Multilevel Finite Element 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.

Multilevel design and construction in nanomembrane rolling for three-dimensional angle-sensitive photodetection

Releasing pre-strained two-dimensional nanomembranes to assemble on-chip three-dimensional devices is crucial for upcoming advanced electronic and optoelectronic applications. However, the release process is affected by many unclear factors, hindering the transition from laboratory to industrial applications. Here, we propose a quasistatic multilevel finite element modeling to assemble three-dimensional structures from two-dimensional nanomembranes and offer verification results by various bilayer nanomembranes. Take Si/Cr nanomembrane as an example, we confirm that the three-dimensional structural formation is governed by both the minimum energy state and the geometric constraints imposed by the edges of the sacrificial layer. Large-scale, high-yield fabrication of three-dimensional structures is achieved, and two distinct three-dimensional structures are assembled from the same precursor. Six types of three-dimensional Si/Cr photodetectors are then prepared to resolve the incident angle of light with a deep neural network model, opening up possibilities for the design and manufacturing methods of More-than-Moore-era devices.

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.

Sparse grid time-discontinuous Galerkin method with streamline diffusion for transport equations

High-dimensional transport equations frequently occur in science and engineering. Computing their numerical solution, however, is challenging due to its high dimensionality. In this work we develop an algorithm to efficiently solve the transport equation in moderately complex geometrical domains using a Galerkin method stabilized by streamline diffusion. The ansatz spaces are a tensor product of a sparse grid in space and discontinuous piecewise polynomials in time. Here, the sparse grid is constructed upon nested multilevel finite element spaces to provide geometric flexibility. This results in an implicit time-stepping scheme which we prove to be stable and convergent. If the solution has additional mixed regularity, the convergence of a 2d-dimensional problem equals that of a d-dimensional one up to logarithmic factors. For the implementation, we rely on the representation of sparse grids as a sum of anisotropic full grid spaces. This enables us to store the functions and to carry out the computations on a sequence regular full grids exploiting the tensor product structure of the ansatz spaces. In this way existing finite element libraries and GPU acceleration can be used. The combination technique is used as a preconditioner for an iterative scheme to solve the transport equation on the sequence of time strips. Numerical tests show that the method works well for problems in up to six dimensions. Finally, the method is also used as a building block to solve nonlinear Vlasov-Poisson equations.

Two-scale concurrent optimization of composites with elliptical inclusions under microstress constraints within the FE2 framework

In order to avoid the adverse effects of structural damage caused by stress concentration, the optimization problem considering stress constraints is very important in structural design. This paper aims at introducing microstress constraints within a multilevel finite element (FE2) analysis framework to perform the two-scale concurrent optimization and reduce the stress concentration while exhibiting better overall stiffness properties for composites with elliptical inclusions. In order to do that, at the macroscale, relying on the solid isotropic material with penalization (SIMP), a density-based optimization algorithm is raised for a maximum stiffness design and free material distribution optimization under a volume constraint, these displacement solutions and material distributions at the macroscopic structural scale provide kinematic constraints for microscale optimization. At the microscale, the parameterized composite microstructure (inclusion orientation and aspect ratio) is optimized under the principal strain direction constraints of the macroscale element’s Gauss integration point. The optimized microstructure in turn updates the macroscopic effective stress directly from the volume average of the microscopic stress field. The clustering strategy is used in order to improve the computation efficiency of the optimization. The novelty of the work lies in the use of microstress constraints to solve the concurrent optimization problems of two-phase composites within the FE2 framework. The effectiveness of the proposed concurrent optimization approach is validated through three numerical examples.

Non-linear vibration analysis of visco-elastically damped composite structures by multilevel finite elements and asymptotic numerical method

Composite materials and structures are inherently in-homogeneous across multiple scales. Multi-scale modelling offers opportunities to apprehend the coupling of material behaviour and characteristics from the micro- to meso- and macro-scales. In this paper, a multi-scale finite element method (FE2) is proposed to compute the modal properties of visco-elastic heterogeneous composite materials in terms of damping frequencies and modal loss factors. In the proposed FE2-based vibration analysis, two finite elements (FE) calculations are carried out in a nested manner, one at the macro-scale and the other at the micro-scale. Unlike conventional analysis, the developed analysis does not require homogenized constitutive properties because these are derived from the micro-scale FE simulations at the representative volume element (RVE) level. The non-linearity at the micro-scale is accounted by using a frequency dependent Young’s modulus. The Asymptotic Numerical Method (ANM) and its automatic differentiation is used to solve the non-linear numerical problem. ANM consists of solving an analytical non-linear problem with a path-following (or continuation) method associated with a high-order perturbation technique. Compared with existing methods, the originality of the proposed approach lies in its ability to account for the frequency dependence of Young’s modulus in visco-elastic microstructure. Using the automatic differentiation makes the proposed approach enough flexible and generic to deal with damped and undamped vibration analyses of composite materials structures.

