There are 174 Subdivisions of the Hexahedron into Tetrahedra

This article answers an important theoretical question: How many different subdivisions of the hexahedron into tetrahedra are there? It is well known that the cube has five subdivisions into 6 tetrahedra and one subdivision into 5 tetrahedra. However, all hexahedra are not cubes and moving the vertex positions increases the number of subdivisions. Recent hexahedral dominant meshing methods try to take these configurations into account for combining tetrahedra into hexahedra, but fail to enumerate them all: they use only a set of 10 subdivisions among the 174 we found in this article. The enumeration of these 174 subdivisions of the hexahedron into tetrahedra is our combinatorial result. Each of the 174 subdivisions has between 5 and 15 tetrahedra and is actually a class of 2 to 48 equivalent instances which are identical up to vertex relabeling. We further show that exactly 171 of these subdivisions have a geometrical realization, i.e. there exist coordinates of the eight hexahedron vertices in a three-dimensional space such that the geometrical tetrahedral mesh is valid. We exhibit the tetrahedral meshes for these configurations and show in particular subdivisions of hexahedra with 15 tetrahedra that have a strictly positive Jacobian

Identifying combinations of tetrahedra into hexahedra: a vertex based strategy

Indirect hex-dominant meshing methods rely on the detection of adjacent tetrahedra an algorithm that performs this identification and builds the set of all possible combinations of tetrahedral elements of an input mesh T into hexahedra, prisms, or pyramids. All identified cells are valid for engineering analysis. First, all combinations of eight/six/five vertices whose connectivity in T matches the connectivity of a hexahedron/prism/pyramid are computed. The subset of tetrahedra of T triangulating each potential cell is then determined. Quality checks allow to early discard poor quality cells and to dramatically improve the efficiency of the method. Each potential hexahedron/prism/pyramid is computed only once. Around 3 millions potential hexahedra are computed in 10 seconds on a laptop. We finally demonstrate that the set of potential hexes built by our algorithm is significantly larger than those built using predefined patterns of subdivision of a hexahedron in tetrahedral elements.Comment: Preprint submitted to CAD (26th IMR special issue

All-Hex Meshing of Multiple-Region Domains without Cleanup

AbstractIn this paper, we present a new algorithm for all-hex meshing of domains with multiple regions without post-processing cleanup. Our method starts with a strongly balanced octree. In contrast to snapping the grid points onto the geometric boundaries, we move points a slight distance away from the common boundaries. Then we intersect the moved grid with the geometry. This allows us to avoid creating any flat angles, and we are able to handle two-sided regions and more complex topologies than prior methods. The algorithm is robust and cleanup-free; without the use of any pillowing, swapping, or smoothing. Thus, our simple algorithm is also more predictable than prior art

Three-Dimensional CFD Simulations of Hydrodynamics for the Lowland Dam Reservoir

This chapter deals with the processes by which a single-phase 3-D CFD model of hydrodynamics in a Sulejow dam reservoir was developed, verified, and tested. A simplified volume of fluid (VOF) model of flow was elaborated to determine the effect of wind on hydrodynamics in the lake. A hexahedral mesh with over 17 million elements and a k-ω SST turbulence model were defined for single-phase simulations in steady-state conditions. The model was verified on the basis of the extensive hydrodynamic measurements. Excellent agreement (average error of less than 10%) between computed and measured velocity profiles was found. The simulation results proved a strong effect of wind on hydrodynamics, especially on the development of the water circulation pattern in the lacustrine zone in the lake

High-quality conforming hexahedral meshes of patient-specific abdominal aortic aneurysms including their intraluminal thrombi

In order to perform finite element (FE) analyses of patient-specific abdominal aortic aneurysms, geometries derived from medical images must be meshed with suitable elements. We propose a semi-automatic method for generating conforming hexahedral meshes directly from contours segmented from medical images. Magnetic resonance images are generated using a protocol developed to give the abdominal aorta high contrast against the surrounding soft tissue. These data allow us to distinguish between the different structures of interest. We build novel quadrilateral meshes for each surface of the sectioned geometry and generate conforming hexahedral meshes by combining the quadrilateral meshes. The three-layered morphology of both the arterial wall and thrombus is incorporated using parameters determined from experiments. We demonstrate the quality of our patient-specific meshes using the element Scaled Jacobian. The method efficiently generates high-quality elements suitable for FE analysis, even in the bifurcation region of the aorta into the iliac arteries. For example, hexahedral meshes of up to 125,000 elements are generated in less than 130 s, with 94.8 % of elements well suited for FE analysis. We provide novel input for simulations by independently meshing both the arterial wall and intraluminal thrombus of the aneurysm, and their respective layered morphologies

HybridOctree_Hex: Hybrid Octree-Based Adaptive All-Hexahedral Mesh Generation with Jacobian Control

