Search results for: finite elementanalysis

Authors: Pavel K. Lopatin, Artyom S. Yegorov

Abstract:

The Algorithm 2 for a n-link manipulator movement amidst arbitrary unknown static obstacles for a case when a sensor system supplies information about local neighborhoods of different points in the configuration space is presented. The Algorithm 2 guarantees the reaching of a target position in a finite number of steps. The Algorithm 2 is reduced to a finite number of calls of a subroutine for planning a trajectory in the presence of known forbidden states. The polynomial approximation algorithm which is used as the subroutine is presented. The results of the Algorithm2 implementation are given.

Keywords: Manipulator, trajectory planning, unknown obstacles.

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Authors: Vineet K. Srivastava, Mukesh K. Awasthi, Mohammad Tamsir

Abstract:

A fully implicit finite-difference method has been proposed for the numerical solutions of one dimensional coupled nonlinear Burgers’ equations on the uniform mesh points. The method forms a system of nonlinear difference equations which is to be solved at each iteration. Newton’s iterative method has been implemented to solve this nonlinear assembled system of equations. The linear system has been solved by Gauss elimination method with partial pivoting algorithm at each iteration of Newton’s method. Three test examples have been carried out to illustrate the accuracy of the method. Computed solutions obtained by proposed scheme have been compared with analytical solutions and those already available in the literature by finding L2 and L∞ errors.

Keywords: Burgers’ equation, Implicit Finite-difference method, Newton’s method, Gauss elimination with partial pivoting.

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Authors: Bijit Kalita, K. V. N. Surendra

Abstract:

Rotating disk is one of the most indispensable parts of a rotating machine. Rotating disk has found many applications in the diverging field of science and technology. In this paper, we have taken into consideration the problem of a heavy spinning disk mounted on a rotor system acted upon by boundary traction. Finite element modelling is used at various loading condition to determine the mixed mode stress intensity factors. The effect of combined shear and normal traction on the boundary is incorporated in the analysis under the action of gravity. The variation near the crack tip is characterized in terms of the stress intensity factor (SIF) with an aim to find the SIF for a wide range of parameters. The results of the finite element analyses carried out on the compressed disk of a belt pulley arrangement using fracture mechanics concepts are shown. A total of hundred cases of the problem are solved for each of the variations in loading arc parameter and crack orientation using finite element models of the disc under compression. All models were prepared and analyzed for the uncracked disk, disk with a single crack at different orientation emanating from shaft hole as well as for a disc with pair of cracks emerging from the same center hole. Curves are plotted for various loading conditions. Finally, crack propagation paths are determined using kink angle concepts.

Keywords: Crack-tip deformations, static loading, stress concentration, stress intensity factor.

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Authors: Sung Wook Yun, Yun Jong Choi, Kyong-min Lee, Poogyeon Park*

Abstract:

Repetitive systems stand for a kind of systems that perform a simple task on a fixed pattern repetitively, which are widely spread in industrial fields. Hence, many researchers have been interested in those systems, especially in the field of iterative learning control (ILC). In this paper, we propose a finite-horizon tracking control scheme for linear time-varying repetitive systems with uncertain initial conditions. The scheme is derived both analytically and numerically for state-feedback systems and only numerically for output-feedback systems. Then, it is extended to stable systems with input constraints. All numerical schemes are developed in the forms of linear matrix inequalities (LMIs). A distinguished feature of the proposed scheme from the existing iterative learning control is that the scheme guarantees the tracking performance exactly even under uncertain initial conditions. The simulation results demonstrate the good performance of the proposed scheme.

Keywords: Finite time horizon, linear matrix inequality (LMI), repetitive system, uncertain initial condition.

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Authors: Watchara Songserm, Teeradej Wuttipornpun

Abstract:

This paper presents a genetic algorithm based permutation and non-permutation scheduling heuristics (GAPNP) to solve a multi-stage finite capacity material requirement planning (FCMRP) problem in automotive assembly flow shop with unrelated parallel machines. In the algorithm, the sequences of orders are iteratively improved by the GA characteristics, whereas the required operations are scheduled based on the presented permutation and non-permutation heuristics. Finally, a linear programming is applied to minimize the total cost. The presented GAPNP algorithm is evaluated by using real datasets from automotive companies. The required parameters for GAPNP are intently tuned to obtain a common parameter setting for all case studies. The results show that GAPNP significantly outperforms the benchmark algorithm about 30% on average.

