Local element operations for curved simplex meshes

Mesh generation and optimization represent a fundamental preprocessing stage in computational fluid dynamics (CFD), where their significance becomes particularly pronounced in hypersonic flow simulations. The complex physical phenomena impose exceptionally rigorous demands on mesh quality to ensure computational accuracy and stability. Although traditional mesh quality inspection can be performed automatically by software, the optimization of low-quality grids heavily relies on manual observation and adjustment, exhibiting low automation efficiency. This presents persistent challenges in CFD research. Addressing the local optimization of the intricate structured grid for CFD simulations, this study establishes an overall mesh quality evaluation model based on the maximum and minimum interior angle metric. Subsequently, it automatically clusters low-quality grids through the K-Means++ clustering method. Finally, it applies the Laplacian smoothing method to the local low-quality blocks partitioned by the K-Means++ clustering method to enhance the overall mesh quality. Taking a distorted wavy structured grid, a space shuttle structured grid, and a hypersonic blunt body structured grid as numerical examples, the results show that this integrated approach establishes a novel intelligent optimization pathway for CFD preprocessing workflows, effectively minimizing the need for manual intervention. Local optimization of low-quality regions demonstrates superior effectiveness compared to the entire mesh optimization. The proposed method demonstrates effective automatic partitioning of low-quality grid regions across different configurations, significantly enhancing mesh optimization efficiency while contributing to enhancing the accuracy of the flow simulation.

We propose a new tetrahedral meshing method, fTetWild, to convert triangle soups into high-quality tetrahedral meshes. Our method builds on the TetWild algorithm, replacing the rational triangle insertion with a new incremental approach to construct and optimize the output mesh, interleaving triangle insertion and mesh optimization. Our approach makes it possible to maintain a valid floating-point tetrahedral mesh at all algorithmic stages, eliminating the need for costly constructions with rational numbers used by TetWild, while maintaining full robustness and similar output quality. This allows us to improve on TetWild in two ways. First, our algorithm is significantly faster, with running time comparable to less robust Delaunay-based tetrahedralization algorithms. Second, our algorithm is guaranteed to produce a valid tetrahedral mesh with floating-point vertex coordinates, while TetWild produces a valid mesh with rational coordinates which is not guaranteed to be valid after floating-point conversion. As a trade-off, our algorithm no longer guarantees that all input triangles are present in the output mesh, but in practice, as confirmed by our tests on the Thingi10k dataset, the algorithm always succeeds in inserting all input triangles.

An automatic 3D mesh generation method for domains with multiple materials

SummaryTwo stable approximation space configurations are treated for the mixed finite element method for elliptic problems based on curved meshes. Their choices are guided by the property that, in the master element, the image of the flux space by the divergence operator coincides with the potential space. By using static condensation, the sizes of global condensed matrices, which are proportional to the dimension of border fluxes, are the same in both configurations. The meshes are composed of different topologies (tetrahedra, hexahedra, or prisms). Simulations using asymptotically affine uniform meshes, exactly fitting a spherical‐like region, and constant polynomial degree distributionk, showL2errors of orderk+1 ork+2 for the potential variable, while keeping orderk+1 for the flux in both configurations. The first case corresponds to RT(k) and BDFM(k+1) spaces for hexahedral and tetrahedral meshes, respectively, but holding for prismatic elements as well. The second case, further incrementing the order of approximation of the potential variable, holds for the three element topologies. The case ofhp‐adaptive meshes is considered for a problem modelling a porous media flow around a cylindrical horizontal well with elliptical drainage area. The effect of parallelism and static condensation in CPU time reduction is illustrated.

We present a technique to extend any distortion (quality) measure for planar meshes to meshes on parameterized surfaces. The resulting distortion (quality) measure is expressed in terms of the parametric coordinates of the nodes. This extended distortion (quality) measure can be used to check the quality and validity of a surface mesh. We also apply it to simultaneously smooth and untangle surface meshes by minimizing the extended distortion measure. The minimization is performed in terms of the parametric coordinates of the nodes and therefore, the nodes always lie on the surface. In addition, we apply the extension technique to define quality measures for high-order meshes on parameterized surfaces. Finally, we include some examples to show the application of the proposed technique. Specifically, we extend several Jacobian based measures, and we smooth and untangle linear and high-order meshes on CAD surfaces.

In numerical computation of neuromodulator such as transcranial direct current stimulation (tDCS) and transcranial magnetic stimulation (TMS), the quality of the tetrahedral mesh is a significant criterion to achieve accurate results in simulating the electric field distribution. Although there are various algorithms and software products have been proposed to achieve this goal, but most of them have challenges in ease of use or in the quality of the final mesh. Even with the production of tetrahedral mesh, there is still a deficiency in the quality of the tetrahedral mesh that leads to reduced accuracy in the final result. In this paper, a new MATLAB toolbox (SimUTab)a is presented that uses several useful algorithms in this field and also employs new methods to generate high-quality meshes for head models. The proposed method reduces the difficulties and makes it easy to use. The user interface implemented in MATLAB makes easily designing electrodes even with unusual shapes and hence provides the basis for examining the effect of electrode shapes on the electric field distribution of the target area. The proposed method is tested and compared with other active pipelines. The results confirm the superiority of the proposed method and show an improvement in the quality of the final mesh.

