Classical Finite Element Method Research Articles

Isogeometric topology and shape optimization for composite structures using level-sets and adaptive Gauss quadrature

The Level-Set Method (LSM) with the smooth and distinct description of structural boundaries offers more superior benefits for topology optimization. However, the classic Finite Element Method (FEM) and ersatz material method in the LSM cause numerical deficiencies in the optimization, such as iterative instability and the crispness loss of boundaries. This work focuses on developing an Isogeometric Topology and Shape Optimization (ITSO) method for the design of composite structures, in which a Non-Uniform Rational B-splines (NURBS)-based Multi-phase Level-Sets (NM-LS) model is constructed to present the distribution of multiple materials. The NM-LS model consists of an immersed representation using level-sets, NURBS parameterized level-sets and its development. Isogeometric Analysis (IGA) with adaptive Gauss quadrature method are applied to solve unknown structural responses, which can effectively remove several numerical artifacts resulting from the linear interpolation of ersatz material model in both design and analysis. Three models of structural geometry, numerical analysis and topology description can be integrated by the same NURBS basis functions, which can effectively improve numerical precision and iterative stability. Finally, several 2D and 3D numerical examples are addressed to demonstrate the effectiveness and efficiency of the proposed ITSO method on the design of composite structures.

Higher order robust numerical computation for singularly perturbed problem involving discontinuous convective and source term

In this article, we considered a singularly perturbed convection–diffusion equation having discontinuous convective and source terms. Due to these discontinuities, an expected behavior appears in the solution known as the interior layer behavior at the point of discontinuity. Layers are some small narrow regions of the domain where the steep variations in the solution occur as singular perturbation parameter ε approaches zero. Classical central finite difference schemes and finite element methods with piecewise polynomial basis functions on the uniform mesh are known to be inadequate to solve such problems unless an unacceptably large number of mesh points are used when ε is very small. Thus, the problem is solved numerically first by a first‐order accurate, simple upwind scheme on a specially designed piecewise uniform Shishkin mesh. Further, to enhance the order of convergence and to achieve better accuracy, we constructed a Richardson extrapolation scheme (RE scheme) whichis almost second‐order accurate. An extensive analysis has been carried out to establish the almost second‐order ε‐uniform convergence of the proposed RE scheme. Lastly, some numerical examples are considered to verify our theoretical results, and comparisons of numerical results are performed with the numerical results that exist in the literature.

On numerical aspects of parameter identification for the Landau-Lifshitz-Gilbert equation in Magnetic Particle Imaging

<p style='text-indent:20px;'>The Landau-Lifshitz-Gilbert equation yields a mathematical model to describe the evolution of the magnetization of a magnetic material, particularly in response to an external applied magnetic field. It allows one to take into account various physical effects, such as the exchange within the magnetic material itself. In particular, the Landau-Lifshitz-Gilbert equation encodes relaxation effects, i.e., it describes the time-delayed alignment of the magnetization field with an external magnetic field. These relaxation effects are an important aspect in magnetic particle imaging, particularly in the calibration process. In this article, we address the data-driven modeling of the system function in magnetic particle imaging, where the Landau-Lifshitz-Gilbert equation serves as the basic tool to include relaxation effects in the model. We formulate the respective parameter identification problem both in the all-at-once and the reduced setting, present reconstruction algorithms that yield a regularized solution and discuss numerical experiments. Apart from that, we propose a practical numerical solver to the nonlinear Landau-Lifshitz-Gilbert equation, not via the classical finite element method, but through solving only linear PDEs in an inverse problem framework.</p>

A numerical approach for solving one nonlinear problem of hydrodynamics in a non-convex polygonal domain

