Assumption Of Scale Separation Research Articles

A mesh‐in‐element method for the theory of porous media

AbstractWhile direct homogenisation approaches such as the FE method are subject to the assumption of scale separation, the mesh‐in‐element (MIEL) approach is based on an approach with strong scale coupling, which is based on a discretization with finite elements. In this contribution we propose a two‐scale MIEL scheme in the framework of the theory of porous media (TPM). This work is a further development of the MIEL method which is based on the works of the authors A. Ibrahimbegovic, R.L. Taylor, D. Markovic, H.G. Matthies, R. Niekamp (in alphabetical order); where we find the physical and mathematical as well as the software coupling implementation aspects of the multi‐scale modeling of heterogeneous structures with inelastic constitutive behaviour, see for example, [Eng Comput, 2005;22(5‐6):664‐683.] and [Eng Comput, 2009;26(1/2):6‐28.]. Within the scope of this contribution, the necessary theoretical foundations of TPM are provided and the special features of the algorithmic implementation in the context of the MIEL method are worked out. Their fusion is investigated in representative numerical examples to evaluate the characteristics of this approach and to determine its range of application.

Elastic properties prediction of two- and three-dimensional multi-material lattices

Advances in multi-material additive manufacturing have opened unprecedented new opportunities for the design and manufacture of lightweight multifunctional structures. The ability to create complex topologies, at a relatively fine resolution, in addition to controlling the material composition on a voxel basis have significantly expanded the design space. To explore this large design space efficiently, accurate and cost-effective modeling tools are essential. In this paper, mechanics-based models for predicting the elastic properties of multi-material 2D and 3D lattice structures are developed or extended. The outcomes are compared with the predictions obtained from finite element models and experimental data. The results reveal that the adapted analytical models demonstrate good accuracy in predicting the elastic modulus of multi-material lattices for relative densities up to approximately 25 % while have considerably less computational cost compared to finite element using solid elements (providing the most accurate results in comparison with experiment). Careful consideration of the accuracy of the predictions is necessary for the use of these models for lattices with high relative density values. Besides, several homogenization-based models were studied to investigate their applicability to multi-material lattice structures when the assumption of scale-separation is considered valid. The capability of these models in predicting the whole elasticity tensor and the potential of multi-material lattices in manipulating the anisotropy are demonstrated. Finally, the introduced prediction frameworks are compared in order to provide an overview of their respective advantages and disadvantages in the case of multi-material lattice structures.

Extended multiscale FEM-based concurrent optimization of three-dimensional graded lattice structures with multiple microstructure configurations

In this work, a novel concurrent design method for three-dimensional graded lattice structures considering multiple microstructure configurations is proposed. Firstly, the Multiple Variable Cutting (M-VCUT) level set method is employed to represent the geometry of lattice microstructures and define the optimization parameters. Then, the extended multiscale finite element method (EMsFEM) is implemented for calculating the equivalent stiffness matrices of microstructures in offline form. Compared with the traditional homogenization-based multiscale approaches, the proposed method eliminates the scale separation assumptions and takes into account the coupling effects in different directions of materials. In addition, a data-driven method and a parallel computing approach are integrated together so that computational efficiency can be greatly improved. After the iteration process, the optimized full-scale graded structures are reconstructed using interpolation technology to generate well-connected microstructures in neighboring cells. Finally, several 3D numerical examples are utilized to validate the effectiveness of the proposed method, where multiple lattice microstructure configurations are considered, including truss-lattices and plate-lattices.

