Quadratic Hexahedral Elements Research Articles

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.

Predicting large-scale pool fire dynamics using an unsteady flamelet- and large-eddy simulation-based model suite

A low-Mach, unstructured, large-eddy-simulation-based, unsteady flamelet approach with a generalized heat loss combustion methodology (including soot generation and consumption mechanisms) is deployed to support a large-scale, quiescent, 5-m JP-8 pool fire validation study. The quiescent pool fire validation study deploys solution sensitivity procedures, i.e., the effect of mesh and time step refinement on capturing key fire dynamics such as fingering and puffing, as mesh resolutions approach O(1) cm. A novel design-order, discrete-ordinate-method discretization methodology is established by use of an analytical thermal/participating media radiation solution on both low-order hexahedral and tetrahedral mesh topologies in addition to quadratic hexahedral elements. The coupling between heat losses and the flamelet thermochemical state is achieved by augmenting the unsteady flamelet equation set with a heat loss source term. Soot and radiation source terms are determined using flamelet approaches for the full range of heat losses experienced in fire applications including radiative extinction. The proposed modeling and simulation paradigm are validated using pool surface radiative heat flux, maximum centerline temperature location, and puffing frequency data, all of which are predicted within 10% accuracy. Simulations demonstrate that under-resolved meshes predict an overly conservative radiative heat flux magnitude with improved comparisons as compared to a previously deployed hybrid Reynolds-averaged Navier–Stokes/eddy dissipation concept-based methodology.

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.

Study on the Accuracy of Structural and FSI Heart Valves Simulations.

The performance of heart valves, either native or artificial, can be evaluated by means of finite element analyses, either from a structural or a fluid-structure interaction (FSI) point of view. The latter captures the coupling between the valve leaflets and the blood in a more realistic way. The selection of the appropriate finite elements approach for the model is the first and fundamental step to achieve accurate simulations. The aim of this work is to investigate the influence of the type, formulation, size, and shape of the elements in heart valves simulations. The effects related to the choice of the finite elements-shell or solid- in structural and FSI simulations were analyzed. In particular, the analysis of grid convergence on both the structure and fluid domains, the influence of the element typology, formulation and damping factor in an idealized three-leaflets valve model loaded with physiological pressure conditions were investigated. Stress values and valve kinematics results confirmed the importance of performing a proper verification process for selecting the most appropriate elements with the optimal accuracy to computational cost ratio. In this regard, our results indicate the quadrangular shell with reduced integration and viscous hourglass control to be the best choice to model heart valves. If a solid discretization is required, quadratic hexahedral elements with full integration are also acceptable. Finally, our results show that the damping coefficient needs to be carefully selected in order to smooth out the high frequency modes of the structure without introducing excessive numerical artificial viscosity.

Electromechanical instability on dielectric polymer surface: Modeling and experiment

We present a dynamic finite element formulation for dielectric elastomers that significantly alleviates the problem of volumetric locking that occurs due to the incompressible nature of the elastomers. We accomplish this by modifying the Q1P0 formulation of Simo et al. [1], and adapting it to the electromechanical coupling that occurs in dielectric elastomers. We demonstrate that volumetric locking has a significant impact on the critical electric fields that are necessary to induce electromechanical instabilities such as creasing and cratering in dielectric elastomers, and that the locking effects are most severe in problems related to recent experiments that involve significant constraints upon the deformation of the elastomers. We then compare the results using the new Q1P0 formulation to that obtained using standard 8-node linear and 27-node quadratic hexahedral elements to demonstrate the capability of the proposed approach. Finally, direct comparison to the recent experimental work on the creasing instability on dielectric polymer surface by Wang et al. [2] is presented. The present formulation demonstrates good agreement to experiment for not only the critical electric field for the onset of the creasing instability, but also the experimentally observed average spacing between the creases.

