Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Geometry meshing”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 487 records · Page 27
All sourcesNASA NTRSDOE OSTI
NASA NTRS2020

A Concept for the Deployment of a Large Lunar Crater Radio Telescope Using Teams of Tethered Robots

Kilometer-scale craters on the far side of the Moon have unique potential as future locations for large radio telescopes, which can observe the universe at wavelengths and frequencies (> 10 m, < 30 MHz) not possible with conventional Earth or orbital-based approaches. Distinct advantages of building a Lunar Crater Radio Telescope (LCRT) on the far side include i) isolation from radio noise due to the Earth’s ionosphere, orbiting satellites, and the Sun, ii) days of uninterrupted dark/cold sky viewing during lunar night, and iii) terrain geometry naturally suited for constructing the largest mesh antenna structure in the Solar System. A key challenge to constructing LCRT on the Moon is related to the complexity of deploying a 1-km diameter antenna and hanging receiver within a lunar crater whose diameter, depth, and slope are 3-5 km, 1 km, and ~30 degrees respectively. In this paper, we first evaluate the trade space for deploying a large, complex structure within a crater, and then provide a more detailed concept evaluation of our favored approach, which employs coordinated teams of tethered rovers to extract and suspend a folded antenna from a lander at the base of a crater. NASA’s Jet Propulsion Laboratory in collaboration with California Institute of Technology (Caltech) have developed a novel robotic system for accessing extremely steep terrains; the Axel rover is a two-wheeled rugged terrain vehicle that is supported by an electro-mechanical tether that provides power, data, and tensile support from a top-side anchor location. Recently, a pair of Axel robots have been used in a DuAxel configuration that allows for four-wheel driving and repeated passive anchoring at different locations. The DuAxel system has unique advantages for deploying an LCRT antenna, including the ability to deploy from a lander near a crater, drive a distance to the crater rim to deploy an Axel, and later, retract the deployed Axel in order to sequentially lift up sections of the antenna. Our proposed concept involves delivering a packaged antenna and receiver to the bottom-center of a crater floor on a lander, then later sending a team of multiple DuAxel rovers to retrieve guide wires from the lander, which are pulled to the top of the crater. We explore this concept in detail and provide some initial quantitative analysis to demonstrate the feasibility of our system with respect to the spatial and mass properties of the antenna as juxtaposed to DuAxel capabilities. Finally, we outline next steps towards validating our concept on the way to a future lunar deployment opportunity.

Hallinan, Greg↗
DOE OSTI2014

Aleph

Aleph is a PIC/DSMC (particle-in-cell/direct simulation Monte Carlo) software package for the simulation of low temperature plasmas. Aleph is intended to be a production quality code capable of handling complex 3D geometries. Unstructured triangular (2D) or tetrahedral (3D) meshes are used to represent simulation domains. Aleph runs either in serial on a single processor or desktop machine, or can be run in parallel using the MPI message-passing library, to enable faster performance on large problems. Aleph employs dynamic load balancing to efficiently harness thousands of processors on high performance computers and enable simulations of up to millions of mesh elements and billions of particles. Multiple decompositions of the same mesh are used: one for the particles, and one for the field solves. Electrostatic fields are computed using an unstructured mesh finite element method (FEM), and the resulting linear systems are solved using solvers from the Trilinos package (http://trilinos.org/).

Source record↗
DOE OSTI2016

Turbulence Model Study for Pressure and Velocity Distributions for Molten Salts in a Crossflow Tube Bundle

