Symmetric Mesh Research Articles

Symmetric Piecewise Developable Approximations

AbstractWe propose a novel method for generating symmetric piecewise developable approximations for shapes in approximately global reflectional or rotational symmetry. Given a shape and its symmetry constraint, the algorithm contains two crucial steps: (i) a symmetric deformation to achieve a nearly developable model and (ii) a symmetric segmentation aided by the deformed shape. The key to the deformation step is the use of the symmetric implicit neural representations of the shape and the deformation field. A new mesh extraction from the implicit function is introduced to construct a strictly symmetric mesh for the subsequent segmentation. The symmetry constraint is carefully integrated into the partition to achieve the symmetric piecewise developable approximation. We demonstrate the effectiveness of our algorithm over various meshes.

Specific pelvic shape in patients with developmental dysplasia of the hip on 3D morphometric homologous model analysis.

To clarify the morphological factors of the pelvis in patients with developmental dysplasia of the hip (DDH), three-dimensional (3D) pelvic morphology was analyzed using a template-fitting technique. Three-dimensional pelvic data of 50 patients with DDH (DDH group) and 3D pelvic data of 50 patients without obvious pelvic deformity (Normal group) were used. All patients were female. A template model was created by averaging the normal pelvises into a symmetrical and isotropic mesh. Next, 100 homologous models were generated by fitting the pelvic data of each group of patients to the template model. Principal component analysis was performed on the coordinates of each vertex (15,235 vertices) of the pelvic homologous model. In addition, a receiver-operating characteristic (ROC) curve was calculated from the sensitivity of DDH positivity for each principal component, and principal components for which the area under the curve was significantly large were extracted (p<0.05). Finally, which components of the pelvic morphology frequently seen in DDH patients are related to these extracted principal components was evaluated. The first, third, and sixth principal components showed significantly larger areas under the ROC curves. The morphology indicated by the first principal component was associated with a decrease in coxal inclination in both the coronal and horizontal planes. The third principal component was related to the sacral inclination in the sagittal plane. The sixth principal component was associated with narrowing of the superior part of the pelvis. The most important factor in the difference between normal and DDH pelvises was the change in the coxal angle in both the coronal and horizontal planes. That is, in the anterior and superior views, the normal pelvis is a triangle, whereas in DDH, it was more like a quadrilateral.

Finite element method for a generalized constant delay diffusion equation

This paper considers the finite element method to solve a generalized constant delay diffusion equation. The regularity of the solution of the considered model is investigated, which is the first time to discover that the solution has non-uniform multi-singularity in time compared with Tan et al. (2022). To overcome the multi-singularity, a symmetrical graded mesh is used to devise the fully discrete finite element scheme for the considered problem based on L1 formula of the Caputo fractional derivative and fractional trapezoidal formula of the Riemann–Liouville fractional integral. Then we investigate the unconditional stability of this scheme. Next, the local truncation errors of the L1 formula and the fractional trapezoidal formula are analyzed in detail, especially the later one is discussed at the first time, under the multi-singularity of the solution and the symmetrical graded mesh. Using these error results, we obtain the convergence of the proposed numerical scheme. Finally, some numerical tests are provided to verify the obtained theoretical results.

Underlying mathematical relations for efficient COMET response expansion coefficient and core calculations

The coarse mesh transport code COMET, a hybrid stochastic-deterministic neutronics solver with formidable computation speed, can provide high fidelity transport solutions to various reactor core problems. In this paper, to take advantage of local rotational and reflection symmetry existing in many reactor cores (e.g., fuel lattices and reflector blocks), the underlying mathematical relations among the response expansion coefficients of symmetric coarse meshes are developed to further improve the computational efficiency in both COMET response expansion coefficient generation and core calculation. This is done by a rigorous derivation of the transformation matrices for the angular and spatial expansion moments resulting from a rotation of a coarse mesh by an arbitrary angle or reflection of a coarse mesh. As a result, the relations for the response expansion coefficients of a coarse mesh with rotational symmetry can be then written as the Kronecker product of those transformation matrices. For coarse meshes with reflection symmetry, the sparse property, the outgoing surface symmetry property and the incident surface symmetry property of response expansion coefficients are rigorously derived. These underlying mathematical relations are implemented into COMET and evaluated on three stylized 3D PWR whole core benchmark problems. The COMET results using the response expansion coefficient library based on the underlying mathematical relations were compared to those using the library directly generated by Monte Carlo for all surfaces. It was found that the eigenvalues as well as the pin and assembly fission densities using the two libraries are in statistical agreement as expected. This indicates that the new COMET using the underlying mathematical relations maintains the high fidelity of the original COMET while improving computational efficiency by 10.7 and 3.7 times for COMET response expansion coefficient generation and core calculation, respectively. This also reduces the size of the library by 10.7 times.

