Tetrahedral Finite Elements Research Articles

Fast tetrahedral mesh generation and segmentation of an atlas-based heart model using a periodic uniform grid

Abstract A fast procedure for generation of regular tetrahedral finite element mesh for objects with complex shape cavities is proposed. The procedure like LBIE-Mesher can generate tetrahedral meshes for the volume interior to a polygonal surface, or for an interval volume between two surfaces having a complex shape and defined in STL-format. This procedure consists of several stages: generation of a regular tetrahedral mesh that fills the volume of the required object; generation of clipping for the uniform grid parts by a boundary surface; shifting vertices of the boundary layer to align onto the surface.We present a sequential and parallel implementation of the algorithm and compare their performance with existing generators of tetrahedral grids such as TetGen, NETGEN, and CGAL. The current version of the algorithm using the mobile GPU is about 5 times faster than NETGEN. The source code of the developed software is available on GitHub.

Supervised learning for bone shape and cortical thickness estimation from CT images for finite element analysis.

Knowledge about the thickness of the cortical bone is of high interest for fracture risk assessment. Most finite element model solutions overlook this information because of the coarse resolution of the CT images. To circumvent this limitation, a three-steps approach is proposed. 1) Two initial surface meshes approximating the outer and inner cortical surfaces are generated via a shape regression based on morphometric features and statistical shape model parameters. 2) The meshes are then corrected locally using a supervised learning model build from image features extracted from pairs of QCT (0.3-1 mm resolution) and HRpQCT images (82 µm resolution). As the resulting meshes better follow the cortical surfaces, the cortical thickness can be estimated at sub-voxel precision. 3) The meshes are finally regularized by a Gaussian process model featuring a two-kernel model, which seamlessly enables smoothness and shape-awareness priors during regularization. The resulting meshes yield high-quality mesh element properties, suitable for construction of tetrahedral meshes and finite element simulations. This pipeline was applied to 36 pairs of proximal femurs (17 males, 19 females, 76 ± 12 years) scanned under QCT and HRpQCT modalities. On a set of leave-one-out experiments, we quantified accuracy (root mean square error = 0.36 ± 0.29 mm) and robustness (Hausdorff distance = 3.90 ± 1.57 mm) of the outer surface meshes. The error in the estimated cortical thickness (0.05 ± 0.40 mm), and the tetrahedral mesh quality (aspect ratio = 1.4 ± 0.02) are also reported. The proposed pipeline produces finite element meshes with patient-specific bone shape and sub-voxel cortical thickness directly from CT scans. It also ensures that the nodes and elements numbering remains consistent and independent of the morphology, which is a distinct advantage in population studies.

Yield design‐based numerical analysis of three‐dimensional reinforced concrete structures

SummaryThe objective of this contribution is to present some new recent developments regarding the evaluation of the ultimate bearing capacity of massive three‐dimensional reinforced concrete structures which cannot be modeled as 1D (beams) or 2D (plates) structural members. The approach is based upon the implementation of the lower bound static approach of yield design through a discretization of the three‐dimensional structure into tetrahedral finite elements, on the one hand, the formulation of the corresponding optimization problem in the context of semi‐definite programming techniques, on the other hand. Another key feature of the method lies in the treatment of the concrete‐embedded reinforcing bars not as individual elements, but by resorting to an extension of the yield design homogenization approach. The whole procedure is first validated on the rather simple illustrative problem of a uniformly loaded simply supported beam, then applied to the design of a bridge pier cap taken as an example of more complex and realistic structure.

A new strain smoothing method for triangular and tetrahedral finite elements

This paper presents a simple and effective strain smoothing method (the strain-smoothed element method) for 3-node triangular and 4-node tetrahedral solid finite elements. While piecewise constant strain fields are constructed through smoothing domains in previous strain smoothing methods, in the proposed method, linear strain fields are constructed within finite elements using constant strains of neighboring finite elements. When the new strain smoothing method is adopted, the triangular and tetrahedral solid finite elements pass all the basic tests; patch, isotropy and zero energy mode tests, and show improved predictive capability. The formulation of the proposed method is presented in detail. Through various numerical examples, we demonstrate the accuracy improvement achieved.

