Two-scale Formulation Research Articles

A multi-scale formulation for polycrystalline materials accounting for cohesive micro-cracks: Homogenisation of the traction-separation law

This contribution investigates the modelling of failure in polycrystalline materials due to the nucleation and propagation of cohesive cracks at the micro-scale. First, a two-scale formulation that enables the analysis of failure, even when the evolution and propagation of cohesive micro-cracks induce material instability, is proposed. The development of this macro-continuous/micro-discontinuous formulation is based on the Method of Multi-Scale Virtual Power, from which dynamic equilibrium and homogenisation equations are derived, ensuring a variationally consistent scale transition. The respective minimal kinematical constraint is retrieved and enforced with the Lagrange multiplier method. The formulation is also extended for periodic boundary conditions. The discretisation of the resulting equilibrium equations, with the finite element method and the generalised-α method, is described in detail.Second, a fracture-based computational homogenisation procedure is developed to obtain homogenised traction-separation laws and fracture properties from microstructural volume elements. It is derived from a crack-based Hill-Mandel principle and employs a novel energetic-based damage variable, within the Park-Paulino-Roesler cohesive model, proposed to define the crack domain and compute the crack homogenised quantities. A strategy for accurately computing the homogenised unit normal vector of the equivalent macroscopic crack is also suggested.Numerical examples demonstrate the formulation’s ability to analyse fracture problems, including a study on the impact of different microscopic boundary conditions. This work also reveals the convergence of the traction-separation law with increasing microstructure model size when using the fracture-based computational homogenisation. Furthermore, it investigates how advanced deformation mechanisms, like slip plasticity and martensitic transformation, affect intergranular crack propagation mechanisms.

A multiscale approach in modeling of chemically reactive porous media

Chemical reactions occur in geotechnical, biological and synthetic porous media. Swelling and shrinkage phenomena can be observed in clays, shales and gels, when they are in close contact with water. In petroleum engineering, wellbore-stability problems are caused by swelling or shrinking of the wellbore. Thus, it is significant to model the Chemo-Hydro-Mechanical (CHM) processes in porous media that include different physical and geometrical heterogeneities. In this paper, a multiscale computational homogenization approach is developed for the analysis of CHM problems. In this manner, the first-order homogenization technique is adopted for the two-scale formulation. The heterogeneities are assumed in the microscale level and a homogeneous domain is considered for the macroscale level, where the constitutive behavior is derived from the microscopic level. The governing equations and constitutive equations of the CHM processes are presented in the coupled manner. The primary variables are taken as the displacement, pore pressure, and solute concentration fields. Moreover, the transient terms are included in the microscale level to increase the accuracy of computations. Consequently, a generalized form of Hill-Mandel principle of macro-homogeneity is defined between the two scales. Appropriate microscopic boundary conditions, i.e. the linear and periodic boundary conditions, are employed to satisfy the averaging constraints. Finally, the accuracy and efficiency of the proposed computational multiscale method are illustrated through several numerical examples.

Geometric Two-Scale Integrators for Highly Oscillatory System: Uniform Accuracy and Near Conservations

In this paper, we consider a class of highly oscillatory Hamiltonian systems which involve a scaling parameter . The problem arises from many physical models in some limit parameter regime or from some time-compressed perturbation problems. The solution of the model exhibits rapid temporal oscillations with -amplitude and -frequency, which makes classical numerical methods inefficient. We apply the two-scale formulation approach to the problem and propose two new time-symmetric numerical integrators. The methods are proved to have the uniform second order accuracy for all at finite times and some near-conservation laws in long times. Numerical experiments on a Hénon–Heiles model, two nonlinear Schrödinger equations, and a charged-particle system illustrate the performance of the proposed methods over the existing ones.

