Fine-scale Problem Research Articles

Spectral variational multiscale enrichment method—A new computational approach to transient dynamics of phononic crystals and acoustic metamaterials

Phononic crystals and acoustic metamaterials hold tremendous potential in controlling mechanical and acoustic waves, thereby enabling remarkable and novel engineering applications such as seismic wave mitigation, impact wave mitigation, elastic cloaking, and acoustic superlens. Given the potential complexity associated with the microstructure of these materials, direct numerical simulation of irregularly shaped and/or large engineering components could be computationally very costly and in many cases prohibitive. Multiscale computational modeling, coupled with model order reduction methodologies present a feasible strategy to simulate the dynamic response of phononic crystals and acoustic metamaterials. In this study, we present the spectral variational multiscale enrichment approach that allows the analysis of a broad range of material microstructures for phononic crystals and acoustic metamaterials. The spectral coarse-scale representation is proposed to capture the salient transient wave phenomena, such as wave dispersion and band gaps that occur in the short wavelength regime. A material phase based model order reduction strategy is devised at the fine-scale. By using the reduced basis for the fine-scale problem, significant numerical efficiency is achieved. Transient elastic wave propagation in two-dimensional phononic crystals and acoustic metamaterials are investigated and the proposed multiscale approach is verified against direct numerical simulations.

Upscaling the poroelastic behavior of the lung parenchyma: A finite-deformation micromechanical model

The lungs are among the most deformable body organs, a mechanical feature that is key to the vital process of breathing. Current micromechanical constitutive models of the lung parenchyma construct the tissue response function either as strain-driven or pressure-driven. However, the lung parenchyma resembles an open-cell foam material consisting of a solid phase and a fluid phase that closely interact with each other. In this work, we introduce a novel finite-deformation micromechanical poroelastic model of the lung parenchyma. Using a two-scale homogenization framework for poroelasticity, we construct the effective coarse-scale response of the tissue by solving a poroelastic fine-scale problem. To this end, we develop a non-linear structural model based on a tetrakaidecahedron (TKD) unit cell that only depends on four microstructural parameters. We validate the TKD model showing that it predicts the effective response of representative volume elements (RVE) constructed from micro-computed-tomography images of the lung under several combinations of deformation and alveolar pressure. Further, we show that the estimation of the effective stress using the TKD model delivers a speed-up in computation time of more than 284,000 × when compared to RVE simulations, at the same time that it delivers higher numerical stability. In addition, we demonstrate through a sensitivity analysis that the model response predominantly depends on the alveolar-wall elasticity and initial tissue porosity, which are parameter values that are inherently connected to measurable microstructural features of the lung tissue. The present TKD model opens the door to large-scale poroelastic simulations of the lung by providing a predictive yet efficient constitutive model of the lung parenchyma. Codes are available for download at https://github.com/dehurtado/PoroelasticTKDModel.

A novel variational multiscale interpolating element-free Galerkin method for generalized Oseen problems

In this paper, a novel variational multiscale interpolating element-free Galerkin (VMIEFG) method is developed for the numerical solution of generalized Oseen problems. The moving Kriging interpolation is used to generate shape functions with Kronecker delta function property, then essential boundary conditions can be enforced easily and directly. The present method is based on a decomposition of the velocity and its weighting function into coarse scales and fine scales. The fine scale problem is solved analytically, and the stabilization parameter appears naturally. Then, the coarse scale problem is solved numerically. The VMIEFG method allows equal-order basis for velocity and pressure, which is fairly easy to implement from an application point of view. Numerical examples involving generalized Stokes problem, Oseen problem and generalized Oseen problem are provided to demonstrate the efficiency of the method.

