Regular Mesh Research Articles

Dr. KID: Direct Remeshing and K-Set Isometric Decomposition for Scalable Physicalization of Organic Shapes.

Dr. KID is an algorithm that uses isometric decomposition for the physicalization of potato-shaped organic models in a puzzle fashion. The algorithm begins with creating a simple, regular triangular surface mesh of organic shapes, followed by iterative K-means clustering and remeshing. For clustering, we need similarity between triangles (segments) which is defined as a distance function. The distance function maps each triangle's shape to a single point in the virtual 3D space. Thus, the distance between the triangles indicates their degree of dissimilarity. K-means clustering uses this distance and sorts segments into k classes. After this, remeshing is applied to minimize the distance between triangles within the same cluster by making their shapes identical. Clustering and remeshing are repeated until the distance between triangles in the same cluster reaches an acceptable threshold. We adopt a curvature-aware strategy to determine the surface thickness and finalize puzzle pieces for 3D printing. Identical hinges and holes are created for assembling the puzzle components. For smoother outcomes, we use triangle subdivision along with curvature-aware clustering, generating curved triangular patches for 3D printing. Our algorithm was evaluated using various models, and the 3D-printed results were analyzed. Findings indicate that our algorithm performs reliably on target organic shapes with minimal loss of input geometry.

Face ShapeNets for 3D Face Recognition

In this paper, we present a deep learning-based method for 3D face recognition. Unlike some previous works, our process does not rely on face representation methods as a proxy step to be accepted by Convolutional Neural Networks (CNNs). Applying 2D CNNs to irregular domains such as 3D meshes is challenging. Therefore, we employed 3D ShapeNets to recognize faces covering the full 3D shape since 3D face datasets are available and 3D data augmentation techniques to enlarge 3D datasets are widespread. The reduced size of 3D datasets is overcome by an appropriate 3D data augmentation to train our model. 3D ShapeNets are commonly used to recognize and analyse objects. To the best of the authors’ knowledge, this is the first time they are used for face recognition. This research work focuses on the preprocessing step. Whatever the nature of the face image is (either 2D or 3D representation), and whatever the acquisition conditions are, a 3D regular mesh of each input face image is first generated. Furthermore, meshes are converted to voxels to get the occupancy grid across all possible views. Finally, 3D ShapeNets are trained and recognition tests are performed. Indeed, 3D ShapeNets prove the efficiency and efficacy of 3D shape analysis task in 3D face recognition. The experimental results show that 3D face recognition using deep 3D CNNs such as 3D ShapeNets leads to significant improvement over the state-of-the-art performance on LFPW, BU3DFE, and FRAV3D datasets, with competitive recognition rates of 94.25%, 97.9% and 98.31% respectively.

Three-dimensional X-FEM modeling of crack coalescence phenomena in the Smart CutTM technology

A new 3D X-FEM approach is proposed to model crack coalescence phenomena in the Smart Cut TM technology. An adaptive mesh is developed based on a 3D fractal mesh allowing a significant gain in computing time compared to regular meshes. In order to calculate stress intensity factors (SIFs), the term corresponding to internal pressure, that is normal loading on crack faces has been determined, added to the interaction integral and the results of SIFs in mode I have been compared to their analytical values with mean error less than 1%. An implicit algorithm is implemented to predict the evolution of the internal pressure for growing cracks of any shape; a constitutive law in pressure correlated with the equation of state of ideal gases has therefore been used. Compared to the analytical solution, numerically evaluated pressure is accurate with maximum errors less than 1% for relevant convergence tolerance. Finally, the coalescence of two cracks is taken into account in the model. Qualitative local stress analysis makes it possible to explain some phenomena experimentally observed in the Smart Cut TM such as neighboring effects of growing cracks, the outgrowth of nearby cracks or influence of the presence of concave areas at a front on growth. • A 3D X-FEM model has been proposed to model crack coalescence phenomena under variable pressure (Smart Cut TM process). • The interaction integral has been adapted to take into account the internal pressure when computing 3D stress intensity factors. • Prediction of the internal pressure of propagating cracks whatever are their shapes. • Analysis of neighboring effects by stress analysis on a two penny-shape crack coalescence model.

SurfFlow: A Flow-Based Approach for Rapid and Accurate Cortical Surface Reconstruction from Infant Brain MRI.

