This document presents a finite element method for predicting the natural modes of acoustic cavities. Two elements are implemented: a 20-noded brick element and an 8-noded axisymmetric element. The method is verified against analytical solutions for a rectangular cavity and experimental results for an empty bottle. Good agreement is found. Axisymmetric models are found to be an order of magnitude more efficient than brick element models for axisymmetric geometries like a bottle, with little loss of accuracy. The addition of an endpiece to models of the bottle has little effect on predicted frequencies.
Numerical Calculation of Solid-Liquid two-Phase Flow Inside a Small Sewage Pumptheijes
Based on a mixture multiphase flow model,theRNG k–εturbulencemodelandfrozen rotor method were used to perform a numerical simulation of steady flow in the internal flow field of a sewage pump that transports solid and liquid phase flows. Resultsof the study indicate that the degree of wear on the front and the back of the blade suction surface from different densities of solid particles shows a completely opposite influencing trend. With the increase of delivered solid-phase density, the isobaric equilibrium position moves to the leading edge point of the blade, but the solid-phase isoconcentration point on the blade pressure surface and suction surface basically remains unchanged. The difference between hydraulic lift and water lift indelivering solid- and liquid-phase flows shows a rising trend with the increase of working flow
The Krylov-TPWL Method of Accelerating Reservoir Numerical Simulationinventionjournals
Because of the large number of system unknowns, reservoir simulation of realistic reservoir can be computationally demanding. Model order reduction (MOR) technique represents a promising approach for accelerating the simulations. In this work, we focus on the application of a MOR technique called Krylov trajectory piecewise-linear (Krylov-TPWL). First, the nonlinear system is represented as a weighted combined piecewise linear system using TPWL method, and then reducing order of each linear model using Krylov subspace. We apply Krylov-TPWL method for a two-phase (oil-water) reservoir model which is solved by full implicit. The example demonstrates that which can greatly reduce the dimension of reservoir model, so as to reduce the calculation time and improve the operation speed.
The primary objective of this project is to extend the conveniences of
deconvolution to non-linear problems of fluid flow in porous media. Unlike
conventional approaches, which are based on an approximate linearization of the
problem, here the solution of the non-linear problem is linearized by a perturbation
approach, which permits term-by-term application of deconvolution. Because the
proposed perturbation solution is more conveniently evaluated in the Laplacetransform
domain and the standard deconvolution algorithms are in the time-domain,
an efficient deconvolution procedure in the Laplace domain is a prerequisite.
For this research objective, a new algorithm is introduced which uses inverse
mirroring at the points of discontinuity and adaptive cubic splines to approximate rate
or pressure versus time data. This algorithm accurately transforms sampled data into
Laplace space and eliminates the Numerical inversion instabilities at discontinuities
or boundary points commonly encountered with the piece-wise linear approximations
of the data.
Applying the algorithm to the field data obtained from published works, we can
unveil the early-time behavior of a reservoir system masked by wellbore-storage
effects. The wellbore-storage coefficient can be variable in the general case. The new
method thus provides a powerful tool to improve pressure-transient-test
interpretation.
Practical use of the algorithm presented in this research has applications in a
variety of Pressure Transient Analysis (PTA) and Rate Transient Analysis (RTA)
problems. A renewed interest in this procedure is inspired from the need to evaluate
production performances of wells in unconventional reservoirs. With this approach,
Numerical Calculation of Solid-Liquid two-Phase Flow Inside a Small Sewage Pumptheijes
Based on a mixture multiphase flow model,theRNG k–εturbulencemodelandfrozen rotor method were used to perform a numerical simulation of steady flow in the internal flow field of a sewage pump that transports solid and liquid phase flows. Resultsof the study indicate that the degree of wear on the front and the back of the blade suction surface from different densities of solid particles shows a completely opposite influencing trend. With the increase of delivered solid-phase density, the isobaric equilibrium position moves to the leading edge point of the blade, but the solid-phase isoconcentration point on the blade pressure surface and suction surface basically remains unchanged. The difference between hydraulic lift and water lift indelivering solid- and liquid-phase flows shows a rising trend with the increase of working flow
The Krylov-TPWL Method of Accelerating Reservoir Numerical Simulationinventionjournals
Because of the large number of system unknowns, reservoir simulation of realistic reservoir can be computationally demanding. Model order reduction (MOR) technique represents a promising approach for accelerating the simulations. In this work, we focus on the application of a MOR technique called Krylov trajectory piecewise-linear (Krylov-TPWL). First, the nonlinear system is represented as a weighted combined piecewise linear system using TPWL method, and then reducing order of each linear model using Krylov subspace. We apply Krylov-TPWL method for a two-phase (oil-water) reservoir model which is solved by full implicit. The example demonstrates that which can greatly reduce the dimension of reservoir model, so as to reduce the calculation time and improve the operation speed.
The primary objective of this project is to extend the conveniences of
deconvolution to non-linear problems of fluid flow in porous media. Unlike
conventional approaches, which are based on an approximate linearization of the
problem, here the solution of the non-linear problem is linearized by a perturbation
approach, which permits term-by-term application of deconvolution. Because the
proposed perturbation solution is more conveniently evaluated in the Laplacetransform
domain and the standard deconvolution algorithms are in the time-domain,
an efficient deconvolution procedure in the Laplace domain is a prerequisite.
For this research objective, a new algorithm is introduced which uses inverse
mirroring at the points of discontinuity and adaptive cubic splines to approximate rate
or pressure versus time data. This algorithm accurately transforms sampled data into
Laplace space and eliminates the Numerical inversion instabilities at discontinuities
or boundary points commonly encountered with the piece-wise linear approximations
of the data.
