responses of metallic and laminated composite straight beam structures. By employing the Carrera Unified Formulation (CUF), 3D displacement fields are expanded as 1D generalized displacement unknowns over the cross-section domain. 2D hierarchical Legendre expansions (HLE) are adopted in the local area for the refinement of cross-section kinematics. In contrast, B-spline functions are used to approximate 1D generalized displacement unknowns, satisfying the requirement of interelement high-order continuity. Consequently, IGA-based weak-form governing equations can be derived using the principle of virtual work and written in terms of fundamental nuclei, which are independent of the class and order of beam theory. Several geometrically linear analyses are conducted to address the enhanced capability of the proposed approach, which is prominent in the detection of shear stresses, higher-order modes and stress wave propagation problems. Besides, 3D-like behaviors can be captured by the present IGA-based CUF-HLE method with reduced computational costs compared with 3D finite element method (FEM) and FEM-based CUF-HLE method.     

Variational formulation of the material failure process in solids by embedded discontinuities model

This work presents a variational formulation of the material failure process, idealized as strain or displacement discontinuities, by weak, strong, or discrete embedded discontinuities into a continuum. It is shown that the solution of the proposed variational formulation may be approximated by different types of finite elements with embedded discontinuities. The developed displacement approximation of a finite element split by the discontinuity leads to a symmetric stiffness matrix, which considers not only the continuity of tractions but also the rigid body relative motions of the portions in which the element is split. The variational formulation of a continuum with more than one discontinuity in its interior is developed. It is shown that this formulation may lead to finite elements with embedded discontinuities that can be classified as displacement, force, mixed, and hybrid models. To show the effectiveness of the proposed formulation, the classical example of a bar under tension is solved using one and 2D finite element approximations. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Multiscale method based on discontinuous Galerkin methods for homogenization problems

We propose a multiscale method for elliptic problems with highly oscillating coefficients based on a coupling of macro and micro methods in the framework of the heterogeneous multiscale method. The macro method, defined on a macroscopic triangulation, aims at recovering the effective (homogenized) solution of an unknown macro model. The unspecified data of this model are computed by micro methods on sampling domains during the macro assembly process. In this Note, we show how to construct such a coupling with a discontinuous macro finite element space. We show that the flux information needed in this formulation in order to impose weak interelement continuity can be recovered from the known micro calculations on the sampling domains. A fully discrete analysis is presented. To cite this article: A. Abdulle, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

含孔洞有限板与加固板的连接问题的分区广义变分原理解法

本文提出了一种新的有限元解法,采用复变函数作为有限单元模式,结合运用分区广义变分原理,解决了经贴焊加固板后的含孔洞有限板应力集中系数的计算问题,得到了级数形式的解析解.计算实践表明,本方法成功地分析了加固板与含孔洞有限板在焊接线上的位移连续和内力平衡问题.由于仅需划分三个单元,故与常规有限元方法比较,本方法可大大节约计算机内存,提高精度,降低计算时间.应力集中系数和焊接线处应力的数值计算结果列于诸表之中,可供工程技术人员设计参考.     

A higher order finite element including transverse normal strain for linear elastic composite plates with general lamination configurations

This paper describes a higher-order global-local theory for thermal/mechanical response of moderately thick laminated composites with general lamination configurations. In-plane displacement fields are constructed by superimposing the third-order local displacement field to the global cubic displacement field. To eliminate layer-dependent variables, interlaminar shear stress compatibility conditions have been employed, so that the number of variables involved in the proposed model is independent of the number of layers of laminates. Imposing shear stress free condition at the top and the bottom surfaces, derivatives of transverse displacement are eliminated from the displacement field, so that C⁰ interpolation functions are only required for the finite element implementation. To assess the proposed model, the quadratic six-node C⁰ triangular element is employed for the interpolation of all the displacement parameters defined at each nodal point on the composite plate. Comparing to various existing laminated plate models, it is found that simple C⁰ finite elements with non-zero normal strain could produce more accurate displacement and stresses for thick multilayer composite plates subjected to thermal and mechanical loads. Finally, it is remarked that the proposed model is quite robust, such that the finite element results are not sensitive to the mesh configuration and can rapidly converge to 3-D elasticity solutions using regular or irregular meshes.     

