Hexahedral Mesh Research Articles

Comprehensive Theoretical Formulation and Numerical Simulation of the Internal Flow in Pressure-Swirl Atomizers Type Screw-Conveyer

The present work shows the development of a comprehensive theoretical formulation for its application in the study of the internal flow of pressure-swirl atomizers with helical channels: “screw-conveyer”, which are characterized by presenting in their inlet channels, an angle of incidence or helix angle ψ. This angle originates a trigonometric factor (cosψ) that must be considered in the geometrical characteristics parameter of pressure-swirl atomizer (Ah), which consequently involves other geometric parameters, such as the annular section coefficient (φ), discharge coefficient (Cd), spray angle (2α), etc., being relevant in the internal flow study and design of the pressure-swirl atomizers type screw-conveyer. This theoretical formulation integrates an internal ideal flow model (Abramovich theory) with a model that considers the influence of the liquid viscosity (Kliachko theory) and hydraulic resistance of Idelchik. For the validation of this theoretical formulation, numerical simulation was used, considering the commercial software Ansys Fluent 2023 R2 furthermore, hexahedral meshes were generated with the ICEM CFD software 2023, for four cases of helix angle ψ (15°, 30°, 45° and 60°), with application of the RNG k-ε turbulence model and VOF multiphase model (volume of fluid) for the location of the liquid-gas interface and spray angle visualization.

Stress‐Aligned Hexahedral Lattice Structures

AbstractMaintaining the maximum stiffness of components with as little material as possible is an overarching objective in computational design and engineering. It is well‐established that in stiffness‐optimal designs, material is aligned with orthogonal principal stress directions. In the limit of material volume, this alignment forms micro‐structures resembling quads or hexahedra. Achieving a globally consistent layout of such orthogonal micro‐structures presents a significant challenge, particularly in three‐dimensional settings. In this paper, we propose a novel geometric algorithm for compiling stress‐aligned hexahedral lattice structures. Our method involves deforming an input mesh under load to align the resulting stress field along an orthogonal basis. The deformed object is filled with a hexahedral grid, and the deformation is reverted to recover the original shape. The resulting stress‐aligned mesh is used as basis for a final hollowing procedure, generating a volume‐reduced stiff infill composed of hexahedral micro‐structures. We perform quantitative comparisons with structural optimization and hexahedral meshing approaches and demonstrate the superior mechanical performance of our designs with finite element simulation experiments.

Numerical investigation on influence of different geometrical cavity flame holders and flame stabilization mechanism in hydrogen fueled multi-strut-based scramjet combustor

A detailed computational study was carried out to explore how the presence of a cavity and cavity angles influence the performance and combustion mechanisms within a scramjet combustor. This research utilized ANSYS Fluent 2021 to create hexahedral meshes and simulate the complex fluid dynamics in scramjet engines equipped with dual strut injectors. The simulations addressed various critical aspects of fluid mechanics, including the continuity, momentum (via the Navier-Stokes equations), and energy equations. These equations were used to model non-reacting flows, while the species transport equations were applied to analyse reacting flows. The turbulence within the combustor was modelled using the standard K-ε model, a common choice for such high-speed aerodynamic evaluations. The study's primary goal was to assess the effects of different geometrical configurations of cavity flame holders on the performance characteristics of an H2-based scramjet combustor overall efficiency, focusing on mixing efficiency, combustion performance, and total pressure loss. Both configurations with and without a cavity were examined. Key findings indicate that incorporating a cavity into the combustor design leads to developing a robust recirculation zone within the cavity area. This recirculation zone is pivotal in enhancing fuel-air mixing and combustion efficiency, with cavity-based combustors showing an earlier onset of combustion and achieving peak combustion efficiencies around 90–95%. The extent of the recirculation region is notably influenced by the proximity of the strut injector to the cavity's length. Additionally, the presence of a shear layer within the cavity generates a weak shock at the cavity's trailing edge. This shock interaction can adversely affect scramjet combustor performance, especially at higher cavity angles (α) and specific geometric configurations, such as an L/D ratio of 4 and α=30°. The L/D ratio of 4 and α=30° combustor model results in a 13.8% increase in turbulence intensity, which raises air-fuel mixing and there by improves mixing and combustion efficiency by 14.23% and 13.34%, compared to the other depth of the cavities. This advantage is critical, especially considering the compact length of the combustor, which is a desirable attribute in scramjet design.