Investigating the Effect of the Separation of Scales in Reduced Order Battery Modelling: Implications on the Validity of the Newman Model

Modelling lithium-ion battery behavior is essential for performance prediction and design improvement. However, this task is challenging due to processes spanning many length scales, leading to computationally expensive models. Reduced order models have been developed to address this, assuming a “separation of scales” between micro- and macroscales. This study compares two approaches: direct microstructure-resolved 3D domain electrochemical modelling and a simplified 1D homogenized model, similar to the Doyle-Fuller-Newman model. The research investigates the validity of the scale separation assumption in continuum electrode-level models by varying scale separation factors, boundary conditions, and geometries. The findings reveal increases in deviation between the 3D models and 1D models for more tortuous, less porous microstructures, especially under higher discharge rates. However, under realistic conditions, with an electrode featuring eight particles across its thickness and typical transport properties, the 3D model predicts only a slight (2%) increase in current compared to the 1D model at a high rate of 7C (approximately j ≈ 350 Am−2). These results suggest that the separation of scales assumption in the DFN model is generally suitable for a wide range of operating conditions. However, 1D models may overlook local variations in electrolyte concentration and potential, crucial for understanding degradation mechanisms.

Multiscale Wave-Turbulence Dynamics in the Atmosphere and Ocean

The atmosphere and oceans present an ongoing first-rate challenge to science and mathematics because they operate on an extremely broad ranges of scales, from molecular to planetary in length and from below seconds to millennia in time. This is the reason why climate simulations still suffer from leading-order uncertainties. Conceptual simplifications, such as scale-separation assumptions and the neglect of many physical processes, have enabled past progress in understanding the interactions of the basic dynamic constituents, i.e. large-scale mean flows, medium-scale waves and vortices, and small-scale turbulence. But present-day research is stretching the validity of this framework. For example, it is recognized that intermediate-scale waves and vortices are key elements linking all relevant players, and are often characterized by nonlinear interactions on comparable scales and also by additional physical nonlinearities due to effects such as air moisture. Motivated by recent advances in mathematical wave–vortex and wave–wave interaction theory, turbulence theory, and the study of internal wave dynamics as well as their numerical parametrization, the workshop gathered leading experts in these fields to foster a synthesis of new approaches and thereby a new level of understanding and numerical treatment of climate dynamics.

Explicit Stabilized Multirate Method for Stiff Stochastic Differential Equations

Stabilized explicit methods are particularly efficient for large systems of stiff stochastic differential equations (SDEs) due to their extended stability domain. However, they loose their efficiency when a severe stiffness is induced by very few fast degrees of freedom, as the stiff and nonstiff terms are evaluated concurrently. Therefore, inspired by [A. Abdulle, M. J. Grote, and G. Rosilho de Souza, Preprint (2020), arXiv:2006.00744] we introduce a stochastic modified equation whose stiffness depends solely on the slow terms. By integrating this modified equation with a stabilized explicit scheme we devise a multirate method which overcomes the bottleneck caused by a few severely stiff terms and recovers the efficiency of stabilized schemes for large systems of nonlinear SDEs. The scheme is not based on any scale separation assumption of the SDE and therefore it is employable for problems stemming from the spatial discretization of stochastic parabolic partial differential equations on locally refined grids. The multirate scheme has strong order 1/2, weak order 1 and its stability is proved on a model problem. Numerical experiments confirm the efficiency and accuracy of the scheme.

Finite-time Response of Dynamo Mean-field Effects in Magnetorotational Turbulence

Abstract Accretion disk turbulence along with its effect on large-scale magnetic fields plays an important role in understanding disk evolution in general, and the launching of astrophysical jets in particular. Motivated by enabling a comprehensive subgrid description for global long-term simulations of accretions disks, we aim to further characterize the transport coefficients emerging in local simulations of magnetorotational disk turbulence. For the current investigation, we leverage a time-dependent version of the test-field method, which is sensitive to the turbulent electromotive force (EMF) generated as a response to a set of pulsating background fields. We obtain Fourier spectra of the transport coefficients as a function of oscillation frequency. These are well approximated by a simple response function, describing a finite-time buildup of the EMF as a result of a time-variable mean magnetic field. For intermediate timescales (i.e., slightly above the orbital frequency), we observe a significant phase lag of the EMF compared to the causing field. Augmented with our previous result on a nonlocal closure relation in space, and incorporated into a suitable mean-field description that we briefly sketch out here, the new framework will allow us to drop the restrictive assumption of scale separation.

