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 4.3333333 6 6 6.4444444 0.706% error x 3 x dx.47305 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.
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.
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
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 3 0 3 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 0.5773... 3 3 Now W = and a = 0.5773... are the W s and a s (x s) for the two-pont Gaussan quadrature as gven n the table.
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 5.55938 9 8.0 9 5.55938 9 x Order N 3 4.5843698 Ponts u Weghts w 0.000000000.00000000 ±0.57735069.00000000 0.000000000 0.88888889 ±0.774596669 0.55555556 ±0.33998044 0.65455 ±0.86363 0.34785485 0.00004% 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 5.0593 9 8.0 9 5.5673475 9 Order N 3 4.47888 Ponts u Weghts w 0.000000000.00000000 ±0.57735069.00000000 0.000000000 0.88888889 ±0.774596669 0.55555556 ±0.33998044 0.65455 ±0.86363 0.34785485 0.00477% error
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 = ±0.5773... and W =.0
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.
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.
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 = -0.5773, s =-0.5773, t =0.5773, s 3 =0.5773, t 3 =-0.5773, and s 4 =t 4 =0.5773 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.
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 0.5773, 0.5773 s, t 0.5773, 0.5773 s3, t3 0.5773, 0.5773 s, t 0.5773, 0.5773 4 4 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 0.5773, 0.5773
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 3 5 5 3 Y 4 4 c c Evaluaton of the Stffness Matrx by Gaussan Quadrature Recall: J 0.5773, 0.5773 3 5 5 3 8.000 0 0.5773 0.5773 0.5773 0.5773 0.5773 0 0.5773 0.5773 0.5773 0.5773 0.5773 0.5773 0 0.5773 4 0.5773 0.5773 0.5773 0.5773 0 4 Smlarly: J J J 0.5773,0.5773.000 0.5773, 0.5773.000 0.5773,0.5773.000
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 0.5773 4 0.3943 N, t 4 s 0.5773 0.3943 4 Smlarly, [B ], [B 3 ], and [B 4 ] must be evaluated lke [B ] at (-0.5773, -0.5773). 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(-0.5773, -0.5773)], as B 0.5773, 0.5773 0.057 0 0.057 0 0 0.057 0 0.3943 0.057 0.057 0.3943 0.057 0.3943 0 0.3943 0 0 0.3943 0 0.057 0.3943 0.3943 0.057 0.3943 Wth smlar expressons for [B(-0.5773, 0.5773)], and so on.
CIVL 8/77 Chapter 0 - Isoparametrc Formulaton 55/56 Evaluaton of the Stffness Matrx by Gaussan Quadrature he matrx [D] s: 0 3 8 0 E 6 [ D] 0 8 3 0 0 ps 0 0 0.5 0 0 Fnally, [k] s: 466 500 866 99 733 500 33 99 500 466 99 33 500 733 99 866 866 99 466 500 33 99 733 500 4 99 33 500 466 99 866 500 733 [ k] 0 733 500 33 99 466 500 866 99 500 733 99 866 500 466 99 33 33 99 733 500 866 99 466 500 99 866 500 733 99 33 500 466 Axsymmetrc Elements Problems 5. o be assgned from your textbook A Frst Course n the Fnte Element Method by D. Logan.
CIVL 8/77 Chapter 0 - Isoparametrc Formulaton 56/56 End of Chapter 0