The paper considers the two-dimensional stationary problem obtained as a result of time discretization of the nonlinear Navier-Stokes equations in rotation form, which describe the incompressible viscous fluid flow in a non-convex polygonal domain. In order to solve a nonlinear problem, it is necessary to find solutions to a sequence of approximate linear problems. For linear problems, the concept of an Rv-generalized solution in weighted sets obeying the weighted analogue of the Ladyzhenskaya-Babuška-Brezzi condition is introduced. A scheme of the weighted finite element method is constructed which suppresses the error arising in the neighborhood of the vertex of the reentrant angle and does not allow it to propagate into the inner part of the computational domain. The mass conservation law is valid directly at the grid nodes, and not only in the (weak) integral sense. Numerical experiments are carried out and a comparative analysis is made in polygonal domains with different values of reentrant angles. The advantage of the proposed approach over the classical finite element method is the order of convergence with respect to the grid step. A set of optimal parameters of the method is determined to obtain the required order of convergence. In this case, the exponent of the weight function depends on the value of the reentrant angle, and the parameters determining the Rv-generalized solution are independent on its value. In contrast to the classical approach, the order of convergence of the proposed approximate method to the exact solution of a nonlinear problem does not depend on the value of the reentrant angle. The constructed approximate method can be used without geometric mesh refinement in the vicinity of the singularity point.

A novel FFT-based homogenization scheme for cohesive zones

Cohesive Zone Models with finite thickness are widely used for the fracture mechanical modeling of layers of material, e.g., adhesives. Within this approach, the whole layer is modeled as a Cohesive Zone. Moreover, computational homogenization techniques are crucial for the development of advanced engineering materials, which are often heterogeneous. Compared to the classical Finite Element Method (FEM), computationally more efficient solvers based on the Fast Fourier Transform (FFT) are expected to reduce the computational effort needed for the homogenization. Originated from an existing method for the computational homogenization of Cohesive Zones using FEM, a novel FFT-based homogenization scheme for Cohesive Zone Models was developed. Our implementation of the FFT solver uses the Barzilai-Borwein scheme and a non-local ductile damage model for the fracture behavior. Finally, the method is applied to the core material of HybrixTM metal sandwich plates, and the good agreement with experimental results in opening mode I is shown.

Partition of Unity Finite Element Method for 2D Vibro-Acoustic Modeling

The Partition of Unity Finite Element Method (PUFEM) shows promise for modeling wave-like problems in the mid-to-high frequency range, allowing to capture several wavelengths in a single element. Despite the increasing attention it received in acoustics and in structural dynamics, its efficacy to deal with coupled problems has not been addressed. The main challenge in this case is to be able to represent different types of physical waves accurately, knowing that the wavelengths can be very different and vary differently, exemplified by the dispersion of flexural waves in a solid. Without a proper handling of the coupling between the coupled media, at best the number of degrees of freedom (DoF) will not be optimal, at worst the coupled model will not converge. Techniques like mesh refinement, wave enrichment and compatible or incompatible meshes might offer a potential solution to the problem, but the model usually needs to be adjusted through a time consuming trial-and-error procedure. To tackle the problem, this paper considers a 2D coupled vibro-acoustic problem, in which the structural and acoustic domains, modeled with PUFEM, are coupled using compatible and incompatible meshes based on different coupling strategies. Numerical analyses show that the proposed method outperforms the classical finite element method by several orders of magnitude in terms of number of DoF. Recommendations are proposed on the technique to choose depending on the frequency range of interest in relation to the critical frequency of the structure to ensure the best convergence rate. Finally, an application example is presented to highlight the performance of the proposed method.

Uniform convergent modified weak Galerkin method for convection-dominated two-point boundary value problems

We propose and analyze a modified weak Galerkin finite element method (MWG-FEM) for solving singularly perturbed problems of convection-dominated type. The proposed method is constructed over piecewise polynomials of degree $k\geq1$ on interior of each element and piecewise constant on the boundary of each element. The present method is parameter-free and has less degrees of freedom compared to the classical weak Galerkin finite element method. The method is shown uniformly convergent for small perturbation parameters. An uniform convergence rate of $\mathcal {O}((N^{-1}\ln N)^k)$ in the energy-like norm is established on the piecewise uniform Shishkin mesh, where $N$ is the number of elements. Various numerical examples are presented to confirm the theoretical results. Moreover, we numerically confirm that the proposed method has the optimal order error estimates of $\mathcal {O}(N^{-(k+1)})$ in a discrete $L^2$-norm and converges at superconvergence order of $\mathcal {O}((N^{-1}\ln N)^{2k})$ in the discrete $L^\infty$-norm.

