Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 18665

Search results for: mixed finite element method

18665 The Finite Element Method for Nonlinear Fredholm Integral Equation of the Second Kind

Authors: Melusi Khumalo, Anastacia Dlamini

Abstract:

In this paper, we consider a numerical solution for nonlinear Fredholm integral equations of the second kind. We work with uniform mesh and use the Lagrange polynomials together with the Galerkin finite element method, where the weight function is chosen in such a way that it takes the form of the approximate solution but with arbitrary coefficients. We implement the finite element method to the nonlinear Fredholm integral equations of the second kind. We consider the error analysis of the method. Furthermore, we look at a specific example to illustrate the implementation of the finite element method. Downloads 129
18664 Computation of Stress Intensity Factor Using Extended Finite Element Method

Authors: Mahmoudi Noureddine, Bouregba Rachid

Abstract:

In this paper the stress intensity factors of a slant-cracked plate of AISI 304 stainless steel, have been calculated using extended finite element method and finite element method (FEM) in ABAQUS software, the results were compared with theoretical values. Downloads 493
18663 Analysis of Thermal Effect on Functionally Graded Micro-Beam via Mixed Finite Element Method

Abstract:

Studies concerning the microstructures are becoming more important as the utilization of various micro-electro mechanical systems (MEMS) are increasing. Thus in recent years, thermal buckling and vibration analysis of microstructures have been subject to many investigations that are utilizing different numerical methods. In this study, thermal effects on mechanical response of a functionally graded (FG) Timoshenko micro-beam are presented in the framework of a mixed finite element formulation. Size effects are taken into consideration via modified couple stress theory. The mixed formulation is based on a function which in turn is derived via Gateaux Differential scientifically. After the resolution of all field equations of the beam, a potential operator is carefully constructed. Then this operator is used for the manufacturing of the functional. Usual procedures of finite element approximation are utilized for the derivation of the mixed finite element equations once the potential is obtained. Resulting finite element formulation allows usage of C₀ type simple linear shape functions and avoids shear-locking phenomena, which is a common shortcoming of the displacement-based formulations of moderately thick beams. The developed numerical scheme is used to obtain the effects of thermal loads on the static bending, free vibration and buckling of FG Timoshenko micro-beams for different power-law parameters, aspect ratios and boundary conditions. The versatility of the mixed formulation is presented over other numerical methods such as generalized differential quadrature method (GDQM). Another attractive property of the formulation is that it allows direct calculation of the contribution of micro effects on the overall mechanical response. Downloads 56
18662 Finite Element Method as a Solution Procedure for Problems in Tissue Biomechanics

Authors: Momoh Omeiza Sheidu

Abstract:

Finite element method as a method of providing solutions to problems in computational bio mechanics provides a framework for modeling the function of tissues that integrates structurally from cell to organ system and functionally across the physiological processes that affect tissue mechanics or are regulated by mechanical forces. In this paper, we present an integrative finite element strategy for solution to problems in tissue bio mechanics as a case study. Downloads 406
18661 A Finite Element Method Simulation for Rocket Motor Material Selection

Authors: T. Kritsana, P. Sawitri, P. Teeratas

Abstract:

This article aims to study the effect of pressure on rocket motor case by Finite Element Method simulation to select optimal material in rocket motor manufacturing process. In this study, cylindrical tubes with outside diameter of 122 mm and thickness of 3 mm are used for simulation. Defined rocket motor case materials are AISI4130, AISI1026, AISI1045, AL2024 and AL7075. Internal pressure used for the simulation is 22 MPa. The result from Finite Element Method shows that at a pressure of 22 MPa rocket motor case produced by AISI4130, AISI1045 and AL7075 can be used. A comparison of the result between AISI4130, AISI1045 and AL7075 shows that AISI4130 has minimum principal stress and confirm the results of Finite Element Method by the used of calculation method found that, the results from Finite Element Method has good reliability. Downloads 334
18660 Forced Vibration of a Planar Curved Beam on Pasternak Foundation

Abstract:

The objective of this study is to investigate the forced vibration analysis of a planar curved beam lying on elastic foundation by using the mixed finite element method. The finite element formulation is based on the Timoshenko beam theory. In order to solve the problems in frequency domain, the element matrices of two nodded curvilinear elements are transformed into Laplace space. The results are transformed back to the time domain by the well-known numerical Modified Durbin’s transformation algorithm. First, the presented finite element formulation is verified through the forced vibration analysis of a planar curved Timoshenko beam resting on Winkler foundation and the finite element results are compared with the results available in the literature. Then, the forced vibration analysis of a planar curved beam resting on Winkler-Pasternak foundation is conducted. Downloads 238
18659 Comparison of Finite-Element and IEC Methods for Cable Thermal Analysis under Various Operating Environments

Authors: M. S. Baazzim, M. S. Al-Saud, M. A. El-Kady

Abstract:

In this paper, steady-state ampacity (current carrying capacity) evaluation of underground power cable system by using analytical and numerical methods for different conditions (depth of cable, spacing between phases, soil thermal resistivity, ambient temperature, wind speed), for two system voltage level were used 132 and 380 kV. The analytical method or traditional method that was used is based on the thermal analysis method developed by Neher-McGrath and further enhanced by International Electrotechnical Commission (IEC) and published in standard IEC 60287. The numerical method that was used is finite element method and it was recourse commercial software based on finite element method. Downloads 216
18658 Numerical Modelling of Dry Stone Masonry Structures Based on Finite-Discrete Element Method

Authors: Ž. Nikolić, H. Smoljanović, N. Živaljić

Abstract:

This paper presents numerical model based on finite-discrete element method for analysis of the structural response of dry stone masonry structures under static and dynamic loads. More precisely, each discrete stone block is discretized by finite elements. Material non-linearity including fracture and fragmentation of discrete elements as well as cyclic behavior during dynamic load are considered through contact elements which are implemented within a finite element mesh. The application of the model was conducted on several examples of these structures. The performed analysis shows high accuracy of the numerical results in comparison with the experimental ones and demonstrates the potential of the finite-discrete element method for modelling of the response of dry stone masonry structures. Downloads 316
18657 Stability of Square Plate with Concentric Cutout

Authors: B. S. Jayashankarbabu, Karisiddappa

Abstract:

The finite element method is used to obtain the elastic buckling load factor for square isotropic plate containing circular, square and rectangular cutouts. ANSYS commercial finite element software had been used in the study. The applied inplane loads considered are uniaxial and biaxial compressions. In all the cases the load is distributed uniformly along the plate outer edges. The effects of the size and shape of concentric cutouts with different plate thickness ratios and the influence of plate edge condition, such as SSSS, CCCC and mixed boundary condition SCSC on the plate buckling strength have been considered in the analysis. Downloads 205
18656 Finite Element Method for Calculating Temperature Field of Main Cable of Suspension Bridge

Authors: Heng Han, Zhilei Liang, Xiangong Zhou

Abstract:

In this paper, the finite element method is used to study the temperature field of the main cable of the suspension bridge, and the calculation method of the average temperature of the cross-section of the main cable suitable for the construction control of the cable system is proposed; By comparing and analyzing the temperature field of the main cable with five diameters, a reasonable diameter limit for calculating the average temperature of the cross section of the main cable by finite element method is proposed. The results show that the maximum error of this method is less than 1℃, which meets the requirements of construction control accuracy; For the main cable with a diameter greater than 400mm, the surface temperature measuring points combined with the finite element method shall be used to calculate the average cross-section temperature. Downloads 18
18655 Analysis of Plates with Varying Rigidities Using Finite Element Method

Abstract:

This paper presents Finite Element Method (FEM) for analyzing the internal responses generated in thin rectangular plates with various edge conditions and rigidity conditions. Comparison has been made between the FEM (ANSYS software) results for displacement, stresses and moments generated with and without the consideration of hole in plate and different aspect ratios. In the end comparison for responses in plain and composite square plates has been studied.