Keywords: Finite capacity MRP, genetic algorithm, linear programming, flow shop, unrelated parallel machines, application in industries.

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Authors: Porojan S., Sandu L., Topală F.

Abstract:

Simulation of occlusal function during laboratory material-s testing becomes essential in predicting long-term performance before clinical usage. The aim of the study was to assess the influence of chamfer preparation depth on failure risk of heat pressed ceramic crowns with and without zirconia framework by means of finite element analysis. 3D models of maxillary central incisor, prepared for full ceramic crowns with different depths of the chamfer margin (between 0.8 and 1.2 mm) and 6-degree tapered walls together with the overlying crowns were generated using literature data (Fig. 1, 2). The crowns were designed with and without a zirconia framework with a thickness of 0.4 mm. For all preparations and crowns, stresses in the pressed ceramic crown, zirconia framework, pressed ceramic veneer, and dentin were evaluated separately. The highest stresses were registered in the dentin. The depth of the preparations had no significant influence on the stress values of the teeth and pressed ceramics for the studied cases, only for the zirconia framework. The zirconia framework decreases the stress values in the veneer.

Keywords: Finite element analysis, full ceramic crown, zirconia framework, stresses.

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Authors: M. Najafi

Abstract:

In the present paper, a large turbo-generator shaft train including a heavy-duty gas turbine engine, a coupling, and a generator is established. The method of analysis is based on finite element simplified model for lateral and torsional vibration calculation. The basic elements of rotor are the shafts and the disks which are represented as circular cross section flexible beams and rigid body elements, respectively. For more accurate results, the gyroscopic effect and bearing dynamics coefficients and function of rotation are taken into account, and for the influence of shear effect, rotor has been modeled in the form of Timoshenko beam. Lateral critical speeds, critical speed map, damped mode shapes, Campbell diagram, zones of instability, amplitudes, phase angles response due to synchronous forces of excitation and amplification factor are calculated. Also, in the present paper, the effect of imbalanced rotor and effects of changing in internal force and temperature are studied.

Keywords: Rotor dynamic analysis, Finite element method, shaft train, Campbell diagram.

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Authors: Tsung-yin Lin, Yi-yu Lee, Ching-hua Hung

Abstract:

This study focuses on the stress analysis of Mandibular Advancement Devices (MADs), which are considered as a standard treatment of snoring that promoted by American Academy of Sleep Medicine (AASM). Snoring is the most significant feature of sleep-disordered breathing (SDB). SDB will lead to serious problems in human health. Oral appliances are ensured in therapeutic effect and compliance, especially the MADs. This paper proposes a new MAD design, and the finite element analysis (FEA) is introduced to precede the stress simulation for this MAD.

Keywords: Finite element analysis, mandibular advancement devices, mechanical stress, snoring.

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Authors: S. A. Moeini, M. T.Ahmadian

Abstract:

Structural behavior of ring stiffened thick walled cylinders made of functionally graded materials (FGMs) is investigated in this paper. Functionally graded materials are inhomogeneous composites which are usually made from a mixture of metal and ceramic. The gradient compositional variation of the constituents from one surface to the other provides an elegant solution to the problem of high transverse shear stresses that are induced when two dissimilar materials with large differences in material properties are bonded. FGM formation of the cylinder is modeled by power-law exponent and the variation of characteristics is supposed to be in radial direction. A finite element formulation is derived for the analysis. According to the property variation of the constituent materials in the radial direction of the wall, it is not convenient to use conventional elements to model and analyze the structure of the stiffened FGM cylinders. In this paper a new cylindrical super-element is used to model the finite element formulation and analyze the static and modal behavior of stiffened FGM thick walled cylinders. By using this super-element the number of elements, which are needed for modeling, will reduce significantly and the process time is less in comparison with conventional finite element formulations. Results for static and modal analysis are evaluated and verified by comparison to finite element formulation with conventional elements. Comparison indicates a good conformity between results.

Keywords: FGMs, Modal analysis, Static analysis, Stiffened cylinders.

