Classical Finite Element Method Research Articles

Brain-inspired spiking neural networks in Engineering Mechanics: a new physics-based self-learning framework for sustainable Finite Element analysis

AbstractThe present study aims to develop a sustainable framework employing brain-inspired neural networks for solving boundary value problems in Engineering Mechanics. Spiking neural networks, known as the third generation of artificial neural networks, are proposed for physics-based artificial intelligence. Accompanied by a new pseudo-explicit integration scheme based on spiking recurrent neural networks leading to a spike-based pseudo explicit integration scheme, the underlying differential equations are solved with a physics-informed strategy. We propose additionally a third-generation spike-based Legendre Memory Unit that handles large sequences. These third-generation networks can be implemented on the coming-of-age neuromorphic hardware resulting in less energy and memory consumption. The proposed framework, although implicit, is viewed as a pseudo-explicit scheme since it requires almost no or fewer online training steps to achieve a converged solution even for unseen loading sequences. The proposed framework is deployed in a Finite Element solver for plate structures undergoing cyclic loading and a Xylo-Av2 SynSense neuromorphic chip is used to assess its energy performance. An acceleration of more than 40% when compared to classical Finite Element Method simulations and the capability of online training is observed. We also see a reduction in energy consumption down to the thousandth order.

Variability and loss of uniqueness of numerical solutions in FEM×DEM modeling with second gradient enhancement

AbstractIn the last decade, a new multi‐scale FEM×DEM approach has been developed using Finite Element Method (FEM) coupled with Discrete Element Method (DEM) as a constitutive law to account for the specificities of the mechanical behavior of granular materials. In FEM×DEM model, a DEM calculation is performed on a particle assembly (volume element—VE) at each Gauss point. Recent publications have demonstrated that FEM×DEM approach naturally captures the discrete and anisotropic nature of granular materials. Despite its advantages, FEM×DEM with classical FEM, suffers from mesh dependency, especially when material enters softening phase and exhibits strain localization. To overcome this limitation, FEM×DEM model has been enriched by incorporating a local second gradient model. Nevertheless, the existence of multiple possible solutions is observed. In this paper, we study the variability and loss of uniqueness of numerical solutions to a boundary value problem. Different VEs with equivalent mechanical properties are generated and used to model the pressuremeter tests by means of FEM×DEM. The modeling results show a great variability of the numerical results, both in shape of borehole and in different modes of shear bands. For the same VE, the loss of uniqueness of numerical solutions is evidenced by a slight modification of loading history at the level of the internal pressure applied to the borehole. Finally, we show that when a certain heterogeneity is introduced by using different VEs within the same BVP, even if the uniqueness of the solution is not guaranteed, the set of possible solutions seems more restrained.

Rate-optimal higher-order adaptive conforming FEM for biharmonic eigenvalue problems on polygonal domains

The a posteriori error analysis of the classical Argyris finite element methods dates back to 1996, while the optimal convergence rates of associated adaptive finite element schemes are established only very recently in 2021. It took a long time to realize the necessity of an extension of the classical finite element spaces to make them hierarchical. This paper establishes the novel adaptive schemes for the biharmonic eigenvalue problems and provides a mathematical proof of optimal convergence rates towards a simple eigenvalue and numerical evidence thereof. This makes the suggested algorithm highly competitive and clearly justifies the higher computational and implementational costs compared to low-order nonconforming schemes. The numerical experiments provide overwhelming evidence that higher polynomial degrees pay off with higher convergence rates and underline that adaptive mesh-refining is mandatory. Five computational benchmarks display accurate reference eigenvalues.

