CIVL 8/7117 Chapter 10 - Isoparametric Formulation 42/56

Transcription

1 CIVL 8/77 Chapter 0 - Isoparametrc Formulaton 4/56 Newton-Cotes Example Usng the Newton-Cotes method wth = ntervals (n = 3 samplng ponts), evaluate the ntegrals: x x cos dx 3 x x dx 3 x x dx % error x 3 x dx Gaussan Quadrature o evaluate the ntegral: ydx where y = y(x), we mght choose (sample or evaluate) y at the mdpont y(0) = y and multply by the length of the nterval, as shown below to arrve at I = y, a result that s exact f the curve happens to be a straght lne.

2 CIVL 8/77 Chapter 0 - Isoparametrc Formulaton 43/56 Gaussan Quadrature Generalzaton of the formula leads to: ydx n Wy x hat s, to approxmate the ntegral, we evaluate the functon at several samplng ponts n, multply each value y by the approprate weght W, and add the terms. Gauss's method chooses the samplng ponts so that for a gven number of ponts, the best possble accuracy s obtaned. Samplng ponts are located symmetrcally wth respect to the center of the nterval. Gaussan Quadrature Generalzaton of the formula leads to: ydx n Wy x In general, Gaussan quadrature usng n ponts (Gauss ponts) s exact f the ntegrand s a polynomal of degree n - or less. In usng n ponts, we effectvely replace the gven functon y = f(x) by a polynomal of degree n-. he accuracy of the numercal ntegraton depends on how well the polynomal fts the gven curve.

3 CIVL 8/77 Chapter 0 - Isoparametrc Formulaton 44/56 Gaussan Quadrature Generalzaton of the formula leads to: ydx n Wy x If the functon f(x) s not a polynomal, Gaussan quadrature s nexact, but t becomes more accurate as more Gauss ponts are used. Also, t s mportant to understand that the rato of two polynomals s, n general, not a polynomal; therefore, Gaussan quadrature wll not yeld exact ntegraton of the rato. Gaussan Quadrature - wo-pont Formula o llustrate the dervaton of a two-pont (n = ) consder: ydxwy x W y x here are four unknown parameters to determne: W, W, x, and x. herefore, we assume a cubc functon for y as follows: yc CxCx Cx 3 0 3

4 CIVL 8/77 Chapter 0 - Isoparametrc Formulaton 45/56 Gaussan Quadrature - wo-pont Formula In general, wth four parameters n the two-pont formula, we would expect the Gauss formula to exactly predct the area under the curve. A C CxCx Cx dxc C 3 0 However, we wll assume, based on Gauss's method, that W = W and that x = x as we use two symmetrcally located Gauss ponts at x = ±a wth equal weghts. he area predcted by Gauss's formula s A W y( a) W y( a) G W C0 Ca Gaussan Quadrature - wo-pont Formula If the error, e = A - A G, s to vansh for any C 0 and C, we must have, n the error expresson: e C 0 e C 0 W W 0 aw a Now W = and a = are the W s and a s (x s) for the two-pont Gaussan quadrature as gven n the table.

5 CIVL 8/77 Chapter 0 - Isoparametrc Formulaton 46/56 Gaussan Quadrature Example Use three-pont Gaussan Quadrature evaluate the ntegrals: x x cos dx 3 x x dx 3 W x cos x Order N Ponts u Weghts w ± ± ± ± % error Gaussan Quadrature Example Use three-pont Gaussan Quadrature evaluate the ntegrals: x x cos dx 3 x x dx 3 W x 3 x Order N Ponts u Weghts w ± ± ± ± % error

6 CIVL 8/77 Chapter 0 - Isoparametrc Formulaton 47/56 Gaussan Quadrature Example In two dmensons, we obtan the quadrature formula by ntegratng frst wth respect to one coordnate and then wth respect to the other as n fstdsdt (,) Wf s, t dt n n Wj W f s, tj j n n j WW f s, t j j Gaussan Quadrature Example For example, a four-pont Gauss rule (often descrbed as a x rule) s shown below wth =, and j =, yelds WW j fs, tj WWfs, t WW fs, t j WWf s t, WWfs, t he four samplng ponts are at s and t = ± and W =.0

7 CIVL 8/77 Chapter 0 - Isoparametrc Formulaton 48/56 Gaussan Quadrature Example In three dmensons, we obtan the quadrature formula by ntegratng frst wth respect to one coordnate and then wth respect to the other two as f (,, s t z) ds dt dz WWW j k fs, tj, zk j k Evaluaton of the Stffness Matrx by Gaussan Quadrature For the two-dmensonal element, we have shown n prevous chapters that [ k] [ B] [ D][ B] hdxdy A where, n general, the ntegrand s a functon of x and y and nodal coordnate values.

