Solving Boundary Value Problem In 2D Using Finite Element .

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International Journal of Engineering Research and Technology. ISSN 0974-3154, Volume 12, Number 12 (2019), pp. 3133-3137 International Research Publication House. http://www.irphouse.comSolving Boundary Value Problem in 2D Using Finite Element and FiniteDifference MethodIffat Ara11Information and Communication Engineering Department,Pabna University of Science and Technology, Pabna, Bangladesh.ORCID: 0000-0001-7244-9999The Finite Difference Method follow three basic steps [5]:AbstractThis paper presents a simple and powerful approach ostatics. In this paper Finite Element and Finitedifference numerical method has been used to solve twodimensional steady heat flow problem with Dirichletboundary conditions in a rectangular domain. FiniteDifference solution with rectangular grid and Finite Elementsolution with triangular grid using spreadsheets isimplemented here. Spreadsheets are used for solvingelectrostatic boundary-value problems. Finally comparisonsare made between the solution obtained from the FiniteDifference and Finite Element Method.(1) Divide the solution region (geometry) into a grid ofnodes. Grid points are typically arranged in arectangular array of nodes.(2) Approximate the PDE and boundary conditions by aset of linear algebraic equations (the finite differenceequations) on grid points within the solution region.(3) Solve this set of linear algebraic equations.Consider the charge-free region depicted in Figure 1. Theregion has prescribed potentials along its boundaries. Theregion is divided into a rectangular grid of nodes, with thenumbering of free nodes as indicated in the figure.Key words: Laplace Equation, Finite Difference Method,Finite Element method.I. INTRODUCTIONIn science and engineering different phenomena aremodeled by partial differential equations (PDEs). PDE is afunction that depends on more than one variable (typically,space variables x, y, z and the temporal variable t). The heatequation, wave equation, and Laplace’s equation are amongthe most prominent PDEs [1, 2]. This paper emphasizesnumerical solutions to PDEs and suggests implementationsthrough spreadsheets. This paper focuses on certainnumerical methods for solving PDEs; in particular, the finitedifference and the finite element methods. The examplespresented in this paper include geometries that aresufficiently nontrivial for hand calculation or analyticalsolution, but reasonably manageable by using spreadsheets.Although specialized software is available for this purpose,oftentimes such sophistication tends to obscure the innerworkings of the numerical methods employed in the solutionof PDEs. Spreadsheets offer a transparent alternativeperhaps proximate to hand calculation for students to betterappreciate the numerical methods for solving PDEs andBVPs [1, 3, 4].Figure 1: Charge-free region showing prescribed potentialsat the boundaries and rectangular grid of free nodes toillustrate the finite difference method.The potential V V(x,y) at an interior point (x,y) within theregion is governed by the two-dimensional Laplace’sequation 2 𝑉 2𝑉 2𝑉 0 π‘₯ 2 𝑦 2(1)For x [0, a] and y [0, b] with a 4, b 2Where V(x,y) is the steady state temperaturedistribution in the domain.Let the location of an interior grid point be identified by apair of integers (i, j), where i and j represent the positionalong the horizontal and vertical directions, respectively. Fora grid having equal horizontal and vertical step sizes, thepotential is given by the finite difference equationII. MATERIALS AND METHODSII.I. The Finite Difference Method1𝑉𝑖,𝑗 (𝑉𝑖 1,𝑗 𝑉𝑖 1,𝑗 𝑉𝑖,𝑗 1 𝑉𝑖,𝑗 1 )4The Finite Difference Method (FDM) is conceptuallysimple. The problems to which the method applies arespecified by a PDE, geometry, and boundary conditions.3133(2)

