Multilevel Finite Element Method Research Articles

A novel multilevel finite element method for a generalized nonlinear Schrödinger equation

In this article, we focus on an efficient multilevel finite element method to solve the time-dependent nonlinear Schrödinger equation which is one of the most important equations of mathematical physics. For the time derivative, we adopt implicit schemes including the backward Euler method and the Crank–Nicolson method. Based on these stable implicit schemes, the proposed method requires solving a nonlinear elliptic problem at each time step. For these nonlinear elliptic equations, a multilevel mesh sequence is constructed. At each mesh level, we first derive a rough approximation by correcting the approximation of the previous mesh level in a special correction subspace. The correction subspace is composed of a coarse finite element space and an additional approximate solution derived from the previous mesh level. Next, we only need to solve a linearized elliptic equation by inserting the rough approximation into the nonlinear term. Then, we derive an accurate approximate solution by performing the aforementioned solving process on the multilevel mesh sequence until we reach the final mesh level. Owing to the special construct of the correction subspace, we derive a multilevel finite element method to solve the nonlinear Schrödinger equation for the first time, and meanwhile we also derive an optimal error estimate with linear computational complexity. Additionally, unlike the existing multilevel methods for nonlinear problems, that typically require bounded second-order derivatives of the nonlinear terms, the nonlinear term in our study requires only one-order derivatives. Numerical results are provided to support our theoretical analysis and demonstrate the efficiency of the presented method.

Companion-based multi-level finite element method for computing multiple solutions of nonlinear differential equations

The utilization of nonlinear differential equations has resulted in remarkable progress across various scientific domains, including physics, biology, ecology, and quantum mechanics. Nonetheless, obtaining multiple solutions for nonlinear differential equations can pose considerable challenges, particularly when it is difficult to find suitable initial guesses. To address this issue, we propose a pioneering approach known as the Companion-Based Multilevel Finite Element Method (CBMFEM). This novel technique efficiently and accurately generates multiple initial guesses for solving nonlinear elliptic semi-linear equations containing polynomial nonlinear terms through the use of finite element methods with conforming elements. As a theoretical foundation of CBMFEM, we present an appropriate and new concept of the isolated solution to the nonlinear elliptic equations with multiple solutions. The newly introduced concept is used to establish the inf-sup condition for the linearized equation around the isolated solution. Furthermore, it is crucially used to derive a theoretical error analysis of finite element methods for nonlinear elliptic equations with multiple solutions. A number of numerical results obtained using CBMFEM are then presented and compared with a traditional method. These not only show the CBMFEM's superiority, but also support our theoretical analysis. Additionally, these results showcase the effectiveness and potential of our proposed method in tackling the challenges associated with multiple solutions in nonlinear differential equations with different types of boundary conditions.

Absolute nodal coordinate formulation – Multilevel finite element framework for the nonlinear multi-scale multibody dynamic analysis of composite structures

In this paper, a multi-scale modeling approach for the dynamic response of flexible multibody system made of composite material is suggested using the absolute nodal coordinate formulation (ANCF) – multilevel finite element (FE2) method. On many occasions, the analysis of composite structures requires a multi-scale modeling approach. FE2 method is one of the famous multi-scale modeling methods. And it is a versatile computational homogenization approach that is commonly applicable to many different materials. Because the present FE2 is capable of computing the mechanical response at two different scales simultaneously, ANCF element will be used at the macroscopic scale. And the microscopic RVE at each macroscopic integration point will be obtained by the finite element method. Since ANCF element uses only the absolute nodes and gradients, the mass matrix will be constant. Due to such characteristics, when combined with FE2, the linearization process will become easier compared against the existing formulation. And Coriolis or centrifugal force will not need to be considered in the dynamic formulation. Therefore, computational cost will be reduced, and the relevant algebraic manipulation will become simplified. Several numerical examples are presented to demonstrate the improved accuracy and diminished computational cost of the present suggestion.

Parallel space-time adaptive numerical simulation of 3D cardiac electrophysiology

The bidomain equations form the state-of-the-art model of cardiac electrophysiology and describe normal or pathological propagation of the excitation wave through cardiac tissue. If based on mechanistic cell models and applied to anatomically realistic geometries, their discretization requires very fine resolutions both in space and time and renders the numerical solution an extremely challenging computational problem. In this paper, we address this challenge by combining space-time adaptive discretization with dynamic load balancing for parallel computing. We use multilevel finite element methods for the spatial adaptation of the mesh, embedded Rosenbrock time integration for an adaptive choice of step sizes, and non-overlapping domain decomposition for the parallelization. In order to further reduce the computational costs, we exploit the sparsity of local matrices during the assembly of global FEM matrices which ensure an optimal usage of memory. In numerical tests, we demonstrate the feasibility of our approach that a substantial reduction factor of computing time is achieved w.r.t the static grid computation and a good parallel efficiency is achieved.

