Orthogonal Mesh Research Articles

Multi-objective optimization and analysis for laser beam cutting of stainless steel (SS304) using hybrid statistical tools GA-RSM

A laser beam machine is a non-traditional manufacturing technique that uses thermal energy to cut nearly all types of materials. The quality of laser cutting is significantly affected by process parameters. The purpose of this study is to use a genetic algorithm (GA) in conjunction with response surface approaches to improve surface roughness in laser beam cutting CO2 with a continuous wave of SS 304 stainless steel. The effects of the machining parameters, such as cutting speed, nitrogen gas pressure, and focal point location, were investigated quantitatively and optimized. The tests were carried out using the Taguchi L9 orthogonal mesh approach. Analysis of variance, main effect plots, and 3D surface plots were used to evaluate the impact of cutting settings on surface roughness. A multi-objective genetic algorithm in MATLAB was used to achieve a minimum surface roughness of 0.93746 μm, with the input parameters being 2028.712 mm/m cutting speed, 11.389 bar nitrogen pressure, and a focal point position of - 2.499 mm. The optimum results of each method were compared, as the results the response surface approach is less promising than the genetic algorithm method.

Metallic Nanomeshes Fabricated by Multimechanism Directed Self-Assembly.

The directed self-assembly of block copolymers (BCPs) is a powerful motif for the continued scaling of feature sizes for nanoscale devices. A multimechanism directed self-assembly (MMDSA) method is described that generates orthogonal meshes from a polystyrene-b-poly-2-vinylpyridine BCP that is subsequently metallized with Pt. The MMDSA process takes advantage of three different mechanisms, trench wall guidance, edge nucleation, and underlayer guidance, to align the mesh with respect to substrate features. The mechanisms and their interactions are investigated via both experiments and dissipative particle dynamics simulations. MMDSA is applied to produce well-aligned conductive nanomeshes and then is extended to fabricate multicomponent metallic structures with 2D/3D hybrid morphologies.

An interpolation-based lattice Boltzmann method for non-conforming orthogonal meshes

We propose an interpolation-based lattice Boltzmann formulation that is applicable to non-conforming orthogonal meshes. This interpolation configuration allows the use of localized and directional refinement that reduces mesh sizes and makes them well adapted for solving flows in various configurations, including aerodynamic shapes in free streams. The novel aspect of the proposed method lies in the interpolation scheme selection algorithm, which uses local mesh topology for choosing a suitable combination of nodes and polynomial terms. The objective of this study consists of validating the scheme accuracy with well documented laminar incompressible flow test cases. The results demonstrate the capability of the proposed interpolation method to successfully solve closed and open flows with either straight or curved walls. Simulations for the flow over a NACA 0012 airfoil show that the aerodynamic coefficients are captured correctly using a mesh with large aspect ratio cells near the wall and in the airfoil wake.

Convergence of a finite volume scheme for immiscible compressible two-phase flow in porous media by the concept of the global pressure

This paper deals with development and analysis of a finite volume (FV) method for the coupled system describing immiscible compressible two-phase flow, such as water-gas, in porous media, capillary and gravity effects being taken into account. We investigate a fully coupled fully implicit cell-centered “phase-by-phase” FV scheme for the discretization of such system. The main goal is to incorporate some of the most recent improvements in the scheme and the convergence of the numerical approximation to the weak solution of such models. The spatial discretization uses a TPFA scheme and a new strategy for handling the upwinding. Based on a priori estimates and compactness arguments, we prove the convergence of the numerical approximation to the weak solution. The particular feature in this convergence analysis of the classical engineering scheme based on the “phase-by-phase” upwinding on an orthogonal mesh relies on the global pressure–saturation fractional flow formulation as was defined relatively recently for immiscible compressible flow in porous media. We have developed and implemented this scheme in a new module in the context of the open source platform DuMuX. Two numerical experiments are presented to demonstrate the efficiency of this scheme. The first test addresses the evolution in 2D of gas migration through engineered and geological barriers for a deep repository for radioactive waste. The second test case is chosen to test the ability of the method to approximate solutions for 3D problems modeling scenarios of CO2 injection in a fully water-saturated domain.