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.

Joint Inversion of DC Resistivity and Gravity Data with Undulating Terrain Based on Deformed Hexahedral Mesh

In the field of mineral resource exploration, accurate imaging of subsurface structures is key to discovering and assessing potential mineral deposits. Traditional single geophysical methods, limited by terrain variations and their own constraints, can lead to divergent solutions and structural inconsistencies, affecting the reliability of exploration outcomes. To address these challenges, this paper presents a joint inversion method for three-dimensional direct current (DC) resistivity and gravity data based on a deformed hexahedral mesh. The article begins by outlining the current state of development of the method under study and proposes a research plan, followed by a detailed explanation of the theoretical basis and algorithmic implementation of the proposed method. Model tests confirm the advantages of the deformed hexahedral mesh in reducing terrain impacts and enhancing model resolution, demonstrating the optimization and complementarity of the resolution between the two methods after joint inversion. Finally, applying this method to actual data from the Huaniu Mountain area shows that joint inversion not only improves the consistency of the ore belt structure but also provides a more precise analysis for the quantitative interpretation of the distribution of underground mineral resources. This confirms the method’s effectiveness and potential in practical geological exploration.

Stability impact on wind turbine blades trailing edge from bonding zone modelling methods

Abstract The trailing edge of wind turbine blade will undergo a large deformation under the action of load, resulting in a higher risk of damage and failure. The finite element analysis modeling method directly affects the simulation result, furthermore impacts the result of blade structure design. Therefore, it is necessary to find out a modeling method, which is capable to reflect the stability of bonding regions more accurately. This paper aims to discuss the influences of different finite element modeling methods on the buckling stability of blade trailing edge. The influences of shell element, tetrahedron element and hexahedron element on the blade stiffness are considered. The bonding region were simulated separately by combining shell element with bonding web, shell element with tetrahedron element and shell element with hexahedron. The influences of different bonding region modeling methods on the trailing edge buckling factor were analyzed by using the eigenvalue buckling analysis method of finite element. The simulation results provided data support for the future validation of the blade trailing edge bonding area modeling method. The results show that, the tetrahedral mesh provides a higher stiffness, followed by the hexahedral mesh, and the adhesive web form has the smallest stiffness. In order to simulate the more realistic stiffness of the adhesive area at the trailing edge of the blade, adhesive web plates can be built in the flange area of the blade root and the core material area of the trial adhesive mold; The direct bonding area can be simulated employing the hexahedron mesh; The tetrahedral mesh has high stiffness and is not suitable for modeling the adhesive area at the trailing edge. Blade simulation modeling can flexibly use different adhesive forms according to actual situations, ensuring the reliability of the simulation model.

X-ray micro-computed tomography for mechanical behaviour analysis of Automated Fiber Placement (AFP) laminates with integrated gaps and overlaps

Automated Fiber Placement (AFP) is a manufacturing technique widely used for the serial production of aerospace parts. A deep understanding of the effect of lay-up defects is crucial for part and lay-up design. Currently, numerical models for structural simulation lack a precise representation of the internal structure of AFP laminates, which is crucial for understanding the impact of defects on mechanical properties. This paper presents a novel approach based on high-resolution micro-computed tomography (micro-CT) scans from specimens manufactured with AFP, which automatically creates a mesoscale numerical model incorporating as-fabricated defect morphologies. The hexahedral mesh, generated from the segmented plies of the micro-CT volume, accounts for ply thickness and out-of-plane fiber orientation. This approach is verified with mechanical testing and digital image correlation (DIC) under tensile loading. The simulation results align closely with experimental testing and accurately illustrate the influence of fiber waviness in various defect configurations, such as gaps and overlaps. The study shows that lay-up defects can lead to knockdown factors of up to 12% in tensile properties, with each defect creating a distinct pattern in the local strain. This model can serve as a benchmark for further numerical simulations and surrogate models of defect configurations under varying loading conditions.

