Chapter 4 ONE - DIMENSIONA INTEGRATION 4. Introduction Since the finite element method is based on integral relations it is logical to expect that one should strive to carry out the integrations as efficiently as possible. In some cases we will employ exact integration. In others we may find that the integrals can become too complicated to integrate exactly. In such cases the use of numerical integration will prove useful or essential. The important topics of local coordinate integration and Gaussian quaatures will be introduced here. They will prove useful when dealing with higher order interpolation functions in complicated element integrals. 4. ocal Coordinate Jacobian We hav e previously seen that the utilization of local element coordinates can greatly reduce the algebra required to establish a set of interpolation functions. ater we will see that some -D elements must be formulated in local coordinates in order to meet the interelement continuity requirements. Thus, we should expect to often encounter local coordinate interpolation. However, the governing integral expressions must be evaluated with respect to a unique global or physical coordinate system. Clearly, these two coordinate systems must be related. The relationship for integration with a change of variable (change of coordinate) was defined in elementary concepts from calculus. At this point it would be useful to review these concepts from calculus. Consider a definite integral (4.) b f (x) dx, a < x < b a where a new variable of integration, r, is to be introduced such that x = x (r ). Here it is required that the function x (r ) be continuous and have a continuous derivative in the interval α r β. The region of r directly corresponds to the region of x such that when r varies between α and β, then x varies between a = x (α ) and b = x ( β ). In that case b f (x) dx = (4.) a β f ( x(r)) dx α 4.3 Draft 5/7/04 004 J.E. Akin 5
6 J. E. Akin or β f (r) J (4.3) where J = dx / is called the Jacobian of the coordinate transformation. 4.3 Exact Polynomial Integration * α If we utilize the unit coordinates, then α = 0 and β =. Then from Sec. 3., the Jacobian is J = in an element domain defined by linear interpolation. By way of comparison, if one employs natural coordinates, then α =, β = +, and from Eq. 3.6 J = /. Generally, we will use interpolation functions that are polynomials. Thus, the element integrals of them and/or their derivatives will also contain polynomial terms. Therefore, it will be useful to consider expressions related to typical polynomial terms. A typical polynomial term is r m where m is an integer. Thus, from the above xe x e r m dx = x r m = r (+m) = e 0 + m ( + m). 0 A similar expression can be developed for the natural coordinates. It gives nm dx = m + 0 if m is odd if m is even. (4.4) (4.5) ater we will tabulate the extension of these concepts to two- and three-dimensional integrals. As an example of the use of Eq. 4.5, consider the integration of the outer product of H e with itself: HeT H e dx. Recall that the integral of a matrix is the matrix resulting from the integration of each of the elements of the original matrix. If linear interpolation is selected for H e on a line element then typical terms will include H, H H, etc. Thus, one obtains : so that I = I = I = H (r) dx = H H dx = H dx = ( r) dx = ( / + /3) = / 3 ( r) rdx = e (/ /3) = / 6 r dx = / 3 e 6 Similarly, if one employs the agrangian quaatic H in Eq. 4.7 one obtains :. (4.6) 4.3 Draft 5/7/04 004 J.E. Akin. All rights reserved.
Finite Elements, -D Integration 7 e 30 4 4 6. (4.7) By way of comparison, if one selects the hierarchical quaatic polynomial in Eq. 4. the above integral becomes e 6 /4 /4 /4 /4 /0 Note that the top left portion of this equation is the same as Eq. 4.7 which was obtained from the linear polynomial. This desirable feature of hierarchical elements was mentioned in Sec. 3.6. Of course, all of the above integrals are so simple one could note that the integral is simply the area under the curve and there are simple algebraic relations for defining the area under power law curves. Figure 4.3. reviews such relations and illustrates the terms needed in Eq. 4.6. Before leaving the subject of simplex integrations one should give consideration to the common special case of axisymmetric geometries, with coordinates ( ρ,z). Recall from calculus that the Theorem of Pappus relates a differential volume and surface area to a differential area and length in the ( ρ,z) plane of symmetry, respectively. That is, dv = π ρ da and ds = π ρdl, where ρ denotes the radial distance to the differential element. Thus, typical axisymmetric surface integrals reduce to. y Area = b h / (n + ) y = c x n h H Area = / 3 b x H Area = / Area = / 4 - ( / )(¼) / 3 = / 6 H * H 0.5 Figure 4.3. Exact integrals by inspection 4.3 Draft 5/7/04 004 J.E. Akin. All rights reserved.
