Finite Element Method Formulation Research Articles

Solution of two-dimensional diffusion–advection problems for non-isotropic media with spatially variable velocity field by the boundary element method

This work presents a boundary element method formulation for the solution of the diffusion–advection problem. The formulation, developed for two-dimensional problems, for non-isotropic media, considers a spatially variable velocity field. The only way to deal with such a kind of problem is employing a steady-state fundamental solution. Consequently, the basic BEM equation presents one domain integral related to the velocity components, and another one related to the time derivative of the variable of interest, which is approximated using a backward finite difference scheme. BEM results are compared with an available analytical solution in the first part of the first example, and with the results provided by a finite element method formulation, taken as reference ones, in the second part of the first example and in the second example. From the comparisons, one can observe a good agreement between the results furnished by the proposed formulation and the analytical and reference solutions.

Third formulation of the space-time finite element method

The paper explains differences existing between the space-time finite elements, which fundamentals were presented by the Kączkowski in 1975 and 1976, and its modification. The replacement of load with concentrated forces and moments (which are to satisfy equilibrium conditions at nodes) was given up, namely in the latter one; instead, a requirement was made that functions approximating the solution would satisfy differential equation in inter-node areas in a specified integral way. This leads to a system of algebraic equations, in which kinematic parameters existing in the solution approximating functions are the unknowns. Displacements and their derivatives (of any order) at nodes are those parameters.

Rotor-dynamic performance of porous hydrostatic thrust bearing operating under magnetic field

Purpose The porous bearings are commonly used in slider thrust bearings owing to their self-lubricating properties and cost effectiveness as compared to conventional hydrodynamic bearings. The purpose of this paper is to numerically investigate usefulness of porous layer in hydrostatic thrust bearing operating with magnetic fluid. The effect of magnetic field and permeability has been analysed on steady-state (film pressure, film reaction and lubricant flow rate) and rotor-dynamic (stiffness and damping) parameters of bearing. Design/methodology/approach Finite element approach is used to obtain numerical solution of flow governing equations (Magneto-hydrodynamics Reynolds equation, Darcy law and capillary equation) for computing abovementioned performance indices. Finite element method formulation converts elliptical Reynolds equation into set of algebraic equation that are solved using Gauss–Seidel method. Findings It has been reported that porosity has limited but adverse effects on performance parameters of bearing. The adverse effects of porosity can be minimized by using a circular pocket for achieving better steady-state response and an annular/elliptical pocket, for having better rotor-dynamic response. The use of magnetic fluid is found to be substantially enhancing the fluid film reaction (53%) and damping parameters (55%). Practical implications The present work recommends use of circular pocket for achieving better steady-state performance indices. However, annular and elliptical pockets should be preferred, when design criteria for the bearing are better rotor-dynamic performance. Originality/value This study deals with influence of magnetic fluid, porosity and pocket shape on rotor-dynamic performance of externally pressurized thrust bearing. Peer review The peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-07-2020-0289/

Numerical Simulation of An In-situ Fluid Injection Experiment into a Fault Using Coupled X-FEM Analysis

Fluid injection into a rock mass from industrial processes can cause perceivable seismic events that may raise public concern and affect the mechanical and physical properties of the rock mass. One such notable source of seismicity is from the additional fluid pressure added to the rock mass causing slip on faults. The aim is to provide a method to predict efficiently the fault mechanics due to anthropogenic fluid injection. This was achieved by extending a two-dimensional fully coupled fluid and mechanical loading extended finite element method (X-FEM) formulation via development of a standalone code in Matlab. This code considers fluid flow along the fault as well as into the rock mass and uses a directly proportional equivalent injected flow rate into the fault as the input. This model was validated by comparing the resultant pressure, normal and tangential displacements at the center of the fault to a previously published in-situ experiment. The main results were that the mechanics of the fault could be simulated with sufficient accuracy using this approach given appropriate assumptions, measurements, and simplifications of the fault and rock properties. The main conclusion is that this coupled X-FEM approach may provide an efficient and accurate method to assist in predicting the fault mechanics due to fluid injection. These results are important, since it shows the applicability of X-FEM in predicting the mechanics of the fault and hence the applicability of this method to predict seismicity due to fluid injection.

