Integrare Numerica - FEM

Chapter 11 Numerical Integration_____________________________________________

112

CHAPTER 11 NUMERICAL INTEGRATION The explicit form of F (s,t) or G (s,t) are usually not available (or, sometimes, they are very difficult to explicit). For this reason, numerical integration has to be used, which is not a severe penalty. The standard practice is to apply the Gaussian integration, because the procedure employs a small number of integration points to achieve the desired level of accuracy. This is important for efficient element calculations, since at each integration point a matrix product should be evaluated. Other methods are also available, such as the Newton – Cotes quadrature. For simplicity, the 1D case will be described first, summarizing some principles. 11.1 ONE-DIMENSIONAL NUMERICAL INTEGRATION To calculate numerically the integral of a function defined over the one-dimensional domain [-1, 1], the general expression is

)()(1

1 1i

n

i sfHdssfI ∫ ∑−

== (11.1)

Fig. 11.1 One dimensional numerical integration

+1 0 -1

22 ),( Hsf

1s s

11),( Hsf

)(sf

2s 2l =∆

______________________Basics of the Finite Element Method Applied in Civil Engineering

113

where si are n selected points in the [-1, 1] interval and Hi are numerical coefficients depending on n, called integration weights. Two procedures can be applied: • The Newton - Cotes quadrature, where the si points are selected at equal

intervals. For n = 2 it becomes the trapezes rule:

)1()1(2

)1()1( fflffI +−=∆+−

=

while for n = 3 it becomes the Simpson’s one-third rule:

[ ])1()0(4)1(31 fffI ++−=

a.

b.

Fig. 11.2 The Newton-Cotes quadrature: n = 2 (a) and n = 3 (b).

+1 0 -1

f(s)

f(+1) f(-1)

s

+1 0 -1

f(s)

f(0) f(+1) f(-1)

s


114

• The Gaussian quadrature, where the points si and the numerical coefficients Hi are located to aim for the best accuracy.

Assuming a polynomial expression for f(s), it is easy to see that for n sampling points, 2n unknowns (Hi and si) should be specified. Hence, a polynomial of (2n - 1) degree can be defined and exactly integrated. A solution for choosing si and Hi can be obtained explicitly in terms of the Legendre polynomials (the process in frequently known as the Gauss - Legendre quadrature). For example let’s consider an n = 2 sampling points integration. The degree of the polynomial expression will be 2n - 1 = 3. The general polynomial form is

33

2210)( sasasaasf +++=

and the definite integral to be calculated

( )dssasasaaI ∫−

+++=1

1

33

2210

Integrating term by term, a four-line algebraic system yields:

( ) 022110

1

10 2asHsHadsa =+=∫

−

( ) 01

1221111 =+=∫

−

sHsHasdsa

( ) 2

1

1

222

2112

22 3

2 asHsHadssa =+=∫−

( ) 01

1

322

3113

33 =+=∫

−

sHsHadssa

Finally, the sample point’s position si and the integration weights Hi are determined by solving the algebraic system. For the peculiar case, the solution for the unknowns is:


115

121 == HH 3

11 =s

31

2 −=s

Similar procedures to find the (si, Hi) pairs can be applied for n = 3, 4 ... . The weight coefficients and the abscissas are tabulated. Generally, a one-dimensional Gaussian rule with n points integrates exactly polynomials up to 2n – 1 order. It is called the degree of the formula. The first one-dimensional Gaussian rules are represented in figure 11.3. The defined integrals have the following approximate values, for:

- one integration point ∫−

≈1

1

)0(2)( fdssf

- two integration points )3/1()3/1()(1

1

ffdssf +−≈∫−

- three integration points )5/3(95)0(

98)5/3(

95)(

1

1

fffdssf ++−≈∫−

11.2 TWO-DIMENSIONAL NUMERICAL INTEGRATION The numerical integrations for rectangular domains are called product rules. They follow the one-dimensional procedure for each independent variable on turn. To apply these rules, the integrant defined over a square area, centered on the origin of the natural coordinate system, should reduce to the canonical form

∫ ∫ ∫∫− − −−

=1

1

1

1

1

1

1

1

),(dd),( dttsfdststsf (11.2)

for which the numerical integration procedure yields:

),(dd),(1

1 1 1

1

1jij

n

i

n

ji tsfHHtstsfI ∫ ∑∑∫

− = =−

== (11.3)


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a.

b.

c.