A finite element method with cover functions for underwater acoustic propagation problems

To improve the performance of the classical finite element method (FEM) in acoustic analyses, in this work the interpolation cover functions are incorporated into the standard linear nodal shape functions to give the finite element with cover functions (FEwC). Since the original linear approximation space of the standard FEM is significantly enriched, more accurate solutions can be obtained. However, this FEwC encounters the intractable linear dependence (LD) problem. In this paper the LD problem is studied systematically and then an effective method is given to completely avoid this problem. The imposition of Dirichlet boundary conditions in FEwC which is different from that in FEM is also illustrated. Besides, to handle the acoustic scattering problems in the unbounded domains, the Dirichlet-to-Neumann (DtN) map technique is employed as the non-reflecting boundary condition. The dispersion error analyses and several typical numerical experiments associated with the joint frequency-domain and time-domain analysis are performed to exhibit the high performance of the FEwC compared with the classic FEM and the edge-based smoothed FEM in solving underwater acoustic propagation problems.

Modeling variable-density flow in saturated-unsaturated porous media: An advanced numerical model

Modeling variable-density flow in unconfined aquifers is a challenging task because of the nonlinear coupling between variably saturated flow and contaminant transport. This results in a highly nonlinear system since the strongly nonlinear Richards flow equation is, in addition, coupled to the advection-dispersion transport equation by viscosity and density variation. The solution of such a nonlinear system is often subject to convergence issues and can be very expansive in terms of computational time, especially for large-scale problems. Conventional numerical algorithms based on the sequential approach and the classical finite difference or finite element methods with the first-order backward Euler time integration scheme are generally inefficient and/or do not provide satisfactory results. In this work, we develop a new efficient and accurate 2D numerical model for the transport of dense contaminants in unsaturated porous media that allows for the simulation of large-scale problems. This research describes a new model that combines advanced spatial discretization methods (mixed hybrid finite element method, discontinuous Galerkin finite element method, and multipoint flux approximation method) with higher-order time integration techniques via the method of lines (MOL). The latter allows one to adapt the time step's size and the order of the time integration to improve the computational efficiency while maintaining accuracy. The robustness and accuracy of the new model are shown by comparison against a widely used commercial code based on the standard finite element method. The applicability of the developed model to a large-scale problem is then investigated by simulating saltwater intrusion under a climate change projection and long-term pumping regimes for the Akkar coastal aquifer in Lebanon using a simplified 2D conceptual model.

Dynamic stability optimization of laminated composite plates: An isogeometric HSDT formulation and PSO algorithm

An isogeometric analysis framework based on the non-uniform rational B-spline (NURBS) and the higher-order shear deformation theory (HSDT) is developed to study the geometry-accurate dynamic instability behavior of the shear-sensitive generally laminated composite plates. The developed approach is combined with the particle swarm optimization (PSO) technique to modify the layup orientations of the laminate corresponding to the minimum dynamic instability opening (DIOI) value. Based on the accuracy assessment results, the convergence rate of the developed method is higher than the classical finite element method. Moreover, the frequency accuracy of cubic NURBS is stabilized for high degrees of freedom denoting their high computational efficiency. The optimization procedure reveals that the mode shapes of the plates with optimal layups are more complex than their non-optimal case. The optimal ply angles are mostly oriented along the maximum-curvature path, perpendicular to the loading direction, and parallel to the supported edges. The mixed-type supports leads to lower DIOI values for optimal layups concerning the homogeneous supports. The pure compressive force expands the fundamental dynamic instability region (DIRI) for the generally laminated plates and leads to wide-range frequency instabilities. However, the combination of compressive force with lateral or shear force increases the lower bound of the DIRI for optimal layups. The minimization of DIOI value does not necessarily increase the lower bound of DIRI for optimal layups.