8 J. E. Akin I Γ = HT d Γ=π Γ HT ρdl = π HT Hdl ρ e = π e 6 ρ e, since ρ = Hρ e. Many workers like to omit the π term and work on a per-unit-radian basis so that they can more easily do both two-dimensional and axisymmetric calculations with a single program. 4.4 Numerical Integration Numerical integration is simply a procedure that approximates (usually) an integral by a summation. To review this subject we refer to Fig. 4.4.. Recall that the integral b a f (x) dx (4.8) can be viewed graphically as the area between the x-axis and the curve y = f (x) inthe region of the limits of integration. Thus, we can interpret numerical integration as an approximation of that area. The trapezoidal rule of numerical integration simply approximates the area by the sum of several equally spaced trapezoids under the curve between the limits of a and b. The height of a trapezoid is found from the integrand, y j = y( x j ), evaluated at equally spaced points, x j and x j+. Thus, a typical contribution is A = h( y j + y j+ ) /, where h = x j+ x j is the spacing. Thus, for n q points (and n q spaces), the well-known approximation is y A = area y = f(x) x A = a b y dx j w j f (x j ) a b y a polynomial -d d y y y 3 x x x 3 y j = f(x j ) x b y y j = f(x j ) h w j =h w =w n =h/ y y y 3 y n x a = x x... x n =b Figure 4.4. One-dimensional numerical integration x 3 4.3 Draft 5/7/04 004 J.E. Akin. All rights reserved.
Finite Elements, -D Integration 9 I h (4.9) y + y + y 3 +... + y n + y n, I Σ n w j f (x j ) j= where w j = h, except w = w n = h /. A geometrical interpretation of this is that the area under curve, I, is the sum of the products of certain heights, f (x j ) times some corresponding widths, w j. In the terminology of numerical integration, the locations of the points, x j, where the heights are computed are called abscissae and the widths, w j, are called weights. Another well-known approximation is the Simpson rule, which uses parabolic segments in the area approximation. For most functions the above rules may require 0 to 40 terms in the summation to yield acceptable accuracy. We want to carry out the summation with the minimum number of terms, n q, in order to reduce the computational cost. What is the minimum number of terms? The answer depends on the form of the integrand f (x). Since the parametric geometry usually involves polynomials we will consider that common special case for f (x). The famous mathematician Gauss posed this question: What is the minimum number of points, n q, required to exactly integrate a polynomial, and what are the corresponding abscissae and weights? If we require the summation to be exact when f (x) is any one of the power functions, x, x,..., x n, we obtain a set of n conditions that allow us to determine the n q abscissae, x i, and their corresponding n q weights, w j. The n q Gaussian quaature points are symmetrically placed with respect to the center of the interval, and will exactly integrate a polynomial of order (n q ). The center point location is included in the abscissae list when n q is odd, but the end points are never utilized in a Gauss rule. The Gauss rule data are usually tabulated for a non-dimensional unit coordinate range of 0 t, or for a natural coordinate range of t +. Table 4. presents the low-order Gauss rule data in natural coordinates, and the alternate unit coordinate data are in Table 4.. A two-point Gauss rule can often exceed the accuracy of a 0-point trapezoidal rule. When computing norms of an exact solution, to be compared to the finite element solution, we often use the trapezoidal rule. That is because the exact solution is usually not a polynomial and the Gauss rule may not be accurate. Sometimes it is desirable to have a numerical integration rule that specifically includes the two end points in the abscissae list when (n ). The obatto rule is such an alternate choice. Its n q points will exactly integrate a polynomial of order (n 3) for n q >. Its data are included in Table 4.3. It is usually less accurate than the Gauss rule but it can be useful. Mathematical handbooks give tables of Gauss or obatto data for much higher values of n q. Some results of Gauss s work are outlined below. et y denote f (x) in the integral to be computed. Define a change of variable x(n) = / (b a) n + / (b + a) (4.0) so that the non-dimensional limits of integration of n become and +. The new value of y(n) is y = f (x) = f [ / (b a) n + / (b + a)] = Φ(n). (4.) Noting from Eq. 4.0 that dx = / (b a) dn, the original integral becomes 4.3 Draft 5/7/04 004 J.E. Akin. All rights reserved.