International Journal of Engineering Research and Technology. ISSN 0974-3154, Volume 12, Number 12 (2019), pp. 3133-3137 International Research Publication House. http://www.irphouse.comThis system of equations may be solved by a variety ofmethods. In this section the Gauss-Seidel method isimplemented in a spreadsheet to solve this system ofequations. The Gauss-Seidel method is a relatively simpleiterative method for solving systems such as thoseencountered in the finite difference formulation.(3) Assemble all elements in the solution region to obtainthe global coefficient matrix.(4) Solve the resulting system of equations.Consider the same charge-free region shown in Figure 1.The region is divided into 16 equal triangular elements asindicated in Figure 4. The elements are identified byencircled numbers 1 through 16. In this discretization thereare 15 global nodes numbered 1 through 15.Boundary Potentiallower side50right side250left side100upper side200Figure 2: Input section of spreadsheet implementation ofthe finite difference method.Gauss-Seidel Iterationiterationv1( 7)v2( 8)v3( 32.1259158.027610120.5315132.1388158.0315Figure 4: Finite element arrangement for electrostaticproblem.For each element e the following quantities are computed𝑃1 𝑦2 𝑦3 , 𝑃2 𝑦3 𝑦1 , 𝑃3 𝑦1 𝑦2 ,𝑄1 π‘₯3 π‘₯2 , 𝑄2 π‘₯1 π‘₯3 , 𝑄3 π‘₯2 π‘₯1 ,(3)Where the subscripts refer to the local nodenumbers 1, 2, and 3 of element e. For example, in Figure 4,element 6 has global nodes 4, 9, and 8, which correspond,respectively to local nodes 1, 2, and 3.With Pi and Qi (i 1, 2, 3) for element e thuscomputed, the entries of the 3 x 3 element coefficient matrixare then given by(𝑒)𝐢𝑖𝑗 Figure 3: The Gauss-Seidel iterations of the differenceequations.14𝐴[𝑃𝑖 𝑃𝑗 𝑄𝑖 𝑄𝑗 ] (i, j 1,2,3)(4)Where1𝐴 [𝑃2 𝑄3 𝑃3 𝑄2 ]2II.II. The Finite Element Method(5)The element coefficient matrix is shown in Figure 6. Theglobal coefficient matrix is then assembled from the elementcoefficient matrices. Since there are 15 nodes, the globalcoefficient matrix will be a 15 x 15 matrix shown in Figure7. For example, node 7, which corresponds to the C7, 7 entryin the global coefficient matrix C, belongs to elements 2, 3,4, 9, 10 and 11; since node 7 is assigned local node number1 in elements 10 and 11, and local node number 2 inelements 2 and 9, and local number 3 in element 3 and 4 (asseen in the middle table of Figure 5), the correspondingglobal coefficient isThe Finite Element Method (FEM) is a numerical techniquefor solving PDEs. FEM was originally applied to problemsin structural mechanics. Unlike FDM, FEM is better suitedfor solution regions having irregularly shaped boundaries.The finite element analysis involves four basic steps [4, 5]:(1) Divide the solution region into a finite number ofelements. The most common elements have triangularor quadrilateral shapes. The collection of all elementsshould resemble the original region as closely aspossible.(2) Derive governing equations for a typical element.This step will determine the element coefficientmatrix.(10)(11)(2)(9)(3)(4)𝐢7,7 𝐢1,1 𝐢1,1 𝐢2,2 𝐢2,2 𝐢3,3 𝐢3,3 . . (6) 0.5 1 1 0.5 0.5 0.5 43134

International Journal of Engineering Research and Technology. ISSN 0974-3154, Volume 12, Number 12 (2019), pp. 3133-3137 International Research Publication House. http://www.irphouse.comNodal Coordinate ofFinite Element MeshesElement Node IdentificationPrescribed PotentialFigure 5: Input section of finite element analysis for electrostatic problem: (left) global node x and y coordinates; (middle)triangular element – global and local node correspondence; (right) nodes in the finite element mesh having prescribed potentials.Figure 6: Computation of element coefficient matrices CeFigure 7: Screenshot showing global coefficient matrix3135