Tunable phononic bandgap materials designed via topology optimization

Topology optimization is used to design phononic bandgap materials that are tunable by mechanical deformation. A periodic media is considered, which due to the assumption of length scale separation, allows the dispersion relations to be obtained by analyzing a single unit cell subjected to Floquet–Bloch boundary conditions. A finite macroscopic deformation is applied to the unit cell to affect its geometry and hence dispersion. We tune the dispersion–deformation relation to our liking by solving a topology optimization problem using nonlinear programming. The adjoint method is employed to compute the sensitivities, and the non-differentiability of degenerate eigenvalues is avoided using symmetric polynomials. Several tunable phononic crystal designs are presented. Also, a verification analysis is performed, wherein the optimized design is interpreted and analyzed using a conforming finite element mesh.

Instability of a dusty Kolmogorov flow

Suspended particles can significantly alter the fluid properties and, in particular, can modify the transition from laminar to turbulent flow. We investigate the effect of heavy particle suspensions on the linear stability of the Kolmogorov flow by means of a multiple-scale expansion of the Eulerian model originally proposed by Saffman (J. Fluid Mech., vol. 13, issue 1, 1962, pp. 120–128). We find that, while at small Stokes numbers particles always destabilize the flow (as already predicted by Saffman in the limit of very thin particles), at sufficiently large Stokes numbers the effect is non-monotonic in the particle mass fraction and particles can both stabilize and destabilize the flow. Numerical analysis is used to validate the analytical predictions. We find that in a region of the parameter space the multiple-scale expansion overestimates the stability of the flow and that this is a consequence of the breakdown of the scale separation assumptions.

Inertia-gravity-wave scattering by three-dimensional geostrophic turbulence

In rotating stratified flows including in the atmosphere and ocean, inertia-gravity waves (IGWs) often coexist with geostrophically balanced turbulent flows. Advection and refraction by such flows lead to wave scattering, redistributing IGW energy in the position–wavenumber phase space. We give a detailed description of this process by deriving a kinetic equation governing the evolution of the IGW phase-space energy density. The derivation relies on the smallness of the Rossby number characterising the geostrophic flow, which is treated as a random field with known statistics, makes no assumption of spatial scale separation, and neglects wave–wave interactions. It extends previous work restricted to near-inertial waves, barotropic flows or waves much shorter than the flow scales. The kinetic equation describes energy transfers that are restricted to IGWs with the same frequency, as a result of the time scale separation between waves and flow. We formulate the kinetic equation on the constant-frequency surface – a double cone in wavenumber space – using polar spherical coordinates, and we examine the form of the two scattering cross-sections involved, which quantify energy transfers between IGWs with, respectively, the same and opposite directions of vertical propagation. The kinetic equation captures both the horizontal isotropisation and the cascade of energy across scales that result from scattering. We focus our attention on the latter to assess the predictions of the kinetic equation against direct simulations of the three-dimensional Boussinesq equations, finding good agreement.

Dimensional reduction analyzing the thermoelastic behavior of wind turbine blades based on the variational asymptotic multiscale method

Accurate and effective prediction of the thermoelastic behavior of composite wind blades is critical to the optimal design and performance evaluation. In this paper, a high-fidelity dimensional reduction model of composite wind blade is established based on the variational asymptotic multiscale method. First, the thermoelastic variational expression of 3-D composite wind blade was derived. Then, the thermoelastic energy functional was extended asymptotically to a series of 2-D energies by using the inherent small parameters of the structure, so that the original 3-D thermoelastic problem of composite wind blades was strictly divided into 2-D airfoil section analysis and 1-D beam analysis. The local displacement and stress fields were recovered based on the global response without a priori assumptions and unnecessary scale separation assumptions. Finally, the numerical examples of composite wind blades with NACA2412 airfoil and MH 104 airfoil are provided to verify the accuracy and effectiveness of the model by comparison with the three-dimensional finite element model (3D-FEM). The results show that the predicted thermoelastic behavior and local 3-D stress field of the composite wind blades are in good agreement with 3D-FEM, but the calculation efficiency is greatly improved.