Large deformation analysis of functionally graded visco-hyperelastic materials

In this paper, a numerical formulation for the analysis of viscoelastic functionally graded materials under finite strains is presented. The general constitutive modeling is described within the context of Lagrangian and isotropic visco-hyperelasticity. The specific models selected are the compressible neo-Hookean hyperelastic law, the Zener rheological model and the isochoric evolution law described in terms of the rate of the viscous right Cauchy-Green stretch tensor. The material coefficients may vary smooth and continuously along one direction according to the power law. The viscous update is performed via the exponential rule. The main novelty of this paper is the use of gradually variable viscoelastic coefficients in the finite strain regime.Four numerical examples involving functionally graded materials and finite viscoelastic strains are originally analyzed to assess the formulation proposed: a bar under uniaxial extension, a block under simple shear, the Cook’s membrane and an elastomeric bridge bearing. Isoparametric solid tetrahedral finite elements of linear, quadratic and cubic orders are employed. The influence of the material viscoelastic parameters on the mechanical behavior is analyzed in detail. Results confirm that mesh refinement provides more accuracy and the present model can reproduce large levels of viscoelastic strains in functionally graded materials.

The effect of myocardial action potential duration on cardiac pumping efficacy: a computational study

Background and aimsAlthough studies on the relation between arrhythmias and the action potential duration (APD) have been carried out, most of them are based only on electrophysiological factors of the heart and lack experiments that consider cardiac mechanical and electromechanical characteristics. Therefore, we conducted this study to clarify the relevance of the shortening of APD of a cell in relation to the mechanical contraction activity of the heart and the associated risk of arrhythmia.MethodsThe human ventricular model used in this study has two dynamic characteristics: electrophysiological conduction and mechanical contraction. The model simulating electrophysiological characteristics was consisted of lumped parameter circuit that can mimic the phenomenon of ion exchange through the cell membrane of myocyte and consisted of 214,319 tetrahedral finite elements. In contrast, the model simulating mechanical contraction characteristics was constructed to mimic cardiac contraction by means of the crossbridge of a myofilament and consisted of 14,720 hermite-based finite elements to represent a natural 3D curve of the cardiac surface. First, we performed a single cell simulation and the electrophysiological simulation according to the change of the APD by changing the electrical conductivity of the IKs channel. Thus, we confirmed the correlation between APD and intracellular Ca2+ concentration. Then, we compared mechanical response through mechanical simulation using Ca2+ data from electrical simulation.ResultsThe APD and the sum of the intracellular Ca2+ concentrations showed a positive correlation. The shortened APD reduced the conduction wavelength of ventricular cells by shortening the plateau and early repolarization in myocardial cells. The decrease in APD reduced ventricular pumping efficiency by more than 60% as compared with the normal group (normal conditions). This change is caused by the decline of ventricular output owing to reduced ATP consumption during the crossbridge of myofilaments and decreased tension.ConclusionThe shortening of APD owing to increased electrical conductivity of a protein channel on myocardial cells likely decreases the wavelength and the pumping efficiency of the ventricles. Additionally, it may increase tissue sensitivity to ventricular fibrillation, including reentry, and cause symptoms such as dyspnea and dizziness.

Bernstein–Bézier bases for tetrahedral finite elements

We present a new set of basis functions for H(curl)-conforming, H(div)-conforming, and L2-conforming finite elements of arbitrary order on a tetrahedron. The basis functions are expressed in terms of Bernstein polynomials and augment the natural H1-conforming Bernstein basis. The basis functions respect the differential operators, namely, the gradients of the high-order H1-conforming Bernstein–Bézier basis functions form part of the H(curl)-conforming basis, and the curl of the high-order, non-gradients H(curl)-conforming basis functions form part of the H(div)-conforming basis, and the divergence of the high-order, non-curl H(div)-conforming basis functions form part of the L2-conforming basis.Procedures are given for the efficient computation of the mass and stiffness matrices with these basis functions without using quadrature rules for (piecewise) constant coefficients on affine tetrahedra. Numerical results are presented to illustrate the use of the basis to approximate representative problems.

