Hexahedral Finite Element Research Articles

Three-dimensional airborne electromagnetic forward modeling based on multiscale hexahedral finite-element method

Slow forward modeling is the main factor that restricts the practical use of three-dimensional (3D) inversion and interpretation of airborne electromagnetic (AEM) data. To improve the modeling efficiency in 3D AEM, we propose a new multiscale finite-element (MsFE) method based on unstructured hexahedral meshes. Compared with the traditional 3D AEM forward modeling, the main advantage of our newly developed method is that it can simulate complex underground structures in the earth quickly. Since we can fit the earth's topography or the anomalous bodies underground using a small number of hexahedral grids, we can quickly model them using MsFE. The main idea of the MsFE forward modeling method is to construct an interpolation operator between a coarse and a dense mesh and use the interpolation operator to map the conventional FE coefficient matrix to the MsFE coefficient matrix and thus reduce the number of unknowns in the modeling process. This will vastly reduce the scale of the linear equations system. We validate our method by simulating a typical mountain peak model and demonstrate its effectiveness by simulating numerous synthetic models and a model from Voisey Bay's Ovoid sulfide deposit, Canada.

Dataset of Finite Element Models of Normal and Deformed Thoracolumbar Spine

Adult spine deformity (ASD) is prevalent and leads to a sagittal misalignment in the vertebral column. Computational methods, including Finite Element (FE) Models, have emerged as valuable tools for investigating the causes and treatment of ASD through biomechanical simulations. However, the process of generating personalised FE models is often complex and time-consuming. To address this challenge, we present a dataset of FE models with diverse spine morphologies that statistically represent real geometries from a cohort of patients. These models are generated using EOS images, which are utilized to reconstruct 3D surface spine models. Subsequently, a Statistical Shape Model (SSM) is constructed, enabling the adaptation of a FE hexahedral mesh template for both the bone and soft tissues of the spine through mesh morphing. The SSM deformation fields facilitate the personalization of the mean hexahedral FE model based on sagittal balance measurements. Ultimately, this new hexahedral SSM tool offers a means to generate a virtual cohort of 16807 thoracolumbar FE spine models, which are openly shared in a public repository.

Hexahedral finite elements with enhanced fixed‐pole interpolation for linear static and vibration analysis of 3D micropolar continuum

AbstractThe spotlight of this research is on the application of the fixed‐pole interpolation, sometimes used in the analysis of three‐dimensional (3D) geometrically non‐linear beams, but for which no attempts have been made to apply it to linear analysis so far. Particular attention is given to the correlation between the linearised forms of the fixed‐pole and helicoidal interpolation with the linked interpolation. Knowledge of this interdependence is crucial for identifying paths for possible enhancement and extension from the Timoshenko beam of arbitrary order to the new hexahedral finite element of arbitrary order for linear analysis of 3D micropolar continuum. After ensuring the convergence of the newly developed micropolar element through a set of patch tests, three numerical examples of a 3D micropolar continuum in static equilibrium and free vibration of 3D micropolar plates with different geometric properties and boundary conditions have been analysed. Based on these results, the newly proposed finite elements have been critically assessed against the conventional elements.

Incompatible-mode geometrically non-linear finite element for micropolar elasticity

In this work a new three-dimensional geometrically non-linear hexahedral micropolar finite element enhanced with incompatible modes is presented. The analytical model is expressed in terms of Biot-like stress and couple-stress tensors and corresponding Biot-like strain and curvature tensors, with a linear, elastic and isotropic constitutive law. The numerical model is derived based on the principle of virtual work, and the residual derivation together with the linearisation and static condensation procedure is given in detail. The newly developed finite element is tested against the analytical solution of the geometrically non-linear micropolar pure bending problem and the element accuracy and robustness is compared against hexahedral Lagrangian finite elements of first and second order on several numerical examples. It is shown that the newly presented element is fast convergent, more robust and more accurate than the available Lagrangian elements. Moreover, the operator split and static condensation provide for a significantly lower computational cost than standard elements.

