Four-node Tetrahedral Elements Research Articles

Modeling the failure process of rock masses using a 3D nodal-based continuous-discontinuous deformation analysis method

A three-dimensional nodal-based continuous-discontinuous deformation analysis method (3D-NCDDAM) is developed in this study for modeling the failure process of rock masses. In the 3D-NCDDAM, four-node tetrahedral elements which can be automatally generated are used to discretize the problem domain. To reduce computational cost, and effectively model the failure process of rock masses at concerned regions, the problem domain is devided into two types of regions, namely, the “non-destructive region” and “possible destructive region”. The non-destructive region is mainly used to reduce the boundary effects on the numerical results, and is regarded transitional zone in which the failure of rock masses is not considered. For the possible destructive region, artificial joint elements are inserted, and the failure of these joint elements is used to represent fracture initiation and propagation in rock masses. Additionally, the contact potential which is initially proposed in finite-discrete element method (FDEM) is incorporated into the 3D-NCDDAM to simuate the opening and slipping of discontinuous structural planes involved in rock masses’ failure process. Note that the difficultly in determing contact types in the traditional 3D discontinuous deformation analysis (3D-DDA) method is avoided in the 3D-NCDDAM. The proposed 3D-NCDDAM can be treated as a hybrid method, which inherite both the advantage of DDA method and FDEM. To validate the 3D-NCDDAM, seven numerical examples are conducted. The numerical results indicte that the 3D-NCDDAM can effectively model the failure process of rock masses in both experimental scale and engineering scale.

An accurate coupled method for analysis of transcranial magneto-acoustic-electrical stimulation

In this paper, a numerical method for simulating transcranial magneto-acoustic-electrical stimulation (TMAES) is presented. The smoothed finite element method (SFEM) is applied to calculate the magnetic field and the Rayleigh integral (RI) is used for solving the ultrasonic field. Three-node triangular and four-node tetrahedral elements are set as the background cells for constructing the smoothing domains (SDs). A gradient smoothing technique (GST) is applied over each SD to obtain the magnetic flux density. The discretized system equations are obtained by using the generalized smoothed Galerkin weakform. To solve the sound pressure, the ultrasonic transducer is discretized into a series of micro-elements. The acoustic pressure in the problem domain is then equal to the superposition of that generated by the individual micro-element. Finally, the current density can be calculated by the magneto-acoustic coupling effect. Numerical examples demonstrate that SFEM can improve accuracy, convergence rate, and efficiency to 10, 1.8, and 4 times those of FEM, respectively. In the calculation of coupled fields, SFEM-RI also improves the accuracy by 2.5 times. Additionally, the presented method possesses the property of insensitivity to mesh distortion.

Thermal Analysis of Concrete Blocks and Stack-Bond Prisms under Different Boundary Conditions

Fire is a significant threat to human lives and the integrity of buildings. To better understand the complex behavior of masonry exposed to high temperatures, thermal analyses were carried out to evaluate the temperature distribution in concrete blocks and stack-bond prisms exposed to high temperature levels. The effects of distinct specimen boundary conditions (restrict or easy access to air circulation inside the voids of the block and prisms) on the thermal response of the masonry materials were investigated. Thersys 2.0 software was used to implement three-dimensional thermal analysis of distinct finite element models. Four-node tetrahedral elements and full integration were used in all models. The modeling approach was validated by experimental data obtained from thermocouples embedded into masonry components. The results indicated that the boundary conditions significantly affected the time required for homogenization of temperature in blocks and prisms. Easy access to air circulation inside the voids of the prisms provided a faster temperature homogenization. In this scenario, the prism reached temperature ranges of (300 ± 0.5% × 300) °C and (600 ± 0.5% × 600) °C after exposure times of 2 h and 2 h 10 min, respectively. When access to air circulation within the voids of the prisms was limited, the same temperature ranges were achieved after exposure times of 5 h 20 min and 6 h, respectively.

