Hexahedral Elements Research Articles

The propagation behavior of hydraulic fracture network in a reservoir with cemented natural fractures

AbstractThere are still some problems in the study of hydraulic fracture (HF) network evolution in cemented naturally fractured reservoirs, such as microseismic mapping showing exaggerated stimulated reservoir volume in some cases. In addition, the dominant role of natural fracture (NF) cementation strength, injection rate, in situ stress difference, NF distribution, and fracture initiation sequence of perforations in synthetically influencing fracture network formation needs to be further studied. For this purpose, a three‐dimensional matrix hexahedral element global coupled 0‐thickness cohesive element hydraulic fracturing model was developed. Results show that each interaction between HF and NF causes HF diameter shrinkage, which increases the propagation pressure of HF. When the cementation strength of the NF is low, the HF tends to deviate toward the tip of the NF to form a complex fracture network. Increasing the injection rate and the number of NFs can significantly enhance the complexity of the HF network, but does not change the HF and NF interaction pattern. The in situ stress differences dominate the morphology of the HF network when the cementation strength of NFs is constant. The stress interference of multiple fractures under segmented fracturing may form “S”‐shaped HFs, and the HFs are difficult to maintain a symmetrical morphology in the direction of the well axis. In addition, some NFs in inactivated damaged zones have developed a certain width geometrically due to the induced effect of HF, but they are still isolated by the low permeability matrix and might only generate some microseismic events.

Quantification of tumorsphere migration with a physics-based machine learning method.

Current analysis techniques available for migration assays only provide quantitative measurements for overall migration. However, the potential of regional migration analyses can open further insight into migration patterns and more avenues of experimentation with the same assays. Previously, we developed an analysis pipeline utilizing the finite element (FE) method to show its potential in analyzing glioblastoma (GBM) tumorsphere migration, especially in characterizing regional changes in the migration pattern. This study aims to streamline and further automate the analysis system by integrating the machine-learning-based U-Net segmentation with the FE method. Our U-Net-based segmentation achieved a 98% accuracy in segmenting our tumorspheres. From the segmentations, FE models made up of 3D hexahedral elements were generated, and the migration patterns of the tumorspheres were analyzed under treatments B and C (under non-disclosure agreements). Our results show that our overall migration analysis correlated very strongly (R2 of 0.9611 and 0.9986 for treatments B and C, respectively) with ImageJ's method of migration area analysis, which is the most common method of tumorsphere migration analysis. Additionally, we were able to quantitatively represent the regional migration patterns in our FE models, which the methods purely based on segmentations could not do. Moreover, the new pipeline improved the efficiency and accessibility of the initial pipeline by implementing machine learning-based automated segmentation onto a mainly open-sourced FE analysis platform. In conclusion, our algorithm enables the development of a high-content and high-throughput in vitro screening platform to elucidate anti-migratory molecules that may reduce the invasiveness of these malignant tumors.

Articulation differences of /s/ observed using inverse tongue atlas modeling

Depending on the vowel context, the consonant /s/ can be articulated by engaging different functional units of the tongue. An inverse finite element (FE) tongue model comprising hexahedral elements generated from a 4D statistical MRI atlas of 22 speakers performing the speech tasks “a geese” (tongue moving forwards) and “a souk” (tongue moving backwards) is used to study this articulatory behavior. The model uses a state-of-the-art inverse tracking controller to simulate the motion of internal tissue points of the different speakers deformed into the atlas space. Motion tracking is successfully carried out by minimizing the L2-norm of velocity error of FEM nodes using the Cottle–Dantzig algorithm. The results show that for “a-geese,” the utterance of /s/ in context of vowel /i/ showed increase in activation of tongue protruder muscles such as genioglossus posterior, floor muscle geniohyoid by ∼5% more than in the case of “a-souk.” In “a-souk” relative activity increased in the tongue retractor muscles mid- and anterior-genioglossus, and superior longitudinal by ∼4%. Our findings are consistent with subject-specific state-of-the-art models and with articulatory expectations. The inverse atlas tongue model can be further used to estimate such articulation behavior on extended datasets of subjects performing more variations of speech tasks.