0 J. E. Akin Table 4. Abscissas and weights for Gaussian quaature nq n f (x) dx = q Σ w i f (x i ) + ± x i w i 0.00000 00000 00000 00000 0000 n q =.00000 00000 00000 00000 000 0.57735 069 8965 76450 949 n q =.00000 00000 00000 00000 000 0.77459 6669 4483 37703 5835 n q = 3 0.55555 55555 55555 55555 556 0.00000 00000 00000 00000 0000 0.88888 88888 88888 88888 889 0.863 635 9405 575 3946 n q = 4 0.34785 4845 37453 85737 306 0.33998 0435 84856 6480 666 0.654 5548 6546 46 694 0.9067 98459 38663 9979 767 n q = 5 0.369 68850 5689 0875 46 0.53846 930 05683 0903 634 0.4786 86704 99366 46804 9 0.00000 00000 00000 00000 0000 0.56888 88888 88888 88888 889 0.9346 954 035 078 30 n q = 6 0.73 4493 7970 34504 030 0.660 93864 6664 5366 400 0.36076 5730 4838 60756 983 0.386 9860 8396 90863 050 0.4679 39345 769 04738 987 0.9490 793 4758 545 690 n q = 7 0.948 4966 68869 6937 06 0.7453 855 99394 43986 3865 0.7970 5394 8976 66790 47 0.40584 553 77397 6690 6607 0.3883 00505 058 94495 037 0.00000 00000 00000 00000 0000 0.4795 9836 73469 38775 50 i= Table 4. Unit abscissas and weights for Gaussian quaature nq n f (x) dx = q Σ w i f (x i ) 0 i= x i w i 0.50000 00000 00000 00000 000 n q =.00000 00000 00000 00000 000 0.3 48654 0587 774 543 n q = 0.50000 00000 00000 00000 000 0.78867 5345 948 885 457 0.50000 00000 00000 00000 000 0.70 6653 7958 348 08 n q = 3 0.7777 77777 77777 77777 778 0.50000 00000 00000 00000 000 0.44444 44444 44444 44444 444 0.8879 83346 074 6885 79 0.7777 77777 77777 77777 778 0.06943 844 0973 738 803 n q = 4 0.739 745 6876 9868 653 0.33000 9478 0757 86759 867 0.3607 5774 373 073 347 0.66999 057 948 340 33 0.3607 5774 373 073 347 0.93056 8557 9706 876 97 0.739 745 6876 9868 653 0.0469 00770 30668 00360 9 n q = 5 0.846 3445 8094 54375 73 0.0307 65344 9475 84544 88 0.393 4335 49683 340 065 0.50000 00000 00000 00000 000 0.8444 44444 44444 44444 444 0.7693 46550 584 5455 86 0.393 4335 49683 340 065 0.95308 999 6933 99639 88 0.846 3445 8094 54375 73 4.3 Draft 5/7/04 004 J.E. Akin. All rights reserved.