A unified two-scale theory for modeling microstructural length scale, strain gradient and strain rate effects on brittle fracture

Microstructure, strain gradient and strain rate are key factors that influence fracture of heterogeneous brittle materials. The present work for the first time unifies effects of these factors in an analytical dynamic fracture criterion based on fundamental micromechanics. By using an energy-based two-scale theory accounting for micro-inertia, we rigorously derive an formulation for microscopic dynamic energy release rate involving additive contributions of macroscopic strain, strain gradient and strain rate. The coefficients of the formulation are correlated to microstructural length scale and are calculated based on integrals of the first order microscopic cell solutions. The two-scale formulation of energy release rate, along with the Griffith law for a single micro-crack, results in a novel dynamic fracture model. The remarkable feature of this modeling approach is that, without extra phenomenological ad hoc hypotheses, all microstructural length scale, strain gradient and strain rate effects are natural consequences of the unified two-scale theory.Capabilities of the dynamic fracture model for predicting coupled effects of microstructure size, strain gradient and strain rate on brittle fracture behaviors are confirmed by numerical simulations. The model is also well validated against experimental results. Especially, finite element simulations based on the model well reproduce free surface velocity profiles, fracture zones and spall strengths measured in series of dynamic spalling fracture experiments.

Learning constitutive models from microstructural simulations via a non-intrusive reduced basis method

In order to optimally design materials, it is crucial to understand the structure–property relations in the material by analyzing the effect of microstructure parameters on the macroscopic properties. In computational homogenization, the microstructure is thus explicitly modeled inside the macrostructure, leading to a coupled two-scale formulation. Unfortunately, the high computational costs of such multiscale simulations often render the solution of design, optimization, or inverse problems infeasible. To address this issue, we propose in this work a non-intrusive reduced basis method to construct inexpensive surrogates for parametrized microscale problems; the method is specifically well-suited for multiscale simulations since the coupled simulation is decoupled into two independent problems: (1) solving the microscopic problem for different (loading or material) parameters and learning a surrogate model from the data; and (2) solving the macroscopic problem with the learned material model. The proposed method has three key features. First, the microscopic stress field can be fully recovered, which is useful for instance for revealing local stress concentrations inside the microstructure. Second, the method is able to accurately predict the stress field for a wide range of material parameters; furthermore, the derivatives of the effective stress with respect to the material parameters are available and can be readily utilized in solving optimization problems. Finally, it is more data efficient, i.e. requiring less training data, as compared to directly performing a regression on the effective stress. To construct the surrogate model, first, a proper orthogonal decomposition is performed on precomputed microscopic stress field snapshots to find a reduced basis for the stress. Second, a regression is employed to infer the coefficients of the reduced basis approximation for any arbitrary parameter value, thus enabling a rapid online evaluation of the microscopic stress. Equipped with the stress field, the effective stress and its partial derivatives can then be derived analytically. For the microstructures in the two test problems considered, the mean approximation error of the effective stress is as low as 0.1% despite using a relatively small training dataset. Embedded into the macroscopic problem, the reduced order model leads to an online computational speed up of approximately three orders of magnitude while maintaining a high accuracy as compared to the FE2 solver.

Comparison of numerical methods for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime

Different efficient and accurate numerical methods have recently been proposed and analyzed for the nonlinear Klein-Gordon equation (NKGE) with a dimensionless parameter ε∈(0,1], which is inversely proportional to the speed of light. In the nonrelativistic limit regime, i.e. 0<ε≪1, the solution of the NKGE propagates waves with wavelength at O(1) and O(ε2) in space and time, respectively, which brings significantly numerical burdens in designing numerical methods. We compare systematically spatial/temporal efficiency and accuracy as well as ε-resolution (or ε-scalability) of different numerical methods including finite difference time domain methods, time-splitting method, exponential wave integrator, limit integrator, multiscale time integrator, two-scale formulation method and iterative exponential integrator. Finally, we adopt the multiscale time integrator to study the convergence rates from the NKGE to its limiting models when ε→0+.

Two-Scale Envelope-Domain Analysis of Injected Chirped Oscillators

The response of chirped oscillators under the injection of independent signals, for spectrum sensing in cognitive radio, and under self-injection, for radio frequency identification, is analyzed in detail. The investigation is performed by means of a semianalytical formulation, based on a realistic modeling of the free-running oscillator, extracted from harmonic-balance simulations or from experimental measurements, through a new characterization technique. In the new formulation, the oscillator is linearized about a free-running solution that varies with the control voltage. This enables its application to oscillators having a frequency characteristic that deviates from the linear one. In the case of injection by independent signals, the two-scale envelope-domain formulation will enable an efficient handling of the difference between the slow chirp frequency and the beat frequency. The input carriers can be detected from their dynamic synchronization intervals or, at lower input-power levels, from the dynamics of the beat frequency. Noise perturbations are introduced into the formulation, which enables an estimation of the minimum detectable signal. In the case of a self-injected oscillator for radio frequency identification, an insightful formulation is derived to predict the propagation and tag-resonance effects on the instantaneous oscillation frequency. The tag-resonance signature gives rise to a distinct modulation of the oscillation frequency during the chirp period, which can be detected from the variation of the oscillator bias current. The analysis methods are illustrated through their application to a chirped oscillator, operating in the band 2–3 GHz.