Applying the algorithm to the field data obtained from published works, we can
unveil the early-time behavior of a reservoir system masked by wellbore-storage
effects. The wellbore-storage coefficient can be variable in the general case. The new
method thus provides a powerful tool to improve pressure-transient-test
interpretation.
Practical use of the algorithm presented in this research has applications in a
variety of Pressure Transient Analysis (PTA) and Rate Transient Analysis (RTA)
problems. A renewed interest in this procedure is inspired from the need to evaluate
production performances of wells in unconventional reservoirs. With this approach,
CFD and Artificial Neural Networks Analysis of Plane Sudden Expansion FlowsCSCJournals
It has been clearly established that the reattachment length for laminar flow depends on two non-dimensional parameters, the Reynolds number and the expansion ratio, therefore in this work, an ANN model that predict reattachment positions for the expansion ratios of 2, 3 and 5 based on the above two parameters has been developed. The R2 values of the testing set output Xr1, Xr2, Xr3, and Xr4 were 0.9383, 0.8577, 0.997 and 0.999 respectively. These results indicate that the network model produced reattachment positions that were in close agreement with the actual values. When considering the reattachment length of plane sudden-expansions the judicious combination of CFD calculated solutions with ANN will result in a considerable saving in computing and turnaround time. Thus CFD can be used in the first instance to obtain reattachment lengths for a limited choice of Reynolds numbers and ANN will be used subsequently to predict the reattachment lengths for other intermediate Reynolds number values. The CFD calculations concern unsteady laminar flow through a plane sudden expansion and are performed using a commercial CFD code STAR-CD while the training process of the corresponding ANN model was performed using the NeuroShellTM simulator.
This paper presents the further developments and working principle of the speed-variable switched differential pump (SvSDP) concept proposed, designed and produced in [1]. The SvSDP system is designed to remove the throttling losses associated with typical valve driven control (VDC) systems. The hydraulic and mechanical system is modelled and linearised. The linearisation point is studied to provide an usable basis for controller design. It is proposed, in this paper, to model the converter and motor using a black box approach, where designed and informative input sequences are used to estimate the mathematical behaviour of the electrical drive based on the equivalent output data. The complete non linear model is verified against available trajectory data from the physical system, obtained from [1]. The linear model is analysed through a relative gain array (RGA) analysis to map the input output couplings present in the system. The results show that the system includes heavy cross-couplings. Results presented in [1] indicate, that it is possible to utilise a input output compensated decoupling to redefine the MIMO system into multiple SISO systems. The SvSDP concept is over-determined in relation to the amount of control inputs compared to possible outputs.
It is proposed in [1] to introduce two new input states and two new output states. The decoupling approach has been investigated in this paper. The decoupling results provided a basis of using decentralised control. The linear control strategies are designed independently based on the notion of decoupling. The first controller is related to the level flow, designed to maintain a desired minimum pressure level. The second load flow controller is related to the cylinder motion. The controller results indicate, that it is possible to achieve a good dynamic tracking performance with an error of maximum 0.5 mm for a given position trajectory.
This paper is also considering the energy consumption issues stated in [1], where two conceptual solutions are proposed, to solve the power loss associated with holding a load at a constant cylinder position. This paper is written as the product of an appendix report describing the whole project.
Recent developments in the field of reduced order modeling - and in particular, active subspace construction - have made it possible to efficiently approximate complex models by constructing low-order response surfaces based upon a small subspace of the original high dimensional parameter space. These methods rely upon the fact that the response tends to vary more prominently in a few dominant directions defined by linear combinations of the original inputs, allowing for a rotation of the coordinate axis and a consequent transformation of the parameters. In this talk, we discuss a gradient free active subspace algorithm that is feasible for high dimensional parameter spaces where finite-difference techniques are impractical. We illustrate an initialized gradient-free active subspace algorithm for a neutronics example implemented with SCALE6.1.
An artificial intelligence based improved classification of two-phase flow patte...ISA Interchange
Flow pattern recognition is necessary to select design equations for finding operating details of the process and to perform computational simulations. Visual image processing can be used to automate the interpretation of patterns in two-phase flow. In this paper, an attempt has been made to improve the classification accuracy of the flow pattern of gas/ liquid two- phase flow using fuzzy logic and Support Vector Machine (SVM) with Principal Component Analysis (PCA). The videos of six different types of flow patterns namely, annular flow, bubble flow, churn flow, plug flow, slug flow and stratified flow are re- corded for a period and converted to 2D images for processing. The textural and shape features extracted using image processing are applied as inputs to various classification schemes namely fuzzy logic, SVM and SVM with PCA in order to identify the type of flow pattern. The results obtained are compared and it is observed that SVM with features reduced using PCA gives the better classification accuracy and computationally less intensive than other two existing schemes. This study results cover industrial application needs including oil and gas and any other gas-liquid two-phase flows.
Study of Velocity and Pressure Distribution Characteristics Inside Of Catalyt...ijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Hardy cross method of pipe network analysissidrarashiddar
Hardy Cross Method of pipe network analysis has revolutionized the municipal water supply design. i.e., EPANET, a public domain software of water supply, uses the Hardy cross method for pipe network analysis. It is an iterative approach to estimate the flows within the pipe network where inflows (supply) and outflows (demand) with pipe characteristics are known.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
CFD and Artificial Neural Networks Analysis of Plane Sudden Expansion FlowsCSCJournals
It has been clearly established that the reattachment length for laminar flow depends on two non-dimensional parameters, the Reynolds number and the expansion ratio, therefore in this work, an ANN model that predict reattachment positions for the expansion ratios of 2, 3 and 5 based on the above two parameters has been developed. The R2 values of the testing set output Xr1, Xr2, Xr3, and Xr4 were 0.9383, 0.8577, 0.997 and 0.999 respectively. These results indicate that the network model produced reattachment positions that were in close agreement with the actual values. When considering the reattachment length of plane sudden-expansions the judicious combination of CFD calculated solutions with ANN will result in a considerable saving in computing and turnaround time. Thus CFD can be used in the first instance to obtain reattachment lengths for a limited choice of Reynolds numbers and ANN will be used subsequently to predict the reattachment lengths for other intermediate Reynolds number values. The CFD calculations concern unsteady laminar flow through a plane sudden expansion and are performed using a commercial CFD code STAR-CD while the training process of the corresponding ANN model was performed using the NeuroShellTM simulator.
