This document discusses numerical dispersion analysis of symplectic and alternating direction implicit (ADI) schemes for computational electromagnetic simulation. It presents Maxwell's equations as a Hamiltonian system that can be written as symplectic or ADI schemes by approximating the time evolution operator. Three high order spatial difference approximations - high order staggered difference, compact finite difference, and scaling function approximations - are analyzed to reduce numerical dispersion when combined with the symplectic and ADI schemes. The document derives unified dispersion relationships for the symplectic and ADI schemes with different spatial difference approximations, which can be used as a reference for simulating large scale electromagnetic problems.
A Novel Cosine Approximation for High-Speed Evaluation of DCTCSCJournals
This article presents a novel cosine approximation for high-speed evaluation of DCT (Discrete Cosine Transform) using Ramanujan Ordered Numbers. The proposed method uses the Ramanujan ordered number to convert the angles of the cosine function to integers. Evaluation of these angles is by using a 4th degree Polynomial that approximates the cosine function with error of approximation in the order of 10^-3. The evaluation of the cosine function is explained through the computation of the DCT coefficients. High-speed evaluation at the algorithmic level is measured in terms of the computational complexity of the algorithm. The proposed algorithm of cosine approximation increases the overhead on the number of adders by 13.6%. This algorithm avoids floating-point multipliers and requires N/2log2N shifts and (3N/2 log2 N)- N + 1 addition operations to evaluate an N-point DCT coefficients thereby improving the speed of computation of the coefficients .
FINITE DIFFERENCE MODELLING FOR HEAT TRANSFER PROBLEMSroymeister007
This report provides a practical overview of numerical solutions to the heat equation using the finite difference method (FDM). The forward time, centered space (FTCS), the backward time, centered space (BTCS), and Crank-Nicolson schemes are developed, and applied to a simple problem in1volving the one-dimensional heat equation. Complete, working Matlab and FORTRAN codes for each program are presented. The results of running the codes on finer (one-dimensional) meshes, and with smaller time steps are demonstrated. These sample calculations show that the schemes realize theoretical predictions of how their truncation errors depend on mesh spacing and time step. The Matlab codes are straightforward and allow us to see the differences in implementation between explicit method (FTCS) and implicit methods (BTCS). The codes also allow us to experiment with the stability limit of the FTCS scheme.
Numerical Solution of Diffusion Equation by Finite Difference Methodiosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
A landing gear assembly consists of various components viz. Lower side stay, Upperside stay, Locking actuators, Extension actuators, Tyres, and Locking pins to name a few. Each unit having a specific operation to deal with, in this project the main unit being studied is the lower brace. The primary objective is to analyse stresses in the element of the lower brace unit using strength of materials or RDM method and Finite Element Method (FEM) and compare both. Using the obtained data a suitable material is proposed for the component. The approach used here is to study the overall behaviour of the element by taking up each aspect, finally summing up the total effect of all the aspects in the functioning of the element.
vFORTRAN is used as a numerical and scientific computing language. The main objective of the lab work is to understand FORTRAN language using which we solve simple numerical problems and compare different methodologies. Through this project we make use of various functions of FORTRAN and solve a simple projectile problem and also LAPACK library to solve a tridiagonal matrix problem. We use DGESV and DGTSV functions to make it possible. The given problems are built and compiled using a free integrated development environment called CODE::BLOCKS [1] which is an open source platform for FORTRAN and C.
A Novel Cosine Approximation for High-Speed Evaluation of DCTCSCJournals
This article presents a novel cosine approximation for high-speed evaluation of DCT (Discrete Cosine Transform) using Ramanujan Ordered Numbers. The proposed method uses the Ramanujan ordered number to convert the angles of the cosine function to integers. Evaluation of these angles is by using a 4th degree Polynomial that approximates the cosine function with error of approximation in the order of 10^-3. The evaluation of the cosine function is explained through the computation of the DCT coefficients. High-speed evaluation at the algorithmic level is measured in terms of the computational complexity of the algorithm. The proposed algorithm of cosine approximation increases the overhead on the number of adders by 13.6%. This algorithm avoids floating-point multipliers and requires N/2log2N shifts and (3N/2 log2 N)- N + 1 addition operations to evaluate an N-point DCT coefficients thereby improving the speed of computation of the coefficients .
FINITE DIFFERENCE MODELLING FOR HEAT TRANSFER PROBLEMSroymeister007
This report provides a practical overview of numerical solutions to the heat equation using the finite difference method (FDM). The forward time, centered space (FTCS), the backward time, centered space (BTCS), and Crank-Nicolson schemes are developed, and applied to a simple problem in1volving the one-dimensional heat equation. Complete, working Matlab and FORTRAN codes for each program are presented. The results of running the codes on finer (one-dimensional) meshes, and with smaller time steps are demonstrated. These sample calculations show that the schemes realize theoretical predictions of how their truncation errors depend on mesh spacing and time step. The Matlab codes are straightforward and allow us to see the differences in implementation between explicit method (FTCS) and implicit methods (BTCS). The codes also allow us to experiment with the stability limit of the FTCS scheme.