FEMS – A Mechanics-oriented Finite Element Modeling Software

This paper is a presentation of a Finite Element Modeling Software named FEMS that integrates mesh generation and adaption features in order to alleviate significantly the difficulty of designing a Finite Element (FE) mesh for a particular problem. FEMS is targeted at engineers and scientists addressing localization problems in mechanics, although it should be suited to many other applications.FEMS is particularly relevant for problems with internal interfaces, both in solid and fluid mechanics, as it has both explicit and implicit interface representation. The former can be generated from signed distance functions using body-fitted meshing capabilities implemented in FEMS, while the latter relies on the level-set method. The choice between the one or the other can be made by the user depending on the severity of deformations in the neighborhood of an interface.During the simulation, FEMS adapts the FE mesh automatically to achieve the best accuracy for a prescribed number of nodes. This is possible for both linear and quadratic interpolation. Additionally, in an updated Lagrangian setting, FEMS triggers mesh adaption automatically to avoid element flipping during node motion.The capabilities of FEMS are demonstrated in this paper for fluid and solid mechanics problems featuring turbulence, multiphase flow, large deformations and plasticity. This wide range of problems that can be handled by FEMS should prove its great interest for the computational mechanics community. Program summaryProgram Title: FEMSCPC Library link to program files:http://dx.doi.org/10.17632/rgv4hkrxjw.1Licensing provisions: GNU General Public License version 3Programming language: C/C++Nature of problem: Partial differential equations in one, two or three dimensions of space related to computational mechanics and used to model large deformations, nonlinear material behavior, incompressibility, heat transfer, turbulent and/or multiphase flow with surface tension.Solution method: Finite element method, higher-order elements, mixed and variational multiscale formulations, level-set method, error estimators, isotropic and anisotropic unstructured mesh adaption, image meshing (from microscopy or tomography sources).Additional comments including restrictions and unusual features: Shared-memory (OpenMP) parallelism, GPU-accelerated, unstructured mesh adaption to the finite element solution, the software is compatible with many element types but its mesh adaption feature is restricted to triangles/tetrahedra.

An extended finite element method formulation for modeling multi-phase boundary interactions in steady state heat conduction problems

The present paper proposes an XFEM formulation for heat transfer analysis of multi-phase materials with explicit treatment of boundary interactions. The existence of interfacial resistance at the boundaries of the material phases produces discontinuities in the temperature field and a standard finite element treatment would require complex domain discretizations and additional surface elements to capture the jumps of the temperature values. To overcome this problem, the proposed method captures these jumps by enriching the temperature field around the phase boundaries with appropriate discontinuous functions. Specifically, a new XFEM enrichment scheme is developed to address the issue of multiple-phase junctions, that is, areas where multiple boundaries with different properties intersect. This approach offers the advantage of bypassing the need for complex meshes required by the standard FE method and thus it significantly simplifies the analysis procedure. The elaborated methodology is first validated with existing results from the literature on heat conduction in polycrystalline materials. Then, a detailed model for heat conduction analysis of polymers reinforced with carbon-nanotubes is introduced, which takes into account the role of the interfacial resistance between different material phases. Even though the proposed method is demonstrated in heat conduction problems, it can be straightforwardly extended to other similar problem types, such us electrical conduction or equivalent mechanical properties.