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.

A machine learning-based multiscale model to predict bone formation in scaffolds.

Computational modeling methods combined with non-invasive imaging technologies have exhibited great potential and unique opportunities to model new bone formation in scaffold tissue engineering, offering an effective alternate and viable complement to laborious and time-consuming in vivo studies. However, existing numerical approaches are still highly demanding computationally in such multiscale problems. To tackle this challenge, we propose a machine learning (ML)-based approach to predict bone ingrowth outcomes in bulk tissue scaffolds. The proposed in silico procedure is developed by correlating with a dedicated longitudinal (12-month) animal study on scaffold treatment of a major segmental defect in sheep tibia. Comparison of the ML-based time-dependent prediction of bone ingrowth with the conventional multilevel finite element (FE2) model demonstrates satisfactory accuracy and efficiency. The ML-based modeling approach provides an effective means for predicting in vivo bone tissue regeneration in a subject-specific scaffolding system.

Effect of Wiring Density and Pillar Structure on Chip Packaging Interaction for Mixed-Signal Cu Low k Chips

Multilevel finite element analysis (FEA) was used to study the effects of wiring density and solder pillar structure on chip-package interaction (CPI) for advanced Cu/low k mixed-signal chips. The mixed signal chip has analog and digital wiring designs with different metal densities, incorporating extreme low-k (ELK), low-k (LK) and oxide dielectrics in 10 wiring layers. The results are compared with uniform signal chips with uniform metal density on each wiring layer. The first principal stress was found to increase by 1.5 to 3x in the mixed signal chip between the analog and the isolation channel of the ELK and LK layers due to the different wiring density. In addition, the energy release rate (ERR) was significantly increased to reach a critical ERR ratio higher than 1 to drive interfacial delamination, raising serious chip-package interaction (CPI) reliability concern for the mixed signal chip. The results were attributed to the Dundurs effect due to the material mismatch and the nonuniform metal density in the mixed signal chip. In the study of the pillar structure effect, intermetallic compound (IMC) growth was found to be important and can substantially increase the critical ERR ratio to degrade the CPI reliability of the mixed signal chip.

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.

A concurrent multiscale framework based on self-consistent clustering analysis for cylinder structure under uniaxial loading condition

Accurate and efficient multiscale simulation of multifilament composite remains challenges due to expensive computation cost of solving the nonlinear representative volume element (RVE) responses in microscale. An effective reduced-order method called self-consistent clustering analysis (SCA) has been proposed recently to solve the dilemma. This paper extends the application of traditional multilevel finite element (FE2) to cylinder structure with domain division, where the inner domain is computed by finite element method (FEM) with SCA instead of direct FE2 method and outer domain is calculated by conventional FEM. The presented framework is named as concurrent modified FEM-SCA. To validate the accuracy and efficiency of presented method, the computation results of SCA, Fast Fourier Transform (FFT) and FEM are compared in microscale at first. Then, under the presented framework, the cylinder structures with diverse RVEs in uniaxial loading are studied, and the results are compared with full-scale direct numerical simulation (DNS) by FEM. Besides, the effects of different element positions, inclusion variations and loading phases on the RVE responses are also studied.

Computational Investigation of Interface Stresses in Duplex Structure Stainless Steels

Duplex structural stainless steels (DP) are highly corrosion-resistant with austenite (γ)-ferrite (δ) compositions and processing routes. These DP steels are used in chemical, petrochemical, marine, power generation, and nuclear industries due to their unique properties. Experimental data of 2D slices obtained from optical micrography were used in a multilevel finite element analysis to investigate the interface stresses between the two phases of DP material. Level #1 (macro-level) finite element analysis involves homogenizing the 2D slices based on their representative constituent properties to identify high-stress regions. Level #2 (micro-level) finite element analysis was conducted with representative austenite/ferrite microstructure identified from level #1 analysis. The results from finite element simulations from both macro- and micro-levels revealed that interface stresses are affected by the phase’s microstructure distributions. Micro-level analysis revealed maximum interface stress is much higher than macro-level analysis. It was also found that stress localization occurred at the ferrite/austenite interfaces based on their distributions.