Simulation of processes in machines using orthogonal meshes

The paper presents an experience of using orthogonal hexahedral meshes for simulation of different processes in machine-building products. Stress state of the details of suspension of a cargo vehicle during the movement on random road profile was simulated. Orthogonal hexahedral meshes for lower suspension levers and housing and piston rod of shock adsorber were used. Thermal state of the elements of drive axle and its housing during the movement of a cargo vehicle with constant speed was calculated using hexahedral orthogonal meshes for details and inner volume of housing of the drive axle. Lubrication fluid circulation in the reducer of a cargo vehicle was modelled. Orthogonal mesh for inner volume of the reducer was used to calculate the parameters of lubrication fluid circulation.

Cable-stayed coverings for large-span public buildings

Structural schemes of coverings using lightweight roof structure from modern composite materials are investigated in this paper. A comparison of two options for cable-stayed covering of a circus auditorium, one with an orthogonal mesh of cables (according to a standard design) and another one, with a mesh of cables formed by two groups of stabilizing cables are considered. The possibilities of reinforcement with composite material are analyzed. Calculation and preliminary selection of cable-stayed sections are also carried out and presented. The variations of the conditional volume with the relative sag arrows, for varying values of the coefficient qk have been investigated and shown on graphs. An optimization problem was solved and the optimal result with detailed calculations are proposed and presented in this research which enable the design of the cable-stayed coating with determination of its design characteristics. As a result, the optimum and most economical sections of the cable-stayed covering were obtained. The calculations also show the inexpediency of using a heavy roof structure for saddle cable-stayed coverings due to the exclusion of stabilizing cables from the operation with a large roof weight.

A finite volume formulation for magnetostatics of discontinuous media within a multi-region OpenFOAM framework

In this article we present a formulation for the numerical computation of static magnetic fields in discontinuous media. Focusing on solving magneto-fluid-structure interaction problems, we developed a finite volume multi-region framework capable of treating an arbitrary number of magnetized, permeable and current carrying bodies. The balance equations are written in terms of the magnetic vector potential leading to a formulation with discontinuous boundary conditions at interfaces between regions. We derived a consistent formulation of the discrete interface boundary conditions which guarantee accurate results for the magnetic field. The computational framework is based on a multi-region implementation that relies on non-conformal field mapping between and an implicit-explicit iterative solution procedure. Several numerical experiments show that the formulation gives good results for orthogonal meshes in terms of magnitude and direction of the magnetic field. We executed a numerical uncertainty analysis for a test case where a volumetric current, a permanent magnet and a ferromagnetic material interact between each other; we report apparent orders of accuracy, grid convergence indexes and grid convergence curves for several points of the computational domain.

A High-Order Conservative Semi-Lagrangian Solver for 3D Free Surface Flows with Sediment Transport on Voronoi Meshes

In this paper, we present a conservative semi-Lagrangian scheme designed for the numerical solution of 3D hydrostatic free surface flows involving sediment transport on unstructured Voronoi meshes. A high-order reconstruction procedure is employed for obtaining a piecewise polynomial representation of the velocity field and sediment concentration within each control volume. This is subsequently exploited for the numerical integration of the Lagrangian trajectories needed for the discretization of the nonlinear convective and viscous terms. The presented method is fully conservative by construction, since the transported quantity or the vector field is integrated for each cell over the deformed volume obtained at the foot of the characteristics that arises from all the vertexes defining the computational element. The semi-Lagrangian approach allows the numerical scheme to be unconditionally stable for what concerns the advection part of the governing equations. Furthermore, a semi-implicit discretization permits to relax the time step restriction due to the acoustic impedance, hence yielding a stability condition which depends only on the explicit discretization of the viscous terms. A decoupled approach is then employed for the hydrostatic fluid solver and the transport of suspended sediment, which is assumed to be passive. The accuracy and the robustness of the resulting conservative semi-Lagrangian scheme are assessed through a suite of test cases and compared against the analytical solution whenever is known. The new numerical scheme can reach up to fourth order of accuracy on general orthogonal meshes composed by Voronoi polygons.