Multiscale Finite Element Modeling of Nonlinear Magnetoquasistatic Problems using Magnetic Induction Conforming Formulations

In this paper we develop magnetic induction conforming multiscale formulations for magnetoquasistatic problems involving periodic materials. The formulations are derived using the periodic homogenization theory and applied within a heterogeneous multiscale approach. Therefore the fine-scale problem is replaced by a macroscale problem defined on a coarse mesh that covers the entire domain and many mesoscale problems defined on finely-meshed small areas around some points of interest of the macroscale mesh (e.g., numerical quadrature points). The exchange of information between these macro and meso problems is thoroughly explained in this paper. For the sake of validation, we consider a two-dimensional geometry of an idealized periodic soft magnetic composite. (An erratum is attached.)

Multiscale model reduction for shale gas transport in poroelastic fractured media

Inherently coupled flow and geomechanics processes in fractured shale media have implications for shale gas production. The system involves highly complex geo-textures comprised of a heterogeneous anisotropic fracture network spatially embedded in an ultra-tight matrix. In addition, nonlinearities due to viscous flow, diffusion, and desorption in the matrix and high velocity gas flow in the fractures complicates the transport. In this paper, we develop a multiscale model reduction approach to couple gas flow and geomechanics in fractured shale media. A Discrete Fracture Model (DFM) is used to treat the complex network of fractures on a fine grid. The coupled flow and geomechanics equations are solved using a fixed stress-splitting scheme by solving the pressure equation using a continuous Galerkin method and the displacement equation using an interior penalty discontinuous Galerkin method. We develop a coarse grid approximation and coupling using the Generalized Multiscale Finite Element Method (GMsFEM). GMsFEM constructs the multiscale basis functions in a systematic way to capture the fracture networks and their interactions with the shale matrix. Numerical results and an error analysis is provided showing that the proposed approach accurately captures the coupled process using a few multiscale basis functions, i.e. a small fraction of the degrees of freedom of the fine-scale problem.

Multiscale finite-element method for linear elastic geomechanics

The demand for accurate and efficient simulation of geomechanical effects is widely increasing in the geoscience community. High resolution characterizations of the mechanical properties of subsurface formations are essential for improving modeling predictions. Such detailed descriptions impose severe computational challenges and motivate the development of multiscale solution strategies. We propose a multiscale solution framework for the geomechanical equilibrium problem of heterogeneous porous media based on the finite-element method. After imposing a coarse-scale grid on the given fine-scale problem, the coarse-scale basis functions are obtained by solving local equilibrium problems within coarse elements. These basis functions form the restriction and prolongation operators used to obtain the coarse-scale system for the displacement-vector. Then, a two-stage preconditioner that couples the multiscale system with a smoother is derived for the iterative solution of the fine-scale linear system. Various numerical experiments are presented to demonstrate accuracy and robustness of the method.

Homogenization of the Navier-Stokes equations by means of the Multi-scale Virtual Power Principle

This work addresses the multi-scale modeling of fluid flow in highly complex media based on the concept of Representative Volume Element (RVE). The Method of Multi-scale Virtual Power developed by the authors is employed to construct a coarse-scale model from standard fluid flow model at a fine-scale. Kinematic conservation principles, duality arguments and the balance of virtual power between scales are employed to set the grounds of the scale transition of physical fields. This allows to derive in a variationally consistent manner (i) the fine-scale problem to be solved at the RVE, and (ii) the homogenization formulae for coarse-scale dual quantities, namely, the force-like and stress-like fields. Examples of application of flow in permeable media are presented to show the potential of the present approach.

Online adaptive local multiscale model reduction for heterogeneous problems in perforated domains