A semi-automatic method for block-structured hexahedral meshing of aortic dissections.

The article presents a semi-automatic approach to generating structured hexahedral meshes of patient-specific aortas ailed by aortic dissection. The condition manifests itself as a formation of two blood flow channels in the aorta, as a result of a tear in the inner layers of the aortic wall. Subsequently, the morphology of the aorta is greatly impacted, making the task of domain discretization highly challenging. The meshing algorithm presented herein is automatic for the individual lumina, whereas the tears require user interaction. Starting from an input (triangle) surface mesh, we construct an implicit surface representation as well as a topological skeleton, which provides a basis for the generation of a block-structure. Thereafter, the mesh generation is performed via transfinite maps. The meshes are structured and fully hexahedral, exhibit good quality and reliably match the original surface. As they are generated with computational fluid dynamics in mind, a fluid flow simulation is performed to verify their usefulness. Moreover, since the approach is based on valid block-structures, the meshes can be made very coarse (around 1000 elements for an entire aortic dissection domain), and thus promote using solvers based on the geometric multigrid method, which is typically reliant on the presence of a hierarchy of coarser meshes.

Numerical approach for the simulation of flow-induced noise around a structure with complex geometry: High-speed train bogie in a cavity

The bogie region is a significant source of aerodynamic noise on high-speed trains. Owing to its complex geometry and flow field, numerical simulations using computational fluid dynamics are especially challenging. The main challenge is to achieve a grid with adequate resolution, especially in the boundary layer, while ensuring computational affordability. This challenge is addressed here by employing a hybrid grid, integrating a structured hexahedral mesh near solid surfaces with an unstructured polyhedral mesh in the remaining volume. To limit the number of cells in the boundary layer region, the delayed detached eddy simulation method is adopted. Additionally, to achieve a further reduction in the cell count, the Reynolds number of the model is decreased by scaling down the model size and lowering the inflow speed. The hybrid grid generation and numerical settings are guided by validated simulations of flow over cylinders. A grid sensitivity study, conducted with a simplified half-width bogie model, reveals the meshing requirements for the full-width model. Aerodynamic results highlight the rear section of the cavity and bogie as primary noise sources, emphasizing the critical role of the detached shear layer from upstream components. Time-resolved surface pressure data are input into the Ffowcs Williams–Hawkings equation for far-field noise calculation. The results indicate that the sound energy is concentrated below 200 Hz in the full-scale model, with the cavity contributing more than the bogie. This study provides a practical numerical approach for simulating a structure with complex geometry, offering insights for realistic model simulations.

A macro BDM H-div mixed finite element on polygonal and polyhedral meshes

A BDM type of H(div) mixed finite element is constructed on polygonal and polyhedral meshes. The flux space is the H(div) subspace of the n-product ΠiPk(Ti)d space such that the divergence is a one-piece Pk−1 polynomial on the big polygon or polyhedron T. Here we assume the 2D polygon can be subdivided into triangles by connecting only one vertex with some vertices of the polygon. For the 3D polyhedron we assume it can be subdivided into tetrahedra, with no added vertex on subdividing its face-polygons, and with either no internal edge or one internal edge. Such mixed finite elements can be more economic on quadrilateral and hexahedral meshes, compared with the standard BDM mixed element on triangular and tetrahedral meshes. Numerical tests and comparisons with the triangular and tetrahedral BDM finite elements are provided.

Arbitrary order spline representation of cohomology generators for isogeometric analysis of eddy current problems

Common formulations of the eddy current problem involve either vector or scalar potentials, each with its own advantages and disadvantages. An impasse arises when using scalar potential-based formulations in the presence of conductors with non-trivial topology. A remedy is to augment the approximation spaces with generators of the first cohomology group. Most existing algorithms for this require a special, e.g., hierarchical, finite element basis construction. Using insights from de Rham complex approximation with splines, we show that additional conditions are here unnecessary. Spanning tree techniques can be adapted to operate on a hexahedral mesh resulting from isomorphisms between spline spaces of differential forms and de Rham complexes on an auxiliary control mesh.