Numerical methods for the two-dimensional Vlasov–Poisson equation in the finite Larmor radius approximation regime

In this paper, we consider numerical methods for solving the two-dimensional Vlasov–Poisson equation in the finite Larmor radius approximation regime. The model describes the behaviour of charged particles under a strong external magnetic field and the finite Larmor radius scaling. We discretise the equation under Particle-in-Cell method, where the characteristics equations are highly oscillatory system in the limit regime. We apply popular numerical integrators including splitting methods, multi-revolution composition methods, two-scale formulation method and limit solver to integrate the characteristics. We then highlight the strengths and drawbacks of each method. Finally, numerical experiments are presented, and comparisons on the accuracy, efficiency and long-time behaviour of the methods are made, pointing to the method with the best performance for this problem.

A multiscale model for reinforced concrete with macroscopic variation of reinforcement slip

A single-scale model for reinforced concrete, comprising the plain concrete continuum, reinforcement bars and the bond between them, is used as a basis for deriving a two-scale model. The large-scale problem, representing the “effective” reinforced concrete solid, is enriched by an effective reinforcement slip variable. The subscale problem on a Representative Volume Element (RVE) is defined by Dirichlet boundary conditions. The response of the RVEs of different sizes was investigated by means of pull-out tests. The resulting two-scale formulation was used in an FE^2 analysis of a deep beam. Load–deflection relations, crack widths, and strain fields were compared to those obtained from a single-scale analysis. Incorporating the independent macroscopic reinforcement slip variable resulted in a more pronounced localisation of the effective strain field. This produced a more accurate estimation of the crack widths than the two-scale formulation neglecting the effective reinforcement slip variable.

Multiscale Particle-in-Cell methods and comparisons for the long-time two-dimensional Vlasov–Poisson equation with strong magnetic field

We applied different kinds of multiscale methods to numerically study the long-time Vlasov–Poisson equation with a strong magnetic field. The multiscale methods include an asymptotic preserving Runge–Kutta scheme, an exponential time differencing scheme, stroboscopic averaging method and a uniformly accurate two-scale formulation. We briefly review these methods and then adapt them to solve the Vlasov–Poisson equation under a Particle-in-Cell discretization. Extensive numerical experiments are conducted to investigate and compare the accuracy, efficiency, and long-time behavior of all the methods. The methods with the best performance under different parameter regimes are identified.

A combination of multiscale time integrator and two-scale formulation for the nonlinear Schrödinger equation with wave operator

In this paper, we consider the nonlinear Schrödinger equation with wave operator (NLSW), which contains a dimensionless parameter 0 < ε ≤ 1 . As 0 < ε ≪ 1 , the solution of the NLSW propagates fast waves in time with wavelength O ( ε 2 ) and the problem becomes highly oscillatory in time. The oscillations come from two parts. One part is from the equation and another part is from the initial data. For the ill-prepared initial data case as described in Bao and Cai (2014) which brings inconsistency in the limit regime, standard numerical methods have strong convergence order reduction in time when ε becomes small. We review two existing methods to solve the NLSW: an exponential integrator and a two-scale method. We comment on their order reduction issues. Then we derive a multiscale decomposition two-scale method for solving the NLSW by first performing a multiscale decomposition on the NLSW which decomposes it into a well-behaved part and an energy-unbounded part, and then applying an exponential integrator for the well-behaved part and a two-scale approach for the energy-unbounded part. Numerical experiments are conducted to test the proposed method which shows uniform second order accuracy without significant order reduction for all 0 < ε ≤ 1 . Comparisons are made with the existing methods.