A general class of enriched methods for the simplicial linear finite elements

Low-order elements are widely used and preferred for finite element analysis, specifically the three-node triangular and four-node tetrahedral elements, both based on linear polynomials in barycentric coordinates. They are known, however, to under-perform when nearly incompressible materials are involved. The problem may be circumvented by the use of higher degree polynomial elements, but their application become both more complex an computationally expensive. For this reason, non-polynomial enriched finite element methods have been proposed for solving engineering problems. In line with previous researches, the main contribution of this paper is to present a general strategy for enriching the standard simplicial linear finite element by non-polynomial functions. A key role is played by a characterization result, given in terms of the non-vanishing of a certain determinant, which provides necessary and sufficient conditions, on the enrichment functions and functionals, that guarantee the existence of families of such enriched elements. We show that the enriched basis functions admit a closed form representation in terms of enrichment functions and functionals. Finally, we provide concrete examples of admissible enrichment functions and perform some numerical tests.

Vulnerability of Pointed Masonry Barrel Vaults Subjected to Differential Settlement Simulated with a GPGPU-Parallelized FDEM

Pointed masonry barrel vaults are widely used in classical historic structures, such as cathedrals and aqueducts, and they are very sensitive to differential settlement. These vaults are assemblages of masonry units and mortar. Since the bonding strength of mortar degrades over ages, dry-joint assumption is widely accepted. Failure behavior of dry-joint pointed masonry barrel vaults subjected to differential settlement is highly complex, discontinuous, and nonlinear. In this study, a 3D GPGPU-parallelized hybrid finite-discrete element method (FDEM), which is an advanced extension of finite element method (FEM) and discrete element method (DEM), is employed to investigate the capacity of pointed masonry barrel vaults subjected to differential settlement. When modeling barrel vaults with 3D FDEM, each masonry unit is discretized into a couple of four-node tetrahedral elements whose deformability is characterized by standard finite element formulation. Thus, structural deformation and interaction forces can be obtained in an accurate manner. Numerical examples are presented and validated with results from literatures. A base case is selected, and the influence of embrace angle ([Formula: see text], sharpness (Sh), stockiness (St), and out-of-plane length ([Formula: see text] on the failure behavior is parametrically investigated. The larger the [Formula: see text] or Sh, the smaller the ultimate settlement. The same applies to St in general, while an excessively large St results in small ultimate settlement due to sliding. The influence of [Formula: see text] can be mitigated should it is large enough compared with the span. It is demonstrated that the 3D GPGPU-parallelized FDEM is a robust tool for analyzing the vulnerability of pointed masonry barrel vaults subjected to differential settlement.

The strength calculation of the transport aircraft control system element by the finite element method

The work is devoted to determining the stress-strain state of the element of the control system of the transport aircraft, which serves to transmit motion and is an element of the mechanism for deflecting the rudder and removing the load from the control wiring. If this element is destroyed, the control system will not function, as a result of which some elements of the electronic installation of the aircraft will be disabled. Analytical methods are not always effective when studying the stress state of complex parts and mechanisms of aircraft construction, although they are also used. To determine the stress-strain state in the work, a numerical method was used - the finite element method, which models the stress-strain state of mechanical systems with great accuracy, taking into account the boundary conditions and the given load. A four-node tetrahedral element was chosen as a finite element. Based on the analysis of the stress-strain state, dangerous places in the structure were identified, where there is a possibility of a crack appearing and, as a consequence, destruction.

An SFEM Abaqus UEL for Nonlinear Analysis of Solids

In this paper, three different smoothed finite element method (SFEM), viz., node-based smoothed finite element method (NS-FEM), face-based smoothed finite element method (FS-FEM) and [Formula: see text]-finite element method ([Formula: see text]-FEM) are adopted for 3D solids undergoing large deformation. The common feature of all these techniques is the introduction of smoothed strain which is written as a weighted average of the compatible strain field over smoothing domains. The choice of smoothing domain is what differentiates them. The spatial discretization can be based on the simplest and automatically genera-table four-node tetrahedral elements and aforementioned techniques have shown to yield accurate results even on a coarser discretization. To take the advantages of the SFEM, it is beneficial to the FEM community to have it implemented in the widely used Abaqus[Formula: see text] software. Such an implementation is challenging because the neighboring SFEM elements are interconnected in the smoothed strain matrices in the elemental level. In this work, the above-mentioned SFEM models are implemented in the commercial software Abaqus using the softwares’ user element (UEL) feature. The challenges during the definition and the assembly of the smoothing domains are effectively addressed in this work. The developed UEL and the associated files can be downloaded from https://github.com/nsundar/3DSFEM. The implementation is validated against benchmark examples and the robustness is demonstrated with complicated real-life problems, viz., tire patch contact with road and simulation of human thumb.