Improved recovered nodal stress for mean-strain finite elements

This paper investigates a method for improving the accuracy of the stress predicted from models using the mean-strain finite elements recently proposed by Krysl and collaborators [IJNME 2016, 2017]. In state-of-the-art finite element programs, the stress values at the integration points are commonly post-processed to obtain nodal values of stress. The mean stresses are element-wise constant, and hence the nodal values obtained from the mean stresses tend to be of lower accuracy. The proposed method post-processes the uniform stress in each element in combination with a linearly-varying stabilization stress field to produce a more accurate representation of the nodal stresses. Selected examples are presented to demonstrate improvements achievable with the proposed methodology for hexahedral and quadratic tetrahedral mean-strain finite elements.

Full field modeling of recrystallization: Effect of intragranular strain gradients on grain boundary shape and kinetics

The effect of stored energy field heterogeneity on the microstructure evolution during static recrystallization (SRX) was assessed by performing three dimensional (3D) full field simulations. These simulations were performed by using the Level set (LS) method in a finite element (FE) framework with unstructured tetrahedral FE mesh. An extension of classical LS-FE for SRX was developed to account for average stored energies in zones adjacent to grain boundaries (GBs) in each grain. Annihilation of dislocations by a passing GB was taken into account. The results of 3D simulations were analyzed to compare the “per interface” LS model to the usual “per grain” LS method (with averaging stored energy per grain). Recrystallization is faster at the beginning of the heat treatment when using the “per interface” model. The analysis of the evolution of grain volume and grain surface scattering reveals that the “per interface” approach results in complex shapes of grains which are closer to those observed in experiments. Local simulations for few grains with refined FE mesh were also performed to study the effect of stored energy heterogeneity along interfaces, and even higher complexity of grain topology was observed compared to the results of “per interface” computations.

Personalized computational modeling of left atrial geometry and transmural myofiber architecture

Atrial fibrillation (AF) is a supraventricular tachyarrhythmia characterized by complete absence of coordinated atrial contraction and is associated with an increased morbidity and mortality. Personalized computational modeling provides a novel framework for integrating and interpreting the role of atrial electrophysiology (EP) including the underlying anatomy and microstructure in the development and sustenance of AF. Coronary computed tomography angiography data were segmented using a statistics-based approach and the smoothed voxel representations were discretized into high-resolution tetrahedral finite element (FE) meshes. To estimate the complex left atrial myofiber architecture, individual fiber fields were generated according to morphological data on the endo- and epicardial surfaces based on local solutions of Laplace’s equation and transmurally interpolated to tetrahedral elements. The influence of variable transmural microstructures was quantified through EP simulations on 3 patients using 5 different fiber interpolation functions. Personalized geometrical models included the heterogeneous thickness distribution of the left atrial myocardium and subsequent discretization led to high-fidelity tetrahedral FE meshes. The novel algorithm for automated incorporation of the left atrial fiber architecture provided a realistic estimate of the atrial microstructure and was able to qualitatively capture all important fiber bundles. Consistent maximum local activation times were predicted in EP simulations using individual transmural fiber interpolation functions for each patient suggesting a negligible effect of the transmural myofiber architecture on EP. The established modeling pipeline provides a robust framework for the rapid development of personalized model cohorts accounting for detailed anatomy and microstructure and facilitates simulations of atrial EP.