Numerical Solution of Diffusion Equation by Finite Difference Methodiosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
A landing gear assembly consists of various components viz. Lower side stay, Upperside stay, Locking actuators, Extension actuators, Tyres, and Locking pins to name a few. Each unit having a specific operation to deal with, in this project the main unit being studied is the lower brace. The primary objective is to analyse stresses in the element of the lower brace unit using strength of materials or RDM method and Finite Element Method (FEM) and compare both. Using the obtained data a suitable material is proposed for the component. The approach used here is to study the overall behaviour of the element by taking up each aspect, finally summing up the total effect of all the aspects in the functioning of the element.
vFORTRAN is used as a numerical and scientific computing language. The main objective of the lab work is to understand FORTRAN language using which we solve simple numerical problems and compare different methodologies. Through this project we make use of various functions of FORTRAN and solve a simple projectile problem and also LAPACK library to solve a tridiagonal matrix problem. We use DGESV and DGTSV functions to make it possible. The given problems are built and compiled using a free integrated development environment called CODE::BLOCKS [1] which is an open source platform for FORTRAN and C.
I am Mathew K. I am a Planetary Science Assignment Expert at eduassignmenthelp.com. I hold a Masters’s Degree in Planetary Science, The University of Chicago, USA. I have been helping students with their homework for the past 8 years. I solve assignments related to Planetary Sciences.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Planetary Science Assignments.
ER Publication,
IJETR, IJMCTR,
Journals,
International Journals,
High Impact Journals,
Monthly Journal,
Good quality Journals,
Research,
Research Papers,
Research Article,
Free Journals, Open access Journals,
erpublication.org,
Engineering Journal,
Science Journals,
9+-
I am Boris M. I am a Computer Science Assignment Help Expert at programminghomeworkhelp.com. I hold MSc. in Programming, McGill University, Canada. I have been helping students with their homework for the past 7 years. I solve assignments related to Computer Science.
Visit programminghomeworkhelp.com or email support@programminghomeworkhelp.com.You can also call on +1 678 648 4277 for any assistance with Computer Science assignments.
Numerical Methods in Mechanical Engineering - Final ProjectStasik Nemirovsky
Final Project for the class of "Numerical Methods in Mechanical Engineering" - MECH 309.
In this project, various engineering problems were analyzed and solved using advanced numerical approximation methods and MATLAB software.
I am Martin J. I am a Digital Signal Processing Assignment Expert at matlabassignmentexperts.com. I hold a Ph.D. in Matlab, Arizona University, USA. I have been helping students with their homework for the past 6 years. I solve assignments related to Digital Signal Processing.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Digital Signal Processing Assignments.
Investigation on the Pattern Synthesis of Subarray Weights for Low EMI Applic...IOSRJECE
In modern radar applications, it is frequently required to produce sum and difference patterns sequentially. The sum pattern amplitude coefficients are obtained by using Dolph-Chebyshev synthesis method where as the difference pattern excitation coefficients will be optimized in this present work. For this purpose optimal group weights will be introduced to the different array elements to obtain any type of beam depending on the application. Optimization of excitation to the array elements is the main objective so in this process a subarray configuration is adopted. However, Differential Evolution Algorithm is applied for optimization method. The proposed method is reliable and accurate. It is superior to other methods in terms of convergence speed and robustness. Numerical and simulation results are presented.
Analysis of stress in circular hollow section by fea and analytical techniqueeSAT Journals
Abstract This study focus on stress calculation in a cantilever beam by FEA &Analytical techniques. To know the value of maximum load bearing capacity of any particular beam this study has been generated. Structural analysis is foremost requirement in a design process. Also when we perform FEA analysis of any structure we cannot blindly trust on its result. If we don’t have any past result data of that structure, it became difficult for us to know the deviation of result. For that purpose we may require analytical calculation result in order to compare result value of FEA. Hence in this study a range of load values are applied on cantilever beam by both techniques. Later graph has been plotted for different load values & verification of results is carried out. Keywords: Structure Analysis, CATIA, FEA and Benchmarking
Linear regression [Theory and Application (In physics point of view) using py...ANIRBANMAJUMDAR18
Machine-learning models are behind many recent technological advances, including high-accuracy translations of the text and self-driving cars. They are also increasingly used by researchers to help in solving physics problems, like Finding new phases of matter, Detecting interesting outliers
in data from high-energy physics experiments, Founding astronomical objects are known as gravitational lenses in maps of the night sky etc. The rudimentary algorithm that every Machine Learning enthusiast starts with is a linear regression algorithm. In statistics, linear regression is a linear approach to modelling the relationship between a scalar response (or dependent variable) and one or more explanatory variables (or independent
variables). Linear regression analysis (least squares) is used in a physics lab to prepare the computer-aided report and to fit data. In this article, the application is made to experiment: 'DETERMINATION OF DIELECTRIC CONSTANT OF NON-CONDUCTING LIQUIDS'. The entire computation is made through Python 3.6 programming language in this article.
