Ross Advanced Applied Finite Element Methods

1. Auflage 1998
ISBN: 978-0-85709-975-4
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 480 Seiten

This book is aimed at senior undergraduates, graduates and engineers. It fills the gap between the numerous textbooks on traditional Applied Mechanics and postgraduate books on Finite Element Methods. - Fills the gap between the applied mechanics and finite element methods - Discusses basic structural concepts and energy theorems, the discrete system, in-plane quadrilateral elements, field problems and mathematical modelling, among other topics - Aimed at senior undergraduates, graduates and engineers

Autoren/Hrsg.

Ross, Carl T. F.

Leseproben

1

Matrix algebra

Publisher Summary

The approach in this chapter is based on technique rather than on rigorous mathematical theories. It commences with various matrix definitions, followed by the laws of matrix algebra. To demonstrate the latter, several examples are worked out in detail and particular attention is paid to the inverse of a matrix and the solution of homogeneous and non-homogeneous simultaneous equations. A vector has both magnitude and direction, and typical vector quantities appear in the form of velocity, displacement, force, weight, etc. A matrix in its most usual forms is an array (or table) of scalar quantities, consisting of rows by columns. The elements of the matrix need not necessarily be scalars, but can take the form of vectors or even matrices. This compact method of representing quantities allows matrices to be particularly suitable for modeling physical problems on digital computers. A null matrix is one that has all its elements equal to zero. A diagonal matrix is a square matrix where all the elements except those of the main diagonal are zero.

The approach in this chapter is based on technique rather than on rigorous mathematical theories. It commences with various matrix definitions, followed by the laws of matrix algebra. To demonstrate the latter, several examples are worked out in detail, and particular attention is paid to the inverse of a matrix and the solution of homogeneous and non-homogeneous simultaneous equations.

If the reader requires a greater depth of understanding of matrix algebra then he/ she should study references [20–23].

1.1 DEFINITIONS

A in its most usual form can be described as a number which is positive or negative or zero. Typical examples of scalars are 1, 2, p, e, -1.57, 2 × 1011, etc., and typical scalar quantities appear in the form of temperature, time, mass, length, etc. Scalars have only magnitude.

A has both magnitude and direction, and typical vector quantities appear in the form of velocity, displacement, force, weight, etc.

A in its most usual forms is an array (or table) of scalar quantities, consisting of rows by columns, as shown in (1.1). The elements of the matrix need not necessarily be scalars, but can take the form of vectors or even matrices. This compact method of representing quantities allows matrices to be particularly suitable for modelling physical problems on digital computers.

A]=[A11A12A13.....A1nA21A22A23  .....A2nA31A32A33.....A3n............Am1Am2Am3.....Amn]. (1.1)

(1.1)

A of a matrix is defined as a horizontal line of quantities.

A of a matrix is defined as a vertical line of quantities.

The quantities 11, 12, 13, etc. are said to be the of the matrix [].

The of a matrix is defined by its number of rows x its number of columns. Thus, the matrix of (1.1) is said to be of order x .

A matrix is where = 1, as in (1.2).

A}={A11A21...Am1}. (1.2)

(1.2)

A is where = 1, as in (1.3).

A?=?A11A12....A1n?. (1.3)

(1.3)

A is where = , as in (1.4).

A]=[A11A13....A1nA21A22....A2n.........An1An2....Ann]. (1.4)

(1.4)

The square matrix of (1.4) is said to be of order .

The of a matrix is obtained by interchanging its rows with its columns, i.e. the transpose of a matrix is obtained by making its first column, its first row, and its second column, its second row, and so on and so forth. For example if

A]=[0-1234-5]

then the transpose of [] is given by

A]T=[03-142-5].

A is a matrix whole elements themselves are matrices, as shown in (1.5).

A]=[A11A12A13......A1nA21A22A23A2nA31A32A33A3n................Am1Am2Am3......Amn].=[abcd] (1.5)

(1.5)

where,

a]=[A11A12A21A22][b]=[A13.......A1nA23.......A2n][c]=[A31A32........Am1Am2][d]=[A33.......A3n........Am3Amn].

The matrix of (1.5) is to be partitioned, as shown by the broken lines. Matrix partitioning is found to be a very useful aid when isolating certain physical features within the matrix.

A matrix is one which has all its elements equal to zero.

A is a square matrix where all the elements except those of the main diagonal are zero, as in (1.6).

A]=[A110.......00A2200A33...........00Ann]. (1.6)

(1.6)

is a diagonal matrix where all the diagonal elements are equal to the same scalar quantity. When the scalar quantity is unity, the matrix is called the or , and is denoted by [I].

An is a matrix which contains all its non-zero elements in and above its main diagonal, as in (1.7).

A11A12A13...A1n0A22A23...A2n00A33...A3n............000Ann]. (1.7)

(1.7)

A is one which contains all its non-zero elements in and below its main diagonal, as in (1.8).

A110.....0A21A22..........An1An2.....Ann]. (1.8)

(1.8)

A has all its non-zero elements contained in a diagonal strip as shown in (1.9) and (1.10). The centre diagonal of the strip is not necessarily the main diagonal.

A11A120000....0A21A22A23000....00A32A33A340....000A43A44A450....0.......................................000.0An,n-1An,n] (1.9)

(1.9)

The bandwidth of the matrix of (1.10) is said to be NW.

(1.10)

(1.10)

A is where all

ij=Aji.

The of a matrix is obtained by summing all the elements on its leading diagonal, as follows:

[A]=?i=1nAii,

and the of a square matrix consists of the elements A11, A22, A33,…, .

1.2 ADDITION AND SUBTRACTION OF MATRICES

If

A]=[1-123-45]

and

B]=[-40-216-7]

then,

A]+[B]=[(1-4)(-1+0)(2-2)(3+1)(-4+6)(5-7)]=[-3-1042-2].

Similarly,

A]-[B]=[(1+4)(-1-0)(2+2)(3-1)(-4-6)(5+7)]=[5-142-1012].

1.3 MATRIX MULTIPLICATION

In the relationship [][] = [], [] is known as the , [] the and [] the . Furthermore, if [] is of order x and [] is of order x then [] is of order x . It should be noted that [] must always have its number of rows equal to the number of columns in [].

If

A]=[12-103-4]

and

B]=[-230-141],

then [] is obtained by multiplying the columns of [] by the rows of [], so that, in general,

ij=?k=1nAik×Bkj,

i.e. to obtain each , the th row of [] must be premultiplied into the th column of [], as follows:

C]=[1×-2+2×(-1)1×3+2×41×0+2×1-1×-2+0×(-1)-1×3+0×4-1×0+0×13×-2-4×(-1)3×3-4×41×0-4×1]=[-41122-30-2-7-4].

Similarly, if

A}={1-23}

and

B?=?-145?

then

A}?B?=[1×(-1)1×41×5-2×(-1)-2×4-2×53×(-1)3×43×5]=[-1452-8-10-31215]

and

B?{A}=(-1×1)+(4×-2)+(5×3)=-1-8+15=6.

Thus, in general, the vector product {}[] will...