The penalty and hybrid methods are being much used in dealing with the general incompatible element, With the penalty method convergence can always be assured, but comparatively speaking its accuracy is lower, and the...The penalty and hybrid methods are being much used in dealing with the general incompatible element, With the penalty method convergence can always be assured, but comparatively speaking its accuracy is lower, and the condition number and sparsity are not so good. With the hybrid method, convergence can be assured only when the rank condition is satisfied. So the construction of the element is extremely limited. This paper presents the mixed hybrid penalty element method, which combines the two methods together. And it is proved theoretically that this new method is convergent, and it has the same accuracy, condition number and sparsity as the compatible element. That is to say, they are optimal to each other.Finally, a new triangle element for plate bending with nine freedom degrees is constructed with this method (three degreesof freedom are given on each corner -- one displacement and tworotations), the calculating formula of the element stiffness matrix is almost the same as that of the old triangle element for plate bending with nine degrees of freedom But it is converged to true solution with arbitrary irregrlar triangle subdivision. If the true solution u?H3 with this method the linear and quadratic rates of convergence are obtianed for three bending moments and for the displacement and two rotations respectively.展开更多
There is a common difficulty in elastic-plastic impact codes such as EPIC[2,3] NONSAP[4], etc.. Most of these codes use the simple linear functions usually taken from static problem to represent the displacement compo...There is a common difficulty in elastic-plastic impact codes such as EPIC[2,3] NONSAP[4], etc.. Most of these codes use the simple linear functions usually taken from static problem to represent the displacement components. In such finite element formulation, the stress components are constant in each element and they are discontinuous in any two neighboring elements. Therefore, the bases of using the virtual work principle in such elements are unreliable. In this paper, we introduce a new method, namely, the compatible stress iterative method, to eliminate the above-said difficulty. The calculated examples show that the calculation using the new method in dynamic finite element analysis of high velocity impact is valid and stable, and the element stiffness can be somewhat reduced.展开更多
The Finite Element Method of Lines （FEMOL） is a semi-analytic approach and takes a position between FEM and analytic methods. First, FEMOL in Fracture Mechanics is presented in detail. Then, the method is applied to...The Finite Element Method of Lines （FEMOL） is a semi-analytic approach and takes a position between FEM and analytic methods. First, FEMOL in Fracture Mechanics is presented in detail. Then, the method is applied to a set of examples such as edge-crack plate, the central-crack plate, the plate with cracks emanating from a hole under tensile or under combination loads of tensile and bending. Their dimensionless stress distribution, the stress intensify factor （SIF） and crack opening displacement （COD） are obtained, and comparison with known solutions by other methods are reported. It is found that a good accuracy is achieved by FEMOL. The method is successfully modified to remarkably increase the accuracy and reduce convergence difficulties. So it is a very useful and new tool in studying fracture mechanics problems.展开更多
The paper describes an approach to teaching mutually-coupled circuits CAD techniques to undergraduate students pursuing a degree course in electrical engineering or physics, and explains how a series of simulated expe...The paper describes an approach to teaching mutually-coupled circuits CAD techniques to undergraduate students pursuing a degree course in electrical engineering or physics, and explains how a series of simulated experiments may be incorporated into the existing subjects. The simulated experiments make use of a two-dimensional open-access software based on the finite-element method. At the laboratory meetings, the students learn how to set up field problems for solution, and how to examine the results. Simulation tasks based on three axisymmetric open-boundary problems are used to introduce different numeric techniques to compute inductance and magnetic forces. The paper takes the reader to a step-by-step simulation journey, and provides all the basic elements required for further exploration of axially-symmetric systems.展开更多
Many problems with underlying variational structure involve a coupling of volume with surface effects.A straight-forward approach in a finite element discretiza- tion is to make use of the surface triangulation that i...Many problems with underlying variational structure involve a coupling of volume with surface effects.A straight-forward approach in a finite element discretiza- tion is to make use of the surface triangulation that is naturally induced by the volume triangulation.In an adaptive method one wants to facilitate'matching'local mesh modifications,i.e.,local refinement and/or coarsening,of volume and surface mesh with standard tools such that the surface grid is always induced by the volume grid. We describe the concepts behind this approach for bisectional refinement and describe new tools incorporated in the finite element toolbox ALBERTA.We also present several important applications of the mesh coupling.展开更多
