Quadratic Tetrahedral Elements Research Articles

Numerical Integration of Some Arbitrary Functions over an Ellipsoid by Discretizing into Hexahedral Elements for Biomaterial Studies

This study mathematically examines chemical and biomaterial models by employing the finite element method. Unshaped biomaterials’ complex structures have been numerically analyzed using Gaussian quadrature rules. It has been analyzed for commercial benefits of chemical engineering and biomaterials as well as biorefinery fields. For the computational work, the ellipsoid has been taken as a model, and it has been transformed by subdividing it into six tetrahedral elements with one curved face. Each curved tetrahedral element is considered a quadratic and cubic tetrahedral element and transformed into standard tetrahedral elements with straight faces. Each standard tetrahedral element is further decomposed into four hexahedral elements. Numerical tests are presented that verify the derived transformations and the quadrature rules. Convergence studies are performed for the integration of rational, weakly singular, and trigonometric test functions over an ellipsoid by using Gaussian quadrature rules and compared with the generalized Gaussian quadrature rules. The new transformations are derived to compute numerical integration over curved tetrahedral elements for all tests, and it has been observed that the integral outcomes converge to accurate values with lower computation duration.

A finite element approach for nonlinear, transient heat conduction problems with convection, radiation or contact boundary conditions

Heat conduction problems with convection and/or radiation boundary conditions arise in nuclear and other engineering fields. This paper presents a finite element approach to solve transient, three-dimensional (3-D), nonlinear heat conduction problems using quadratic tetrahedral elements with area/volume coordinates and quadrature integrations. Boundary conditions modeled include surface convection, to-ambient radiation, and surface-to-surface radiation (using the radiosity matrix method, RMM). Instead of the commonly used hemi-cube method, L′Huilier’s theorem is used here to calculate view factors. Contact constraints are also imposed. Interface condition is set up for the interfaces of different material. The one- and two-step backward differentiation are used for time integration. Nonlinearities introduced by temperature-dependent material thermal properties and the Stefan–Boltzmann law for radiation are handled using the Picard and/or Newton–Raphson methods. The convergence rate of the numerical scheme is verified using an analytical solution obtained using the method of manufactured solutions (MMS). Several practical cases are simulated, including that of two spheres in contact with each other, and results are compared to those obtained using commercial codes.

Robust hybrid/mixed finite elements for rubber-like materials under severe compression

A new family of hybrid/mixed finite elements optimized for numerical stability is introduced. It comprises a linear hexahedral and quadratic hexahedral and tetrahedral elements. The element formulation is derived from a consistent linearization of a well-known three-field functional and related to Simo–Taylor–Pister (STP) elements. For the quadratic hexahedral and tetrahedral elements we derive (static reduced) discontinuous hybrid elements, as well as continuous mixed finite elements with additional primary unknowns for the hydrostatic pressure and the dilation, whereas the linear hexahedral element is of the discontinuous type. The elements can readily be used in combination with any isotropic, invariant-based hyperelastic material model and can be considered as being locking-free. In a representative numerical benchmark test the elements numerical stability is assessed and compared to STP-elements and the family of discontinuous hybrid elements implemented in the commercial finite element code Abaqus/Standard. The new elements show a significant advantage concerning the numerical robustness.

A reduced unified continuum formulation for vascular fluid–structure interaction

We recently derived the unified continuum and variational multiscale formulation for fluid–structure interaction (FSI) using the Gibbs free energy as the thermodynamic potential. Restricting our attention to vascular FSI, we now reduce this formulation in arbitrary Lagrangian–Eulerian (ALE) coordinates by adopting three common modeling assumptions for the vascular wall. The resulting semi-discrete formulation, referred to as the reduced unified continuum formulation, achieves monolithic coupling of the FSI system in the Eulerian frame through a simple modification of the fluid boundary integral. While ostensibly similar to the semi-discrete formulation of the coupled momentum method introduced by Figueroa et al., its underlying derivation does not rely on an assumption of a fictitious body force in the elastodynamics sub-problem and therefore represents a direct simplification of the ALE method. Furthermore, uniform temporal discretization of the entire FSI system is performed via the generalized-α scheme. In contrast to the predominant approach yielding only first-order accuracy for pressure, we collocate both pressure and velocity at the intermediate time step to achieve uniform second-order temporal accuracy. In conjunction with quadratic tetrahedral elements, our methodology offers higher-order temporal and spatial accuracy for quantities of clinical interest, including pressure and wall shear stress. Furthermore, without loss of consistency, a segregated predictor multi-corrector algorithm is developed to preserve the same block structure as for the incompressible Navier–Stokes equations in the implicit solver’s associated linear system. Block preconditioning of a monolithically coupled FSI system is therefore made possible for the first time. Compared to alternative preconditioners, our three-level nested block preconditioner, which achieves improved representation of the Schur complement, demonstrates robust performance over a wide range of physical parameters. We present verification of our methodology against Womersley’s deformable wall theory and additionally develop practical modeling techniques for clinical applications, including tissue prestressing. We conclude with an assessment of our combined FSI technology in two patient-specific cases.

