Finite Element Method Formulation Research Articles

Application of Finite Element Method to Problems with Mixed Boundary Conditions

In this study, we explore the effectiveness of the Finite Element Method (FEM) employing linear shape functions to address problems governed by Dirichlet, Neumann, and Robin boundary conditions. We use the derived weak formulation of FEM to solve various types of partial differential equations (PDEs) with mixed boundary conditions. Convergence and stability analyses are carried out to evaluate the performance of this approach, and different types of errors, like absolute error, dissipation, dispersion, and total mean square error, are investigated. This method is applied, and both the exact and approximate solutions are tabulated in three distinct cases: a one-dimensional Burgers-Huxley equation with Dirichlet boundary conditions; a diffusionreaction equation with Neumann boundary conditions; and a uniformly propagating shock problem with Robin boundary conditions. Approximate solutions are compared to exact ones through 2D and 3D graphical representations, and tabular data offers a thorough error analysis. Additionally, error maps provide strong evidence for the accuracy of the suggested approach, demonstrating its capacity to precisely and quickly solve challenging problems with a variety of boundary conditions. J. Bangladesh Math. Soc. 44.2 (2024) 047–064

Geometrically nonlinear analysis of planar frame structures with positional finite element method

Problems in structural engineering involving geometric nonlinearity are widely studied from numerical and experimental perspectives. From the numerical modeling point of view, to investigate the resistance mechanisms that develop in structures when subjected to large displacements, a mathematical formulation capable of numerically representing the phenomena that occur during the nonlinear behavior is necessary. This work addresses problems of planar frame structures involving geometric nonlinearity with a formulation available in the literature of the Finite Element Method (FEM) based on positions and generalized vectors as degrees of freedom. In this formulation, nodal positions are used as degrees of freedom of the problem, and the Saint-Venant-Kirchhoff constitutive law for plane stress state is adopted. Unlike the FEM formulation for displacements, the positional formulation adopted requires strategies for connecting non-collinear finite elements, involving penalization techniques to satisfy boundary conditions. As detailed in the literature the connections between non-collinear elements are performed with uniaxial and flexural springs. The strain energies of the springs are calculated based on the nodal positions of the structure, and their contributions to the Hessian matrix and internal force vector of the structure are performed with a penalization technique. To validate the implementation developed the solutions of the positional FEM are compared to analytical and numerical solutions obtained by the finite element software DIANA FEA. In addition, parametric analyses varying the connections stiffness values are carried out to determine minimum stiffness values for the connection springs to allow the representation of the nonlinear equilibrium trajectory of planar frame structures.

A Treatment for Overlapping Fractures in the Context of Mixed Finite Elements for Flow in Fractured Porous Media

Accurately modeling flow in discrete fracture networks (DFNs) can be important for several fields of engineering such as groundwater management, petroleum engineering, and geotechnical engineering where these DFNs can play a key role in the flow patterns in a formation. These fracture networks are often complex and can have fractures in very close proximity, which, in turn, can lead to challenges in creating fitting meshes for finite element analyses. This work proposes a methodology to handle overlapping fractures in the context of simulating flow in fractured porous media using a Mixed Finite Element Method formulation. With this capability, fractures in close proximity can simply occupy the same geometric location in the mesh, which avoids the creation of elements with poor aspect ratio. In this work, the porous media flow is modeled with traditional Darcy's equations and the fractures and their interaction with the porous media are modeled using the Discrete-Fracture-Matrix (DFM) representation. A mixed finite element formulation is adopted to solve the flow problem in both porous media and fractures, which has key features such as local mass conservation and improved velocity approximation. The mesh generation is done using the DFNMesh algorithm, which is a mesh generator for DFNs that can handle overlapping fractures by using pre-defined user settings. The proposed methodology is analyzed using a set of numerical examples, which show the importance of the treatment even for very simple cases and that it can handle overlapping fractures and provide accurate results for the flow in complex fractured porous media problems.

