Finite Element Method Solution Research Articles

The enriched modified local green's function method applied to elasto static problems

The Modified Local Green's Function Method (MLGFM) has been applied to solve a large number of solid mechanics problems. It employs the finite element method (FEM) as an auxiliary tool to automatically develop the Green's functions projections (GFp) without knowing the fundamental solutions using conventional Lagrange shape functions to approximate the solution space. Also, improvements have been made in Finite Element Methods aiming to develop approximate solution capabilities of given problems by enriching their solution spaces. A good example of that is the hierarchical finite element approach. In this paper, the Modified Local Green's Function Method is enriched using a hierarchical approach based on Lobatto shape functions. Some standard set of elasto static problems are employed and compared with the MLFGM, classic FEM and analytical solutions.

Prediction of 3D grinding temperature field based on meshless method considering infinite element

A three-dimensional numerical model to calculate the grinding temperature field distribution is presented. The finite block method, which is developed from meshless method, is used to deal with the stationary and the transient heat conduction problems in this paper. The influences of workpiece feed velocity, cooling coefficient, and the depth of cut on temperature distribution are considered. The model with temperature-dependent thermal conductivity and specific heat is presented. The Lagrange partial differential matrix from the heat transfer governing equation is obtained by using Lagrange series and mapping technique. The grinding wheel-workpiece contact area is assumed as a moving distributed square heat source. The Laplace transformation method and Durbin’s inverse technique are employed in the transient heat conduction analysis. The results of the developed model are compared with others’ finite element method solutions and analytical solutions where a good agreement is demonstrated. And the finite block method was proved a better convergence and accuracy than finite element method by comparing the ABAQUS results. In addition, the three-dimensional infinite element is introduced to perform the thermal analysis, and there is a great of advantages in the simulation of large boundary problems.

Conjugate Heat Transfer Methodology for Thermal Design and Verification of Gas Turbine Cooled Components

Gas turbine design has been characterized over the years by a continuous increase of the maximum cycle temperature, justified by a corresponding increase of cycle efficiency and power output. In such way, turbine components heat load management has become a compulsory activity, and then, a reliable procedure to evaluate the blades and vanes metal temperatures is, nowadays, a crucial aspect for a safe components design. In the framework of the design and validation process of high pressure turbine cooled components of the BHGE NovaLTTM 16 gas turbine, a decoupled methodology for conjugate heat transfer prediction has been applied and validated against measurement data. The procedure consists of a conjugate heat transfer analysis in which the internal cooling system (for both airfoils and platforms) is modeled by an in-house one-dimensional thermo-fluid network solver, the external heat loads and pressure distribution are evaluated through 3D computational fluid dynamics (CFD) analysis and the heat conduction in the solid is carried out through a 3D finite element method (FEM) solution. Film cooling effect has been treated by means of a dedicated CFD analysis, implementing a source term approach. Predicted metal temperatures are finally compared with measurements from an extensive test campaign of the engine in order to validate the presented procedure.

Conductive Layer Modeling by Improved Second-Order Artificial Material Single-Layer Method

An improved second-order artificial material single-layer (AMSL) method is proposed to predict the electromagnetic field in the presence of conductive thin layers by the finite-element method (FEM). The AMSL method is based on the replacement of the material physical constants of a conductive shield region with those of an artificial material. The new AMSL physical constants are analytically extracted by equating the equivalent transmission line equations governing the field propagation inside the shield with the FEM solution. This new formulation of the method is a significant improvement of the original AMSL as it is much more robust and accurate for near field solutions, and, above all, it is much more easily implementable in commercial software tools.

A mesh‐independent method for planar three‐dimensional crack growth in finite domains

SummaryThe discrete crack mechanics (DCM) method is a dislocation‐based crack modeling technique where cracks are constructed using Volterra dislocation loops. The method allows for the natural introduction of displacement discontinuities, avoiding numerically expensive techniques. Mesh dependence in existing computational modeling of crack growth is eliminated by utilizing a superposition procedure. The elastic field of cracks in finite bodies is separated into two parts: the infinite‐medium solution of discrete dislocations and an finite element method solution of a correction problem that satisfies external boundary conditions. In the DCM, a crack is represented by a dislocation array with a fixed outer loop determining the crack tip position encompassing additional concentric loops free to expand or contract. Solving for the equilibrium positions of the inner loops gives the crack shape and stress field. The equation of motion governing the crack tip is developed for quasi‐static growth problems. Convergence and accuracy of the DCM method are verified with two‐ and three‐dimensional problems with well‐known solutions. Crack growth is simulated under load and displacement (rotation) control. In the latter case, a semicircular surface crack in a bent prismatic beam is shown to change shape as it propagates inward, stopping as the imposed rotation is accommodated.

