Multi-scale modelling of sandwich structures using the Arlequin method Part I: Linear modelling

The paper presents an Arlequin based multi-scale method for studying problems related to the mechanical behaviour of sandwich composite structures. Towards this end, different models are mixed and glued to each other. Several coupling operators are tested in order to assess the usefulness of the proposed approach. A new coupling operator is proposed and tested on the different glued Arlequin zones. A free–clamped sandwich beam with soft core undergoing a concentrated effort on the free edge is used as a typical example (benchmark) in the validation procedure. Numerical simulations were conducted as the preliminary evaluation of the various coupling operators and the discrepancies between local and global models in the gluing zone have been addressed with sufficient care.     

Numerical modeling of atomistic-to-continuum coupling based on bridging domain method

Nano-submodeling is an approach that enables insertion of nano-refined submodel (atomistic) in the global model (continuum). In this work analysis of the spurious effects that may arise in the concurrent atomistic-to-continuum coupling is performed. The coupling is based on the overlapping domain decomposition (ODD) method called bridging domain [1, 2] (similar is Arlequin [3] method) where different models are overlapped and the displacements compatibility is enforced via Lagrange multipliers (LM). Some coupling options such as energy weighting, coupling zone geometry and LM field interpolation are tested. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling issues of a holistic finite element approach in the virtual engineering design process

In the paper methods for the coupling of different meshed local domains of a product model, which occur in an adaptivity process are compared. The properties of the applied coupling methods are demonstrated on finite element models from structural mechanics to point out the advantages and disadvantages of the methods in a model adaptivity process in the field of automotive engineering. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Domain Formation in Membranes Near the Onset of Instability

The formation of microdomains, also called rafts, in biomembranes can be attributed to the surface tension of the membrane. In order to model this phenomenon, a model involving a coupling between the local composition and the local curvature was proposed by Seul and Andelman in 1995. In addition to the familiar Cahn–Hilliard/Modica–Mortola energy, there are additional ‘forces’ that prevent large domains of homogeneous concentration. This is taken into account by the bending energy of the membrane, which is coupled to the value of the order parameter, and reflects the notion that surface tension associated with a slightly curved membrane influences the localization of phases as the geometry of the lipids has an effect on the preferred placement on the membrane. The main result of the paper is the study of the \(\Gamma \)-convergence of this family of energy functionals, involving nonlocal as well as negative terms. Since the minimizers of the limiting energy have minimal interfaces, the physical interpretation is that, within a sufficiently strong interspecies surface tension and a large enough sample size, raft microdomains are not formed.     

On the symmetric boundary element method and the symmetric coupling of boundary elements and finite elements

The finite element method and the boundary element method areamong the most frequently applied tools in the numerical treatmentof partial differential equations. However, their propertiesappear to be complementary: while the boundary element methodis appropriate for the most important linear partial differentialequations with constant coefficients in bounded or unboundeddomains, the finite element method seems to be more appropriatefor inhomogeneous or even nonlinear problems. but is somehowrestricted to bounded domains. The symmetric coupling of thetwo methods inherits the advantages of both methods. This paper treats the symmetric coupling of finite elementsand boundary elements for a model transmission problem in twoand three dimensions where we have two domains: a bounded domainwith nonlinear, even plastic material behaviour, is surroundedby an unbounded, exterior, domain with isotropic homogeneouslinear elastic material. Practically. the coupling is performedsuch that the boundary element method contributes a macro-element,like a large finite element, within a standard finite elementanalysis program. Emphasis is on two-dimensional problems wherethe approach using the Poincaré-Steklov operator seemsto be impossible at first glance. E-mail: cc{at}numerik.uni-kiel.de     

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).     

A spectral method for the Stokes problem in three-dimensional unbounded domains

We present a method for solving the Stokes problem in unbounded domains. It relies on the coupling of the transparent boundary operator and a spectral method in spherical coordinates. It is done explicitly by the use of vector-valued spherical harmonics. A uniform inf-sup condition is proved, which provides an optimal error estimate.