Large Deformation Analysis of Hyperelastic Continuum with Hexahedral Adaptive Finite Elements

The use of hyperelastic materials capable of large deformations, such as elastomeric bearings used to reduce seismic effects, is quite common in civil engineering. Such environments are, in most cases, addressed by numerical solution techniques such as the finite element method. In case of large deformations, nonlinear analysis is used in the solution. In the study presented here, large deformations of a hyperelastic continuum expressed by the Mooney-Rivlin material model are calculated using hexahedral adaptive finite elements. A code was written in MATLAB using the total Lagrangian formulation for the nonlinear adaptive finite element solution. Comparisons were made with Abaqus software to check the consistency of the results obtained from this program. It has been observed that local refinements in the adaptive element mesh occur in the regions where they are needed. Considering the variation of maximum displacement and maximum stress with the number of elements, it has been observed that mesh refinement creates a convergent solution.

Research of generalized mixed hexahedral finite element method based on volume coordinates

The sensitivity of isoparametric elements to mesh distortions has not been sufficiently addressed. Based on the generalized H-R variational principle, the formula of the hexahedral volume coordinate mixed method (HVMM) is derived which applies volume coordinates as local coordinates, maintaining a linear relationship with the Cartesian coordinate system. The influence of the stress boundary conditions on the stress results is also discussed. Numerical results demonstrate the superiority of the proposed element against mesh distortion, and the accuracy of stress results is higher than that of the hexahedral volume coordinate method element (HVCC8) and other displacement elements.

Development, calibration and validation of a comprehensive customizable lumbar spine FE model for simulating fusion constructs.

Instrumentation alters the biomechanics of the spine, and therefore prediction of all output quantities that have critical influence post-surgically is significant for engineering models to aid in clinical predictions. Geometrical morphological finite element models can bring down the development time and cost of custom intact and instrumented models and thus aids in the better inference of biomechanics of surgical instrumentation on patient-specific diseased spine segments. A comprehensive hexahedral morphological lumbosacral finite element model is developed in this work to predict the range of motions, disc pressures, and facet contact forces of the intact and instrumented spine. Facet contact forces are needed to predict the impact of fusion surgeries on adjacent facet contacts in bending, axial rotation, and extension motions. Extensive validation in major physiological loading regimes of the pure moment, pure compression, and combined loading is undertaken. In vitro, experimental corridor results from six different studies reported in the literature are compared and the generated model had statistically significant comparable values with these studies. Flexion, extension and bending moment rotation curves of all segments of the developed model were favourable and within two separately established experimental corridor windows as well as recent simulation results. Axial torque moment rotation curves were comparable to in vitro results for four out of five lumbar functional units. The facet contact force results also agreed with in vitro experimental results. The current model is also computationally efficient with respect to contemporary models since it uses significantly smaller number of elements without losing the accuracy in terms of response prediction. This model can further be used for predicting the impact of different instrumentation techniques on the lumbar vertebral column.

Unified three-dimensional finite elements for large strain analysis of compressible and nearly incompressible solids

This work proposes a displacement-based finite element model for large strain analysis of isotropic compressible and nearly-incompressible hyperelastic materials. Constitutive law is written in terms of invariants of the right Cauchy-Green tensor; coupled and decoupled formulations of strain energy functions are presented, whereas a penalty function is used to impose an incompressibility constraint. Based on a total Lagrangian formulation, the nonlinear governing equations are thus obtained by employing the principle of virtual displacements. Analytic expression of both internal forces vector and tangent matrix of linear and high-order hexahedral finite elements are derived by adopting a three-dimensional formalism based on the Carrera Unified Formulation. Popular benchmark problems in hyperelasticity are analyzed to establish the capabilities of the present implementation of fully-nonlinear solid elements in the case of compressible and nearly-incompressible beams, cylindrical shells, and curved structures.