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Authors: Jonathan H. Perez, Fumihiko Tanaka, Daisuke Hamanaka, Toshitaka Uchino

Abstract:

A finite element analysis was conducted to determine the effect of moisture diffusion and hygroscopic swelling in rice. A parallel simple stochastic modeling was performed to predict the number of grains cracked as a result of moisture absorption and hygroscopic swelling. Rice grains were soaked in thermally (25 oC) controlled water and then tested for compressive stress. The destructive compressive stress tests revealed through compressive stress calculation that the peak force required to cause cracking in grains soaked in water reduced with time as soaking duration was extended. Results of the experiment showed that several grains had their value of the predicted compressive stress below the von Mises stress and were interpreted as grains which become cracked and/or broke during soaking. The technique developed in this experiment will facilitate the approximation of the number of grains which will crack during soaking.

Keywords: Cracking, Finite element analysis, hygroscopic swelling, moisture diffusion, von Mises stress.

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Authors: K. Noah, F.-S. Lien

Abstract:

In Computational Fluid Dynamics (CFD), there are a variety of numerical methods, of which some depend on macroscopic model representatives. These models can be solved by finite-volume, finite-element or finite-difference methods on a microscopic description. However, the lattice Boltzmann method (LBM) is considered to be a mesoscopic particle method, with its scale lying between the macroscopic and microscopic scales. The LBM works well for solving incompressible flow problems, but certain limitations arise from solving compressible flows, particularly at high Mach numbers. An improved lattice Boltzmann model for compressible flow problems is presented in this research study. A higher-order Taylor series expansion of the Maxwell equilibrium distribution function is used to overcome limitations in LBM when solving high-Mach-number flows. Large eddy simulation (LES) is implemented in LBM to simulate turbulent jet flows. The results have been validated with available experimental data for turbulent compressible free jet flow at subsonic speeds.

Keywords: Compressible lattice Boltzmann metho-, large eddy simulation, turbulent jet flows.

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Authors: Ruediger Schmidt, Thang Duy Vu

Abstract:

Two geometrically nonlinear plate theories, based either on first- or third-order transverse shear deformation theory are used for finite element modeling and simulation of the transient response of smart structures incorporating piezoelectric layers. In particular the time histories of nonlinear vibrations and sensor voltage output of a thin beam with a piezoelectric patch bonded to the surface due to an applied step force are studied.

Keywords: Nonlinear vibrations, piezoelectric patches, sensor voltage output, smart structures.

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Authors: Xijian Wang

Abstract:

This paper is concerned with the numerical minimization of energy functionals in BV ( ) (the space of bounded variation functions) involving total variation for gray-scale 1-dimensional inpainting problem. Applications are shown by finite element method and discontinuous Galerkin method for total variation minimization. We include the numerical examples which show the different recovery image by these two methods.

Keywords: finite element method, discontinuous Galerkin method, total variation minimization, inpainting

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Authors: S. Alih, A. Khelil

Abstract:

In the present work, behavior of inoxydable steel as reinforcement bar in composite concrete is being investigated. The bar-concrete adherence in reinforced concrete (RC) beam is studied and focus is made on the tension stiffening parameter. This study highlighted an approach to observe this interaction behavior in bending test instead of direct tension as per reported in many references. The approach resembles actual loading condition of the structural RC beam. The tension stiffening properties are then applied to numerical finite element analysis (FEA) to verify their correlation with laboratory results. Comparison with laboratory shows a good correlation between the two. The experimental settings is able to determine tension stiffening parameters in RC beam and the modeling strategies made in ABAQUS can closely represent the actual condition. Tension stiffening model used can represent the interaction properties between inoxydable steel and concrete.

Keywords: Inoxydable steel, Finite element modeling, Reinforced concrete beam, Tension-stiffening.

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Authors: Shi Qi Koo, Ahmad Beng Hong Kueh

Abstract:

In the present work, the finite element formulation for the investigation of the effects of a localized interfacial degeneration on the dynamic behavior of the [90°/0°] laminated composite plate employing the state-space technique is performed. The stiffness of the laminate is determined by assembling the stiffnesses of subelements. This includes an introduction of an interface layer adopting the virtually zero-thickness formulation to model the interfacial degeneration. Also, the kinematically consistent mass matrix and proportional damping have been formulated to complete the free vibration governing expression. To simulate the interfacial degeneration of the laminate, the degenerated areas are defined from the center propagating outwards in a localized manner. It is found that the natural frequency, damped frequency and damping ratio of the plate decreases as the degenerated area of the interface increases. On the contrary, the loss factor increases correspondingly.