Earthquake response of linear-elastic arch-frames using exact curved beam formulations

PurposeThis study aims to obtain earthquake responses of linear-elastic multi-span arch-frames by using exact curved beam formulations. For this purpose, the dynamic stiffness method (DSM) which uses exact mode shapes is applied to a three-span arch-frame considering axial extensibility, shear deformation and rotational inertia for both columns and curved beams. Using exact free vibration properties obtained from the DSM approach, the arch-frame model is simplified into an equivalent single degree of freedom (SDOF) system to perform earthquake response analysis.Design/methodology/approachThe dynamic stiffness formulations of curved beams for free vibrations are validated by using the experimental data in the literature. The free vibrations of the arch-frame model are investigated for various span lengths, opening angle and column dimensions to observe their effects on the dynamic behaviour. The calculated natural frequencies via the DSM are presented in comparison with the results of the finite element method (FEM). The mode shapes are presented. The earthquake responses are calculated from the modal equation by using Runge-Kutta algorithm.FindingsThe displacement, base shear, acceleration and internal force time-histories that are obtained from the proposed approach are compared to the results of the finite element approach where a very good agreement is observed. For various span length, opening angle and column dimension values, the displacement and base shear time-histories of the arch-frame are presented. The results show that the proposed approach can be used as an effective tool to calculate earthquake responses of frame structures having curved beam elements.Originality/valueThe earthquake response of arch-frames consisting of curved beams and straight columns using exact formulations is obtained for the first time according to the best of the author’s knowledge. The DSM, which uses exact mode shapes and provides accurate free vibration analysis results considering each structural members as one element, is applied. The complicated structural system is simplified into an equivalent SDOF system using exact mode shapes obtained from the DSM and earthquake responses are calculated by solving the modal equation. The proposed approach is an important alternative to classical FEM for earthquake response analysis of frame structures having curved members.

Discontinuous Galerkin methods through the lens of variational multiscale analysis

In this article, we present a theoretical framework for integrating discontinuous Galerkin methods in the variational multiscale paradigm. Our starting point is a projector-based multiscale decomposition of a generic variational formulation that uses broken Sobolev spaces and Lagrange multipliers to accommodate the non-conforming nature at the boundaries of discontinuous Galerkin elements. We show that existing discontinuous Galerkin formulations, including their penalty terms, follow immediately from a specific choice of multiscale projector. We proceed by defining the “fine-scale closure function”, which captures the closure relation between the remaining fine-scale term in the discontinuous Galerkin formulation and the coarse-scale solution via a single integral expression for each basis function in the coarse-scale test space. We show that the projectors that correspond to discontinuous Galerkin methods lead to fine-scale closure functions with more compact support and smaller amplitudes compared to the fine-scale closure function of the classical (conforming) finite element method. This observation provides a new perspective on the natural stability of discontinuous Galerkin methods for hyperbolic problems, and may open the door to rigorously designed variational multiscale based fine-scale models that are suitable for DG methods.

A new finite element approach to model microscale strain localization within olivine aggregates

Abstract. This paper presents a new mesoscopic full field approach for the modeling of microstructural evolutions and mechanical behavior of olivine aggregates. The mechanical framework is based on a reduced crystal plasticity (CP) formulation which is adapted to account for non-dislocation glide strain-accommodating mechanisms in olivine polycrystals. This mechanical description is coupled with a mixed velocity–pressure finite element (FE) formulation through a classical crystal plasticity finite element method (CPFEM) approach. The microstructural evolutions, such as grain boundary migration and dynamic recrystallization, are also computed within a FE framework using an implicit description of the polycrystal through the level-set approach. This numerical framework is used to study the strain localization, at the polycrystal scale, on different types of pre-existing shear zones for thermomechanical conditions relevant to laboratory experiments. We show that both fine-grained and crystallographic textured pre-existing bands favor strain localization at the sample scale. The combination of both processes has a large effect on strain localization, which emphasizes the importance of these two microstructural characteristics (texture and grain size) on the mechanical behavior of the aggregate. Table 1 summarizes the list of the acronyms used in the following.

Local extreme learning machines and domain decomposition for solving linear and nonlinear partial differential equations