Verification of a High-Order FEM-based CFD Code using the Method of Manufactured Solutions

A high-order computational fluid dynamics (CFD) code capable of solving compressible turbulent flow problems was developed. The CFD code employs the Flowfield Dependent Variation (FDV) scheme implemented in a Finite Element Method (FEM) framework. The FDV scheme is basically derived from the Lax-Wendroff Scheme (LWS) involving the replacement of LWS’s explicit time derivatives with a weighted combination of explicit and implicit time derivatives. The code utilizes linear, quadratic and cubic isoparametric quadrilateral and hexahedral Lagrange finite elements with corresponding piecewise shape functions that have formal spatial accuracy of second-order, third-order and fourth-order, respectively. In this paper, the results of observed order-of-accuracy of the implemented FDV FEM-based CFD code involving grid and polynomial order refinements on uniform Cartesian grids are reported. The Method of Manufactured Solutions (MMS) is applied to governing 2-D Euler and Navier-Stokes equations for flow cases spanning both subsonic and supersonic flow regimes. Global discretization error analyses using discrete 𝐿2 norm show that the spatial order-of-accuracy of the FDV FEM-based CFD code converges to the shape function polynomial order plus one, in excellent agreement with theory. Uniquely, this procedure establishes the wider applicability of MMS in verifying the spatial accuracy of a high-order CFD code.

Mechanical Response of Thin Composite Beams Made of Functionally Graded Material Using Finite Element Method

Functionally Graded Material (FGM) is a new generation of composite materials, it can be used for different engineering fields according to the loading environment, but the study of its mechanical behavior requires sophisticated numerical and analytical models. Several investigations in these models are available in the literature, however, most of those investigations are based on simplifying assumptions. In this paper, we present a three-dimensional finite element modeling of functionally graded material (FGM) beams subjected to static loading. Material properties are assumed to vary continuously along the beam thickness according to the power-law distribution with linear elastic behavior. The FGM beams are discretized by hexahedral finite elements type C3D20R (continuum stress/displacement, three-dimensional 20-node, reduced integration). We studied several numerical examples of FGM beams and compare the obtained numerical results with those of analytical models in the literature.

3D Structural Topology Optimization Using ESO, SESO and SERA: Comparison and an Extension to Flexible Mechanisms

This article investigates the study of Topology Optimization (TO) in 3D elasticity problems to determine the optimal topology by applying the evolutionary methods of Smoothing Evolutionary Structural Optimization (SESO), Sequential Element Rejection and Admission (SERA), and Evolutionary Structural Optimization (ESO). These procedures were implemented in MATLAB code as an extension of Top3d implemented for SIMP by using the eight-node hexahedral finite element formulation in three-dimensional elastostatic structures. The approaches conducted in the present study are demonstrated with numerical examples involving the compliance minimization criterion. Further, a brief synthesis of flexible mechanisms was studied to emphasize the performance of complaint mechanisms measured in terms of two design specifications/functionalities: mechanical and geometrical advantages, which are the highlights of this article. To show the gains of the proposed methods, numerical results obtained are compared with Solid Isotropic Material with Penalization (SIMP) models.

Higher-Order Hexahedral Finite Elements for Structural Dynamics: A Comparative Review

The finite element method (FEM) is widely used in many engineering applications. The popularity of FEM led to the development of several variants of formulations, and hexahedral meshes surged as one of the most computationally effective. After briefly reviewing the reasons and advantages behind the formulation of increasing order elements, including the serendipity variants and the associated reduced integration schemes, a systematic comparison of the most common hexahedral formulations is presented. A numerical benchmark was used to assess convergency rates and computational efficiencies when solving the eigenvalue problem for linear dynamic analysis. The obtained results confirmed the superior performances of the higher-order brick element formulations. In terms of computational efficiency, defined as the ratio between achievable accuracy and computational execution time, quadratic or cubic formulations exhibited the best results for the stages of FE model assembly and solution computation, respectively.