Keywords: Dynamic finite element, localized interface degeneration, proportional damping, state-space modeling.

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Authors: M. Azadbakht, Y. Fadakar

Abstract:

Since fuel must be injected with appropriate pressure and time for accurate performance of diesel engines, then proper function of engine is influenced by accurate function of injector pump. At first total pump was designed by SolidWorks 2012 software. Then the total relationship of rotor, roller, internal cam ring, pole shoe and plunger in injector pump in MF285 tractor and their performance was shown. During suction state rollers connect with dents in internal cam ring and in pressure course pole shoes have drawer move in rotor and perform tappet action between rollers and plungers. The maximum stress was obtained by using analysis of finite element method. The maximum stress in contact surface of roller and internal cam ring and on roller surface. The maximum amount of this stress is 288.12 MPa. According to conducted analyses, the minimum value for safety factor is related to roller surface and it equals to 2.0477.

Keywords: Rotary injector pump, MF285 tractor, finite element, stress.

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Authors: Wah Wah Aung, Cho Cho San

Abstract:

The paper deals with a mathematical model for fluid dynamic flows on road networks which is based on conservation laws. This nonlinear framework is based on the conservation of cars. We focus on traffic circle, which is a finite number of roads that meet at some junctions. The traffic circle with junctions having either one incoming and two outgoing or two incoming and one outgoing roads. We describe the numerical schemes with the particular boundary conditions used to produce approximated solutions of the problem.

Keywords: boundary conditions, conservation laws, finite difference Schemes, traffic flow.

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Authors: A. Pourkamali Anaraki, M. Shahabizadeh, B. Babaee

Abstract:

The plastic forming process of sheet plate takes an important place in forming metals. The traditional techniques of tool design for sheet forming operations used in industry are experimental and expensive methods. Prediction of the forming results, determination of the punching force, blank holder forces and the thickness distribution of the sheet metal will decrease the production cost and time of the material to be formed. In this paper, multi-stage deep drawing simulation of an Industrial Part has been presented with finite element method. The entire production steps with additional operations such as intermediate annealing and springback has been simulated by ABAQUS software under axisymmetric conditions. The simulation results such as sheet thickness distribution, Punch force and residual stresses have been extracted in any stages and sheet thickness distribution was compared with experimental results. It was found through comparison of results, the FE model have proven to be in close agreement with those of experiment.

Keywords: Deep drawing, Finite element method, Simulation.

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Authors: Migbar Assefa

Abstract:

The performance of a machine tool is eventually assessed by its ability to produce a component of the required geometry in minimum time and at small operating cost. It is customary to base the structural design of any machine tool primarily upon the requirements of static rigidity and minimum natural frequency of vibration. The operating properties of machines like cutting speed, feed and depth of cut as well as the size of the work piece also have to be kept in mind by a machine tool structural designer. This paper presents a novel approach to the design of machine tool column for static and dynamic rigidity requirement. Model evaluation is done effectively through use of General Finite Element Analysis software ANSYS. Studies on machine tool column are used to illustrate finite element based concept evaluation technique. This paper also presents results obtained from the computations of thin walled box type columns that are subjected to torsional and bending loads in case of static analysis and also results from modal analysis. The columns analyzed are square and rectangle based tapered open column, column with cover plate, horizontal partitions and with apertures. For the analysis purpose a total of 70 columns were analyzed for bending, torsional and modal analysis. In this study it is observed that the orientation and aspect ratio of apertures have no significant effect on the static and dynamic rigidity of the machine tool structure.

Keywords: Finite Element Modeling, Modal Analysis, Machine tool structure, Static Analysis.

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Authors: Davod Khojasteh Salkuyeh

Abstract:

An important task in solving second order linear ordinary differential equations by the finite difference is to choose a suitable stepsize h. In this paper, by using the stochastic arithmetic, the CESTAC method and the CADNA library we present a procedure to estimate the optimal stepsize hopt, the stepsize which minimizes the global error consisting of truncation and round-off error.

Keywords: Ordinary differential equations, optimal stepsize, error, stochastic arithmetic, CESTAC, CADNA.