Conforming weighted delaunay triangulations

Given a set of points together with a set of simplices we show how to compute weights associated with the points such that the weighted Delaunay triangulation of the point set contains the simplices, if possible. For a given triangulated surface, this process provides a tetrahedral mesh conforming to the triangulation, i.e. solves the problem of meshing the triangulated surface without inserting additional vertices. The restriction to weighted Delaunay triangulations ensures that the orthogonal dual mesh is embedded, facilitating common geometry processing tasks. We show that the existence of a single simplex in a weighted Delaunay triangulation for given vertices amounts to a set of linear inequalities, one for each vertex. This means that the number of inequalities for a given triangle mesh is quadratic in the number of mesh elements, making the naive approach impractical. We devise an algorithm that incrementally selects a small subset of inequalities, repeatedly updating the weights, until the weighted Delaunay triangulation contains all constrained simplices or the problem becomes infeasible. Applying this algorithm to a range of triangle meshes commonly used graphics demonstrates that many of them admit a conforming weighted Delaunay triangulation, in contrast to conforming or constrained Delaunay that require additional vertices to split the input primitives.

Table Organization Optimization in Schools for Preserving the Social Distance during the COVID-19 Pandemic

The COVID-19 pandemic has supposed a challenge for education. The school closures during the initial coronavirus outbreak for reducing the infections have promoted negative effects on children, such as the interruption of their normal social relationships or their necessary physical activity. Thus, most of the countries worldwide have considered as a priority the reopening of schools but imposing some rules for keeping safe places for the school lessons such as social distancing, wearing facemasks, hydroalcoholic gels or reducing the capacity in the indoor rooms. In Spain, the government has fixed a minimum distance of 1.5 m among the students’ desks for preserving the social distancing and schools have followed orthogonal and triangular mesh patterns for achieving valid table dispositions that meet the requirements. However, these patterns may not attain the best results for maximizing the distances among the tables. Therefore, in this paper, we introduce for the first time in the authors’ best knowledge a Genetic Algorithm (GA) for optimizing the disposition of the tables at schools during the coronavirus pandemic. We apply this GA in two real-application scenarios in which we find table dispositions that increase the distances among the tables by 19.33% and 10%, respectively, with regards to regular government patterns in these classrooms, thus fulfilling the main objectives of the paper.

Modelling of 3D viscous fingering: Influence of the mesh on coreflood experiments

The modeling of Viscous Fingering (VF) is of interest to the study of multiphase flow in porous media, especially in the O&G industry, where this phenomenon is one of the root causes of high-water production and low sweep efficiency. In this study, 3D coreflood experiments found in the literature were simulated with CFD, and nine different structured and unstructured meshes were compared to determine which one gives the best description of this phenomenon. The meshes included the structured orthogonal mesh, commonly used in reservoir simulation, up to unstructured hexahedral, polyhedral, and tetrahedral meshes, among others. On the other hand, the CFD model, which was based on the Volume of Fluid (VOF) model and the porous media model, was validated against experimental data for the oil recovery profile with an absolute error below 7% for the best cases. It was found that VF is severely affected by the type of mesh and the Courant–Friedrichs–Lewy (CFL) number used. It was concluded that the unstructured polyhedral and structured hexahedral meshes gave the most reliable description of the finger growth dynamics and oil recovery compared to experimental data. Finally, a mesh independence test using the Grid Convergence Index (GCI) methodology, along with a CFL number analysis, allowed to determine that the modeling of this phenomenon should be made using an average CFL number of 0.25.

The symmetry-preserving difference schemes and exact solutions of some high-dimensional differential equations

A direct and effective method is employed to construct the discrete models of high-dimensional generalized Zakharov–Kuznetsov and diffusion–convection equations. Through the compatibility condition, we construct their potential systems, whose local symmetries can be projected into the local and nonlocal symmetries of the original equations. Furthermore, based on the resulting Lie symmetries, the invariant difference models and symmetry-preserving difference models of the original two equations can be derived by using the orthogonal meshes which are uniform in space. Finally, some exact solutions of these two equations are also obtained with their graphic analysis.