Mixed finite element formulation based on the discontinuous stress approximation

The paper is devoted to the development of the mixed form of finite element method for the calculation of constructions and structures. We consider the approach based on the approximation of discontinuous stress fields and the use of the penalty function method to satisfy the equilibrium equations. It is shown that the continuity of both normal and tangential stresses only on the adjacent sides of the finite elements contributes to the expansion of the class of statically admissible stress fields. At the same time, the consistent approximation is provided, both of the main part of the functional of additional energy, and its penalty part. Using the given approach, solutions of some problems were obtained and analyzed, in particular, the calculation of the cantilever beam-wall. The results obtained were compared with the decisions on the FEM in displacements and stresses, as well as with the exact solutions. It is shown that solutions based on the approximation of discontinuous stress fields in mixed form FEM converge to exact values from below and are more flexible in comparison with the solutions obtained by FEM in displacements. In addition, the accuracy of determining stresses is higher here.

A Comparative Analysis of Neutron Transport Calculations Based on Variational Formulation and Finite Element Approaches

The application of continuous and discontinuous approaches of the finite element method (FEM) to the neutron transport equation (NTE) has been investigated. A comparative algorithm for analyzing the capability of various types of numerical solutions to the NTE based on variational formulation and discontinuous finite element method (DFEM) has been developed. The developed module is coupled to the program discontinuous finite element method for neutron (DISFENT). Each variational principle (VP) is applied to an example with drastic changes in the distribution of neutron flux density, and the obtained results of the continuous and discontinuous finite element (DFE) have been compared. The comparison between the level of accuracy of each approach using new module of DISFENT program has been performed based on the fine mesh solutions of the multi-PN (MPN) approximation. The obtained results of conjoint principles (CPs) have been demonstrated to be very accurate in comparison to other VPs. The reduction in the number of required meshes for solving the problem is considered as the main advantage of this principle. Finally, the spatial additivity to the context of the spherical harmonics has been implemented to the CP, to avoid from computational error accumulation.

Different interface delamination effects on laminated composite plate structure under free vibration analysis based on classical laminated plate theory

This paper presents a dynamic characterization of the delaminated composite plate structure. The governing Equation of the delaminated composite plate developed using finite element method (FEM) formulation grounded on a classical laminated plate theory. The FEM formulation designed for the delaminated composite plate under bending validated by comparing the natural frequencies evaluated using the present FEM formulation through MATLAB coding and experimental results. Various parametric studies have investigated. Simulation study conducted for different interfaces with different delamination location problems solved by the FEM. It shows that percentage of natural frequency decrease in the delaminated composite plate when compared to the intact composite plate. The significant variation in the natural frequency depends on the delamination location and delamination interface. This study is useful for the designers to tailor the structures with various interface delamination present in the arrangements. Artificial neural network algorithm implemented to study the effect of delamination in laminated composite plate structures

Dynamic response of an embedded flexible foundation of general shape in a transversely isotropic and multilayered half-space

A new semi-analytical method is developed for solving the dynamic response of an embedded flexible foundation of general shape in a transversely isotropic and multilayered half-space. It is based on the hybrid Green's functions (GFs) and finite-element method (FEM) via the modified subtraction method. The GFs in the multilayered half-space are derived by virtue of the dual variable and position (DVP) method whilst the FEM is utilized to discretize the domain of the embedded foundation. Making use of the modified subtraction method, the GFs and FEM formulations are combined to derive the dynamic impedance matrix of the embedded foundation. After verifying the accuracy and stability of the proposed algorithm, numerical studies are carried out to investigate the effect of the foundation modulus, half-space anisotropy, and layering on the dynamic impedance matrix of the embedded foundation. Results show that these factors could significantly affect the dynamic impedance matrix of the foundation, which could provide reference for practical engineering design.