Multiharmonic multiscale modelling in 3-D nonlinear magnetoquasistatics: Composite material made of insulated particles

The use of the classical finite element method (FEM) to solve problems with magnetic composites leads to huge linear systems that are impossible to solve. Instead, homogenization and multiscale methods are often used with the composite material replaced by a homogeneous material with the homogenized constitutive law obtained by solving cell-problems representing the mesoscale material structure. For non-linear time-dependent problems, FEM is often used with a time-transient method (TTM) and the solution is obtained one time-step at a time. However, in cases where a steady-state solution is of interest, the multiharmonic method can be faster and more cost effective for the same accuracy of the time discretization. In addition, when solving magnetoquasistatic multiscale problems with TTM, the dynamic hysteresis in the homogenized fields can slow down or even impede the convergence of the macro-scale problem due to the possibly non-continuously differentiable homogenized material laws. This work presents a novel robust modelling approach for non-linear magnetoquasistatic problems combining multiharmonic method with the multiscale method.

A Hybrid Discrete-Finite Element method for continuous and discontinuous beam-like members including nonlinear geometric and material effects

This paper introduces a novel formulation, called Hybrid Discrete-Finite Element (HybriDFEM) method, for modelling one-directional continuous and discontinuous planar beam-like members, including nonlinear geometric and material effects. In this method, the structure is modelled as a series of distinct rigid blocks, connected to each other through contact pairs distributed along the interfaces. Each of those contact pairs are composed of two nonlinear multidirectional springs in series, which can represent either the deformation of the blocks themselves, or the deformation of their interface. Unlike the Applied Element Method, in which contact pairs are composed of one single spring, the current approach allows capturing phenomena such as sectional deformations or relative deformations between two blocks composed of different materials. This method shares similarities with the Discrete Element Methods in its ability to model contact interfaces between rigid or deformable units, but does not require a numerical time-domain integration scheme. More importantly, its formulation resembles that of the classical Finite Elements Method, allowing one to easily couple the latter with HybriDFEM. Following the presentation of its formulation, the method is benchmarked against analytical solutions selected from the literature, ranging from the linear-elastic response of a cantilever beam to the buckling and rocking response of continuous flexible columns, and rigid block stackings. One final example showcases the coupling of a HybriDFEM element with a linear beam finite element.

An efficient and easy-to-implement recovery-based a posteriori error estimator for isogeometric analysis of the Stokes equation

In this article, we present an efficient recovery-based a posteriori error estimator for adaptive isogeometric analysis (IGA) of the Stokes equation, which is straightforward to implement. The increased regularity of splines in IGA versus Lagrange polynomials used in the classical finite element method plays a significant role for the success of the presented recovery technique. We consider LR B-splines (Johannessen et al., 2014) for the adaptive mesh refinement, and the isogeometric Taylor–Hood element, Sub-Grid element (Buffa et al., 2011), and Raviart–Thomas element (divergence-conforming) (Buffa et al., 2010; Evans and Hughes, 2013; Johannessen et al., 2015) for the mixed discretization of the Stokes equation. Three benchmark problems with analytical solutions are tested, and they demonstrate clearly the great performance of the recovery-based error estimator compared to classical residual-based error estimator.

Automatic variationally stable analysis for finite element computations: Transient convection-diffusion problems

We present an application of stable finite element (FE) approximations of convection-diffusion initial boundary value problems (IBVPs) using a weighted least squares FE method, the automatic variationally stable finite element (AVS-FE) method [1]. The transient convection-diffusion problem leads to issues in classical FE methods as the differential operator can be considered a singular perturbation in both space and time. The stability property of the AVS-FE method, allows us significant flexibility in the construction of FE approximations in both space and time. Thus, in this paper, we take two distinct approaches to the FE discretization of the convection-diffusion problem: i) considering a space-time approach in which the temporal discretization is established using finite elements, and ii) a method of lines approach in which we employ the AVS-FE method in space whereas the temporal domain is discretized using the generalized-α method. We also consider another space-time technique in which the temporal direction is partitioned, thereby leading to finite space-time “slices” in an attempt to reduce the computational cost of the space-time discretizations.We present numerical verifications for these approaches, including numerical asymptotic convergence studies highlighting optimal convergence properties. Furthermore, in the spirit of the discontinuous Petrov-Galerkin (DPG) method by Demkowicz and Gopalakrishnan [2–6], the AVS-FE method also leads to readily available a posteriori error estimates through a Riesz representer of the residual of the AVS-FE approximations. Hence, the norm of the resulting local restrictions of these estimates serves as error indicators in both space and time for which we present multiple numerical verifications in mesh adaptive strategies.