A simple and efficient numerical method for the Allen–Cahn equation on effective symmetric triangular meshes

&lt;abstract&gt;&lt;p&gt;In this paper, we propose a novel, simple, efficient, and explicit numerical method for the Allen–Cahn (AC) equation on effective symmetric triangular meshes. First, we compute the net vector of all vectors starting from each node point to its one-ring neighbor vertices and virtually adjust the neighbor vertices so that the net vector is zero. Then, we define the values at the virtually adjusted nodes using linear and quadratic interpolations. Finally, we define a discrete Laplace operator on triangular meshes. We perform several computational experiments to demonstrate the performance of the proposed numerical method for the Laplace operator, the diffusion equation, and the AC equation on triangular meshes.&lt;/p&gt;&lt;/abstract&gt;

Semi-analytical calculation of pore-related parameters of wire/woven screens

Wire/woven screens have a wide range of applications, from being used as simple mechanical screening device to nanoscreen wicking with nanofluids. The vast number of applications makes important to study these screens with high accuracy, to reduce errors in characterisation and performance predictions. Previous works to date focused either on the study of these screens as a two-dimensional surface (e.g. in ventilation openings as insect-proof screens) or as three-dimensional structures under important assumptions (symmetric mesh, thickness of two times their diameter, linear evolution of the pore area along the thickness). These incomplete modellings introduce errors in applications such as the estimation of permeability of the porous media (two-dimensional porosity is identical for two meshes with the same projected area of pore but different thickness) or computational simulations of ventilation in buildings/greenhouses, where these parameters are imposed as boundary condition. The present investigation shows a method to calculate three-dimensional pore related structural properties semi-analytically for the first time and for any plain square mesh. We found that when sweeping the mesh with a plane parallel to it there are up to six different zones or stretches which can be integrated by a piece-wise approach (here named Discretisation Method). Results demonstrated high accuracy in the calculation of three-dimensional porosity and constriction factor (a parameter that is calculated by integration over the pore volume). Due to the mathematical complexity in the method, a software (AeroScreen v1.0) is available to obtain pore-related structural parameters from diameters, separations and thickness of the screen.

Effect of Heat Treatment on the Vibration Isolation Performance of Axially Symmetric NiTi Wire Mesh Damper

In this paper, superelastic (SE) NiTi wire is used to fabricate axially symmetric wire mesh dampers (WMDs) with the expectation of a higher damping capacity. However, the phase transformation damping of the NiTi WMD could be suppressed by the cold-work-induced dislocation. Therefore, the NiTi WMDs were heat-treated and then tested by a hydraulic universal testing machine. The NiTi WMD is found to achieve higher damping capacity when heat-treated at 200 °C. However, the WMD heat-treated at 250 °C suffers from a sharp decline in the loss factor in exchange for an improvement in the stiffness. The sine sweep test was then conducted to examine the dependency of the WMD’s vibration isolation performance upon the heat treatment temperature and the excitation acceleration. The NiTi WMD outperforms the 304 stainless steel (SS 304) WMD in damping capacity only when the excitation acceleration magnitude is less than 1.5 g. The stiffness of NiTi WMD can be improved without significantly compromising its damping capacity by heat treatment at 200 °C for 30 min. The present work carries out comprehensive measurements of the NiTi WMD’s response to dynamic mechanical test and sine sweep test and addresses how heat treatment influences the stiffness and damping capacity of the SE NiTi WMD.

Failure properties and stability monitoring of strip and column cemented gangue backfill bodies under uniaxial compression in constructional backfill mining.