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Authors: U. Amin, A. Talib, S. A. Qureshi, M. J. Hossain, G. Ahmad

Abstract:

The choice evaluation between oil-immersed and dry-type transformers is often controlled by cost, location, and application. This paper compares the electrical performance of liquid- filled and dry-type transformers, which will assist the customer to choose the right and efficient ones for particular applications. An accurate assessment of the time-average flux density, electric field intensity and voltage distribution in an oil-insulated and a dry-type transformer have been computed and investigated. The detailed transformer modeling and analysis has been carried out to determine electrical parameter distributions. The models of oil-immersed and dry-type transformers are developed and solved by using the finite element method (FEM) to compare the electrical parameters. The effects of non-uniform and non-coherent voltage gradient, flux density and electric field distribution on the power losses and insulation properties of transformers are studied in detail. The results show that, for the same voltage and kilo-volt-ampere (kVA) rating, oil-immersed transformers have better insulation properties and less hysteresis losses than the dry-type.

Keywords: Finite element method, flux density, transformer, voltage gradient.

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Authors: Jaime R. S. Fonseca

Abstract:

Nevertheless the widespread application of finite mixture models in segmentation, finite mixture model selection is still an important issue. In fact, the selection of an adequate number of segments is a key issue in deriving latent segments structures and it is desirable that the selection criteria used for this end are effective. In order to select among several information criteria, which may support the selection of the correct number of segments we conduct a simulation study. In particular, this study is intended to determine which information criteria are more appropriate for mixture model selection when considering data sets with only categorical segmentation base variables. The generation of mixtures of multinomial data supports the proposed analysis. As a result, we establish a relationship between the level of measurement of segmentation variables and some (eleven) information criteria-s performance. The criterion AIC3 shows better performance (it indicates the correct number of the simulated segments- structure more often) when referring to mixtures of multinomial segmentation base variables.

Keywords: Quantitative Methods, Multivariate Data Analysis, Clustering, Finite Mixture Models, Information Theoretical Criteria, Simulation experiments.

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Authors: Alberto Hananel

Abstract:

The aim of this work is to modelize the occlusion of a person with temporomandibular disorders as an evolutionary equation and approach its solution by the construction and characterizing of discrete variational splines. To formulate the problem, certain boundary conditions have been considered. After showing the existence and the uniqueness of the solution of such a problem, a convergence result of a discrete variational evolutionary spline is shown. A stress analysis of the occlusion of a human jaw with temporomandibular disorders by finite elements is carried out in FreeFem++ in order to prove the validity of the presented method.

Keywords: Approximation, evolutionary PDE, finite element method, temporomandibular disorders, variational spline.

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Authors: Thanh-Duoc Phan, Saden H. Zahiri, S. H. Masood, Mahnaz Jahedi

Abstract:

In this article, using finite element analysis (FEA) and an X-ray diffractometer (XRD), cold-sprayed titanium particles on a steel substrate is investigated in term of cooling time and the development of residual strains. Three cooling-down models of sprayed particles after deposition stage are simulated and discussed: the first model (m1) considers conduction effect to the substrate only, the second model (m2) considers both conduction as well as convection effect to the environment, and the third model (m3) which is the same as the second model but with the substrate heated to a near particle temperature before spraying. Thereafter, residual strains developed in the third model is compared with the experimental measurement of residual strains, which involved a Bruker D8 Advance Diffractometer using CuKa radiation (40kV, 40mA) monochromatised with a graphite sample monochromator. For deposition conditions of this study, a good correlation was found to exist between the FEA results and XRD measurements of residual strains.

Keywords: cold gas dynamic spray, X-ray diffraction, explicit finite element analysis, residual strain, titanium, particle impact, deformation behavior.

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Authors: Weifeng Wang, Xianwei Zeng, Jianping Ding

Abstract:

When the characteristic length of an elastic solid is down to the nanometer level, its deformation behavior becomes size dependent. Surface energy /surface stress have recently been applied to explain such dependency. In this paper, the effect of strain-independent surface stress on the deformation of an isotropic elastic solid containing a nanosized elliptical hole is studied by the finite element method. Two loading cases are considered, in the first case, hoop stress along the rim of the elliptical hole induced by pure surface stress is studied, in the second case, hoop stress around the elliptical opening under combined remote tension and surface stress is investigated. It has been shown that positive surface stress induces compressive hoop stress along the hole, and negative surface stress has opposite effect, maximum hoop stress occurs near the major semi-axes of the ellipse. Under combined loading of remote tension and surface stress, stress concentration around the hole can be either intensified or weakened depending on the sign of the surface stress.