Multilevel correction adaptive finite element method for solving nonsymmetric eigenvalue problems

Large-scale nonsymmetric eigenvalue problems are common in various fields of science and engineering computing. However, their efficient handling is challenging, and research on their solution algorithms is limited. In this study, a new multilevel correction adaptive finite element method is designed for solving nonsymmetric eigenvalue problems based on the adaptive refinement technique and multilevel correction scheme. Different from the classical adaptive finite element method, which requires solving a nonsymmetric eigenvalue problem in each adaptive refinement space, our approach requires solving a symmetric linear boundary value problem in the current refined space and a small-scale nonsymmetric eigenvalue problem in an enriched correction space. Since it is time-consuming to solve a large-scale nonsymmetric eigenvalue problem directly in adaptive spaces, the proposed method can achieve nearly the same efficiency as the classical adaptive algorithm when solving the symmetric linear boundary value problem. In addition, the corresponding convergence and optimal complexity are verified theoretically and demonstrated numerically.

An Improved Model Reduction Method on AIMs for N-S Equations Using Multilevel Finite Element Method and Hierarchical Basis

An Improved Model Reduction Method on AIMs for N-S Equations Using Multilevel Finite Element Method and Hierarchical Basis

Structural-Genome-Driven computing for composite structures

The aim of this work is to propose a new approach, named Structural-Genome-Driven (SGD) computing, for composite structures, where the structural-genome database is collected through the multi-level finite element method (FE2). Compared to Material-Genome-Driven (MGD) analysis, proposed by Kirchdoerfer and Ortiz, 2016, the new SGD method benefits from not only the model reduction, but also the ability to easily deal with complex microscopic geometry and material properties. Several numerical tests are considered to demonstrate the efficiency and robustness of the proposed method.

A Fourier-related FE2 multiscale model for instability phenomena of long fiber reinforced materials

This work aims to develop a computationally efficient multiscale model to accurately simulate and analyze the instability phenomena of long fiber reinforced composites. The multilevel finite element method (FE2) based on the homogenization theory is adopted to realize the real-time interaction between the microscopic and the macroscopic levels. To increase the computational efficiency in the microscopic level and control the nonlinear calculations, the Fourier-related analysis, is carried out on the representative volume elements (RVEs), where all fast varying unknowns are replaced by slowly varying unknowns. The established nonlinear multiscale system is solved by the Asymptotic Numerical Method (ANM), which is more reliable and less time consuming than other classical iterative methods. This model is used to study the structural instability and the fiber micro-buckling of long fiber reinforced composites. The effects of fiber wavelength on the buckling and post-buckling of macro structure are discussed.

A Multiscale Analysis on the Superelasticity Behavior of Architected Shape Memory Alloy Materials.

In this paper, the superelasticity effects of architected shape memory alloys (SMAs) are focused on by using a multiscale approach. Firstly, a parametric analysis at the cellular level with a series of representative volume elements (RVEs) is carried out to predict the relations between the void fraction, the total stiffness, the hysteresis effect and the mass of the SMAs. The superelasticity effects of the architected SMAs are modeled by the thermomechanical constitutive model proposed by Chemisky et al. 2011. Secondly, the structural responses of the architected SMAs are studied by the multilevel finite element method (FE), which uses the effective constitutive behavior of the RVE to represent the behavior of the macroscopic structure. This approach can truly couple the responses of both the RVE level and structural level by the real-time information interactions between two levels. Through a three point bending test, it is observed that the structure inherits the strong nonlinear responses—both the hysteresis effect and the superelasticity—of the architected SMAs at the cellular level. Furthermore, the influence of the void fraction at the RVE level to the materials’ structural responses can be more specifically and directly described, instead of using an RVE to predict at the microscopic level. Thus, this work could be referred to for optimizing the stiffness, the hysteresis effect and the mass of architected SMA structures and extended for possible advanced applications.

Space–time adaptive linearly implicit peer methods for parabolic problems

In this paper a linearly implicit peer method is combined with a multilevel finite element method for the discretization of parabolic partial differential equations. Following the Rothe method it is first discretized in time and then in space. A spatial error estimator based on the hierarchical basis approach is derived. It is shown to be a reliable and efficient estimator up to some small perturbations. The efficiency index of the estimator is shown to be close to the ideal value one for two one-dimensional test problems. Finally we compare the performance of the overall method, based on second, third, and fourth order peer methods with that of some Rosenbrock methods. We conclude that the presented peer methods offer an attractive alternative to Rosenbrock methods in this context.