Empirical pressure drop correlations for various compositions of molten salts in crossflow across tube bundles/banks are limited due to the scarcity of experimental data for these fluids. This lack of experimental data presents a limitation when designing heat exchangers for Molten Salt Reactors (MSRs) that utilize cross-flow configurations with turbulent molten salts flowing in the shell side. In an effort to provide a starting point for future design and experimental work, a computational fluid dynamics (CFD) campaign is being conducted to estimate pressure drops over a range of Reynolds numbers, pitch to diameter ratios, and for both staggered and in-line tube banks. In an effort to continue this work, the study of various turbulence models were conducted to determine the applicability for steady RANS simulations. An in-line tube bank with FLiBe (LiF-BeF{sub 2}), a frequently selected molten salt in MSRs, was selected for these simulations. The estimated pressure drops and pressure and velocity distributions for each turbulence model was compared to determine the applicability. The k-ε turbulence model family was found to be the most appropriate model for steady Reynolds averaged Navier Stokes within this type of domain. Difference turbulence models were investigated for simulation of molten salt experiencing crossflow within an in-line tube bundle. The flow was simulated using a steady RANS formulation with different families of turbulence models. The different turbulence models provide a view of this specific flow behavior potential requirements and applicable for steady RANS. Of particular note, the k-ε models were able to converge easily within this domain. The predicted behavior for both models were observed to be mostly similar for both velocity and pressure distributions. Whereas, k-ω SST and SA models did not converge well and had observable differences for the predictions of pressure and velocity distributions. The global behaviors were generally similar, but the variations occurring suggest k-ω SST and SA turbulence models are not useful in the steady RANS formulation of this domain. When comparing to the empirical pressure drop, each turbulence model resulted in overpredictions ranging from 8.7% and 19.75% difference. The standard low-Re k-ε model having the lowest difference is likely the better model for simplified predictions for pressure drops. Further investigations will be conducted to determine the convergence issues for the k-ω SST and SA turbulence models. These might be due to the flow geometry or potentially a revision of the meshing is needed. The flow behavior may not be resolved using these models in a steady formulation due to inherent unsteadiness of the flow. (authors)

42 ENGINEERING↗
DOE OSTI2017

Verification of the Coupled Neutronics/Thermal-Hydraulics Code TANSY against the MSFR Benchmark

The MSRs is a liquid-fuel reactor in which uranium, fission products and actinides are dissolved in the molten salt. The molten salt reactors are featured by circulating fuel, low pressure and high temperature. The MSRs meets many of the future goals of nuclear technology, in particular for what concerns as improved sustainability and unique characteristics in terms of actinide burning. The fluid nature of fuel in MSRs gives extra flexibility in reactor design, fuel fabrication and recycling. But because of the application of fluid fuel, MSRs have several special characteristics compared with the traditional solid-fuel reactors. Firstly, the transport of the delayed neutron precursors by the fluid flow reduces the contribution of delayed neutrons to the chain reaction.. Secondly, the fact that the fuel is dissolved in the coolant rather than separated from the coolant by the claddings results in a much stronger coupling phenomenon between the neutronics and thermal-hydraulics. Thirdly, the shape of the fuel is determined by the container because of the characteristics of fluid, the unstructured meshes are required to model the complex geometry accurately. However most of the conventional reactor codes were developed for solid-fuel reactor and therefore not capable of taking liquid fuel flow into account. To address all these particular modeling issues of MSRs, a multi-physics code named TANSY has been developed in our laboratory. In this paper, the benchmark problem based on the MSFR design was calculated to verify the code.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
NASA NTRS2016

Tangle-Free Mesh Motion for Ablation Simulations

Problems involving mesh motion-which should not be mistakenly associated with moving mesh methods, a class of adaptive mesh redistribution techniques-are of critical importance in numerical simulations of the thermal response of melting and ablative materials. Ablation is the process by which material vaporizes or otherwise erodes due to strong heating. Accurate modeling of such materials is of the utmost importance in design of passive thermal protection systems ("heatshields") for spacecraft, the layer of the vehicle that ensures survival of crew and craft during re-entry. In an explicit mesh motion approach, a complete thermal solve is first performed. Afterwards, the thermal response is used to determine surface recession rates. These values are then used to generate boundary conditions for an a posteriori correction designed to update the location of the mesh nodes. Most often, linear elastic or biharmonic equations are used to model this material response, traditionally in a finite element framework so that complex geometries can be simulated. A simple scheme for moving the boundary nodes involves receding along the surface normals. However, for all but the simplest problem geometries, evolution in time following such a scheme will eventually bring the mesh to intersect and "tangle" with itself, inducing failure. This presentation demonstrates a comprehensive and sophisticated scheme that analyzes the local geometry of each node with help from user-provided clues to eliminate the tangle and enable simulations on a wide-class of difficult problem geometries. The method developed is demonstrated for linear elastic equations but is general enough that it may be adapted to other modeling equations. The presentation will explicate the inner workings of the tangle-free mesh motion algorithm for both two and three-dimensional meshes. It will show abstract examples of the method's success, including a verification problem that demonstrates its accuracy and correctness. The focus of the presentation will be on the algorithm; specifics on how the techniques may be used in spacecraft design will be not discussed.

