In this paper, numerical methods algorithms are designed and implemented for the solution of partial differential two dimensional Laplaceequations. The domain used in the problem is sufficiently nontrivial for analytical solution.To determine the solution Finite difference method (FDM)and Finite element method (FEM) are used. To investigate the performance of these methodsalgorithms are developed in MATLAB. The electric potential over the entire domain for both methods are computed. The solution obtained using FEM is compared with FDMsolution. The results found using FDM is considered more accurate in this problem, increased accuracyof FEM can be obtained using higher order elements or by dividing the solution domain into more number of triangular elements.

Keyword:Finite element method,Finite difference method, Partial differential equation, Laplace equation Dirichlet boundary conditions, Electrostatics

Introduction

In the area of mathematics, formation of differential equations and their solutions are the most important features to almost every numerical.Partial differential equations aretoo important and useful several fields of science, engineering e.g. electromagnetic theory, stress equation for beam, heat transfer, etc.

The Laplace equation and Poisson equation are a powerful tool for demonstrating the performance of electrostatic systems. These equations can be solved analytically only for simple region. Therefore, numerical techniques must be applied in order to model the performance of complex domains with practical value.

Numerical techniques, such asfinite element method finite difference method, etc are best suited to produce approximatebut acceptable values of quantities to be determined at discrete number of points in the given area. Instead of solving the problem for complete domain in one stage, the solutions are found for each constituent unit and finally combined to compute the complete solution.

Finite Element Method andFinite Difference Method and were used widely to analyze the stresses in various load conditions. Besides instrumental in structural analysis, they have been generally used to solvepartial differential equations involved in many domains of Electronics, Electrical, Civil and Mechanical Engineering. In electrostatics, Laplace orPoisson equation are used to calculate the electric potential and electric field.

A static electric field E in vacuum due topvi.e. volume charge density is given as

?(?.E=?_v/?_0 #(1))

Equation 1 is known as Gauss’s law in a differential form.

The del operator ? in Cartesian co-ordinate system is given by,

?(?=d/dx u_x+d/dy u_y+d/dz u_z#(2))

The electric fieldE in terms of gradient? and electric potential V is given by

?(E=-?V#(3))

From equation 1 and 3 we write

?(?^2 V=?-??_v/?_0 #(4))

where ?^2is Laplacian operator.

Equation 4 is called as Poisson’s equation 2-3.

When volume charge density pvbecomes zero, then equation 4 becomes Laplace equation as 4,

?(?^2 V=0#(5))

In Cartesian coordinate system, ?^2 operating on electric potential (V) for a two -dimensional Laplace equation is given as 5,

?(?^2 V=(d^2 V)/(dx^2 )+(d^2 V)/(dy^2 )=0#(6))

The solution of equation (6) is obtained using FDM and FEM.

The paper is organized into four sections. Introduction is given in section 1.Section 2 defines the electrostatic 2-D problem and its formulation. Section 3 discusses numerical methods solution for the given boundary value problem and compare the results for developed algorithms and finally section 4 concludes the paper.

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Problem definition and formulation.

Determine the electric potential at the free nodes in the electrostatic system of Figure 1 with given boundary conditions using .Laplace equations.

A electric potential for the domain shown in Fig 1 can be formulated in 2-D Laplace equation as

?((?^2 V)/(?x^2 )+(?^2 V)/(?y^2 )=0#(7))

For the given problem,

x=(2,0)andy=(0,2.5)

The electric potential distribution in the region and the Dirichlet boundary conditions are,

V(x,0)=0,V(0,y)=0,

V(x,2.5)=15 ,V(2,y)=0

Finite Difference Method

The FDM is a simple numerical technique used to solved Laplace or Poisson’s equations.

A problem is uniquely defined by three steps:

1. Partial differential equation such as Laplace’s or Poisson’s equations.

2. A solution region.

3. Boundary and/or initial conditions.

The solution to such problem can be obtained using finite difference method that evaluates by discretizing the region into a mesh of nodes, approximates the differential equation with boundary condition by set of linear equations known as difference equation and solving these set of equation by either iteration methodor band matrix method.

A 2D region along with the given boundary conditions is divided into rectangular grid of 26 nodes with numbering is shown in figure 1.2(a)

Fig. 1.2(a): Rectangular region with 10 x 10 grid for Finite Difference Method

Finite Element method:

A FEM when used to a 2-D problem in electrostaticsgenerally involves a second order differential equation of a single dependent variable subjected to set of boundary conditions. The 2-D geometry of the region can be of arbitrary shape. An perfect illustration of this region can be realized using discretization methods with either triangular or quadrilateral shapes called finite elements 6, 7.

The solution of such 2-D problems using FEM can be solved using following steps 8-10:

a) 2D domain Discretization.

b) Derivation of the weak formulation of the governing differential equation.

c) Proper choice of interpolation functions.

d) Derivation of the element matrices and vectors.

e) Assembly of the global matrix system.

f) Imposition of boundary conditions.

g) Solution of the global matrix system.

h) Postprocessing of the results.

A 2D solution domain is divided into 32 three-node triangular finite elements having 26 nodes as shown in Fig….

3. Numerical Solution for 2D Electrostatic Boundary Value Problem

The given 2D electrostatic boundary value problem is solved using FDM and FEM algorithms developed in MATLAB and results are compared at free nodes.