CutFEM-based MEG forward modeling improves source separability and sensitivity to quasi-radial sources: A somatosensory group study.

Source analysis of magnetoencephalography (MEG) data requires the computation of the magnetic fields induced by current sources in the brain. This so-called MEG forward problem includes an accurate estimation of the volume conduction effects in the human head. Here, we introduce the Cut finite element method (CutFEM) for the MEG forward problem. CutFEM's meshing process imposes fewer restrictions on tissue anatomy than tetrahedral meshes while being able to mesh curved geometries contrary to hexahedral meshing. To evaluate the new approach, we compare CutFEM with a boundary element method (BEM) that distinguishes three tissue compartments and a 6-compartment hexahedral FEM in an n = 19 group study of somatosensory evoked fields (SEF). The neural generators of the 20 ms post-stimulus SEF components (M20) are reconstructed using both an unregularized and a regularized inversion approach. Changing the forward model resulted in reconstruction differences of about 1 centimeter in location and considerable differences in orientation. The tested 6-compartment FEM approaches significantly increase the goodness of fit to the measured data compared with the 3-compartment BEM. They also demonstrate higher quasi-radial contributions for sources below the gyral crowns. Furthermore, CutFEM improves source separability compared with both other approaches. We conclude that head models with 6 compartments rather than 3 and the new CutFEM approach are valuable additions to MEG source reconstruction, in particular for sources that are predominantly radial.

Конечно-элементное моделирование нелинейных задач упругости в абсолютных узловых координатах на неструктурированных шестигранных сетках

We consider a finite element approach to solving the problem of elasticity theory in terms of absolute nodal coordinate formulation (ANCF), in which large body displacements are described in a global reference frame without using any local coordinate system. The main feature of the method is the absence of gyroscopic effects and, as a result, constancy of the mass matrix and the vector of the generalized force of gravity. In contrast to the traditional ANCF approach, the sets of nodal degrees of freedom of the finite element are formed only on the basis of absolute coordinates of nodes, which allows us to solve the problem using, in particular, unstructured hexahedral meshes. To construct the stiffness matrix, we apply a second-order automatic differentiation algorithm, which ensures its symmetrical form (Hessian matrix) and is analytically accurate in calculating the derivative. This approach also makes it possible to carry out calculations for models of hyperelastic materials without the corresponding Piola-Kirchhoff tensor. It has been shown that within the framework of discretization of the equation of motion, along with the well-known Newmark numerical integration scheme, it is possible to use the HTT-α scheme, which is unconditionally stable, second-order accurate and dissipative for high frequencies. We present several examples of solving static and dynamic elasticity problems for compressible and incompressible models of hyperelastic materials, in which the functions of internal energy density of the body are specified in terms of the deformation gradient.

Integer‐Sheet‐Pump Quantization for Hexahedral Meshing

AbstractSeveral state‐of‐the‐art algorithms for semi‐structured hexahedral meshing involve a so called quantization step to decide on the integer DoFs of the meshing problem, corresponding to the number of hexahedral elements to embed into certain regions of the domain. Existing reliable methods for quantization are based on solving a sequence of integer quadratic programs (IQP). Solving these in a timely and predictable manner with general‐purpose solvers is a challenge, even more so in the open‐source field. We present here an alternative robust and efficient quantization scheme that is instead based on solving a series of continuous linear programs (LP), for which solver availability and efficiency are not an issue. In our formulation, such LPs are used to determine where inflation or deflation of virtual hexahedral sheets are favorable. We compare our method to two implementations of the former IQP formulation (using a commercial and an open‐source MIP solver, respectively), finding that (a) the solutions found by our method are near‐optimal or optimal in most cases, (b) these solutions are found within a much more predictable time frame, and (c) the state of the art run time is outperformed, in the case of using the open‐source solver by orders of magnitude.