MOMENT FINITE ELEMENT FOR SOLVING 3D ELASTICITY PROBLEMS

The implementation of a new 8-node finite element for solving 3D elasticity static problems is considered. The idea of this finite element is based on the projection of a rare mesh FEM scheme into a space of lower dimension. The resulting scheme of the 8-node hexahedral finite element has a number of features compared to the scheme of the classical 8-node polyline element. This is one integration point in the element compared to eight in the classical element, as well as the presence of four parameters that allow you to adjust the convergence of the computational process. In a given finite element, the stresses as well as their moments (three bending and one torsional) are assumed to be constant within the element. Also, the continuity of displacement fields is preserved only in the nodes of finite elements. This circumstance is not a disadvantage and is characteristic of many new numerical FEM schemes (suffice it to mention the recently actively developed direction of the discontinuous Galerkin method). The issues of using this finite element in solving the well-known problem of increased shear stiffness (shear locking) of a number of well-known FEM schemes are discussed. The problem of hourglass instability, which is typical for FEM schemes with reduced integration and a number of other numerical schemes, in particular, the Wilkins scheme, which is popular in solving dynamic elasticity and plasticity problems, is also considered. The implementation of a technique for the numerical solution of three-dimensional static problems of the theory of elasticity on the basis of a given finite element within the framework of the traditional FEM technique using a vector-matrix notation is described. The results of solving test static problems of elasticity theory are presented.

Double-precision hardware accelerator for incompressible Navier–Stokes equations solver based on discontinuous Galerkin method

We present the first known hardware accelerator for a complete solver based on the modal discontinuous Galerkin method (DGM) of full incompressible Navier–Stokes equations. The solver is designed for hexahedral finite elements and, at the current stage, the modal basis is limited to second-order polynomials giving 27 degrees of freedom per finite element. The solution algorithm employs the projection method for the treatment of the incompressibility constraint, but the Poisson pressure equation is replaced with a parabolic equation solved in pseudo-time. Our implementation requires about 105 floating-point operations (FLOP) for a single finite element per time step, and its arithmetic intensity is 1.8 FLOP/byte on average. We also designed the hardware based on a set of simple, simultaneously working microprogrammable computing units that contain only a floating-point arithmometer and a small memory. The design fits a medium-size field-programmable gate array (FPGA, we used Xilinx XCZU15EG) and, for all tested cases excluding one, achieves performance over 100 000 mesh elements per second (122 000 in peak) with a reasonable hardware utilisation: memory bandwidth between 8 and 9 GB/s (over 0.4 of the machine peak) and computational performance between 13 and 15 GFLOPS (almost 0.7 of the peak). The main reasons limiting hardware utilisation are: internal hardware limits in FPGA that prevent allocation of all DDR memory bandwidth for computations; and the differences in arithmetic intensities of different parts of the algorithm — the arithmetic intensities vary between 1.1 and 3.3 FLOP/byte.

FE mesh density influence on blast loading analysis

To examine the influence of mesh density on the blast loads of anti-landmines (ALMs) and improvised explosive devices (IEDs) under an armored vehicle, models with different FE mesh densities were numerically analysed. The multi-material arbitrary Lagrangian-Eulerian (MM-ALE) algorithm was used to simulate under armored vehicle explosions. Explicit dynamic analysis was performed in LS-DYNA. Blast loading simulation was performed using the ConWep (conventional weapon) loading model, which is implemented in LS-DYNA. The material characteristics of high-strength steel were used to model the protective plates on the armored vehicle. Protective plates were modeled in a V-shape and as a deformable solid with the Johnson-Cook plasticity model. In this paper, the FE model with different mesh densities was numerically tested to analyse the influence of mesh density on the obtained results. For the purposes of the modeling, different types of finite elements were used for the FE models. The floor of the vehicle was modeled with 3D hexahedral eight-noded finite elements, and the protective plates were modeled with four-noded plate finite elements. The maximal value of plastic strain and the maximal value of the displacement of the central node on the protective plate was chosen as the parameters to compare the obtained results.

A specific eight-node hexahedral finite element for the analysis of 3D fibre-reinforced composites

This contribution deals with the development of an eight-node hexahedral finite element to estimate the effective elastic properties of fibre-reinforced composite materials and structures. This three-dimensional (3D) composite element is a combination of the standard eight-node hexahedral element, modelling a unit of the matrix domain, and 3D two-node truss elements that represent fibres crossing each matrix element. The so-called ‘Projected Fibre Approach’ is then considered to express the truss elements variables in terms of their corresponding matrix elements. Consequently, the obtained system of equations size is equivalent to that of a non-reinforced medium. In particular, we show that the proposed finite element formulation allows readily studying 3D composite structures with complex shape of the reinforcement like stitched sandwich panels.