Variational coupling of non-matching discretizations across finitely deforming fluid-structure interfaces.

This paper presents a stabilized monolithic method for coupling incompressible viscous fluids with finitely deforming elastic solids across non-matching interfacial meshes. Governing equations for the solid are written in the finite deformation Lagrangian frame using velocity field as the primary unknown, while the governing equations for the fluid are written in an Arbitrary Lagrangian-Eulerian (ALE) frame to accommodate large motions of the fluid-solid interfaces. Interface coupling terms are derived by embedding Discontinuous Galerkin (DG) ideas in the Variational Multiscale (VMS) framework and locally resolving the fine-scale variational equations from the fluid and the solid subdomains along the common interface. The unique attribute of the proposed Variational Multiscale Discontinuous Galerkin (VMDG) method is a systematic procedure for deriving analytical expression for the traction Lagrange multiplier at the fluid-solid interface. The structure of the interface stabilization tensor emerges naturally and is shown to be a function of the boundary operators that are associated with the domain interior operators from the fluid and the solid subdomains. The derived stabilization tensor possesses the features of area-averaging and stress-averaging and evolves spatially and temporally with the evolving nonlinear fields at the interface. An analysis of the mathematical properties of the interface stabilization tensor is presented and subsequently verified numerically. The method is implemented using four-node tetrahedral elements for the fluid as well as the solid subdomains while employing equal-order interpolations for the various interacting fields. Benchmark problems are presented for the verification of the method for finitely deforming fluid-structure interfaces.

A three-dimensional hybrid immersed smoothed point interpolation method for fluid-structure interactions

A three-dimensional hybrid immersed smoothed point interpolation method (3D HIS-PIM) has been developed to handle fluid-structure interactions (FSI) with simple geometries. Incompressible viscous fluid is solved by semi-implicit characteristic-based split method with standard Galerkin discretization, smoothed point interpolation method is employed to solve the motion and deformation of nonlinear solids based on gradient smoothing technique, and the virtual fluid is introduced into the entire solid domain to calculate the FSI force. The proposed method can avoid the fluid mesh adjustments, and allows the use of four-node tetrahedral elements for both fluid and solid domains to simplify the pre-processing for mesh generations. The FSI force condition is imposed with a hybrid force, which uses a form of body force for pressure term and boundary force for shear force. The boundary force accords with the physical law and the body force can ensure the numerical stability. The proposed method works well for FSI problems with moving or largely deformable solids, and can achieve more accurate results using a hybrid force than the fully body force in the original method. A large range of mesh size ratios between fluid and solid meshes is allowed due to the treatment of FSI boundary conditions.

Transcranial stimulation analysis using the smoothed finite element method

This paper presents a series of smoothed finite element methods (SFEM) for transcranial stimulation simulations. The problem domain is first discretized into a set of four-node tetrahedral elements, and linear shape functions are employed to interpolate the field variables. Then, the smoothing domains are further formed in conjunction with the nodes, edges or faces of elements. In order to improve the accuracy of low-order interpolation, the magnetic flux density and gradient of electric potential are smoothed using the gradient smoothing technique (GST) over each smoothing domain. Based on the generalized smoothed Galerkin weakform, the discretized system equations are finally obtained. Numerical examples, including transcranial direct current stimulation (tDCS) and transcranial magnetic stimulation (TMS) problems, demonstrate that the SFEM possesses the following important properties: (1) better accuracy; (2) faster convergence; (3) higher computational efficiency; (4) more robust in transcranial stimulation simulations.

