Finite Element Method Formulation Research Articles

An isogeometric FE-BE method to investigate fluid–structure interaction effects for an elastic cylindrical shell vibrating near a free surface

In this study, an isogeometric finite element-boundary element (FE-BE) framework is proposed to investigate frequency-dependent hydrodynamic characteristics of elastic structures vibrating at free surface or in the vicinity of free surface. The overall numerical framework consists of two parts. In the first part, in vacuo dynamic characteristics are obtained by an isogeometric finite element method (IGAFEM) formulation. Then, in the second part, frequency-dependent generalized added mass and generalized hydrodynamic damping coefficients are obtained by an isogeometric boundary element method (IGABEM) formulation, assuming that the surrounding fluid is ideal, i.e., inviscid, incompressible and its motion is irrotational. To include free surface effects, the free surface boundary conditions are satisfied with an appropriate Green’s function in the mathematical formulation. To show the applicability of the presented framework, an elastic cylindrical shell which was studied previously in the literature is investigated considering that the structure is partially or fully submerged into fluid of infinite depth. It is shown that the results obtained in this study compare very well with the results available in literature.

Inverse Method for Estimation of Thermophysics Properties of a Bimaterials

The aim of this study is to identify simultaneously the thermal conductivity tensor and the volume heat coefficient of a bimaterial (checkerboard) whose heat conduction obeys Fourier's law. This approach is validated by numerical simulation. The simulated temperature fields are obtained by the direct resolution heat conduction solved numerically by the finite element method formulation. To identify parameters, an inverse method is developed by using the finite element temperature approach (FEU-T) model fitting method based on the Levemberg-Marquardt algorithm. We validated the numerical procedure by using noiseless temperature fields at different time and space steps. The influence of the noise on the temperature fields is also studied and shows the efficiency of the inverse method. The results show that this procedure is not very sensitive to the number of elements (or space steps) and the number of time steps.

A fast matrix-free approach to the high-order control volume finite element method with application to low-Mach flow

A fast matrix-free formulation of the control volume finite element method is presented, requiring much less memory and computational work than previous efforts. The method is implemented and evaluated as a solver for low-Mach flow, including the evaluation of a preconditioning strategy for the pressure Poisson equation. The efficiency and scaling with polynomial order is evaluated on simple turbulent flows of interest, with appropriate solution quality metrics, and compared with a reference node-centered finite volume discretization. For a turbulent channel flow test, we show improvement in computational work for a given accuracy with the high-order scheme. The performance on a GPU accelerated platform is also investigated, with benefit shown for the matrix-free discretization.

Uncertainty analysis in solid mechanics with uniform and triangular distributions using stochastic perturbation-based Finite Element Method

In this paper theoretical formulation and computational implementation of the Stochastic perturbation-based Finite Element Method (SFEM) for uncertainty analysis in solid mechanics with symmetric non-Gaussian input parameters are presented. Theoretical foundations of the method are based on the general order Taylor expansions of all uncertain input parameters and state functions including even orders only. The first four probabilistic characteristics of the structural responses have been derived for symmetrical triangular and uniform probability distributions of random input including probability distribution truncation effect. The Stochastic Finite Element Method implementation has been completed for the displacement version of the FEM using statistically optimized nodal polynomial response bases, and their coefficients are determined using the Least Squares Method using the weighted and non-weighted schemes. Structural responses of several mechanical systems are analyzed using their basic probabilistic characteristics, which have been validated using the probabilistic semi-analytical approach, and also the crude Monte-Carlo simulation. A relatively good coincidence of three probabilistic numerical techniques confirms the applicability of the Stochastic perturbation-based Finite Element Method to study boundary and initial problems in mechanics with uncertainties having uniform and/or triangular probability distributions.