Permanent magnets with high energy products are widely used in a variety of electromagnetic devices. Such devices can be found in marine, aerospace, and robotic applications which require the minimization of weight an...Permanent magnets with high energy products are widely used in a variety of electromagnetic devices. Such devices can be found in marine, aerospace, and robotic applications which require the minimization of weight and volume of the electromagnetic device. During the magnetizing process, the magnet may not be perfectly magnetized. Therefore, it needs to be demagnetized. Because of high coercivity of some permanent magnets, the demagnetization process requires the intense magnetic fields in close proximity with the magnetic material. The fields must be produced for a short period of time （millisecond range） and they also must be bidirectional in order to overcome the coercivity of magnetic material. Different parameters have been known to affect the demagnetizer operation such as the core shape, core material, turn number, cross section of air gap, magnet type and so on. The amplitude and waveform of the fixture current is obtained from PSPICE simulations and also from experimental measurement. A 2D finite element analysis is developed to simulate the magnetic fields and the mechanical forces. In this paper, the main parameters affecting the optimal design of the demagnetizer are discussed. Simulation results show that the core structure, air gap width, and turn numbers are the most important parameters when designing such a device.展开更多
The paper starts with a brief overview to the necessity of sheet metal forming simulation and the complexity of automobile panel forming, then leads to finite element analysis (FEA) which is a powerful simulation too...The paper starts with a brief overview to the necessity of sheet metal forming simulation and the complexity of automobile panel forming, then leads to finite element analysis (FEA) which is a powerful simulation tool for analyzing complex three-dimensional sheet metal forming problems. The theory and features of the dynamic explicit finite element methods are introduced and the available various commercial finite element method codes used for sheet metal forming simulation in the world are discussed,and the civil and international status quo of automobile panel simulation as well. The front door outer panel of one certain new automobile is regarded as one example that the dynamic explicit FEM code Dynaform is used for the simulation of the front door outer panel forming process. Process defects such as ruptures are predicted. The improving methods can be given according to the simulation results. Foreground of sheet metal forming simulation is outlined.展开更多
With the development of aeronautic and astronautic techniques, radiation becomes much more significant while the structure is exposed to the higher and higher temperature. Most of the current finite element software p...With the development of aeronautic and astronautic techniques, radiation becomes much more significant while the structure is exposed to the higher and higher temperature. Most of the current finite element software packages treat it using the net-radiation method or absorbed radiation method based on the assumption of isothermal surface with uniform radiation heat flux, which brings the conflict between the precision and the quantity of grids. Using integral method to compute the variable radiation heat flux in higher-order finite element, the precision can be improved greatly while using the same quantity of grids, because it is more consistent with the distribution of real temperature. In this paper, the integral is only processed on the same integral points as those used for solving the finite element equations, so it may be of high efficiency. In an academic testing model, the result is contrast to which get in ANSYS, proving the high precision of the method. Then an actual sandwich panel used in the thermal protection system is analyzed with the method, and the error is comparatively low to the analytical answer while the computation being of high efficiency.展开更多
The dynamic behaviour of power line cables have been a source of interest to researchers ever since the phenomenon was first noticed in the 1920s. Conductor oscillation is mostly caused by the dynamic forces of nature...The dynamic behaviour of power line cables have been a source of interest to researchers ever since the phenomenon was first noticed in the 1920s. Conductor oscillation is mostly caused by the dynamic forces of nature such as wind loading. This imposes a periodic force on the conductors which is highly undesirable. It is therefore important for engineers to account for the possible effect of the wind loading when designing the power line. Investigations have shown that modeling the exact dynamic behaviour of a conductor is very difficult. Based on this fact, getting the exact analytical solution to conductor vibration is difficult, which is almost impossible, hence the numerical approximation becomes an option. This paper presents the developed finite element method used to analyse the dynamic behaviour of transmission line conductors. The developed FEM （finite element method） is implemented on MATLAB. The numerical analysis using MATLAB that is presented in this paper is used to simulate the response of the conductor when subjected to external loading in the time domain. The simulation is used to analyse the transverse vibration of the conductor. The formulation of the stiffness matrix and load vector is done and the results obtained are used to evaluate the conductor＇s internal energy dissipation. This finite element solution is compared with the results documented in literature. This numerical simulation is also used to investigate the effects of varying the axial tension on energy dissipation within the strands. Hence, this evolved in physically appropriate energy characterization process that can be used to evaluate the conductor self-damping with respect to line contact.展开更多