The FDM algorithm developed and implemented in MATLAB can be briefly explained as,

Step1: Specify array size and boundary conditions

Step2: Calculate step size and specify volume charge density, if needed.

Step 3: Specify boundary conditions.

Step 4: Use central difference method to calculate electric potential distribution and iterate it until all the values are achieved.

Step 5: obtained the results.

The finite difference formula for interior node is

V_0=1/4 (V_1+V_2+V_3+V_4 )

The electric potential at inner point is the average of its 4 adjacent nodes.

Using finite difference analysis ,the electric potential at the free nodes are found for different number of iteration as given in Table. It can be observed from Table that convergence is achieved after 12 iterations.

Number of Iterations Potential at free nodes

5 10 13 14 15 18 19 20

2 5.1563 6.6211 5.8008 1.6406 2.4609 2.1643 0.5658 0.1414

4 6.0846 7.5172 6.2262 2.2401 3.0362 2.4736 0.6579 0.1645

6 6.2298 7.6499 6.2892 2.3320 3.1222 2.5198 0.6717 0.1679

8 6.2515 7.6698 6.2986 2.3458 3.1351 2.5267 0.6738 0.1684

10 6.2548 7.6727 6.3000 2.3478 3.1370 2.5278 0.6741 0.1685

12 6.2553 7.6732 6.3003 2.3481 3.1373 2.5279 0.6741 0.1685

14 6.2553 7.6732 6.3003 2.3482 3.1373 2.5279 0.6741 0.1685

16 6.2554 7.6732 6.3003 2.3482 3.1373 2.5279 0.6741 0.1685

20 6.2554 7.6732 6.3003 2.3482 3.1373 2.5279 0.6741 0.1685

25 6.2554 7.6732 6.3003 2.3482 3.1373 2.5279 0.6741 0.1685

30 6.2554 7.6732 6.3003 2.3482 3.1373 2.5279 0.6741 0.1685

The FEM algorithm developed in MATLAB involves following steps 10-11

Step1: A Rectangular Geometry is defined

Step 2: Define the number of elements in the x and y directions

Step 3: Define the given Electric potential i.e. top side

Step 4: Generate and plot the triangular mesh for given nodes/elements

Step 5: Define the overall boundary conditions

Step 6: Initialize the global K matrix and right-hand side vector

Step7: Form and assemble the element matrices vectors into the global K matrix and right -hand side vector

Step 8: Apply Dirichlet boundary conditions

Step 9: Obtain the solution of the global matrix system and plot it

For the given electrostatic problem the nodal coordinates of finite element grid(mesh),element and node identification(globle-local node correspondence) and given electric potential(Boundary conditions) at fixed nodes are shown in Table …..,……,…….. respectively.

Node number x-coordinate y-coordinate

1 1 0

2 1.5 0

3 2 0

4 1 0.5

5 1.5 0.5

6 2 0.5

7 0 1

8 0.5 1

9 1 1

10 1.5 1

11 2 1

12 0 1.5

13 0.5 1.5

14 1 1.5

15 1.5 1.5

16 2 1.5

17 0 2

18 0.5 2

19 1 2

20 1.5 2

21 2 2

22 0 2.5

23 0.5 2.5

24 1 2.5

25 1.5 2.5

26 2 2.5

The Using finite difference analysis ,the electric potential at the free nodes are found

Elements Local Nodes

1 1 2 4

2 2 5 4

3 2 3 5

4 3 6 5

5 4 5 9

6 5 10 9

7 5 6 10

8 6 11 10

9 7 8 12

10 8 13 12

11 8 9 13

12 9 14 13

13 9 10 14

14 10 15 14

15 10 11 15

16 11 16 15

17 12 13 17

18 13 18 17

19 13 14 18

20 14 19 18

21 14 15 19

22 15 20 19

23 15 16 20

24 16 21 20

25 17 18 22

26 18 23 22

27 18 19 23

28 19 24 23

29 19 20 24

30 20 25 24

31 20 21 25

32 21 26 25

Nodes number Boundary Conditions

1 0

3 0

4 0

6 0

7 0

8 0

9 0

11 0

12 0

16 0

17 0

21 0

22 15

23 15

24 15

25 15

26 15

Using finite element analysis ,the electric potential at the free nodes are found and is given in Table.

Free Node number FEM solution

5 0.194616

10 0.681158

13 2.348372

14 3.137996

15 2.530014

18 6.255493

19 7.673598

20 6.300903

The validity of theFEM solution is checked with FDM solution and is given in Table. And its graphical presentation is shown in Fig.

Node number FDM FEM

5 0.168529 0.194616

10 0.674117 0.681158

13 2.348174 2.348372

14 3.13734 3.137996

15 2.527938 2.530014

18 6.255355 6.255493

19 7.673248 7.673598

20 6.300297 6.300903

The FDM and FEM solutions give closer results at different free nodes.

Conclusion

This paper presentsFinite Difference, Finite Element numerical techniques to solve two dimensional differential equation system for electrostatic field of engineering with Dirichlet boundary conditions. The electrical potential over the complete domain is evaluated. The results found using FDM is considered more accurate in this problem, increased accuracy of FEM can be obtained using higher order elements or by dividing the solution domain into more number of triangular elements.The FEM has two main advantages over the FDM.The electric voltages are found only at discrete points in the solution domain using FDM,they can be found at any location in the solution domain in FEM. Also , the complex geometries is easily handle using FEM.

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