This paper presents the further developments and working principle of the speed-variable switched differential pump (SvSDP) concept proposed, designed and produced in [1]. The SvSDP system is designed to remove the throttling losses associated with typical valve driven control (VDC) systems. The hydraulic and mechanical system is modelled and linearised. The linearisation point is studied to provide an usable basis for controller design. It is proposed, in this paper, to model the converter and motor using a black box approach, where designed and informative input sequences are used to estimate the mathematical behaviour of the electrical drive based on the equivalent output data. The complete non linear model is verified against available trajectory data from the physical system, obtained from [1]. The linear model is analysed through a relative gain array (RGA) analysis to map the input output couplings present in the system. The results show that the system includes heavy cross-couplings. Results presented in [1] indicate, that it is possible to utilise a input output compensated decoupling to redefine the MIMO system into multiple SISO systems. The SvSDP concept is over-determined in relation to the amount of control inputs compared to possible outputs.
It is proposed in [1] to introduce two new input states and two new output states. The decoupling approach has been investigated in this paper. The decoupling results provided a basis of using decentralised control. The linear control strategies are designed independently based on the notion of decoupling. The first controller is related to the level flow, designed to maintain a desired minimum pressure level. The second load flow controller is related to the cylinder motion. The controller results indicate, that it is possible to achieve a good dynamic tracking performance with an error of maximum 0.5 mm for a given position trajectory.
This paper is also considering the energy consumption issues stated in [1], where two conceptual solutions are proposed, to solve the power loss associated with holding a load at a constant cylinder position. This paper is written as the product of an appendix report describing the whole project.
Recent developments in the field of reduced order modeling - and in particular, active subspace construction - have made it possible to efficiently approximate complex models by constructing low-order response surfaces based upon a small subspace of the original high dimensional parameter space. These methods rely upon the fact that the response tends to vary more prominently in a few dominant directions defined by linear combinations of the original inputs, allowing for a rotation of the coordinate axis and a consequent transformation of the parameters. In this talk, we discuss a gradient free active subspace algorithm that is feasible for high dimensional parameter spaces where finite-difference techniques are impractical. We illustrate an initialized gradient-free active subspace algorithm for a neutronics example implemented with SCALE6.1.
An artificial intelligence based improved classification of two-phase flow patte...ISA Interchange
Flow pattern recognition is necessary to select design equations for finding operating details of the process and to perform computational simulations. Visual image processing can be used to automate the interpretation of patterns in two-phase flow. In this paper, an attempt has been made to improve the classification accuracy of the flow pattern of gas/ liquid two- phase flow using fuzzy logic and Support Vector Machine (SVM) with Principal Component Analysis (PCA). The videos of six different types of flow patterns namely, annular flow, bubble flow, churn flow, plug flow, slug flow and stratified flow are re- corded for a period and converted to 2D images for processing. The textural and shape features extracted using image processing are applied as inputs to various classification schemes namely fuzzy logic, SVM and SVM with PCA in order to identify the type of flow pattern. The results obtained are compared and it is observed that SVM with features reduced using PCA gives the better classification accuracy and computationally less intensive than other two existing schemes. This study results cover industrial application needs including oil and gas and any other gas-liquid two-phase flows.
Study of Velocity and Pressure Distribution Characteristics Inside Of Catalyt...ijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Hardy cross method of pipe network analysissidrarashiddar
Hardy Cross Method of pipe network analysis has revolutionized the municipal water supply design. i.e., EPANET, a public domain software of water supply, uses the Hardy cross method for pipe network analysis. It is an iterative approach to estimate the flows within the pipe network where inflows (supply) and outflows (demand) with pipe characteristics are known.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
Optimization of Closure Law of Guide Vanes for an Operational Hydropower Plan...Dr. Amarjeet Singh
This paper addresses the optimization of twostage closure law of guide vanes in an operational
hydropower plant of Nepal. The mathematical model
has been established in commercial software Bentley
Hammer, whose correctness has been validated by
comparing the results with the data of experimental
load rejection test. The validated mathematical model
has been employed to find the parameters of optimum
closure pattern, which minimizes the non-linear
objective function of maximum water pressure and
maximum rotational speed of turbine.
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Evaluation of the Sensitivity of Seismic Inversion Algorithms to Different St...IJERA Editor
Seismic wavelet estimation is an important step in processing and analysis of seismic data. Inversion methods as Narrow-Band and theConstrained Sparse-Spike ones require information about it so that the inversion solution, once it is not a unique problem, may be restricted by comparing the real seismic trace with the synthetic generated by convolution of the estimated reflectivity and wavelet. Besides helping in seismic inversion, a good estimate of the wavelet enables an inverse filter with less uncertainty to be computed in the deconvolution step and while tying well logs, a better correlation between the seismic trace and well log can be achieved. Depending on the use or not of well log information, the methods of wavelet estimation can be divided into two classes: statistical and deterministic. This work aimed to test the sensitivity of acoustic post-stack seismic inversion algorithms to wavelets statistically estimated by two distinct methods
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
Split Second Analysis Covering High Pressure Gas Flow Dynamics At Pipe Outlet...AEIJjournal2
A detailed investigation covering piped gas flow characteristics in high pressure flow conditions. Such flow
analysis can be resolved using established mathematical equations known as the Fanno condition, which
usually cover steady state, or final flow conditions. However, in real life, such flow conditions are
transient, varying with time. This paper uses CFD analysis providing a split second “snapshot” at what
happens at the pipe outlet, and therefore, a closer understanding at what happens at the pipe’s outlet in
high pressure gas flow condition.