A Large-Scale Comparison of Tetrahedral and Hexahedral Elements for Solving Elliptic PDEs with the Finite Element Method

The Finite Element Method (FEM) is widely used to solve discrete Partial Differential Equations (PDEs) in engineering and graphics applications. The popularity of FEM led to the development of a large family of variants, most of which require a tetrahedral or hexahedral mesh to construct the basis. While the theoretical properties of FEM basis (such as convergence rate, stability, etc.) are well understood under specific assumptions on the mesh quality, their practical performance, influenced both by the choice of the basis construction and quality of mesh generation, have not been systematically documented for large collections of automatically meshed 3D geometries. We introduce a set of benchmark problems involving most commonly solved elliptic PDEs, starting from simple cases with an analytical solution, moving to commonly used test problem setups, and using manufactured solutions for thousands of real-world, automatically meshed geometries. For all these cases, we use state-of-the-art meshing tools to create both tetrahedral and hexahedral meshes, and compare the performance of different element types for common elliptic PDEs. The goal of this benchmark is to enable comparison of complete FEM pipelines, from mesh generation to algebraic solver, and exploration of relative impact of different factors on the overall system performance. As a specific application of our geometry and benchmark dataset, we explore the question of relative advantages of unstructured (triangular/ tetrahedral) and structured (quadrilateral/hexahedral) discretizations. We observe that for Lagrange-type elements, while linear tetrahedral elements perform poorly, quadratic tetrahedral elements perform equally well or outperform hexahedral elements for our set of problems and currently available mesh generation algorithms. This observation suggests that for common problems in structural analysis, thermal analysis, and low Reynolds number flows, high-quality results can be obtained with unstructured tetrahedral meshes, which can be created robustly and automatically. We release the description of the benchmark problems, meshes, and reference implementation of our testing infrastructure to enable statistically significant comparisons between different FE methods, which we hope will be helpful in the development of new meshing and FEA techniques.

Prediction of Optimal Numbers of Quadrature Points for Element Matrices with Deep Learning

Element stiffness matrices in the FEM are usually calculated with numerical quadrature, such as the Gauss-Legendre quadrature. The accuracy of the quadrature for an element depends on its shape. Deep learning can find the optimal number of quadrature points for an element, which can improve the computational efficiency. In this paper, the DL-based prediction of the optimal number of quadrature points of finite elements is applied to quadratic elements, such as quadratic hexahedral elements and quadratic tetrahedral elements, where non-corner nodes effect the convergence of the numerical quadrature. Basic properties of the proposed method for quadratic elements are investigated in detail through some numerical examples.

A machine learning approach to predict the stress results of quadratic tetrahedral elements

Industries are always looking for an effective and efficient way to reduce the computation time of simulation because of the huge expenditure involved. From basics of Finite Element Method (FEM), it is known that, linear order finite elements consume less computation time and are less accurate compared to higher order finite elements say quadratic elements. An approach to get the benefit of less computation cost of linear elements and the good accuracy of quadratic elements can be of a good thought. The methodology to get the accurate results of quadratic elements with the advantage of less simulation run time of linear elements is presented here. Machine Learning (ML) algorithms are found to be effective in making predictions based on some known data set. The present paper discusses a methodology to implement ML model to predict the results equivalent to that of quadratic elements based on the solutions obtained from the linear elements. Here, a ML model is developed using python code, the stress results from Finite Element (FE) model of linear tetrahedral elements is given as the input to it to predict the stress results of quadratic tetrahedral elements. Abaqus is used as the FEM tool to develop the FE models. A python script is used to extract the stresses and the corresponding node numbers. The results showed that the developed ML model is successful in prediction of the accurate stress results for the set of test data. The scatter plots showed that the Z-score method was effective in removing the singularities. The proposed methodology is effective to reduce the computation time for simulation.