Uniformly accurate numerical schemes for the nonlinear Dirac equation in the nonrelativistic limit regime

We apply the two-scale formulation approach to propose uniformly accurate (UA) schemes for solving the nonlinear Dirac equation in the nonrelativistic limit regime. The nonlinear Dirac equation involves two small scales \epsilon and \epsilon^2 with epsilon \to 0 in the nonrelativistic limit regime. The small parameter causes high oscillations in time which brings severe numerical burden for classical numerical methods. We transform our original problem as a two-scale formulation and present a general strategy to tackle a class of highly oscillatory problems involving the two small scales \epsilon and\epsilon^2. Suitable initial data for the two-scale formulation is derived to bound the time derivatives of the augmented solution. Numerical schemes with uniform (with respect to \epsilon\in (0;1]) spectral accuracy in space and uniform first order or second order accuracy in time are proposed. Numerical experiments are done to confirm the UA property.

Cohesive surface model for fracture based on a two-scale formulation: computational implementation aspects

The paper describes the computational aspects and numerical implementation of a two-scale cohesive surface methodology developed for analyzing fracture in heterogeneous materials with complex micro-structures. This approach can be categorized as a semi-concurrent model using the representative volume element concept. A variational multi-scale formulation of the methodology has been previously presented by the authors. Subsequently, the formulation has been generalized and improved in two aspects: (i) cohesive surfaces have been introduced at both scales of analysis, they are modeled with a strong discontinuity kinematics (new equations describing the insertion of the macro-scale strains, into the micro-scale and the posterior homogenization procedure have been considered); (ii) the computational procedure and numerical implementation have been adapted for this formulation. The first point has been presented elsewhere, and it is summarized here. Instead, the main objective of this paper is to address a rather detailed presentation of the second point. Finite element techniques for modeling cohesive surfaces at both scales of analysis (FE $$^2$$ approach) are described: (i) finite elements with embedded strong discontinuities are used for the macro-scale simulation, and (ii) continuum-type finite elements with high aspect ratios, mimicking cohesive surfaces, are adopted for simulating the failure mechanisms at the micro-scale. The methodology is validated through numerical simulation of a quasi-brittle concrete fracture problem. The proposed multi-scale model is capable of unveiling the mechanisms that lead from the material degradation phenomenon at the meso-structural level to the activation and propagation of cohesive surfaces at the structural scale.

Uniformly Accurate Forward Semi-Lagrangian Methods for Highly Oscillatory Vlasov--Poisson Equations

This work is devoted to the numerical simulation of a Vlasov--Poisson equation modeling charged particles in a beam submitted to a highly oscillatory external electric field. A numerical scheme is constructed for this model. This scheme is uniformly accurate with respect to the size of the fast time oscillations of the solution, which means that no time step refinement is required to simulate the problem. The scheme combines the forward semi-Lagrangian method with a class of uniformly accurate (UA) time integrators to solve the characteristics. These UA time integrators are derived by means of a two-scale formulation of the characteristics, with the introduction of an additional periodic variable. Numerical experiments are performed to show the efficiency of the proposed methods compared to conventional approaches.

A New Heterogeneous Multiscale Method for Time-Harmonic Maxwell's Equations

In this paper, we suggest a new heterogeneous multiscale method (HMM) for the time-harmonic Maxwell equations in locally periodic media. The method is constructed by using a divergence-regularization in one of the cell problems. This allows us to introduce fine-scale correctors that are not subject to a cumbersome divergence-free constraint and which can hence easily be implemented. To analyze the method, we first revisit classical homogenization theory for time-harmonic Maxwell equations and derive a new homogenization result that makes use of the divergence-regularization in the two-scale homogenized equation. We then show that the HMM is equivalent to a discretization of this equation. In particular, writing both problems in a fully coupled two-scale formulation is the crucial starting point for a corresponding numerical analysis of the method. With this approach we are able to prove rigorous a priori error estimates in the $\mathbf{H}({curl})$- and the $H^{-1}$-norm, and we derive reliable and efficient localized residual-based a posteriori error estimates. Numerical experiments are presented to verify the a priori convergence results.