In this paper, we develop and analyze an adaptive multiscale approach for heterogeneous problems in perforated domains. We consider commonly used model problems including the Laplace equation, the elasticity equation, and the Stokes system in perforated regions. In many applications, these problems have a multiscale nature arising because of the perforations, their geometries, the sizes of the perforations, and configurations. Typical modeling approaches extract average properties in each coarse region, that encapsulate many perforations, and formulate a coarse-grid problem. In some applications, the coarse-grid problem can have a different form from the fine-scale problem, e.g. the coarse-grid system corresponding to a Stokes system in perforated domains leads to Darcy equations on a coarse grid. In this paper, we present a general offline/online procedure, which can adequately and adaptively represent the local degrees of freedom and derive appropriate coarse-grid equations. Our approaches start with the offline procedure, which constructs multiscale basis functions in each coarse region and formulates coarse-grid equations. We presented the offline simulations without the analysis and adaptive procedures, which are needed for accurate and efficient simulations. The main contributions of this paper are (1) the rigorous analysis of the offline approach, (2) the development of the online procedures and their analysis, and (3) the development of adaptive strategies. We present an online procedure, which allows adaptively incorporating global information and is important for a fast convergence when combined with the adaptivity. We present online adaptive enrichment algorithms for the three model problems mentioned above. Our methodology allows adding and guides constructing new online multiscale basis functions adaptively in appropriate regions. We present the convergence analysis of the online adaptive enrichment algorithm for the Stokes system. In particular, we show that the online procedure has a rapid convergence with a rate related to the number of offline basis functions, and one can obtain fast convergence by a sufficient number of offline basis functions, which are computed in the offline stage. The convergence theory can also be applied to the Laplace equation and the elasticity equation. To illustrate the performance of our method, we present numerical results with both small and large perforations. We see that only a few (1 or 2) online iterations can significantly improve the offline solution.

Defect corrected averaging for highly oscillatory problems

The accurate solution of partial differential equations with highly oscillatory source terms over long time scales constitutes a challenging problem. There exists a variety of methods dealing with problems where there are processes, equations or variables on fine and coarse scales. Multiscale methods have in common, that they neither fully resolve the fine scale, nor completely ignore it. On the one hand, these methods strive, without significantly sacrificing accuracy or essential properties of the system, to be much more efficient than methods that fully resolve the fine scale. On the other hand, these methods should be considerably more accurate than methods that completely ignore the fine scale. Our defect corrected averaging procedure is based on a modified coarse scale problem, that approximates the solution of the fine scale problem in stroboscopic points. Nevertheless, our approximation process is clearly different from the stroboscopic averaging method. We give an error estimate for the solution of the modified problem. The computational efficiency of the approximation is furthermore improved by the application of preconditioning techniques. Tests on numerical examples show the efficiency and reliability of our approach.

Two-Level Discretization Techniques for Ground State Computations of Bose-Einstein Condensates

This work presents a new methodology for computing ground states of Bose--Einstein condensates based on finite element discretizations on two different scales of numerical resolution. In a preprocessing step, a low-dimensional (coarse) generalized finite element space is constructed. It is based on a local orthogonal decomposition of the solution space and exhibits high approximation properties. The nonlinear eigenvalue problem that characterizes the ground state is solved by some suitable iterative solver exclusively in this low-dimensional space, without significant loss of accuracy when compared with the solution of the full fine scale problem. The preprocessing step is independent of the types and numbers of bosons. A postprocessing step further improves the accuracy of the method. We present rigorous a priori error estimates that predict convergence rates $H^3$ for the ground state eigenfunction and $H^4$ for the corresponding eigenvalue without pre-asymptotic effects; $H$ being the coarse scale discretization parameter. Numerical experiments indicate that these high rates may still be pessimistic.

A multiscale approach for modeling progressive damage of composite materials using fast Fourier transforms

Composite materials possess a highly complex material behavior, and thus advanced simulation techniques are necessary to compute their mechanical response. In this regard, especially modeling failure and progressive damage presents a challenging task. Conventional macro mechanical methods and even closed form estimates are in many cases not sufficient to predict the appropriate mechanical material response. Full-field simulations must be resorted to, but these are known to be very expensive from the computational point of view. In this contribution we propose a more efficient multiscale approach similar to FE2. Nonlinear material effects caused by progressive damage behavior are captured directly on the discretized material level using simple isotropic continuum damage laws. In contrast to conventional FE2 methods which use the Finite Element Method (FEM) to solve both scales numerically, the fine scale problem (material level) is rewritten in an integral form of Lippmann–Schwinger type and solved efficiently using the fast Fourier transformation (FFT). The calculation is carried out on a regular voxel grid that can be obtained from 3D images like tomographies. The fine scale problem is integrated in a standard Finite Element framework which is used to solve the macroscopic BVP (component level). In the work at hand, the scale coupling technique and the computation of the macroscopic tangent are described, and in some numerical examples the convergence behavior of the macroscopic Newton algorithm is investigated. Thereby the simulations were considered until localization and softening on the material scale occurred. It is shown that the proposed method presents an effective way to determine the exact physical macroscopic response considering arbitrary microstructures and loading conditions.