Finite element solution of heat conduction in complex 3D geometries

This paper presents the use of the finite element method (FEM) to solve heat conduction problems in complex 3-dimensional geometries not amenable to analytical solutions. Heat conduction is important across engineering domains, but closed-form solutions only exist for basic shapes. For intricate real-world component geometries, numerical techniques like FEM must be applied. The paper outlines the mathematical formulation of FEM, starting from the heat conduction governing equations. The domain is discretized into a mesh of interconnected finite elements. Element equations are derived and assembled into a global matrix system relating nodal temperatures. Boundary conditions are imposed and the matrix equations solved to find the temperature distribution. An example problem analyzes steady state conduction in an L-shaped block with 90-degree corners and surface convection. Results show FEM can capture localized gradients and discontinuities difficult to model otherwise. Detailed temperature contours provide insight. FEM enables robust thermal simulation of complex 3D geometries with localized effects, expanding analysis capabilities beyond basic analytical shapes. Proper application of FEM is critical for accurate results.

Phase field modeling of ductile fracture with isotropic hardening and radius return method

Phase field model has been investigated for brittle fracture in many static and dynamic scenarios, but its applications to ductile fracture is not as common as brittle fracture, especially implementing in software LS-DYNA with explicit scheme. In this study, an efficient LS-DYNA implementation of the phase field modeling of ductile fracture is presented and both with and without the split of elastic strain energy have been considered for the damage evolution. In more detail, plasticity formulation of ductile material with isotropic hardening is briefly presented first and then the governing equations of the classical phase field model are derived, which gives the displacement-phase coupled problem. For with the split of elastic strain energy, the shear component of elastic strain energy is considered for the damage evolution. The influence of degradation function on stress–strain curve is also investigated by using three kinds of function (polynomial function, algebraic fraction function and sigmoid function), which leads to linear and nonlinear finite element method (FEM) formulation of the phase field model and Newton–Raphson method is used to solve the nonlinear FEM formulation of the phase field model. A tensile bar test shows the influence of critical energy release rate and degradation function on stress–strain curve. Mode Ⅰ failure of three-point bending test, Mode Ⅱ failure of single-edge notched plate and mixed-mode failure of asymmetrical double-notched plate verify the proposed model in this study. From these simulations, with the split of elastic strain energy shows improvements on plastic deformation than without the split of elastic strain energy.

Experimental and numerical assessment of fracture toughness variations in steam boiler components after a long period of operation

Experimental and numerical assessment of fracture toughness variations in steam boiler components after a long period of operation

Modeling gas flow in low-permeability formations: An efficient combination of mixed finite elements and high order time integration schemes

Modeling gas flow in low-permeability formations: An efficient combination of mixed finite elements and high order time integration schemes

A recurrence algorithm of time-dependent analysis for concrete bridges

A recurrence algorithm of time-dependent analysis for concrete bridges

A coupled model of wellbore-formation thermal phenomena and salt creep in offshore wells

A coupled model of wellbore-formation thermal phenomena and salt creep in offshore wells

Is the Jaumann Stress Rate Applicable? — Discussions on Choosing Proper Rate Expressions of a Finite Deformation Constitutive Model

The rate form constitutive expression plays an important role in finite deformation theory and is widely used in the finite element method (FEM) software. However, the choice of objective stress and strain rates in constitutive expressions has caused many controversies. This paper aims to clarify these controversies by distinguishing the related concepts: constitutive behavior, constitutive model and constitutive expression, and discuss the influence of constitutive model choices and constitutive expression choices separately. As a certain constitutive model may correspond to many different rate form constitutive expressions, the conversion relationships among them are derived, which can help clarify many controversies. First, it is proven that three different stress–strain curves in the controversial simple shear example actually correspond to three constitutive models. For each constitutive model, the corresponding different rate form constitutive expressions yield consistent calculation results. We also demonstrate that the applicability of a constitutive model is completely independent of the choice of objective stress rates in the constitutive expressions. Second, we find that for FEM simulation, the element tangent stiffness matrix can serve as an indicator to evaluate the validity of different constitutive expressions and their corresponding FEM formulations. Moreover, even though non-work-conjugate stress-strain rates are adopted, such as the Jaumann stress rate/deformation rate, different rate form constitutive expressions can correctly reflect the constitutive model with the proper tangent modulus tensors. All these conclusions have been verified by examples. For convenience, we recommend using the constitutive expressions in terms of the second Piola–Kirchhoff stress versus Green’s strain tensor and Truesdell stress rate versus deformation rate for experimental measurements and FEM simulations. Of course, other objective stress rates are all applicable and have no effect on the calculation results as long as the correct conversion relationships are used.