Numerical analysis of thermal and hydro dynamical processes in upper fuel channel part of boiling water reactor

This article presents results of 3D numerical analysis of the thermal and hydro dynamical processes which take place inside the fuel channel of the RBMK-1500 reactor during interaction between coolant and fuel assembly. Numerical modelling was made for three different operational regimes of reactor: for minimal (0.91 MW), average (2.53 MW) and for the maximal power (3.50 MW) levels. Simulation was made using ANSYS CFX program package in order to evaluate coolant’s pressure, temperature, flow velocity and steam mass fraction changes at the distance grids (spacers) and at the heat transfer intensification grids (intensifiers) along and across the fuel channel. In order to create CFX mesh, a tetrahedral finite element type with 10 million elements was chosen. The k-e turbulence model and homogeneous model of the heat transfer (HEM) were applied for numerical simulation. The obtained results help better to understand the processes which have a direct influence on the reactor safety. Information about the peculiarities of steam generation process inside the fuel channel (temperature change, pressure drop, flow velocity, steam mass fraction) enables to identify the most loaded points of the fuel channel and to make necessary design and operational improvements (location of spacers and intensifiers; core loading and etc.). The results can be used for predicting of hydrogen formation process and for the forecast of the decrease of zirconium alloy strength.DOI: http://dx.doi.org/10.5755/j01.mech.23.5.16457

A theoretical procedure for the stress state assessment of hyper-static crankshafts

Crankshafts with many rods and crankpins for mechanical and industrial applications can be designed with first-order-approximation theoretical procedures in the literature. Numerical boundary element method (BEM) or finite element method (FEM) procedures can be used, with the latter models having a 3D tetrahedral, wire beam or hexahedral finite elements. To the best of the author's knowledge, the literature does not contain accurate evaluations of hyper-static reactions at the restraints of a crankshaft with four crankpins and five supports. The principle of virtual work is implemented to allow calculation of the reactions at the bearings of the shaft and to determine the internal actions (bending, torsion, shear and axial force) for each section of the shaft. The developed procedure does not give better results than a numerical BEM or 3D FEM code, but it is less expensive and less time consuming when implemented in a mathematical commercial code. Furthermore, the procedure gives results that have a better approximation with respect to the theoretical literature models. The theoretical model was validated through comparison with the results of a finite-element linear beam model developed using a commercial FEM code.

Tetrahedral Finite Element Modeling of UNS S31600 Stainless Steel Corrosion Behavior under Superfinish Turning Conditions

Tetrahedral Finite Element Modeling of UNS S31600 Stainless Steel Corrosion Behavior under Superfinish Turning Conditions

The finite cell method: Polygonal and tetrahedral cells

AbstractPartial differential equations arising in many physical problems are most commonly solved by using the finite element method (FEM). Despite being very versatile, the FEM has one crucial drawback when heterogeneous material systems are considered, i.e. it relies on geometry‐conforming discretization. This meshing process often constitutes a bottleneck in the simulation pipeline and therefore needs to be alleviated. One idea is to exploit the benefits of the fictitious domain concept. Here, Cartesian grids can be deployed to straightforwardly discretize an extended domain. In conjunction with higher order shape functions the recently introduced method is referred to as the finite cell method (FCM).The main objective of the contribution at hand is to extend the FCM to arbitrary unstructured meshes. In two‐dimensional application polygonal finite elements based on generalized barycentric coordinates are deployed, while the three‐dimensional implementation is based on tetrahedral finite elements. A further extension to pyramidal and pentahedral (wedge) elements is, however, straightforward.There are two distinct advantages of using unstructured meshes: (i) a local mesh refinement can easily be conducted and (ii) existing computational models can be re‐used for parametric studies. The latter point is important if virtual defects need to be introduced in the ideal model to study their effect on the material behaviour. These possibilities make the unstructured FCM a powerful numerical tool for the investigation of complex highly heterogeneous materials. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Novel Nonlinear Parameter Estimation Method of Soft Tissues

The elastic parameters of soft tissues are important for medical diagnosis and virtual surgery simulation. In this study, we propose a novel nonlinear parameter estimation method for soft tissues. Firstly, an in-house data acquisition platform was used to obtain external forces and their corresponding deformation values. To provide highly precise data for estimating nonlinear parameters, the measured forces were corrected using the constructed weighted combination forecasting model based on a support vector machine (WCFM_SVM). Secondly, a tetrahedral finite element parameter estimation model was established to describe the physical characteristics of soft tissues, using the substitution parameters of Young’s modulus and Poisson’s ratio to avoid solving complicated nonlinear problems. To improve the robustness of our model and avoid poor local minima, the initial parameters solved by a linear finite element model were introduced into the parameter estimation model. Finally, a self-adapting Levenberg–Marquardt (LM) algorithm was presented, which is capable of adaptively adjusting iterative parameters to solve the established parameter estimation model. The maximum absolute error of our WCFM_SVM model was less than 0.03 Newton, resulting in more accurate forces in comparison with other correction models tested. The maximum absolute error between the calculated and measured nodal displacements was less than 1.5 mm, demonstrating that our nonlinear parameters are precise.