A hierarchical multilevel finite element method for mechanical analyses of periodical composite structures

A hierarchical multilevel finite element method (HMM) is developed for simulating mechanical behaviors of periodical composite structures. This method is established based on the idea of constructing shape function numerically which is proposed in the multiscale finite element method (MsFEM). In order to improve the computational accuracy, a hierarchical shape function and a modal shape function are also constructed numerically on sub-grids domain in addition to the original shape function. Then totally three kinds of numerical shape function are introduced in the proposed HMM. Correspondingly, the macroscopic displacement of coarse element can be divided into three parts which are contributed by nodal (original), hierarchical and modal shape functions, respectively. By virtue of these functions, the equivalent mass matrix, stiffness matrix and external loading vector of one single coarse element can be determined easily. The overall macroscopic quantities of whole composite structure are obtained by assembling those of each coarse element. Finally the original problems can be solved at the macroscopic scale, which will save amount of computational cost. To verify the validation and efficiency of the developed HMM, the static analysis, generalized eigenvalue analysis and transient response analysis are investigated.

Correct Multilevel Discrete-Continual Finite Element Method of Structural Analysis

The distinctive paper is devoted to correct multilevel discrete-continual finite element method (DCFEM) of structural analysis based on precise analytical solutions of resulting multipoint boundary problems for systems of ordinary differential equations with piecewise-constant coefficients. Corresponding semianalytical (discrete-continual) formulations are contemporary mathematical models which currently becoming available for computer realization. Major peculiarities of DCFEM include universality, computer-oriented algorithm involving theory of distributions, computational stability, optimal conditionality of resulting systems and partial Jordan decompositions of matrices of coefficients, eliminating necessity of calculation of root vectors.

A multi-level method for transmission eigenvalues of anisotropic media

In this paper, we propose a multi-level finite element method for the transmission eigenvalue problem of anisotropic media. The problem is non-standard and non-self-adjoint with important applicati...

Wavelet-Based Multilevel Discrete-Continual Finite Element Method for Local Deep Beam Analysis

High-accuracy solution of the problem of deep beam analysis is normally required in some pre-known domains (regions with the risk of significant stresses that could potentially lead to the destruction of structure, regions which are subject to specific operational requirements). The distinctive paper is devoted to correct wavelet-based multilevel discrete-continual finite element method for local analysis of deep beams with regular (in particular, constant or piecewise constant) physical and geometrical parameters (properties) in one direction. Initial discrete-continual operational formulation of the considering problem and corresponding operational formulation with the use of wavelet basis are presented. Due to special algorithms of averaging within multigrid approach, reduction of the problem is provided. Resultant multipoint boundary problem of structural mechanics for system of ordinary differential equations with piecewise-constant coefficients is given.

A multi-level method for transmission eigenvalues of anisotropic media

In this paper, we propose a multi-level finite element method for the transmission eigenvalue problem of anisotropic media. The problem is non-standard and non-self-adjoint with important applications in inverse scattering theory. We employ a suitable finite element method to discretize the problem. The resulting generalized matrix eigenvalue problem is large, sparse and non-Hermitian. To compute the smallest real transmission eigenvalue, which is usually an interior eigenvalue, we devise a multi-level method using Arnoldi iteration. At the coarsest mesh, the eigenvalue is obtained using Arnoldi iteration with an adaptive searching technique. This value is used as the initial guess for Arnoldi iteration at the next mesh level. This procedure is then repeated until the finest mesh level. Numerical examples are presented to show the viability of the method.

Wavelet-Based Multilevel Discrete-Continual Finite Element Method for Local Plate Analysis

High-accuracy solution of the problem of plate analysis is normally required in some pre-known domains (regions with the risk of significant stresses that could potentially lead to the destruction of structure, regions which are subject to specific operational requirements). The distinctive paper is devoted to correct wavelet-based multilevel discrete-continual finite element method for local analysis of plates with regular (in particular, constant or piecewise constant) physical and geometrical parameters (properties) in one direction. Initial discrete-continual operational formulation of the considering problem and corresponding operational formulation with the use of wavelet basis are presented. Due to special algorithms of averaging within multigrid approach, reduction of the problem is provided. Resultant multipoint boundary problem of structural mechanics for system of ordinary differential equations with piecewise-constant coefficients is given.