We present a neural network-based method for solving linear and nonlinear partial differential equations, by combining the ideas of extreme learning machines (ELM), domain decomposition and local neural networks. The field solution on each sub-domain is represented by a local feed-forward neural network, and Ck continuity conditions are imposed on the sub-domain boundaries. Each local neural network consists of a small number of hidden layers, while its last hidden layer can be wide. The weight/bias coefficients in all the hidden layers of the local neural networks are pre-set to random values and fixed throughout the computation, and only the weight coefficients in the output layers of the local neural networks are training parameters. The overall neural network is trained by a linear or nonlinear least squares computation, not by the back-propagation type algorithms. We introduce a block time-marching scheme together with the presented method for long-time simulations of time-dependent linear/nonlinear partial differential equations. The current method exhibits a clear sense of convergence with respect to the degrees of freedom in the neural network. Its numerical errors typically decrease exponentially or nearly exponentially as the number of training parameters, or the number of training data points, or the number of sub-domains increases. Extensive numerical experiments have been performed to demonstrate the computational performance of the presented method. We also demonstrate its capability for long-time dynamic simulations with some test problems. We compare the presented method with the deep Galerkin method (DGM) and the physics-informed neural network (PINN) method in terms of the accuracy and computational cost. The current method exhibits a clear superiority, with its numerical errors and network training time considerably smaller (typically by orders of magnitude) than those of DGM and PINN. We also compare the current method with the classical finite element method (FEM). The computational performance of the current method is on par with, and often exceeds, the FEM performance in terms of the accuracy and computational cost.

Multi-resolution approach based on a variables separation method in unsteady thermal problem for composites

In this work, a multi-resolution strategy is developed to solve unsteady diffusion problems with arbitrary heat source location. It is based on a variable separation approach which allows us to compute an explicit solution with respect to the heat source location and also to reuse an initial basis for new configurations. Thus, the change of sets of parameters (type of heat source, number of layers ...) does not imply the resolution of the complete thermal problem for each time step, but it is reduced to update 1D functions. For this purpose, the temperature is written as a sum of separated functions of the axial coordinate x, the transverse coordinate z, the time t and the volumetric heat source location xs. The derived non-linear problem implies an iterative process in which four 1D problems are solved successively at each iteration. In the thickness direction, a fourth-order expansion in each layer is considered. For the axial description, classical Finite Element method is used. The presented approach is assessed on various laminated beams under different loadings and comparisons with reference solutions with a fixed heat source location are proposed. These test cases show the possibilities and the accuracy of the method.

An Analytical Approach for the Nonlinear Free Vibration Analysis of Thin-Walled Circular Cylindrical Shells

This paper presents a new formulation combining the nonlinear theory of Novozhilov with the classical finite element method for the purpose of evaluating the vibratory characteristics of thin, closed and isotropic cylindrical shells. The theory developed in this paper is able to include the shell curvature effect in the circumferential direction of the orthogonal displacements and considers the impact of initial geometric imperfections on the dynamic response of the system. The formulation first takes a general form by expressing the shell displacements as an alliance between the generalized coordinates and spatial functions. Nonlinear kinematic relationships are inferred from Novozhilov’s theory. The equations of motion as well as the expressions of the mass, linear and nonlinear stiffness matrices are derived through the Lagrange method by considering the coupling between the different modes. An application of this model is illustrated in a further step, by adopting the displacement functions derived from exact solutions of linear Sanders’ theory equilibrium equations for thin cylindrical shells. The governing equations of motion are solved with the help of a direct iterative method. Linear and nonlinear frequencies are validated by comparison with the results in the literature. The relative nonlinear frequencies are determined as a function of vibration amplitudes and then compared with published results for several cases of shells. Excellent agreement is observed between the results derived from this theory and those found in the literature. The effect of different parameters including axial and circumferential wave number, length-to-radius ratio, thickness-to-radius ratio and various boundary conditions, on the nonlinear frequencies of cylindrical shells is investigated.

Robust topology optimization under material and loading uncertainties using an evolutionary structural extended finite element method

This research presents a novel algorithm for robust topology optimization of continuous structures under material and loading uncertainties by combining an evolutionary structural optimization (ESO) method with an extended finite element method (XFEM). Conventional topology optimization approaches (e.g. ESO) often require additional post-processing to generate a manufacturable topology with smooth boundaries. By adopting the XFEM for boundary representation in the finite element (FE) framework, the proposed method eliminates this time-consuming post-processing stage and produces more accurate evaluation of the elements along the design boundary for ESO-based topology optimization methods. A truncated Gaussian random field (without negative values) using a memory-less translation process is utilized for the random uncertainty analysis of the material property and load angle distribution. The superiority of the proposed method over Monte Carlo, solid isotropic material with penalization (SIMP) and polynomial chaos expansion (PCE) using classical finite element method (FEM) is demonstrated via two practical examples with compliances in material uncertainty and loading uncertainty improved by approximately 11% and 10%, respectively. The novelty of the present method lies in the following two aspects: (1) this paper is among the first to use the XFEM in studying the robust topology optimization under uncertainty; (2) due to the adopted XFEM for boundary elements in the FE framework, there is no need for any post-processing techniques. The effectiveness of this method is justified by the clear and smooth boundaries obtained.