Java3D is an Application Programming Interface used for writing 3D graphics applications and applets. This paper gives a short introduction of java3D, analyses the mathematics of Hermite, Bezier, FourPoints, B-Splines curve, and describes implementation of curve creation and curve
operations using Java3D API.
I am Mathew K. I am a Planetary Science Assignment Expert at eduassignmenthelp.com. I hold a Masters’s Degree in Planetary Science, The University of Chicago, USA. I have been helping students with their homework for the past 8 years. I solve assignments related to Planetary Sciences.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Planetary Science Assignments.
ER Publication,
IJETR, IJMCTR,
Journals,
International Journals,
High Impact Journals,
Monthly Journal,
Good quality Journals,
Research,
Research Papers,
Research Article,
Free Journals, Open access Journals,
erpublication.org,
Engineering Journal,
Science Journals,
9+-
I am Boris M. I am a Computer Science Assignment Help Expert at programminghomeworkhelp.com. I hold MSc. in Programming, McGill University, Canada. I have been helping students with their homework for the past 7 years. I solve assignments related to Computer Science.
Visit programminghomeworkhelp.com or email support@programminghomeworkhelp.com.You can also call on +1 678 648 4277 for any assistance with Computer Science assignments.
Numerical Methods in Mechanical Engineering - Final ProjectStasik Nemirovsky
Final Project for the class of "Numerical Methods in Mechanical Engineering" - MECH 309.
In this project, various engineering problems were analyzed and solved using advanced numerical approximation methods and MATLAB software.
I am Martin J. I am a Digital Signal Processing Assignment Expert at matlabassignmentexperts.com. I hold a Ph.D. in Matlab, Arizona University, USA. I have been helping students with their homework for the past 6 years. I solve assignments related to Digital Signal Processing.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Digital Signal Processing Assignments.
Investigation on the Pattern Synthesis of Subarray Weights for Low EMI Applic...IOSRJECE
In modern radar applications, it is frequently required to produce sum and difference patterns sequentially. The sum pattern amplitude coefficients are obtained by using Dolph-Chebyshev synthesis method where as the difference pattern excitation coefficients will be optimized in this present work. For this purpose optimal group weights will be introduced to the different array elements to obtain any type of beam depending on the application. Optimization of excitation to the array elements is the main objective so in this process a subarray configuration is adopted. However, Differential Evolution Algorithm is applied for optimization method. The proposed method is reliable and accurate. It is superior to other methods in terms of convergence speed and robustness. Numerical and simulation results are presented.
Analysis of stress in circular hollow section by fea and analytical techniqueeSAT Journals
Abstract This study focus on stress calculation in a cantilever beam by FEA &Analytical techniques. To know the value of maximum load bearing capacity of any particular beam this study has been generated. Structural analysis is foremost requirement in a design process. Also when we perform FEA analysis of any structure we cannot blindly trust on its result. If we don’t have any past result data of that structure, it became difficult for us to know the deviation of result. For that purpose we may require analytical calculation result in order to compare result value of FEA. Hence in this study a range of load values are applied on cantilever beam by both techniques. Later graph has been plotted for different load values & verification of results is carried out. Keywords: Structure Analysis, CATIA, FEA and Benchmarking
Linear regression [Theory and Application (In physics point of view) using py...ANIRBANMAJUMDAR18
Machine-learning models are behind many recent technological advances, including high-accuracy translations of the text and self-driving cars. They are also increasingly used by researchers to help in solving physics problems, like Finding new phases of matter, Detecting interesting outliers
in data from high-energy physics experiments, Founding astronomical objects are known as gravitational lenses in maps of the night sky etc. The rudimentary algorithm that every Machine Learning enthusiast starts with is a linear regression algorithm. In statistics, linear regression is a linear approach to modelling the relationship between a scalar response (or dependent variable) and one or more explanatory variables (or independent
variables). Linear regression analysis (least squares) is used in a physics lab to prepare the computer-aided report and to fit data. In this article, the application is made to experiment: 'DETERMINATION OF DIELECTRIC CONSTANT OF NON-CONDUCTING LIQUIDS'. The entire computation is made through Python 3.6 programming language in this article.