Finite Elements, -D Integration Table 4.3. Abscissas and weight factors for obatto integration nq n f (x) dx q Σ w i f (x i ) + i = ± x i w i 0.00000 00000 00000 n q =.00000 00000 00000.00000 00000 00000 n q =.00000 00000 00000.00000 00000 00000 n q = 3 0.33333 33333 33333 0.00000 00000 00000.33333 33333 33333.00000 00000 00000 n q = 4 0.6666 66666 66667 0.447 35954 99958 0.83333 33333 33333.00000 00000 00000 n q = 5 0.0000 00000 00000 0.65465 36707 07977 0.54444 44444 44444 0.00000 00000 00000 0.7.00000 00000 00000 n q = 6 0.06666 66666 66667 0.76505 5339 9465 0.37847 4956 97847 0.853 564 80645 0.55485 83770 35486 (b a) Φ(n) dn. (4.) Gauss showed that the integral in Eq. 4. is given by nq n Φ(n) dn = q Σ W i Φ (n i ), i= where W i and n i represent tabulated values of the weight functions and abscissae associated with the n q points in the non-dimensional interval (, ). The final result is n (b a) q Σ W i Φ(n i ) = i= n q Σ f ( x(n i ) ) W i. i= (4.3) Gauss also showed that this equation will exactly integrate a polynomial of degree (n q ). For a higher number of space dimensions (which range from to + ), one obtains a multiple summation. Since Gaussian quaature data are often tabulated in references for the range n +, it is popular to use the natural coordinates in defining element integrals. However, one can convert the tabulated data to any convenient system such as the unit coordinate system where 0 r. The latter may be more useful on triangular regions. As an example of Gaussian quaatures, consider the following one-dimensional integral : x x ( + x dx = ) F(x) dx. If two Gauss points are selected (n q = ), then the tabulated values from Table 4. give W = W = and r = 0. 57735 = r. The change of variable gives x(r) = (r + 3) /, so that x(r ) =. 788675 and x(r ) =. 35. Therefore, from Eq. 4.3 4.3 Draft 5/7/04 004 J.E. Akin. All rights reserved.
J. E. Akin ( ) W F( x(r ) ) + W F( x(r ) ) = () () sym. (. 788675) + (. 788675) + () sym. (. 35) + (. 35). 00000 3. 00000 3. 00000 5. 66667 which is easily shown to be in good agreement with the exact solution. As another example consider a typical term in Eq. 4.8. Specifically, from Eqs. 4.6 and 4.8 I 33 =, H 3 dx = e + ( n ) dn. Since the polynomial terms to be integrated are fourth order, we should select (n q ) = 4, or n q = 3 for an integer number of Gaussian points. Then, I 33 = I 33 = e H 3 dx = e + H 3 (n) dn 0. 55556. 00000 ( 0. 77459) + 0. 88889. 00000 (0. 0) + 0. 55556. 00000 (+0. 77459) I 33 = e (0. 08889 + 0. 88889 + 0. 08889) = 0. 5333 e, which agrees well with the exact value of 6 / 30, when using 6 digits. 4.5 Variable Jacobians When the parametric space and physical space have the same number of dimensions then the Jacobian is a square matrix. Otherwise, we need to use more calculus to evaluate the integrals. For example, we often find the need to execute integrations along a twodimensional curve defined by our one-dimensional parametric representation. Consider a planar boundary curve in the xy-plane such as that shown in Fig. 4.4.. We may need to know its length and first moments (centroid), which are defined as = ds, x = x(r) ds, y = y(r) ds, respectively, where ds denotes the physical length of a segment,, of the parametric length. To evaluate these quantities we need to convert to an integral in the parametric space. For example, = ds = 0 ds. To relate the physical and parametric length scales, we use the planar relation that 4.3 Draft 5/7/04 004 J.E. Akin. All rights reserved.
Finite Elements, -D Integration 3 r y r x = x (r) y = y (r) ds = J(r) ds dy dx x J + ( dx / ) ( dy / ) Figure 4.4. A variable curve metric or Jacobian ds = dx + dy. Since both x and y are defined in terms of r, we can extend this identity to the needed quantity ds = dx + dy where dx / can be found from the spatial interpolation functions, etc. for dy /. Thus, our physical length is defined in terms of the parametric coordinate, r, and the spatial data for the nodes defining the curve location in the xy-plane (i.e., x e and y e ): where dx(r) = n b = i= dx + 0 Σ dh i(r) xi e, dy(r) dy = n b i= Σ dh i(r) yi e. Note that this does preserve the proper units for, and it has what we could refer to as a variable Jacobian. The preceding integral is trivial only when the planar curve is a straight line (n b = ). Then, from linear geometric interpolation dx / = x e x e and dy / = y e y e are both constant, and the result simplifies to = xe xe + ye ye = 0 which, by inspection, is exact. For any other curve shape, these integrals become unpleasant to evaluate, and we would consider their automation by means of numerical integration. 4.3 Draft 5/7/04 004 J.E. Akin. All rights reserved.