The FORTRAN programming language was used in early days of writing finite element field computation to write programs Much of those codes were developed in an ad hoc way. Today, modem software developers face problems...The FORTRAN programming language was used in early days of writing finite element field computation to write programs Much of those codes were developed in an ad hoc way. Today, modem software developers face problems in understanding, modifying and utilizing those codes. As modem software engineers are very concerned with object oriented design, if those codes are converted into an object oriented language, they could be redesigned and deployed in an object oriented system. Those legacy codes often need to be converted not only into an object oriented programming language such as Java but also into functional oriented languages such as C. Conversion of those legacy codes into modem languages gives many advantages. The purpose of this paper is to compare the performances of such converted legacy finite element codes originally written in FORTRAN, the relevant converted C program and Java program. Sample finite element programs written in FORTRAN are converted for purposes of comparison into modem languages such as C and Java. Performances are compared based on the execution time. In addition to that, the memory sizes of the execution file of FORTRAN and C programs are also compared. Java being interpretive there is no execution file to compare.展开更多
This paper describes the research undertaken on the strength and stiffness of fluctuation on the wall thickness of steel beverage cans using the Monte Carlo stochastic finite element method. Sample distributions were...This paper describes the research undertaken on the strength and stiffness of fluctuation on the wall thickness of steel beverage cans using the Monte Carlo stochastic finite element method. Sample distributions were firstly assumed and then proven using the data observations of the wall thickness, the APDL language was then applied, and the stresses and displacements of the can were calculated by using the ANSYS software. It is concluded that the structural reliability of a steel making beverage can be estimated accurately.展开更多
The electric field intensity (EFI) is important characteristic quantity for evaluating the internal insulation state of cable joints. Based on finite element method, this paper proposes two EFI research methods, field...The electric field intensity (EFI) is important characteristic quantity for evaluating the internal insulation state of cable joints. Based on finite element method, this paper proposes two EFI research methods, field-circuit coupling method and equivalent circuit method. The average EFI of the inner surface of the outer semi-conducting shield can be calculated from the current in the measuring circuit. The relative error between these two methods is about 15%, which roughly proves the consistency of the two methods. Further practical application research enables online monitoring of cable joints.展开更多
Building a reasonable and accurate finite element model is the first and critical step for structural analysis of complicated bridge. In this article, modeling assistant for continuous suspension with multi-pylon is d...Building a reasonable and accurate finite element model is the first and critical step for structural analysis of complicated bridge. In this article, modeling assistant for continuous suspension with multi-pylon is developed based on .Net platform, with VB.Net, C# language and OpenGL graphic technique. With parameterized modeling method, finite element model of this kind of bridge can be built quickly and accurately, and multi-type element modeling with uniform parameters is realized. With advanced graphic technique, three-dimensional model graph can be real-timely previewed for intuitive data check. With an example of practice project, the accuracy and feasibility of this modeling method and practicality of this software are verified.展开更多
The maximum internal tangential stress is a critical parameter for the design of the PDC （polycrystalline diamond compact） die that has been widely applied to offshore oil drilling. A new simple equation for the cal...The maximum internal tangential stress is a critical parameter for the design of the PDC （polycrystalline diamond compact） die that has been widely applied to offshore oil drilling. A new simple equation for the calculation of the stress is developed, and verified by the test data from Kingdream Corp. of China, the largest bit Company in China. An opti- mum method for the design of the PDC die is presented and demonstrated in detail, and software for the design and FEM analysis of the die is developed on the basis of the method. This software has been used in oil industry in recent years.展开更多
The purpose of this article is to develop a new methodology to evaluate the statistical characteristic of the response of structures subjecting to random excitation, by combining the Finite Element Method (FEM) with t...The purpose of this article is to develop a new methodology to evaluate the statistical characteristic of the response of structures subjecting to random excitation, by combining the Finite Element Method (FEM) with the Transforming Density Function (TDF). Uncertainty modeling of structure with random variables encourages the coupling of advanced TDF for reliability analysis to analyze problems of stochastic mechanical systems. The TDF is enthusiastically applicable in the situation where the relationship between input and output of structures is available in explicit analytical form. However, the situation is much more involved when it is necessary to perform the evaluation of implicit expression between input and output of structures through numerical models. For this aim, we propose a new technique that combines the FEM software, and the TDF method to evaluate the most important statistical parameter the Probability Density Function (PDF) of the response where the expression between input and output of structures is implicit. Once the PDF is evaluated, all other statistical parameters are derived easily. This technique is based on the numerical simulations of the FEM and the TDF by making a middleware between Finite Element software and Matlab. Some problems, range from simple to complex, of structures are analyzed using our proposed technique. Its accuracy is validated through Monte-Carlo simulation.展开更多