A method for calculation of lateral displacements of buildings under distributed loads

Lateral displacement is a vary important parameter that we need to calculate when structures are subjected to lateral loads like earthquake and wind loads. In this study, a method is proposed for lateral displacement calculation of structures with different structural systems in different planes. This method is based on the continuum system calculation model. The method suggested in the literature for only top displacement in the case of uniform loading, is developed in this study for the calculation of displacements at each storey level and also in both uniform and triangular loading conditions. At the end of the study, twenty-eight storey building with shear wall-frame bearing system, which was taken from the literature, was solved with the presented method and Finite Element Method. The shear walls were modelled with three different element types for the analysis with the Finite Element Method with structural engineering program used. The results, obtained from the Continuum Method and Finite Element Method were presented in tables and by figures. Thus, the compatibility of the proposed method with the classical Finite Element Method was investigated. From the compared results of the method and the literature or finite element models it was seen that the method used for the case of uniform or triangular distributed loading gives very close results.

A machine learning method of accelerating multiscale analysis for spatially varying microstructures

Multiscale computing for heterogeneous materials provides a powerful and fidelity approach to handling situations where a suitable macroscopic constitutive model is unavailable. However, there have been very few instances in the industrial sector where concurrent multiscale modeling techniques have been adopted, primarily because they require significant computational resources. In particular, when dealing with large numbers of representative volume elements (RVEs) with different microstructure morphologies throughout the macrostructure. In this paper, the novelty lies in providing an artificial neural network (ANN) model-based multiscale approach for accelerating multiscale analysis of the elliptical inclusion-reinforced composites with spatially varying microstructures, which give up the assumption of global periodicity. To tackle this challenge, firstly, a series of RVEs with different microstructure morphologies characterized by inclusion orientations and aspect ratios, under different load conditions, are simulated to generate training data. Second, the back propagation neural network (BPNN) is trained offline using data acquired from the RVE simulation for learning the underlying and possibly RVE morphologies. The trained BPNN replaces the complex microscopic mechanical solution problems over each macroscopic integration point at each iteration, to update the macroscopic stress-strain responses and it significantly reduces the computational cost of multiscale simulations. Then, the trained neural network is implemented on ABAQUS via user-defined materials (UMAT) and its effectiveness is verified by comparison with the classical multiscale finite element method (FE2) method. Finally, the outstanding computational capability of the proposed approach is further demonstrated through two numerical examples.

Inclined crack identification in bidirectional FG beams on an elastic foundation using the h-version of the finite element method

This article examines the impact of inclined transverse cracking on the natural frequency of Euler-Bernoulli-model imperfect FG beams on an elastic foundation, taking into account the pinned-pinned and clamped-clamped boundary conditions. The equations are diffused using the classic finite element method (h-version). Material properties are considered to vary in both directions of the beam; width and thickness, using the power-law formula. The approximate porosity model is adopted with a uniform distribution. Cracked element stiffness is determined by reducing the cross-section of a bi-directional FG beam. The numerical results are compared with those of previous convergence studies for dimensionless fundamental frequencies. Case studies were carried out to evaluate the effect of the porosity values, the crack depth, the crack location, slope angle, and the Winkler-elastic foundation parameters on the first three natural frequencies of a beam with different support conditions and material gradient. In addition, the results demonstrated that the two-way distribution function plays an important role in the convergence of frequencies.