Forming of bioabsorbable clips using magnesium alloy strips with enhanced characteristics

The paper shows the feasibility of manufacturing surgical clips using bioabsorbable AZ31B magnesium alloy strips characterized by enhanced forming characteristics, and, at the same time, improved corrosion resistance to human fluids that make them suitable for temporary applications. The novel process chain used for the clip manufacturing includes an extrusion-cutting step, to fabricate strips with a peculiar shear texture and extremely refined microstructure, and a dedicated bending operation to achieve the clip final shape. The strip forming operation setup was numerically designed by a multi-level finite element approach to provide the clip required shape without fracture occurrence.

A cascadic multigrid method for nonsymmetric eigenvalue problem

In this paper, a cascadic multigrid method is proposed to solve nonsymmetric eigenvalue problems. Based on the multilevel correction method, the proposed method transforms a nonsymmetric eigenvalue problem solving on the finest finite element space to linear smoothing steps on a sequence of multilevel finite element spaces and some nonsymmetric eigenvalue problems solving on a very low dimensional space. Choosing the sequence of finite element spaces and the number of smoothing steps appropriately, we obtain the optimal convergence rate with the optimal computing complexity. Some numerical examples are provided to validate the theoretical results and the efficiency of this proposed scheme.

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

Multiscale CUF-FE2 nonlinear analysis of composite beam structures

In this paper, a new paradigm for Carrera’s Unified Formulation (CUF) based on multiscale structural modelling is accomplished by bridging micromechanics and the advanced CUF one-dimensional/beam structural theories by means of the Multilevel Finite Element (also known as FE2) framework. Under the framework of the FE2 method, the analysis is divided into a macroscopic/structural problem and a microscopic/material problem. At the macroscopic level, several higher-order refined beam elements can be easily implemented via CUF by deriving a fundamental nucleus that is independent of the approximation order over the thickness and the number of nodes per element (they are free parameters of the formulation). The unknown macroscopic constitutive law is obtained by numerical homogenisation of a Representative Volume Element (RVE) at the microscopic level. Vice versa, the microscopic deformation gradient is calculated from the macroscopic model. Information is passed between the two scales in a FE2 sense. The resulting nonlinear problem is solved through the Asymptotic Numerical Method (ANM) that is more reliable and less Newton-Raphson’s one. The developed models are used as a first attempt to investigate the microstructure effect on the macrostructure geometrically nonlinear response. The proposed paradigm is used for investigating the effect of microscale imperfections (not straight carbon fibres) on the macroscale response (instability). Results are assessed in terms of accuracy and computational costs towards full FEM solutions. Three factors have been considered for an imperfection sensitivity parametric analysis: the defect wavelength as well as the amplitude and the size of RVE.

A type of cascadic multigrid method for coupled semilinear elliptic equations

This paper is to introduce a type of cascadic multigrid method for coupled semilinear elliptic equations. Instead of solving the coupled semilinear elliptic equation on a very fine finite element space directly, the new scheme needs to solve a decoupled linear system by some smoothing steps on each of multilevel finite element spaces and solve a coupled semilinear elliptic equation on a coarse space. By choosing the appropriate number of smoothing steps on different finite element spaces, the corresponding optimal convergence rate and optimal computation work can be derived. Besides, the adaptive cascadic multigrid method for coupled semilinear elliptic equations and its analysis are also presented theoretically and numerically. Moreover, the requirement of bounded second order derivatives of the nonlinear term in the existing multigrid methods is reduced to the Lipschitz continuation property in the presented new scheme. Some numerical experiments are presented to validate our theoretical analysis.

Multigrid method for coupled semilinear elliptic equation

This paper introduces a kind of multigrid finite element method for the coupled semilinear elliptic equations. Instead of the common way of directly solving the coupled semilinear elliptic problems on some fine spaces, the presented method transforms the solution of the coupled semilinear elliptic problem into a series of solutions of the corresponding decoupled linear boundary value problems on the sequence of multilevel finite element spaces and some coupled semilinear elliptic problems on a very low dimensional space. The decoupled linearized boundary value problems can be solved by some multigrid iterations efficiently. The optimal error estimate and optimal computational work are proved theoretically and demonstrated numerically. Moreover, the requirement of bounded second‐order derivatives of the nonlinear term in the existing multigrid method is reduced to a Lipschitz continuous condition in the proposed method.

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.