A simulation was carried out on an unsteady flow of a viscous, incompressible and electrically conducting fluid past an infinite vertical porous plate. A generic computer program using the Galerkin finite element meth...A simulation was carried out on an unsteady flow of a viscous, incompressible and electrically conducting fluid past an infinite vertical porous plate. A generic computer program using the Galerkin finite element method is employed to solve the coupled non-linear differential equations for velocity and temperature fields. The diffusion equation, the energy equation, the momentum equations and other relevant parameters are transformed into interpretable postfix codes. Numerical calculations are carried out on the flow fields both in the presence of cooling and heating of the plate by free convection currents. The effects of the dimensionless parameters, namely, the Prandtl number, the Eckert number, the modified Grashof number, the Schmidt number and the time on the temperature and velocity distributions are discussed.展开更多
文摘The penalty and hybrid methods are being much used in dealing with the general incompatible element, With the penalty method convergence can always be assured, but comparatively speaking its accuracy is lower, and the condition number and sparsity are not so good. With the hybrid method, convergence can be assured only when the rank condition is satisfied. So the construction of the element is extremely limited. This paper presents the mixed hybrid penalty element method, which combines the two methods together. And it is proved theoretically that this new method is convergent, and it has the same accuracy, condition number and sparsity as the compatible element. That is to say, they are optimal to each other.Finally, a new triangle element for plate bending with nine freedom degrees is constructed with this method (three degreesof freedom are given on each corner -- one displacement and tworotations), the calculating formula of the element stiffness matrix is almost the same as that of the old triangle element for plate bending with nine degrees of freedom But it is converged to true solution with arbitrary irregrlar triangle subdivision. If the true solution u?H3 with this method the linear and quadratic rates of convergence are obtianed for three bending moments and for the displacement and two rotations respectively.
文摘There is a common difficulty in elastic-plastic impact codes such as EPIC[2,3] NONSAP[4], etc.. Most of these codes use the simple linear functions usually taken from static problem to represent the displacement components. In such finite element formulation, the stress components are constant in each element and they are discontinuous in any two neighboring elements. Therefore, the bases of using the virtual work principle in such elements are unreliable. In this paper, we introduce a new method, namely, the compatible stress iterative method, to eliminate the above-said difficulty. The calculated examples show that the calculation using the new method in dynamic finite element analysis of high velocity impact is valid and stable, and the element stiffness can be somewhat reduced.
文摘The Finite Element Method of Lines （FEMOL） is a semi-analytic approach and takes a position between FEM and analytic methods. First, FEMOL in Fracture Mechanics is presented in detail. Then, the method is applied to a set of examples such as edge-crack plate, the central-crack plate, the plate with cracks emanating from a hole under tensile or under combination loads of tensile and bending. Their dimensionless stress distribution, the stress intensify factor （SIF） and crack opening displacement （COD） are obtained, and comparison with known solutions by other methods are reported. It is found that a good accuracy is achieved by FEMOL. The method is successfully modified to remarkably increase the accuracy and reduce convergence difficulties. So it is a very useful and new tool in studying fracture mechanics problems.
基金the Brazilian National Council of Technological and Scientific Development(CNPq)and Brazilian Federal Agency for Postgraduate Studies(CAPES).
文摘The paper describes an approach to teaching mutually-coupled circuits CAD techniques to undergraduate students pursuing a degree course in electrical engineering or physics, and explains how a series of simulated experiments may be incorporated into the existing subjects. The simulated experiments make use of a two-dimensional open-access software based on the finite-element method. At the laboratory meetings, the students learn how to set up field problems for solution, and how to examine the results. Simulation tasks based on three axisymmetric open-boundary problems are used to introduce different numeric techniques to compute inductance and magnetic forces. The paper takes the reader to a step-by-step simulation journey, and provides all the basic elements required for further exploration of axially-symmetric systems.