Keywords: Surface stress, finite element method, stress concentration, nanosized elliptical hole

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Authors: Gustavo Rodrigues, Hans Weber, Larissa Driemeier

Abstract:

The prevision of post-impact conditions and the behavior of the bodies during the impact have been object of several collision models. The formulation from Hertz’s theory is generally used dated from the 19^th century. These models consider the repulsive force as proportional to the deformation of the bodies under contact and may consider it proportional to the rate of deformation. The objective of the present work is to analyze the behavior of the bodies during impact using the Finite Element Method (FEM) with elastic and plastic material models. The main parameters to evaluate are, the contact force, the time of contact and the deformation of the bodies. An advantage of using the FEM approach is the possibility to apply a plastic deformation to the model according to the material definition: there will be used Johnson–Cook plasticity model whose parameters are obtained through empirical tests of real materials. This model allows analyzing the permanent deformation caused by impact, phenomenon observed in real world depending on the forces applied to the body. These results are compared between them and with the model-based Hertz theory.

Keywords: Collision, finite element method, Hertz’s Theory, impact models.

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Authors: Sam Teek Ling, Norhashidah Hj. Mohd. Ali

Abstract:

We investigate the formulation and implementation of new explicit group iterative methods in solving the two-dimensional Poisson equation with Dirichlet boundary conditions. The methods are derived from a fourth order compact nine point finite difference discretization. The methods are compared with the existing second order standard five point formula to show the dramatic improvement in computed accuracy. Numerical experiments are presented to illustrate the effectiveness of the proposed methods.

Keywords: Explicit group iterative method, finite difference, fourth order compact, Poisson equation.

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Authors: Ho Young Son, Bu Seog Ju, Woo Young Jung

Abstract:

This study presents the seismic safety evaluation of weir structure subjected to strong earthquake ground motions, as a flood defense structure in civil engineering structures. The seismic safety analysis procedure was illustrated through development of Finite Element (FE) and InFinite Element (IFE) method in ABAQUS platform. The IFE model was generated by CINPS4, 4-node linear one-way infinite model as a sold continuum infinite element in foundation areas of the weir structure and then nonlinear FE model using friction model for soil-structure interactions was applied in this study. In order to understand the complex behavior of weir structures, nonlinear time history analysis was carried out. Consequently, it was interesting to note that the compressive stress gave more vulnerability to the weir structure, in comparison to the tensile stress, during an earthquake. The stress concentration of the weir structure was shown at the connection area between the weir body and stilling basin area. The stress both tension and compression was reduced in IFE model rather than FE model of weir structures.

Keywords: Weir, Finite Element, Infinite Element, Nonlinear, Earthquake.

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Authors: Majid Mirzaei, Saeid Samiezadeh , Abbas Khodadadi, Mohammad R. Ghazavi

Abstract:

This paper presents a novel method for prediction of the mechanical behavior of proximal femur using the general framework of the quantitative computed tomography (QCT)-based finite element Analysis (FEA). A systematic imaging and modeling procedure was developed for reliable correspondence between the QCT-based FEA and the in-vitro mechanical testing. A speciallydesigned holding frame was used to define and maintain a unique geometrical reference system during the analysis and testing. The QCT images were directly converted into voxel-based 3D finite element models for linear and nonlinear analyses. The equivalent plastic strain and the strain energy density measures were used to identify the critical elements and predict the failure patterns. The samples were destructively tested using a specially-designed gripping fixture (with five degrees of freedom) mounted within a universal mechanical testing machine. Very good agreements were found between the experimental and the predicted failure patterns and the associated load levels.

Keywords: Bone, Osteoporosis, Noninvasive methods, Failure Analysis

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Authors: S.S. Razavipour, M. Jahangiri, H. Sadeghipoor

Abstract:

In this paper, the computation of the electrical field distribution around AC high-voltage lines is demonstrated. The advantages and disadvantages of two different methods are described to evaluate the electrical field quantity. The first method is a seminumerical method using the laws of electrostatic techniques to simulate the two-dimensional electric field under the high-voltage overhead line. The second method which will be discussed is the finite element method (FEM) using specific boundary conditions to compute the two- dimensional electric field distributions in an efficient way.

Keywords: Electrical field, unloaded transmission lines, finite element method, electrostatic images technique.

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