In this example air was selected for simulation purposes. In HVAC applications, such gas flow conditions
can occur in typical applications such as; air compressors releasing high pressure air through a pipe, or
compressor over pressure refrigerant gas being released into the atmosphere via a discharge pipe.
Investigation has shown that rather than a steady mass flow rate condition occurring at the pipe outlet,
calculated by the Fanno flow condition, a spiked increase in flow rate occurs at the beginning,and then
stabilizing after a few seconds, with relatively minor ripples in flow rate. Other observations were also
made and commented.
CFD results in mass flow rate were compared with the mathematically derived results, differences were
recorded. The CFD analysis showed how the k-omega turbulence model performed well, with the processor
stabilizing at an early stage.
Soft Computing Technique Based Enhancement of Transmission System Lodability ...IJERA Editor
Due to the growth of electricity demands and transactions in power markets, existing power networks need to be enhanced in order to increase their loadability. The problem of determining the best locations for network reinforcement can be formulated as a mixed discrete-continuous nonlinear optimization problem (MDCP). The complexity of the problem makes extensive simulations necessary and the computational requirement is high. This paper compares the effectiveness of Evolutionary Programming (EP) and an ordinal optimization (OO) technique is proposed in this paper to solve the MDCP involving two types of flexible ac transmission systems (FACTS) devices, namely static var compensator (SVC) and thyristor controlled series compensator (TCSC), for system loadability enhancement. In this approach, crude models are proposed to cope with the complexity of the problem and speed up the simulations with high alignment confidence. The test and Validation of the proposed algorithm are conducted on IEEE 14–bus system and 22-bus Indian system.Simulation results shows that the proposed models permit the use of OO-based approach for finding good enough solutions with less computational efforts.
A New Two-Dimensional Analytical Model of Small Geometry GaAs MESFETIJMERJOURNAL
ABSTRACT : In this paper, a simple and exact analytical model for Small Geometry GaAs MESFET is developed to determine the potential distribution along the channel of the device. The model is based on the exact solution of two-dimensional Poisson’s equation in the depletion region under the gate. Then the obtained model is used to study the channel potential and threshold voltage of the device. Using the analytical model, the effect of the device parameter and bias conditions on performance of the device is investigated. The obtained results are graphically exhibited and discussed. In order to verification of the analytical results, TCAD device simulator is used and good accordance is observed.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
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.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
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Author: Robbie Edward Sayers
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(C) 2024 Robbie E. Sayers
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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.
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TECHNICAL COMMUNICATION 89/302
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4. Abstract
A finite element method for acoustic cavity modal analysis is presented. The approach
utilizes the VAST finite element program eigensolution routines operating on equivalent
acoustic mass and stiffness matrices generated from a program ACOUSTIC. Both closed
and open cavities can be treated, within the assumption of rigid boundaries. Two ele-
ments have been implemented in ACOUSTIC: a 20-noded isoparametric brick element and
an 8-noded isoparametric axisymmetric element. The performance of the analysis system
has been evaluated via comparison to both analytical and experimental results, and good
agreement has been found.
Ce texte dfrit une m6thode i El6ments finis pour !'analyse modale de cavitis
acoustques. Cente approche exploite les routines de solutions roprlS du programme I
el6ments finis VAST, appliqu6es aux matrices de masse et de ngidit6 produites par le
programme ACOUSTIC. Cette mthode pernet de traiter des cavit6s ouvertes ou fermtes,
en se fondant sur lypothtse de limites rigides. Deux 6l6ments ont 6t6 implant6s dans
ACOUSTIC: un l61ment de brique isopamm6trique 1 20 noeuds et un dl(nent asym6brique
isoparam zrique 1 8 noeuds. Le performance du systbne d'analyse a 6t6 6valu6e par
comparaison aux prvsions analytiques et aux r6sultats expdrimentaux et on a constat6 une
bonne corrElation.
ii
5. Contents
Abstract II
Table of Contents iii
List of Figures Iv
1 Introduction 1
2 Theory 2
3 Verification and Applications 4
4 Conclusions 7
Appendix A - Program ACOUSTIC 14
References 15
Accession For
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-Aail a/or
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6. List of Figures
1 Quadratic brick and axisymmetric finite elements................... 8
2 Geometry and finite element model of the rectangular cavity. ........... 8
3 Three brick element models of the bottle and end-piece................ 9
4 Axisymmetric models of the bottle and end-piece equivalent to the solid models. 10
5 Axial distribution of pressure for the first three modes as determined with
axisymmetric Model 1 ..................................... 11
6 Experimental aet-up for determination of the higher cavity modes of the bottle. 11
7 Power spectrum obtained from blowing over the bottle lip........... .. 12
8 Frequency response curve obtained from the loudspeaker/bottle system. . . 12
9 Tank modelled with 20-noded brick elements...................... 13
iv
7. 1 Introduction
The prediction of natural modes of vibration of acoustic cavities is an area of interest to
those involved in acoustical design or measurement. Cavity resonances can lead to annoying
human environmental conditions, interfere with acoustical measurements, and can in some
cases produce significant acoustic radiation. A familiar example of the last phenomenon
is the Helmholtz resonance tonals produced by blowing over the open end of an empty
bottle. For some simple shapes, natural modes can be easily estimated from the linear
frequency-wavelength relation. For more complex geometry, or for non-homogeneous fluids,
numerical methods must be used for accurate modal prediction. The finite element method
is well-suited for the prediction of cavity modes, since the governing differential equation is
easily cast in a weighted residual form.