The strip and column cemented gangue backfill bodies (CGBBs) are the main supporting components in the design of constructional backfill mining for coal mining, which determines the stability of goaf. Previous researches have mostly focused on the mechanical properties of column CGBB, but the mechanical properties of strip CGBB are still unclear. Herein, the uniaxial compression experiments for strip and column CGBBs were conducted to compare the failure properties. The acoustic emission (AE) and two types of resistivity monitoring were used to monitor the damage evolution. The effect of the length-height ratio on the mechanical characteristic of strip CGBB was analyzed by discrete element simulation. The results show that the strength and peak strain of strip CGBB under uniaxial compression is higher than those of column CGBB and the strip CGBB shows better ductility. The stress of column CGBB decreases significantly faster than that of strip CGBB at the post-peak stage. The strength and ductility of strip CGBB increase with the increase of length-height ratio. The strip CGBB is destroyed from both ends to the middle under uniaxial compression, and the core bearing area is reduced correspondingly. The AE signal evolution of CGBBs under uniaxial compression before the peak stress contains three stages, and the AE signals of strip CGBB at the peak stress will not rise sharply compared with column CGBB. The resistivity monitoring effect of the horizontally symmetrical conductive mesh is better than that of the axial. The horizontal resistivity increases gradually with the increase of stress under uniaxial compression, and increases sharply at the peak stress, and then drops after the peak stress. The damage constitutive models and the stability monitoring models of the CGBBs are established based on the experimental results. This work would be instructive for the design and stability monitoring of CGBB.

Regenerative furnace simulation under flameless combustion regime with natural gas and syngas

The increase in pollutant emissions has generated the development of clean technologies and encouraged the use of alternative fuels like syngas. Flameless combustion is one of the most promising clean technologies due to lower pollutants emissions and high thermal efficiency. In the present study, regenerative furnace simulation under flameless combustion was carried out using a fuel mixture of 70% natural gas and 30% syngas with high H2 content (in vol.); by mean of CFD approach. A symmetrical mesh with 405,632 hexahedral cells was used in the calculations. The κ-ε standard, discrete ordinates and Eddy Dissipation Concept models were used for physical phenomena of turbulence, radiation and turbulence - chemistry interaction, respectively. The combustion chambers thermal uniformity factor was 0.3 and 0.1 for vertical and horizontal planes, respectively. The results confirm that it is possible to achieve the flameless combustion regime in a regenerative furnace using the addition of syngas, preserving the temperature and reaction zone uniformity.

High surface accuracy and pretension design for mesh antennas based on dynamic relaxation method

Cable truss structures are widely used in large satellite antennas with ultrahigh electromagnetic performance. The electromagnetic performance is directly determined by the mesh reflector's surface accuracy, which triggers research on the pretension design. In this paper, a novel pretension design method for both high surface accuracy and uniform tension distribution is proposed based on the dynamic relaxation (DR) method. In this approach, members of the cable network are regarded as string elements with 3 degrees of freedom to resist the axial tensile force, and members of the supporting truss are regarded as Euler-Bernoulli beams with 6 degrees of freedom to resist the axial and transverse forces and the bending and torsion moments. The DR method is adopted to find the static equilibrium of the whole mesh antenna, including both the cable mesh and the supporting truss. Thus, the deformation of the cables and rods and their interactive effects are fully captured. Then, to find an expected high-precision mesh surface with a uniform tension distribution, the inverse iteration algorithm (IIA) is adopted to adjust the original cable lengths of the cable network in each DR calculation cycle until the convergence criterion is met. Finally, this approach is effectively applied to the pretension design of symmetric and asymmetric mesh antennas, and the results indicate that the method is effective in obtaining both high surface accuracy and uniform tension distribution with high computational efficiency.

Principal symmetric meshes

The isolines of principal symmetric surface parametrizations run symmetrically to the principal directions. We describe two discrete versions of these special nets/quad meshes which are dual to each other and show their usefulness for various applications in the context of fabrication and architectural design. Our discretization of a principal symmetric mesh comes naturally with a family of spheres, the so-called Meusnier and Mannheim spheres. In our representation of principal symmetric meshes, we have direct control over the radii of theses spheres and the intersection angles of the parameter lines. This facilitates tasks such as generating Weingarten surfaces including constant mean curvature surfaces and minimal surfaces. We illustrate the potential of Weingarten surfaces for paneling doubly curved freeform facades by significantly reducing the number of necessary molds. Moreover, we have direct access to curvature adaptive tool paths for cylindrical CNC milling with circular edges as well as flank milling with rotational cones. Furthermore, the construction of curved support structures from congruent circular strips is easily managed by constant sphere radii. The underlying families of spheres are in a natural way discrete curvature spheres in analogy to smooth Möbius and Laguerre geometry which further leads to a novel discrete curvature theory for principal symmetric meshes.