A New 3D Marine Controlled-Source Electromagnetic Modeling Algorithm Based on Two New Types of Quadratic Edge Elements

This study proposes a new algorithm for a higher-order vector finite element method based on two new types of second-order edge elements to solve the electromagnetic field diffusion problem in a 3D anisotropic medium. To avoid source singularity in the quasistatic variant of Maxwell’s function, a secondary field formulation was adopted. The modeling domain was discretized using two types of quadratic edge hexahedral elements, which were obtained using the edge unification method to reduce variables on each side of two conventional quadratic edge elements. Compared with the traditional quadratic element, the number of unknowns that needed to be solved was significantly reduced. The sparse linear equation of the finite element system was solved using an open-source direct solver called MUMPS. The numerical results demonstrated that the proposed algorithm has the same level of accuracy as the conventional vector finite element method and has a significant advantage over it in terms of computational cost.

The LSPIM-based numerical manifold method (NMM) for modeling transition elements

In many structural applications, the transition region, where the solids connect to the shells, is of considerable interest. Due to the complex behavior of this area, accurate modeling is necessary in practical problems. A numerical manifold method (NMM) based on the least square point interpolation method (LSPIM) free from linear dependence, is proposed to simulate this region. First, the basic formulations of local approximation based on LSPIM and the method to construct composite shape function are briefly introduced. This new composite shape function possesses the Kronecker-delta, inter-element compatibility, and higher-order completeness properties. Then, the formulations of the hexahedral element, 3D shell and 3D transition elements are derived based on the proposed method. Several examples are carried out to validate the accuracy and robustness of the suggested method for simulation of transition regions.

A computationally efficient isoparametric tangled finite element method for handling inverted quadrilateral and hexahedral elements

The finite element method (FEM) requires elements to be tangle-free i.e. the Jacobian must be positive throughout the element. In particular, quadrilateral and hexahedral elements are required to be convex. However, generating high quality tangle-free meshes, especially 3D hexahedral meshes, remains an open challenge. Recently, the tangled finite element method (TFEM) was proposed to handle concave (tangled) quadrilateral elements. However, even in 2D, it was found to be computationally expensive and programmatically complex.Here, we present a computationally efficient isoparametric-TFEM (i-TFEM) framework for inverted 2D quadrilateral and 3D hexahedral elements. i-TFEM employs the properties of isoparametric elements to make the formulation computationally much more efficient. In i-TFEM, the constraint on full invertibility (convexity) is replaced by partial invertibility by modifying the elemental stiffness matrices of the concave elements, and by incorporating certain piecewise-compatibility conditions. The proposed i-TFEM is simple, efficient, and provides accurate solutions with optimal convergence rate even in the presence of severely inverted elements. Moreover, i-TFEM requires minimal changes to the standard FEM framework and reduces to standard FEM for meshes without tangled elements. The accuracy and efficiency of i-TFEM are demonstrated by solving elastostatics problems on several 2D and 3D tangled meshes.

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.

Second-order pyramid element formulations suitable for lumped-mass explicit methods in nonlinear solid mechanics

Pyramid elements can facilitate meshing by simplifying transitions from hexahedral to tetrahedral regions and a second-order basis promotes robust accuracy in unstructured meshes. Over the last decade, second-order C0 hexahedral, tetrahedral, and wedge elements that demonstrate important benefits have emerged for lumped-mass explicit dynamic methods in nonlinear solid mechanics. Whereas various second- and higher-order pyramid element formulations have been developed, none have addressed the necessities for lumped-mass explicit methods. Standard row-summation lumping for the popular 13- and 14-node pyramid elements, for example, produces negative lumped masses at the vertex nodes. Degeneration of hexahedral elements into pyramids is also unsatisfactory for second-order lumped-mass elements, except perhaps for low-interest fill regions. Such issues are resolved herein by developing 19-node second-order C0 pyramid element formulations that provide all-positive row-sum nodal lumped masses and are also compatible with the other suitable second-order lumped-mass elements. Other specific requirements of explicit methods are also addressed, and several uniform and selectively reduced pyramid quadrature rules are investigated. Application to a variety of benchmark problems demonstrates robust performance, including no apparent ill effects arising from using rational shape functions to treat the apex and good accuracy with pyramids placed in critical regions of highly nonlinear inelastic analyses.