Java3D is an Application Programming Interface used for writing 3D graphics applications and applets. This paper gives a short introduction of java3D, analyses the mathematics of Hermite, Bezier, FourPoints, B-Splines curve, and describes implementation of curve creation and curve
operations using Java3D API.
The Six Highest Performing B2B Blog Post FormatsBarry Feldman
If your B2B blogging goals include earning social media shares and backlinks to boost your search rankings, this infographic lists the size best approaches.
Each technological age has been marked by a shift in how the industrial platform enables companies to rethink their business processes and create wealth. In the talk I argue that we are limiting our view of what this next industrial/digital age can offer because of how we read, measure and through that perceive the world (how we cherry pick data). Companies are locked in metrics and quantitative measures, data that can fit into a spreadsheet. And by that they see the digital transformation merely as an efficiency tool to the fossil fuel age. But we need to stretch further…
A Novel Space-time Discontinuous Galerkin Method for Solving of One-dimension...TELKOMNIKA JOURNAL
In this paper we propose a high-order space-time discontinuous Galerkin (STDG) method for
solving of one-dimensional electromagnetic wave propagations in homogeneous medium. The STDG
method uses finite element Discontinuous Galerkin discretizations in spatial and temporal domain
simultaneously with high order piecewise Jacobi polynomial as the basis functions. The algebraic
equations are solved using Block Gauss-Seidel iteratively in each time step. The STDG method is
unconditionally stable, so the CFL number can be chosen arbitrarily. Numerical examples show that the
proposed STDG method is of exponentially accuracy in time.
EXACT SOLUTIONS OF A FAMILY OF HIGHER-DIMENSIONAL SPACE-TIME FRACTIONAL KDV-T...cscpconf
In this paper, based on the definition of conformable fractional derivative, the functional
variable method (FVM) is proposed to seek the exact traveling wave solutions of two higherdimensional
space-time fractional KdV-type equations in mathematical physics, namely the
(3+1)-dimensional space–time fractional Zakharov-Kuznetsov (ZK) equation and the (2+1)-
dimensional space–time fractional Generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony
(GZK-BBM) equation. Some new solutions are procured and depicted. These solutions, which
contain kink-shaped, singular kink, bell-shaped soliton, singular soliton and periodic wave
solutions, have many potential applications in mathematical physics and engineering. The
simplicity and reliability of the proposed method is verified.
Using Subspace Pursuit Algorithm to Improve Performance of the Distributed Co...Polytechnique Montreal
This paper applies a compressed algorithm to improve the spectrum sensing performance of cognitive radio technology.
At the fusion center, the recovery error in the analog to information converter (AIC) when reconstructing the
transmit signal from the received time-discrete signal causes degradation of the detection performance. Therefore, we
propose a subspace pursuit (SP) algorithm to reduce the recovery error and thereby enhance the detection performance.
In this study, we employ a wide-band, low SNR, distributed compressed sensing regime to analyze and evaluate the
proposed approach. Simulations are provided to demonstrate the performance of the proposed algorithm.
A High Order Continuation Based On Time Power Series Expansion And Time Ratio...IJRES Journal
In this paper, we propose a high order continuation based on time power series expansion and time rational representation called Pad´e approximants for solving nonlinear structural dynamic problems. The solution of the discretized nonlinear structural dynamic problems, by finite elements method, is sought in the form of a power series expansion with respect to time. The Pad´e approximants technique is introduced to improve the validity range of power series expansion. The whole solution is built branch by branch using the continuation method. To illustrate the performance of this proposed high order continuation, we give some numerical comparisons on an example of forced nonlinear vibration of an elastic beam.
Discrete wavelet transform-based RI adaptive algorithm for system identificationIJECEIAES
In this paper, we propose a new adaptive filtering algorithm for system identifica- tion. The algorithm is based on the recursive inverse (RI) adaptive algorithm which suffers from low convergence rates in some applications; i.e., the eigenvalue spread of the autocorrelation matrix is relatively high. The proposed algorithm applies discrete-wavelet transform (DWT) to the input signal which, in turn, helps to overcome the low convergence rate of the RI algorithm with relatively small step-size(s). Different scenarios has been investigated in different noise environments in system identification setting. Experiments demonstrate the advantages of the proposed DWT recursive inverse (DWT-RI) filter in terms of convergence rate and mean-square-error (MSE) compared to the RI, discrete cosine transform LMS (DCT-LMS), discretewavelet transform LMS (DWT-LMS) and recursive-least-squares (RLS) algorithms under same conditions.
journals like IJERST to provide a valuable resource for researchers, students, and professionals who are interested in the latest developments and insights in the field of Information Technology and Computer Engineering.