Fully-coupled hydro-mechanical cracking using XFEM in 3D for application to complex flow in discontinuities including drainage system

The presence of uplift pressures in cracked plain concrete hydraulic structures is a major concern for their durability, serviceability and stability. To assess the performance of cracked structures several mechanical and hydraulic response parameters must be computed. This paper presents the development, implementation and application of a new nonlinear combined segregated fully-coupled hydromechanical model for application to non-planar 3D hydraulic fractures where complex flow is occurring. The eXtended Finite Element Method (XFEM) formulation is used, as it facilitates the computation of the crack aperture as well as the application of water pressure on crack surfaces for the simulation of hydraulic fracture initiation and propagation. To take into consideration the effects of drainage in numerical modeling and simulation, two coupled (multi-physics) finite element models are used; (1) one for the uplift pressures where a hydraulic mesh with refinement around drains is used, and (2) one for the mechanical response coupled with the computation of the unsaturated interstitial seepage problem. The two subproblems are solved using a partitioned procedure, as they have different resolution requirements. Finally, the mechanical mesh and the hydraulic mesh have non-matching discrete interfaces that must be coupled while respecting the equilibrium of the applied loads. Applications of the proposed hydromechanical constitutive model and numerical solution strategy to different problems are presented on a wedge splitting test and a full scale gravity dam including multiple drainage configurations. A case study adapted from a real dam is also given with a complex 3D non-planar discontinuity surface.

FEM-Based Mechanics Modeling of Bio-Inspired Compliant Mechanisms for Medical Applications

Compliant mechanisms are widely used in the design of medical robotics and devices because of their monolithic structure and high flexibility. Many compliant mechanisms derive their design ideas from nature, since the structure of biological organisms sometimes offers a better solution than the conventional mechanisms. However, the bio-inspired structures usually have very complex geometries which cannot be easily modeled and analyzed using traditional methods. In this paper, we present a novel finite element method (FEM) based modeling framework in MATLAB to analyze the mechanics of different bio-inspired compliant mechanisms. Since the basic linear FEM formulation can only be employed to model small displacements of compliant mechanisms, a non-linear FEM formulation that integrates the modeling of large displacements, tendon-driven mechanisms and contact problems was implemented in the proposed framework to overcome the limitations. Simulations and experiments were also conducted to evaluate the performance of the modeling framework. Results have demonstrated the accuracy and plausibility of the proposed non-linear FEM formulation. Furthermore, the proposed framework can also be used to achieve structural optimization of bio-inspired compliant mechanisms.

A high-order FEM formulation for free and forced vibration analysis of a nonlocal nonlinear graded Timoshenko nanobeam based on the weak form quadrature element method

The purpose of this paper is to provide a high-order finite element method (FEM) formulation of nonlocal nonlinear nonlocal graded Timoshenko based on the weak form quadrature element method (WQEM). This formulation offers the advantages and flexibility of the FEM without its limiting low-order accuracy. The nanobeam theory accounts for the von Kármán geometric nonlinearity in addition to Eringen’s nonlocal constitutive models. For the sake of generality, a nonlinear foundation is included in the formulation. The proposed formulation generates high-order derivative terms that cannot be accounted for using regular first- or second-order interpolation functions. Hamilton’s principle is used to derive the variational statement which is discretized using WQEM. The results of a WQEM free vibration study are assessed using data obtained from a similar problem solved by the differential quadrature method (DQM). The study shows that WQEM can offer the same accuracy as DQM with a reduced computational cost. Currently the literature describes a small number of high-order numerical forced vibration problems, the majority of which are limited to DQM. To obtain forced vibration solutions using WQEM, the authors propose two different methods to obtain frequency response curves. The obtained results indicate that the frequency response curves generated by either method closely match their DQM counterparts obtained from the literature, and this is despite the low mesh density used for the WQEM systems.

