3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254 – 4143 Ed. 33 Vol. 9 N.º 1 Marzo - Junio

ERROR ESTIMATION OF BILINEAR GALERKIN FINITE

ELEMENT METHOD FOR 2D THERMAL PROBLEMS

S. M. Afzal Hoq

Department of Mechanical Engineering

International Islamic University Malaysia, Kualalumpur, (Malaysia).

E-mail: afzalhoqsu@gmail.com ORCID: https://orcid.org/0000-0002-5917-2507

Abdurahim Okhunov

Department of Science and Engineering

International Islamic University Malaysia, Kualalumpur, (Malaysia).

E-mail: abdurahimokhun@iium.edu.my ORCID: https://orcid.org/0000-0001-7092-5699

C. P. Tso

Faculty of Engineering and Technology

Multimedia University, (Malaysia).

E-mail: cptso@mmu.edu.my ORCID: https://orcid.org/0000-0003-3884-5992

Recepción:

03/02/2020

Aceptación:

14/02/2020

Publicación:

13/03/2020

Citación sugerida:

Hoq, S. M. A., Okhunov, A., y Tso, C. P. (2020). Error estimation of bilinear Galerkin nite element

method for 2D thermal problems. 3C Tecnología. Glosas de innovación aplicadas a la pyme, 9(1), 79-93. http://doi.

org/10.17993/3ctecno/2020.v9n1e33.79-93

3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254 – 4143 Ed. 33 Vol. 9 N.º 1 Marzo - Junio

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ABSTRACT

This study demonstrates a two-dimensional steady state heat conduction Laplace partial dierential

equation solution using the bilinear Galerkin nite element method. Heat transfer analysis is of vital

importance in many engineering applications and devising computationally inexpensive numerical

methods while maintaining accuracy is one of the primary concerns. The method uses structured mesh

grid over a two-dimensional rectangular domain and solved using a stiness matrix for the bilinear

elements, calculated using the proposed modied numerical scheme. Several numerical experiments are

conducted by controlling the number of nodes and changing element sizes of the presented scheme, and

comparison made between analytical solution and software generated solution.

KEYWORDS

Galerkin method, Bilinear element, Heat conduction, Error analysis, Partial dierential equation.

3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254 – 4143 Ed. 33 Vol. 9 N.º 1 Marzo - Junio

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1. INTRODUCTION

Finite element Analysis is a robust numerical approach for solving partial dierential equations (PDEs)

used in various branches of science such as solid mechanics, uid mechanics, electromagnetics,

thermodynamics etc. (Kreyszig, 2011; Deb, Babuška, & Oden, 2001). One of the most powerful

techniques for solving PDEs with weak formulation is using a weighed residual method called the Galerkin

nite element method (GFEM). The formulation requires generating a basis function (Ainsworth &

Oden, 1997) based on the elemental boundary conditions. This trial function is substituted in the

partial dierential equation and the rst derivative of the trial function is taken for each nodal variable

(Burden & Faires, 2001) to construct the residual function. The weighed form of the residual function

for the whole domain is integrated by setting it equal to zero. Green’s theorem can be applied over the

boundary if necessary (Afzal, Sulaeman, & Okhunov, 2016). The corresponding numerical model is set

up by discretizing the rectangular domain into smaller elements where each element consists of nodal

coordinates and nodal variables, which are used to perform the Galerkin approximation of the PDEs.

This is followed by the generation of an element matrix and vector matrix of the boundary by integrating

the total number of elements. The set of linear equations represented by the matrices are consecutively

solved using the Galerkin approach. For comparison purposes, an exact solution already available for

the non-linear PDE (time independent and no heat source) used in 2D heat conduction rectangular

domain with both Dirichlet and Neumann boundary condition, is presented. Finally, a stiness matrix

applicable for homogenous rectangular domain consisting of structured mesh grid elements is presented,

the solution scheme of which signicantly reduces the CPU performance cost.

2. MATHEMATICAL CONSTRUCTION

A homogeneous domain represented by a time independent heat conduction problem with zero heat

extraction is considered and can be mathematically formulated by Laplace equation with Neumann

3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254 – 4143 Ed. 33 Vol. 9 N.º 1 Marzo - Junio

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or Dirichlet boundary conditions as shown in Figure 1 (Gerald & Wheatly, 2003; Gockenbach, 2002;

Burden & Faires, 2001; Kreyszig, 2011).

Figure 1. Physical domain of Ω bounded by Γ.

Two dimensional time independent heat-conduction problem can be represented by the following partial

dierential equation.

(1)

where,

and

The boundary conditions are given by:

(2)

where u

and f

are Dirichlet and Neumann boundary conditions respectively.

The weighed residual w is applied on Eqs (1) and (2) to generate its strong formulation as given:

3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254 – 4143 Ed. 33 Vol. 9 N.º 1 Marzo - Junio

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(3)

Consequently, the weak formulation is generated using integration by part over eq. (3) as given below:

(4)

The stiness matrix is generated by discretising the solution domain into smaller elements and performing

integration for each element.

