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• International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064

Index Copernicus Value (2013): 6.14 | Impact Factor (2013): 4.438

Volume 4 Issue 6, June 2015

www.ijsr.net Licensed Under Creative Commons Attribution CC BY

Study of 1D Bar Problem by Finite Element Method

on Parallel Architectures

Pikle Nileshchandra K1, Umesh B. Chavan

2

Department of Information Technology Walchand College of Engineering, Sangli, India

Abstract: Finite element method (FEM) is one of the most commonly used numerical technique to find approximate solution for various problems in the field of mechanical engineering, civil engineering etc. In this paper we are presenting the overview of solving

mechanical 1D bar problem by Finite Element Method (FEM). This method include two steps, first we have to generate stiffness matrix

and after generating stiffness matrix solve the system of linear equations by suitable method either by direct method or by iterative

method. As Iterative methods are more computationally efficient in this paper we emphasize Conjugate gradient (CG) method and

Preconditioned Conjugate Gradient method (PCG). Later on we will focus on the parallelization of these methods on parallel

architectures such as CPU-GPU architecture or Message passing interface (MPI).

Keywords: Finite Element Method (FEM), Element By Element Finite Element Method (EBEFEM), Conjugate Gradient (CG), Preconditioned Conjugate Gradient (PCG), Graphics Processing Unit (GPU), Compute Unified Device Architecture (CUDA).

1. Introduction

Finite Element Method is one of the best methods for

solving partial differential equations (PDE) from various

domains of engineering such as Mechanical Engineering

civil engineering etc. Various structural analysis problems[1]

like determining effects of loads on structures like vehicles,

bridges, buildings, etc. are routinely carried out using the

Finite Element Method. A typical Finite Element simulation

of a practical problem usually involves the assembly and

solution of hundreds of thousands of simultaneous linear

algebraic equations which can be written in the of the form

Ax=b (1)

Solving FEM problem divided into following steps

1. Divide problem into number of finite elements. 2. Generate local stiffness matrix. 3. Assemble to form the linear systems of equations of the

form Ku = f.

4. Solve this linear system of equations by preconditioned conjugate gradient method.

Figure 1: Steps in Finite Element Method

where K is stiffness matrix, u is a load vector, L is total

length of bar and f is a applied force refer figure. One end of

bar is fixed and force (f) is applied at other end of bar. Bar is

divided into n number of finite elements area of cross

sections (A) at each element is given. According to the

hook's law stiffness (k) is given by

k=

(2)

where l is segmented length and is computed by Total

length/number of elements (L/n), E is modulus of elasticity.

The memory requirements and the computational time

required to solve such equations increases as the number of

equations increases. To deal with such large numerical

problems in the Finite Element Analysis, parallel computing

on high performance computer is gradually becoming a main

stream tool. Many parallel algorithms and programs for

finite element computation have been developed on parallel

computers, utilizing vast numbers of CPUs or GPU cores to

achieve high speed up and scalability.

2. Literature Survey

The theory and implementation of the Finite Element

Method is discussed various books, see for example Seshu

[1]. The element by element algorithm is discussed in

Hughes et. al [2] and Hughes [3]. A serial implementation of

the element by element method is discussed in King and

Sonnad [4]. A parallel implementation of the element-by-

element method using CUDA is presented in Kiss et. al [5]

where they solved the problem of heat conduction in an in

homegeneous media. Sheth [6] has demonstrated a proof of

concept implementation of the element by element method

using CUDA to solve plane linear elastic problems. Mafi and

Sirouspour [7] have also implemented the element by

element Finite Element Method using GPU solve problems

in nonlinear finite deformation analysis. High Performance

Conjugate Gradient (HPCG) algorithm explained in [8] on

GPU.

Paper ID: SUB155753 2065

• International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064

Index Copernicus Value (2013): 6.14 | Impact Factor (2013): 4.438

Volume 4 Issue 6, June 2015

www.ijsr.net Licensed Under Creative Commons Attribution CC BY

Problem solving by FEM

Figure 2: 1Dimensional Bar Problem

a) Generating Local Stiffness Matrix According to hook's law stress directly proportional to strain

f=ku (3)

then stiffness is given by equation (2) where A is area of

cross section, E young's modulus, L is length of the bar, f is

force acting on bar, u is displacement.

let and be the force acting on bar in X and Y direction

respectively.

=

(5)

=

(6)

Then is reaction to force i.e. opposite to

therefor = -

=

( ) (7)

=

( ) (8)

[

]=k [1 11 1

] [

] (9)

b) Generating global stiffness matrix (Assembling) In the second step all local stiffness matrices generated from

previous steps are assembled to produce the stiffness matrix

(K). Assembled matrix looks like as follows

K=[

1 1 0 0

1 1+k2 0

0 0

] (10)

This matrix is sparse, symmetric and tridiagonal. When the

arrangement of the elements get changed the stiffness matrix

is also get changed so there is no guarantee that stiffness

matrix generated will be symmetric and tridiagonal. By

putting stiffness matrix in equation (2) we will generate a

system of linear equations of the form shown in equation

(1).Then next step is solving this system.

c) Solving the system of linear equations There are several methods of solving system of linear

equations divided direct and iterative method. Direct

methods like gauss-elimination, LU decomposition,

cholesky's decom- position[9] are somewhat costlier in

terms of memory and number of iterations. As in this case

the matrix generated is tridiagonal and symmetric Thomas

Algorithm[10] can be used to reduce space complexity.

Because in Thomas algorithm no need to store whole

stiffness matrix, only diagonal is stored so space is reduced

from 2to 3. Like other direct methods solution is found by forward or backward substitution which takes n iterations in

Thomas algorithm. On the other hand iterative methods are

more suitable in terms of space and time complexity. One of

the best iterative methods is conjugate gradient method. It

converges towards the solution faster than the other

methods. The main drawback of direct method is we have to

followiterations forequations. In CG method we can iterate less than , it depends on up to what approximation we can tolerate the error. If the solution to this equation

represents the deformation takes place in the bar at various

positions. The stiffness matrix generated from assembling

process is a very large sparse matrix, to solve this very large

memory space required. So another method called Element

by Element Finite Element Method (EBEFEM) [11] is used

to solve the 1D bar problem. EBEFEM is solved by using

preconditioned conjugate gradient method.

3. Element by Element Finite Element Method

Element-by-element (EBE) algorithm implements the

conjugate gradient method at the element level. It reduces

solving the linear set of equations Ax=b for the whole system to

() ()=b

(11)

That is instead of generating the global stiffness matrix,

solution is find out at element level. This system is

explained by law[12]. In equation (11) ()is an element level stiffness- matrix ()vector of unknowns at element level and ()is right hand side known vector. The reason behind such element level computations is, easy for

parallelization. On parallel architecture each processor will

compute these computations locally hence speedup will be

achived. Further, Liu and Yang [13] combined a similar

model proposed by Hughes, Levit et al. [1] with this model

and implemented an element by element (EBE) Jacobi

preconditioned conjugate gradient (PCG) method.

4. Graphics Processing Unit

Graphics processing unit (GPU) most of the times it is also

called as visual processing unit (VPU).GPU is not a

standalone device it is works with CPU. Most of the

computers now a days comes with GPUs.GPU accelerated

computing combined with CPU together used to accelerate

the scientific, analytical, etc. applications. Generally GPUs

considers two architectures memory architecture and host

device architecture. In GPUs thousands of cores are present

as compare to CPUs now a days 4, 8, 16 cores are present.

Though GPUs having these many cores it cannot be used as

stand

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