A two-scale finite element formulation of Stokes flow in porous media

Seepage through saturated porous material with an open pore system is modeled as a non-linear Stokes flow through a rigid matrix. Based on variationally consistent homogenization, the resulting macroscale problem becomes a Darcy-type flow. The prolongation of the Darcy flow fulfills a macrohomogeneity condition, which in a Galerkin context implies a symmetric macroscale problem. The homogenization is of 1st order and periodic boundary conditions are adopted on a Representative Volume Element. A nonlinear nested multiscale technique, in which the subscale problem is used as a constitutive model, is devised. In the presented numerical investigation, the effects of varying physical parameters as well as of the discretization are considered. In particular, it is shown that the two-scale results agree well with those of the fully resolved fine-scale problem.

A Comparison Study Between an Adaptive Quadtree Grid and Uniform Grid Upscaling for Reservoir Simulation

An adaptive quadtree grid generation algorithm is developed and applied for tracer and multiphase flow in channelized heterogeneous porous media. Adaptivity was guided using two different approaches. In the first approach, wavelet transformation was used to generate a refinement field based on permeability variations. The second approach uses flow information based on the solution of an initial-time fine-scale problem. The resulting grids were compared with uniform grid upscaling. For uniform upscaling, two commonly applied methods were used: renormalization upscaling and local-global upscaling. The velocities obtained by adaptive grid and uniformly upscaled grids, were downscaled. This procedure allows us to separate the upscaling errors, on adaptive and uniform grids, from the numerical dispersion errors resulting from solving the saturation equation on a coarse grid. The simulation results obtained by solving on flow-based adaptive quadtree grids for the case of a single phase flow show reasonable agreement with more computationally demanding fine-scale models and local-global upscaled models. For the multiphase case, the agreement is less evident, especially in piston-like displacement cases with sharp frontal movement. In such cases a non-iterative transmissibility upscaling procedure for adaptive grid is shown to significantly reduce the errors and make the adaptive grid comparable to iterative local-global upscaling. Furthermore, existence of barriers in a porous medium complicates both upscaling and grid adaptivity. This issue is addressed by adapting the grid using a combination of flow information and a permeability based heuristic criterion.

The variational multiscale element free Galerkin method for MHD flows at high Hartmann numbers

The aim of the paper is the development of an efficient numerical algorithm for the solution of magnetohydrodynamics (MHD) flow problems with either fully insulating walls or partially insulating and partially conducting walls. Toward this, we first extend the influence domain of the shape function for the element free Galerkin (EFG) method to have arbitrary shape. When the influence factor approaches 1, we find that the EFG shape function almost has the Delta property at the node (i.e. the value of the EFG shape function of the node is nearly equal to 1 at the position of this node) as well as the property of slices in the influence domain of the node (i.e. the EFG shape function in the influence domain of the node is nearly constructed by different functions defined in different slices). Therefore, for MHD flow problems at high Hartmann numbers we follow the idea of the variational multiscale finite element method (VMFEM) to combine the EFG method with the variational multiscale (VM) method, namely the variational multiscale element free Galerkin (VMEFG) method is proposed. Subsequently, in order to validate the proposed method, we compare the obtained approximate solutions with the exact solutions for some problems where such exact solutions are known. Finally, several benchmark problems of MHD flows are simulated and the numerical results indicate that the VMEFG method is stable at moderate and high values of Hartmann number. Another important feature of this method is that the stabilization parameter has appeared naturally via the solution of the fine scale problem. Meanwhile, because this proposed method is a type of meshless method, it can avoid the need for meshing, a very demanding task for complicated geometry problems.