The infant brain undergoes rapid changes in volume, shape, and structural organization during the first postnatal year. Accurate cortical surface reconstruction (CSR) is essential for understanding rapid changes in cortical morphometry during early brain development. However, existing CSR methods, designed for adult brain MRI, fall short in reconstructing cortical surfaces from infant MRI, owing to the poor tissue contrasts, partial volume effects, and rapid changes in cortical folding patterns. Here, we introduce an infant-centric CSR method in light of these challenges. Our method, SurfFlow, utilizes three seamlessly connected deformation blocks to sequentially deform an initial template mesh to target cortical surfaces. Remarkably, our method can rapidly reconstruct a high-resolution cortical surface mesh with 360k vertices in approximately one second. Performance evaluation based on an MRI dataset of infants 0 to 12 months of age indicates that SurfFlow significantly reduces geometric errors and substantially improves mesh regularity compared with state-of-the-art deep learning approaches.

The imposition of nonconforming Neumann boundary condition in the material point method without boundary representation

Since its original formulation, the material point method (MPM) has been repeatedly further developed to become a powerful tool for simulating large deformation problems. However, the imposition of boundary condition in the MPM remains a challenge since most material points are not located at the boundary and regular background mesh is most commonly used. Therefore, the existing methods to impose boundary condition always require a boundary tracking or reconstruction algorithm, which is not always efficient or accurate. In the current work, a novel strategy to apply nonconforming Neumann boundary condition is proposed that does not require the exact boundary position. The original problem with a nonconforming traction boundary condition is transformed into an equivalent problem with a prescribed virtual stress field. Importantly, both the original and the transformed problems produce the exact same response within the material domain. To solve the equivalent problem, a modified governing equation without boundary representation is subsequently constructed, where the volume integral terms are computed by both particle quadrature and cell quadrature. Several numerical examples are investigated to assess the accuracy and demonstrate the capability of the proposed approach in simulating different engineering problems. These numerical examples include mesh refinement to illustrate the method’s good convergence and a 3D complex surface to illustrate the method’s capability.

Methods of visual analysis in the design of the stray light protection of optical devices

We provide an analysis of the existing methods for calculating parasitic illumination, methods for its analysis, and visualization. To search for and analyze the sources of stray light in optical devices, a software model of the beam propagation criterion in an optical device is proposed, which makes it possible to select beams that satisfy a given criterion. We consider the possibility of using the method of bidirectional progressive stochastic ray tracing with inverse photon maps to calculate the stray illumination of an image and analyze the causes of its occurrence. To analyze the sources of stray illumination on the radiation detector, it was proposed to use progressive photon maps, which calculated the caustic and secondary components of the illumination and fixed the image illumination on a static regular mesh attached to the surfaces of the optical device. Special visualization tools allow the display of this illumination over the image of an optical device and determination of the elements of the device that have the greatest impact on the level of stray illumination of the image. We present examples of calculation, analysis, and visualization of parasitic illumination of a number of optical systems of lenses with real mechanical structures.

Towards improving the 2D-MITC4 element for analysis of plane stress and strain problems

In this paper, we improve the performance of the 2D-MITC4 element for analysis of plane stress and plane strain problems. While the element shows excellent convergence behavior in regular meshes, it exhibits accuracy loss in distorted meshes, as many other elements do. We focus on improving performance in distorted meshes for general use in engineering practice. We simplify the assumed strain field of the original 2D-MITC4 element and introduce the geometry dependent Gauss integration scheme, in which integration points vary according to element geometry distortion. Consequently, a new 2D-MITC4 element is developed. The new 2D-MITC4 element passes the basic tests: patch, zero energy mode, and isotropy tests. Through various numerical examples, improved convergence behaviors of the new element are demonstrated, especially in distorted meshes.

On the Resolution of Triangular Meshes

AbstractIt is generally agreed that the resolution of a regular quadrilateral mesh is the side length of quadrilateral cells. There is less agreement on the resolution of triangular meshes, exacerbated by the fact that the numbers of edges or cells on triangular meshes are approximately three or two times larger than that of vertices. However, the geometrical resolution of triangular meshes, that is, maximum wavenumbers or smallest wavelengths that can be represented on such meshes, is a well defined quantity, known from solid state physics. These wavenumbers are related to a smallest common mesh cell (primitive unit cell), and the set of mesh translations that map it into itself. They do not depend on whether discrete degrees of freedom are placed on vertices, cells or edges. For equilateral triangles the smallest wavelength equals twice the triangle height. Resolutions of quadrilateral and triangular meshes approximately agree if they have the same numbers of vertices.