2-D FEM Formulation for Closed Waveguides With Magnetically Biased Graphene Sheets

Graphene is one of the most popular 2-D-like materials, whose properties have been thoroughly studied by the scientific community. Its conductivity is complex, frequency-dependent, and anisotropic when a magnetic bias is present. In this letter, we propose an ad-hoc 2-D finite element method (FEM) formulation that allows for the modal computation of enclosed waveguides containing arbitrary surfaces made of this novel material with or without magnetic bias. The tailored formulation takes advantage of 2-D mixed elements for approximating the vectorial electric field in the waveguide cross-section, using the smallest set needed for the correct modeling of the tensorial conductivity. Different results are shown, comparing them against others obtained through different techniques to validate the proposed formulation, which inherits the power of 2-D FEM for addressing general geometries and the advantages of using well-established numerical eigenvalue computation libraries.

ひずみ勾配塑性論に基づく弾粘塑性動的有限要素解析

One mechanical property of metallic materials that has yet to be clarified is the size dependence of material strength on the micrometer scale. Conventional elasto-viscoplasticity theories do not represent any size dependent behavior predicted by dislocation theory-based gradient crystal plasticity models. This study presents a formulation of dynamic finite element method that incorporates strain gradient plasticity theory, which can take into account the size dependence of mechanical properties, and shows several computational results. We analyze dynamic tensile deformation of a plane strain block with geometrical initial imperfections. It is confirmed that the size dependent property from the strain gradient effect manifests in plastic flow localization behavior, but does not affect the position where diffuse neck occurs.

Passive Safety Solutions on Transit Buses: Experimental and Numerical Analyses

A bus rollover accounted for one of the most severe crashes that would lead to a large number of fatalities and injured occupants. This study intends to investigate the structural behavior of passenger buses submitted to a rollover test in accordance with the United Nations Economic Commission for Europe (UNECE) Regulation No. 66. It aims at establishment of passive safety solutions of the bus. Although this regulation only applies to Vehicles M3 Class II/III, it intends to provide solutions applied on Vehicle M3 Class I, city busses. The rollover simulation is performed by means of an explicit dynamic analysis via Finite Element Method (FEM) formulations validated by experimental data. As a result, the force-displacement curve is characterized and the internal energy released by the studied bus section during loading is obtained and compared to the value recommended by R66 regulation. The outcome of this work contributes to assess the bus structural integrity and propose the passive safety solutions if necessary.

Topology Optimization – unconventional approaches using the Generalized Finite Element Method and the Stable Generalized Finite Element Method

The Structural Optimization process has an increasing importance in industry and academic fields, assisting in the development of designs at the initial stages of a project. Nowadays, the structural optimization methodology can be conducted by Topology Optimization Method (TOM), which is an efficiently combination of the Finite Element Method (FEM) with an optimization algorithm, in order to find the optimized material distribution inside a given design domain subjected to a set of constraints. Application of the FEM in TOM suffers from a series of instability problems, being one of them the checkerboard pattern. This paper investigates the impact of the Generalized Finite Element Method (GFEM) and Stable Generalized Finite Element Method (SGFEM) in the implementation of the TOM algorithm. This work shows that these unconventional FEM formulations are able to solve most of the checkerboard pattern problem when combined with an enriched mesh designed specifically to each example evaluated. Significant improvement in results of the topology optimization is achieved when compared to the conventional formulation of TOM.

Characterization of Edge-Coupled Broadside- Coupled and Complementary Split-Ring Resonator Periodic Media Based on Numerical Solutions of Eigenvalue Problems

We study the behavior of 3-D periodic composite media consisting of several types of split-ring resonators. By initially introducing the H-D periodic eigenmode finite element method (FEM) formulation, the dual counterpart of our formerly proposed E-B formulation, we are able to extract accurate dispersion diagrams and field distributions of electromagnetically complex structures and compare the returned characteristics of the supported modes. Using Babinet’s principle, we design the corresponding complementary resonators and evaluate their dual characteristics. Particular attention is given to the explanation of the behavior of these composite media under wave incidence perpendicular to the scatterers’ plane. Additionally, we perform a cross-check with excitation problem results, to interpret the response predicted by analytical models existing in literature and confirmed by the eigen-analysis of the structures. Useful conclusions are drawn concerning the potential utilization of the studied resonators in more complex structures in the field of microwave applications.