In vitro and in silico characterization of open-cell structures of trabecular bone

This work aimed to perform a detailed in vitro and in silico characterization of open-cell structures, which resemble trabecular bone, to elucidate osteoporosis failure mechanisms. Experimental and image-based computational methods were used to estimate Young′s modulus and porosities of different open-cell structures (Sawbones; Malmö, Sweden). Three different open-cell structures with different porosities were characterized. Additionally, some open-cell structures were scanned using a microcomputed tomography system (μCT) to non-destructively predict specimen Young′s modulus of the structures by developing voxel-based and tetrahedral finite element (FE) models. A 3D reconstruction and FE analyses were used. The experimental and computational results with different element types (linear and quadratic tetrahedrons and voxel-based meshes) were compared with Sawbones data (Sawbones; Malmö, Sweden) revealing important differences in Young′s modulus and porosities. The specimens with high and low volume fractions were best represented by linear and quadratic tetrahedrons, respectively. These results could be used to develop new osteoporosis-prevention strategies.

Development of ANCF tetrahedral finite elements for the nonlinear dynamics of flexible structures

In this paper, methods for developing isoparametric tetrahedral finite elements (FE) based on the absolute nodal coordinate formulation (ANCF) are presented. The proposed ANCF tetrahedral elements have twelve coordinates per node that include three position and nine gradient coordinates. The fundamental differences between the coordinate parametrizations used for conventional finite elements and the coordinate parametrizations employed for the proposed ANCF tetrahedral elements are discussed. Two different parametric definitions are introduced: a volume parametrization based on coordinate lines along the sides of the tetrahedral element in the straight (un-deformed) configuration and a Cartesian parametrization based on coordinate lines directed along the global axes. The volume parametrization facilitates the development of a concise set of shape functions in a closed form, and the Cartesian parametrization serves as a unique standard for the element assembly. A linear mapping based on the Bezier geometry is used to systematically define the cubic position fields of ANCF tetrahedral elements: the complete polynomial-based eight-node mixed-coordinate and the incomplete polynomial-based four-node ANCF tetrahedral elements. An element transformation matrix that defines the relationship between the volume and Cartesian parametrizations is developed and used to convert the parametric gradients to structure gradients, thereby allowing for the use of a standard FE assembly procedure. A general computational approach is employed to formulate the generalized inertia, external, and elastic forces. The performance of the proposed ANCF tetrahedral elements is evaluated by comparison with the conventional linear and quadratic tetrahedral elements and also with the ANCF brick element. In the case of small deformations, the numerical results obtained show that all the tetrahedral elements considered can correctly produce rigid body motion. In the case of large deformations, on the other hand, the solutions of all the elements considered are in good agreement, provided that appropriate mesh sizes are used.

Finite deformation analysis of visco-hyperelastic materials via solid tetrahedral finite elements

An alternative solid finite element formulation for large deformation analysis of viscoelastic materials is proposed. This new approach is based on positions and makes possible a robust implementation of an isoparametric solid tetrahedral that presents no locking when dealing with complex stress, strain and strain rate for general structural analysis. A consistent way to write internal variables that accounts for finite viscoelastic strains is proposed. In this alternative methodology the neo-Hookean hyperelastic law is taken into account together with the Zener viscoelastic model. The evolution law is described in terms of a rate equation involving the viscous right Cauchy-Green stretch tensor. The study is dedicated to homogeneous materials under isothermal and quasi-static conditions. The nonlinear solution procedure is performed via the Newton-Raphson iterative technique and the backward-Euler method.Four illustrative examples involving large viscoelastic strains are analyzed: uniaxial tension, simple shear, buckling of a clamped column and partially loaded block. The present formulation can reproduce creep, stress relaxation and viscoelastic rate dependent stiffening at large strains, which are usually observed in polymeric materials. Even for very complex stress, strain and strain rates the mesh refinement of the proposed methodology leads to more accurate results, avoiding general locking problems. The effect of the viscosity parameter on the material response and the evolution of viscous stretches over time are also highlighted in the results.