3. BASIS FUNCTION AND STIFFNESS MATRIX

The 2D computational domain is represented by bilinear rectangular elements where each element is

constructed with a node at each of its four edges as shown in Figure 2. The nodal points are represented

by (x

, y

), (x

, y

), (x

, y

) and (x

, y

), while the nodal variables are represented by u

, u

and u

. The

nodal variable u at any given location (x,y) for a bilinear element is approximated by the basis function,

which is written as:

u=a

x+a

y+ a

xy (5)

Figure 2. Bilinear element.

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The above-mentioned basis function can be represented in matrix form:

(6)

Considering each bilinear element composed of four nodal points and nodal variables, the corresponding

basis function in matrix form is represented as:

(7)

The corresponding shape function is formulated as:

(8)

The individual shape function is based on the four geometric bilinear coordinates for each element

located in a structured mesh grid:

(9)

(10)

(11)

(12)

3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254 – 4143 Ed. 33 Vol. 9 N.º 1 Marzo - Junio

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Where the rectangle Area, A is given by:

(13)

As it can be observed from Eq. (5) to (11), the trial function u is dependent on the nodal variables at the

corners of the bilinear nite elements and the shape functions. The rst derivative of this trial function

gives the weighed function w in Eq. (4)

(14)

In order to determine the elemental stiness matrix, the rst integral of the weak formulation in Eq. (4)

is carried out sequentially given below:

(15)

(16)

The matrix generated is a 4x4 matrix as given by:

3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254 – 4143 Ed. 33 Vol. 9 N.º 1 Marzo - Junio

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(17)

The diagonal values as follows:

(18)

(19)

(20)

(21)

Since:

, k

and k

the matrix is symmetric about its diagonal.

(22)

3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254 – 4143 Ed. 33 Vol. 9 N.º 1 Marzo - Junio

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4. RESULTS AND DISCUSSION

In this section a set of numerical experiments are carried out for the evaluation and analysis of the

results. A rectangular 2D domain is developed and steady state heat conduction with no heat source is

applied. The domain measures 5 units and 10 units in the x and y direction respectively. GFEM is used

by implementing the weighed residual approximation approach on each bilinear element for solving

the Laplace PDE governing the 2D heat conduction problem. Both Dirichlet and Neumann boundary

conditions are specied for the solution domain as given in Figure 3. The boundary conditions are below:

Figure 3. Steady state heat conduction problem of rectangular domain.

Solving the above mentioned problem required calculating the stiness matrix of the bilinear elements

using the Galerkin nite element approach. The accuracy of the approach largely depends on the element

size and number of elements considered. It has been observed increasing the number of elements and

decreasing the element size increases the accuracy of the solution by achieving higher convergence rates.

3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254 – 4143 Ed. 33 Vol. 9 N.º 1 Marzo - Junio

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Comparison is made with the exact solution of the Eq. (23) below and Figure 4 shows the graphical

analysis.

(23)

Figure 4. Comparison between Galerkin FE solution and exact solution.

The Figure 5 represents the solution domain discretized into a numbers of smaller bilinear elements

each consisting of nodes at the corners.

Figure 5. Physical domain with bilinear mesh.

3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254 – 4143 Ed. 33 Vol. 9 N.º 1 Marzo - Junio

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The errors analysis of the Galerkin approach solution and analytical solution are given by the following

equations:

(24)

(25)

(26)

Table 1 below lists the set of errors analysis for dierent numerical experiments using various number of

elements and element sizes. The equation form

represents the L

norm, represents

norm and represents L

∞

norm (He, Lin, & Lin, 2010).

Table 1. L

∞

and H

errors of the Galerkin method solution.

3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254 – 4143 Ed. 33 Vol. 9 N.º 1 Marzo - Junio

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It can be observed from Table 1 that increasing the number of elements while decreasing the element

size has a converging on both the exact and GFEM solution on all three of the calculated norms,

, L

and L

∞

with convergence rates of O(h

), O(h) and O(h

) respectively. This validates convergence

towards the expected exact solution and accuracy of the presented Galerkin nite element scheme.

Figure 6 below show the graphical illustration of the convergence of decreasing the size and increasing

the number of elements on the rate of errors respectively for the generated scheme.

Figure 7. Error analysis vs number of element.

Figures 8 and 9 shows the 2D rectangular solution domain where, the surface plot and contour plot are

taken from the results. The temperature distribution is governed by time independent heat conduction

with no heat source.

Figure 8. Surface plot of temperature variation in the GFEM solution domain using Matlab.

3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254 – 4143 Ed. 33 Vol. 9 N.º 1 Marzo - Junio

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Figure 9. Contour plot using Matlab.

5. CONCLUSION

It is shown that GFEM is an inexpensive and powerful technique to accurately model and has

demonstrated solving a time dependent and external heat source independent 2D heat conduction

Laplace partial dierential equation, governing a rectangular planar domain, using bilinear elements.

Increasing the number of elements while decreasing the element size representing the solution domain

shows optimal convergence towards the exact solution, thereby validating the accuracy of the scheme.

ACKNOWLEDGEMENT

The work has been nancially supported by the ministry of higher education, Malaysia, under the grant

FRGS 19-039-0647 and is gratefully acknowledged.

3C Tecnología. Glosas de innovación aplicadas a la pyme. ISSN: 2254 – 4143 Ed. 33 Vol. 9 N.º 1 Marzo - Junio

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