Superconvergence of an Ultra-Weak Discontinuous Galerkin Method for Nonlinear Second-Order Initial-Value Problems

In this paper, we develop and analyze an ultra-weak discontinuous Galerkin (UWDG) method for nonlinear second-order initial-value problems for ordinary differential equations of the form [Formula: see text]. Our main concern is to study the convergence and superconvergence properties of the proposed scheme. With a suitable choice of the numerical fluxes, we prove the optimal error estimates with order [Formula: see text] in the [Formula: see text]-norm for the solution, when piecewise polynomials of degree at most [Formula: see text] are used. We use these results to prove that the UWDG solution is superconvergent with order [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text] towards a special projection of the exact solution. We further prove that the [Formula: see text]-degree UWDG solution and its derivative are [Formula: see text] superconvergent at the end of each step. Our proofs are valid for arbitrary regular meshes using piecewise polynomials with degree [Formula: see text]. Finally, numerical experiments are provided to verify that all theoretical findings are sharp. The main advantage of our method over the standard DG method for systems of first-order equations is that the UWDG method can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system, which reduces memory and computational costs.

ARPIST: Provably accurate and stable numerical integration over spherical triangles

Numerical integration on spherical triangles, including the computation of their areas, is a core computation in geomathematics. The commonly used techniques sometimes suffer from instabilities and significant loss of accuracy. We describe a new algorithm, called ARPIST, for accurate and stable integration of functions on spherical triangles. ARPIST is based on an easy-to-implement transformation to the spherical triangle from its corresponding linear triangle via radial projection to achieve high accuracy and efficiency. More importantly, ARPIST overcomes potential instabilities in computing the Jacobian of the transformation, even for poorly shaped triangles that may occur at poles in regular longitude-latitude meshes, by avoiding potential catastrophic rounding errors. We compare our proposed technique with L’Huilier’s Theorem for computing the area of spherical triangles, and also compare it with the recently developed LSQST method (Beckmann et al., 2014) and a radial-basis-function-based technique (Reeger and Fornberg, 2016) for integration of smooth functions on spherical triangulations. Our results show that ARPIST enables better or comparable accuracy over previous methods while being easier to implement, significantly faster, and more tolerant of poor element shapes.

Battling Gibbs phenomenon: On finite element approximations of discontinuous solutions of PDEs

In this paper, we want to clarify the Gibbs phenomenon when continuous and discontinuous finite elements are used to approximate discontinuous or nearly discontinuous PDE solutions from the approximation point of view. For a simple step function, we explicitly compute its continuous and discontinuous piecewise constant or linear projections on discontinuity matched or non-matched meshes. For the simple discontinuity-aligned mesh case, piecewise discontinuous approximations are always good. For the general non-matched case, we explain that the piecewise discontinuous constant approximation combined with adaptive mesh refinements is a good choice to achieve accuracy without overshoots. For discontinuous piecewise linear approximations, non-trivial overshoots will be observed unless the mesh is matched with discontinuity. For continuous piecewise linear approximations, the computation is based on a “far-away assumption”, and non-trivial overshoots will always be observed under regular meshes. We calculate the explicit overshoot values for several typical cases. Numerical tests are conducted for a singularly-perturbed reaction-diffusion equation and linear hyperbolic equations to verify our findings in the paper. Also, we discuss the L 1 -minimization-based methods and do not recommend such methods due to their similar behavior to L 2 -based methods and more complicated implementations.

Robust Quantization for Polycube Maps

An important part of recent advances in hexahedral meshing focuses on the deformation of a domain into a polycube; the polycube deformed by the inverse map fills the domain with a hexahedral mesh. These methods are appreciated because they generate highly regular meshes. In this paper we address a robustness issue that systematically occurs when a coarse mesh is desired: algorithms produce deformations that are not one-to-one, leading to collapse of large portions of the model when trying to apply the (undefined) inverse map. The origin of the problem is that the deformation requires to perform a mixed integer optimization, where the difficulty to enforce the integer constraints is directly tied to the expected coarseness. Our solution is to introduce sanity constraints preventing the loss of bijectivity due to the integer constraints.