Discrete-wavelet-transform recursive inverse algorithm using second-order est...TELKOMNIKA JOURNAL
The recursive-least-squares (RLS) algorithm was introduced as an alternative to LMS algorithm with enhanced performance. Computational complexity and instability in updating the autocolleltion matrix are some of the drawbacks of the RLS algorithm that were among the reasons for the intrduction of the second-order recursive inverse (RI) adaptive algorithm. The 2nd order RI adaptive algorithm suffered from low convergence rate in certain scenarios that required a relatively small initial step-size. In this paper, we propose a newsecond-order RI algorithm that projects the input signal to a new domain namely discrete-wavelet-transform (DWT) as pre step before performing the algorithm. This transformation overcomes the low convergence rate of the second-order RI algorithm by reducing the self-correlation of the input signal in the mentioned scenatios. Expeirments are conducted using the noise cancellation setting. The performance of the proposed algorithm is compared to those of the RI, original second-order RI and RLS algorithms in different Gaussian and impulsive noise environments. Simulations demonstrate the superiority of the proposed algorithm in terms of convergence rate comparedto those algorithms.
A Review: Compensation of Mismatches in Time Interleaved Analog to Digital Co...IJERA Editor
The execution of today's correspondence frameworks is exceedingly subject to the utilized Analog-to-Digital converters (ADCs), and with a specific end goal to give more flexibility and exactness to the developing correspondence innovations, superior-ADCs are needed. In this respect, the time-interleaved operation of an exhibit of ADCs (TI-ADC) might be a sensible result. A TI-ADC can build its throughput by utilizing M channel ADCs or sub converters in parallel and examining the data motion in a period-interleaved way. In any case, the execution of a TI-ADC gravely suffers from the bungles around the channel ADCs. In this paper we survey the advancement in the configuration of low-intricacy advanced remedy structures and calculations for time-interleaved ADCs in the course of the most recent five years. We devise a discrete-time model, state the outline issue, and finally infer the calculations and structures. Specifically, we examine proficient calculations to outline time-differing remedy filters and additionally iterative structures using polynomial based filters. Thusly, the remuneration structure may be utilized to repay time-differing recurrence reaction befuddles in time-interleaved ADCs, and in addition to remake uniform examples from nonuniformly tested indicators. We examine the recompense structure, research its execution, and exhibit requisition zones of the structure through various illustrations. At long last, we give a standpoint to future examination questions.
Design of Resonators for Coupled Magnetic Resonance-based Wireless Power Tran...Quang-Trung Luu
Quang-Trung Luu, Duc-Hung Tran, Bao-Huy Nguyen, Yem Vu-Van, and Cao-Minh Ta, "Design of Resonators for Coupled Magnetic Resonance-based Wireless Power Transmission System," In Proc. Vietnam Conference on Control and Automation, Da Nang, Nov. 2013.
Sparse data formats and efficient numerical methods for uncertainties in nume...Alexander Litvinenko
Description of methodologies and overview of numerical methods, which we used for modeling and quantification of uncertainties in numerical aerodynamics
Ill-posedness formulation of the emission source localization in the radio- d...Ahmed Ammar Rebai PhD
To contact the authors : tarek.salhi@gmail.com and ahmed.rebai2@gmail.com
In the field of radio detection in astroparticle physics, many studies have shown the strong dependence of the solution of the radio-transient sources localization problem (the radio-shower time of arrival on antennas) such solutions are purely numerical artifacts. Based on a detailed analysis of some already published results of radio-detection experiments like : CODALEMA 3 in France, AERA in Argentina and TREND in China, we demonstrate the ill-posed character of this problem in the sens of Hadamard. Two approaches have been used as the existence of solutions degeneration and the bad conditioning of the mathematical formulation problem. A comparison between experimental results and simulations have been made, to highlight the mathematical studies. Many properties of the non-linear least square function are discussed such as the configuration of the set of solutions and the bias.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
NO1 Uk best vashikaran specialist in delhi vashikaran baba near me online vas...Amil Baba Dawood bangali
Contact with Dawood Bhai Just call on +92322-6382012 and we'll help you. We'll solve all your problems within 12 to 24 hours and with 101% guarantee and with astrology systematic. If you want to take any personal or professional advice then also you can call us on +92322-6382012 , ONLINE LOVE PROBLEM & Other all types of Daily Life Problem's.Then CALL or WHATSAPP us on +92322-6382012 and Get all these problems solutions here by Amil Baba DAWOOD BANGALI
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Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Final project report on grocery store management system..pdf
Numerical disperison analysis of sympletic and adi scheme
1. Numerical Dispersion Analysis of Symplectic and
ADI Schemes
Xin-gang Ren#
, Zhi-xiang Huang*
, Xian-liang Wu,
Si-long Lu, Yi-cai Mei, Hong-mei Du, Hui Wang
Key Lab of Intelligent Computing & Signal Processing
Ministry of Education, Anhui University
Hefei, China
#xingangahu@126.com *zxhuang@ahu.edu.cn
Xian-liang Wu, Jing Shen
Department of Physics and Electronic Engineering
Hefei Normal University
Hefei, China
xlwu@ahu.edu.cn
Abstract-In this paper, Maxwell’s equations are taken as a
Hamiltonian system and then written as Hamiltonian canonical
equations by using the functional variation method. The
symplectic and ADI schemes, which can be extracted by applying
two types of approximation to the time evolution operator, are
explicit and implicit scheme in computational electromagnetic
simulation, respectively. Since Finite-difference time-domain
(FDTD) encounter low accuracy and high dispersion, the more
accurate simulation methods can be derived by evaluating the
curl operator in the spatial direction with kinds of high order
approaches including high order staggered difference, compact
finite difference and scaling function approximations. The
numerical dispersion of the symplectic and ADI schemes
combining with the three high order spatial difference
approximations have been analyzed. It has been shown that
symplectic scheme combining with compact finite difference and
ADI scheme combining with scaling function performance better
than other methods. Both schemes can be usefully employed for
simulating and solving the large scale electromagnetic problems.