Biomechanical performance of short and long cephalomedullary nail constructs for stabilizing different levels of subtrochanteric fracture

IntroductionThe aim of this study was to assess biomechanical performance of short and long Cephalomedullary nail constructs consisting of different number of distal screw for stabilizing different levels of subtrochanteric fracture. Materials and methodsThe femur obtained from computed tomography scanner was used to create a transverse fracture at 15 mm (level A), 35 mm (level B), and 55 mm (level C) below the lesser trochanter. Short and long Cephalomedullary nails were virtually inserted to the fractured femur. Four-node tetrahedral element was used to build up finite element (FE) models for biomechanical analysis. The analysis focused on post-operative stage of partial weight-bearing. ResultsStress on the implant localized at the surface between lag screw/nail and distal screw/nail. Short Cephalomedullary nail exhibited higher stress than long Cephalomedullary nail. The stress in short Cephalomedullary nail could be reduced by using two distal screws fixation and the fracture at level A produced less stress than that of level B and C. Either short or long nail with two distal screws is sufficient to withstand the stress magnitude produced from the physiologic load. When single dynamic distal screw was used, stress on implant, elastic strain at fracture gap, and bone stress reached the high values. Elastic strain of the fracture gap at level C were less than that of level A and B, but no statistically significant difference. There was no proximal cancellous bone damage observed from the FE analysis. ConclusionsLong Cephalomedullary nail with at least two distal locking screws remains a proper implant for subtrochanteric fracture fixation in overall locations. However, short Cephalomedullary nail with two distal screws may be a candidate for a high subtrochanteric fracture. Single dynamic screw insertion is strongly not recommended with either short or long nail regarding implant failure.

A selective smoothed finite element method for 3D explicit dynamic analysis of the human annulus fibrosus with modified composite-based constitutive model

The human annulus fibrosus (HAF) is the major component in response to external forces for the intervertebral disk (IVD), which maintains the stability and flexibility of human spine. It can be assumed to be an anisotropic nearly incompressible hyperelastic composite consisting of collagen fibers and matrix in the numerical simulations of biomechanics. However, due to the geometric complexity and material nonlinearity of HAF, the conventional Finite Element Method (FEM) often gets into difficulties in mesh generation and uncertainty of accuracy control. In this paper, a modified composite-based constitutive model, which considers the slight compressibility of ground substance and the shear interaction between collagen fibers and matrix, is developed to describe the mechanical behavior of HAF. In addition, based on the gradient smoothing techniques, the selective 3D-edge-based and node-based smoothed finite element method (Selective 3D-ES/NS-FEM) is developed to alleviate volume locking and improve the accuracy of linear four-node tetrahedral (TET4) elements. Combined with the modified constitutive model, the Selective 3D-ES/NS-FEM is applied into the explicit dynamic analysis of HAF undergoing large deformation. By comparing with the experiment data in the literatures and the numerical results produced by conventional FEM, the presented approach is proved to possess excellent accuracy and efficiency in predicting the nonlinear mechanical behavior of HAF, as well as the orientation change of the collagen fibers. Moreover, the Selective 3D-ES/NS-FEM is demonstrated to have robust capability in handling element distortion, even with the simplest TET4 mesh. This study is significant to the biomechanical research of HAF, and has potential value for guiding the prevention and treatment of low back pain.

Mechanical analysis of cluster-grain separator device of a new head of paddy harvesting machine

This paper presents stress and strain analysis of outer shaft shoulder (OSS) and inner shaft shoulder (ISS) of a new paddy harvesting head (PHH) using finite element method (FEM). Snapping grain unit is a part of this head, with OSS and ISS as its members. The analysis was performed using the ABAQUS software with Dynamics Explicit Solution Method. Eight-node block and four-node tetrahedral elements were used to mesh the parts of the head. After the analyses, the stress and strain curves, and their maximum values were estimated for various parts. Analytical method was used to verify the FEM results and to calculate the factor of safety (FS) of the OSS and ISS components and also to estimate the number of cycles to failure of each component. The maximum amount of force applied to each tooth of shoulder was 4.29 Nmm-1. The maximum stress in both shoulders was obtained as 44.43 MPa. Other results showed, fatigue factors of safety for all components were less than their relevant yield factor of safety. Therefore, the fatigue in components would occur first. The study showed the predicted life for OSS and ISS components is more than 106 cycles, thus the components have an infinite-life. A fitted regression line to the data showed that the calculated stresses from analytical method lie within bounds of 7.89 % of the predicted values with a coefficient of determination of 0.98. Hence, it could be concluded that there is a good agreement between the analytical and FEM results.

