Multiscale Finite Element Method for Coupling Analysis of Heterogeneous Magneto-Electro-Elastic Structures in Thermal Environment

Magneto-electro-elastic (MEE) materials, a new type of composite intelligent materials, exhibit excellent multifield coupling effects. Due to the heterogeneity of the materials, it is challenging to use the traditional finite element method (FEM) for mechanical analysis. Additionally, the MEE materials are often in a complex service environment, especially under the influence of the thermal field with thermoelectric and thermomagnetic effects, which affect its mechanical properties. Therefore, this paper proposes the efficient multiscale computational method for the multifield coupling problem of heterogeneous MEE structures under the thermal environment. The method constructs a multi-physics field with numerical base functions (the displacement, electric potential, and magnetic potential multiscale base functions). It equates a single cell of heterogeneous MEE materials to a macroscopic unit and supplements the macroscopic model with a microscopic model. This allows the problem to be solved directly on a macroscopic scale. Finally, the numerical simulation results demonstrate that compared with the traditional FEM, the multiscale finite element method (MsFEM) can achieve the purpose of ensuring accuracy and reducing the degree of freedom, and significantly improving the calculation efficiency.

Extended multiscale finite element method for mechanical analysis of heterogeneous materials

An extended multiscale finite element method （EMsFEM） is developed for solving the mechanical problems of heterogeneous materials in elasticity.The underlying idea of the method is to construct numerically the multiscale base functions to capture the small-scale features of the coarse elements in the multiscale finite element analysis.On the basis of our existing work for periodic truss materials, the construction methods of the base functions for continuum heterogeneous materials are systematically introduced. Numerical experiments show that the choice of boundary conditions for the construction of the base functions has a big influence on the accuracy of the multiscale solutions, thus,different kinds of boundary conditions are proposed. The efficiency and accuracy of the developed method are validated and the results with different boundary conditions are verified through extensive numerical examples with both periodic and random heterogeneous micro-structures.Also, a consistency test of the method is performed numerically. The results show that the EMsFEM can effectively obtain the macro response of the heterogeneous structures as well as the response in micro-scale,especially under the periodic boundary conditions.

TWO-LEVEL MULTISCALE FINITE ELEMENT METHODS FOR THE STEADY NAVIER-STOKES PROBLEM

In this article, on the basis of two-level discretizations and multiscale finite element method, two kinds of finite element algorithms for steady Navier-Stokes problem are presented and discussed. The main technique is first to use a standard finite element discretization on a coarse mesh to approximate low frequencies, then to apply the simple and Newton scheme to linearize discretizations on a fine grid. At this process, multiscale finite element method as a stabilized method deals with the lowest equal-order finite element pairs not satisfying the inf-sup condition. Under the uniqueness condition, error analyses for both algorithms are given. Numerical results are reported to demonstrate the effectiveness of the simple and Newton scheme.

THE TWO-LEVEL STABILIZED FINITE ELEMENT METHOD BASED ON MULTISCALE ENRICHMENT FOR THE STOKES EIGENVALUE PROBLEM

In this paper,we first propose a new stabilized finite element method for the Stokes eigenvalue problem.This new method is based on multiscale enrichment,and is derived from the Stokes eigenvalue problem itself.The convergence of this new stabilized method is proved and the optimal priori error estimates for the eigenfunctions and eigenvalues are also obtained.Moreover,we combine this new stabilized finite element method with the two-level method to give a new two-level stabilized finite element method for the Stokes eigenvalue problem.Furthermore,we have proved a priori error estimates for this new two-level stabilized method.Finally,numerical examples confirm our theoretical analysis and validate the high effectiveness of the new methods.

Generalized Multiscale Finite Element Methods.Nonlinear Elliptic Equations

In this paper we use the Generalized Multiscale Finite Element Method(GMsFEM)framework,introduced in[26],in order to solve nonlinear elliptic equations with high-contrast coefficients.The proposed solution method involves linearizing the equation so that coarse-grid quantities of previous solution iterates can be regarded as auxiliary parameters within the problem formulation.With this convention,we systematically construct respective coarse solution spaces that lend themselves to either continuous Galerkin(CG)or discontinuous Galerkin(DG)global formulations.Here,we use Symmetric Interior Penalty Discontinuous Galerkin approach.Both methods yield a predictable error decline that depends on the respective coarse space dimension,and we illustrate the effectiveness of the CG and DG formulations by offering a variety of numerical examples.

Time-Lapse 3-D Seismic Wave Simulation via the Generalized Multiscale Finite Element Method

Numerical solution of time-lapse seismic monitoring problems can be challenging due to the presence of finely layered reservoirs.Repetitive wave modeling using fine layered meshes also adds more computational cost.Conventional approaches such as finite difference and finite element methods may be prohibitively expensive if the whole domain is discretized with the cells corresponding to the grid in the reservoir subdomain.A common approach in this case is to use homogenization techniques to upscale properties of subsurface media and assign the background properties to coarser grid;however,inappropriate application of upscaling might result in a distortion of the model,which hinders accurate monitoring of the fluid change in subsurface.In this work,we instead investigate capabilities of a multiscale method that can deal with fine scale heterogeneities of the reservoir layer and more coarsely meshed rock properties in the surrounding domains in the same fashion.To address the 3-D wave problems,we also demonstrate how the multiscale wave modeling technique can detect the changes caused by fluid movement while the hydrocarbon production activity proceeds.