     

How single node dynamics enhances synchronization in neural networks with electrical coupling

The stability of the completely synchronous state in neural networks with electrical coupling is analytically investigated applying both the Master Stability Function approach (MSF), developed by Pecora and Carroll (1998), and the Connection Graph Stability method (CGS) proposed by Belykh et al. (2004). The local dynamics is described by Morris–Lecar model for spiking neurons and by Hindmarsh–Rose model in spike, burst, irregular spike and irregular burst regimes. The combined application of both CGS and MSF methods provides an efficient estimate of the synchronization thresholds, namely bounds for the coupling strength ranges in which the synchronous state is stable. In all the considered cases, we observe that high values of coupling strength tend to synchronize the system. Furthermore, we observe a correlation between the single node attractor and the local stability properties given by MSF. The analytical results are compared with numerical simulations on a sample network, with excellent agreement.     

Radial basis function collocation method for the numerical solution of the two-dimensional transient nonlinear coupled Burgers’ equations

This paper examines the numerical solution of the transient nonlinear coupled Burgers’ equations by a Local Radial Basis Functions Collocation Method (LRBFCM) for large values of Reynolds number (Re) up to 10³. The LRBFCM belongs to a class of truly meshless methods which do not need any underlying mesh but works on a set of uniform or random nodes without any a priori node to node connectivity. The time discretization is performed in an explicit way and collocation with the multiquadric radial basis functions (RBFs) are used to interpolate diffusion-convection variable and its spatial derivatives on decomposed domains. Five nodded domains of influence are used in the local support. Adaptive upwind technique [1] and [54] is used for stabilization of the method for large Re in the case of mixed boundary conditions. Accuracy of the method is assessed as a function of time and space discretizations. The method is tested on two benchmark problems having Dirichlet and mixed boundary conditions. The numerical solution obtained from the LRBFCM for different value of Re is compared with analytical solution as well as other numerical methods [15], [47] and [49]. It is shown that the proposed method is efficient, accurate and stable for flow with reasonably high Reynolds numbers.     

Local a posteriori estimates for pointwise gradient errors in finite element methods for elliptic problems

We prove local a posteriori error estimates for pointwise gradient errors in finite element methods for a second-order linear elliptic model problem. First we split the local gradient error into a computable local residual term and a weaker global norm of the finite element error (the ``pollution term'). Using a mesh-dependent weight, the residual term is bounded in a sharply localized fashion. In specific situations the pollution term may also be bounded by computable residual estimators. On nonconvex polygonal and polyhedral domains in two and three space dimensions, we may choose estimators for the pollution term which do not employ specific knowledge of corner singularities and which are valid on domains with cracks. The finite element mesh is only required to be simplicial and shape-regular, so that highly graded and unstructured meshes are allowed.

     

Stress continuity in DEM-FEM multiscale coupling based on the generalized bridging domain method

Concurrent multiscale method is a spatial and temporal combination of two different scale models for describing the micro/meso and macro mixed behaviors observed in strain localization, failure and phase transformation processes, etc. Most of the existing coupling schemes use the displacement compatibility conditions to glue different scale models, which leads to displacement continuity and stress discontinuity for the obtained multiscale model. To overcome stress discontinuity, this paper presented a multiscale method based on the generalized bridging domain method for coupling the discrete element (DE) and finite element (FE) models. This coupling scheme adopted displacement and stress mixed compatibility conditions. Displacements that were interpolated from FE nodes were prescribed on the artificial boundary of DE model, while stresses at numerical integration points that were extracted from DE contact forces were applied on the material transition zone of FE model (the coupling domain and the artificial boundary of FE model). In addition, this paper proposed an explicit multiple time-steps integration algorithm and adopted Cundall nonviscous damping for quasi-static problems. DE and FE parameters were calibrated by DE simulations of a biaxial compression test and a deposition process. Numerical examples for a 2D cone penetration test (CPT) show that the proposed multiscale method captures both mesoscopic and macroscopic behaviors such as sand soil particle rearrangement, stress concentration near the cone tip, shear dilation, penetration resistance vibration and particle rotation, etc, during the cone penetration process. The proposed multiscale method is versatile for maintaining stress continuity in coupling different scale models.     