A Multiscale Adaptive Local-Global Method for Modeling Flow in Stratigraphically Complex Reservoirs

Summary A robust and efficient simulation technique is developed on the basis of the extension of the mimetic finite-difference method (MFDM) to multiscale hierarchical-hexahedral (corner-point) grids by use of the multiscale mixed finite-element method (MsMFEM). The implementation of the mimetic subgrid-discretization method is compact and generic for a large class of grids and, thereby, is suitable for discretizations of reservoir models with complex geologic architecture. Flow equations are solved on a coarse grid where basis functions with subgrid resolution account accurately for subscale variations from an underlying fine-scale geomodel. The method relies on the construction of approximate velocity spaces that are adaptive to the local properties of the differential operator. A variant of the method for computing velocity basis functions is developed that uses an adaptive local-global (ALG) algorithm to compute multiscale velocity basis functions by capturing the principal characteristics of global flow. Both local and local-global methods generate subgrid-scale velocity fields that reproduce the impact of fine-scale stratigraphic architecture. By using multiscale basis functions to discretize the flow equations on a coarse grid, one can retain the efficiency of an upscaling method, while at the same time produce detailed and conservative velocity fields on the underlying fine grid. The accuracy and efficacy of the multiscale method is compared with those of fine-scale models and of coarse-scale models with no subgrid treatment for several two-phase-flow scenarios. Numerical experiments involving two-phase incompressible flow and transport phenomena are carried out on high-resolution corner-point grids that represent explicitly example stratigraphic architectures found in real-life shallow-marine and turbidite reservoirs. The multiscale method is several times faster than the direct solution of the fine-scale problem and yields more accurate solutions than coarse-scale modeling techniques that resort to explicit effective properties. The accuracy of the multiscale simulation method with ALG-velocity basis functions is compared with that of the local velocity basis functions. The multiscale simulation results are consistently more accurate when the local-global method is employed for computing the velocity basis functions.

Two-Stage Algebraic Multiscale Linear Solver for Highly Heterogeneous Reservoir Models

Summary An efficient Two-Stage Algebraic Multiscale Solver (TAMS) that converges to the fine-scale solution is described. The first (global) stage is a multiscale solution obtained algebraically for the given fine-scale problem. In the second stage, a local preconditioner, such as the Block ILU (BILU), or the Additive Schwarz (AS) method is used. Spectral analysis shows that the multiscale solution step captures the low-frequency parts of the error spectrum quite well, while the local preconditioner represents the high-frequency components accurately. Combining the two stages in an iterative scheme results in efficient treatment of all the error components associated with the fine-scale problem. TAMS is shown to converge to the reference fine-scale solution. Moreover, the eigenvalues of the TAMS iteration matrix show significant clustering, which is favorable for Krylov-based methods. Accurate solution of the nonlinear saturation equations (i.e., transport problem) requires having locally conservative velocity fields. TAMS guarantees local mass conservation by concluding the iterations with a multiscale finite-volume step. We demonstrate the performance of TAMS using several test cases with strong permeability heterogeneity and large-grid aspect ratios. Different choices in the TAMS algorithm are investigated, including the Galerkin and finite-volume restriction operators, as well as the BILU and AS preconditioners for the second stage. TAMS for the elliptic flow problem is comparable to state-of-the-art algebraic multigrid methods, which are in wide use. Moreover, the computational time of TAMS grows nearly linearly with problem size.

Multiscale modeling using goal-oriented adaptivity and numerical homogenization. Part I: Mathematical formulation and numerical results