This communication presents a brief description of the development and verification of
a system for finite element prediction of acoustic cavity modes. Familiarity with the VAST I
finite element program and finite element methods in general is assumed.
The approach utilizes two programs, ACOUSTIC and VAST. The two programs have
been made as compatible as is practical at this time; the ACOUSTIC program reads VAST
format input files, and forms and assembles the element 'stiffness' and 'mass' matrices in
the format required for a natural frequency analysis using VAST. The VAST bandwidth
minimization routines have been imbedded in the ACOUSTIC program to increase the
overall solution efficiency.
Within the program ACOUSTIC, two common structural solid elements have been
adapted to model the acoustic mode problem: a brick element and an axisymmetric element.
The former allows accurate modelling of general geometries, while the axisymmetric element
is limited to domains meeting the requirement of circumferential invariance, in both the
geometry and the pressure distribution. The restriction on the pressure distribution could
be removed by the inclusion of higher order harmonics in the formulation.
A number of analyses have been undertaken to verify the implementation of the acoustic
modal analysis capability, several of which are reported here. Numerical predictions are
compared to analytical values for the case of a rectangular cavity, and to experimental
results for an empty bottle. The latter example also highlights the effectiveness of the
axisymmetric formulation for geometries which satisfy that restriction. As an illustration
of a more practical analysis, the natural freqencies of a complex tank are calculated.
The assumption ofacoustically hard boundaries inherent in this formulation is in general
quite realistic when the cavity fluid is air. For high density fluids such as water, this
assumption may not be as reliable, and work is progressing on the inclusion of the structural
compliance in the prediction of the natural modes, both of the structure and cavity. In the
limit of very low frequency, the added fluid mass approach can be utilized to obtain the
effect of the fluid on the structure. The current work thus represents a further step in the
8. development of a capability to solve the general fluid/structure interaction problem.
2 Theory
Linear cavity acoustics are governed by the simple wave equation which reduces to a
Helmholtz form for the spatial distribution of pressure,
V2 p+ ((1)'p = )
At acoustically hard boundaries, the normal pressure gradient vanishes,
Op(2L--= 0 (2)
an
A variational form of equation (1), suitable for finite element solution, can be obtained by
defining a weighted residual statement with a virtual pressure as a weight function and
integrating over the cavity volume,
[Vp O+ 6p] (3=0 3)
Integrating by parts the first term of equation (3) and using the boundary condition of
equation (2), the equivalent integral form becomes
6 f[(Vp)2 - ('0)2p] dV = 0 (4)
The first term involving the pressure gradient can be interpreted as a generalized stiffness
associated with the potential energy of the fluid. The second term is a generalized mass
term associated with the kinetic energy of the fluid.
Following a standard finite element development (see Reference 2), the pressure distribu-
tion in the cavity is approximated by a set of nodal values {p}i and interpolation functions
[Nji. Within volume elements, the pressure field is defined as
p = [Nj,{p), (5)
Within each element, the energy integrals of equation (4) become
f {p}r[k],{ dv (6)
f{pT[mJidp)i du (7)
2
9. with
[k], = [B]T[B]i, (8)
[M]j = uITlT[N] (9)
and
[B+l = [Jl-1 v .[N], (10)
where (J] is the Jacobian matrix relating local and global coordinate derivatives. The
program ACOUSTIC isused to form and assemble the individual element matrices into the
global stiffness and mass matrices [K] and [M] respectively. The VAST system is then used
to solve the eigenvalue problem
11K] - w[M]f{p) =0 (1)
To minimize roundoff error, the c2 factor in equation (9) is not explicitly used in the actual
element matrix formulation for homogeneous fluids. In its place, a non-dimensional sound
speed parameter (c1 .f/C)2 is used, in which the reference speed in input by the program
user. The eigenvalues calculated with the VAST program then must be multiplied by that
factor to obtain the actual frequency predictions. Note that this option is also applicable
to models which include fluid of variable density.
The two elements implemented to date in the ACOUSTIC program are a 20-noded
isoparametric brick element and an 8-noded isoparametric axisymmetric element, Figure 1.
Both of these elements allow quadratic variation of both geometry and pressure within an
element, providing quite a high degree of modelling versatility. The element connectivity
must follow that defined for these elements (types 2 and 15) in the VAST analysis system.
A 2T-point Gauss integration is used for the brick element; a 9-point surface integration
with explicit circumferential integration is used for the axisymmetric elements. The imple-
mentation of the 20-noded element and the assembly routine has drawn largely from the
routines developed for added fluid mass calculation within VAST.3
For completely closed cavities for which the boundary condition of equation (2) is en-
forced everywhere, the stiffness matrix [K] in equation (11) is singular. This singularity can
be expected, since the one boundary 'condition is insufficient to provide a unique solution to
the second order governing differential equation. To obtain natural modes of closed cavities,
the shifting technique must be invoked in the VAST decomposition module. The choice of
an appropriate shifting parameter is important from the point of view of both accuracy
and computational efficiency. Since the shift should be of the same order as the calculated
eigenvalues, it is suggested that an estimate of the fundamental frequency, in Hz, divided
by the reference sound speed, be used. This will in general be much lower than the default
value used in the VAST program.