A non-overlapping domain decomposition for low-frequency time-harmonic Maxwell’s equations in unbounded domains

In this paper, we are concerned with a non-overlapping domain decomposition method for solving the low-frequency time-harmonic Maxwell’s equations in unbounded domains. This method can be viewed as a coupling of finite elements and boundary elements in unbounded domains, which are decomposed into two subdomains with a spherical artificial boundary. We first introduce a discretization for the coupled variational problem by combining Nédélec edge elements of the lowest order and curvilinear elements. Then we design a D-N alternating method for solving the discrete problem. In the method, one needs only to solve the finite element problem (in a bounded domain) and calculate some boundary integrations, instead of solving a boundary integral equation. It will be shown that such iterative algorithm converges with a rate independent of the mesh size. The work of Qiya Hu was supported by Natural Science Foundation of China G10371129.     

Low-amplitude instability as a premise for the spontaneous symmetry breaking in the new integrable semidiscrete nonlinear system

The new integrable semidiscrete multicomponent nonlinear system characterized by two coupling parameters is presented. Relying upon the lowest local conservation laws the concise form of the system is given and its selfconsistent symmetric parametrization in terms of four independent field variables is found. The comprehensive analysis of quartic dispersion equation for the system low-amplitude excitations is made. The criteria distinguishing the domains of stability and instability of low-amplitude excitations are formulated and a collection of qualitatively distinct realizations of a dispersion law are graphically presented. The loop-like structure of a low-amplitude dispersion law of reduced system emerging within certain windows of adjustable coupling parameter turns out to resemble the loop-like structure of a dispersion law typical of beam-plasma oscillations. Basing on the peculiarities of low-amplitude dispersion law as the function of adjustable coupling parameter it is possible to predict the windows of spontaneous symmetry breaking even without an explicit knowledge of the system Lagrangian function. Having been rewritten in terms of properly chosen modified field variables the reduced four wave integrable system can be qualified as consisting of two coupled nonlinear lattice subsystems, namely the self-dual ladder network and the vibrational ones.     

A Renormalization Group Method by Harmonic Extensions and the Classical Dipole Gas

In this paper, we develop a new renormalization group method, which is based on conditional expectations and harmonic extensions, to study functional integrals of small perturbations of Gaussian fields. In this new method, one integrates Gaussian fields inside domains at all scales conditioning on the fields outside these domains, and by the variation principle solves local elliptic problems. It does not rely on an a priori decomposition of the Gaussian covariance. We apply this method to the model of classical dipole gas on the lattice, and show that the scaling limit of the generating function with smooth test functions is the generating function of the renormalized Gaussian free field.     

Local limit theorems for occupancy models

We present a rather general method for proving local limit theorems, with a good rate of convergence, for sums of dependent random variables. The method is applicable when a Stein coupling can be exhibited. Our approach involves both Stein's method for distributional approximation and Stein's method for concentration. As applications, we prove local central limit theorems with rate of convergence for the number of germs with d neighbors in a germ‐grain model, and the number of degree‐d vertices in an Erd?s‐Rényi random graph. In both cases, the error rate is optimal, up to logarithmic factors.     

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)     

Multiscale analysis and simulation of a reaction–diffusion problem with transmission conditions

This paper considers a transmission problem for partial differential equations obtained in the recent paper [M. Neuss-Radu, W. Jäger, Effective transmission conditions for reaction-diffusion processes in domains separated by an interface, SIAM J. Math. Anal. 39 (2007) 687–720] as the effective system modeling a reaction–diffusion process in two domains separated by a membrane. For the analysis of this problem an appropriate function space, which includes the coupling conditions for the concentrations on the interface, is introduced. The transmission conditions for the flux are included in the variational formulation with respect to this function space.The solution of the transmission problem is approximated by using the Galerkin method. Numerically the problem is reduced to a system of ordinary differential equations, where the coupling of the micro- and macro-variables leads to a special structure, distinguishing the variables relevant for the transmission. Results of numerical simulations are illustrating the effect of the microscopic process in the membrane on the macroscopic reaction–diffusion process in the bulk domains.     

Modeling of deformation-induced phase transformation during very high cycle fatigue (VHCF) using the boundary element method

A microstructural simulation model is proposed which accounts for damage accumulation in shear bands and deformation-induced martensite formation in a metastable austenitic stainless steel (AISI304). The model is numerically solved using the two-dimensional (2-D) boundary element method. By using this method, sliding displacements can be directly evaluated in shear bands and austenite grains as well as generated martensite domains with their individual mechanical properties and shape deformation can be considered. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)