Continuous constitutive model for bimodulus materials with meshless approach

One of the difficulties in dealing with bimodular materials including composites, rock and asphalt-mixture material is the discontinuity of Young's modulus when the principal stress changes sign, i.e. from a tensile stress state to a compressive stress state. According to the general elastic theory proposed by Ambartsumyan, there are two kinds of domains in which the coefficients of elasticity are constant. The discontinuity of Young's modulus causes divergence in the computational procedure. In order to overcome this difficulty, two continuous modes for bimodular materials are proposed in this paper. The non-linear equilibrium equations have been formulated with a continuous constitute equation of stress and strain. The meshless finite block method is successful in solving the nonlinear problems for bimodular materials. The numerical solutions of the meshless finite block method in a strong form are obtained using an iterative technique. The degree of accuracy and convergence of the proposed technique is demonstrated by directly comparing the achieved results with the finite element method and analytical solutions.

A posteriori error estimation and adaptivity based on VMS for the Stokes problem

SummaryIn this paper, we present residual‐based error estimators applied to the Stokes problem. Implicit and explicit error estimators are developed to predict the error of the numerical solution obtained by a stabilized finite element formulation. Both error estimators arise from the variational multiscale framework, in which the variational form is split into coarse scales (finite element method solution) and fine scales (committed error). This leads to a local problem set on each element in which the error and the residuals are involved. The different ways of linking the error to the residuals produce two error estimators. Local and global error estimates measured in the H1‐seminorm are provided for triangles and quadrilaterals. As an application, using the local error estimates, an adaptive mesh refinement strategy is implemented in order to optimize the computational resources.

THE FINITE ELEMENT METHOD (FEM): AN APPLICATION TO FLUID MECHANICS AND HEAT TRANSFER

In this paper the finite element method (FEM) is used to solve three problems that are of the paramount importance in Chemical Engineering. The first problem is related with the bidimensional flow of an ideal fluid around a cylindrical body, and the objective is to determine the velocity distribution of the flow. To model the flow, the potential formulation is used to obtain an analytical solution, and then, the approximated solution obtained by using FEM is compared with the analytical solution. From this comparison, it is deduced that both solutions have a good agreement. The second problem is the calculation of the temperature profile in a two-dimensional body with specified boundary conditions. This problem is modeled by the two-dimensional Laplace equation, and from the problem data and using variables separation, an analytical solution was obtained. Then, FEM was used to obtain an approximate solution and compared with analytical ones. Besides, from this comparison, it is concluded that both solutions agree. Finally, in the third problem the temperature distribution in a bidimensional body with internal heat generation is studied. This problem is modeled by Poisson equation in two dimensions, but due to the boundary conditions and the complications that arise by adding some heat sources in the final FEM discretization, the problem does not have an analytical solution. However, the analysis of FEM solution indicates that this solution is correct.

Finite-Element Method Solution of Non-Fickian Transport in Porous Media: The CTRW-FEM Package.

An exposition is given of a finite-element method (FEM) software package to calculate solutions for the continuous time random walk (CTRW) integro-differential equation for non-Fickian (and Fickian) conservative or reactive transport in disordered media. The solutions encompass one-dimensional/two-dimensional (1D/2D) breakthrough curves and spatial concentration profiles for general geometry and grid. The velocity field, used as input to the 2D solutions, may also be calculated by applying a compatible Darcy flow 2D solver. The software enables both "forward" modeling (1D/2D) and "inverse" (best-fit) modeling (1D) of experimental data. Various inlet and outlet boundary conditions are implemented. The CTRW-FEM Package is freely downloadable and contains a User's Guide, source and executable files, and accompanying files with easy-to-use format, including descriptions of input/output files and pre- and post-processing. Several examples are provided to clearly demonstrate how to work with the CTRW-FEM Package in many typical analyses. The CTRW-FEM Package also allows users to adapt the software to specific needs, such as, for example, first- and second-order chemical reactions.

Explicit Vibration Formulas Applicable to Timber Plates with Free and Simply Supported Edges

Advances in research and development have led to the application of high-strength timber plates to flooring-slab construction. It is envisaged that floors built from timber plates will have different vibration behavior from conventional joisted floors. In this paper, explicit formulas are developed based on a matrix formulation that links both the symmetric and antisymmetric vibration modes of orthotropic plates. The developed solution can be used for the dynamic analyses of orthotropic timber floor slabs. The solution presented in this paper overcomes limitations associated with previous solutions by eliminating the need for approximate series expansions for solving the characteristic equations. Its application is demonstrated by considering responses of plates with all or two simply supported edges. The new solutions agree closely with classical and Finite-Element Method (FEM) solutions, as well as with the experimental results for Cross-Laminated Timber (CLT) plates. The advantage of the new approach versus alternatives is a combination of accuracy and computational efficiency.

