Optimal Dual Schemes for Adaptive Grid Based Hexmeshing

Hexahedral meshes are a ubiquitous domain for the numerical resolution of partial differential equations. Computing a pure hexahedral mesh from an adaptively refined grid is a prominent approach to automatic hexmeshing, and requires the ability to restore the all hex property around the hanging nodes that arise at the interface between cells having different size. The most advanced tools to accomplish this task are based on mesh dualization. These approaches use topological schemes to regularize the valence of inner vertices and edges, such that dualizing the grid yields a pure hexahedral mesh. In this article, we study in detail the dual approach, and propose four main contributions to it: (i) We enumerate all the possible transitions that dual methods must be able to handle, showing that prior schemes do not natively cover all of them; (ii) We show that schemes are internally asymmetric, therefore not only their construction is ambiguous, but different implementative choices lead to hexahedral meshes with different singular structure; (iii) We explore the combinatorial space of dual schemes, selecting the minimum set that covers all the possible configurations and also yields the simplest singular structure in the output hexmesh; (iv) We enlarge the class of adaptive grids that can be transformed into pure hexahedral meshes, relaxing one of the tight topological requirements imposed by previous approaches. Our extensive experiments show that our transition schemes consistently outperform prior art in terms of ability to converge to a valid solution, amount and distribution of singular mesh edges, and element count. Last but not least, we publicly release our code and reveal a conspicuous amount of technical details that were overlooked in previous literature, lowering an entry barrier that was hard to overcome for practitioners in the field.

A posteriori error analysis of an enriched Galerkin method of order one for the Stokes problem

We derive optimal reliability and efficiency of a posteriori error estimates for the steady Stokes problem, with a nonhomogeneous Dirichlet boundary condition, solved by a stable enriched Galerkin scheme (EG) of order one on triangular or quadrilateral meshes in ℝ2, and tetrahedral or hexahedral meshes in ℝ3.

Fast Computation by MLFMM-FFT with NURBS in Large Volumetric Dielectric Structures

A refinement for the computation of the rigorous part of the multi-level fast multipole method (MLFMM) of analyzing volumetric objects is presented. A scheme based on the fast Fourier technique (FFT) is proposed with the objective of reducing the computational resources required to accurately analyze large homogeneous and non-homogeneous dielectric volumes. In order to reduce the memory requirements, the storage of the near-field terms of the method of moments (MoM) matrix is performed only for the positions corresponding to a parallelepiped with the size of the level 1 block of the MLFMM, computed with the vacuum permittivity, taking advantage of the Toeplitz symmetry present in regular hexahedral meshes. The FFT avoids applying the near-field MoM matrix in the iterative solution process. The application of this approach results in huge improvements in terms of memory usage, but also a speeds up the iterative solution process because the use of three-dimensional (3D) FFTs is very efficient for computing convolutions when the number of unknowns of the problems becomes very large as happens in volumetric problems. We also propose a new approach for the numerical treatment of the transition of the dielectric permittivity between different dielectrics or between a dielectric and a free space. To validate the computation technique, the radar cross section (RCS) of several dielectric bodies is computed using the classical MLFMM approach and it is compared with the presented FFT-based-MLFMM solution. The results demonstrate that the efficient memory and computation time usage of the proposed approach.

Hexahedral Mesh Quality Improvement with Geometric Constraints

Hexahedral mesh is of great value in the analysis of mechanical structure, and the mesh quality has an important impact on the efficiency and accuracy of the analysis. This paper presents a quality improvement method for hexahedral meshes, which consists of node classification, geometric constraints based single hexahedron regularization and local hexahedral mesh stitching. The nodes are divided into different types and the corresponding geometric constraints are established in single hexahedron regularization to keep the geometric shapes of original mesh. In contrast to the global optimization strategies, we perform the hexahedral mesh stitching operation within a few local regions surrounding elements with undesired quality, which can effectively improve the quality of the mesh with less consuming time. A number of mesh quality improvements for hexahedral meshes generated by a variety of methods are introduced to demonstrate the effectiveness of our method.

Assessment of the Dimensional and Geometric Precision of Micro-Details Produced by Material Jetting

Additive Manufacturing (AM) technology has been increasing its penetration not only for the production of prototypes and validation models, but also for final parts. This technology allows producing parts with almost no geometry restrictions, even on a micro-scale. However, the micro-Detail (mD) measurement of complex parts remains an open field of investigation. To be able to develop all the potential that this technology offers, it is necessary to quantify a process’s precision limitations, repeatability, and reproducibility. New design methodologies focus on optimization, designing microstructured parts with a complex material distribution. These methodologies are based on mathematical formulations, whose numerical models assume the model discretization through volumetric unitary elements (voxels) with explicit dimensions and geometries. The accuracy of these models in predicting the behavior of the pieces is influenced by the fidelity of the object’s physical reproduction. Despite that the Material Jetting (MJ) process makes it possible to produce complex parts, it is crucial to experimentally establish the minimum dimensional and geometric limits to produce parts with mDs. This work aims to support designers and engineers in selecting the most appropriate scale to produce parts discretized by hexahedral meshes (cubes). This study evaluated the dimensional and geometric precision of MJ equipment in the production of mDs (cubes) comparing the nominal design dimensions. A Sample Test (ST) with different sizes of mDs was modeled and produced. The dimensional and geometric precision of the mDs were quantified concerning the nominal value and the calculated deviations. From the tests performed, it was possible to conclude that: (i) more than 90% of all analyzed mDs exhibit three dimensions (xyz) higher than the nominal ones; (ii) for micro-details smaller than 423 m, they show a distorted geometry, and below 212 m, printing fails.