3 Finite Element Parametric Geometry

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1 3 Finit Elmnt Paramtric Gomtry 3. Introduction Th intgral of a matrix is th matrix containing th intgral of ach and vry on of its original componnts. Practical finit lmnt analysis rquirs intgrating matrics, from particular applications, ovr arbitrarily shapd curvilinar gomtry. To modl a curvilinar gomtry rquirs using paramtric intrpolation of th gomtry. Hr th mphasis is on how to numrically intgrat matrics ovr mshd domains, but th matrics illustratd nxt com from basic calculus dfinitions of gomtric quantitis usd in many nginring applications. Bginning with th nxt chaptr th matrics will com from physical applications. In a two-dimnsional strss analysis it is oftn ncssary to intgrat th prssur acting on a curvd dg in or to find th rsultant forc componnts. Likwis, a thrmal study oftn has a normal hat flux pr unit lngth ntring along a curvd dg. It has to b intgratd ovr th lngth to dtrmin th total hat flow into th domain. It will b shown, in th following chaptrs, that on a thr-nod curvd dg, th scond application givs th intgral F b = H bt q(r) Γ b ds = H bt (r) q(r) ds whr H b dnots th lin intrpolation functions of th adjacnt lmnt that ar just valuatd on its dg, and s is th lngth along th curv. Usually, th flux is input at th nods of th dg and intrpolatd along it as q(r) = H b (r) q b = 3 b H k k= (r) q k b so that th rsultant flux intgral bcoms a squar matrix tims th input data: F b = Γ b H bt (r) H b b ds (r) q = HbT (r) Γ b H b (r) ds qb S b q b. Intgrals of th intrpolation functions and thir products occur in all practical finit lmnt applications. Thy also appar in valuating common, but important, gomtric proprtis. Thus, th many possibl physical applications will b dlayd whil th calculus dtails of valuating th most common matrix intgrals. 3.2 Paramtric curvs Gomtric ntitis hav a numbr of intgral proprtis that appar in strss, fluid, and dynamic applications. Thy includ th masur of th ntity, its first and scond momnts, and its cntroid. Consi a gnral planar curv; its gomtric proprtis ar dfind by th scalar intgrals; Pag of

2 Lngth L = ds First momnt M x = y ds, First momnt M y = x ds Scond momnt I xx = y 2 ds, Scond momnt I yy = x 2 ds Product momnt I xy = x y ds X-cntroid x = M y /L, Y-cntroid y = M x /L Lt th curv b modld by a msh of n m control nods whr its coordinats ar input, and n curvd lin lmnts ach having n n nods pr lmnt in thir connction list. Now any of th abov intgral proprtis is th summation of intgrals ovr ach lmnt n I = f(x, y)ds = n = f(x, y) ds = f(x, y) s(r) L =. Th x- coordinat data, x, and y-coordinat data, y, can b intrpolatd in a non-dimnsional spac,, such as in unit paramtric coordinats r, n n x(r) = H k (r) x k = H(r) x = x T H(r) T k= If ach lmnt is straight, thn th Jacobian n n y(r) = H k (r) k= J (r) = s(r) will b a constant and th intgrals ar simplr. Howvr, for a gnral curv it is ncssary to rlat th physical diffrntial lngth to th paramtric diffrntial lngth (s Figur.6): y k ( Ls ) 2 = ( x ) 2 + ( y ) 2 s(r) = ( x(r) 2 ) + ( y(r) 2 ) which rquirs valuating th paramtric ivativ of ach spatial coordinat. For a two-nod lin lmnt with th linar intrpolations (so n n = 2) th position is: or altrnatly x(r) = H(r) x = ( r)x + rx 2 Likwis, y(r) x(r) = x + r(x 2 x ) = x + r x and x(r) = x. = y and for that particular lmnt typ th Jacobian is a constant: s = ( x ) 2 + ( y ) 2 L, and th lngth, for a singl lmnt, is L = dl = s(r) = L = L, Pag 2 of