Droba, Justin↗
NASA NTRS2014

Using SpaceClaimTD Direct for Modeling Components with Complex Geometries for the Thermal Desktop-Based Advanced Stirling Radioisotope Generator Model

SpaceClaim/TD Direct has been used extensively in the development of the Advanced Stirling Radioisotope Generator (ASRG) thermal model. This paper outlines the workflow for that aspect of the task and includes proposed best practices and lessons learned. The ASRG thermal model was developed to predict component temperatures and power output and to provide insight into the prime contractor's thermal modeling efforts. The insulation blocks, heat collectors, and cold side adapter flanges (CSAFs) were modeled with this approach. The model was constructed using mostly TD finite difference (FD) surfaces/solids. However, some complex geometry could not be reproduced with TD primitives while maintaining the desired degree of geometric fidelity. Using SpaceClaim permitted the import of original CAD files and enabled the defeaturing/repair of those geometries. TD Direct (a SpaceClaim add-on from CRTech) adds features that allowed the "mark-up" of that geometry. These so-called "mark-ups" control how finite element (FE) meshes are to be generated through the "tagging" of features (e.g. edges, solids, surfaces). These tags represent parameters that include: submodels, material properties, material orienters, optical properties, and radiation analysis groups. TD aliases were used for most tags to allow analysis to be performed with a variety of parameter values. "Domain-tags" were also attached to individual and groups of surfaces and solids to allow them to be used later within TD to populate objects like, for example, heaters and contactors. These tools allow the user to make changes to the geometry in SpaceClaim and then easily synchronize the mesh in TD without having to redefine the objects each time as one would if using TDMesher. The use of SpaceClaim/TD Direct helps simplify the process for importing existing geometries and in the creation of high fidelity FE meshes to represent complex parts. It also saves time and effort in the subsequent analysis.

Power Systems↗
DOE OSTI2015

Gyrokinetic particle simulation of microturbulence for general magnetic geometry and experimental profiles

Developments in gyrokinetic particle simulation enable the gyrokinetic toroidal code (GTC) to simulate turbulent transport in tokamaks with realistic equilibrium profiles and plasma geometry, which is an essential step in the code–experiment validation process. These novel developments include numerical equilibrium representation using B-splines, a new Poisson solver based on finite difference using field-aligned mesh and magnetic flux coordinates, a new zonal flow solver for general geometry, and improvements on the conventional four-point gyroaverage with nonuniform background marker loading. The gyrokinetic Poisson equation is solved in the perpendicular plane instead of the poloidal plane. Exploiting these new features, GTC is able to simulate a typical DIII-D discharge with experimental magnetic geometry and profiles. The simulated turbulent heat diffusivity and its radial profile reflects good agreement with other gyrokinetic codes. The newly developed nonuniform loading method provides a modified radial transport profile to that of the conventional uniform loading method.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS2014

Using SpaceClaim/TD Direct for Modeling Components with Complex Geometries for the Thermal Desktop-Based Advanced Stirling Radioisotope Generator Model

SpaceClaim/TD Direct has been used extensively in the development of the Advanced Stirling Radioisotope Generator (ASRG) thermal model. This paper outlines the workflow for that aspect of the task and includes proposed best practices and lessons learned. The ASRG thermal model was developed to predict component temperatures and power output and to provide insight into the prime contractors thermal modeling efforts. The insulation blocks, heat collectors, and cold side adapter flanges (CSAFs) were modeled with this approach. The model was constructed using mostly TD finite difference (FD) surfaces solids. However, some complex geometry could not be reproduced with TD primitives while maintaining the desired degree of geometric fidelity. Using SpaceClaim permitted the import of original CAD files and enabled the defeaturing repair of those geometries. TD Direct (a SpaceClaim add-on from CRTech) adds features that allowed the mark-up of that geometry. These so-called mark-ups control how finite element (FE) meshes were generated and allowed the tagging of features (e.g. edges, solids, surfaces). These tags represent parameters that include: submodels, material properties, material orienters, optical properties, and radiation analysis groups. TD aliases were used for most tags to allow analysis to be performed with a variety of parameter values. Domain-tags were also attached to individual and groups of surfaces and solids to allow them to be used later within TD to populate objects like, for example, heaters and contactors. These tools allow the user to make changes to the geometry in SpaceClaim and then easily synchronize the mesh in TD without having to redefine these objects each time as one would if using TD Mesher.The use of SpaceClaim/TD Direct has helped simplify the process for importing existing geometries and in the creation of high fidelity FE meshes to represent complex parts. It has also saved time and effort in the subsequent analysis.