An edge-based smoothed finite element method for the assessment of human exposure to extremely low frequency electric fields

This paper presents an edge-based smoothed finite element method (ES-FEM) for the assessment of human exposure to extremely low frequency (ELF) electric fields. The problem domain is first discretized into a set of four-node tetrahedral elements, and linear shape functions are employed to interpolate the field variables. Then, the smoothing domains are further formed with respect to the edges of elements. In order to improve the accuracy of lower order interpolation, a gradient smoothing technique (GST) is introduced to perform the electric potential gradient smoothing operation over each smoothing domain. Based on the generalized smoothed Galerkin weakform, the discretized system equations are finally obtained. Numerical examples, including both benchmark and practical complex cases, demonstrate that the present method possesses the following interesting properties in assessing human exposure to ELF electric fields: (1) super accuracy and faster convergence rate; (2) higher computational efficiency; (3) insensitive to mesh distortion.

A Four-Node Tetrahedral Finite Element Based on Space Fiber Rotation Concept

Abstract The paper presents a four-node tetrahedral solid finite element SFR4 with rotational degrees of freedom (DOFs) based on the Space Fiber Rotation (SFR) concept for modeling three-dimensional solid structures. This SFR concept is based on the idea that a 3D virtual fiber, after a spatial rotation, introduces an enhancement of the strain field tensor approximation. Full numerical integration is used to evaluate the element stiffness matrix. To demonstrate the efficiency and accuracy of the developed four-node tetrahedron solid element and to compare its performance with the classical four-node tetrahedral element, extensive numerical studies are presented.

Solution Bounds and Nearly Exact Solutions for 3D Nonlinear Problems of Large Deformation of Solids Using S-Fem

In this work, a three-dimensional (3D) nonlinear smoothed finite element method (S-FEM) solver is developed for large deformation problems. Node-based and face-based S-FEM using automatically generable four-noded tetrahedral elements (NS-FEM-Te4 and FS-FEM-Te4) are adopted to find the solution bounds in strain energy. The lower bound solutions are obtained using FEM-Te4 and FS-FEM-Te4, while the upper bound solutions are obtained using NS-FEM-Te4. A combined [Formula: see text]S-FEM-Te4 with a scaling factor [Formula: see text] that controls the combination is constructed to find nearly exact solutions for the nonlinear solids mechanics problems through adjusting [Formula: see text]. This is achieved using the property that a successive change of scaling factor [Formula: see text] can make the model transform from “overly-stiff” to “overly-soft”. Considering the properties of FS-FEM and NS-FEM, a selective FS/NS-FEM-TE4 is also used to solve 3D nonlinear large deformation problems, which produces a lower bound in strain energy. Hyperelastic Mooney–Rivlin and Ogden materials are both used in this study. Numerical examples reveal that S-FEM-Te4 is an effective method for obtaining solution bounds together with the standard FEM, and the FS-FEM-Te4, NS-FEM-Te4 and selective FS/NS-FEM-TE4 are robust with the high accuracy and computational efficiency for large deformation nonlinear problems.

A Stabilization Method of F-barES-FEM-T4 for Dynamic Explicit Analysis of Nearly Incompressible Materials

A stabilization method of the F-bar aided edge-based smoothed finite element method with four-node tetrahedral elements (F-barES-FEM-T4), which is named SymF-barES-FEM-T4, is proposed. The proposed formulation aims to remove the instability associated with energy divergence of F-barES-FEM-T4 in dynamic analysis. The stiffness matrix of F-barES-FEM-T4 is asymmetric due to the adoption of F-bar method, meanwhile that of SymF-barES-FEM-T4 is symmetrized so that the dynamics system of the formulation is stabilized. The differences between F-barES-FEM-T4 and SymF-barES-FEM-T4 in their formulations are only the definition of the [Formula: see text]-matrix and the corresponding volume in the internal force calculation. Therefore, SymF-barES-FEM-T4 is applicable in dynamic explicit schemes as is the case with F-barES-FEM-T4. A few demonstrations of dynamic explicit analysis for nearly incompressible materials reveal that SymF-barES-FEM-T4 conserves the total energy and does not cause energy divergence in the free vibration problems. They also reveal that SymF-barES-FEM-T4 has a moderate capability in the suppression of pressure checkerboarding: worse than F-barES-FEM-T4 but better than Selective ES/NS-FEM-T4.