Fig. 11.3 One dimensional Gaussian integration rules: (a) one integration point; (b) two integration points; (c) three integration points

s

f(s)

s1= 0

f(s2) f(s3) f(s1)

532 =s532 −=s+1 -1 0

+1 0 -1 s

)(sf

f(0) Approximate area 2f(0)

0,577...1 =s 0,577...2 −=s 0 +1 -1

)( 2sf

s

)( 1sf

f(s)


117

The most evident way to calculate the defined integral is to evaluate firstly the inner term, keeping t constant

)(),(),(1

1 1

ttsfHdstsfI i

n

i Φ=== ∫ ∑−

(11.4)

Evaluating the outer integral in a similar mode, we have:

),(),()()(1 111

1

1 1jij

n

j

n

iiji

n

ii

n

jjj

n

jj tsfHHtsfHHtHdttI ∑∑∑∑∫ ∑

= ===− =

==Φ=Φ=

(11.5) In the above example it was assumed to have the same number of integrating points on each direction, n = n1 = n2. Obviously, this is not compulsory and sometimes the use of a different number of integration points on each direction could be favorable. The two-dimensional Gaussian product rule with n = n1 = n2 = 2 is illustrated in figure 11.4.

Fig. 11.4 Two-dimensional Gaussian product rule (n=2).Integration points

position

t

s

+1

-1 +1

-1

0,577...=s

0,577...−=t

0,577...−=s

0,577...=t ( )12 , ts ( )12 , ts

( )22 , ts ( )11, ts


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11.3 THREE-DIMENSIONAL NUMERICAL INTEGRATION The numerical integration over a cube with the edge length equal to 2, centered in the origin of coordinates system, is carried out using the same rules as for the 1D and the 2D space. Similarly, for an integrant defined over the cubic volume that can be reduced to the canonical form, we have:

),,(),,(1 1 1

1

1

1

1

1

1kjik

n

i

n

jj

n

ki rtsfHHHdsdtdrrtsf ∑∑∑∫ ∫ ∫

= = =− − −

= (11.6)

The integration points’ position and the integration weights are determined by extending the one-dimensional procedures. 11.4 REQUIRED ORDER OF NUMERICAL INTEGRATION The computing time for numerical integration can be quite significant - especially in 3D analyses, where the BTEB matrices are large. Therefore, it is worthwhile to determine:

- the minimum integration order required to provide convergence; - the integration order requirement to preserve the same convergence rate that would result by exact integration.

For C0 class problems, the first derivatives may be, at limit, constant values (such as the components of internal deformation energy ∫= VE T dσε ). Consequently, with the aim of accomplishing convergence, integrals such as

∫ EBdVBT should evaluate correctly the element’s volume. Thus,

∫ dsdtdrJdet has to be exactly calculated in local coordinates. Supposing that the polynomial equivalence of

∫ ∫ ∫ ∫=s t r

T dsdtdrFdV J EBB det

is defined as being of order m, than the required order of integration is


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12+=

mn (11.7)

Because the Jacobian is variable for distorted elements, the evaluation of m is not an easy task. Therefore, the numerical integration essentially relies on experiment. 11.5 CONCLUDING REMARKS The versatility of isoparametric finite elements and the advantages of numerical integration are evident. In computer programming, to increase the element order from linear to quadratic or cubic, is only a matter of changing the computation subroutines for the shape functions and their derivatives, as well as for the Jacobian. A larger number of DOF leads only to the growth of characteristic matrices B and k. The geometrical shape of the finite elements can be extremely divers, especially for those with higher order, curved edges. For a given class of problems, the general matrices always have the same form. To evaluate the elemental matrix k and load vector r, only the shape functions, their derivatives and the integration order should be specified. The same shape functions can be used in many different types of problems.

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Integrare Numerica - FEM