Investigation of the Eigenvector of Stochastic Finite Element Methods of Functionally Graded Beams with Random Elastic Modulus

This paper presents a stochastic finite element method to calculate the variation of eigenvalues and eigenvectors of functionally graded beams. The modulus of functionally graded material is assumed to have spatial uncertainty as a one-dimensional random field. The formulation of the stochastic finite element method for the functionally graded beam due to the randomness of the elastic modulus of the beam is given using the first-order perturbation approach. This approach was validated with Monte Carlo simulation in previous studies using spectral representation to generate the random field. The statistics of the beam responses were investigated using the first-order perturbation method for different fluctuations of the elastic modulus. A comparison of the results of the stochastic finite element method with the first-order perturbation approach and the Monte Carlo simulation showed a minimal difference.

Matrix-oriented FEM formulation for reaction-diffusion PDEs on a large class of 2D domains

Matrix-oriented FEM formulation for reaction-diffusion PDEs on a large class of 2D domains

Static and free vibration response of a box-girder bridge using the finite element technique

PurposeIn this study, a finite element model of a box-girder bridge along with the railway sub-track system is developed to predict the static behavior due to different combinations of the Indian railway system and free vibration responses resulting in different natural frequencies and their corresponding mode shapes.Design/methodology/approachThe modeling and evaluation of the bridge and sub-track system were performed using non-closed form finite element method (FEM)-based ANSYS software.FindingsFrom the analysis, the worst possible cases of deformation and stress due to different static load combinations were determined in the static analysis, while different natural frequencies were determined in the free vibrational analysis that can be used for further analysis because of the dynamic effect of the train vehicle.Research limitations/implicationsThe scope of the current investigation is confined to the structure's static and free vibration analysis. However, this study will help the designers obtain relevant information for further analysis of the dynamic behavior of the bridge model.Originality/valueIn static analysis, the maximum deformation of the bridge deck was found to be 10.70E-03m due to load combination 5, whereas the maximum natural frequency for free vibration analysis is found to be 4.7626 Hz.

Modeling and analysis of a modular-structured tendon-driven continuum robot for the three-dimensional end effector coordinate

This article introduces the design and control of a tendon-driven continuum robot (CR) segment with a modular structure consisting of a series of backbone discs connected in series. Thanks to this developed design; The backbone length can be changed by increasing or decreasing the number of discs, the number of segments can be increased thanks to the hollow flexible central backbone, and additional apparatus such as a camera or holder can be sent to the end effector. Multi-part and multi-degree-of-freedom CRs are difficult to control because their kinematics and dynamics are non-linear. To overcome this challenge, the development of a finite element method (FEM) based model is presented. In this framework, a nonlinear FEM formulation was applied to model large displacements and contact problems in CRs with motion control by a tendon driven mechanism. The study was supported by a series of experiments to evaluate the accuracy and performance of the developed model. A Feed-forward Artificial Neural Network (FNN) model was developed using the data obtained from the FEM model. The FNN model predicts the X, Y, Z coordinates of the end effector in the 3D plane, based on the tendon drive distance of the CR. The results demonstrated the accuracy and reliability of the proposed nonlinear FEM formulation. At the same time, the FNN algorithm developed with optimum parameters demonstrated that the X, Y and Z position of the end effector has a high estimation accuracy of R=0.9995 on average, depending on the amount of tendon drive.

Controlled-source electromagnetic modeling using a versatile secondary electric-field formulation and efficient multigrid-based preconditioner