Keywords: ANSYS, finite element method, plates, static analysis

18654 Design of a Vehicle Door Structure Based on Finite Element Method

Authors: Tawanda Mushiri, Charles Mbohwa

Abstract:

The performance of door assembly is very significant for the vehicle design. In the present paper, the finite element method is used in the development processes of the door assembly. The stiffness, strength, modal characteristic, and anti-extrusion of a newly developed passenger vehicle door assembly are calculated and evaluated by several finite element analysis commercial software. The structural problems discovered by FE analysis have been modified and finally achieved the expected door structure performance target of this new vehicle. The issue in focus is to predict the performance of the door assembly by powerful finite element analysis software, and optimize the structure to meet the design targets. It is observed that this method can be used to forecast the performance of vehicle door efficiently when it’s designed. In order to reduce lead time and cost in the product development of vehicles more development will be made virtually. Downloads 327
18653 Wave Interaction with Defects in Pressurized Composite Structures

Authors: R. K. Apalowo, D. Chronopoulos, V. Thierry

Abstract:

A wave finite element (WFE) and finite element (FE) based computational method is presented by which the dispersion properties as well as the wave interaction coefficients for one-dimensional structural system can be predicted. The structural system is discretized as a system comprising a number of waveguides connected by a coupling joint. Uniform nodes are ensured at the interfaces of the coupling element with each waveguide. Then, equilibrium and continuity conditions are enforced at the interfaces. Wave propagation properties of each waveguide are calculated using the WFE method and the coupling element is modelled using the FE method. The scattering of waves through the coupling element, on which damage is modelled, is determined by coupling the FE and WFE models. Furthermore, the central aim is to evaluate the effect of pressurization on the wave dispersion and scattering characteristics of the prestressed structural system compared to that which is not prestressed. Numerical case studies are exhibited for two waveguides coupled through a coupling joint. Downloads 197
18652 Identification of the Orthotropic Parameters of Cortical Bone under Nanoindentation

Abstract:

A good understanding of the mechanical properties of the bone implies a better understanding of its various diseases, such as osteoporosis. Berkovich nanoindentation tests were performed on the human cortical bone to extract its orthotropic parameters. The nanoindentation experiments were then simulated by the finite element method. Different configurations of interactions between the tip indenter and the bone were simulated. The orthotropic parameters of the material were identified by the inverse method for each configuration. The friction effect on the bone mechanical properties was then discussed. It was found that the inverse method using the finite element method is a very efficient method to predict the mechanical behavior of the bone. Downloads 301
18651 A Finite Element/Finite Volume Method for Dam-Break Flows over Deformable Beds

Authors: Alia Alghosoun, Ashraf Osman, Mohammed Seaid

Abstract:

A coupled two-layer finite volume/finite element method was proposed for solving dam-break flow problem over deformable beds. The governing equations consist of the well-balanced two-layer shallow water equations for the water flow and a linear elastic model for the bed deformations. Deformations in the topography can be caused by a brutal localized force or simply by a class of sliding displacements on the bathymetry. This deformation in the bed is a source of perturbations, on the water surface generating water waves which propagate with different amplitudes and frequencies. Coupling conditions at the interface are also investigated in the current study and two mesh procedure is proposed for the transfer of information through the interface. In the present work a new procedure is implemented at the soil-water interface using the finite element and two-layer finite volume meshes with a conservative distribution of the forces at their intersections. The finite element method employs quadratic elements in an unstructured triangular mesh and the finite volume method uses the Rusanove to reconstruct the numerical fluxes. The numerical coupled method is highly efficient, accurate, well balanced, and it can handle complex geometries as well as rapidly varying flows. Numerical results are presented for several test examples of dam-break flows over deformable beds. Mesh convergence study is performed for both methods, the overall model provides new insight into the problems at minimal computational cost. Downloads 91
18650 Dynamic Modeling of Orthotropic Cracked Materials by X-FEM

Abstract:

In this paper, dynamic fracture behaviors of cracked orthotropic structure are modeled using extended finite element method (X-FEM). In this approach, the finite element method model is first created and then enriched by special orthotropic crack tip enrichments and Heaviside functions in the framework of partition of unity. The mixed mode stress intensity factor (SIF) is computed using the interaction integral technique based on J-integral in order to predict cracking behavior of the structure. The developments of these procedures are programmed and introduced in a self-software platform code. To assess the accuracy of the developed code, results obtained by the proposed method are compared with those of literature. Downloads 405
18649 Modeling Revolution Shell Structures by MATLAB Programming-Axisymmetric and Nonaxisymmetric Shells