This paper is the first in this series to develop a numerical homogenization method for heterogeneous media and integrate it with goal-oriented finite element mesh adaptivity. We describe the physical application, Step and Flash Imprint Lithography, in brief and present the mathematical ideas and numerical verification. The method requires the Moore–Penrose pseudoinverse of element stiffness matrices. Algorithms for efficiently computing the pseudoinverse of sparse matrices will be presented in the second paper.The purpose of numerical homogenization is to reduce the number of degrees of freedom, find locally optimal effective material properties, and perform goal-oriented mesh refinement. Traditionally, a finite element mesh is designed after obtaining material properties in different regions. The mesh has to resolve material discontinuities and rapid variations in the solution. In our approach, however, we generate a sequence of coarse meshes (possibly 1-irregular), and homogenize material properties on each coarse mesh element using a locally posed constrained convex quadratic optimization problem. This upscaling is done using the Moore–Penrose pseudoinverse of the linearized fine-scale element stiffness matrices, and a material-independent interpolation operator.Numerical verification is done using a two dimensional conductivity problem with known analytical limit. Finally, we present results for two and three dimensional geometries. The results show that this method uses orders of magnitude fewer degrees of freedom to give fast and approximate solutions of the original fine-scale problem.

Accuracy assessment of contextual classification results for vegetation mapping

A new procedure for quantitatively assessing the geometric accuracy of thematic maps, obtained from classifying hyperspectral remote sensing data, is presented. More specifically, the methodology is aimed at the comparison between results from any of the currently popular contextual classification strategies. The proposed procedure characterises the shapes of all objects in a classified image by defining an appropriate reference and a new quality measure. The results from the proposed procedure are represented in an intuitive way, by means of an error matrix, analogous to the confusion matrix used in traditional thematic accuracy representation. A suitable application for the methodology is vegetation mapping, where lots of closely related and spatially connected land cover types are to be distinguished. Consequently, the procedure is tested on a heathland vegetation mapping problem, related to Natura 2000 habitat monitoring. Object-based mapping and Markov Random Field classification results are compared, showing that the selected Markov Random Fields approach is more suitable for the fine-scale problem at hand, which is confirmed by the proposed procedure.

A variational multiscale method for incompressible turbulent flows: Bubble functions and fine scale fields

This paper presents a residual-based turbulence model for the incompressible Navier–Stokes equations. The method is derived employing the variational multiscale (VMS) framework. A multiscale decomposition of the continuous solution and a priori unique decomposition of the admissible spaces of functions lead to two coupled nonlinear problems termed as the coarse-scale and the fine-scale sub-problems. The fine-scale velocity field is assumed to be nonlinear and time-dependent and is modeled via the bubble functions approach applied directly to the fine-scale sub-problem. A significant contribution in this paper is a systematic and consistent derivation of the fine-scale variational operator, commonly termed as the stabilization tensor that possesses the right order in the advective and diffusive limits, and variationally projects the fine-scale solution onto the coarse-scale space. A direct treatment of the fine-scale problem via bubble functions offers several fine-scale approximation options with varying degrees of mathematical sophistication that are investigated via benchmark problems. Numerical accuracy of the proposed method is shown on a forced-isotropic turbulence problem, statistically stationary turbulent channel flow problems at Re T = 395 and 590, and non-equilibrium turbulent flow around a cylinder at Re = 3,900.

A stabilized mixed finite element method for the incompressible shear-rate dependent non-Newtonian fluids: Variational Multiscale framework and consistent linearization

This paper presents a stabilized mixed finite element method for shear-rate dependent incompressible fluids. The viscosity of the fluid is considered a function of the second invariant of the rate-of-deformation tensor, thus making the shear-stress shear-strain relation nonlinear. The weak form of the generalized Navier–Stokes equations is cast in the Variational Multiscale (VMS) framework that leads to a two-level description of the problem. Consistent linearization of the fine-scale problem with respect to the fine-scale velocity field and the use of bubble functions to expand the fine-scale trial and test functions lead to an analytical expression for the fine-scale velocity along with a definition of the stabilization tensor τ . The ensuing nonlinear stabilized form is presented and the consistent tangent tensor is derived. Numerical convergence of the proposed method on structured and unstructured meshes that are composed of linear triangles and bilinear quadrilaterals are presented. Shear-thinning and shear-thickening effects are investigated via the backward facing step problem and the effects of geometric parameters on the flow characteristics are highlighted. Time dependent features are investigated via the transient vortex-shedding problem and the accuracy and stability properties of the new method are shown.