Quadratic tetrahedral micropolar element for the vibration analysis of three-dimensional micro-structures

The micropolar continuum theory (MPT) is able to describe the influence of micro-structures on the mechanical behavior of materials where the classical theories of elasticity are unable to interpret. In this paper, the free vibration analysis of three-dimensional (3D) micropolar structures with different geometries is presented. To this aim, first, based on the 3D linear elasticity, MPT is formulated. Then a novel size-dependent quadratic tetrahedral element is developed by the user element subroutine (UEL) within commercial finite element software ABAQUS to study the free vibration behavior of micropolar structures with various geometries. The length scale parameter effects on the dimensionless natural frequencies of micropolar structures with various geometries are studied.

Sensitivity of the stress field of the proximal femur predicted by CT-based FE analysis to modeling uncertainties.

Proximal femur anatomy and bone mineral density vary widely among individuals, precluding the use of one predefined finite element (FE) model to determine the stress field for all proximal femurs. This variability poses a challenge in current prosthetic hip design approach. Given the numerous options for generating computed tomography (CT)-based FE models, selecting the best methods for defining the mechanical behavior of the proximal femur is difficult. In this study, a combination of computational and experimental approaches was used to explore the susceptibility of the predicted stress field of the proximal femur to different combinations of density-elasticity relationships, element type, element size, and calibration error. Our results suggest that FE models with first-order voxelized elements generated by the Keyak and Falkinstein density-elasticity relationship or quadratic tetrahedral elements generated by the Morgan density-elasticity relationship lead to accurate estimations of the mechanical behavior of human femurs. Other combinations of element size, element type, and mathematical relationships produce less accurate results, especially in the cortical bone of the femoral neck and calcar region. The voxelized model was more susceptible to variation of element size and density-elasticity relationships than FE models with quadratic tetrahedral elements. Regardless of element type, the stress fields predicted by the Keyak and Falkinstein and the Morgan relationships were the most robust to calibration error when deriving material density from CT-generated Hounsfield data. These results provide insight into the implementation of a robust platform for designing patient-specific implants capable of maintaining or modifying the stress in bones.

Extending a 10‐node composite tetrahedral finite element for solid mechanics

AbstractWe propose to extend the composite tetrahedral finite element first introduced by Thoutireddy et al. (2002) and recently reformulated by Ostien et al. (2016). We generalize the gradient operator and mass matrix to curved domains through analytical expressions weighted by subtetrahedra Jacobians. Optimal integration weights are constructed to increase the accuracy of Gaussian quadrature. We preserve the variational structure of the formulation through a new five‐field functional with additional, independent fields for the Jacobian and the pressure. This approach not only obviates volumetric locking but also yields symmetry. A deleterious soft mode, common to both the quadratic and composite tetrahedral element with constant pressure formulations is effectively stabilized through a novel convex energy penalty function. Numerous numerical examples spanning a patch test to the impact of a Taylor bar demonstrate the accuracy, robustness, and convergence of the extended composite tetrahedral element for application to structural metals.

Variationally derived closure models for large eddy simulation of incompressible turbulent flows

AbstractWe present a variationally consistent method for deriving residual‐based closure models for incompressible Navier–Stokes equations. The method is based on the fine‐scale variational structure facilitated by the variational multiscale framework where fine scales are driven by the residuals of the Euler–Lagrange equations of the resolved scales in the balance of momentum and conservation of mass equations. A bubble‐functions based approach is applied directly to the fine‐scale variational equation to derive analytical expressions for the closure model. Variational consistency of the model lends itself to rigorous linearization that results in quadratic rate of convergence of the method in the iterative solution strategy for the nonlinear equations. The method is shown to work for a family of linear and quadratic hexahedral and tetrahedral elements as well as composite discretizations that are comprised of hexahedral and tetrahedral elements in the same computational domain. Numerical tests with the proposed model are presented for various classes of turbulent flow problems to show its generality and range of applicability. The test cases investigated include Taylor–Green vortex stretching, statistically stationary wall‐bounded channel flows, and modeling the effects of the geometry of the leading edge of the plate on the instability of the boundary layer that leads to flow separation and flow reversal over flat plates of finite thickness.

A numerical study into element type and mesh resolution for crystal plasticity finite element modeling of explicit grain structures

A large number of massive crystal-plasticity-finite-element (CPFE) simulations are performed and post-processed to reveal the effects of element type and mesh resolution on accuracy of predicted mechanical fields over explicit grain structures. A CPFE model coupled with Abaqus/Standard is used to simulate simple-tension and simple-shear deformations to facilitate such quantitative mesh sensitivity studies. A grid-based polycrystalline grain structure is created synthetically by a phase-field simulation and converted to interface-conformal hexahedral and tetrahedral meshes of variable resolution. Procedures for such interface-conformal mesh generation over complex shapes are developed. FE meshes consisting of either hexahedral or tetrahedral, fully integrated as linear or quadratic elements are used for the CPFE simulations. It is shown that quadratic tetrahedral and linear hexahedral elements are more accurate for CPFE modeling than linear tetrahedral and quadratic hexahedral elements. Furthermore, tetrahedral elements are more desirable due to fast mesh generation and flexibility to describe geometries of grain structures.