Numerical analysis of interaction between sea ice and propeller based on coupled DEM-FEM model

When a ship navigates through ice-covered seas, the interaction between sea ice and propellers damages the propeller blades. A coupled discrete element method (DEM)-finite element method (FEM) model was developed to investigate the interaction between an ice block and a propeller. The ice block and ice-class propeller were simulated using the sphere-based DEM and hexahedral solid elements of the FEM, respectively. According to the mechanical mechanism of the ice-blade interaction, the compressive strength of sea ice is the most important parameter for the simulation. The bonding strength between the element was determined by the element size and compressive strength of the sea ice material through uniaxial tests. Then, the ice-propeller interaction was simulated using the proposed DEM-FEM model. The convergence of the proposed numerical model was verified. For simulating the ice-propeller milling process, four parameters, i.e., carriage speed, cut depth, rotational speed, and ice strength, were investigated to analyze the ice load and dynamic response of the propeller. Finally, considering the randomness of the ice-propeller interaction with the real condition, a numerical simulation of ice-propeller interaction during backward sailing of the ship was established, and the random characteristics of the ice load on the blades were analyzed.

Three-dimensional natural frequency investigation of bidirectional FG truncated thick hollow cone

PurposeThe authors examined the numerical natural frequency analysis of a 2D functionally graded (FG) truncated thick hollow cone using 3D elasticity theory.Design/methodology/approachThe material properties of the 2D-FGM (two dimensional-functionally graded materials) cone are graded along the radial and axial axes of the cone using a power–law distribution. The eigenvalue problem was solved using finite element analysis (FEA) employing graded hexahedral elements, and the verification of the finite element approach was assessed by comparing the current solution to earlier experimental studies.FindingsThe effects of semivertex angle, material distribution and the cone configuration on the natural frequencies have been analyzed. For various semivertex angles, thickness, length and power law exponents, many results in the form of natural frequencies and mode shapes are presented for the 2D-FGM cone. As a result, the effects of the given parameters were addressed, and the results were compared, demonstrating the direct efficiency of raising the power–law exponents and cone thickness on the rise of natural frequencies.Originality/valueFor the first time, the numerical natural frequency analysis of a 2D-FG truncated thick hollow truncated cone based on 3D equations of elasticity has been investigated. The material properties of the truncated cone have been distributed along two directions, which has not been considered before in any research for the truncated thick cone. The reason for using these innovative volume fraction functions is the lack of accurate coverage by functions that are available in the literature (Asemi et al., 2011; Babaei et al. 2021).

Penalty C0 8-node quadrilateral and 20-node hexahedral elements for consistent couple stress elasticity based on the unsymmetric finite element method

In this paper, the penalty unsymmetric finite element framework for the consistent couple stress theory is derived from the virtual work principle. The C1 continuity requirement is satisfied in weak form by using the penalty function method to constrain the independently introduced rotations for approximating the mechanical rotations, enabling the utilization of C° continuous interpolations for designing the element displacement without the loss of convergence property. Within the proposed framework, 8-node quadrilateral element and 20-node hexahedral solid element are constructed for analyzing the size-dependent mechanical responses of consistent couple stress elasticity materials. In these developments, the quadratic serendipity isoparametric shape functions are enriched by the rotation degrees of freedom for determining the test functions, whilst the metric stress functions that are derived from the concerned equilibrium equations are used to design the trial functions. A series of numerical benchmarks are examined for verifying their effectiveness and accuracy. It is shown that the elements can efficiently capture the size dependences, exhibiting good accuracy and low susceptibility to mesh distortion.