Degeneracy-Discriminating Modal FEM Computation in Higher Order Rotationally Symmetric Waveguides

An accurate and efficient method is proposed to characterize waveguides with arbitrary cross sections exhibiting discrete rotational symmetries, known in the literature as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$C_{N}$ </tex-math></inline-formula> symmetries. This new method is based on a novel tailored 2-D finite element method (FEM) formulation exploiting rotationally periodic boundary conditions involved in the minimum sector that defines the waveguide cross section. While retaining the proven accuracy and convergence rate of FEM, it allows for a reduction in dimensionality of the problem and classifies the modes with respect to their rotational symmetry. Most importantly, it discriminates reliably and unambiguously degenerate modes (no matter how high their order is) with the exact same computed cutoff frequency value. This is especially interesting in modal analysis of devices involving circular polarization, since the formulation inherently provides different mode sets for clockwise and counterclockwise circularly polarized modes. The method has been tested on a wide variety of waveguides and computed higher order modes have been compared against analytical solutions and general-purpose FEM.

An Efficient Mixed Finite Element Perfectly Matched Layer with Optimal Parameters Selection for Two-Dimensional Time Domain Soil-Structure Interaction Analysis

Perfectly matched layer (PML) is known as one of the best methods to simulate infinite domains in many fields such as soil-structure interaction (SSI). The performance of PML is significantly affected by PML parameters selection. However, the way to select PML parameters still remains unclear. This study proposes a method for PML parameters determination for elastic wave propagation in two-dimensional (2D) media. The scaling and attenuation functions are developed in order to increase the accuracy and effectiveness of the PML. The proposed scheme is applied for a mixed PML in time domain. The finite element method (FEM) formulations of the PML are presented so that it can be easily applied to the existing codes. ABAQUS, a popular FEM code, is used for numerical applications in this study. The proposed PML is imported into ABAQUS by using a user-defined element (UEL) written in Fortran language. Six numerical analyses of SSI are implemented to prove the efficiency of the proposed PML. The numerical analyses cover many realistic problems, including free field, surface structure, and embedded structure problems. The results demonstrate the efficiency of the proposed PML in terms of the accuracy and computational cost.

General Consistency of Strong Discontinuity Kinematics in Embedded Finite Element Method (E-FEM) Formulations.

This paper performs an in-depth study of the theoretical basis behind the strong discontinuity methods to improve local fracture simulations using the Embedded Finite Element Method (E-FEM). The process starts from a review of the elemental enhancement functions found in current E-FEM literature, providing the reader a solid context of E-FEM fundamentals. A set of theoretical pathologies is then discussed, which prevent current frameworks from attaining full kinematic consistency and introduce unintended mesh dependencies. Based on this analysis, a new proposal of strong discontinuity enhancement functions is presented considering generalised fracture kinematics in a full tridimensional setting and a more robust definition of internal auxiliary functions. Element-level simulations are performed to compare the outputs within a group of selected E-FEM approaches, including the novel proposal. Simulations show that the new element formulation grants a wider level of basic kinematic coherence between the local fracture outputs and element kinematics, demonstrating an increase in robustness that might drive the usefulness of E-FEM techniques for fracture simulations to a higher level.

Buckling analysis of members restrained by sandwich panels

An analytic method is presented for the analysis of flexural restraint of members by sandwich panels. Using the method, which is based on the solutions of the fourth order differential equations, the restraint effect of sandwich panels can be approximated in practical cases. The reliability of the method is shown based on tests and finite element analyses. New results are shown using the analytic method for buckling cases and for P-δ analysis in the elastic range. The exact finite element method (FEM) formulation is given for more complicated cases.