Analytical and FEM solutions for free vibration of joined cross-ply laminated thick conical shells using shear deformation theory

In this study, the free vibration analysis of two joined laminated conical shells is investigated. Five equilibrium equations for each conical shell have been derived in a particular coordinate system; using Hamilton’s principle and first-order shear deformation theorem. The analytical solutions are obtained in the form of power series based on separation of variables method. The boundary conditions at both ends of the joined shells and the continuity conditions, at the conical shells contact, are extracted from energy formulations. As a result, the non-dimensional natural frequencies are studied for cross-ply laminated shells. The effects of semi-vertex angle, circumferential modes, number of layers, thickness, length of shells and different boundary conditions on non-dimensional frequencies are considered. As a comparing result, the non-dimensional frequencies and mode shapes are extracted using finite element method (FEM). The results are compared and verified with the previous available results in other researches. The results reveal a good agreement among analytical solutions, FEM and other results. The output of this paper can be used for analyzing cylindrical–conical shells in addition to joined conical shells.

New Strategy for Finite Element Mesh Generation for Accurate Solutions of Electroencephalography Forward Problems.

The finite element method (FEM) is a numerical method that is often used for solving electroencephalography (EEG) forward problems involving realistic head models. In this study, FEM solutions obtained using three different mesh structures, namely coarse, densely refined, and adaptively refined meshes, are compared. The simulation results showed that the accuracy of FEM solutions could be significantly enhanced by adding a small number of elements around regions with large estimated errors. Moreover, it was demonstrated that the adaptively refined regions were always near the current dipole sources, suggesting that selectively generating additional elements around the cortical surface might be a new promising strategy for more efficient FEM-based EEG forward analysis.

Thermoelastic analysis of temperature-dependent functionally graded rectangular plates using finite element and finite difference methods

ABSTRACTTwo-dimensional thermoelastic analyses of fully clamped functionally graded rectangular plates (FGRPs) by applying a constant in-plane heat flux from one edge were studied by using Finite Difference Method (FDM) and Finite Element Method (FEM) with temperature-dependent material properties. Some differences about 1.5 to 2 times between FDM and the FEM solutions in terms of normal strain, equivalent strain, shear stress, and equivalent stress levels are observed. However, similar distribution characteristics from FDM and FEM analyses are obtained for temperature, displacement, strain, and stress components.

Some considerations regarding the thermal field at welded joints

Mathematical modelling and finite element analysis of the thermal processes, welding of complex metals in terms of chemical composition and structure, allowed to investigate and deepen the heat transfer phenomena and the establishment of a new technological variant of welding, The thermal field, in welding, is depending on the three modes of heat transfer: conductive, convective, manifested by heat losses in the atmosphere and the movement of the fluid in the molten metal bath and by radiation. For the connection of different materials thermo-physical it is proposed that the welding is achieved by moving the thermal source, along the joint, closer to the material characterized by higher thermal conductivity values. The mathematical model and finite element method solution is a modern and modern solution for the optimization of welding technologies, correlating the power developed by the electric arc and the welding speed, so that the final results are as close as possible to the temperature values and dimensions of the welding bath default.

Effect of Nonstructural Masses on Civil Structures by CUF-Based Finite Element Models

This paper proposes an application of already established higher-order models and, namely, investigates the free vibration analysis of civil engineering structures subjected to nonstructural masses. The refined one-dimensional theories adopted are based on the Carrera unified formulation (CUF). The stiffness and mass matrices are obtained by means of the principle of virtual displacements in conjunction with the finite element method (FEM). According to CUF, in fact, their formulation is performed in terms of fundamental nuclei, which depend neither on the adopted class of beam theory nor on the FEM approximation along the axis. In order to account for nonstructural localised masses, the fundamental nucleus of the mass matrix has been opportunely modified. In this work, Lagrange polynomials have been employed in the framework of CUF to develop pure translational displacement-based refined beam models. The models obtained using this approach are referred, in the literature, to as component wise (CW), which allows to model each structural component as a 1D element. The free vibration analysis has been carried out for different cases of nonstructural mass distribution in typical civil buildings. The results obtained are compared with classical Solid/Shell FEM solutions from the commercial code MSC Nastran. The capabilities of the CW models are demonstrated, since this formulation is able to replicate the 3D solid results with enhanced performances in terms of computational efforts. Moreover, the importance of taking into account the correct distribution of the nonstructural masses is shown to be of fundamental importance in vibration analysis of civil buildings.