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.

CFD Strategies for Transonic Flows with OpenFOAM Using High Skewed Tetrahedral Meshes

The usage of OpenFOAM flow solver program can be challenging for some industrial applications as it is highly sensitive to the quality of the solution mesh. Therefore, users are inherently encouraged to use low skewed meshes which have six or more faces. The implementation of hexahedral elements helps to accurately model a physical problem and speed up the convergence by allowing the solution meshes to be aligned orthogonally along a curved geometry. However, tetrahedral elements are mostly the first choice in order to easily mesh complex and curved structures encountered in industrial problems, since they do not cause any distortions. However, in some cases, the implementation of highly skewed tetrahedral elements may be a necessity and thus a challenging issue for OpenFOAM users. Although the solution approaches can be varying due to the nature and necessities of the flow problems, a generalized solution strategy has been put forward as the starting way for compressible, turbulent, external flow problems, and then it is aimed to reach the correct results with required minor modifications throughout the solution. This suggested solution strategy has been applied to some basic CFD benchmark cases created with highly skewed tetrahedral meshes and they have been successfully validated with experimental data. As a result, it is shown that the sensitivity of OpenFOAM to highly-skewed tetrahedral meshes can be managed with properly chosen solution strategies if external compressible flows are supposed to be solved numerically.

An Interactive Modeling Method for Complex Hexahedral Mesh Generation with Volumetric Subdivision

The existing methods are difficult to construct hexahedral meshes. To address this problem, an interactive construction method of complex hexahedral mesh model based on volumetric subdivision is presented. The user firstly needs to interactively construct the model skeleton, by placing cubes at the nodes of the skeleton structure, and performing interactive operations such as rotation, translation, and scaling of the node cubes. Then, through the connection between the nodes and the topological split operation, the initial control mesh can be constructed. Further, interpolatory Catmull-Clark volumetric subdivision method is used to generate hexahedral meshes with different resolutions. Finally, the padding operation is used to eliminate the degraded elements at the boundary and improve the quality of the hexahedral mesh elements to obtain the final hexahedral mesh. The results of numerical example show that the method can easily and efficiently generate hexahedral meshes interactively, and eliminates the intermediate steps of generating volume meshes from surface meshes. It has some practical applications in finite element analysis, isogeometric analysis and geometric modeling in animation.

Investigation of continuous surface cap model (CSCM) for numerical simulation of strain-hardening fibre-reinforced cementitious composites against low-velocity impacts

This study calibrated the parameters of the continuous surface cap model (CSCM) for ultra-high toughness cementitious composites (UHTCC) with compressive strength from 25 MPa to 75 MPa, and the calibrated parameters could describe the mechanical behaviour of UHTCC, including the deviatoric response, volumetric response, tensile strain-hardening behaviour, strain rate effect, as well as the reduction of tensile strain-hardening behaviour at higher strain rate. The drop-weight impact tests on the reinforced concrete slab and reinforced UHTCC slab were proceeded to validate the soundness of the calibrated CSCM. The results show that the calibrated CSCM could accurately predict the impact response of reinforced UHTCC slab, and the most accurate simulation results were achieved in the model of the hexahedral elements with a mesh size of 10 mm. The calibrated CSCM parameters for UTHCC are expected to extend the application of UHTCC in the field of impact engineering.