Structural uncertainty through the lens of model building

An important epistemic issue in climate modelling concerns structural uncertainty: uncertainty about whether the mathematical structure of a model accurately represents its target. How does structural uncertainty affect our knowledge and predictions about the climate? How can we identify sources of structural uncertainty? Can we manage the effect of structural uncertainty on our knowledge claims? These are some of the questions that an epistemology of structural uncertainty faces, and these questions are also important for climate scientists and policymakers. I develop three desiderata for an epistemological account of structural uncertainty. In my view, an account of structural uncertainty should (1) identify sources of structural uncertainty, (2) explain how these sources limit the applicability of a model, and (3) show how the severity of structural uncertainty depends on the questions that can be asked of a model. I argue that analyzing structural uncertainty by paying attention to the details of model building can satisfy these desiderata. I focus on parametrizations, which are representations of important processes occurring at scales that are not resolved by climate models. Parametrizations are often thought to be ad-hoc, but I show that some important parametrizations are theoretically justified by explicit or implicit scale separation assumptions. These assumptions can also be supported empirically. Analyzing these theoretical and empirical justificatory roles of the scale separation assumptions can provide insights into how parametrizations contribute to structural uncertainty. I conclude by sketching how my approach can satisfy the desiderata I set out at the beginning, highlighting its importance for policy-relevant scientific statements about the climate.

On the model formulations for the interaction of nonlinear waves and current

We study the model formulations of wave–current interactions in the framework of Euler equations. This work is intrigued by a recent paper from Wang et al. (2018) (hereafter WMY), which proposes such a model for the evolution of nonlinear broadband surface waves under the influence of a prescribed steady and irrotational current without vertical shear. We show that WMY’s model can be derived from a more general model accounting for an arbitrary steady and irrotational current. Under further assumption of scale separation between waves and current (i.e. horizontally slowly-varying current), WMY’s model is equivalent to an earlier model, in contrast to WMY’s claim that their model includes additional higher-order effects in wave steepness. We demonstrate the usefulness of such models in a numerical study on wave blocking by opposing current, where the nonlinear effect on the caustic location and wave amplitude amplification is elucidated. We further show that the model formulation in the framework of Euler equations form a Hamiltonian system conserving the total energy of waves and current, justifying the theoretical significance of the model equations. Finally, we generalize the formulation to nonlinear wave evolution in the presence of a rotational current with constant vorticity, which overcomes a limitation of such models that has been overlooked in previous work.

LES Eulerian diffuse-interface modeling of fuel dense sprays near- and far-field

Engine fuel spray modeling still remains a challenge, especially in the dense near-nozzle region. This region is difficult to experimentally access and also to model due to the complex and rapid liquid and gas interaction. Modeling approaches based on Lagrangian particle tracking have failed in this area, while Eulerian modeling has proven to be particularly useful. Interface resolved methods are still limited to primary atomization academic configurations due to excessive computational requirements. To overcome those limitations, the single-fluid diffuse interface model known as Σ-Y, arises as a single-framework for spray simulations. Under the assumption of scale separation at high Reynolds and Weber numbers, liquid dispersion is modeled as turbulent mixing of a variable density flow. The concept of surface area density is used for representing liquid structures, regardless of the complexity of the interface.In this work, a LES based implementation of the Σ-Y model in the OpenFOAM CFD library is applied to simulate the ECN Spray A configuration. Model assessment is performed for both near- and far-field spray development regions using different experimental diagnostics available from ECN database. The CFD model is able to capture near-nozzle fuel mass distribution and, after Σ equation constant calibration, interfacial surface area. Accurate predictions of spray far-field evolution in terms of liquid and vapor tip penetration and local velocity can be simultaneously achieved. Model accuracy is lower when compared to mixture fraction axial evolution, despite radial distribution profiles are well captured.