Advanced Modeling and Simulation of Rockfall Attenuator Barriers Via Partitioned DEM-FEM Coupling

Attenuator barriers, in contrast to conventional safety nets, tend to smoothly guide impacting rocks instead of absorbing large amounts of strain energy arresting them. It has been shown that the rock’s rotation plays an important role in the bearing capacity of these systems. Although experimental tests have to be conducted to gain a detailed insight into the behavior of both the structures and the rock itself, these tests are usually costly, time-consuming, and offer limited generalizability to other structure/environment combinations. Thus, in order to support the engineer’s design decision, reinforce test results and confidently predict barrier performance beyond experimental configurations this work describes an appropriate numerical modeling and simulation method of this coupled problem. For this purpose, the Discrete Element Method (DEM) and the Finite Element Method (FEM) are coupled in an open-source multi-physics code. In order to flexibly model rocks of any shape, sphere clusters are used which employ simple and efficient contact algorithms despite arbitrarily complicated shapes. A general summary of the FEM formulation is presented as well as detailed derivations of finite elements particularly pertinent to rockfall simulations. The presented modeling and coupling method is validated against experimental testing conducted by the company Geobrugg. Good agreement is achieved between the simulated and experimental results, demonstrating the successful practical application of the proposed method.

A Concise and Geometrically Exact Planar Beam Model for Arbitrarily Large Elastic Deformation Dynamics.

The potential of large elastic deformations in control applications, e.g., robotic manipulation, is not yet fully exploited, especially in dynamic contexts. Mainly because essential geometrically exact continuum models are necessary to express these arbitrarily large deformation dynamics, they typically result in a set of nonlinear, coupled, partial differential equations that are unsuited for control applications. Due to this lack of appropriate models, current approaches that try to exploit elastic properties are limited to either small deflection assumptions or quasistatic considerations only. To promote further exploration of this exciting research field of large elastic deflection control, we propose a geometrically exact, but yet concise a beam model for a planar, shear-, and torsion-free case without elongation. The model is derived by reducing the general geometrically exact the 3D Simo–Reissner beam model to this special case, where the assumption of inextensibility allows expressing the couple of planar Cartesian parameters in terms of the curve tangent angle of the beam center line alone. We further elaborate on how the necessary coupling between position-related boundary conditions (i.e., clamped and hinged ends) and the tangent angle parametrization of the beam model can be incorporated in a finite element method formulation and verify all derived expressions by comparison to analytic initial value solutions and an energy analysis of a dynamic simulation result. The presented beam model opens the possibility of designing online feedback control structures for accessing the full potential that elasticity in planar beam dynamics has to offer.

Immersed‐interface finite element method based on a nonconformal Petrov–Galerkin formulation

AbstractImmersed interface methods are becoming increasingly necessary to solve problems of practical importance. To be considered as replacement for conventional, interface‐fitted, finite element method (FEM), an immersed‐interface finite element method (IIFEM) has to exhibit equivalent accuracy and robustness. The paper develops such an IIFEM based on nonconformal Petrov–Galerkin (ncPG) formulation for two‐dimensional linear elasticity. First, a bimaterial patch test (BPT) is developed which, when satisfied, guarantees consistency of an FEM formulation for problems with two materials in a single element. Then, nodal trial functions that satisfy the BPT are derived for triangular elements with immersed interface. These trial functions satisfy the Hadamard interface conditions for elements with immersed interface but are nonconformal on element edges with an immersed interface. Conventional conformal test functions are used in the Petrov–Galerkin formulation. The proposed ncPG‐IIFEM is applied to a benchmark multimaterial problem. The solution using ncPG‐IIFEM converges optimally and the error in displacement as well as stress are similar to interface‐fitted FEM. The ncPG‐IIFEM also leads to a well‐conditioned linear system with the condition number of the global stiffness matrix similar to interface‐fitted FEM. Overall, the proposed ncPG‐IIFEM is shown to be straightforward in application, optimal in convergence, and robust.

C-FLEX Advanced Finite Element Analysis Program for Flexible Pavement Analysis and Design

