Four-node Tetrahedral Elements Research Articles

A four-node tetrahedral element with continuous nodal stress

Formulation of a partition of unity (PU) based four-node tetrahedral element with continuous nodal stress (Tetr4-CNS) and its applications to the analysis of linear elasticity problems in three-dimension are presented in this paper. By simply using the same mesh as the classical tetrahedral element (Tetr4), Tetr4-CNS element is able to obtain continuous nodal stress without recourse to stress smoothing operation in the post-processing process, and to construct high order global approximation without adding extra nodes or nodal DOFs. Moreover, it is free from the linear dependence problem which cripples many of the PU-based methods. A series of numerical tests are carried out to evaluate the performance of the Tetr4-CNS element. The numerical results show that accuracy through the proposed element is superior to that through Tetr4 element and eight-node hexahedral element (Hexa8). More importantly, the proposed element has excellent mesh distortion tolerant capabilities.

Large deformation explicit dynamics for nearly incompressible materials using F-bar aided edge-based smoothed finite element method with four-node tetrahedral elements

Large deformation explicit dynamics for nearly incompressible materials using F-bar aided edge-based smoothed finite element method with four-node tetrahedral elements

A mixed tetrahedral element with nodal rotations for large-displacement analysis of inelastic structures

Summary A novel mixed four-node tetrahedral finite element, equipped with nodal rotational degrees of freedom, is presented. Its formulation is based on a Hu–Washizu-type functional, suitable to the treatment of material nonlinearities. Rotation and skew-symmetric stress fields are assumed as independent variables, the latter entering the functional to impose rotational compatibility and suppress spurious modes. Exploiting the choice of equal interpolation for strain and symmetric stress fields, a robust element state determination procedure, requiring no element-level iteration, is proposed. The mixed element stability is assessed by means of an original and effective numerical test. The extension of the present formulation to geometric nonlinear problems is achieved through a polar decomposition-based corotational framework. After validation in both material and geometric nonlinear context, the element performances are investigated in demanding simulations involving complex shape memory alloy structures. Supported by the comparison with available linear and quadratic tetrahedrons and hexahedrons, the numerical results prove accuracy, robustness, and effectiveness of the proposed formulation. Copyright © 2016 John Wiley & Sons, Ltd.

Stabilized tetrahedral elements for crystal plasticity finite element analysis overcoming volumetric locking

Image-based CPFE modeling involves computer generation of virtual polycrystalline microstructures from experimental data, followed by discretization into finite element meshes. Discretization is commonly accomplished using three-dimensional four-node tetrahedral or TET4 elements, which conform to the complex geometries. It has been commonly observed that TET4 elements suffer from severe volumetric locking when simulating deformation of incompressible or nearly incompressible materials. This paper develops and examines three locking-free stabilized finite element formulations in the context of crystal plasticity finite element analysis. They include a node-based uniform strain (NUS) element, a locally integrated B-bar (LIB) based element and a F-bar patch (FP) based element. All three formulations are based on the partitioning of TET4 element meshes and integrating over patches to obtain favorable incompressibility constraint ratios without adding large degrees of freedom. The results show that NUS formulation introduces unstable spurious energy modes, while the LIB and FP elements stabilize the solutions and are preferred for reliable CPFE analysis. The FP element is found to be computationally efficient over the LIB element.

A finite element head and neck model as a supportive tool for deformable image registration

A finite element (FE) head and neck model was developed as a tool to aid investigations and development of deformable image registration and patient modeling in radiation oncology. Useful aspects of a FE model for these purposes include ability to produce realistic deformations (similar to those seen in patients over the course of treatment) and a rational means of generating new configurations, e.g., via the application of force and/or displacement boundary conditions. The model was constructed based on a cone-beam computed tomography image of a head and neck cancer patient. The three-node triangular surface meshes created for the bony elements (skull, mandible, and cervical spine) and joint elements were integrated into a skeletal system and combined with the exterior surface. Nodes were additionally created inside the surface structures which were composed of the three-node triangular surface meshes, so that four-node tetrahedral FE elements were created over the whole region of the model. The bony elements were modeled as a homogeneous linear elastic material connected by intervertebral disks. The surrounding tissues were modeled as a homogeneous linear elastic material. Under force or displacement boundary conditions, FE analysis on the model calculates approximate solutions of the displacement vector field. A FE head and neck model was constructed that skull, mandible, and cervical vertebrae were mechanically connected by disks. The developed FE model is capable of generating realistic deformations that are strain-free for the bony elements and of creating new configurations of the skeletal system with the surrounding tissues reasonably deformed. The FE model can generate realistic deformations for skeletal elements. In addition, the model provides a way of evaluating the accuracy of image alignment methods by producing a ground truth deformation and correspondingly simulated images. The ability to combine force and displacement conditions provides flexibility for simulating realistic anatomic configurations.