Modelling three dimensional dynamic problems using the four-node tetrahedral element with continuous nodal stress

A partition of unity (PU) based four-node tetrahedral element with continuous nodal stress (Tetr4-CNS) was recently proposed for static analysis of three-dimensional solids. By simply using the same mesh as the classical four-node tetrahedral (Tetr4) element, high order global approximation function in the Tetr4-CNS element can be easily constructed without extra nodes or nodal DOFs. In this paper, the Tetr4-CNS element is further applied in the analysis of three dimensional dynamic problems. A series of free vibration and forced vibration problems are solved using the Tetr4-CNS element. The numerical results show that, for regular meshes, accuracy obtained using the Tetr4-CNS element is superior to that obtained using the Tetr4 and eight-node hexahedral (Hexa8) elements. For distorted meshes, the Tetr4-CNS element has better mesh-distortion tolerance than both the Tetr4 and Hexa8 elements.

Application of a non-conforming tetrahedral element in the context of the three-dimensional strain gradient elasticity

In the size-dependent continuum theories such as the strain gradient theory, the higher-order derivatives of displacement field appear in the energy functional of the structure which leads to the employment of C1 continuous shape functions within the finite element discretization procedure. Although a wide range of one- and two-dimensional small-scale finite elements were developed to analyze the structural behavior of micro- and nano-structures, a few studies can be found on the development of size-dependent three-dimensional (3D) finite elements. Hence, the main purpose of this work is the introduction of a four-node tetrahedral element to analyze the size-dependent mechanical behavior of micro- and nano-structures based on the three-dimensional strain gradient theory (SGT). In the proposed element, the values of displacement components and their related first-order derivatives are considered as degrees of freedom at each node. To present the governing equations, the matrix form of kinetic and strain energies and the work of external forces are derived based on the 3D elasticity theory and strain gradient model. To show the efficiency of the proposed model, the size-dependent linear vibration analysis of circular and elliptical micro- and nano-plates is presented. Various numerical results including comparative and convergence studies are reported to check the accuracy and performance of the introduced finite element.

A modified, hybrid nodal-integral/finite-element method for 3D convection-diffusion problems in arbitrary geometries

A modified, hybrid nodal-integral/finite-element method (NI-FEM) is developed to solve the three-dimensional (3D), steady-state, convection-diffusion problems in arbitrary geometries. The hybrid NI-FEM takes advantage of the high efficiency of the conventional nodal-integral method (NIM) and the flexible mesh generation of the finite element method (FEM), which is applicable to arbitrary geometries. In this method, the computational domain is discretized into, for 3D problems, four-node tetrahedral elements and eight-node cuboid elements. The cuboid elements are used to discretize the interior region and regions adjacent to boundaries that are parallel to planes formed by two of the axes, while the tetrahedral elements are used to discretize the remaining irregular regions. The conventional NIM is used to develop difference equations for the transverse-averaged variables on the interface between two adjacent cuboid elements. The FEM is applied to develop algebraic equations for the node temperatures of tetrahedral elements. On the interface between two different kinds of elements, the transverse-averaged variable for the cuboid element is obtained by averaging the node values of its adjacent tetrahedral elements, while the heat flux for the tetrahedral element is calculated using the corresponding transverse-averaged variables of its adjacent cuboid element. The hybrid NI-FEM is developed to be solved using a matrix formulation for the entire domain rather than an iterative procedure, detailed derivations of which for the 3D case are presented in this paper. Using the NI-FEM developed here, 2D and 3D convection-diffusion test problems are solved, and the numerical results are compared to the exact (manufactured) solutions to determine the order, accuracy and efficiency of the method. Numerical scheme is found to be of, as expected, second order.