8 CIVL 8/77 Chapter 0 - Isoparametrc Formulaton 49/56 Evaluaton of the Stffness Matrx by Gaussan Quadrature We have shown that [k] for a quadrlateral element can be evaluated n terms of a local set of coordnates s-t, wth lmts from - to wthn the element. [ k] [ B] [ D][ B] h J dsdt Each coeffcent of the ntegrand [B] [D] [B] [J] evaluated by numercal ntegraton n the same manner as f(s, t) was ntegrated. Evaluaton of the Stffness Matrx by Gaussan Quadrature A flowchart to evaluate [k] for an element usng four-pont Gaussan quadrature s shown here.

9 CIVL 8/77 Chapter 0 - Isoparametrc Formulaton 50/56 Evaluaton of the Stffness Matrx by Gaussan Quadrature he explct form for four-pont Gaussan quadrature (now usng the sngle summaton notaton wth =,, 3, 4), we have [ k] [ B] [ D][ B] h J dsdt,,, B s t D B s t J s t WW Bs, t D Bs, t Js, t WW Bs3, t3 D Bs3, t3 Js3, t3 W3W3 Bs4, t4 D Bs4, t4 Js4, t4 W4W4 where s =t = , s = , t =0.5773, s 3 =0.5773, t 3 = , and s 4 =t 4 = and W =W =W 3 =W 4 =.0 Evaluaton of the Stffness Matrx by Gaussan Quadrature Evaluate the stffness matrx for the quadrlateral element shown below usng the four-pont Gaussan quadrature rule. Let E = 30 x 0 6 ps and = 0.5. he global coordnates are shown n nches. Assume h = n.

10 CIVL 8/77 Chapter 0 - Isoparametrc Formulaton 5/56 Evaluaton of the Stffness Matrx by Gaussan Quadrature Usng the four-pont rule, the four ponts are: s, t , s, t , s3, t , s, t , Wth W = W = W 3 = W 4 =.0 Evaluaton of the Stffness Matrx by Gaussan Quadrature k Bs, t D Bs, t Js, t,,, B s t D B s t J s t,,, B s3 t3 D B s3 t3 J s3 t3,,, B s4 t4 D B s4 t4 J s4 t4 Frst evaluate [J] at each Gauss, for example: J ,

11 CIVL 8/77 Chapter 0 - Isoparametrc Formulaton 5/56 Evaluaton of the Stffness Matrx by Gaussan Quadrature Recall: 0 t t s s t 0 s st J Xc Yc 8 st s 0 t s st t 0 X x x x x Y y y y y c 3 4 c 3 4 For ths example: X Y 4 4 c c Evaluaton of the Stffness Matrx by Gaussan Quadrature Recall: J , Smlarly: J J J , , ,

12 CIVL 8/77 Chapter 0 - Isoparametrc Formulaton 54/56 Evaluaton of the Stffness Matrx by Gaussan Quadrature he shape functons are computed as: N, s 4 t N, t 4 s Smlarly, [B ], [B 3 ], and [B 4 ] must be evaluated lke [B ] at ( , ). We then repeat the calculatons to evaluate [B] at the other Gauss ponts. Evaluaton of the Stffness Matrx by Gaussan Quadrature Usng a computer program wrtten specfcally to evaluate [B], at each Gauss pont and then [k], we obtan the fnal form of [B( , )], as B , Wth smlar expressons for [B( , )], and so on.

13 CIVL 8/77 Chapter 0 - Isoparametrc Formulaton 55/56 Evaluaton of the Stffness Matrx by Gaussan Quadrature he matrx [D] s: E 6 [ D] ps Fnally, [k] s: [ k] Axsymmetrc Elements Problems 5. o be assgned from your textbook A Frst Course n the Fnte Element Method by D. Logan.

14 CIVL 8/77 Chapter 0 - Isoparametrc Formulaton 56/56 End of Chapter 0