Abstract:

The objective of this work is setting numerically operational finite element CAXI_L for the axisymmetric and nonaxisymmetric shells. This element is based on the Reissner-Mindlin theory and mixed model formulation. The MATLAB language is used for the programming. In order to test the elaborated program, some applications are carried out. Downloads 215
18648 Dynamic Analysis of Viscoelastic Plates with Variable Thickness

Abstract:

In this study, the dynamic analysis of viscoelastic plates with variable thickness is examined. The solutions of dynamic response of viscoelastic thin plates with variable thickness have been obtained by using the functional analysis method in the conjunction with the Gâteaux differential. The four-node serendipity element with four degrees of freedom such as deflection, bending, and twisting moments at each node is used. Additionally, boundary condition terms are included in the functional by using a systematic way. In viscoelastic modeling, Three-parameter Kelvin solid model is employed. The solutions obtained in the Laplace-Carson domain are transformed to the real time domain by using MDOP, Dubner & Abate, and Durbin inverse transform techniques. To test the performance of the proposed mixed finite element formulation, numerical examples are treated. Downloads 246
18647 A FE-Based Scheme for Computing Wave Interaction with Nonlinear Damage and Generation of Harmonics in Layered Composite Structures

Authors: R. K. Apalowo, D. Chronopoulos

Abstract:

A Finite Element (FE) based scheme is presented for quantifying guided wave interaction with Localised Nonlinear Structural Damage (LNSD) within structures of arbitrary layering and geometric complexity. The through-thickness mode-shape of the structure is obtained through a wave and finite element method. This is applied in a time domain FE simulation in order to generate time harmonic excitation for a specific wave mode. Interaction of the wave with LNSD within the system is computed through an element activation and deactivation iteration. The scheme is validated against experimental measurements and a WFE-FE methodology for calculating wave interaction with damage. Case studies for guided wave interaction with crack and delamination are presented to verify the robustness of the proposed method in classifying and identifying damage. Downloads 54
18646 Postbuckling Analysis of End Supported Rods under Self-Weight Using Intrinsic Coordinate Finite Elements

Abstract:

A formulation of postbuckling analysis of end supported rods under self-weight has been presented by the variational method. The variational formulation involving the strain energy due to bending and the potential energy of the self-weight, are expressed in terms of the intrinsic coordinates. The variational formulation is accomplished by introducing the Lagrange multiplier technique to impose the boundary conditions. The finite element method is used to derive a system of nonlinear equations resulting from the stationary of the total potential energy and then Newton-Raphson iterative procedure is applied to solve this system of equations. The numerical results demonstrate the postbluckled configurations of end supported rods under self-weight. This finite element method based on variational formulation expressed in term of intrinsic coordinate is highly recommended for postbuckling analysis of end-supported rods under self-weight. Downloads 63
18645 A Coupled Extended-Finite-Discrete Element Method: On the Different Contact Schemes between Continua and Discontinua

Authors: Shervin Khazaeli, Shahab Haj-zamani

Abstract:

Recently, advanced geotechnical engineering problems related to soil movement, particle loss, and modeling of local failure (i.e. discontinua) as well as modeling the in-contact structures (i.e. continua) are of the great interest among researchers. The aim of this research is to meet the requirements with respect to the modeling of the above-mentioned two different domains simultaneously. To this end, a coupled numerical method is introduced based on Discrete Element Method (DEM) and eXtended-Finite Element Method (X-FEM). In the coupled procedure, DEM is employed to capture the interactions and relative movements of soil particles as discontinua, while X-FEM is utilized to model in-contact structures as continua, which may consist of different types of discontinuities. For verification purposes, the new coupled approach is utilized to examine benchmark problems including different contacts between/within continua and discontinua. Results are validated by comparison with those of existing analytical and numerical solutions. This study proves that extended-finite-discrete element method can be used to robustly analyze not only contact problems, but also other types of discontinuities in continua such as (i) crack formations and propagations, (ii) voids and bimaterial interfaces, and (iii) combination of previous cases. In essence, the proposed method can be used vastly in advanced soil-structure interaction problems to investigate the micro and macro behaviour of the surrounding soil and the response of the embedded structure that contains discontinuities. Downloads 405
18644 Mixed Mode Fracture Analyses Using Finite Element Method of Edge Cracked Heavy Spinning Annulus Pulley

