8. chapter 8 - types of finite elements _a4

7/27/2019 8. Chapter 8 - Types of Finite Elements _a4

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Chapter 8 Types of Finite Elements___________________________________________

80

CHAPTER

TYPES OF FINITE ELEMENTS

This chapter deals with general aspects regarding the finite elements

properties, which are necessary to be fulfilled in order to achieve a

convergent approximate solution. Also, some examples of shape functionsdefinition will be presented.

8.1 SHAPE FUNCTIONS PROPERTIES

First of all, a trivial statement should be emphasized. The real domain which

makes the object of the F.E.A. (a structural component, a solid subjected to

heat transfer or a ground region experiencing seepage) is always a

continuum with an infinite number of points and a diversity of material

properties. Theoretically, in order to achieve the exact solution of a

continuum problem, a discrete model with an infinite number of point-size

elements, following the exact geometry, materials distribution and boundary

conditions, should be created. Because such a refined decomposition wouldlead to the best solution, it is obvious that the element size reductionis the

most opportune procedure to improve the finite element model.

Unfortunately, the discrete models may have only a restricted number of

elements, due to the bounded computer memory and the limited available

computing time. Thus, two questions arise: how good the approximation is

and how can the discrete solution be improved to reach, as close as possible,

the exact solution? For the first question the answer is rather difficult or

impossible because the exact solution is usually unknown. Experimental

results or error estimations are necessary. Regarding the second one, the

answer is that the approximation process should be a convergent one.

As it was already shown, in the F.E. approach the unknown function u(x,y,z)is expressed over an element in terms of its nodal values, due to a set of

shape functionsNi, such as


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______________________Basics of the Finite Element Method Applied in Civil Engineering

81

=iiuNzyxu ),,( (8.1)

with uithe nodal values or nodal parameters.

The total number of available parameters ui depends on the number of

elements in the mesh (m) and the number of nodal points per element (n). If,

for an increasing number of parameters (due to the augmentation of nodal

points nand elements number m), a better approximation of the true solution

is obtained, the approximation process is convergent (see fig. 8.1). Of

course, increasing the number of elements equals to elements size

reduction.

Fig. 8.1 Convergence to the exact solution by augmenting the parameters number

8.1.1 Classes of continuity

When performing a F.E.A., i.e. solving a field problem by a discrete

procedure, some continuity classes can by defined regarding the unknown

function u and its derivatives, over the elements domain and across the

boundaries between adjoined elements:

C0 - the unknown function u is continuous across the boundaries

between elements; the first derivatives

x

u are continuous

over the element but not across the boundary between elements; the

higher order derivatives, if they exist, do not necessarily have a finitevalue;

exact solution

z)y,u(x,

(u)ai


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C1 - the unknown function u and its first derivatives

xu are

continuous across the boundaries between elements; the second

derivative is not continuous but it is finite;

Cn

- the unknown function u and its "nth

" order derivatives

n

n

x

x

x

uu ..., are continuous across the boundary between

elements.

The necessary class of continuity is related to the encountered derivative

order of the unknown function u within the general expression of the

variational principle

+

=

ee

dx

uuGdD

x

u

x

uuFE

D n

n

e ,...),(),...,( (8.2)

For a given order of derivatives n, the approximate solution converges to the

exact one by reducing the elements size, if the following conditions are

fulfilled:

- the compatibility condition meaning that at the common frontier

between two elements a Cn-1class of continuity exists;- the completeness condition meaning that inside the element, a C

n

class of continuity is available.

Stress and stain field problems of massive structures are usually C0 class

problems (see figure 5.2). The nodal parameters are the values of the

unknown function (the displacement) in the nodal points of the element.

Structural problems related to shell or membrane structures are C1 class

problems. This time, the nodal parameters are both the values of the

unknown function and its first derivatives in the nodal points of the element

(displacements and curvature).

For massive structures, the stress field in 2D plane stress is

E=


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83

with

T

y

u

x

u

= ... and

=

2

100

01

01

12

EE

For plates or shells, the so called generalized stresses are

gg

xy

y

x

g

m

m

m

E =

= (8.3)

with the generalized strains

=

yx

w

y

wx

w

g

2

2

2

2

2

(8.4)

and the elasticity matrix

=

2

100

01

01

)1(122

3

EtgE (*) (8.5)

Thus, for shell elements, the functional contains second order derivatives

and requires a C1 continuity. In order to ensure the continuity of the first

order of derivatives the displacements (lateral deflection) wand the slopes

yw

xw , are used as nodal values. The DOF per node are

yw

xww

,, .

* These last relationships will be developed later


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84

elements

elements

elements

elem.i elem.j

discontinuous but finite

not continuous

u

x

u

2x

u

2

element

boundary

Fig. 8.2 Continuity provided by C0class finite elements across their boundaries

8.1.2 Shape functions requirements for convergence

Three other criteria for convergence attainment withdraw from the solid

mechanics are stated:

Criterion 1 - the shape function should not permit (or cause) element

straining when the nodal displacements correspond to a rigid body

movement.