Finite element dependence of stress evaluation for human trabecular bone

Numerical simulation using finite element models (FEM) has become more and more suitable to estimate the mechanical properties of trabecular bone. The size and kind of elements involved in the models, however, may influence the results. The purpose of this study is to analyze the influence of hexahedral elements formulation on the evaluation of mechanical stress applied to trabeculae bone during a compression test simulation.Trabecular bone cores were extracted from 18 L2 vertebrae (12 women and 6 men, mean age: 76±11, BV/TV=7.5±1.9%). Samples were micro-CT scanned at 20μm isotropic voxel size. Micro-CT images have been sub-sampled (20, 40 and 80μm) to create 5.6mm cubic FEM. For each sample, a compression test FEM has been created, using either 8-nodes linear hexahedral elements with full or reduced integration or 20-nodes quadratic hexahedral elements fully integrated, resulting in nine models per samples. Bone mechanical properties have been assumed isotropic, homogenous and to follow a linear elastic behavior law (Young modulus: 8GPa, Poisson ratio: 0.3).Despite micro-architecture modifications (loss of connectivity, trabeculae thickening) due to voxel size increase, apparent mechanical properties calculated with low resolution models are significantly correlated with high resolution results, no matter the element formulation. However, stress distributions are more sensitive to both resolution and element formulation modifications. With linear elements, increasing voxel size leads to an alteration of stress concentration areas due to stiffening errors. On the opposite, the use of reduced integration induces severe smoothing and underestimation of stress fields resulting in stress raisers loss. Notwithstanding their high computational cost, quadratic elements are most appropriate for stress prediction in low resolution trabecular bone FEM. These observations are dependent on trabecular bone micro-architecture, and are more significant for low density sample displaying low trabecular thickness.In conclusion, we found that element formulation is almost important as element size when evaluating trabecular bone mechanical behavior at trabeculae scale. Therefore, element type should be chosen carefully when evaluating trabecular bone behavior using FEM.

The 3D Incompatible Element Modeling of Flexible Body in Multibody System Dynamics

An efficient 3D incompatible element for the simulation of Flexible Multibody System (FMBS) is derived based on the principle of virtual work and the theory of continuum mechanics. The presented incompatible element uses absolute coordinates and internal parameters for the degrees of freedom and fixed the locking problem under the state of pure bending. Considering the geometric nonlinearities and using the Total Lagrangian Formulation, system dynamic equations are derived and then solved by the Backward Difference Formula (BDF) in conjunction with Newton-Raphson algorithm. Eigenvalue analysis and dynamic analysis of several numerical examples are presented to show the convergence of the 3D incompatible element and excellent performance in terms of accuracy and time efficiency comparing with standard linear and quadratic hexahedral elements.

Numerical Simulation of Convection-Diffusion Problems by the Control-Volume-Based Finite-Element Method

This work presents a numerical study of the tri-dimensional convection-diffusion equation by the control-volume-based on finite-element method using quadratic hexahedral elements. Considering that the equation governing this problem in its main variable may represent several properties, including temperature, turbulent kinetic energy, viscous dissipation rate of the turbulent kinetic energy, specific dissipation rate of the turbulent kinetic energy, or even the concentration of a contaminant in a given medium, among others, the wide applicability of this problem is thus evidenced. Three cases of temperature distributions will be studied specifically in this work, in addition to one case of pollutant dispersion upon analysis of the concentration of a contaminant in a fixed flow point. Some comparisons will be carried out against works found in the open literature, while others will be done according to each phenomenon characteristics.

On a compact mixed-order finite element for solving the three-dimensional incompressible Navier-Stokes equations

Our work is an extension of the previously proposed multivariant element. We assign this refined element as a compact mixed-order element in the sense that use of this element offers a much smaller bandwidth. The analysis is implemented on quadratic hexahedral elements with a view to analysing a three-dimensional incompressible viscous flow problem using a method formulated within the mixed finite element context. The idea of constructing such a stable element is to bring the marker-and-cell (MAC) grid lay-out to the finite element context. This multivariant element can thus be classified as a discontinuous pressure element. We have several reasons for advocating the proposed multivariant element. The primary advantage gained is its ability to reduce the bandwidth of the matrix equation, as compared with its univariant counterparts, so that it can be effectively stored in a compressed row storage (CRS) format. The resulting matrix equation can be solved efficiently by a multifrontal solver owing to its reduced bandwidth. The coding is, however, complicated by the appearance of restricted degrees of freedom at mid-face nodes. Through analytic study this compact multivariant element has a marked advantage over the multivariant element of Gupta et al. in that both bandwidth and computation time have been drastically reduced. © 1997 John Wiley & Sons, Ltd.

An adaptive multigrid technique for three‐dimensional elasticity

AbstractA program for finite element analysis of 3D linear elasticity problems is described. The program uses quadratic hexahedral elements. The solution process starts on an initial coarse mesh; here error estimators are determined by the standard Babuška‐Rheinboldt method and local refinement is performed by partitioning of indicated elements, each hexahedron into eight new elements. Then the discrete problem is solved on the second mesh and the refinement process proceeds in the following way‐on the ith mesh only the elements caused by refinement on the (i‐1)th mesh can be refined. The control of refinement is the task of the user because the dimension of the discrete problem grows very rapidly in 3D. The discrete problem is being solved by the frontal solution method on the initial mesh and by a newly developed and very efficient local multigrid method on the refined meshes. The program can be successfully used for solving problems with structural singularities, such as re‐entrant corners and moving boundary conditions. A numerical example shows that such problems are solved with the same efficiency as regular problems.