Multiscale Finite ElementMethods for Flows on Rough Surfaces

In this paper,we present the Multiscale Finite Element Method(MsFEM)for problems on rough heterogeneous surfaces.We consider the diffusion equation on oscillatory surfaces.Our objective is to represent small-scale features of the solution via multiscale basis functions described on a coarse grid.This problem arises in many applications where processes occur on surfaces or thin layers.We present a unified multiscale finite element framework that entails the use of transformations that map the reference surface to the deformed surface.The main ingredients of MsFEM are(1)the construction of multiscale basis functions and(2)a global coupling of these basis functions.For the construction of multiscale basis functions,our approach uses the transformation of the reference surface to a deformed surface.On the deformed surface,multiscale basis functions are defined where reduced(1D)problems are solved along the edges of coarse-grid blocks to calculate nodalmultiscale basis functions.Furthermore,these basis functions are transformed back to the reference configuration.We discuss the use of appropriate transformation operators that improve the accuracy of the method.The method has an optimal convergence if the transformed surface is smooth and the image of the coarse partition in the reference configuration forms a quasiuniform partition.In this paper,we consider such transformations based on harmonic coordinates(following H.Owhadi and L.Zhang[Comm.Pure and Applied Math.,LX(2007),pp.675-723])and discuss gridding issues in the reference configuration.Numerical results are presented where we compare the MsFEM when two types of deformations are used formultiscale basis construction.The first deformation employs local information and the second deformation employs a global information.Our numerical results showthat one can improve the accuracy of the simulations when a global information is used.

A multiscale 3D finite element analysis of fluid/solute transport in mechanically loaded bone

The transport of fluid, nutrients, and signaling molecules in the bone lacunar-canalicular system （LCS） is critical for osteocyte survival and function. We have applied the fluorescence recovery after photobleaching （FRAP） approach to quantify load-induced fluid and solute transport in the LCS in situ, but the measurements were limited to cortical regions 30-50 μm underneath the periosteum due to the constrains of laser penetration. With this work, we aimed to expand our understanding of load-induced fluid and solute transport in both trabecular and cortical bone using a multiscaled image-based finite element analysis （FEA） approach. An intact murine tibia was first re-constructed from microCT images into a three-dimensional （3D） linear elastic FEA model, and the matrix deformations at various locations were calculated under axial loading. A segment of the above 3D model was then imported to the biphasic poroelasticity analysis platform （FEBio） to predict load-induced fluid pressure fields, and interstitial solute/fluid flows through LCS in both cortical and trabecular regions. Further, secondary flow effects such as the shear stress and/or drag force acting on osteocytes, the presumed mechano-sensors in bone, were derived using the previously developed ultrastructural model of Brinkman flow in the canaliculi. The material properties assumed in the FEA models were validated against previously obtained strain and FRAP transport data measured on the cortical cortex. Our results demonstrated the feasibility of this computational approach in estimating the fluid flux in the LCS and the cellular stimulation forces （shear and drag forces） for osteocytes in any cortical and trabecular bone locations, allowing further studies of how the activation of osteocytes correlates with in vivo functional bone formation. The study provides a promising platform to reveal potential cellular mechanisms underlying the anabolic power of exercises and physical activities in treating patients with skeletal deficiencies.

A Multiscale Multilevel Monte Carlo Method for Multiscale Elliptic PDEs with Random Coefficients

We propose a multiscale multilevel Monte Carlo(MsMLMC)method to solve multiscale elliptic PDEs with random coefficients in the multi-query setting.Our method consists of offline and online stages.In the offline stage,we construct a small number of reduced basis functions within each coarse grid block,which can then be used to approximate the multiscale finite element basis functions.In the online stage,we can obtain the multiscale finite element basis very efficiently on a coarse grid by using the pre-computed multiscale basis.The MsMLMC method can be applied to multiscale RPDE starting with a relatively coarse grid,without requiring the coarsest grid to resolve the smallestscale of the solution.We have performed complexity analysis and shown that the MsMLMC offers considerable savings in solving multiscale elliptic PDEs with random coefficients.Moreover,we provide convergence analysis of the proposed method.Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation.

Multiscale Modeling of Magnetic Distribution of Ribbon Magnetic Cores

The multiscale finite element method(MsFEM)combined with conventional finite element method(CFEM)is proposed to solve static magnetic field in the ribbon magnetic core with non-periodical corners considered.Firstly,a simple 2-dimensional electrostatic problem is used to introduce the MsFEM implementation process.The results are compared to analytical method,as well as conventional FEM.Then,an exam-ple of magneto-static problem is considered for a ribbon magnetic core built sheet by sheet as well as corners taken into considera-tion.Conventional FEM and MsFEM are used to compute the magneto-static field by adopting scalar magnetic potential.Both magnetic potential and magnetic flux density on a certain path are compared.It is shown that the results obtained by MsFEM agree well with the one from conventional FEM.Moreover,MsFEM combined with FEM is potentially a general strategy for mul-tiscale modeling of ribbon magnetic cores with complex and non-periodical structures considered,like corners and T-joints,which can effectively reduce the computational cost.