Dynamic Analysis of Isotropic Homogeneous Beams Using the Method of Initial Functions and Comparison with Classical Beam Theories and FEM

The main aim of this study is the dynamic analysis of isotropic homogeneous beams using the method of initial functions (MIFs) and comparison with classical beam theories and FEM. Also, this research employs the state space methodology, with a special emphasis on isotropy, to analyse simply supported beam systems. A mathematical model for the dynamic response of beams is constructed using the method of initial functions. The novelty of this study lies in its approach to dynamic analysis, where isotropic homogeneous beams are explored without making assumptions, thus ensuring increased precision using the method of initial functions. Importantly, the approach remains free from restrictive assumptions and relies solely on mathematical formulations, yielding results superior to classical beam theories such as Euler–Bernoulli, Timoshenko, and Rayleigh beam theories. In this work, the application of MIFs of various orders (4th, 6th, 8th, and 10th) to calculate natural frequencies is explored, enabling a thorough examination of the beam’s dynamic characteristics. In addition, parameters such as normal stresses, shear stresses, and deflections in different directions are considered to provide a comprehensive understanding of beam behaviour. To validate the findings, a detailed comparison with a finite element method (FEM) is conducted, achieving excellent agreement between the analytical results and FEM solutions. Furthermore, the influence of Poisson’s ratio (μ) on natural frequencies is investigated by varying its value from 0.18 to 0.30. The research also explores the deviation of plane stress values of the beam section from those estimated using the FEM for the corresponding components.

Error analysis of a weak Galerkin finite element method for singularly perturbed differential-difference equations

A weak Galerkin finite element method is applied to singularly perturbed delay reaction-diffusion problems. A robust uniform convergence has been proved both in the energy and balanced norms using higher-order piecewise discontinuous polynomials on Shishkin meshes. The error analysis for singularly perturbed reaction-diffusion problems with negative or positive shift in the balanced norm has appeared for the first time. The proposed method uses piecewise polynomials of order k ≥ 1 on interior of each element and piecewise constant polynomials on the end points of each element. By the Schur complement technique, the interior degrees of foredoom (DOF) can be eliminated from the discrete system resulting from the numerical scheme, and thus the degrees of freedom of the proposed method comparable with the classical finite element methods, and it is remarkably less than that of the discontinuous Galerkin method. Finally, we give various numerical experiments to verify the theoretical findings.

Structure Dynamic Analysis of C-9 Parachute Using Absolute Nodal Coordinate Formulation Thin Shell Element

In parachute fluid structure interaction (FSI) simulation, traditional parachute structure discrete methods including classic finite element method and mass-spring-damp method suffers from its large computational cost and deficiency in describing material property. This paper develops a generic shell element based on absolute nodal coordinate formulation (ANCF) to model parachute canopy to reduce computation resources and retain membrane strain/stress analyzing capability. We conduct multiple validation cases of ANCF plate and shell element and compare with current literature, in which large deformation configuration and stress distribution have been verified. A 1/4 C-9 parachute canopy inflation process by uniform pressure is simulated by discretizing into 7 ANCF shell and 84 shell element, for which the number of element discretization reduces to 1/10 than classic FEM and computational cost is considerably reduced using band sparse matrix solving technique.

How to achieve the fast computation for voxel-based irregular structures by few finite elements?

We propose machine learning (ML) based smart interpolation functions to enhance the finite element computation (named as smart-I finite element) for the heterogeneous structures (e.g., alloy material, concrete material and porous rock) that is composed of many microstructures known as voxel-based irregular shape (VI-shape). This element can directly describe the geometries with different VI-shapes and material properties without the complex meshing process by traditional finite element method (FEM). To achieve the smart interpolation in such element, a machine learning algorithm containing multiple parallel artificial neural networks (ANNs) and DeepONets assembled by the convolutional architectures is constructed for the fast prediction of interpolation functions. On this basis, the smart-I finite element is combined with classic FEM element to derive the global equilibrium equations and obtain the full-field variable distribution. Numerical tests with various material, geometrical as well as loading parameters prove that, the smart-I finite element enhanced FEM (SFEE-FEM) is able to conduct the accurate computation with a small number of elements, and thus greatly reduces the time cost. The smart-I finite element proposed by this work breaks through the limitation of traditional element shape with regular boundary (e.g., straight line), and may pave a new way to implement the FEM calculation on voxel-based geometries efficiently.