Short communication: Landlab v2.0: a software package for Earth surface dynamics

Abstract. Numerical simulation of the form and characteristics of Earth's surface provides insight into its evolution. Landlab is an open-source Python package that contains modularized elements of numerical models for Earth's surface, thus reducing time required for researchers to create new or reimplement existing models. Landlab contains a gridding engine which represents the model domain as a dual graph of structured quadrilaterals (e.g., raster) or irregular Voronoi polygon–Delaunay triangle mesh (e.g., regular hexagons, radially symmetric meshes, and fully irregular meshes). Landlab also contains components – modular implementations of single physical processes – and a suite of utilities that support numerical methods, input/output, and visualization. This contribution describes package development since version 1.0 and backward-compatibility-breaking changes that necessitate the new major release, version 2.0. Substantial changes include refactoring the grid, improving the component standard interface, dropping Python 2 support, and creating 31 new components – for a total of 58 components in the Landlab package. We describe reasons why many changes were made in order to provide insight for designers of future packages. We conclude by discussing lessons about the dynamics of scientific software development gained from the experience of using, developing, maintaining, and teaching with Landlab.

Scaling Limits of Discrete Optimal Transport

We consider dynamical transport metrics for probability measures on discretizations of a bounded convex domain in $\mathbb R^d$. These metrics are natural discrete counterparts to the Kantorovich metric $\mathbb{W}_2$, defined using a Benamou--Brenier-type formula. Under mild assumptions we prove an asymptotic upper bound for the discrete transport metric $\mathcal{W}_\mathcal{T}$ in terms of $\mathbb{W}_2$, as the size of the mesh $\mathcal{T}$ tends to 0. However, we show that the corresponding lower bound may fail in general, even on certain one-dimensional and symmetric two-dimensional meshes. In addition, we show that the asymptotic lower bound holds under an isotropy assumption on the mesh, which turns out to be essentially necessary. This assumption is satisfied, e.g., for tilings by convex regular polygons, and it implies Gromov--Hausdorff convergence of the transport metric.

Parameterized synthesis of self-stabilizing protocols in symmetric networks

Self-stabilization in distributed systems is a technique to guarantee convergence to a set of legitimate states without external intervention when a transient fault or bad initialization occurs. Recently, there has been a surge of efforts in designing techniques for automated synthesis of self-stabilizing algorithms that are correct by construction. Most of these techniques, however, are not parameterized, meaning that they can only synthesize a solution for a fixed and predetermined number of processes. In this paper, we report a breakthrough in parameterized synthesis of self-stabilizing algorithms in symmetric networks, including ring, line, mesh, and torus. First, we develop cutoffs that guarantee (1) closure in legitimate states, and (2) deadlock-freedom outside the legitimate states. We also develop a sufficient condition for convergence in self-stabilizing systems. Since some of our cutoffs grow with the size of the local state space of processes, scalability of the synthesis procedure is still a problem. We address this problem by introducing a novel SMT-based technique for counterexample-guided synthesis of self-stabilizing algorithms in symmetric networks. We have fully implemented our technique and successfully synthesized solutions to maximal matching, three coloring, and maximal independent set problems for ring and line topologies.

Analysis of Numerical Errors of the Hess Smith Panel Method With Asymmetric Meshes

AbstractAlthough less accurate in comparison with most of the existing high-order panel methods, the Class Hess Smith panel method is still popular for potential flow calculations due to its simplicity, robustness, and especially efficiency. In the application where the method is adopted for the problems of concern, the method is usually first validated and analyzed for numerical accuracy prior to the application. This is a common practice as also seen in almost all the literature. However, the objects chosen for validation and accuracy analysis are often geometrically symmetric (most typically, spheres) and discretized into symmetric meshes. However, such analysis with symmetric meshes does not actually reveal the numerical errors as the body involved in the problem in concern or the flow around the body may not be symmetrical. In this paper, the numerical errors of the Hess Smith method are systematically analyzed with asymmetrical meshes. The values of various quantities associated with isolated panels and also the entire body are obtained and compared with the Hess Smith panel method does exhibit a very different error pattern.