Multiscale formulation for material failure accounting for cohesive cracks at the macro and micro scales

This contribution presents a two-scale formulation devised to simulate failure in materials with heterogeneous micro-structure. The mechanical model accounts for the nucleation of cohesive cracks in the micro-scale domain. The evolution and propagation of cohesive micro-cracks can induce material instability at the macro-scale level. Then, a cohesive crack is nucleated in the macro-scale model which considers, in a homogenized sense, the constitutive response of the intricate failure mode taking place at the smaller length scale.The two-scale semi-concurrent model is based on the concept of Representative Volume Element (RVE). It is developed following an axiomatic variational structure. Two hypotheses are introduced in order to build the foundations of the entire theory, namely: (i) a mechanism for transferring kinematical information from macro-to-micro scale along with the concept of “Kinematical Admissibility”, and (ii) a Multiscale Variational Principle of internal virtual power equivalence between the involved scales of analysis. The homogenization formulae for the generalized stresses, as well as the equilibrium equations at the micro-scale, are consequences of the variational statement of the problem.The present multiscale technique is a generalization of a previous model proposed by the authors (Sánchez et al., 2013; Toro et al., 2014) and could be viewed as an application of a recent contribution (Blanco et al., 2014). The main novelty in this article lies on the fact that failure modes in the micro-structure involve a set of multiple cohesive cracks, connected or disconnected, with arbitrary orientation, conforming a complex tortuous failure path. Following the present multiscale modeling approach, the tortuosity effect is introduced as a kinematical concept and has a direct consequence in the homogenized mechanical response.Numerical examples are presented showing the potentialities of the model to simulate complex and realistic fracture problems in heterogeneous materials. In order to validate the multiscale technique in a rigorous manner, comparisons with the so-called DNS (Direct Numerical Solution) approach are also presented.

Variational approach to homogenization of doubly-nonlinear flow in a periodic structure

This work deals with the homogenization of an initial- and boundary-value problem for the doubly-nonlinear system (0.1)Dtw−∇⋅z→=g(x,t,x/ε)(0.2)w∈α(u,x/ε)(0.3)z→∈γ→(∇u,x/ε). Here ε is a positive parameter; α and γ→ are maximal monotone with respect to the first variable and periodic with respect to the second one. The inclusions (0.2) and (0.3) are here formulated as null-minimization principles, via the theory of Fitzpatrick [MR 1009594]. As ε→0, a two-scale formulation is derived via Nguetseng’s notion of two-scale convergence, and a (single-scale) homogenized problem is then retrieved.

VARIATIONALLY CONSISTENT HOMOGENIZATION OF STOKES FLOWIN POROUS MEDIA

Seepage through a strongly heterogeneous material, consisting of open saturated pores, is modeled as a Stokes flow contained in a rigid matrix. Through homogenization of the problem, a two-scale formulation is derived. The subscale problem is that of a Stokes flow whereas the macroscale problem pertains to a Darcy flow. The prolongation of the macroscale Darcy flow fulfills the variational consistent macrohomogeniety condition and is valid for both linear and nonlinear subscale flows. The subscale problem is solved using the finite element method. Numerical results concerning both linear and nonlinear flow are presented.

Two-scale div-curl lemma

The div-curl lemma, one of the basic results of the theory of compen- sated compactness of Murat and Tartar, does not take over to the case in which the two factors two-scale converge in the sense of Nguetseng. A suitable modifica- tion of the differential operators however allows for this extension. The argument follows the lines of a well-known paper of F. Murat of 1978, and uses a two-scale extension of the Fourier transform. This result is also extended to time-dependent functions, and is applied to a two-scale formulation of the Maxwell system of electromagnetism, that accounts for the energy embedded in both coarse- and fine-scale oscillations.

The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains

In this contribution we analyze a generalization of the heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains. The method was originally introduced by E and Engquist (Commun Math Sci 1(1):87–132, 2003) for homogenization problems in fixed domains. It is based on a standard finite element approach on the macroscale, where the stiffness matrix is computed by solving local cell problems on the microscale. A-posteriori error estimates are derived in L 2(Ω) by reformulating the problem into a discrete two-scale formulation (see also, Ohlberger in Multiscale Model Simul 4(1):88–114, 2005) and using duality methods afterwards. Numerical experiments are given in order to numerically evaluate the efficiency of the error estimate.