Novel quadratic Bézier triangular and tetrahedral elements using existing mesh generators: Applications to linear nearly incompressible elastostatics and implicit and explicit elastodynamics

SummaryIn this paper, we present novel techniques of using quadratic Bézier triangular and tetrahedral elements for elastostatic and implicit/explicit elastodynamic simulations involving nearly incompressible linear elastic materials. A simple linear mapping is proposed for developing finite element meshes with quadratic Bézier triangular/tetrahedral elements from the corresponding quadratic Lagrange elements that can be easily generated using the existing mesh generators. Numerical issues arising in the case of nearly incompressible materials are addressed using the consistent B‐bar formulation, thus reducing the finite element formulation to one consisting only of displacements. The higher‐order spatial discretization and the nonnegative nature of Bernstein polynomials are shown to yield significant computational benefits. The optimal spatial convergence of the B‐bar formulation for the quadratic triangular and tetrahedral elements is demonstrated by computing error norms in displacement and stresses. The applicability and computational efficiency of the proposed elements for elastodynamic simulations are demonstrated by studying several numerical examples involving real‐world geometries with complex features. Numerical results obtained with the standard linear triangular and tetrahedral elements are also presented for comparison.

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.

A three-dimensional coupled thermo-hydro-mechanical model for deformable fractured geothermal systems

A fully coupled thermal-hydraulic-mechanical (THM) finite element model is presented for fractured geothermal reservoirs. Fractures are modelled as surface discontinuities within a three-dimensional matrix. Non-isothermal flow through the rock matrix and fractures are defined and coupled to a mechanical deformation model. A robust contact model is utilised to resolve the contact tractions between opposing fracture surfaces under THM loadings. A numerical model has been developed using the standard Galerkin method. Quadratic tetrahedral and triangular elements are used for spatial discretisation. The model has been validated against several analytical solutions, and applied to study the effects of the deformable fractures on the injection of cold water in fractured geothermal systems.Results show that the creation of flow channelling due to the thermal volumetric contraction of the rock matrix is very likely. The fluid exchanges heat with the rock matrix, which results in cooling down of the matrix, and subsequent volumetric deformation. The cooling down of the rock matrix around a fracture reduces the contact stress on the fracture surfaces, and increases the fracture aperture. Stress redistribution reduces the aperture, as the area with lower contact stress on the fracture expands. Stress redistribution reduces the likelihood of fracture propagation under pure opening mode, while the expansion of the area with lower contact stress may increase the likelihood of shear fracturing.

Finite element simulations of interactions between multiple hydraulic fractures in a poroelastic rock

A fully coupled three-dimensional finite-element model for hydraulic fracturing in permeable rocks is utilised to investigate the interaction between multiple simultaneous and sequential hydraulic fractures. Fractures are modelled as surface discontinuities within a three-dimensional matrix. This model simultaneously accounts for laminar flow within the fracture, Darcy flow within the rock matrix, poroelastic deformation of the rock, and the propagation of fractures using a linear elastic fracture mechanics framework. The leakoff of fracturing fluid into the surrounding rocks is defined as a function of the pressure gradient at the fracture surface, the fluid viscosity, and the matrix permeability. The coupled equations are solved numerically using the finite element method. Quadratic tetrahedral and triangle elements are used for spatial discretisation of volumes and surfaces, respectively. The model is validated against various analytical solutions for plane-strain and penny-shaped hydraulic fractures. Several cases of simultaneous fracturing of multiple hydraulic fractures are simulated in which the effects of the various parameters (the in situ stresses, the distance between fractures, the permeability of the matrix, the Biot poroelastic coefficient, and the number of the fractures in a group) are investigated. The results show that the stress induced by the opening of the fractures, and the stress induced by the fluid leakoff, each have the effect of locally altering the magnitudes and orientations of the principal stresses, hence altering the propagation direction of the fractures. Opening of a fracture induces excessive compression (also known as the “stress shadow”) that causes adjacent fractures to curve away from each other. This excessive compression competes against the differential in situ stresses, which tend to cause the fracture to grow in the plane normal to the minimum in situ stress. The stress shadow effect is reduced by increasing the distance between fractures, and is increased by increasing the leakoff, which may be due to increased permeability of the rock, or an increase in the Biot coefficient.