Optimal [formula omitted](div) flux approximations from the Primal Hybrid Finite Element Method on quadrilateral meshes

In this work, we propose and analyze an element-wise H(div,Ω)-conforming computational strategy to post-process the flux vector from the approximated solution of a second-order elliptic equation, obtained by the Primal Hybrid Finite Element Method on regular meshes of convex quadrilaterals generated by bilinear isomorphisms. The recovery strategy makes use of the Arnold–Boffi–Falk (ABFt, t≥0) spaces, leading to locally conservative approximated fluxes with continuous normal components. The proposed method is proven to provide optimal order approximations in H(div,Ω), in the sense that both the flux and its divergence achieve optimal order ht+1 in the L2(Ω)-norm, on quadrilateral meshes. We also show that, within the recovery strategy, it is possible to achieve higher-order approximations for the divergence of the flux by increasing the index t of the local spaces, with almost no extra overall computational cost. Numerical results on affine and bilinear quadrilateral meshes are presented, confirming the theoretical predictions.

Inverse Determination of Acoustic Properties of Acoustic Ventilation Cloth (Woven and Non-Woven) by Particle Swarm Optimization and Estimation of Its Effect on the Frequency Response of the Microspeaker

In this work, microspeaker frequency response was determined based on measurement and simulation. The vents after the rear chamber of the microspeaker were covered with ventilation cloths. Two types of ventilation cloths (acoustic cloths) are commonly used in electroacoustic products, non-woven and woven. Non-woven cloths (5 nos.) consist of irregular meshes and woven cloths (5 nos.) consist of regular meshes. The equivalent circuit model of the microspeaker was formulated by considering five types each of non-woven and woven ventilation cloths. The acoustic properties of the ventilation cloths were estimated from the measured frequency response of the microspeaker and by subsequent use of particle swarm optimization algorithm. Estimated value of the acoustic impedances of the ventilation cloths were used in an equivalent circuit model of the microspeaker for the simulated frequency responses and subsequently compared with the anechoic chamber measurements. Based on the results, the equivalent circuit adequately simulated the measured frequency response of the microspeaker and the estimations of the acoustic impedances of the ventilation cloths were in good agreement with the measured frequency responses of the microspeaker.

Evolutionary $\Gamma$-Convergence of Entropic Gradient Flow Structures for Fokker--Planck Equations in Multiple Dimensions

We consider finite-volume approximations of Fokker--Planck equations on bounded convex domains in $\mathbb{R}^d$ and study the corresponding gradient flow structures. We reprove the convergence of the discrete to continuous Fokker--Planck equation via the method of evolutionary $\Gamma$-convergence, i.e., we pass to the limit at the level of the gradient flow structures, generalizing the one-dimensional result obtained by Disser and Liero. The proof is of variational nature and relies on a Mosco convergence result for functionals in the discrete-to-continuum limit that is of independent interest. Our results apply to arbitrary regular meshes, even though the associated discrete transport distances may fail to converge to the Wasserstein distance in this generality.

ARCHER - a new Three-Dimensional method of characteristics neutron transport code for Pebble-bed HTR with coarse mesh finite difference acceleration

A new deterministic code ARCHER using Three-Dimensional (3-D) Method of Characteristics (MOC) has been developed by Institute of Nuclear and new Energy Technology (INET), Tsinghua University, aiming to obtain high-fidelity transport solution of pebble-bed High Temperature gas-cooled Reactor (HTR) with explicit pebble-bed geometry. However, 3-D MOC usually suffers from the slow convergence rate without efficient acceleration methods. In this work, the Coarse Mesh Finite Difference (CMFD) based on regular mesh is implemented to accelerate 3-D MOC calculation, which is available for the problems with cylinder and cuboid geometry. The complex geometry of reactor core with randomly stacked spherical fuel elements and irregular helium flow area, as well as the topological relationships between fine Flat Source Region (FSR) mesh for MOC and coarse CMFD mesh, are the key but challenging tasks for high-fidelity transport solution. Constructive Solid Geometry (CSG) and Boolean operation are constructed to describe problem geometry, which has the ability to deal with complex topological relationships between meshes. In addition, tree grid structure is utilized to reduce the time complexity of ray tracing and obtain the grid mapping relationship between coarse mesh and fine mesh. Spatial domain decomposition parallel based on MPI is utilized to alleviate the memory pressure and pursue high computational performance. Ray parallel is further achieved through OpenMP with shared memory. Several problems have been used to verify the geometric modeling capability and computational performance of this new 3-D MOC code for pebble-bed HTR.