Power Systems↗
DOE OSTI2022

Reducing Memory Consumption in Calico with Shared Memory

This document details the work to reduce memory consumption in Calico. Calico is SimTools’ Constructive Solid Geometry (CSG) and geometry painting library. It is primarily used to paint material volume fractions in the Eulerian meshes of the physics codes. Calico provides point-in-body checks for the geometry supplied by an Oso model, which are then aggregated by the host codes. In addition, Calico can be used to build Oso models and is used by Ingen for that purpose. Oso models, and thus Calico, provide support for various CSG primitives such as spheres, cylinders, surfaces generated by rotating tabular curve data, and STL files as well as binary combinations of those primitives. Prior to refactoring Calico will run out of memory on CTS-1 machines when 36 MPI ranks are used per node when reading STL models on the order of 1.5 GB. This limitation is a bottleneck in designer workflow. This problem has been alleviated through the use of data structures to both reduce memory consumption and to leverage MPI-3 shared memory. This report details the data structures targeted for refactoring in Calico, the methods and implementation details for reducing memory consumption and leveraging shared memory, and results for one test problem. Results show a memory reduction when loading a 1 GB STL file by a factor of 27.5, from 93.4 to 3.4 GB.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1996

Meshing of a Spiral Bevel Gearset with 3D Finite Element Analysis

Recent advances in spiral bevel gear geometry and finite element technology make it practical to conduct a structural analysis and analytically roll the gearset through mesh. With the advent of user specific programming linked to 3D solid modelers and mesh generators, model generation has become greatly automated. Contact algorithms available in general purpose finite element codes eliminate the need for the use and alignment of gap elements. Once the gearset is placed in mesh, user subroutines attached to the FE code easily roll the gearset through mesh. The method is described in detail. Preliminary results for a gearset segment showing the progression of the contact lineload is given as the gears roll through mesh.

Bibel, George D.↗
DOE OSTI2022

MCNP6.3 Unstructured Mesh Verification: GodivR and CANDU Models

A geometric cell of the Monte Carlo N-Particle (MCNP)1 transport code is traditionally created by using Boolean operators on defined surfaces. This constructive solid geometry (CSG) capability has been available in the MCNP code since its beginning. However, a CSG model approach is limited when it comes to constructing a representative geometry for a complex model in its ability to capture a correct model representation. Starting with the version 6.0, the MCNP code has the ability of embedding an unstructured mesh (UM) model into a CSG cell to create a hybrid geometry [1]. The MCNP UM feature provides the flexibility of defining very complex geometries because computer aided design (CAD) and mesh generation software packages can be utilized to construct UM models for MCNP simulations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Numerical Model of IProTech PIP WEC Device

iProTech PIP wave energy converter (WEC) is a slack moored, single hull device with no moving parts in the water, joints or bearings. This submission includes data of the simulation, reports, and code for the iProTech PIP (WEC) project. The organization of the data included in the provided archive is detailed below and in the data description of the archive. The data teamer-iprotech-nrel folder includes and explains matlab and python code developed to hydrodynamically model the PIP WEC device in WEC-Sim. The subfolders cover the following steps: 1) report: explanatory information on device geometry 2) pip_mesher: python code to generate mesh panels from device profile data 3) wec-sim_models: matlab code to run WEC-Sim The data uploaded is a snapshot as of 11/02/2121 of code residing in a Github repository administered by David Ogden of NREL.