Thermal stress analysis of variable angle tow composite plates through high-order structural models

This work focuses on thermo-elastic analysis of Variable Angle Tow (VAT) composite plates. Emphasis is given to the through-the-thickness 3D stress component distributions as a consequence of constant temperature profiles. High-order structural models are used, in order to take into account the volumetric and the distortion contributions. The last contribute arise from boundary conditions or asymmetric staking sequences. Governing equations of the uncoupled thermo-elastic problem are derived in the domain of the Carrera Unified Formulation (CUF), which in turn is coupled with a classical finite element method to obtain opportune numerical solutions. The advantage of using CUF lies in the fact that different theories of structures can be implemented automatically and a critical study on the use of standard to high-order plate finite elements can be performed with ease. In this manner we classify models based on their efficiency and depending on the degree of accuracy needed, both in terms of displacements and stresses. Different numerical problems are considered and it is demonstrated that layerwise approximations are needed whenever shear stresses trends are of interest. Boundary Conditions influence more the choice of the adequate expansion theory otherwise.

A MGT thermoelastic problem with two relaxation parameters

In this paper, we consider, from both analytical and numerical viewpoints, a thermoelastic problem. The so-called MGT model, with two different relaxation parameters, is used for both the displacements and the thermal displacement, leading to a linear coupled system made by two third-order in time partial differential equations. Then, using the theory of linear semi-groups the existence and uniqueness to this problem is proved. If we restrict ourselves to the one-dimensional case, the exponential decay of the energy is obtained assuming some conditions on the constitutive parameters. Then, using the classical finite element method and the implicit Euler scheme, we introduce a fully discrete approximation of a variational formulation of the thermomechanical problem. A main a priori error estimates result is shown, from which we conclude the linear convergence under suitable additional regularity conditions. Finally, we present some one-dimensional numerical simulations to demonstrate the convergence of the fully discrete approximation, the behavior of the discrete energy decay and the dependence on a coupling parameter.

Three-field mixed hp-finite element method for the solution of the Guyer–Krumhansl heat conduction model

On the basis of numerous experimental studies the Guyer–Krumhansl heat conductivity model can be considered as one of the most promising theoretical models to simulate the low temperature processes and the thermal behavior of such complex structures as materials with inhomogeneities and interfaces, as well as porous materials. However, the classical h-version finite element methods do not provide convergent and accurate results for the solution of the Guyer–Krumhansl heat conductivity model. In recent paper, a new three-field variational formulation is derived treating the temperature, the heat flow and its current density as independent variables. Both the temperature- and the heat flow boundary condition are weakly imposed, i.e., built in the variational form.Based on this variational background, a new, hp-version mixed finite element method is constructed. The h- and p-convergence behaviors of the temperature and the heat flow are analyzed on the transient region for two representative model problems: (i) a rapid heating process with exponentially changing rate and (ii) a ramp-type heating process. The relative and absolute errors are measured in maximum norm. From the computational experiments it follows that the mixed hp-finite element method gives reliable, robust (uniformly stable) results not only for the h- but also for the p-approximation in the case of both the temperature and the heat flow.

A well-balanced and exactly divergence-free staggered semi-implicit hybrid finite volume / finite element scheme for the incompressible MHD equations