An efficient inelastic approach using SCM/RPHM coupling to study reinforced concrete beams, columns and frames under fire conditions

This work has as its main objective the study of the behavior of reinforced concrete beams, columns and structural frames in a fire situation. To do so an efficient numerical formulation was developed, implemented and evaluated. When exposed to high temperatures, the characteristics of the materials deteriorate, resulting in a considerable loss of strength and stiffness of the structure. The CS-ASA (Computational System for Advanced Structural Analysis) was used to achieve the objective. This computer system was expanded for advanced analysis of structures in fire conditions, taking advantage of the existing features and adding new ones. Two new computational modules were created: CS-ASA/FA (Fire Analysis) and CS-ASA/FSA (Structural Fire Analysis). The first one was used to determine the temperature field in the structural elements’ cross-section through thermal analysis by the Finite Element Method (FEM) in permanent and transient regimes. The second was created to perform the second-order inelastic analysis of structures under fire using the FEM formulations based on the Refined Plastic Hinge Method (RPHM) and the Strain Compatibility Method (SCM) coupling, which can be considered a unique feature of the present study. The use of SCM allows for a more realistic analysis against the design codes prescriptions. Consequently, even under high temperatures, SCM is used for evaluation of both bearing capacity and stiffness parameters. The results of the nonlinear analysis in a fire situation for eight structural elements and systems with different geometries, boundary, heating and loading conditions are in good agreement with the numerical and experimental results found in the literature.

Axisymmetric Scaled Boundary Finite Element Formulation for Wave Propagation in Unbounded Layered Media

Wave propagation in unbounded layered media with a new formulation of Axisymmetric Scaled Boundary Finite Element Method (AXI-SBFEM) is derived. Dividing the general three-dimensional unbounded domain into a number of independent two-dimensional ones, the problem could be solved by a significant reduction in required storage and computational time. The equations of the corresponding Axisymmetric Scaled Boundary Finite Element (AXI-SBFE) are derived in detail. For an arbitrary excitation frequency, the dynamic stiffness could be solved by a numerical integration method. The dynamic response of layered unbounded media has been verified with the literature. Numerical examples indicate the applicability and high accuracy of the new method.

A coupled thermomechanical crystal plasticity model applied to Quenched and Partitioned steel

Quenched and Partitioned (QP) steel is a class of Advanced High Strength Steel (AHSS) that exhibits high strength and good ductility. These desirable properties are a consequence of the Transformation Induced Plasticity (TRIP) effect where retained austenite transforms into martensite under applied stress and strain. Accurate microstructural modeling of these advanced deformation mechanisms is critical for automakers to exploit QP steels for automotive weight reduction applications. In this work, the stress and martensitic transformation response of a QP1180 steel alloy are characterized using in-situ High Energy X-Ray Diffraction (HEXRD) uniaxial tension experiments. The initial microstructure is characterized using Electron Backscatter Diffraction (EBSD) and Scanning Electron Microscope (SEM) data. A novel thermodynamically consistent rate-dependent crystal plasticity formulation is used to simulate the large deformation behavior of QP steels that exhibit the TRIP effect. The TRIP effect is captured through a martensite variant selection and evolution scheme, which is governed by a driving force that accounts for various effects (i.e., temperature, orientation, hydrostatic pressure, and martensite surface energy). The constitutive model is implemented into a thermo-mechanical Crystal Plasticity (CP) Finite Element Method (FEM) formulation to study the effect of texture, phase morphology and temperature on the bulk material properties of QP1180 steel. A new method for incorporating thermal boundary conditions in CPFEM models is proposed. The constitutive model is calibrated using experimental stress-strain and martensite evolution measurements. Significant variance in transformation rate was observed, consistent with preferential transformation in the ⟨100⟩ crystal direction. Numerical experiments are conducted to study the effect of thermal and mechanical boundary conditions. Significant differences are observed between the simulations of interrupted and non-interrupted uniaxial tension as a result of the differences in the evolution of temperature. Additional numerical experiments were conducted that demonstrate the fidelity of the constitutive model for elevated temperatures and strain-rates, for several thermal boundary conditions. These simulations showed that the predicted RA and temperature evolution behaviors under quasi-static deformation were not accurately captured, either with adiabatic or isothermal boundary conditions. Finally, simulations showed that the RA and temperature evolution behavior under moderate strain-rates are accurately captured by adiabatic boundary conditions.