16 TIDAL AND WAVE POWER↗
DOE OSTI2023

FINITE ELEMENT MODEL MESH REFINEMENT EFFECTS ON QUALIFICATION OF NUCLEAR GRADE GRAPHITE COMPONENTS

The American Society of Mechanical Engineers (ASME) provides the full and simplified design-by-analysis probabilistic assessments for determining acceptance of nuclear grade graphite core components. The assessments can be characterized by three parts: (1) a component stress distribution, often determined by a finite element (FE) model; (2) a Weibull probability density function (pdf) that characterizes the experimental tensile strength distribution; and (3) the post-processor, which combines the FE model and the Weibull strength distribution in accordance with the full and simplified assessments to determine component acceptance. It is known that the level of mesh refinement in FE models can affect the modeled component’s calculated stress distribution. Depending on the component geometry, the stress distribution may converge with sufficient refinement. It was previously unknown whether the acceptance decision resulting from the full and simplified assessments might change even with sufficient mesh refinement. This study explores that question using experimental strength results for a dog-bone geometry for two graphite grades, IG-110 and PCEA. The simplified assessment has two criteria that must be met, the first limits the combined membrane stress by the allowable stress and the second limits the peak equivalent stress by the allowable stress scaled by the ratio of flexural to tensile strength. In the application of the simplified assessment, convergence of the peak equivalent stress required extreme mesh refinement, however, the acceptance decision was not affected. It is hypothesized that more complex geometries with stress concentrations may present mesh refinement effects on the simplified assessment acceptance decision. Mesh refinement did affect the acceptance decision in the full assessment for the applied pressure loadings in this study. This work suggests component stress distribution convergence is not a sufficient criteria for POF convergence in the full assessment and that mesh refinement should continue until the POF has converged, especially where the resulting POF is bordering the SRC acceptable POF limit.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2026

A meshing framework for digital twins for extrusion based additive manufacturing

Additive manufacturing (AM) allows for manufacturing of complex three-dimensional geometries not typically realizable with standard manufacturing practices. The internal microstructure of AM components has a significant impact on mechanical, vibrational, and shock properties and permits richer design space when this is controllable. Due to complex interactions of internal geometry of an extrusion-based AM component, it is common practice to assume homogeneous behavior or to perform characterization testing on specific toolpath configurations. To avoid testing or material waste, it is necessary to develop a consistently accurate numerical simulation framework with relevant boundary value problems that can handle the complicated geometry of internal material microstructure present in AM components. Herein, a framework is proposed to directly create computational meshes suitable for finite element analysis (FEA) of the fine-scale features generated from extrusion-based AM tool paths to maintain a strong process–structure–property-performance linkage. This mesh can be manually or automatically analyzed using standard FEA simulations such as quasi-static preloading or modal analysis. The framework allows an in-silico assessment of a target AM geometry where fine-scale features greatly impact quantities of design interest such as in soft elastomeric lattices where toolpath infill can greatly influence the self-contact of a structure in compression, which we use as a motivating exemplar. This approach greatly reduces both time and resource waste present in traditional build and test design cycles for non-intuitive design spaces, and acts as a tool for use in the production of a key component of a digital twin, a mesh suitable for finite element analysis. In conclusion, it also further allows for the exploration of toolpath infill to optimize component properties beyond simple linear properties such as density and stiffness.

Additive manufacturing↗
NASA NTRS2019

Extrusion of Complex Surface Meshes Utilizing Face Offsetting and Mean Curvature Smoothing

Grids for three-dimensional Computational Fluid Dynamics problems frequently require a prismatic layer of cells, typically extruded in the off-body direction from a two-dimensional surface mesh to properly resolve boundary layers. When the surface geometry is complex, the extrusion process can fail, resulting in the physical boundary layer being thicker than the prismatic layer, leading to under-resolved boundary layers. To address the shortcomings of existing grid generation tools, a new tool has been developed as part of the Mesh Tools suite, that is capable of extruding complex surface meshes. The new tool uses a face offsetting method to preserve surface curvature and a mean curvature smoothing algorithm to prevent the cells from self-intersecting in concave regions. Testing of the new tool found that not only were complex surface meshes able to be extruded to the desired thickness, but extrusion of simple surface meshes was also improved, due to the face offsetting method.

mesh extrusion↗
DOE OSTI2018

Aerodynamic Drag Scoping Work.