The Coarse Mesh Condensation Multiscale Method for parallel computation of heterogeneous linear structures without scale separation

A Coarse Mesh Condensation Multiscale Method (CMCM) is proposed to solve large heterogeneous linear structures without scale separation assumption. The technique aims to approximate the full field solution in heterogeneous structures by performing parallel calculations on subdomains. In the linear case, treated in this paper, direct linear relationships can be established between a reduced number of parameters describing Dirichlet boundary conditions on the subdomains boundaries and the degrees of freedom of a coarse mesh. The problem associated with the coarse mesh can be solved in one iteration and allows reconstructing the fine mesh solution in all subdomains. The accuracy of the method is analyzed through benchmarks involving subdomains crossed by the interfaces. Applications to large industrial finite element applications are presented, including one involving around 1.3 billion degrees of freedom.

Eulerian CFD modeling of nozzle geometry effects on ECN Sprays A and D: assessment and analysis

Diesel spray modeling is a multi-scale problem with complex interactions between different flow regions, that is, internal nozzle flow, near-nozzle region and developed spray, including evaporation and combustion. There are several modeling approaches that have proven particularly useful for some spray regions although they have struggled at other areas, while Eulerian modeling has shown promise in dealing with all characteristics at a reasonable computational effort for engineering calculations. In this work, the [Formula: see text]–Y single-fluid diffuse-interface model, based on scale separation assumptions at high Reynolds and Weber numbers, is used to simulate the engine combustion network Sprays A and D within a Reynolds-averaged Navier–Stokes turbulence modeling approach. The study is divided into two parts. First of all, the larger diameter Spray D is modeled from the nozzle flow till evaporative spray conditions, obtaining successful prediction of numerous spray metrics, paying special attention to the near-nozzle region where spray dispersion and interfacial surface area can be validated against measurements conducted at the Advanced Photon Source at Argonne National Laboratory, including both the ultra-small-angle X-ray scattering and the X-ray radiography. Afterwards, an analysis of the modeling predictions is made in comparison with previous results obtained for Spray A, considering the nozzle geometry effects in the modeling behavior.

A novel subdomain level set method for structural topology optimization and its application in graded cellular structure design

A novel subdomain structural topology optimization method is proposed for the minimum compliance problem based on the level sets with the parameterization of radial basis function (RBF). In this method, the level set function evolves on each subdomain separately and independently according to the requirements of objective functions and additional constraints. This makes the parameterization in the proposed subdomain method much faster and more cost-effective than that in the classical global method, as well as the evolution of the level set function since it can be achieved on each subdomain in parallel. In addition, the microstructures on arbitrary two adjacent subdomains can be connected perfectly, without any mismatch around the interfaces of the microstructures. Several typical examples are conducted to verify the correctness and effectiveness of the developed subdomain method. The effects of some factors on the optimized results are also investigated in detail, such as the RBF types, the connectivity types of microstructures, and the size of subdomain division. Without scale separation assumption, several layered graded cellular structures are successfully designed by employing the proposed method under the condition of corresponding repetition constraints. To improve the computational efficiency, a multi-node extended multiscale finite element method (EMsFEM) is used to solve the structural static equilibrium equation for the three-dimensional layered structure optimization problems. Furthermore, a MATLAB code is also provided in the Appendix for readers to reproduce the results of the two-dimensional problems in this work.