A coupled smoothed finite element method (S-FEM) for structural-acoustic analysis of shells

In this paper, a coupled smoothed finite element method (S-FEM) is developed to deal with the structural-acoustic problems consisting of a shell configuration interacting with the fluid medium. Three-node triangular elements and four-node tetrahedral elements that can be generated automatically for any complicated geometries are adopted to discretize the problem domain. A gradient smoothing technique (GST) is introduced to perform the strain smoothing operation. The discretized system equations are obtained using the smoothed Galerkin weakform, and the numerical integration is applied over the further formed edge-based and face-based smoothing domains, respectively. To extend the edge-based smoothing operation from plate structure to shell structure, an edge coordinate system is defined local on the edges of the triangular element. Numerical examples of a cylinder cavity attached to a flexible shell and an automobile passenger compartment have been conducted to illustrate the effectiveness and accuracy of the coupled S-FEM for structural-acoustic problems.

Linear tetrahedral element for problems of plastic deformation

Linear tetrahedra perform poorly in problems with plasticity, nearly incompressible materials, and in bending. While higher-order tetrahedra can cure or alleviate some of these weaknesses, in many situations low-order tetrahedral elements would be preferable to quadratic tetrahedral elements: e.g. for contact problems or fluid-structure interaction simulations. Therefore, a low-order tetrahedron that would look on the outside as a regular four-node tetrahedron, but that would possess superior accuracy is desirable. An assumed-strain, nodally integrated, four-node tetrahedral element is presented (NICE-T4). Several numerical benchmarks are provided showing its robust performance in conjunction with material nonlinearity in the form of von Mises plasticity. In addition we compare the computational cost of the nodally integrated NICE-T4 with the isoparametric quadratic tetrahedron. Because of the reduced number of quadrature points, the NICE-T4 element is competitive in nonlinear analyses with complex material models.

A smoothed finite element method for analysis of anisotropic large deformation of passive rabbit ventricles in diastole.

The smoothed FEM (S-FEM) is firstly extended to explore the behavior of 3D anisotropic large deformation of rabbit ventricles during the passive filling process in diastole. Because of the incompressibility of myocardium, a special method called selective face-based/node-based S-FEM using four-node tetrahedral elements (FS/NS-FEM-TET4) is adopted in order to avoid volumetric locking. To validate the proposed algorithms of FS/NS-FEM-TET4, the 3D Lame problem is implemented. The performance contest results show that our FS/NS-FEM-TET4 is accurate, volumetric locking-free and insensitive to mesh distortion than standard linear FEM because of absence of isoparametric mapping. Actually, the efficiency of FS/NS-FEM-TET4 is comparable with higher-order FEM, such as 10-node tetrahedral elements. The proposed method for Holzapfel myocardium hyperelastic strain energy is also validated by simple shear tests through the comparison outcomes reported in available references. Finally, the FS/NS-FEM-TET4 is applied in the example of the passive filling of MRI-based rabbit ventricles with fiber architecture derived from rule-based algorithm to demonstrate its efficiency. Hence, we conclude that FS/NS-FEM-TET4 is a promising alternative other than FEM in passive cardiac mechanics.