文摘Many problems with underlying variational structure involve a coupling of volume with surface effects.A straight-forward approach in a finite element discretiza- tion is to make use of the surface triangulation that is naturally induced by the volume triangulation.In an adaptive method one wants to facilitate'matching'local mesh modifications,i.e.,local refinement and/or coarsening,of volume and surface mesh with standard tools such that the surface grid is always induced by the volume grid. We describe the concepts behind this approach for bisectional refinement and describe new tools incorporated in the finite element toolbox ALBERTA.We also present several important applications of the mesh coupling.
文摘Permanent magnets with high energy products are widely used in a variety of electromagnetic devices. Such devices can be found in marine, aerospace, and robotic applications which require the minimization of weight and volume of the electromagnetic device. During the magnetizing process, the magnet may not be perfectly magnetized. Therefore, it needs to be demagnetized. Because of high coercivity of some permanent magnets, the demagnetization process requires the intense magnetic fields in close proximity with the magnetic material. The fields must be produced for a short period of time （millisecond range） and they also must be bidirectional in order to overcome the coercivity of magnetic material. Different parameters have been known to affect the demagnetizer operation such as the core shape, core material, turn number, cross section of air gap, magnet type and so on. The amplitude and waveform of the fixture current is obtained from PSPICE simulations and also from experimental measurement. A 2D finite element analysis is developed to simulate the magnetic fields and the mechanical forces. In this paper, the main parameters affecting the optimal design of the demagnetizer are discussed. Simulation results show that the core structure, air gap width, and turn numbers are the most important parameters when designing such a device.
文摘The paper starts with a brief overview to the necessity of sheet metal forming simulation and the complexity of automobile panel forming, then leads to finite element analysis (FEA) which is a powerful simulation tool for analyzing complex three-dimensional sheet metal forming problems. The theory and features of the dynamic explicit finite element methods are introduced and the available various commercial finite element method codes used for sheet metal forming simulation in the world are discussed,and the civil and international status quo of automobile panel simulation as well. The front door outer panel of one certain new automobile is regarded as one example that the dynamic explicit FEM code Dynaform is used for the simulation of the front door outer panel forming process. Process defects such as ruptures are predicted. The improving methods can be given according to the simulation results. Foreground of sheet metal forming simulation is outlined.
文摘With the development of aeronautic and astronautic techniques, radiation becomes much more significant while the structure is exposed to the higher and higher temperature. Most of the current finite element software packages treat it using the net-radiation method or absorbed radiation method based on the assumption of isothermal surface with uniform radiation heat flux, which brings the conflict between the precision and the quantity of grids. Using integral method to compute the variable radiation heat flux in higher-order finite element, the precision can be improved greatly while using the same quantity of grids, because it is more consistent with the distribution of real temperature. In this paper, the integral is only processed on the same integral points as those used for solving the finite element equations, so it may be of high efficiency. In an academic testing model, the result is contrast to which get in ANSYS, proving the high precision of the method. Then an actual sandwich panel used in the thermal protection system is analyzed with the method, and the error is comparatively low to the analytical answer while the computation being of high efficiency.
文摘The dynamic behaviour of power line cables have been a source of interest to researchers ever since the phenomenon was first noticed in the 1920s. Conductor oscillation is mostly caused by the dynamic forces of nature such as wind loading. This imposes a periodic force on the conductors which is highly undesirable. It is therefore important for engineers to account for the possible effect of the wind loading when designing the power line. Investigations have shown that modeling the exact dynamic behaviour of a conductor is very difficult. Based on this fact, getting the exact analytical solution to conductor vibration is difficult, which is almost impossible, hence the numerical approximation becomes an option. This paper presents the developed finite element method used to analyse the dynamic behaviour of transmission line conductors. The developed FEM （finite element method） is implemented on MATLAB. The numerical analysis using MATLAB that is presented in this paper is used to simulate the response of the conductor when subjected to external loading in the time domain. The simulation is used to analyse the transverse vibration of the conductor. The formulation of the stiffness matrix and load vector is done and the results obtained are used to evaluate the conductor＇s internal energy dissipation. This finite element solution is compared with the results documented in literature. This numerical simulation is also used to investigate the effects of varying the axial tension on energy dissipation within the strands. Hence, this evolved in physically appropriate energy characterization process that can be used to evaluate the conductor self-damping with respect to line contact.