This memo summarizes the aerodynamic drag scoping work done for Goodyear in early FY18. The work is to evaluate the feasibility of using Sierra/Low-Mach (Fuego) for drag predictions of rolling tires, particularly focused on the effects of tire features such as lettering, sidewall geometry, rim geometry, and interaction with the vehicle body. The work is broken into two parts. Part 1 consisted of investigation of a canonical validation problem (turbulent flow over a cylinder) using existing tools with different meshes and turbulence models. Part 2 involved calculating drag differences over plate geometries with simple features (ridges and grooves) defined by Goodyear of approximately the size of interest for a tire. The results of part 1 show the level of noise to be expected in a drag calculation and highlight the sensitivity of absolute predictions to model parameters such as mesh size and turbulence model. There is 20-30% noise in the experimental measurements on the canonical cylinder problem, and a similar level of variation between different meshes and turbulence models. Part 2 shows that there is a notable difference in the predicted drag on the sample plate geometries, however, the computational cost of extending the LES model to a full tire would be significant. This cost could be reduced by implementation of more sophisticated wall and turbulence models (e.g. detached eddy simulations - DES) and by focusing the mesh refinement on feature subsets with the goal of comparing configurations rather than absolute predictivity for the whole tire.

42 ENGINEERING↗
DOE OSTI2019

Multi-rate time integration on overset meshes

Overset meshes are an effective tool for the computational fluid dynamic simulation of problems with complex geometries or multiscale spatio-temporal features. When the maximum allowable timestep on one or more meshes is significantly smaller than on the remaining meshes, standard explicit time integrators impose inefficiencies for time-accurate calculations by requiring that all meshes advance with the smallest timestep. With the targeted use of multi-rate time integrators, separate meshes can be time-marched at independent rates to avoid wasteful computation while maintaining accuracy and stability. This work applies time-explicit multi-rate integrators to the simulation of the compressible Navier-Stokes equations discretized on overset meshes using summation-by-parts (SBP) operators and simultaneous approximation term (SAT) boundary conditions. We introduce a class of multi-rate Adams-Bashforth (MRAB) schemes that offer significant stability improvements and computational efficiencies for SBP-SAT methods. We present numerical results that confirm the efficacy of MRAB integrators, outline a number of implementation challenges, and demonstrate a reduction in computational cost enabled by MRAB. We also look at the use of our method in the setting of a large-scale distributed-memory parallel implementation where we discuss concerns involving load balancing and communication efficiency.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2013

Recent Development in the CESE Method for the Solution of the Navier-Stokes Equations Using Unstructured Triangular or Tetrahedral Meshes With High Aspect Ratio

In the multidimensional CESE development, triangles and tetrahedra turn out to be the most natural building blocks for 2D and 3D spatial meshes. As such the CESE method is compatible with the simplest unstructured meshes and thus can be easily applied to solve problems with complex geometries. However, because the method uses space-time staggered stencils, solution decoupling may become a real nuisance in applications involving unstructured meshes. In this paper we will describe a simple and general remedy which, according to numerical experiments, has removed any possibility of solution decoupling. Moreover, in a real-world viscous flow simulation near a solid wall, one often encounters a case where a boundary with high curvature or sharp corner is surrounded by triangular/tetrahedral meshes of extremely high aspect ratio (up to 106). For such an extreme case, the spatial projection of a space-time compounded conservation element constructed using the original CESE design may become highly concave and thus its centroid (referred to as a spatial solution point) may lie far outside of the spatial projection. It could even be embedded beyond a solid wall boundary and causes serious numerical difficulties. In this paper we will also present a new procedure for constructing conservation elements and solution elements which effectively overcomes the difficulties associated with the original design. Another difficulty issue which was addressed more recently is the wellknown fact that accuracy of gradient computations involving triangular/tetrahedral grids deteriorates rapidly as the aspect ratio of grid cells increases. The root cause of this difficulty was clearly identified and several remedies to overcome it were found through a rigorous mathematical analysis. However, because of the length of the current paper and the complexity of mathematics involved, this new work will be presented in another paper.

Chang, Sin-Chung↗