Frequency-domain controlled-source electromagnetic (CSEM) surveying is a widely used geophysical electromagnetic exploration method. To avoid the singularity near the transmitting sources and improve the accuracy of three-dimensional (3D) CSEM forward modeling, we adopt the vector finite element method and secondary electric field formulation combined with hexahedral grids to solve the frequency-domain Maxwell equations. Primary field computation applies the fast Hankel transform using digital linear filters to quickly calculate the electromagnetic field generated by different electromagnetic dipoles or loops in the layered medium. Therefore, our forward modeling algorithm is highly versatile and compatible with various electromagnetic sources and scenarios of CSEM application (e.g., airborne, offshore, or marine). To improve the convergence rate, we use an algebraic multigrid solver based on the established Hiptmair–Xu decomposition. Using direct solvers requires a vast amount of memory, while conventional iterative solvers usually fail or converge only very slowly. Our new iterative scheme does not require the divergence corrections commonly applied in finite difference or finite volume electromagnetic modeling to accelerate the convergence. We implement complete parallelism in our finite element programming to efficiently solve the CSEM forward problems on massively parallel clusters. We use different numerical methods to obtain high-precision comparison results for one-dimensional and typical 3D models to demonstrate our algorithm's effectiveness and correctness. Finally, a public model of 2 million cells is used to simulate the responses of marine CSEM surveying. The proposed new solver shows robust scalability when using nearly 1000 cores on the cluster, demonstrating that our algorithm is particularly suitable for large-scale 3D forward modeling.

Modeling of Magnetoelectric Effects in Composite Structures by FEM–BEM Coupling

In this paper, a coupling of the Finite Element Method (FEM) and the Boundary Element Method (BEM) is used to model the behavior of magnetoelectric effects in composite structures. This coupling of numerical methods makes it possible not to have to consider a free space domain, and thus to use a single mesh for the magnetic, mechanical and electrical problems. This results in a consequent reduction of the number of unknowns which is accompanied by shorter computation times compared to a classical FEM approach. A mixed magnetic vector potential, reduced magnetic scalar potential formulation is used for the magnetic problem, and classical FEM formulations are used for electrical and mechanical problems. The resulting global algebraic system is solved by a block Gauss-Seidel solver.

The solution of the wave-diffusion equation by a caputo derivative-based finite element method formulation

The solution of the wave-diffusion equation by a caputo derivative-based finite element method formulation

Stabilization in 3‐D FEM and solution of the MHD equations

In this study, we consider the numerical solution of convection‐diffusion typed equations defined in 3‐D domain using the finite element method (FEM) with the stabilized version in order to fill the gap in literature on this area in the sense of different applications. The formulation processed is performed step by step. For this purpose, we started with FEM formulation of the 3‐D Laplace equation. Then, the obtained formulation is extended to the convection‐diffusion equation with standard Galerkin FEM. In order to solve the stability problems for the convection dominated case, the most popular stabilized formulation known as the streamline upwind Petrov‐Galerkin (SUPG) type stabilization method is considered. As a further step, obtained stabilized formulation is used in the solutions of the MHD equations by transforming the coupled system of differential equations to the decoupled convection‐diffusion type equations defined on different 3‐D domains. For the comparison purpose, MHD equations are also formulated with finite volume method (FVM). Obtained numerical results are displayed in terms of a table and figures as the 2‐D slices of the 3‐D plots in order to visualize the effect and accuracy of the proposed numerical scheme over the different test problems.

Accelerated nonlinear finite element method for analysis of isotropic hyperelastic materials nonlinear deformations

Accelerated nonlinear finite element method for analysis of isotropic hyperelastic materials nonlinear deformations

An image-based numerical homogenization strategy for the characterization of viscoelastic composites

An image-based strategy to characterize viscoelastic composites is presented. The homogenized relaxation moduli are computed in the direct time domain at independent discrete instants by applying effective constant strain fields to a representative volume element of the heterogeneous material. The relaxation master curves are then adjusted via an optimization problem using the generalized Maxwell model represented by the Prony series. In the proposed strategy, we consider image-based modeling, in which every voxel can be interpreted as an element in the Finite Element Method (FEM) formulation, thus dramatically reducing the time-consuming task of meshing the domain. Here, a matrix-free solver is employed to handle the multiple simulations at each instant, which significantly lowers the memory requirements of the computations. Nevertheless, the procedure admits employment of other sorts of memory-efficient solvers. The proposed strategy enables the evaluation of viscoelastic properties of samples depicted by images with hundreds of millions of voxels using personal computers. In that sense, it can be an interesting option for those working with image acquisition devices. Two- and three-dimensional examples are analyzed to showcase effectiveness and to validate both the formulation and the implementation. The results captured well the anisotropic nature of the studied microstructures and the expected variation in the relaxation behavior given different ratios of elastic to viscoelastic constituents.