Nonlinear three–dimensional finite–element modelling of reinforced–concrete beams: Computational challenges and experimental validation

Abstract Three–dimensional nonlinear finite–element models have been developed to investigate the loading–unloading–reloading behaviour of two reinforced–concrete beams under four–point bending using explicit dynamics in ABAQUS. The damaged–plasticity model proposed by Lubliner and collaborators was employed for the plain concrete, and elastic–perfectly plastic models were employed for the steel reinforcement. A perfect bond was assumed between the steel rebars and concrete, whereby the bond–slip behaviour, as well as damage along crack patterns, were modelled through concrete damage. The influence of the shape of the tension–softening law on the numerical load–deflection response was studied by considering bi–linear, exponential and linear post–failure stress–displacement and stress–strain relationships. The effect of modelling steel rebars with truss or beam elements was also investigated. Structured meshes of linear hexahedral elements either with incompatible modes or with reduced integration, and unstructured meshes of either linear or ‘modified’ quadratic tetrahedral elements were considered. In terms of load–deflection curves, both the structured and the unstructured meshes gave results in very good agreement with test results. In terms of crack patterns, results predicted by the structured meshes exhibited some mesh bias, which was less pronounced with the unstructured meshes. In the post–yield phase, if a geometrically nonlinear model is used, discrepancies were found when truss elements were used for steel rebars, whereas good agreement was found if the bending stiffness of the rebars is included using beam elements. This is a non–obvious result that may be important to consider when studying the progressive collapse of RC structures.

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 element analysis of the compressive and shear responses of structural foams using computed tomography

This paper aim to establish a framework for finite element modelling of the compressive and shear responses of closed-cell foams using computed tomography images, and to explore the deformation and failure mechanisms at cell level in closed-cell foams under compression and shear. Results show that quadratic tetrahedral mesh is more suited than cubic voxel mesh in modelling closed-cell foams. The mesh density with one quadratic tetrahedral element across the thickness of cell walls can yield results with good accuracy. Representative volume elements for closed-cell foams need contain at least 5.2 cells and 3.9 cells along each edge for compression and shear tests, respectively. Compressive buckling and shear buckling appear during the initial elastic regime of compression and shear, respectively. Buckling acts as a failure initiator, which changes the deformation modes of cell walls from in-plane compression, tension or shear to bending. Subsequent large deflection of cell walls and material yielding lead to further degradation in the global stiffness of foams. Shear buckling causes cell walls to lose load-carrying capacity along the direction of the compressive component of the shear load, therefore is less detrimental than compressive buckling in reducing the load-bearing capacity of cell walls. Consequently, the global stiffness of closed-cell foams degrades less rapidly in shear tests compared to compression tests. The Young’s modulus of foam base materials slightly affects foam strengths. This is because cell walls undergo bending after buckling, and the maximum bending load that the buckled cell walls can bear is determined by the yield strength of base materials.

Stabilized mixed three‐field formulation for a generalized incompressible Oldroyd‐B model

SummaryThis paper presents a generalization of the incompressible Oldroyd‐B model based on a thermodynamic framework within which the fluid can be viewed to exist in multiple natural configurations. The response of the fluid is viewed as a combination of an elastic component and a dissipative component. The dissipative component leads to the evolution of the underlying natural configurations, while the response from the natural configuration to the current configuration is considered elastic and therefore non‐dissipative. For an incompressible fluid, it is necessary that both the elastic behavior as well as the dissipative behavior is isochoric. This is achieved by ensuring that the determinant of the stretch tensor associated with the elastic response meets the constraint that its determinant is unity.A new stabilized mixed method is developed for the velocity, pressure and the kinematic tensor fields. Analytical models for fine scale fields are derived via the solution of the fine‐scale equations facilitated by the Variational Multiscale framework that are then variationally embedded in the coarse‐scale variational equations. The resulting method inherits the attributes of the classical SUPG and GLS methods, while a significant new contribution is that the form of the stabilization tensors is explicitly derived. A family of linear and quadratic tetrahedral and hexahedral elements is developed with equal‐order interpolations for the various fields. Numerical tests are presented that validate the incompressibility of the elastic stretch tensor, show optimal L2 convergence for the conformation tensor field, and present stable response for high Weissenberg number flows. Copyright © 2016 John Wiley & Sons, Ltd.