I. INTRODUCTION
The finite-difference time domain (FDTD) method, which
was firstly proposed by K.S.Yee[1]and has widely been used
for solving the electromagnetic problems. The central
difference was used to approximate the temporal and spatial
derivate in time domain Maxwell’s equations. Since the
FDTD method is based on the explicit difference, the Courant
stability condition must be satisfied in order to guarantee
numerical stability. The time step size must be very small in
order to obtain a high accuracy when a large scale size
structure was simulated. The numerical dispersion error will
be accumulated with an increased simulation time step. The
high order FDTD scheme is proposed to reduce the dispersion
but will encounter a low stability. Researchers have carried
out many improvements to overcome the shortcoming. The
one is the symplectic scheme which have been proved to
enhance the stability and reduce the dispersion error because
of energy conservation of Hamiltonian system[2]. Besides the
alternating-direction implicit (ADI) scheme which is
unconditionally stable is proposed to manipulate the large
scale size problems and theoretically there is no limitation on
the time step size [3].But as the size of time step is increased,
numerical dispersion errors will become large. To overcome
this shortcoming, the more accurate curl operator
approximations have been employed to result in a low
numerical dispersion. Three types of approximations are
considered in the following context. The high order staggered
difference which is also an explicit scheme can obtain a better
dispersion than the central difference [4, 5]. The compact
finite difference which is an implicit scheme approximate will
result in the solution of a tri-diagonal matrix and lead to a low
numerical dispersion [6]. The multiresoultion time domain
(MRTD) method which is based on the scaling function has
been broadly accepted as a high accurate method to improve
the numerical dispersion [7].
In this paper, Maxwell’s equations firstly are written as
Hamiltonian canonical equations by using the functional
variation method. The symplectic and ADI schemes have been
extracted by applying two types of approximation to the time
evolution operator. The high order staggered difference,
compact finite difference and scaling function approximations
are taken to approximate the spatial curl operators to obtain
low numerical dispersion errors. The unified dispersion
relationships are derived for the symplectic and ADI schemes,
respectively. The numerical dispersion is studied by applying
different curl operator approximation. The result can be used
as a reference when simulate and solve the large scale
electromagnetic problems.
II. FORMULATION
Maxwell’s equations can be rewritten in the form of
Hamilton function as[2, 8]:
6. exp 0t t) ' ' )A (4)
However, the exponential operator exp( )t'A cannot be
evaluated at any t' . Fortunately, there are mainly two
approximations will deduce to lots of simulation methods
which have been widely used, one is the use of symplectic
propagator technique which will extract the explicit
symplectic scheme and the other is the use of
Lie-Trotter-Suzuki approximation which will extract the
implicit ADI scheme.
1) Symplectic scheme
The operator exp( )t'A is approximated with the
symplectic propagator technique by splitting matrix A into
two noncommuting operators ,B C , i.e. A B C and , then
a m-stage and p-order approximation can be obtained in the
following product form of the exponential operator[9,14]:
9. 1
6 6
1
exp ( )( ) ( )
m
p
l l
l
t d t c t O t
' ' ' '–A I B I C
(7)
The value of the symplectic integrator coefficients can
be found in Ref[10]. Especially, one can find that the
symplectic scheme can be reduced to the conventional
FDTD method when the symplectic integrator coefficients
are chosen as 1 2 1/ 2c c ; 1 1d , 2 0d . Here we use
the coefficient as shown in Table.I, then a fourth order
accuracy will be gained in the temporal differential
approximation.