Coupled Edge-based Smoothing Finite Element Method for Structural Acoustic Problems

The coupled finite element method(coupled FEM) and coupled finite element method/boundary element method(coupled FEM/BEM) are always adopted to solve the structural-acoustic problems. and it is known that the finite element methods behave "over-stiffness" which induce the large numerical error for the structural-acoustic problems. In order to soften the "over-stiffness" and improve the results of FEM, a coupled ES-FEM model is extended to solve the structural-acoustic problems consisting of a plate structure interacting with the fluid medium. Three-node triangular elements and four-node tetrahedral elements are used to discretize the two-dimensional(2D) plate and three-dimensional(3D) fluid, respectively, as they can be generated easily and even automatically for complicated geometries. Numerical studies have verified that the gradient smoothing technique can provide a proper softening effect to the model, and hence significantly improve the accuracy of the solution for the coupled systems, which will lay the foundation for further applications.

Analysis of coupled structural-acoustic problems based on the smoothed finite element method (S-FEM)

Abstract The smoothed finite element method (S-FEM) developed recently shows great efficiency in solving solid mechanics and acoustics problems. In this paper, coupled structural-acoustic problems are studied using the S-FEM method. Three-node triangular elements and four-node tetrahedral elements are used to discretize the two-dimensional and three-dimensional domains, respectively. The gradient field of the problem is smoothed using gradient smoothing operations over the edge-based and face-based smoothing domains in two-dimensional plate and three-dimensional fluid, respectively. Because the gradient smoothing technique can provide a proper softening effect to the “overly-stiff” FEM model, significant improvements are achieved on the accuracy of solution for the coupled systems. Typical numerical examples have been conducted to verify the effectiveness of the S-FEM for coupled structural-acoustic problems.

Finite element modeling to estimate the apparent material properties of trabecular bone

The in-vivo micro-structure and corresponding material property of trabecular bone is important to simulate the mechanical behavior of macroscopic bone structure. In order to simulate the mechanical behavior of bone numerically, the apparent Young’s modulus of trabecular bone should be available. Generally a high-resolution finite-element model based on micro-CT-images could be was used to estimate this value. However, all the previous works regarding this issue have employed either eight-noded voxel elements or four-noded tetrahedral elements, which usually produces large amount of error in estimating an apparent material property. Therefore, rigorous studies on the accuracy of element type for predicting the material properties of cancellous bone have been made in this paper. Micro-CT-data were extracted from a femoral neck to construct three-dimensional finite-element models with three different element types and compression analyses were performed numerically (up to 1.3% strain) to estimate the apparent modulus. Compression tests using the specimens extracted from a cadaver were also performed to validate the simulated apparent material properties using different element types. As a result, ten-noded tetrahedral elements are recommended to obtain reliable material properties of cancellous bone instead of eight-noded voxel elements or four-noded tetrahedral elements.

A CONTROL-VOLUME FINITE ELEMENT METHOD FOR THE PREDICTION OF THREE-DIMENSIONAL DIFFUSION-TYPE PHENOMENA IN ANISOTROPIC MEDIA

The formulation and testing of a control-volume finite element method (CVFEM) for the prediction of three-dimensional, linear and nonlinear, diffusion-type phenomena in anisotropic media in irregular calculation domains are presented and discussed in this paper. In this CVFEM, the calculation domain is discretized into four-node tetrahedral elements. Contiguous, non-overlapping, polyhedral control volumes are then associated with each node, and the governing differential equation is integrated over these control volumes. The dependent variable is interpolated linearly in each four-node tetrahedral element. Centroidal values of the diffusion coefficients are stored and assumed to prevail over the corresponding tetrahedral element. The source term is linearized, and nodal values of its coefficients are stored and assumed to prevail over the polyhedral sub-control volumes. Using these interpolation functions, the discretized equations, which are algebraic approximations to the integral conservation equations, are derived. The discretized equations, which in general, are nonlinear and coupled, are solved using an iterative procedure. The proposed CVFEM for the solution of anisotropic diffusion- type problems appears to be the first such method that is based on tetrahedral elements and vertex-centered polyhedral control volumes. These features make it particularly attractive for amalgamation with adaptive-grid schemes and applications to problems with complex irregular geometries, such those encountered in the general areas of drying, ground-water flows, conduction in composite materials, injection molding in heterogeneous porous media, and solidification. The proposed three-dimensional CVFEM and its computer implementation were tested using several steady conduction-type problems, for which analytical solutions were constructed using a special technique. In all cases, the agreement between the numerical and analytical solutions was excellent.