Numerical convergence of the family of flux-continuous schemes with variable quadrature (qu1,qu2) for single phase flow in porous media

Finite-volume schemes, which honor pressure and flux-continuity conditions, is developed using double quadrature (qu1,qu2), referred as double family scheme. The scheme is applicable to solve the elliptic pressure equation used in reservoir simulation. Schemes are applicable on both regular cartesian and unstructured triangular meshes. The scheme is defined over a control-volume distributed formulation. The scheme can be applied to both diagonal and full permeability tensor elliptic pressure equation with discontinuous coefficients. The scheme removes the first order errors, which are introduced by standard reservoir simulation schemes when applied to full tensor flow. The scheme is quantified with help of a quadrature rule. When the scheme is applied to highly heterogeneous and anisotropic porous media it does not honor maximum principle resulting in unstable solution with oscillatory behavior. The numerical solution is termed non-monotonicity for high anisotropy ratios with results showing oscillations in the numerical pressure solution. In this paper a double (qu1,qu2) quadrature flux continuous schemes is presented, which with specific choice of quadrature (qu1,qu2) helps in improved stability of the numerical solutions. Numerical convergence of the scheme is also demonstrated with help of a number of numerical test cases and schemes impact on monotonicity behavior is also demonstrated with numerical examples.

Analysis of Numerical Artifacts Using Tetrahedral Meshes in Low Frequency Numerical Dosimetry

Anatomical realistic voxel models of human beings are commonly used in numerical dosimetry to evaluate the human exposure to low-frequency electromagnetic fields. The downside of these models is that they do not correctly reproduce the boundaries of curved surfaces. The stair-casing approximation errors introduce computational artifacts in the evaluation of the induced electric field and the use of post-processing filtering methods is essential to mitigate these errors. With a suitable exposure scenario, this paper shows that tetrahedral meshes make it possible to remove stair-casing errors. However, using tetrahedral meshes is not a sufficient condition to completely remove artifacts, because the quality of the tetrahedral mesh plays an important role. The analyses carried out show that in real exposure scenarios, other sources of artifacts cause peak values of the induced electric field even with regular meshes. In these cases, the adoption of filtering techniques cannot be avoided.

Thermo-Mechanical Calculation of Printed Circuit Heat Exchanger Using Homogenization

The Finite-Element Method (FEM) is mainly used to design Compact Heat Exchanger structures. However, the narrow channel structures require fine mesh to accurately compute stress and strain. Combined with the dimensions of the overall component, this leads to an excessively large numerical model and therefore long computing time. This is especially true for transient thermal analyses, which require taking into account the full geometry. It is therefore interesting to reduce the size of the mesh. Periodic homogenization is an efficient method for achieving this goal. It can be applied to the core of the structure when periodic patterns (identified on basis of the arrangement of the channels) exist. A method based on this technique, with some improvements, is proposed in this paper for thermal loading without internal pressure. In addition to the Equivalent Homogenous Medium (EHM) replacing the periodic patterns, some explicit channels are interposed between the EHM and the cover plates. This has two advantages. The first is to smooth the transition of stiffness between the cover plates and the homogenized medium. The second one is to be able to directly compute stress and strain on the most critical channels located in this area. This paper assesses the method’s effectiveness for thermal loadings in order to conduct thermal stress analyses. First, the equivalent elastic constants of the EHM are obtained with a numerical Finite Elements Method. Then, two 2D cases using EHM are compared against a 2D explicit model (i.e. explicit geometry for the core channels). These cases are chosen to validate the EHM itself and its effectiveness for a real section of the heat exchanger. Results show very good agreement with a relative difference lower than 1%. In addition, the sensitivity to the number of layers added between the EHM and the cover plates is analysed. It is recommended interposing at least two layers of patterns to obtain converged results for the considered configuration. Finally, a triangular mesh is considered to reduce the size of the model. No difference with a regular quadrangular mesh can be observed whereas the computing time is reduced. This method can be used to perform the design of any CHE under thermal loading as long as the channel arrangement shows periodicity.

Some Estimates for Virtual Element Methods in Three Dimensions

Abstract Some estimates on virtual element methods (VEMs), including inverse inequalities, norm equivalence, and interpolation error estimates, are developed for star-shaped polyhedral meshes whose faces admit virtual regular and quasi-uniform triangulations. This mesh regularity covers the usual ones used for theoretical analysis of VEMs, and the proofs are carried out in a straightforward way.