2) ADI scheme
The matrix operator A was divided into series of real
antisymmetric operators
1
s
i
i
¦A A . Then the formulation of
Lie-Trotter-Suzuki approximation can be expressed as[11]:
10. 1 1
exp exp( ) lim exp( )
nss
i
i
n
i i
t
t t
nof
'ª º
' ' « »
¬ ¼
¦ –
A
A A (8)
Especially, if we set the parameters 2, 2s n and apply
the Pade approximation, Eq.(9) will be reduced to a simple
form[12]:
11. 1
1 2
2
22
1
exp ( ) ( )( )
2 2
( )( ) ( )
2 2
t
t
t
t
O t
t
'
'
'
'
˜ '
'
AI
A A I
I A
AI
I
I A
(9)
It can be proved that Eq.(10) is the time evolution operator
of the implicit and unconditional stable ADI scheme.
B. The Spatial difference approximation
There are kinds of methods to approximate the spatial
derivate, but three types of high accurate method will be
considered in the following including high order staggered
difference, compact finite difference and scaling function
approximations. Firstly, , ,| ( , , ; )n
i j kf f i x j y k z n t' ' ' '
was denoted to approximate the exact solution ( , , )f x y z at
point ( , , )i x j y k z' ' ' in the n-th time step.
1) The high order staggered difference
The high order accuracy discretized scheme can be express
as[11]:
/2
(2 1) 2 (2 1) 2
1
1
| [ | | ]
M
n n n
l s l s l s
s
f C f f
[ [
w
w '
¦ (10)
Where , ,x y z[ and
1 2
2 2
2 2
( 1) [( 1)!!]
2 (2 1) ( 1)!( )!
s
s M MM
M
C
s s s
The coefficients of the fourth order accuracy are 1
9
8
C
and 2
1
24
C . A low dispersion error will be achieved by
applying the high order staggered difference so it can be done
with the large scale problem, while the low
Courant–Friedrichs–Levy (CFL) number is the drawback. The
12. fourth order accuracy scheme will be taken into account.
2) The compact finite difference
The compact finite difference expressed as[6]:
1/2 1/2
1 1 1| | |n n n l l
l l l
f ff f f
D D E
[ [ [ [
w w w
w w w '
(11)
where , ,x y z[ , and a fourth order accuracy of the spatial
difference can be given by setting the compact finite
difference coefficients 1/ 22, 12 /11D E in our numerical
experiment.
3) The scaling function
The multiresolution time domain (MRTD) which based on
Daubechies scaling functions is proposed to enhance stability
and reduce the numerical dispersion. The mainly idea is that
electromagnetic field component, taking xE for example,
expansion with the Daubechies compact support scaling
function ( )xI can be written as[7]:
1 2
, ,
( , , , ) ( 1 2, , ) ( ) ( ) ( ) ( )n
x x i j k n
n
i j k
E x y z t E i j k x y z h tI I I
f
f
¦ (12)
where ( ) ( 1 2)nh t h t t n' , ( )h t is the Haar wavelet scaling
function. The other field components can be obtained with a
similar way. Substituting expression of the field components
into the Maxwell equation with the application of the Galerkin
method and the vanishing moment L , then the spatial
difference can be expressed in a similar way of the high order
staggered difference as:
1
0
1
| ( )[ | | ]
sL
n n n
l l s l s
s
f a l f f
[ [
w
w '
¦ (13)
Where 2 1sL L ,
( 1 2)
( ) ( )
x
a l x l dx
x
I
I
f
f
w
w³
The coefficients of the Daubechies compact support scaling
function are listed in Table .II. [13], in case of 0,2,4s
corresponding to 1 2 3, ,D D D . Here, 2D will be used to
analyze the numerical dispersion.
TABLE I
COEFFICIENTS OF THE SYMPLECTIC INTEGATOR PROPAGATORS
cl dl
1 0.17399689146541 0.62337932451322
2 -0.12038504121430 -0.12337932451322
3 0.89277629949778 -0.12337932451322
4 -0.12038504121430 0.62337932451322
5 0.89277629949778 0
TABLE II
COEFFICIENTS OF THE DAUBECHIES SCALING FUNCTION
D1 D2 D3
a(0) 1 1.229166667 1.291812928
a(1) -0.093750000 -0.137134347
a(2) 0.010416667 0.028761772
a(3) -0.003470141
a(4) 0.000008027
C. The numerical dispersion relationship
The phase velocity of the simulation wave will slightly
differ from the phase velocity of the natural media when the
electromagnetic problem is simulated by a numerical method.
The phase velocity will be varied with the frequency, direction
of propagation, spatial and temporal increment. The numerical
dispersion of the symplectic and ADI schemes were briefly
given in following.
The numerical dispersion relationship of the symplectic
scheme can be expressed as[10]:
2 2 2 2
1
1
cos( ) 1 [4 ( )]
2
m
p
p x y z
p
t g sZ K K K' ¦ (14)
Where
1 1 2 2
1 1 1 2 2
1 1 2 2
1 1 1 2 2
p i j i j ip jp
i j i j ip jp m
i j i j ip jp
i j i j ip jp m
g c d c d c d
d c d c d c
d d d d d d d d
d d
¦
¦
The space increment is ' ( x y z' ' ' ' ), and the
temporal increment is t' , CFL number is s c t' ' .