4 J. E. Akin To complete this section we outline an algorithm to automate the calculation of the y-centroid :. Recover the n b points describing the curve, x e, y e.. Recover the n q -point quaature data, r j and w j. 3. Zero the integrals : = 0, y = 0. 4. oop over all the quaature points: j n q. At each local quaature point, t j : A. Find the length scales (i.e., Jacobian): (i) Compute local derivatives of the n b interpolation functions D_H j H r r=r j (ii) Get x- and y-derivatives from curve data dx D_H j x e = nb j Σ H i (r j ) xi e, r i= dy j D_H j y e (iii) Find length scale at point r j ds j B. Evaluate the integrand at point r j : dx = + j dy j (i) Evaluate the n b interpolation functions : H j H( r j ) (ii) Evaluate y from curve data y j = H j y e = Σ N H i (r j ) yi e C. Form products and add to previous values = + ds j w j, y = y+y j ds j w j. 4. Evaluate items computed from the completed integrals : y = y/,etc. for x. Note that to automate the integration we simply need: ) storage of the quaature data, r j and w j, ) access to the curve data, x e, y e, 3) a subroutine to find the parametric derivative of the interpolation functions at any point, r, and 4) a function program to evaluate the integrand(s) at any point, r. This usually requires (see Step 4B) a subroutine to evaluate the interpolation functions at any point, r. The evaluations of the interpolation products, at a point r j, can be thought of as a dot product of the data array x e and the evaluated interpolation quantities H j and D_H j. i= 4.3 Draft 5/7/04 004 J.E. Akin. All rights reserved.
Finite Elements, -D Integration 5 4.6 Exercises. In Tables 4. and 4.3 the sum of the weights is exactly, but in Table 4. the sum is exactly. Explain why.. Note that the Gauss abscissas are always interior to the domain and symmetrically placed. The obatto rules for n q always includes the two end points. Discuss an advantage or disadvantage of including the end points. 3. For a one-dimensional quaatic element use Gaussian quaatures to numerically evaluate the matrices: a) C e = e HT dx, b) M e = e HT H dx, c) S e = e dh T dh dx dx dx, d) Ue = e dh HT dx dx. 4. Use the exact integrals for S e and M e from Problem 3 to hard code the equivalent of Fig..0. for a three node quaatic line element. Note that such an element will also allow for Q(x) = x as well as the original linear source term. In that case, of course, the exact solution changes. 5. Use obatto quaature to evaluate the matrices in Problem 3. Note that M e becomes a diagonal matrix. 4.7 Bibliography [] Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions, National Bureau of Standards (964). [] Carey, G.F. and Oden, J.T., Finite Elements Computational Aspects, Englewood Cliffs: Prentice-Hall (984). [3] Hinton, E. and Owen, D.R.J., Finite Element Programming, ondon: Academic Press (977). [4] Hughes, T.J.R., The Finite Element Method, Englewood Cliffs: Prentice-Hall (987). [5] Smith, I.M. and Griffiths, D.V., Programming the Finite Element Method, 3rd Edition, Chichester: John Wiley (998). [6] Stroud, A.H. and Secrest, D., Gaussian Quaature Formulas, Englewood Cliffs: Prentice-Hall (966). [7] Zienkiewicz, O.C. and Taylor, R.., The Finite Element Method, 5th Edition, ondon: Butterworth-Heinemann (000). 4.3 Draft 5/7/04 004 J.E. Akin. All rights reserved.