An ABAQUS toolbox for multiscale finite element computation

In this paper, we propose to implement, in the framework of a commercial finite element software, a computational multilevel finite element method for the modeling of composite materials and structures. In the present approach, the unknown constitutive relationship at the macroscale is obtained by solving a local finite element problem at the microscale. The main advantages of the proposed computational approach are that it can greatly save computer memory and CPU time, and it has good accuracy at the same time while it allows to easily building nonlinear behavior for high order mechanical theories to deal with problems which cannot be handled by classical multiscale or homogenization theories. The linear and the non-linear cases are introduced and implemented in ABAQUS. A Python script and user-defined FORTRAN subroutines have been developed for this purpose. Finally numerical results show that the method presented in this paper is effective and reliable.

Sparse tensor finite elements for elliptic multiple scale problems

Locally periodic, elliptic multiscale problems in a bounded Lipschitz domain D ⊂ R n with K ⩾ 2 separated scales are reduced to an elliptic system of K coupled, anisotropic elliptic one-scale problems in a cartesian product domain of total dimension Kn (e.g. [2,3,11,26]). In [23,31], it has been shown how these coupled elliptic problems could be solved by sparse tensor wavelet Finite Element Methods in log-linear complexity with respect to the number N of degrees of freedom required by multilevel solvers for elliptic one-scale problems in D with the same convergence rate. In the present paper, the high dimensional one-scale limiting problems are discretized by a sparse tensor product Finite Element Method (FEM) with standard, one-scale FE basis functions as used in engineering FE codes. Sparse tensorization and multilevel preconditioning is achieved by a BPX multilevel iteration. We show that the resulting sparse tensor multilevel FEM resolves all physical length scales throughout the domain, with efficiency (i.e., accuracy versus work and memory) comparable to that of multigrid solvers for elliptic one-scale problems in the physical domain D. In particular, our sparse tensor FEM gives numerical approximations of the correct homogenized limit as well as compressed numerical representations of all first order correctors, throughout the physical domain with performance independent of the physical problem’s scale parameters. Numerical examples with standard FE shape functions and BPX multilevel preconditioners for elliptic problems with K = 2 separated, physical scales in spatial dimension n = 2 confirm the theoretical results. In particular, the present approach allows to avoid the construction of wavelet FE bases necessary in previous work [23,31] while achieving resolution of all scales throughout the physical domain in log-linear complexity, with the logarithmic exponent behaving linearly in the number K of scales.

Damage analysis on a graded porous biocomposite structure under flexural load using FEM

A multi-level finite element (FE) method was developed to firstly study the mechanical responses and progressive damage behaviors of graded porous structures made of hydroxyapatite particle reinforced polymer biocomposite (HA/PEEK) under flexure load. In order to bridge the mechanical properties of the base material, damage and failure criteria of the building blocks for the graded porous structure, and macroscopic response of the structure, three-level FE models from micro-, mesoto macro-scale were established. A graded porous biocomposite implant was considered as a continuum structure with a solid section and a porous section with a gradient change in porosity from 29.5 to 80.9% in six equal steps. The macro-scale FE model of the implant was then constructed from the meso-scale unit cubic-shaped cell model, which was made of beam elements with a fixed length but a gradient change in their cross-section areas depending on the porosity level. The base material of the structure, HA/PEEK, was obtained from our previous work through a micro-scale unit cell model consisting of the particle and polymer matrix for predicting its stress–strain relation with dependence on variation of HA volume fraction. By adoption of the proposed method, a virtual three-point bending test was simulated to predict force–displacement relations of the implant and its damage initiation and propagation in both graded and uniform porous structures under flexural load. The trend of the predicted results agreed with those reported in open literature.

Model reduction on inertial manifolds for N–S equations approached by multilevel finite element method

Approximate Inertial Manifolds (AIMs) is approached by multilevel finite element method, which can be referred to as a Post-processed nonlinear Galerkin finite element method, and is applied to the model reduction for fluid dynamics, a typical kind of nonlinear continuous dynamic system from viewpoint of nonlinear dynamics. By this method, each unknown variable, namely, velocity and pressure, is divided into two components, that is the large eddy and small eddy components. The interaction between large eddy and small eddy components, which is negligible if standard Galerkin algorithm is used to approach the original governing equations, is considered essentially by AIMs, and consequently a coarse grid finite element space and a fine grid incremental finite element space are introduced to approach the two components. As an example, the flow field of incompressible flows around airfoil is simulated numerically and discussed, and velocity and pressure distributions of the flow field are obtained accurately. The results show that there exists less essential degrees-of-freedom which can dominate the dynamic behaviors of the discretized system in comparison with the traditional methods, and large computing time can be saved by this efficient method. In a sense, the small eddy component can be captured by AIMs with fewer grids, and an accurate result can also be obtained.