Evaluation of stress distributions in the geometrical nonlinear regime of functionally graded structures

In this study, stresses distributions in the large deflection field of Functionally Graded (FG) structures are investigated. Numerical simulations are carried out using one-dimensional (1D) models combining the Finite Element Method (FEM) and Carrera Unified Formulation (CUF) to evaluate the three-dimensional (3D) displacement field over the cross-section, employing Lagrange and Taylor polynomials. Then, the geometrical nonlinear relations are solved with the Newton-Raphson linearization and constrained with an arc-length method. The nonlinear behavior of cantilever beams undergoing bending and compression loading conditions is calculated. The results show the influence of the adopted theory to approximate the displacement field over the cross-section on FG beams. According to the results of this study, it is emphasized that geometrical nonlinearities should be taken into consideration in the evaluation of stress distributions in FG structures.

The effect of finite electrical conductivity of small-scale beam resonators on their vibrational response under electrostatic fields

Electrostatic actuation is one of the most commonly used methods for excitation and measurement in micro and nanoscale resonators. In the dynamical behavior analyses of such systems, the resonating beam is often assumed to be a perfect conductor. In this paper, the effect of electrical resistivity on the vibrational response of these systems, including the natural frequency and damping, is investigated. The governing coupled nonlinear partial differential equations of motion are derived and a new finite element method formulation is presented by developing a new electromechanical element. The numerical natural frequencies are compared with experimental measurements and the achieved correlation is better than that in the prior studies. Results indicate that there is a jump in the frequency and damping of the system at a critical resistivity. As the system size decreases and the applied voltage approaches the pull-in voltage, the electrical resistivity completely dominates the response nature of the system. An experiment is also conducted, and good agreement with the theory is observed regarding the effect of electrical resistivity.

NURBS-based non-periodic finite element framework for Kohn-Sham density functional theory calculations

A real-space non-periodic computational framework is developed for Kohn-Sham density functional theory (DFT). The electronic structure calculation framework is based on the finite element method (FEM) where the underlying basis is chosen as non-uniform rational B-splines (NURBS) which display continuous higher-order derivatives. The framework is formulated within a unified presentation that can simultaneously address both all-electron and pseudopotential settings in radial and three-dimensional cases. The canonical Kohn-Sham equation and the Poisson equation are discretized on different meshes in order to ensure that the underlying variational structural of Kohn-Sham DFT is preserved within the weak formulation of FEM. The discrete generalized eigenvalue problem emanating from the Kohn-Sham equation is solved efficiently based on the Chebyshev-filtered subspace iteration method. Numerical investigations in the radial case demonstrate all-electron and local pseudopotential capabilities on single atoms. In the three-dimensional case, all-electron and nonlocal pseudopotential computations on single atoms and small molecules are followed by local and nonlocal pseudopotential studies on larger systems. At all stages, special care is taken to demonstrate optimal convergence rates towards the ground state energy with chemical accuracy. Comparisons with classical Lagrange basis sets indicate the significantly higher per-degree-of-freedom accuracy displayed by NURBS. Specifically, cubic NURBS discretizations can offer a faster route to a prescribed accuracy than even sixth-order Lagrange discretizations on comparable meshes, thereby indicating considerable efficiency gains which are possible with these higher-order basis sets within effective numerical implementations.

A category theoretical interpretation of discretization in Galerkin finite element method

The Galerkin finite element method (FEM) is used widely in finding approximative solutions to field problems in engineering and natural sciences. When utilizing FEM, the field problem is said to be discretized. In this paper, we interpret discretization in FEM through category theory, unifying the concept of discreteness in FEM with that of discreteness in other fields of mathematics, such as topology. This reveals structural properties encoded in this concept: we propose that discretization is a dagger mono with a discrete domain in the category of Hilbert spaces made concrete over the category of vector spaces. Moreover, we discuss parallel decomposability of discretization, and through examples, connect it to different FEM formulations and choices of basis functions.