A modification of the hexahedral mesh generator based on voxel geometry representation

We consider a modification of the previously developed voxel-based mesh algorithm to generate models given in STL-geometry format. Proposed hexahedral mesh generator belongs to the family of grid methods, and is general-purpose in terms of a capability to use as source data both volume (voxel) and STL-surface representation of model geometry. For now, the algorithm works with CAD models described in the well-known STL format. However, it also allows to handle higher-order surface patches defined in an arbitrary format if appropriate procedures for projection and intersection operations will be specified. To define the initial position of mesh nodes, a “signed distance field” volume data file, obtained from the STL-geometry, is used. A special projection technique was developed to adapt constructed orthogonal mesh on the model's boundary. It provides an approximation of sharp edges and corners and is performed before running any other operations with the mesh. Finally, to improve the quality of the mesh, additional procedures were implemented, including boundary layers insertion, bad quality cells splitting, and optimization-based smoothing technique. The algorithm has been tested on a sufficient number of models, some of which are given as examples.

Numerical analysis of the indoor climate in a Zenith-type greenhouse in the Valle de Santiago region, Guanajuato

The objective of this project was to carry out an analysis of the indoor climate of the greenhouse located in the Valle de Santiago region, Guanajuato. It is a zenith-type greenhouse with two wings with a symmetrical face. An orthogonal mesh was made of 50 nodes (25 nodes for each height) Total Den located in the cultivable area and taking the value at the midpoint of each rectangle considering two heights 0.25m and 1.30m with respect to the ground. Humidity and temperature readings were taken in each of the nodes for three weeks and subsequently a data analysis was made and a comparison with the data collected in the different situations; also the temperature was analyzed with the double sum of Riemann and the rule of the middle point. In conclusion, it was determined that the greenhouse yields heat on warm days, while it receives heat on cold days. This behavior coincides with previous studies; however, it occurs that with the hydroponic method there is a greater growth of the crop.

New approach to the generation of orthogonal body-fitted meshes and its application to non-steady plane-strain ideal plastic flow

For rigid-perfect plastic solids, ideal plastic flow happens when all material elements follow minimum work paths and the two principal stretch lines are materially embedded. To describe its kinematics effectively, Lagrange convective orthogonal curvilinear system has to be constructed. Therefore, a numerical procedure for generating orthogonal body-fitted meshes has been developed and applied to non-steady plane-strain ideal plastic flow. The mapping between the Cartesian system and the orthogonal system is based on Laplace equations and Cauchy–Riemann condition. The numerical procedure takes into consideration of all the prescribed boundary curves instead of one as the boundary conditions, which has been adopted in earlier studies when using conventional characteristic line method. After each iteration step, the obtained boundary points are adjusted towards the orthogonality conditions until error tolerance requirement is met. For demonstration purpose, the numerical procedure is applied to the optimization of a bulk part under forging. The obtained orthogonal body-fitted meshes are in good agreement with previous methods. Then the optimal initial scale factor is also calculated based on this method and compared with earlier studies. Finally, the evolution of the intermediate shapes and frictional external boundary conditions is also calculated when there are no elastic dead zones.

Accurate Calculation of Partial Inductances for the Orthogonal PEEC Formulation

The Partial Element Equivalent Circuit (PEEC) method is promising numerical technique for three-dimension electromagnetic modeling across various application fields. In the framework of the PEEC method, the partial elements modeling the magnetic and electric field coupling between elementary volumes and surfaces are computed by double-folded volume and surface integrals. Assuming the quasi-static hypothesis and an orthogonal mesh, the integrals have been computed by the analytical formulas derived in literature, which significantly reduces the computational time in comparison to the numerical integration. However, the existing analytical formulas are affected by significant numerical errors for certain PEEC structural mesh necessary to model the skin and proximity effects with a higher accuracy. To utilize the full potential of the PEEC method, the calculation of partial elements has to be carefully addressed, which has not been investigated in a comprehensive way so far. Accordingly, this paper presents a systematic accuracy analysis of the existing closed-form analytical formulas and methods for calculating the self and mutual inductances between two rectangular conductors. Additionally, a new strategy to select a proper analytical formula depending on the dimensions and positions of two conductors is proposed, which allows the mutual inductance extraction with a relative error of less than 0.1%. The new method is systematically validated on examples of 3-D dense PEEC systems using the quadruple precision arithmetic as reference.