3
10. To model openings or planes of antisymmetry in a cavity, the nodal pressures can be set
to zero by imposing boundary conditions. The imposition of one or more such constraints
will render the stiffness matrix non-singular, and shifting will not be required. Numerically,
these constraints are imposed by placing large stiffness terms on the diagonal term associated
with the node in the assembled stiffness matrix. While it is computationally more efficient
to remove these nodes from the assembled system, the modal displacement vectors produced
by VAST are then incomplete, and interpretation ofmodes becomes quite difficult. It should
be noted that acoustic stiffness terms are in general much smaller than structural stiffness
terms, and a lower spring stiffnem value can be used in VAST.
For detailed models, particularily those modelled with the brick elements, bandwidth
minimization can be very effective in reducing both storage and computing time in the
VAST frequency analysis. For this purpose, the bandwidth rinimization module from
the VAST program has been adapted for use in the ACOUSTIC program. This module
provides both the GPS and NSA algorithms for nodal reordering. The reordering remains
transparent to the user.
3 Verification and Applications
The performance of the 20-noded element was evaluated via comparison to results pre-
sented in Reference 3 for a rectangular cavity. An identical element was used in that study.
The geometry of the cavity is shown in Figure 2. A half-model was used, which provides
xz-plane symmetric modes for the closed cavity, and antisymmetric xz-plane modes when
the nodal pressures in that plane are set to zero. The analytical values, in Hz, can be
obtained from the relation
f ( +..)(Th!' + (12(12)
where I, m, n= 0,1,2... and subscripted variables are actual lengths. A comparison of the
results is presented in Table 1, for an 8 element model. In accordance with standard
practice, the mode description refers to the number of half-wavelengths in the x, y, and z
directions respectively. For the symmetric modes (m--O), the finite element results agree
exactly with those of Reference 3, but the antisymmetr'- modes in that reference were
apparently calculated for a slightly different geometry (y--0.132 m in place of 0.128 m).
The sound speed used in all analyses was 330 m/sec.
The results of this analysis indicate that the 20-noded element gives quite accurate
results for discretisations down to about 2 elements per wavelength for this simple geometry.
A second example considered in the application of the acoustic elements was the pre-
diction of the cavity modes of an empty bottle. For this case, experimentally determined
4
11. Mode Predicted Theory Ref 3
1,0,0 699 699 699
0,1,0 1294 1289 1255
2,0,0 1404 1398 1404
1,1,0 1471 1466 1437
0,0,1 1505 1500 1506
1,0,1 1660 1654 1660
Table 1: Comparison of numerically predicted and analytical frequencies of a rectangular
cavity.
natural frequencies could be compared to preditions from the finite element analysis. Vari-
ous discretization levels were employed, and an exterior endpiece was added to all but one
model, as an approximation to the open end." As suggested earlier, an alternative to the
endpiece is simply to define the pressure at the opening to be zero, and it was part of the
intent of this study to determine if that model would provide sufficient accuracy.
The results obtained from the brick models indicated that all of the lowest modes of
interest were in fact axisymmetric, which could probably be anticipated given the axisym-
metric nature of the volume. A number of axisymmetric models were thus developed, and
the results of those analyses compared to the brick model results. Three of the brick mod-
els are shown in Figure 3. The corresponding axisymmetric models are shown in Figure 4.
Model 1 represents the actual bottle volume. In the analyses, the end nodes in this model
were assigned zero pressure.
A summary of the results of the numerical analyses is presented in Table 2. The first
line for each model refers to the brick element model, the second line to the axisymmetric
model. Included also is the number of elements and nodes in the model and the computer
time demands for both the matrix formulation (program ACOUSTIC) and total analysis
time for three modes. As is evident, all of the models produce essentially the same results,
although the axisymmetric analyses are an order of magnitude more efficient than those
which utilize the brick element. The addition of the endpiece does little if anything to
improve the overall accuracy of the modal predictions. Other geometries tested during
the course of the program development showed similar results. The analysis times are for
bandwidth-minimized models. The axial pressure distributions for the first 3 modes are
shown in Figure 5; these were obtained using the axisymmetric model of only the bottle
volume (Model 1).
To obtain results for comparison, two experiments were carried out using the same
5
12. Model Frequencies (Hz) Elems Nodes CPU (sec)
F1 F2 F3 [K],[M] Total
Model 1 212 1361 2080 54 296 121 290
212 1357 2078 6 33 4 12
Model 2 209 1353 2050 93 488 248 645
209 1350 2054 9 59 5 17
Model 3 209 1354 2053 119 608 411 1076
1209 1351 2056 11 75 5 20
Exp 202 1381 2123 1 1 1 d
Table 2: Comparison of predicted and experimentally obtained natural frequencies of an
empty bottle.
bottle as was modelled. First, the fundamental cavity mode was determined by measuring
the predominant frequencies emmitted when air was blown over the open end. To excite
the higher cavity modes, the set-up shown in Figure 6 was used. A typical power spectrum
obtained by blowing over the bottle is shown in Figure 7. Note the strong harmonics of
the fundamental frequency, approximately 202 Hz, which are present in the signature. A
portion of the frequency response plot obtained from the loudspeader set-up is shown in
Figure 8. Note that the scale on the frequency axis is not the same as for Figure 7. The
first mode identified using the loudspeaker system was the Helmholtz fundamental obtained
in the first test. The experimentally obtained results are given in the final line of Table 2.