Finite-element analysis for model parameters distributed on a hierarchy of geometric simplices

A resurgence of interest in the problem of electrical or electromagnetic scattering from thin conductors, either on a line or in a plane, has been motivated in recent years by time-lapse fracture monitoring in the near surface, enhanced geothermal reservoirs, and unconventional hydrocarbon plays. However, finite-element modeling of small electrical features in large computational domains results in a disproportionately large number of elements concentrated in a volumetrically insignificant fraction of the mesh, and it focuses computational resources away from areas of interest elsewhere, such as receiver locations. We have developed a novel hierarchical electrical model is proposed for unstructured tetrahedral finite-element meshes, in which the usual volume-based conductivity on tetrahedra is augmented by facet- and edge-based conductivity on the infinitesimally thin regions between elements. Doing so allows a slender borehole casing of arbitrary shape to be coarsely approximated by set of connected edges on which a conductivity-area product is explicitly defined. Similarly, conductive fractures are approximated by a small number of connected facets that, together, may warp and bend with the mesh topology at no added cost of localized mesh refinement. Benchmarking tests of the direct current (DC) resistivity problem indicate excellent agreement between the facet/edge representations and independent analytic solutions. Consistency tests are also favorable between the facet/edge and volume representations. Building on prior work in DC modeling of a single horizontal well, a multilateral well casing and fracture set is simulated, yielding estimates of borehole casing voltage and surface electric fields measurable with existing sensor technology. Finally, the implications on broadband electromagnetic simulation with the proposed hierarchical conductivity model are discussed — in particular, its utility for describing variations in the magnetic permeability and electrical conductivity.

The effect of screw tunnels on the biomechanical stability of vertebral body after pedicle screws removal: a finite element analysis

Posterior reduction and pedicle screw fixation is a widely used procedure for thoracic and lumbar vertebrae fractures. Usually, the pedicle screws would be removed after the fracture healing and screw tunnels would be left. The aim of this study is to evaluate the effect of screw tunnels on the biomechanical stability of the lumbar vertebral body after pedicle screws removal by finite element analysis (FEA). First, the CT values of the screw tunnels wall in the fractured vertebral bodies were measured in patients whose pedicle screws were removed, and they were then compared with the values of vertebral cortical bone. Second, an adult patient was included and the CT images of the lumbar spine were harvested. Three dimensional finite element models of the L1 vertebra with unilateral or bilateral screw tunnels were created based on the CT images. Different compressive loads were vertically acted on the models. The maximum loads which the models sustained and the distribution of the force in the different parts of the models were recorded and compared with each other. The CT values of the tunnels wall and vertebral cortical bone were 387.126±62.342 and 399.204±53.612, which were not statistically different (P=0.149). The models of three dimensional tetrahedral mesh finite element of normal lumbar 1 vertebra were established with good geometric similarity and realistic appearance. After given the compressive loads, the cortical bone was the first one to reach its ultimate stress. The maximum loads which the bilateral screw tunnels model, unilateral screw tunnel model, and normal vertebral model can sustain were 3.97 Mpa, 3.83 Mpa, and 3.78 Mpa, respectively. For the diameter of the screw tunnels, the model with a diameter of 6.5 mm could sustain the largest load. In addition, the stress distributing on the outside of the cortical bone gradually decreased as the thickness of the tunnel wall increased. Based on the FEA, pedicle screw tunnels would not decrease the biomechanical stability and strength of the vertebral body. A large diameter of screw tunnel and thick tunnel wall were helpful for the biomechanical stability of the vertebral body.