A three dimensional immersed smoothed finite element method (3D IS-FEM) for fluid–structure interaction problems

A three-dimensional immersed smoothed finite element method (3D IS-FEM) using four-node tetrahedral element is proposed to solve 3D fluid–structure interaction (FSI) problems. The 3D IS-FEM is able to determine accurately the physical deformation of the nonlinear solids placed within the incompressible viscous fluid governed by Navier-Stokes equations. The method employs the semi-implicit characteristic-based split scheme to solve the fluid flows and smoothed finite element methods to calculate the transient dynamics responses of the nonlinear solids based on explicit time integration. To impose the FSI conditions, a novel, effective and sufficiently general technique via simple linear interpolation is presented based on Lagrangian fictitious fluid meshes coinciding with the moving and deforming solid meshes. In the comparisons to the referenced works including experiments, it is clear that the proposed 3D IS-FEM ensures stability of the scheme with the second order spatial convergence property; and the IS-FEM is fairly independent of a wide range of mesh size ratio.

A coupled edge-/face-based smoothed finite element method for structural–acoustic problems

The edge-based smoothed finite element method (ES-FEM) and the face-based smoothed finite element method (FS-FEM) developed recently have shown great efficiency in solving solid mechanics problems with triangular and tetrahedral meshes. In this paper, a coupled ES-/FS-FEM model is extended to solve the structural–acoustic problems consisting of a plate structure interacting with the fluid medium. Three-node triangular elements and four-node tetrahedral elements are used to discretize the two-dimensional (2D) plate and three-dimensional (3D) fluid, respectively, as they can be generated easily and even automatically for complicated geometries. The field variable in each element is approximated using the linear shape functions, which is exactly the same as that in the standard FEM. The gradient field of the problem is obtained particularly using the gradient smoothing operation over the edge-based and face-based smoothing domains in 2D and 3D, respectively. The gradient smoothing technique can provide a proper softening effect to the model, effectively solve the problems caused by the well-known “overly-stiff” phenomenon existing in the standard FEM, and hence significantly improve the accuracy of the solution for the coupled systems. Intensive numerical studies have been conducted to verify the effectiveness of the coupled ES-/FS-FEM for structural–acoustic problems.

COMPUTATIONAL MODELING OF DYNAMIC BRITTLE FAILURE USING THREE-DIMENSIONAL EXPLICIT FEM

Ceramic materials show good mechanical properties. However, the brittleness makes it difficult to predict their dynamic behavior of ceramic materials, and further the discrete nature of brittle failure requires a three-dimensional computation. In this paper, a computational model was developed for analyzing high velocity impact situations of brittle materials, incorporating with node separation scheme to describe the distinctive behavior of a brittle failure. This material behavior model was applied in a three-dimensional explicit code with Lagrangian description, which was developed to compute problems of dynamic brittle failure using four-node tetrahedral elements. To effectively handle the complex contact situations, involving multiple collisions and self-contact of severely deformed and fragmented bodies, careful considerations were paid to the global and local contact searching. In addition, an edge contact algorithm was developed to handle contacts between sharp edges. To validate the developed computational model and three-dimensional explicit code, oblique penetration into sandwiched ceramic plate by long rod was simulated, and the results were compared with experiments.

A novel alpha finite element method ( αFEM) for exact solution to mechanics problems using triangular and tetrahedral elements

The paper presents an alpha finite element method ( αFEM) for computing nearly exact solution in energy norm for mechanics problems using meshes that can be generated automatically for arbitrarily complicated domains. Three-node triangular ( αFEM-T3) and four-node tetrahedral ( αFEM-T4) elements with a scale factor α are formulated for two-dimensional (2D) and three-dimensional (3D) problems, respectively. The essential idea of the method is the use of a scale factor α ∈ [0,1] to obtain a combined model of the standard fully compatible model of the FEM and a quasi-equilibrium model of the node-based smoothed FEM (N-SFEM). This novel combination of the FEM and N-SFEM makes the best use of the upper bound property of the N-SFEM and the lower bound property of the standard FEM. Using meshes with the same aspect ratio, a unified approach has been proposed to obtain a nearly exact solution in strain energy for linear problems. The proposed elements are also applied to improve the accuracy of the solution of nonlinear problems of large deformation. Numerical results for 2D (using αFEM-T3) and 3D (using αFEM-T4) problems confirm that the present method gives the much more accurate solution comparing to both the standard FEM and the N-SFEM with the same number of degrees of freedom and similar computational efforts for both linear and nonlinear problems.