The numerical results compare well with those measured experimentally. The largest
discrepancy of about 4 percent occurs for the lowest mode. The exact sound speed was
not determined during the experiment, and it is probable that some of the discrepancy is
due to the potential difference in that value, which was again taken as 330 m/sec in the
numerical calculations. The harmonics observed in the spectrum obtained from blowing
over the bottle do not appear in the results of the second experiment. This is apparently a
result of the different forms of excitation mechanisms in the two tests. Blowing air over the
bottle leads to a nonlinear interaction of shear layer instability and cavity oscillation s- ,
which can support harmonics of the fundamental which are not true cavity modes. The
excitation of the cavity mode by the loudspeaker is a linear process, and only true cavity
modes should be identified through this approach, within the limits of the experiment.
As a final example, the acoustic analysis system has been utilized to predict the cavity
modes of a large tank of complex geometry. The finite element model, with hidden lines
removed, is shown in Figure 9. The model contains 172 20-noded elements, and a total of
1220 nodes. The three circular structures in the model represent internal detail in the main
6
13. tank.
The results of three acoustic modal analyses of this model are summarized in Table 3.
The initial analysis was of the model, filled with water, with no openings. The effect of
including surface openings and the introduction of a small volume of air into the model
were then investigated. The natural frequency predictions have been non-dimensionalized
against the fundamental mode prediction for the closed model.
Mode Model
Closed Open Open with air
1 1.000 0.557 0.528
2 1.165 1.116 1.104
3 1.671 1.635 1.420
4 2.112 2.076 1.945
5 2.315 2.208 2.064
Table 3: Non-dimensionalized frequency predictions for the tank model.
These results show quite a wide separation of frequencies for the tank. The introduction
of the surface openings appears to have a large effect on the fundamental frequency, while
the addition of air into the model also decreases the fundamental frequency, but to a much
lesser extent.
4 Conclusions
A capability for predicting acoustic cavity modes by finite element analysis has been
implemented. The two elements developed provide both a general purpose and axisym-
metric modelling capability. The element formulation, bandwidth reduction and assembly
algorithms have been included in the program ACOUSTIC, which reads and writes data
files in a format compatible with the VAST finite element analysis system. Several sample
verification analyses have been reported on here, and the system appears to perform its
intended function.
7
14. Lz
128 DIMENSIONS IN MM
Y
Figure 2: Geometry and finite element model of the rectangular cavity.
15. MODEL 1
MODEL 2
MODEL 3
Figure 3: Three brick element models of the bottle and end-piece.
9
17. MODE 1 (212 HZ)
MODE 2 (1361 HZ)
MODE 3 (2080 HZ)
Figure 5: Axial distribution of pressure for the first three modes as determined with ax-
isymmetric Model 1.
MICROPHONE
SPEAKER
FREQUENCY RESPONSE
FREQUENCY SOURCE
Figure 6: Experimental set-up for determination of the higher cavity modes of the bottle.
11
18. dBV Xt 200 Hz
10
dB
/DIV
-61
0 500 Hz 1000
Figure 7: Power spectrum obtained from blowing over the bottle lip.
2.5
/Div
112
....
, I I
100 300 Hz 500
Figure 8: Freqency reponse curve obtained from the loudspeaker/bottle system.
12
19. Figure 9: Tank modelled with 20-noded brick elements.
13
20. Appendix A - Program ACOUSTIC
Following the standard VAST format, the program ACOUSTIC requires the following
input files:
1. PREFX.USE
2. PREFX.GOM
The format of then files can be found in Reference 1. The .USE file must contain a
master control line, the IELEMA header and control line, and the IBANRD header and
control line, in this order. Since it is likely that a natural frequency analysis will be carried
out using the same .USE file, that file ends up looking essentially the same as would be
required for a complete VAST run. Note that substructuring of the model is not currently
permitted.
For the ACOUSTIC program, the master control line switches IELEMS, IBANRD,
IASSEM, and ISTIFM must be set. The program will restart at either bandwidth min-
imization or matrix assembly. For assembly (IASSEM = 1), the master control variable
ISTIFM is used to control the form of the assembled stiffness matrix. If ISTIFM = 1,
ACOUSTIC produces a PREFX.T46 file, anticipating that boundary conditions are to be
imposed. If ISTIFM = 0, ACOUSTIC produces the PREFX.T48 file directly, which will
then require shifting to decompose. If boundary conditions are required, they are put under
the ISTIFM section in the .USE file (or boundary condition file) and VAST is used to create
the PREFX.T48 file.
There is one small difference in the GOM file used by the ACOUSTIC program; the
element group sound speed, in length units/second, replaces the material modulus specifi-
cation. The Poison's ratio and material density are not required. As is the case for material
property variations, elements with different sound speeds must be in different groups. The
element type codes are the same as for the VAST structural elements; IEC = 2 and IEC =
15 for the 20-noded and axisymmetric elements respectively.
The output files created by ACOUSTIC are as follows:
1. PREFX.LPA - formatted output data file
2. PREFX.T31 - element mass matrices
3. PREFX.T32 - element stiffness matrices
4. PREFX.T41 - geometry control file for VAST compatibility
5. PREFX.T45 - bandwidth reduction mapping file
14
21. 6. PREFX.T46 - assembled stiffnes matrix before boundary conditions are imposed
7. PREFX.T48 - assembled stiffness matrix
8. PREFX.T49 - assembled mam matrix
All files are binary with the exception of the .LPA file, which resembles the .LPT file
produced by VAST. The print control parameters in the geometry and bandwidth control
lines in the .USE file are active. The .T41 file produced by ACOUSTIC does not contain
the geometric data as in the case for files created by VAST. Unfortunately, this data is
required for the eigenvector plotting routines of the VASTG 9 postprocessor. In general,
plotting of the pressure eigenvectors is not particularily useful, since they are interpreted
as x-axis displacement in VASTG. If plotting is desired, the element formulation section
of VAST must be executed to obtain a correct .T41 file.