Topology optimization of periodic lattice structures taking into account strain gradient

We present a topology optimization for lattice structures in the case of non-separated scales, i.e. when the characteristic dimensions of the periodic unit cells in the lattice are not much smaller than the dimensions of the whole structure. The present method uses a coarse mesh corresponding to a homogenized medium taking into strain gradient through a non-local numerical homogenization method. Then, the topological optimization procedure only uses the values at the nodes of the coarse mesh, reducing drastically the computational times. We show that taking into account the strain gradient within the topological optimization procedure brings significant increase in the resulting stiffness of the optimized lattice structure when scales are not separated, as compared to using a homogenized model based on the scale separation assumption.

Considering Realistic Microstructure Heterogeneity: Variational Multiscale Modeling of Li-Ion Batteries

The simulation of battery systems needs to consider structural heterogeneity in battery electrodes as the electrodes are highly irregular in shape and size while containing pores or even cracks at different length scales, which may result in non-uniform transport and kinetics throughout the electrodes. A more realistic battery model necessitates considering realistic microstructure heterogeneity as well as incorporating ongoing microscale phenomena (e.g., the predictability of diffusion, interfacial side reactions, and stress evolution) into numerical problems governing macroscale behaviors. However, developing such model with detailed 3D microstructure can be very computationally expensive for direct simulations, as shown in Figure 1. Here, we propose to reduce the computational cost by developing a more realistic battery model via the variational multiscale method (VMM). The VMM couples micro- and macro- scales by decomposing the problem into micro- and macro- scale problem. In detail, we apply the VMM to decompose the response (e.g., ) into coarse-scale (e.g., 1D macro-scale) and fine-scale (e.g., 3D micro-scale), as shown in Figure 1. The associated governing PDEs will accordingly be decomposed into coarse-scale components and fine-scale components. Without introducing the assumption of scale separation, one of the key features of VMM lies in that the relevant governing PDEs will be decomposed into coarse-scale and fine-scale components. Therefore, two sets of solution will be obtained on different domains, i.e., is the macro solution at, and is the micro solution at. The developed model aims at obtaining accurate micro-solution at micro-scale in a subdomain where it is known a priori that important physicochemical phenomena are likely to occur. The effect of micro-scale is then considered in the solution of the problem. Due to consideration of integrals or the weak formulation as distribution (i.e., average of functions over certain areas of the computational domain), we can perform integration operation on the function that represents the solution to the problem, regardless of existence of discontinuity points in the domain. Thus, one of the main advantageous of developed model is that it resolves the heterogeneities of micro-scale without considering the minor details (such as discontinuity points) but rather our focus in invested in the distribution of the function in the a priori areas of the subdomain, which reduces the computational costs of the numerical analysis. This is of significant importance, especially for cases that the effect of ongoing micro-scale phenomena is more significant than the minor details of the micro-scale. Figure 1

Hydrodynamic optical soliton tunneling.

A notion of hydrodynamic optical soliton tunneling is introduced in which a dark soliton is incident upon an evolving, broad potential barrier that arises from an appropriate variation of the input signal. The barriers considered include smooth rarefaction waves and highly oscillatory dispersive shock waves. Both the soliton and the barrier satisfy the same one-dimensional defocusing nonlinear Schrödinger (NLS) equation, which admits a convenient dispersive hydrodynamic interpretation. Under the scale separation assumption of nonlinear wave (Whitham) modulation theory, the highly nontrivial nonlinear interaction between the soliton and the evolving hydrodynamic barrier is described in terms of self-similar, simple wave solutions to an asymptotic reduction of the Whitham-NLS partial differential equations. One of the Riemann invariants of the reduced modulation system determines the characteristics of a soliton interacting with a mean flow that results in soliton tunneling or trapping. Another Riemann invariant yields the tunneled soliton's phase shift due to hydrodynamic interaction. Soliton interaction with hydrodynamic barriers gives rise to effects that include reversal of the soliton propagation direction and spontaneous soliton cavitation, which further suggest possible methods of dark soliton control in optical fibers.