Meshless numerical simulation of steel connections: application to the T‐stub component

AbstractFinite element analysis is a widely used simulation technique in steel design to analyze the structural behavior of structural components, such as beams, columns or connections. The major steps involved in finite element simulations are the definition of the basic parameters and the discretization in elements of the component, also known as preprocessing; the analysis of the model and the post‐processing by the representation and the interpretation of the obtained results. On the particular case of bolted connections, the most time‐consuming of these three phases is that of preprocessing, because on traditional finite element technology is required that the real geometry of the CAD model is simplified to be meshed and analyzed; this task is specifically severe and requires paying close attention on the case of components affected to concentration stresses. Nowadays, different techniques have been proposed and developed with this objective: reduce costs in terms of time and specialized human resources because the workflow is easier and simpler. One of these methods are the known as the meshfree methods, that do not require connection between nodes and are based on the interaction of the points of the geometry with the neighbors, so the processor analyzes taken as base the real CAD geometry and no simplifications are needed.On this paper is exposed the meshless method based on the theory of external approximations as an alternative to the classic finite element method. This method is based on the theory where all parts of the assembly are considered as different elements, and the approximation of displacements inside is arbitrary and not only by polynomials. The convergence of the simulation model is based on the external equilibrium of the assembly. The method is applied to a practical case: the simulation of the T‐stub connection. The T‐stub component corresponds to one T‐shaped steel profile bolted through their flanges and loaded through their web until collapse, and its behavior is similar to the tension zone of beam to column bolted end plate connections. The analysis of strength, stiffness and ductility have been widely investigated experimentally, by simulation and theoretically, and they are obtained from the force‐displacement characteristic curve. Information presented by Bursi and Jaspart on the paper “Benchmarks for finite element modelling of bolted steel connections” are considered to elaborate the models and to compare the results.

FEM/XFEM modeling of the 3D fracturing process in transversely isotropic geomaterials

This paper studies the fracturing process in geomaterials in order to define the mechanical properties of elements belonging to the lithosphere, such as the kaolonite-rich Opalinus Clay (OPA) formation, due to its potential application as host rock for radioactive waste. For the sedimentary genesis, this rock can be considered a transversely isotropic geomaterial, as also documented in the experimental literature. The fracturing problem is here tackled computationally for a Semi-Circular Bending test (SCB), by resorting to a classical Finite Element Method (FEM) combined with a cohesive zone modeling, or a more advanced eXtended Finite Element Method (XFEM). The selected approaches are successfully compared despite the different computational demand and complex nonlinear nature of the problem. A parametric investigation checks, also, for the possible influence of the notch shape on the fracturing behaviour of the specimen, both in terms of peak load and direction of propagation of the crack within the specimen.

Modeling plasticity of cubic crystals using a nonlocal lattice particle method

In this paper, a novel nonlocal crystal plasticity lattice particle method (CPLPM) is proposed and verified for modeling plasticity of cubic crystals. The proposed CPLPM is fundamentally different from the classical crystal plasticity finite element method, in which the underlying lattice structures are explicitly incorporated in the model formulation and lattice rotation is used to represent the material anisotropy of crystals. As a first attempt to develop a comprehensive CPLPM framework, constitutive relations based on the Schmid’s law and rate-dependent crystal plasticity for infinitesimal deformations are formulated for single crystals of face-centered and body-centered cubic lattice structures. Slip-system-dependent plastic deformation is automatically handled using the underlying crystal topology incorporated in LPM. Numerical solution algorithms for global iteration and plasticity state variables update are discussed in detail. The validity and prediction accuracy of CPLPM are established using several numerical examples, e.g., perfectly plastic deformation of single crystals, plastic slip near crack tip, and the calibration and prediction of experimental results. The performance of the CPLPM algorithm is also discussed considering different viscous properties.