A stabilized mixed formulation for finite strain deformation for low-order tetrahedral solid elements

A stabilized mixed finite element formulation for four-noded tetrahedral elements is introduced for robustly solving small and large deformation problems. The uniqueness of the formulation lies within the fact that it is general in that it can be applied to any type of material model without requiring specialized geometric or material parameters. To overcome the problem of volumetric locking, a mixed element formulation that utilizes linear displacement and pressure fields was implemented. The stabilization is provided by enhancing the rate of deformation tensor with a term derived using a bubble function approach. The element was implemented through a user-programmable element of the commercial finite element software ANSYS. Using the ANSYS platform, the performance of the element was evaluated by comparing the predicted results with those obtained using mixed quadratic tetrahedral elements and hexahedral elements with a B-bar formulation. Based on the quality of the results, the new element formulation shows significant potential for use in simulating complex engineering processes.

Fracture Mechanics Parameter Computation Using Tetrahedral Finite Element with Drilling Degrees of Freedoms (Virtual Crack Closure-Integral Method for Generalized Tetrahedral Finite Element)

This paper presents a virtual crack closure-integral method (VCCM) for a generalized fournode tetrahedral finite element that has drilling nodal degrees of freedoms (DOFs) . The generalized tetrahedral finite element has superior accuracy over the liner tetrahedral element. VCCM computes the energy release rate and the stress intensity factor at the crack front. VCCM computes the fracture mechanics parameters accurately only from the nodal reaction forces and displacements. Furthermore, in mixed mode problem, it is easy to split energy release rate into each mode of crack deformation. In this paper, a VCCM formulation for the generalized tetrahedral finite element is developed for the case of mode I crack deformation. The contributions of nodal drilling DOFs on the computation of energy release rate are described in detail. Similar discussions can be extended to mixed mode problems. Finally, some numerical examples are presented and discussions on the accuracy of proposed VCCM are given.

The influence of circumferential edge constraint on the acoustical properties of open-cell polyurethane foam samples

In this paper, a three-dimensional Laplace domain finite element model with four-node tetrahedral porous elements is developed and applied for the investigations. Without the incident acoustic field, the exact acoustical properties of a laterally infinite planar foam, which is backed with a rigid plane, are accurately predicted by assuming the circumferential edge constraints as rollers. It is found that the rigidity of the circumferential edge constraint has a profound effect on the predictions. If the circumferential edge constraints are all fixed, the amplitude of the first sound absorption peak is increased, and the peak is moved to a lower frequency as compared with the exact solutions. Measurements obtained by the two-microphone impedance tube method are bounded by the predictions of samples with roller as well as with fixed constraints. They are much closer to that predicted with fixed edge constraints. Upon increasing the diameter-to-thickness ratio of the sample, the influence of the fixed edge constraint is reduced. The measured acoustic results can be the representatives of the laterally infinite planar foam from which the sample was cut only if the roller edge constraint of the sample is managed or the diameter-to-thickness ratio of the sample is increased.

Overstrain in flush optimal-chamfered cross-bored cylinders

A three-dimensional finite-element method computer program was developed to establish the elastic—plastic, residual, and service stress distributions in cylinders with flush and non-protruding optimal-chamfered cross-bores under internal pressure. Eight-noded brick and four-noded tetrahedral isoparametric elements and the displacement formulation were used. The incremental theory of plasticity with a 5 per cent yield condition and von Mises yield criterion were assumed. The incipient and 5 per cent overstrain (ov) pressures were established for various thickness ratios and cross-bore to main bore radius ratios. For the optimum chamfer angle geometrical configuration, the stresses were determined for varying ov. The maximum and minimum effective stresses were located 7.5° from the meridional and transverse planes, respectively. Meridional plane through thickness yielding occurred at an ov of 41 per cent. The service stress gradients at the cross-bore chamfer end increased with ov for ovs >30 per cent. Stress reversals were eliminated for overstrain >27 per cent. Alternative autofrettage and yield condition procedures were proposed.