The lumped mass FEM for a time-fractional cable equation

We consider the numerical approximation of a time-fractional cable equation involving two Riemann–Liouville fractional derivatives. We investigate a semidiscrete scheme based on the lumped mass Galerkin finite element method (FEM), using piecewise linear functions. We establish optimal error estimates for smooth and middly smooth initial data, i.e., v∈Hq(Ω)∩H01(Ω), q=1,2. For nonsmooth initial data, i.e., v∈L2(Ω), the optimal L2(Ω)-norm error estimate requires an additional assumption on mesh, which is known to be satisfied for symmetric meshes. A quasi-optimal L∞(Ω)-norm error estimate is also obtained. Further, we analyze two fully discrete schemes using convolution quadrature in time based on the backward Euler and the second-order backward difference methods, and derive error estimates for smooth and nonsmooth data. Finally, we present several numerical examples to confirm our theoretical results.

Computational Design of Acoustic Materials Using an Adaptive Optimization Algorithm

We consider the problem of design of the acoustic structure of arbitrary geometry with prescribed desired properties. We use optimization approach for the solution of this problem and minimize the Tikhonov functional on adaptively refined meshes. These meshes are refined locally only in places where the acoustic structure should be designed. Our special symmetric mesh refinement strategy together with interpolation procedure allows the construction of the symmetric acoustic material with prescribed properties. Efficiency of the presented adaptive optimization algorithm is illustrated on the construction of the symmetric acoustic material in two dimensions.

Weight function determinations for shear cracks in reinforced concrete beams based on finite element method

An efficient approach of evaluating the weight functions for the generally load applied faces of shear cracked RC beams with varying shear span/beam depth ratios is presented using the corresponding weight functions for the RC beams with two shear span/beam depth ratios. The shear cracked RC beam under applied load can be considered as an oblique edge crack elastic geometry under mixed load conditions if the nonlinear bridging force is evaluated according to a crack bridging model separately. With pure bend load conditions, both Mode I and II weight functions for the oblique edge crack geometries with three shear span/beam depth ratios for different crack length/beam depth ratios are evaluated through applying the Virtual Crack Extension technique with symmetric mesh around the crack-tip into finite element analysis, where all weigh function components exhibit orderly trends with respect to the changing shear span/beam depth. As a result, the weight functions for the shear crack RC beams with the other shear span/beam depth ratios can be calculated following the orderly trends. Since the crack-tip singular behaviors of fracture problems is observed only in the primary crack-face nodal weight functions, to facilitate application, the primary crack-face weight functions are formulated accurately as a function of crack length and distance away from the crack-tip employing the fitting and interpolation methods.

A decomposition technique of generalized degrees of freedom for mixedmode crack problems

SummaryThe numerical manifold method (NMM) builds up a unified framework that is used to describe continuous and discontinuous problems; it is an attractive method for simulating a cracking phenomenon. Taking into account the differences between the generalized degrees of freedom of the physical patch and nodal displacement of the element in the NMM, a decomposition technique of generalized degrees of freedom is deduced for mixed mode crack problems. An analytic expression of the energy release rate, which is caused by a virtual crack extension technique, is proposed. The necessity of using a symmetric mesh is demonstrated in detail by analysing an additional error that had previously been overlooked. Because of this necessity, the local mathematical cover refinement is further applied. Finally, four comparison tests are given to illustrate the validity and practicality of the proposed method. The aforementioned aspects are all implemented in the high‐order NMM, so this study can be regarded as the development of the virtual crack extension technique and can also be seen as a prelude to an h‐version high‐order NMM. Copyright © 2017 John Wiley &amp; Sons, Ltd.

Computational Fluid Dynamics (CFD) simulating heated air from wood burning inside a poultries barn

Animal welfare is essential for livestock yield gains, which has been ethically justifiable and socially acceptable. Thus, our research aimed to test computational fluid dynamics model (CFD) to simulate heated air within a poultry barn. Data were simulated in CFD software based on Navier-Stokes equations (geometry of 3 m x 6 m, considering a symmetric mesh). As boundary conditions, a temperature of 38oC was considered for walls where heating pipe outlets are, to the west side, besides a heat flow of zero, in symmetry to the same side. As for the sides east, south and north, walls were taken as isolated, i.e. heat flow equals to zero. The aviary heating system did not achieve a homogeneous temperature distribution, heat flow, heated air pressure and speed. Heated-air convection cells were spotted in the upper part of the building, being little used for thermal comfort by the birds.