The U.S. Army Engineer Research and Development Center (ERDC) of the U.S. Army Corps of Engineers has initiated an effort to modernize the Department of Defense (DOD) pavement design and evaluation procedures initially developed in the 1950s. Flexible pavement analyses and evaluations are currently performed based on the elastic layered WESLEA and WESDEF software programs. To modernize the current pavement design and evaluation procedures used by the DOD, an advanced axisymmetric Finite Element Method (FEM) based analysis program, named C-FLEX, was developed and is introduced in this paper. The C-FLEX program is designed to feature accurate material models for all pavement layers with the capability to model the cross-anisotropic and nonlinear elastic properties of unbound base/subbase and subgrade layers, the viscoelastic behavior of the asphalt mixture, as well as mechanical reinforcement using geosynthetics in flexible pavements. The FEM formulation in C-FLEX and the program architecture and implementation details are introduced and discussed in this paper. The different analysis schemes and proper models used to characterize the cross-anisotropy and stress-dependent material nonlinearity are also described in detail. The viscoelastic analysis scheme and the geosynthetic characterization are currently under development and so are not included in this paper. Furthermore, two conventional flexible pavements with different layer properties are analyzed to verify the solutions and reliability of the C-FLEX program. Based on this development effort with ERDC, the C-FLEX program is envisioned to eventually serve as the flexible pavement analysis engine for the DOD’s new mechanistic design and evaluation platform.

Coupling of non‐ordinary state‐based peridynamics and finite element method for fracture propagation in saturated porous media

AbstractIn the present manuscript fracture propagation in a saturated porous medium is modeled based on the classical Biot theory, where solid skeleton and fluid flow are represented by separate two layers. The non‐ordinary state‐based peridynamics (NOSBPD) layer is employed to capture deformation including fracturing of the solid skeleton, while the fluid flow is controlled by the finite element method (FEM) layer. The interaction between the layers is realized by considering the effects of pore pressure from the FEM layer on the NOSBPD layer and, vice versa, the effect of the volumetric strain, porosity, and permeability variations from the NOSBPD layer on the FEM layer. The coupling terms retain their parent characteristics, that is, the interaction term in the momentum balance equation is approximated by the local FEM formulation whereas the interaction term in the mass balance equation is approximated by the nonlocal NOSBPD formulation. By doing so, the model retains the flexibility of coupling two independent discretizations. The coupled system is solved by a fully implicit solution scheme. The accuracy of the proposed method has been verified against available closed‐form solutions and published numerical approaches for the pressure‐ and fluid‐driven facture propagation problems.

On Computational Solution of the Dynamic and Static Behaviour of a Coupled Thermoelastic Timoshenko Beam

The Timoshenko beam theory caters for transverse shear deformations, which are more pronounced in short beams. Previous works were examined, and Hamilton’s principle was used in deriving the governing equation. This research considers two dimensions (2-D): heat and displacement response. A more comprehensive mathematical expression that incorporates this 2-D model on the vibration of a coupled Timoshenko thermoelastic beam and axial deformation effect is formulated. The significance of this model will be expressed through its finite element method (FEM) formulation. The results compared favourably with those of previous works. It was re-established that the amplitude of deflections, as well as cross-sectional rotations, increases considerably as the aspect ratio of the beam decreases. In this way, for larger aspect ratios, the response of the beam is like the quasi-static heating condition. This is expected since the increase in the aspect ratio of the beam reduces its structural stiffness and consequently its natural frequencies. So, the amplitude and temporal period of its vibrations become greater. The beam under the applied thermal loading experiences thermally-induced vibrations. Also, the dynamic solution is substantially influenced by the coupling between strain and temperature fields. The results also reveal that the aspect ratio of the beam could have a significant impact on the vibratory response of the beam. Specifically, it is proportional to the amplitude and temporal period of the thermally-induced vibrations of the beam.

A coupled formulation of finite and boundary element methods for flexoelectric solids

A numerical method for analyzing the mechanical deformation and electric polarization of flexoelectric solids is proposed that considers the electric interactions among flexoelectric solids, electric charges, and their surrounding medium. The proposed method uses finite and boundary elements simultaneously. Finite elements are used to model flexoelectric solids, and boundary elements are applied along the boundary of the solids and an arbitrary surface encompassing the solids and electric charges. To account for the couple stress and flexoelectric terms, a C1 continuous finite element formulation is introduced. Next, a coupled formation between the finite elements and boundary elements is developed. The coupled method provides a converged solution for any system composed of flexoelectric solids and electric charges without prescribing electric boundary conditions on the boundary of solids. Several numerical examples prove that the method is convergent and effective for systems containing multiple solids and electric charges.