Three dimensional modeling and numerical analysis of hydrogen effects on the thermomechanical response of Nickel–Titanium orthodontic applications

This paper deals with effects of the hydrogen presence on the thermo-mechanical behavior of NiTi shape memory alloys (SMAs), widely considered for manufacturing orthodontic archwires. A three dimensional phenomenological model is formulated, based on thermodynamics of irreversible processes. It takes into account the experimentally observed effects on the main physical mechanisms (martensitic transformation and reorientation of martensite). The assumption of an additive decomposition between a reversible lattice and a trapped hydrogen concentrations is adopted. The complete problem including chemical, thermal and mechanical balanced equations and their weak form are solved by the finite element method. A 3D hexahedral continuum finite element with five degrees of freedom per node (displacements, temperature and total hydrogen concentration) is developed. Its implementation is carried out in the Abaqus finite element software. Numerical results are presented and compared to experimental ones which show the significant effects of the hydrogen presence in NiTi SMA.

Axial Compression and Buckling Analysis of Columnar Structures with Tetra-Anti-Chiral Configuration

Abstract The present work is focused on the investigation of tetra-anti-chiral structures by means of numerical and analytical methods. Specimens were evaluated under compressive load using analytical and numerical methods. The paper summarizes a theoretical solution for the estimate of Poisson’s ratio and the plateau force. The models can handle structures with various configurations, such as the radius of the connection node, lengths, and thickness of the ligaments. A section dedicated to the evaluation of the buckling load is included to extend the investigation of the behavior under compressive loads. The theoretical model is based on Euler’s formula, and a series of amendments are performed to adapt the formula to the analysis of chiral structures. Throughout the paper, theoretical results are compared with results from the simulations to validate the principles stated. Two sets of numerical models were developed: a fully 3D model using hexahedral finite elements and a 2.5D model using a beam finite element model. An overall comparison of results is presented, showing a good agreement between datasets. The present work might set the background for future activities, allowing for a selection of individual investigation methods.

An 8-Nodes 3D Hexahedral Finite Element and an 1D 2-Nodes Structural Element for Timoshenko Beams, Both Based on Hermitian Intepolation, in Linear Range

The following article presents the elaboration and results obtained from a 3D finite element, of the 8-node hexahedron type with 6 degrees of freedom (DOF) per node (48 DOF per element) based on third degree Hermitian polynomials, and of a 2-node structural element, with 6 DOF per node (12 DOF per element), based on third degree Hermitian polynomials and the theory of Timoshenko for beams. This article has two purposes; the first one is the formulation of a finite element capable of capturing bending effects, and the second one is to verify whether it is possible to obtain the deformation of the beam’s cross section of a structural member of the beam type, based on the deformations of its axis. The results obtained showed that the 8-node hexahedron FE was able to reproduce satisfactory results by simulating some cases of beams with different contour and load conditions, obtaining errors between 1% and 4% compared to the ANSYS software, educational version. Regarding the structural element of the beam type, it reproduced results that were not as precise as the FE Hexa 8, presenting errors of between 6% and 7% with regard to the axis but with error rounding between 10% and 20%.

On the choice of finite element for applications in geodynamics

Abstract. Geodynamical simulations over the past decades have widely been built on quadrilateral and hexahedral finite elements. For the discretization of the key Stokes equation describing slow, viscous flow, most codes use either the unstable Q1×P0 element, a stabilized version of the equal-order Q1×Q1 element, or more recently the stable Taylor–Hood element with continuous (Q2×Q1) or discontinuous (Q2×P-1) pressure. However, it is not clear which of these choices is actually the best at accurately simulating “typical” geodynamic situations. Herein, we provide a systematic comparison of all of these elements for the first time. We use a series of benchmarks that illuminate different aspects of the features we consider typical of mantle convection and geodynamical simulations. We will show in particular that the stabilized Q1×Q1 element has great difficulty producing accurate solutions for buoyancy-driven flows – the dominant forcing for mantle convection flow – and that the Q1×P0 element is too unstable and inaccurate in practice. As a consequence, we believe that the Q2×Q1 and Q2×P-1 elements provide the most robust and reliable choice for geodynamical simulations, despite the greater complexity in their implementation and the substantially higher computational cost when solving linear systems.