Numerical simulations of the acoustic and electrical properties of digital rocks based on tetrahedral unstructured mesh

Abstract Unconventional reservoirs typically exhibit strong heterogeneity leading to a significant scale effect in digital rock physics simulations. To ensure the reliability of the simulation results, improving computational efficiency and increasing sample sizes are crucial. In this study, we present a numerical finite element simulation method for the acoustic and electrical properties of digital rock cores based on tetrahedral unstructured meshes. We calculated the elastic moduli and electrical resistivity of the Fontainebleau sandstone digital rock samples. A comparison was made between the tetrahedral mesh and the traditional voxel-based hexahedral mesh in terms of the accuracy and efficiency of finite element numerical simulations. The results indicate that this numerical simulation method based on the tetrahedral mesh exhibits high accuracy comparable to experimental results, and its computational efficiency is significantly improved compared to the traditional hexahedral mesh method. These findings highlight the advantages of this finite element simulation method in improving the computational scale and efficiency of digital rock simulations. It effectively addresses common computational resource constraints in dealing with large-scale core systems and facilitates better integration with engineering construction, well-logging instrument simulations, and production applications.

Surrogate-based integrated design optimization for aerodynamic/stealth performance enhancements

The shape optimization considering both aerodynamic and radar stealth performances has been a considerable attention area for next-generation aircraft, with challenges involving balancing design contradictions and improving optimization efficiency. This paper develops a surrogate-based integrated design optimization method coupling Computational Fluid Dynamics (CFD) and Computational Electromagnetics (CEM) for aerodynamic/stealth performance enhancements. The topology-based 3D hexahedral mesh generation and 2D triangular mesh generation with adaptive refinement provide efficient preprocessing for CFD/CEM solvers. The surrogate-based optimization algorithm with parallel infill-sampling strategy and adaptive design space expansion is employed to search the global optimum and improve optimization efficiency while filtering out numerical noise. The Weighted Sum method is employed to address design contradictions in different disciplines. Optimizations considering aerodynamic, stealth, and aerodynamic/stealth performance of a flying wing aircraft are conducted to verify the effectiveness of the developed method. The optimization results indicate that exclusive consideration of a singular discipline leads to a significant deterioration in the performance of the other discipline. Balanced performance is achieved through coupling optimization. The numerical noise in the stealth discipline is markedly higher than in the aerodynamics discipline. The results demonstrate that the developed method is capable of addressing aerodynamic/stealth design optimization problems and significantly reducing the number of CFD/CEM evaluations while maintaining global optimization effectiveness.

Application of polyhedral meshes in large-eddy simulation of building array flow fields within neutral and unstable boundary layers

Urbanization has intensified the urban heat island phenomenon, a critical concern for human habitation in recent decades. Analyzing flow fields around building arrays is vital for mitigating issues such as heatstroke and air pollution, thereby enhancing human comfort in urban environments. However, challenges in accuracy and computational efficiency persist in simulating urban thermal environments. This study evaluates how boundary layer meshes in polyhedral meshes affect wind and temperature distributions in neutral and unstable boundary layers, utilizing large-eddy simulation (LES). A total of ten cases were examined, including eight with local mesh refinement and two without. The study demonstrates that local mesh refinement is an effective strategy to reduce mesh count by approximately 60 %, thereby reducing computational time by 60 % using the same computational configuration. The performance of the polyhedral meshes was found to be comparable to that of the hexahedral meshes. For the mean velocity and temperature, the discrepancies between the hexahedral and polyhedral cases were within 10 %, while for the second-order statistics, the discrepancies were within 20 %. Within the neutral boundary layer, boundary layer meshes on the ground and building surfaces had minimal impact on the wind field. However, these meshes enhanced the accuracy of low percentile wind speed estimations by approximately 20 % within the unstable boundary layer. Moreover, this study lays the foundation for research on the thermal environment in actual urban areas and contributes to the guidelines for LES with non-isothermal inflow conditions.