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85

Criterion 2- the shape functions should lead to a constant strain field overthe elements domain if the nodal loads or displacements are compatible with

constant strains.

Criterion 3 the shape functions should lead to finite strain values on

element boundaries.

These three criteria are usually satisfied by all the known finite elements and

have to be verified only for the new ones developed.

8.2 TYPES OF FINITE ELEMENTS*

As it was stated before, in a finite element approximation the unknownfunction ucan be a scalar (as temperature, hydraulic head) or a vector with

more components (as displacement). Consequently, the number of DOF per

node depends on the physical meaning of the problem. There are elements

with 1 DOF/node, 2 DOF/node, 3 DOF/node or 6 DOF/node.

The shape functions which define the distribution of the unknown over the

elements domain are usually polynomials. Regarding the polynomial order,

the elements can be linear, quadratic, cubic, etc. This designation has

nothing to do with the elements geometric shape but emphasizes the degree

of the polynomial which defines the shape function.

Linear finite elements are the simplest ones, using first degree polynomialsas shape functions. Also, the approximation depends on accepted dimension

of the problem. The shape functions have the following forms:

- for the one dimensional space (1D):

xaaxu 21)( += (8.6)

- for the two dimensional space (2D):

xyayaxaayxu 4321),( +++= (8.7)

- for the three dimensional space (3D):

* The paragraph is dealing only with elements for general modeling purpose.

Special elements will be exemplified later.


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86

xyzazxayzaxyazayaxaazyxu 87654321),,( +++++++= (8.8)

Concerning the geometric shape of linear elements, the 1D element is a

straight line between two nodes, the 2D element is a quadrilateral plane with

straight edges which can degenerate into a triangle by merging two nodes

and the 3D element is a hexahedron (a six plane-faced volume, also called

brick element) which can degenerate into a prism by merging two edges

or into a tetrahedron by merging four nodes (see figure 8.3).

Fig. 8.3 Types of linear C0class finite elements: a. 2D space; b. 3D space

In order to determine the polynomial coefficients ai, the u= uicondition for

the nodal points should be applied. Therefore, the number of nodes nshould

equal the number of independent parameters ai, i.e. 2 nodes for the 1D (line)

linear element, 4 nodes for the 2D (plane) linear element and 8 nodes for the

3D (hexahedron) linear element. Note that for the 2D and 3D spaces, the

unknown function (if displacement) becomes a vector with components

along the Cartesian system coordinates.

Another type of C0 class of element is the quadratic element. Its shape

functions are second order polynomials and the approximation of theunknown function over the element (in 2D space) has the following form:

1

2

12

3

4

1

2

3

12

3

45

8

16

7

2

3

4

5

6

1

2

3

4

a.

b.


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87

yxaxyayaxaxyayaxaayxu2

8

2

7

2

6

2

54321),( +++++++= (8.9)

In the same way, polynomial shape functions for 1D and 3D elements can

be written. In order to determine the independent parameters, the quadratic

element should have a suitable number of nodes: 3 for 1D element, 8 for 2D

element and 20 for 3D elements (see figure 8.4). The presence of the mid-

side nodes enables the defining of a curved element shape, for edges in 2D

or faces in 3D.

Fig. 8.4 Types of quadratic C0class finite elements

Concluding, the finite elements can be classified by their class of continuity

and their shape functionspolynomial order. The first aspect affects the

number of DOF per node while the second one the number of nodes per

element.

8.3 COMPLETENESS REQUIREMENTS

In order to get a monotonically convergence toward the exact solution by

increasing the DOF number, two additional conditions should be fulfilled:

1. The elements compatibility in the mesh meaning that elements

remain in contact at their boundaries (no overlapping or gaps should

be encountered); for thin plane or curved shells also the tangent to

the deformed surface of adjacent elements is common (C1 class

continuity is required);

2. No preferential directions for the element behavior should be

defined*.

1

2

3

12

3

4

5

6

7

8

1

2

3

4

5 6

789

10

. 11

12

13

14

15

1617

18

19

20

.


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The second condition is equivalent with the completeness criterion, statingthat the necessary condition for a monotonicallyconvergent solution is the

use of complete polynomialsas shape functions. No preferential directions

within the element means that under the same boundary conditions

(prescribed displacements, distributed loads, etc) the element has to have the

same stress and strain field, regardless the element orientation in a certain

coordinate system:

)','(),( yxAyxAA == (8.10)

as it is shown in figure 8.5.

Fig. 8.5 The completeness criterion

If a polynomial of order p is used to define the shape functions N, then it

must be a complete polynomial (the terms are defined by the Pascal

triangle). Thus, the complete polynomials are

xyayaxaayxd 4321),( +++= for linear,2

8

2

7

2

6

2

54321),( yaxyayxaxaxyayaxaayxd +++++++= for quadratic

and so on. Similar developments are available for 3D problems.

1x1y

A (x,y)

(x1,y1)

p

* The heterogeneity of material properties is excluded.

8. chapter 8 - types of finite elements _a4

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