Homogenization of carbon/polymer composites with anisotropic distribution of particles and stochastic interface defects

The main objective is to investigate an effect of anisotropic distribution of the reinforcing particles in a cubic representative volume element (RVE) of the carbon–polymer composite including stochastic interphases on its homogenized elastic characteristics. This is done using a probabilistic homogenization technique implemented using a triple approach based on the stochastic perturbation method, Monte Carlo simulation as well as on the semi-analytical approach. On the other hand, the finite element method solution to the uniform deformations of this RVE is carried out in the system ABAQUS. This composite model consists of two neighboring scales–the micro-contact scale relevant to the imperfect interface and the micro-scale—having 27 particles inside a cubic volume of the polymeric matrix. Stochastic interface defects in the form of semi-spheres with Gaussian radius are replaced with the interphase having probabilistically averaged elastic properties, and then such a three-component composite is subjected to computational homogenization on the microscale. The computational experiments described here include FEM error analysis, sensitivity assessment, deterministic results as well as the basic probabilistic moments and coefficients (expectations, deviations, skewness and kurtosis) of all the components of the effective elasticity tensor. They also include quantification of anisotropy of this stiffness tensor using the Zener, Chung–Buessem and the universal anisotropy indexes. A new tensor anisotropy index is proposed that quantifies anisotropy on the basis of all not null tensor coefficients and remains effective also for tensors other than cubic (orthotropic, triclinic and also monoclinic). Some comparison with previous analyses concerning the isotropic case is also included to demonstrate the anisotropy effect as well as the numerical effort to study randomness in composites with anisotropic distribution of reinforcements and inclusions.

The technique of domain superposition to solve piecewise homogeneous elastic problems

The Boundary Element Method, a numerical technique that is not based on domain discretization, faces difficulties to model piecewise homogeneous problems that do not appear in many other applications. Thus, in this paper it is presented an alternative methodology for solution of this kind of problem, previously tested successfully for the Laplace Equation, applied here to static cases of linear elasticity. It is substantially different from the classic sub-region technique, since it is based on the sum of elastic energy retained in each distinct sector. Several examples that include cases with irregular domains are simulated showing the robustness and adequacy of the proposed technique. In the absence of analytical solutions, Finite Element Method solutions are used as reference for error evaluation.

Numerical solution of Richards’ equation of water flow by generalized finite differences

Richards equation is a degenerate elliptic parabolic nonlinear expression which models flow in unsaturated porous media. Due to its importance in engineering, a number of linearization schemes for approximating its solution have been proposed. Among the more efficient are combinations of Newtonian iterations for the spatial discretization using finite elements, and an implicit θ-method for the time integration. However, when the finite element formulation is used, numerical oscillations near the infiltration front are presented. To overcome this problem, this paper presents a novel generalized finite differences scheme and an adaptive step size Crank-Nicolson method, which can be applied for solving Richards’ equation on nonrectangular structured grids. The proposed method is tested on an illustrative numerical example on a road embankment and the results are compared with a finite element method solution.

Integrated calculation method of acoustic radiation and propagation for floating bodies in shallow water.

Based on the three-dimensional sono-elasticity theory for ship structures, the Green's function is incorporated and an integrated calculation method of acoustic radiation and propagation for floating bodies in shallow water considering the sound velocity profile is proposed. The near-field and arbitrary far-field acoustic radiation problem can be efficiently calculated. A numerical example of a rigid sphere is given and the results are compared with the finite element method solution to validate the reliability and demonstrate improvements in efficiency. The method is applied to an elastic capsular shell, verifying the applicability of the method for any elastic floating bodies.

Segmentation of the Globus Pallidus Internus Using Probabilistic Diffusion Tractography for Deep Brain Stimulation Targeting in Parkinson Disease.

Although globus pallidus internus deep brain stimulation is a widely accepted treatment for Parkinson disease, there is persistent variability in outcomes that is not yet fully understood. In this pilot study, we aimed to investigate the potential role of globus pallidus internus segmentation using probabilistic tractography as a supplement to traditional targeting methods. Eleven patients undergoing globus pallidus internus deep brain stimulation were included in this retrospective analysis. Using multidirection diffusion-weighted MR imaging, we performed probabilistic tractography at all individual globus pallidus internus voxels. Each globus pallidus internus voxel was then assigned to the 1 ROI with the greatest number of propagated paths. On the basis of deep brain stimulation programming settings, the volume of tissue activated was generated for each patient using a finite element method solution. For each patient, the volume of tissue activated within each of the 10 segmented globus pallidus internus regions was calculated and examined for association with a change in the Unified Parkinson Disease Rating Scale, Part III score before and after treatment. Increasing volume of tissue activated was most strongly correlated with a change in the Unified Parkinson Disease Rating Scale, Part III score for the primary motor region (Spearman r = 0.74, P = .010), followed by the supplementary motor area/premotor cortex (Spearman r = 0.47, P = .15). In this pilot study, we assessed a novel method of segmentation of the globus pallidus internus based on probabilistic tractography as a supplement to traditional targeting methods. Our results suggest that our method may be an independent predictor of deep brain stimulation outcome, and evaluation of a larger cohort or prospective study is warranted to validate these findings.