The only input required by ACOUSTIC is the 5 character file prefix variable PREFX
and the reference sound speed. The program offers a list of defaults for the latter; a choice
can be made depending on the units of the geometry. For models which include fluids of
different density, a median value of reference sound speed is recommended. The value is
echoed in the .LPA file, for future reference in factoring the calculated eigenvalues. The
program can also be run in tatch mode.
References
1. 'Vibration and Strength Analysis Program (VAST): User's Manual Version 04', DREA
Contract Report CR/86/429, Martec Ltd., Halifax, Canada, 1986.
2. Petyt, M., et al, 'A Finite Element Method for Determining the Acoustic Modes
of Irregular Shaped Cavities', Journal of Sound and Vibration, Vol. 45, 1976, pp.
495-502.
3. Norwood, M., 'Application of the Finite Element Method for the Calculation of the
Hydrodynamic Added Mass of Marine Propeller Blades', DREA Informal Communi-
cation, 1977.
4. Bernhard, R.J., Holger, D.K., 'Finite Element Analysis of Helmholtz Resonators Using
Substructuring', in Finite Element Applications in Acoustics, Eds. Kamal, M.M.,
Wolf, J.A., A.S.M.E. Vibrations Conference, 1981.
5. Blake, W.K., Mechanics of Flow-Induced Sound and Vibration, Volume 1, Academic
Press, 1986.
15
22. 6. Cummings, A., 'Acoustics of a Cider Bottle', Applied Acoustics, Vol. 5, 1972, pp.16 1-
170.
7. Cummings, A., 'Acoustics of a Wine Bottle', Journal of Sound and Vibration, Vol.
31, 1973, pp. 331-343.
8. Howe, M.S., 'On the Helmholtz Resonator', Journal of Sound and Vibration, Vol. 45,
1976, pp.427-440.
9. 'Vibration and Strength Analysis Graphics Program VAST-G User's Manual', DREA
Contract Report CR/86/430, Martec Ltd., Halifax, Canada, 1986.
16
23. UNCLOLSSIFIED
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P11 talm elaion f This. MArot. Koywordg)
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(Seaniy uleftlton of tle. body of sbtmc and Indexing arawastlen must be omrod"whnme eversi" decumet Is dmlled)
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Defence Research Establishment Atlantic IUnclassified
3TITLE (the omplsW docum tWe as Irilonad on goe til page. he eleloto hd be babded by Nosappftu abuwmton
MCRor U) Inperonhese ate fte tie.)
Prediction of Acoustic Cavity Modes by Finite Element Methods
4. AUTHORS (LM newsn. fim nmm. ud i bstLe I wuary, she ran. eo4 Owe Me4 MinuL.)
Vernon, Thomas A. and Tang S.
DATE OF PtUBLCATION (mont and yar of puaticalon df docmient) NO. OF PAGES (tOWed st Ob. NO. OF REFS 0t" eked in
kduemalbn Minis Aroues dowat)
March 1989 Appenluee. aft-) 22 19
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Technical Communication
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Defence Research Establishment Atlantic
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writsn. Ploss specify whether projest Or grant)
lAH
IDS. ORIGIATORSI DOCUMENT NUMER (me ofi'd ~ jobme.OTHER DOCUMENT NOS. (any othe numbers wfddi may be
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aidly. This number mus be unlqei 10 t doument) sos
Drea Technical Communication 89/302
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3 Dlstdoln Ilmi to defnI depertmntsenwdefense esfors; fuwher lbuton only - apppoe
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UNCLASS IFIlED
IECLRNMCLmD~ATIN OF FOR
17
24. Unclassified
SECRITY CLASSIFICATIONi OPPOAM
11. ABSTRACT (obiefad fagigl amy Iofst dussusat k =Wdoelqe swlumipe asf bedy o1 fte desm kgi. " a hill*
desirable asog dMens of gloifieadietwei be usiat Wu poopop of e*obstruct sholSbeg. whito usdswee of ft
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~-?--A finite element method for acoustic cavity modal analysis is
presented. The approach utilizes the VAST finite element program eigen-
solution routines operating on equivalent acoustic mass and stiffness
matrices generated from the program ACOUSTIC. Both closed and open
cavities can be treated, within the assumption of rigid boundaries. Two
elements have been implemented in ACOUSTIC: a 20-noded isoparametric brick
element and an 8-noded isoparametric auisyummetric element. The perform-
ance of the analysis system has been evaluated via comparison to both
analytical and experimental results, and good agreement has been found.
14. KEYWORDS. DESCRIPTORS or WETPERS bs'egIVi mauwo otrsdon pbrmSwitass oasr a decmtsaffWA w be
helipfu inm ieguin as deussa They souiM be seoce* as nemwisy gloesiinvasi requsa, tatM afLes uchn e~imIs
medal# dosluuui. Ira* Naie. axisgy Project cede sus. Pegrsb loousmaway OWabe uchdad If possible kerawrds iSasid bealce
frust apublihed Owasssm 9.& Thesmus of Emguorma w4nScoawific Tarsu lST) awl Owmatsomro-idssmlied If is aeso pessible 0
aelac uima Sims whocwe tinchlmessed.a ctuailimm of SuchsOAbebe radcmad a wis as title.)
,Finite elements
Frequency analysis
Vibration
Cavities
Modal analysis
Acoustics
Numerical methods
Eigenvalues
Natural modes
Unclassified
INICUITY CLASSSPICATION OFPOWM
18