Parameters [K ( , ,x y z[ ) are determined by the spatial
difference approximation scheme. Parameters [K have been
defined for the high order staggered difference, compact finite
difference and Daubechies scaling function, respectively.
1) The high order staggered difference
39 1
sin( ) sin( )
8 2 24 2
k k[ [
[
[ [
K
' '
(15)
2) The compact finite difference
sin( )
2
2 cos( ) 1
k
k
[
[
[
[
E
K
D [
'
'
(16)
3) The Daubechies scaling function
3 5
(0)sin( ) (1)sin( ) (2)sin( )
2 2 2
k k k
a a a
[ [ [
[
[ [ [
K
' ' '
(17)
If k represents the numerical wave-number, then the
numerical wave-number in , ,x y z direction can be defined
as cos sinxk k I T , sin sinyk k I T and coszk k T .
The numerical dispersion formula of the ADI scheme can
be given by the relation:
2 2 2 2 2 2 2 2 2 2 2 2 2
2
2 2 2 2 2 2 2
4 [ ( ][1 ]
sin ( )
[(1 )(1 )(1 )]
x y y z z x x y z
x y z
s s s
t
s s s
K K K K K K K K K K
Z
K K K
'
(18
where 2 2 2 2
x y zK K K K .
The numerical dispersion of the symplectic finite difference
time domain (S-FDTD), symplectic compact finite difference
time domain (S-CFDTD) and symplectic multiresolution time
domain (S-MRTD) can be obtained by substituting (15), (16),
(17) into (14), respectively. The numerical dispersion of ADI
finite difference time domain (ADI-FDTD), ADI compact
finite difference time domain (ADI-CFDTD) and ADI
multiresolution time domain (ADI-MRTD) were obtained by
substituting(15), (16), (17) into (18), respectively.
III. NUMERICAL VALIDATION
The relative phase velocity error of the aforementioned
13. symplectic and ADI schemes are analyzed firstly as a
function of the propagation angleI as shown in Fig.1, in case
PPW=10, CFL=0.4 and 3T S . Then for a better
understanding of the dispersion, the relative phase velocity
error was taken as a function of PPW and CFL number with
a fixed propagation angle 6I S and 3T S . The
results reveal that the S-CFDTD scheme has the lowest
numerical dispersion curve, and the dispersion curve of
ADI-MRTD scheme is better than other ADI schemes
especially at a low PPW number and small propagation angle.
That means both ADI-MRTD and S-CFDTD have a high
computational precision and can be used to simulate the
large scale size electromagnetic problems.
IV. CONCLUSION
In this paper, Maxwell’s equations are taken as a
Hamiltonian system and then written as Hamiltonian
canonical equations by using the functional variation method.
The symplectic and ADI schemes, which can be extracted by
applying two types of approximations to the time evolution
operator, are explicit and implicit scheme in computational
electromagnetic simulation, respectively. Then the unified
dispersion relationships are derived for the symplectic and
ADI scheme, respectively. The numerical dispersion is studied
by applying three types of high order spatial difference
approximations. It has been shown in the dispersion curves
that symplectic scheme combining with compact finite
difference and ADI scheme combining with scaling function
performance a better dispersion than other methods. Both
schemes can be usefully employed for simulating and solving
the large scale electromagnetic problems.
ACKNOWLEDGMENT
The authors gratefully acknowledge the support of the NSFC
of China (60931002, 61101064), Distinguished Natural
Science Foundation (1108085J01), and Universities Natural
Science Foundation of Anhui Province (No. KJ2011A002,
KJ2011A242), and Financed by the 211 Project of Anhui
University.
0 10 20 30 40 50 60 70 80 90
-90
-85
-80
-75
-70
-65
-60
-55
-50
-45
-40
Propagation Angle I(q)
RelativePhaseVelocityError(dB)
ADI-FDTD
ADI-MRTD
ADI-CFDTD
S-FDTD
S-MRTD
S-CFDTD
Fig.1. Numerical dispersion curves as a function of I, for T S/3, PPW=10
and CFL=0.4.
0.2
0.4
0.6
0.8
1
5
10
15
20
-120
-100
-80
-60
-40
-20
CFL(c't/'x)PPW(O/'x)
RelativePhaseVelocityError(dB)
S-CFDTD
ADI-FDTD
S-MRTTD
ADI-MRTD
ADI-CFDTD
S-FDTD
Fig.2. Numerical dispersion curves as a function of PPW and CFL, for
I=S/6 and T S/3.
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