Coupling strategies for the numerical simulation of blood flow in deformable arteries by 3D and 1D models

The fluid structure interaction mechanism in vascular dynamics can be described by either 3D or 1D models, depending on the level of detail of the flow and pressure patterns needed for analysis. A successful strategy that has been proposed in the past years is the so-called geometrical multiscale approach, which consists of coupling both 3D and 1D models so as to use the former only in those regions where details of the fluid flow are needed and describe the remaining part of the vascular network by the simplified 1D model.In this paper we review recently proposed strategies to couple the 3D and 1D models, and within the 3D model, to couple the fluid and structure sub-problems. The 3D/1D coupling strategy relies on the imposition of the continuity of flow rate and total normal stress at the interface. On the other hand, the fluid–structure coupling strategy employs Robin transmission conditions. We present some numerical results and show the effectiveness of the new approaches.     

Analysis of arbitrarily shaped planar cracks in two-dimensional hexagonal quasicrystals with thermal effects. Part I: Theoretical solutions

An extended displacement discontinuity (EDD) boundary integral equation method is proposed for analysis of arbitrarily shaped planar cracks in two-dimensional (2D) hexagonal quasicrystals (QCs) with thermal effects. The EDDs include the phonon and phason displacement discontinuities and the temperature discontinuity on the crack surface. Green's functions for unit point EDDs in an infinite three-dimensional medium of 2D hexagonal QC are derived using the Hankel transform method. Based on the Green's functions and the superposition theorem, the EDD boundary integral equations for an arbitrarily shaped planar crack in an infinite 2D hexagonal QC body are established. Using the EDD boundary integral equation method, the asymptotic behavior along the crack front is studied and the classical singular index of 1/2 is obtained at the crack edge. The extended stress intensity factors are expressed in terms of the EDDs across crack surfaces. Finally, the energy release rate is obtained using the definitions of the stress intensity factors.     

Generation and propagation ofSH-type waves due to stress discontinuity in a linear viscoelastic layered medium

In this paper the generation and propagation ofSH-type waves due to stress discontinuíty in a linear viscoelastic layered medium is studied. Using Fourier transforms and complex contour integration technique, the displacement is evaluated at the free surface in closed form for two special types of stress discontinuity created at the interface. The numerical result for displacement component is evaluated for different values of nondimensional station (distance) and is shown graphically. Graphs are compared with the corresponding graph of classical elastic case.     

三维非局部弹性场中裂纹问题的分析方法

通过求解得到了三维非局部弹性力学对称情形的单位集中不连续位移基本解·基于该基本解和三维局部（经典）弹性力学的不连续位移边界积分方程———边界元方法·提出了三维非局部弹性力学中的平片裂纹Ⅰ型问题的通用解法,并给出了算例     

Development and application of multiscale models of acute viral infections in intervention research

The objective of this study is to advance two frontiers in multiscale modelling of acute viral infections, which are (a) the mathematical technology or technical frontier, where we present a new method for development of multiscale models of acute viral infections using influenza A virus (IAV) as a paradigm in which a new set of metrics to measure both individual level and community level infectiousness are introduced, and (b) the scientific applications frontier, where we demonstrate the implementation of multiscale modelling in evaluating the comparative effectiveness of IAV health interventions from efficacy data. The multiscale model is developed by integrating the within-host scale and the between-host scale. Using the example of IAV as a paradigm, we demonstrate the utility and process by which multiscale modelling can be used to evaluate the comparative effectiveness of health interventions that operate at different scale domains. The multiscale modelling is general enough to be applicable to other acute viral infections.     

Linking macroscopic deformation processes to microstructure evolution using an FE-FFT-based micro-macro transition and non-conserved phase-fields

The purpose of this work is the multiscale FE-FFT-based prediction of macroscopic material behavior, micromechanical fields and bulk microstructure evolution in polycrystalline materials subjected to macroscopic mechanical loading. The macroscopic boundary value problem (BVP) is solved using implicit finite element (FE) methods. In each macroscopic integration point, the microscopic BVP is embedded, the solution of which is found employing fast Fourier transform (FFT), fixed-point and Green's function methods. The mean material response is determined by the stress-strain relation at the micro scale or rather the volume average of the micromechanical fields. The evolution of the microstructure is modeled by means of non-conserved phase-fields. As an example, the proposed methodology is applied to the modeling of stress-induced martensitic phase transformations in metal alloys. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)