Efficient Numerical Computation of Full-Wave Partial Elements Modeling Magnetic Materials in the PEEC Method

This article presents the analytical computation of full-wave partial elements modeling magnetic materials in the framework of the partial element equivalent circuit method. Assuming an orthogonal mesh, the free-space Green’s function is expanded in Taylor’s series. The resulting series expansion coefficients occurring in magnetic material modeling are computed analytically up to the second order. Extensive tests are performed confirming the accuracy of the proposed analytical solutions. As expected, significant speedups are observed with respect to the standard Gaussian integration.

A fast Fourier transform based direct solver for the Helmholtz problem

SummaryThis article is devoted to the efficient numerical solution of the Helmholtz equation in a two‐ or three‐dimensional (2D or 3D) rectangular domain with an absorbing boundary condition (ABC). The Helmholtz problem is discretized by standard bilinear and trilinear finite elements on an orthogonal mesh yielding a separable system of linear equations. The main key to high performance is to employ the fast Fourier transform (FFT) within a fast direct solver to solve the large separable systems. The computational complexity of the proposed FFT‐based direct solver is operations. Numerical results for both 2D and 3D problems are presented confirming the efficiency of the method discussed.

A 3D Unstructured Mesh FDTD Scheme for EM Modelling

The Yee finite difference time domain (FDTD) algorithm is widely used in computational electromagnetics because of its simplicity, low computational costs and divergence free nature. The standard method uses a pair of staggered orthogonal cartesian meshes. However, accuracy losses result when it is used for modelling electromagnetic interactions with objects of arbitrary shape, because of the staircased representation of curved interfaces. For the solution of such problems, we generalise the approach and adopt an unstructured mesh FDTD method. This co-volume method is based upon the use of a Delaunay primal mesh and its high quality Voronoi dual. Computational efficiency is improved by employing a hybrid primal mesh, consisting of tetrahedral elements in the vicinity of curved interfaces and hexahedral elements elsewhere. Difficulties associated with ensuring the necessary quality of the generated meshes will be discussed. The power of the proposed solution approach is demonstrated by considering a range of scattering and/or transmission problems involving perfect electric conductors and isotropic lossy, anisotropic lossy and isotropic frequency dependent chiral materials.

3D PIC-MCC simulation of corona discharge in needle-plate electrode with external circuit

This paper investigates the external circuit effect in corona discharge by means of numerical simulation, while the effect is usually disregarded in previous plasma simulations. A general numerical solution method of serial external circuit, which can compute surface charges on any shape of electrodes adaptively and solve electrode potential and circuit current simultaneously, is developed to support the investigation, due to lack of theoretical solution in irregular electrode systems. Brief description and validation of the scheme are presented in section . In general, the accuracy of numerical solution improves with mesh refinement in orthogonal mesh, due to stair step approximation. Simulations are performed in our self-developed 3D PIC-MCC (particle-in-cell, Monte Carlo collision) code with photoionization and external circuit. The simulation model including recently developed photoionization model is described briefly in section . Then the simulation results in a needle-plate electrode in air, such as spatiotemporal evolution of corona discharge and waveforms of electrode voltage and circuit current, are presented. Physical mechanisms of formation of branched corona discharge and self-consistent adjustment of electrode voltage and circuit current are discussed. The waveforms of voltage and current in this simulation are generally consistent with experimental result. The external circuit has negative feedback effect on the discharge, and can adjusts electrode voltage self-consistently. The application of external circuit improves the applicability of particle simulation method to the simulation of discharge phenomenon in actual devices.