We present a new software package, "HybridOctree_Hex," for adaptive all-hexahedral mesh generation based on hybrid octree and quality improvement with Jacobian control. The proposed HybridOctree_Hex begins by detecting curvatures and narrow regions of the input boundary to identify key surface features and initialize an octree structure. Subsequently, a strongly balanced octree is constructed using the balancing and pairing rules. Inspired by our earlier preliminary hybrid octree-based work, templates are designed to guarantee an all-hexahedral dual mesh generation directly from the strongly balanced octree. With these pre-defined templates, the sophisticated hybrid octree construction step is skipped to achieve an efficient implementation. After that, elements outside and around the boundary are removed to create a core mesh. The boundary points of the core mesh are connected to their corresponding closest points on the surface to fill the buffer zone and build the final mesh. Coupled with smart Laplacian smoothing, HybridOctree_Hex takes advantage of a delicate optimization-based quality improvement method considering geometric fitting, Jacobian and scaled Jacobian, to achieve a minimum scaled Jacobian that is higher than $0.5$. We empirically verify the robustness and efficiency of our method by running the HybridOctree_Hex software on dozens of complex 3D models without any manual intervention or parameter adjustment. We provide the HybridOctree_Hex source code, along with comprehensive results encompassing the input and output files and statistical data in the following repository: https://github.com/CMU-CBML/HybridOctree_Hex

Twisted Pair Transmission Line Coil -- A Flexible, Self-Decoupled and Extremely Robust Element for 7T MRI

This study evaluates the performance of a twisted pair transmission line coil as a transceive element for 7T MRI in terms of physical flexibility, robustness to shape deformations, and interelement decoupling. Each coil element was created by shaping a twisted pair of wires into a circle. One wire was interrupted at the top, while the other was interrupted at the bottom, and connected to the matching circuit. Electromagnetic simulations were conducted to determine the optimal number of twists per length (in terms of B$_1^+$ field efficiency, SAR efficiency, sensitivity to elongation and interelement decoupling properties) and for investigating the fundamental operational principle of the coil through fields streamline visualization. A comparison between the twisted pair coil and a conventional loop coil in terms of B$_1^+$ fields, maxSAR10g, and stability of $S_{11}$ when the coil was deformed, was performed. Experimentally measured interelement coupling between individual elements of multichannel arrays was also investigated. Increasing the number of twists per length resulted in a more physically robust coil. Poynting vector streamline visualization showed that the twisted pair coil concentrated most of the energy in the near field. The twisted pair coil exhibited comparable B$_1^+$ fields and improved maxSAR10g to the conventional coil but demonstrated exceptional stability with respect to coil deformation and a strong self-decoupling nature when placed in an array configuration. The findings highlight the robustness of the twisted pair coil, showcasing its stability under shape variations. This coil holds great potential as a flexible RF coil for various imaging applications using multiple-element arrays, benefiting from its inherent decoupling.Comment: Revised version; 20 pages, 16 figures, preprin

Flutter mitigation of turbofan blades using viscoelastic patches

Flutter as a self-feeding aeroelastic instability presents one of the biggest challenges in aero-engine designsto improve its aerodynamic and structural performance. This work presents a detailed feasibility study ofusing different viscoelastic patches as Constrained Layer Damping (CLD) enhancement for an aero-enginefan blade to reduce potential flutter risks. The static and dynamic responses of the different materials andconfigurations (thicknesses, layers and locations) are evaluated on both cruise and take-off/landingconditions. It is found that a double bi-layer 3M® ISD110 is the optimal choice of material for the CLDtreatment for the fan blade. The study also shows that an optimal CLD treatment of 15 % total surface areaof the blade at the root demonstrated a 36 % reduction in resonance amplitudes across the first six modes

Development of a Nomogram to Predict the Maximum Contact Stress Between a Bridge I-Girder and a Support Roller

The incremental launching method (ILM) is one of the methods of bridge construction located in environmentally sensitive areas. During the bridge launching, there are significant contact stress issues at the contact regions between the launching system and the steel bridge girders. The Iowa River Bridge (IRB) is a case study of bridge construction that used the ILM during a steel I-girders installation. Contact stress can cause severe defects during launch, which can occur within the material where the material could be damaged. Hertz contact theory is applied for calculating contact stress between two solid surfaces, which is initially derived from the contact between cylinder and plane surface. However, Hertz contact theory can calculate only the contact area and stress between two elastic solids with specific modeling in equilibrium. The solutions of the launching girder bridge construction\u27s contact stress are not usually possible with closed-form Hertz contact theory solutions because of the complex geometries, loads, and material properties. Typically, the issues, including complicated structural systems, need to rely on numerical modeling such as the finite element analysis (FEA) from ANSYS® . The primary objective of this study is to provide an estimate the relationship of the maximum contact stress between an I-girder and a roller using a nomogram. The nomogram is built based on a parametric study with various roller dimensions and vertical loads by numerical modeling. A total of 180 numerical models were used to develop the contract stress nomogram. The maximum contact stress from the nomogram can be useful tool in designing a bridge girder on a support roller