ОГЛЯД МОЖЛИВОСТЕЙ СІТКОВОГО ПРЕПРОЦЕСОРА ANSYS MESHING ДЛЯ СТВОРЕННЯ ВИСОКОЯКІСНИХ СКІНЧЕННО-ЕЛЕМЕНТНИХ СІТОК

The finite element method is one of the most widely used numerical methods for solving problems of solid mechanics, heat transfer, hydrodynamics and electrodynamics. It refers to approximate methods for solving partial differential equations, as well as integral equations that arise in solving applied problems of mechanics. One of the steps of the method’s implementation is discretization: the process of replacing a real physical object with its discrete model, consisting of a set of elements of a certain geometric shape and finite sizes. As a result of this transition, a reduction in the overall dimension of the problem being solved is achieved, which makes it possible to practically implement this method on a computer in the form of a package of applied programs. One of the strategic issues of the method under consideration is the accuracy of the resulting solution, which depends on the degree of discretization of the computational model. Building a finite element mesh is one of the most time-consuming steps and its effectiveness is largely determined by the methods used to build the mesh and, of course, the practical experience of the user with a particular calculation program. The issue of creating a high-quality and economical finite element mesh is a key issue in solving resource-intensive problems of solid mechanics and computational fluid dynamics in conditions of limited computing resources. The quality of the created mesh largely determines the outcome of the computer simulation process and affects the accuracy of the resulting solution, its stability and convergence, as well as the required computing resources and time costs. The article provides an overview of the methods for constructing finite element meshes implemented in the ANSYS Workbench system, describes individual functions that improve the quality of the mesh, as well as reduce the dimension of the finite element model. The capabilities of the Mesh Metric tool used to assess the quality of the mesh are described and practical recommendations for its use are given. A number of practical examples show that the use of methods that make it possible to create finite element meshes containing predominantly hexahedral elements makes it possible to increase individual metric data that determine the quality of the mesh and reduce the model dimension.

Wall Shear Stress Analysis and Optimization in Tissue Engineering TPMS Scaffolds.

When designing scaffolds for bone tissue engineering (BTE), the wall shear stress (WSS), due to the fluid flow inside the scaffold, is an important factor to consider as it influences the cellular process involved in new tissue formation. The present work analyzed the average WSS in Schwartz diamond (SD) and gyroid (SG) scaffolds with different surface topologies and mesh elements using computational fluid dynamics (CFD) analysis. It was found that scaffold meshes with a smooth surface topology with tetrahedral elements had WSS levels 35% higher than the equivalent scaffold with a non-smooth surface topology with hexahedral elements. The present work also investigated the possibility of implementing the optimization algorithm simulated annealing to aid in the design of BTE scaffolds with a specific average WSS, with the outputs showing that the algorithm was able to reach WSS levels in the vicinity of 5 mPa (physiological range) within the established limit of 100 iterations. This proved the efficacy of combining CFD and optimization methods in the design of BTE scaffolds.

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.

FFT-based Inverse Homogenization for Cellular Material Design

• An FFT-based inverse homogenization approach is proposed for efficiently designing cellular materials. • Rotating architected materials with negative Poisson's ratio are generated. • Architected cellular materials with maximum stiffness and isotropy are sought out. • The inverse homogenization method for scaffold design with specified mechanical performance is validated. Besides constituting components, the properties of composites are highly relevant to their microstructures. The work proposed a fast Fourier transform (FFT)-based inverse homogenization method implemented by the bi-directional evolutionary structural optimization (BESO) technique to explore the vast potential of cellular materials. The periodic boundary condition of self-repeated representative volume elements can be naturally satisfied in the FFT-based homogenization scheme. The objective function of the optimization problem is the specific moduli or the quadratic difference between the effective value and the target, which are obtained in terms of mutual strain energies. Its sensitivity to the design variable, namely the elemental density, is derived from the adjoint variable method and used as the criterion to remove or add material in local elements. Numerical examples show that the proposed method generates a series of architected cellular materials with maximum modulus, negative Poisson's ratio, and specific elasticity tensor. FFT-based homogenization in the method demands less memory usage but has high efficiency. Thus, it can achieve topology optimization of unit cell with one million hexahedral elements. This approach contributes to the extended application of FFT-based homogenization and can guide the microstructure design of mechanical metamaterials. :