Texture evolution and slip mode of a Ti-5.5Mo-7.2Al-4.5Zr-2.6Sn-2.1Cr dual-phase alloy during cold rolling based on multiscale crystal plasticity finite element model

The complex micromechanical response among grains remains a persistent challenge to understand the deformation mechanism of titanium alloys during cold rolling.Therefore,in this work,a multiscale crystal plasticity finite element method of dual-phase alloy was proposed and secondarily developed based on LS-DYNA software.Afterward,the texture evolution and slip mode of a Ti-5.5Mo-7.2Al-4.5Zr-2.6Sn-2.1Cr alloy,based on the realistic 3D microstructure,during cold rolling(20%thickness reduction)were systematically investigated.The relative activity of the■slip system in theαphase gradually increased,and then served as the main slip mode at lower Schmid factor(<0.2).In contrast,the contribution of the■slip system to the overall plastic deformation was relatively limited.For theβphase,the relative activity of the<111>{110}slip system showed an upward tendency,indicating the important role of the critical resolved shear stress relationship in the relative activity evolutions.Furthermore,the abnormally high strain of very fewβgrains was found,which was attributed to their severe rotations compelled by the neighboring pre-deformedαgrains.The calculated pole figures,rotation axes,and compelled rotation behavior exhibited good agreement to the experimental results.

Size effect of lattice material and minimum weight design

The size effects of microstructure of lattice materials on structural analysis and minimum weight design are studied with extented multiscale finite element method（EMsFEM） in the paper. With the same volume of base material and configuration, the structural displacement and maximum axial stress of micro-rod of lattice structures with different sizes of microstructure are analyzed and compared.It is pointed out that different from the traditional mathematical homogenization method, EMsFEM is suitable for analyzing the structures which is constituted with lattice materials and composed of quantities of finite-sized micro-rods.The minimum weight design of structures composed of lattice material is studied with downscaling calculation of EMsFEM under stress constraints of micro-rods. The optimal design results show that the weight of the structure increases with the decrease of the size of basic sub-unit cells. The paper presents a new approach for analysis and optimization of lattice materials in complex engineering constructions.

Numerical Investigation on the Boundary Conditions for the Multiscale Base Functions

We study the multiscale finite element method for solving multiscale elliptic problems with highly oscillating coefficients,which is designed to accurately capture the large scale behaviors of the solution without resolving the small scale characters.The key idea is to construct the multiscale base functions in the local partial differential equation with proper boundary conditions.The boundary conditions are chosen to extract more accurate boundary information in the local problem.We consider periodic and non-periodic coefficients with linear and oscillatory boundary conditions for the base functions.Numerical examples will be provided to demonstrate the effectiveness of the proposed multiscale finite element method.

HETEROGENEOUS MULTISCALE METHOD FOR OPTIMAL CONTROL PROBLEM GOVERNED BY ELLIPTIC EQUATIONS WITH HIGHLY OSCILLATORY COEFFICIENTS

In this paper, we investigate heterogeneous multiscale method （HMM） for the optimal control problem with distributed control constraints governed by elliptic equations with highly oscillatory coefficients. The state variable and co-state variable are approximated by the multiscale discretization scheme that relies on coupled macro and micro finite elements, whereas the control variable is discretized by the piecewise constant. By applying the well- known Lions＇ Lemma to the discretized optimal control problem, we obtain the necessary and sufficient optimality conditions. A priori error estimates in both L^2 and H^1 norms are derived for the state, co-state and the control variable with uniform bound constants. Finally, numerical examples are presented to illustrate our theoretical results.

Multilevel Markov Chain Monte Carlo Method for High-Contrast Single-Phase Flow Problems

In this paper we propose a general framework for the uncertainty quantification of quantities of interest for high-contrast single-phase flow problems.It is based on the generalized multiscale finite element method(GMsFEM)and multilevel Monte Carlo(MLMC)methods.The former provides a hierarchy of approximations of different resolution,whereas the latter gives an efficient way to estimate quantities of interest using samples on different levels.The number of basis functions in the online GMsFEM stage can be varied to determine the solution resolution and the computational cost,and to efficiently generate samples at different levels.In particular,it is cheap to generate samples on coarse grids but with low resolution,and it is expensive to generate samples on fine grids with high accuracy.By suitably choosing the number of samples at different levels,one can leverage the expensive computation in larger fine-grid spaces toward smaller coarse-grid spaces,while retaining the accuracy of the final Monte Carlo estimate.Further,we describe a multilevel Markov chain Monte Carlo method,which sequentially screens the proposal with different levels of approximations and reduces the number of evaluations required on fine grids,while combining the samples at different levels to arrive at an accurate estimate.The framework seamlessly integrates the multiscale features of the GMsFEM with the multilevel feature of the MLMC methods following the work in[26],and our numerical experiments illustrate its efficiency and accuracy in comparison with standard Monte Carlo estimates.