We present a new exactly divergence-free and well-balanced hybrid finite volume/finite element scheme for the numerical solution of the incompressible viscous and resistive magnetohydrodynamics (MHD) equations on staggered unstructured mixed-element meshes in two and three space dimensions. The equations are split into several subsystems, each of which is then discretized with a particular scheme that allows to preserve some fundamental structural features of the underlying governing PDE system also at the discrete level. The pressure is defined on the vertices of the primary mesh, while the velocity field and the normal components of the magnetic field are defined on an edge-based/face-based dual mesh in two and three space dimensions, respectively. This allows to account for the divergence-free conditions of the velocity field and of the magnetic field in a rather natural manner. The non-linear convective and the viscous terms in the momentum equation are solved at the aid of an explicit finite volume scheme, while the magnetic field is evolved in an exactly divergence-free manner via an explicit finite volume method based on a discrete form of the Stokes law in the edges/faces of each primary element. The latter method is stabilized by the proper choice of the numerical resistivity in the computation of the electric field in the vertices/edges of the 2D/3D elements. To achieve higher order of accuracy, a piecewise linear polynomial is reconstructed for the magnetic field, which is guaranteed to be exactly divergence-free via a constrained L2 projection. Finally, the pressure subsystem is solved implicitly at the aid of a classical continuous finite element method in the vertices of the primary mesh and making use of the staggered arrangement of the velocity, which is typical for incompressible Navier-Stokes solvers. In order to maintain non-trivial stationary equilibrium solutions of the governing PDE system exactly, which are assumed to be known a priori, each step of the new algorithm takes the known equilibrium solution explicitly into account so that the method becomes exactly well-balanced. We show numerous test cases in two and three space dimensions in order to validate our new method carefully against known exact and numerical reference solutions. In particular, this paper includes a very thorough study of the lid-driven MHD cavity problem in the presence of different magnetic fields and the obtained numerical solutions are provided as free supplementary electronic material to allow other research groups to reproduce our results and to compare with our data. We finally present long-time simulations of Soloviev equilibrium solutions in several simplified 3D tokamak configurations, showing that the new well-balanced scheme introduced in this paper is able to maintain stationary equilibria exactly over very long integration times even on very coarse unstructured meshes that, in general, do not need to be aligned with the magnetic field.

A Coupled Overlapping Finite Element Method for Analyzing Underwater Acoustic Scattering Problems

It is found that the classic finite element method (FEM) requires much time for adequate meshes to acquire satisfactory numerical solutions, and is restricted to acoustic problems with low and middle frequencies. In this work, a coupled overlapping finite element method (OFEM) is employed by combining the overlapping finite element and the modified Dirichlet-to-Neumann (mDtN) boundary condition to solve underwater acoustic scattering problems. The main difference between the OFEM and the FEM lies in the construction of the local field approximation. In the OFEM, virtual nodes are utilized to form the partition of unity functions while no degree of freedom is assigned to these virtual nodes, which suppresses the linear dependence issue in other generalized finite element methods. Moreover, the user-defined enrichment functions can be flexibly utilized in the local field, and thus the numerical dispersions can be significantly mitigated. To truncate the infinite problem domain and satisfy the Sommerfeld radiation condition, an artificial boundary is constructed by incorporating the mDtN technique. Several numerical examples are studied and it is shown that the proposed method can greatly diminish the numerical error and is insensitive to distorted meshes, indicating that the proposed method is promising in predicting underwater acoustic scattering.

Thermal-mechanical buckling analysis and optimization of the stringer stiffened cylinder using smeared stiffener based reduced-order models

In this work, the thermal-mechanical buckling analysis and optimization of stringer stiffened cylindrical shells are achieved using the smeared stiffener based reduced-order models. A novel reduced-order modeling method is proposed for nonlinear buckling analysis of the stringer stiffened cylinder subjected to an axially compressive load and initial temperature field. The reduced-order model is constructed based on the improved Koiter perturbation theory and the smeared stiffener method. The axially compressive load, initial temperature field and perturbation loads are converted to be the degrees of freedom in the smeared stiffener based reduced-order model. The thermoelastic geometrically nonlinear response of the stringer stiffened cylinder can be obtained using the proposed method accurately and efficiently. Optimization problems are designed to achieve the optimal stiffening configuration for minimum weight and maximum buckling load, where the solutions of reduced-order models are selected as inputs. Stringer stiffened cylinders with various ratios of radius to thickness and length to radius are considered to illustrate the good performance of the proposed method. Compared to the fully nonlinear analysis based on the classical finite element method, the proposed method can achieve the nonlinear buckling load of the stringer stiffened cylinder much more efficiently and realize the optimization of the stiffening configuration at an affordable computational cost.