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. Downloads 64
18643 3D Finite Element Analysis of Yoke Hybrid Electromagnet

Abstract:

The objective of this paper is to analyze a 4-pole hybrid magnetic levitation system by using 3D finite element and analytical methods. The magnetostatic analysis of the system is carried out by using ANSYS MAXWELL-3D package. An analytical model is derived by magnetic equivalent circuit (MEC) method. The purpose of magnetostatic analysis is to determine the characteristics of attractive force and rotational torques by the change of air gap clearances, inclination angles and current excitations. The comparison between 3D finite element analysis and analytical results are presented at the rest of the paper. Downloads 575
18642 Analysis of the Suspension Rocker of Formula SAE Prototype by Finite Element Method

Abstract:

This work aims to study the rocker. This is a device of the suspension of Formula SAE vehicle that receives efforts from the motion scrolling of the vehicle and transmits them to the chassis frame minimized by a momentum ratio and smoothed by the set spring - damper. A review of parameters used in vehicle dynamics and a geometric analysis of the forces and stresses caused by such was carried out. The main function of the rocker is to reduce the force transmitted to the frame due to movement of rolling and subsequent application of the suspension. This functions is taken as satisfactory, since the force applied to the wheel and which would be transmitted to the chassis is reduced from 3833.9N to 3496.48N. From these values can be further more detailed simulations using the finite element method aimed at mass reduction or even rocker manufacturing feasibility aluminum. Then, the analysis by the finite element method was applied. This analysis uses the theory of discretization of systems and examines the strength of the component based on the distortion energy, determining the maximum straining experienced by the component and the region of higher demand. Downloads 457
18641 Reliability Estimation of Bridge Structures with Updated Finite Element Models

Authors: Ekin Ozer

Abstract:

Assessment of structural reliability is essential for efficient use of civil infrastructure which is subjected hazardous events. Dynamic analysis of finite element models is a commonly used tool to simulate structural behavior and estimate its performance accordingly. However, theoretical models purely based on preliminary assumptions and design drawings may deviate from the actual behavior of the structure. This study proposes up-to-date reliability estimation procedures which engages actual bridge vibration data modifying finite element models for finite element model updating and performing reliability estimation, accordingly. The proposed method utilizes vibration response measurements of bridge structures to identify modal parameters, then uses these parameters to calibrate finite element models which are originally based on design drawings. The proposed method does not only show that reliability estimation based on updated models differs from the original models, but also infer that non-updated models may overestimate the structural capacity. Downloads 127
18640 Modeling Thin Shell Structures by a New Flat Shell Finite Element

Abstract:

In this paper, a new computationally-efficient rectangular flat shell finite element named 'ACM_RSBEC' is presented. The formulated element is obtained by superposition of a new rectangular membrane element 'RSBEC' based on the strain approach and the well known plate bending element 'ACM'. This element can be used for the analysis of thin shell structures, no matter how the geometrical shape might be. Tests on standard problems have been examined. The convergence of the new formulated element is also compared to other types of quadrilateral shell elements. The presented shell element ‘ACM_RSBEC’ has been demonstrated to be effective and useful in analysing thin shell structures. Downloads 289
18639 An Implementation of Meshless Method for Modeling an Elastoplasticity Coupled to Damage

Abstract:

The modeling of mechanical problems including both material and geometric nonlinearities with Finite Element Method (FEM) remains challenging. Meshless methods offer special properties to get rid of well-known drawbacks of the FEM. The main objective of Meshless Methods is to eliminate the difficulty of meshing and remeshing the entire structure by simply insertion or deletion of nodes, and alleviate other problems associated with the FEM, such as element distortion, locking and others. In this study, a robust numerical implementation of an Element Free Galerkin Method for an elastoplastic coupled to damage problem is presented. Several results issued from the numerical simulations by a DynamicExplicit resolution scheme are analyzed and critically compared with Element Finite Method results. Finally, different numerical examples are carried out to demonstrate the efficiency of this method. Downloads 213
18638 B Spline Finite Element Method for Drifted Space Fractional Tempered Diffusion Equation

Authors: Ayan Chakraborty, BV. Rathish Kumar

Abstract:

Off-late many models in viscoelasticity, signal processing or anomalous diffusion equations are formulated in fractional calculus. Tempered fractional calculus is the generalization of fractional calculus and in the last few years several important partial differential equations occurring in the different field of science have been reconsidered in this term like diffusion wave equations, Schr$\ddot{o}$dinger equation and so on. In the present paper, a time-dependent tempered fractional diffusion equation of order $\gamma \in (0,1)$ with forcing function is considered. Existence, uniqueness, stability, and regularity of the solution has been proved. Crank-Nicolson discretization is used in the time direction. B spline finite element approximation is implemented. Generally, B-splines basis are useful for representing the geometry of a finite element model, interfacing a finite element analysis program. By utilizing this technique a priori space-time estimate in finite element analysis has been derived and we proved that the convergent order is $\mathcal{O}(h²+T²)$ where $h$ is the space step size and $T$ is the time. A couple of numerical examples have been presented to confirm the accuracy of theoretical results. Finally, we conclude that the studied method is useful for solving tempered fractional diffusion equations. Downloads 95
18637 A Mixed Finite Element Formulation for Functionally Graded Micro-Beam Resting on Two-Parameter Elastic Foundation

Abstract:

Micro-beams are one of the most common components of Nano-Electromechanical Systems (NEMS) and Micro Electromechanical Systems (MEMS). For this reason, static bending, buckling, and free vibration analysis of micro-beams have been the subject of many studies. In addition, micro-beams restrained with elastic type foundations have been of particular interest. In the analysis of microstructures, closed-form solutions are proposed when available, but most of the time solutions are based on numerical methods due to the complex nature of the resulting differential equations. Thus, a robust and efficient solution method has great importance. In this study, a mixed finite element formulation is obtained for a functionally graded Timoshenko micro-beam resting on two-parameter elastic foundation. In the formulation modified couple stress theory is utilized for the micro-scale effects. The equation of motion and boundary conditions are derived according to Hamilton’s principle. A functional, derived through a scientific procedure based on Gateaux Differential, is proposed for the bending and buckling analysis which is equivalent to the governing equations and boundary conditions. Most important advantage of the formulation is that the mixed finite element formulation allows usage of C₀ type continuous shape functions. Thus shear-locking is avoided in a built-in manner. Also, element matrices are sparsely populated and can be easily calculated with closed-form integration. In this framework results concerning the effects of micro-scale length parameter, power-law parameter, aspect ratio and coefficients of partially or fully continuous elastic foundation over the static bending, buckling, and free vibration response of FG-micro-beam under various boundary conditions are presented and compared with existing literature. Performance characteristics of the presented formulation were evaluated concerning other numerical methods such as generalized differential quadrature method (GDQM). It is found that with less computational burden similar convergence characteristics were obtained. Moreover, formulation also includes a direct calculation of the micro-scale related contributions to the structural response as well. Downloads 58
18636 Electromagnetic Assessment of Submarine Power Cable Degradation Using Finite Element Method and Sensitivity Analysis

Authors: N. Boutra, N. Ravot, J. Benoit, O. Picon

Abstract:

Submarine power cables used for offshore wind farms electric energy distribution and transmission are subject to numerous threats. Some of the risks are associated with transport, installation and operating in harsh marine environment. This paper describes the feasibility of an electromagnetic low frequency sensing technique for submarine power cable failure prediction. The impact of a structural damage shape and material variability on the induced electric field is evaluated. The analysis is performed by modeling the cable using the finite element method, we use sensitivity analysis in order to identify the main damage characteristics affecting electric field variation. Lastly, we discuss the results obtained. Downloads 242