This paper studies the coupling between anisotropic mesh adaptation and goal-oriented error estimate. The former is very well suited to the control of the interpolation error. It is generally interpreted as a local geometric error estimate. On the contrary, the latter is preferred when studying approximation errors for PDEs. It generally involves non local error contributions. Consequently, a full and strong coupling between both is hard to achieve due to this apparent incompatibility. This paper shows how to achieve this coupling. This is done in three steps. First, a new ap riorierror estimate is proved in a formal framework adapted to goal-oriented mesh adaptation for output functionals. Second, the error estimate is applied to the set of steady compressible Euler equations which are solved by a stabilized Galerkin finite element discretization. A goal-oriented error estimation is derived. Third, rewritten in the continuous mesh framework, the previous estimate is minimized on the set of continuous meshes thanks to a calculus of variations. The optimal mesh is then derive. 3D examples of steady flows around supersonic and transsonic aircrafts are presented to validate the current approach and to demonstrate its efficiency.

In this paper we present a new algorithm which turns an unstructured triangle mesh into a quad-dominant mesh with edges aligned to the principal directions of the underlying geometry. Instead of computing a globally smooth parameterization or integrating curvature lines along a tangent vector field, we simply apply an iterative relaxation scheme which incrementally aligns the mesh edges to the principal directions. The quad-dominant mesh is eventually obtained by dropping the not-aligned diagonals from the triangle mesh. A post-processing stage is introduced to further improve the results. The major advantage of our algorithm is its conceptual simplicity since it is merely based on elementary mesh operations such as edge collapse, flip, and split. The resulting meshes exhibit a very good alignment to surface features and rather uniform distribution of mesh vertices. This makes them very well-suited, e.g., as Catmull-Clark Subdivision control meshes.

Tetrahedral mesh improvement using moving mesh smoothing, lazy searching flips, and RBF surface reconstruction

In this paper, a tetrahedral mesh-based approach is investigated for 3D image segmentation on a given image volume. We present a series of algorithms to generate high quality, feature-sensitive, and adaptive meshes to partition a 3D volume, where the 3D Canny edge detector is utilized to preserve important feature boundaries in the generated tetrahedral mesh. Each cluster of voxels within a tetrahedron is called a supervoxel, which is treated as a node in a graph and weighted edges are formed between adjacent tetrahedra (or supervoxels). A graph cut method is then applied on the weighted graph to partition the tetrahedral mesh, resulting in a segmentation of the original image volume. Thanks to the significantly reduced number of elements (supervoxels) as compared to that of voxels in a 3D image volume, the supervoxel-based segmentation method has tremendously improved the segmentation speed. In addition, the segmented image feature is naturally composed of high-quality, adaptive tetrahedral mesh that can be readily used for finite element analysis without further post-generation of tetrahedral mesh.

We present a practical approach to isotropic tetrahedral meshing of 3D domains bounded by piecewise smooth surfaces. Building upon recent theoretical and practical advances, our algorithm interleaves Delaunay refinement and mesh optimization to generate quality meshes that satisfy a set of user-defined criteria. This interleaving is shown to be more conservative in number of Steiner point insertions than refinement alone, and to produce higher quality meshes than optimization alone. A careful treatment of boundaries and their features is presented, offering a versatile framework for designing smoothly graded tetrahedral meshes.

Adaptive triangular element generation and optimization-based smoothing, Part 1: On the plane

Chapter 13 - GPU Accelerated Derivative-Free Mesh Optimization

Quality tetrahedral mesh smoothing via boundary-optimized Delaunay triangulation

Due to environmental pollution and resource shortages, the electric vehicle industry has been developing swiftly, and the market demand for batteries, as an essential part of electric vehicles, has also surged. Proper disassembly of end-of-life vehicle batteries (ELV batteries) is necessary to achieve the integrity and closure of their life cycle, promote the development of green remanufacturing, effectively reduce the pollution of the environment caused by metal ion leakage, and reduce people's dependence on natural resources to a certain extent. To schedule the disassembly operations of ELV batteries more rationally and further promote their disassembly quality and efficiency, this paper proposes a dual-objective disassembly sequence planning (DSP) optimisation model, which aims to minimise the hazard index and energy cost during ELV battery disassembly operations. Since the proposed model is a complex NP-hard optimisation problem, this study develops an efficient metaheuristic algorithm for solving this model based on the northern goshawk optimisation algorithm. The main algorithm adds two types of discrete recombination operators and a local search operator. At the same time, the predatory behaviour of the goshawk is optimised by combining the characteristics of the disassembly sequence planning problem to improve its performance. Finally, the disassembly of the battery of a Tesla Model 1 is used as a case study to demonstrate the effectiveness and feasibility of the proposed method.