International Journal of Engineering Research and Technology. ISSN 0974-3154, Volume 12, Number 12 (2019), pp. 3133-3137 International Research Publication House. http://www.irphouse.comDefining the vector of potentials vf and vp, where thesubscripts f and p refer to nodes with free (unknown)potentials and prescribed potentials, respectively, the globalcoefficient matrix is then partitioned accordingly and theunknown potentials are obtained from 1𝑣𝑓 𝐢𝑓𝑓𝐢𝑓𝑝 𝑣𝑝(7)The matrices Cfp ( free and prescribed node matrix) and Cff( free node matrix) are formed by extracting the appropriaterows and columns from the global coefficient matrix C. Inthis case node 7, 8, 9 are the free node while 1, 2, 3, 4, 5, 6,10, 11,12, 13, 14, and 15 are the nodes with prescribedpotential, The result is shown in figure 8.Microsoft Excell function MINVERSE and MMULT areused in the implementation of equation (7), the final resultsare shown in Figure 9.Figure 8: The matrix Cff and Cfp are computed from GlobalCoefficient matrix.𝑨 ( 𝟏) π‘ͺ𝒇𝒇 ( 570.071429Node7890.0178570.0714290.267857𝒗𝒇 𝑨 ( 𝟏) 𝒃(Vf p o t en t ial at f r een o d e)NodePotential7120.53578132.14299158.0357𝒃 π‘ͺ𝒇𝒑 𝒗𝒑(Vp vector of prescribed potential)Node789Vector (b)350250500Figure 9: Final calculations section of finite element solution to electrostatic problem.III. RESULTS AND DISCUSSIONIV. CONCLUSIONIn Sections II.I and II.II spreadsheet implementations ofFDM and FEM were presented. As indicated in Table 1, thepotentials at the free nodes computed by both methodscompared fairly well. The node numbers in the table forFDM correspond to those in Figure 1, while those for FEMcorrespond to the node numbers shown in Figure 4.This paper presented spreadsheet implementations of twonumerical methods for solving electrostatics problems. Asindicated in above table the potentials at the free nodes werecomputed by using spreadsheet. Numerical solutionscompared fairly well. The better agreement should beobtained between the finite element numerical solutionresults by using spreadsheet with a triangular grid and finitedifference solution with a rectangular grid.Table 1: Comparison of results obtainedfrom FDM and FEM.Finite DifferenceSpreadsheets may be considered as a viable alternative toenhancing education in other subjects and engineeringfields.Finite 3136Teh, K. and Morgan, L. The application of Excel inteaching finite element analysis to final yearengineering students, Proceedings of the 2005ASEE/AaeE (Australian Association for EngineeringEducation) Global Colloquium on EngineeringEducation, Paper # 50, 2005.

International Journal of Engineering Research and Technology. ISSN 0974-3154, Volume 12, Number 12 (2019), pp. 3133-3137 International Research Publication House. http://www.irphouse.com[2]Asmar, N. Partial differential equations with Fourierseries and boundary value problems, 2nd edition,Prentice Hall, 2004.[3]Yamani, A. and Kharab, A. Use of a spreadsheetprogram in electromagnetics, IEEE Transactions onEducation, 44(3): 292-297, 2001.[4]Lau, Mark A. and Kuruganty, Sastry P. SpreadsheetImplementations for Solving Boundary-ValueProblems in Electromagnetics, Spreadsheets inEducation (eJSiE): Vol. 4: Iss. 1, Article 1, 2010.[5]Sadiku, M. Elements of electromagnetics, 4th edition,Oxford University Press, 2006.[6]Parag V.Patil, Dr. J.S.V.R. Krishna Prasad SolutionOf Laplace Equation Using Finite Element MethodPratibha: International Journal Of Science,Spirituality, Business And Technology (Ijssbt), Vol.2, No.1, November 20133137

The Finite Element Method The Finite Element Method (FEM) is a numerical technique for solving PDEs. FEM was originally applied to problems in structural mechanics. Unlike FDM, FEM is better suited for solution regions having irregularly shaped boundaries. The finite element analysis involves four basic steps [4, 5]:

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