Mesh morphing-based pre-processing and numerical simulation of blockage accident in lead–bismuth fast reactor fuel assembly

This paper presents a CFD modeling approach for a 19-rod bundle fuel assembly with spiral semi-cylindrical ribs, which potentially can be used for liquid metal cooled fast reactor, as well as a numerical simulation of flow and thermal behavior under partially blocked conditions. The ribs on fuel rod surface are integrally formed with the cladding to enhance heat transfer while preventing detachment. However, the complex channel geometry presents challenges for grid generation, as using a large number of tetrahedral or polyhedral elements would consume significant computational resources. Therefore, this paper proposes a mesh morphing-based pre-processing method to establish a structured hexahedral mesh for such complex geometry. Using this approach, the 19-rods assembly is analyzed, and the result has been verified by conventional mesh scheme as well as experiment.

Hybrid meshless-FEM method for 3-D magnetotelluric modelling using non-conformal discretization

SUMMARY We propose a novel method for 3-D magnetotelluric (MT) forward modelling based on hybrid meshless and finite-element (FE) methods. This method divides the earth model into a central computational region and an expansion one. For the central region, we adopt scatter points to discretize the model, which can flexibly and accurately characterize the complex structures without generating unstructured mesh. The meshless method using compact support radial basis function is applied to simulate this area's electromagnetic field. While in the expansion region, to avoid the heavy time consumption and numerical error of the meshless method caused by non-uniform nodes, we adopt a node-based finite-element method with regular hexahedral mesh for stability. Finally, the two discretized systems are coupled at the interface nodes according to the continuity conditions of vector and scalar potentials. Considering that the normal electric field is discontinuous at the interface with resistivity discontinuity, while the shape functions for the meshless method are continuous, we further adopt the visibility criterion in constructing the support region. Numerical experiments on typical models show that using the same degree of freedom, the hybrid meshless-finite element method (FEM) algorithm has higher accuracy than the node-based FEM and meshless method. In addition, the electric field discontinuity at interfaces is well preserved, which proves the effectiveness of the visibility criterion method. In general, compared to the conventional grid-based method, this new approach does not need the complex mesh generation for complex structures and can achieve high accuracy, thus it has the potential to become a powerful 3-D MT forward modelling technique.

Unveiling interactions between intervertebral disc morphologies and mechanical behavior through personalized finite element modeling.

Introduction: Intervertebral Disc (IVD) Degeneration (IDD) is a significant health concern, potentially influenced by mechanotransduction. However, the relationship between the IVD phenotypes and mechanical behavior has not been thoroughly explored in local morphologies where IDD originates. This work unveils the interplays among morphological and mechanical features potentially relevant to IDD through Abaqus UMAT simulations. Methods: A groundbreaking automated method is introduced to transform a calibrated, structured IVD finite element (FE) model into 169 patient-personalized (PP) models through a mesh morphing process. Our approach accurately replicates the real shapes of the patient's Annulus Fibrosus (AF) and Nucleus Pulposus (NP) while maintaining the same topology for all models. Using segmented magnetic resonance images from the former project MySpine, 169 models with structured hexahedral meshes were created employing the Bayesian Coherent Point Drift++ technique, generating a unique cohort of PP FE models under the Disc4All initiative. Machine learning methods, including Linear Regression, Support Vector Regression, and eXtreme Gradient Boosting Regression, were used to explore correlations between IVD morphology and mechanics. Results: We achieved PP models with AF and NP similarity scores of 92.06\% and 92.10\% compared to the segmented images. The models maintained good quality and integrity of the mesh. The cartilage endplate (CEP) shape was represented at the IVD-vertebra interfaces, ensuring personalized meshes. Validation of the constitutive model against literature data showed a minor relative error of 5.20%. Discussion: Analysis revealed the influential impact of local morphologies on indirect mechanotransduction responses, highlighting the roles of heights, sagittal areas, and volumes. While the maximum principal stress was influenced by morphologies such as heights, the disc's ellipticity influenced the minimum principal stress. Results suggest the CEPs are not influenced by their local morphologies but by those of the AF and NP. The generated free-access repository of individual disc characteristics is anticipated to be a valuable resource for the scientific community with a broad application spectrum.