文摘The FORTRAN programming language was used in early days of writing finite element field computation to write programs Much of those codes were developed in an ad hoc way. Today, modem software developers face problems in understanding, modifying and utilizing those codes. As modem software engineers are very concerned with object oriented design, if those codes are converted into an object oriented language, they could be redesigned and deployed in an object oriented system. Those legacy codes often need to be converted not only into an object oriented programming language such as Java but also into functional oriented languages such as C. Conversion of those legacy codes into modem languages gives many advantages. The purpose of this paper is to compare the performances of such converted legacy finite element codes originally written in FORTRAN, the relevant converted C program and Java program. Sample finite element programs written in FORTRAN are converted for purposes of comparison into modem languages such as C and Java. Performances are compared based on the execution time. In addition to that, the memory sizes of the execution file of FORTRAN and C programs are also compared. Java being interpretive there is no execution file to compare.
文摘This paper describes the research undertaken on the strength and stiffness of fluctuation on the wall thickness of steel beverage cans using the Monte Carlo stochastic finite element method. Sample distributions were firstly assumed and then proven using the data observations of the wall thickness, the APDL language was then applied, and the stresses and displacements of the can were calculated by using the ANSYS software. It is concluded that the structural reliability of a steel making beverage can be estimated accurately.
文摘The electric field intensity (EFI) is important characteristic quantity for evaluating the internal insulation state of cable joints. Based on finite element method, this paper proposes two EFI research methods, field-circuit coupling method and equivalent circuit method. The average EFI of the inner surface of the outer semi-conducting shield can be calculated from the current in the measuring circuit. The relative error between these two methods is about 15%, which roughly proves the consistency of the two methods. Further practical application research enables online monitoring of cable joints.
基金National Science and Technology Support Program of China(No.2009BAG15B01)Key Programs for Science and Technology Development of Chinese Transportation Industry(No.2008-353-332-190)
文摘Building a reasonable and accurate finite element model is the first and critical step for structural analysis of complicated bridge. In this article, modeling assistant for continuous suspension with multi-pylon is developed based on .Net platform, with VB.Net, C# language and OpenGL graphic technique. With parameterized modeling method, finite element model of this kind of bridge can be built quickly and accurately, and multi-type element modeling with uniform parameters is realized. With advanced graphic technique, three-dimensional model graph can be real-timely previewed for intuitive data check. With an example of practice project, the accuracy and feasibility of this modeling method and practicality of this software are verified.
文摘The maximum internal tangential stress is a critical parameter for the design of the PDC （polycrystalline diamond compact） die that has been widely applied to offshore oil drilling. A new simple equation for the calculation of the stress is developed, and verified by the test data from Kingdream Corp. of China, the largest bit Company in China. An opti- mum method for the design of the PDC die is presented and demonstrated in detail, and software for the design and FEM analysis of the die is developed on the basis of the method. This software has been used in oil industry in recent years.
文摘The purpose of this article is to develop a new methodology to evaluate the statistical characteristic of the response of structures subjecting to random excitation, by combining the Finite Element Method (FEM) with the Transforming Density Function (TDF). Uncertainty modeling of structure with random variables encourages the coupling of advanced TDF for reliability analysis to analyze problems of stochastic mechanical systems. The TDF is enthusiastically applicable in the situation where the relationship between input and output of structures is available in explicit analytical form. However, the situation is much more involved when it is necessary to perform the evaluation of implicit expression between input and output of structures through numerical models. For this aim, we propose a new technique that combines the FEM software, and the TDF method to evaluate the most important statistical parameter the Probability Density Function (PDF) of the response where the expression between input and output of structures is implicit. Once the PDF is evaluated, all other statistical parameters are derived easily. This technique is based on the numerical simulations of the FEM and the TDF by making a middleware between Finite Element software and Matlab. Some problems, range from simple to complex, of structures are analyzed using our proposed technique. Its accuracy is validated through Monte-Carlo simulation.
文摘A simulation was carried out on an unsteady flow of a viscous, incompressible and electrically conducting fluid past an infinite vertical porous plate. A generic computer program using the Galerkin finite element method is employed to solve the coupled non-linear differential equations for velocity and temperature fields. The diffusion equation, the energy equation, the momentum equations and other relevant parameters are transformed into interpretable postfix codes. Numerical calculations are carried out on the flow fields both in the presence of cooling and heating of the plate by free convection currents. The effects of the dimensionless parameters, namely, the Prandtl number, the Eckert number, the modified Grashof number, the Schmidt number and the time on the temperature and velocity distributions are discussed.