3 as xpctd. A thr-nod quaatic lin lmnt will b a straight lin if th nods ar collinar, but it will only hav a constant Jacobian if th intrior nod is at th midpoint of th lin. Likwis, a four-nod cubic curvd lin lmnt will fall on a straight lin whn its nods ar collinar, but th Jacobian bcoms constant only whn th two intrior nods occur at th third points. In gnral, lin lmnts and triangular lmnts hav constant Jacobians whn th physical nods ar placd in th sam qual spacing as th paramtric lmnt (calld an affin mapping). Likwis, for a singl lmnt straight lin, along th x-axis, th first momnt is valuatd using intrpolation: hr so x(r) M y = x ds = H(r) x = [ H(r) x(r) ] x = [L H(r) ] x H(r) = [( r) (r)] = [(r r2 2 ) (r2 2 )] = [ 2 M y = x ds = L [ ] {x 2 2 x} = L (x + x 2 )/2, 2 and th x-cntroid of a singl straight lin lmnt is 2 ] as also xpctd. x = M y = L (x +x 2 )/2 = x +x 2 L L 2, Th scond momnt of a gnral curvd lin mad up of a msh of curvd lmnts is with a typical lmnt contribution is n I yy = x 2 ds = x 2 s(r) I yy = x 2 s(r) = (H(r) x ) T H(r) x = s(r) = ( n n ) (n n n n ) (n n ) = x T [ H(r) T H(r) s(r) ] x. In othr words, this scond momnt is th product of th nodal x-coordinats, x, pr- and postmultiplying a squar matrix, say m I yy = x T m x m H(r) T H(r) s(r) which is asily valuatd in closd form whn th Jacobian happns to b constant (s Tabl 2.) Pag 3 of

4 m = L 6 [2 2 ]. Thus, for a straight lin sgmnt th scond momnt contribution is I yy = (x 2 x ) [x x 2 ] [ ] {x x } 2 I yy = (x 2 x ) (2x x x 2 + 2x 2 2 ) = (x 3 2 x 3 ) which is xact for a straight lin along th x-axis, as xpctd. For a gnral curvd lmnt any proprty must b valuatd by numrical intgration: whr n q m H(r) T H(r) s(r) + H(r q ) T H(r q ) ( x(r 2 q) ) + ( y(r 2 q) ) w q x(r q ) q= = H(r q) In words, th matrix to b valuatd is zrod. Thn at ach tabulatd coordinat location th paramtric ivativ of th intrpolation functions ar valuatd, thir innr product (dot product) is takn with th lmnt s x-coordinats to form x(r q ) ; thn th innr product is takn with th lmnt s y-coordinats; thos two trms ar squard, addd togthr and th squar root is takn to form th numrical valu of s(r q ). Likwis, th intrpolation functions ar valuatd at th quaatur point, is transposd, and multiplid tims itslf to form th squar matrix at th point. Finally, th squar matrix numrical valus ar multiplid by th gnralizd Jacobian dtrminant, s(r q ), multiplid by th numrical wight, w q, and addd to th prvious contributions to th matrix intgral. In practical finit lmnt studis th sam typs of squar and column matrics ar valuatd in th sam way for ach lmnt, but thy ar not pr- or post-multiplid by spatial coordinats. x. 3.2 Paramtric aras Th xtnsion of dtrmining th abov paramtric curv proprtis to computing th gomtric proprtis of a planar ara is straight forward, but illustrats why it is ncssary to us paramtric intrpolation to match curvilinar shaps. Th gomtric proprtis of a planar ara ar dfind by th scalar intgrals; Ara A = da First momnts M x = y da, M y = x da Scond momnts I xx = y 2 da, I yy = x 2 da, I zz = I xx + I yy Product momnt I xy = x y da X-cntroid x = M y /A, y = M x /A Lt th ara with curvilinar boundaris b modld by a msh of n m control nods whr its coordinats ar input, and n lmnts ach having n n nods pr lmnt in thir connction list. Th Pag 4 of

5 intrpolation functions usd must b capabl of dfining th msh such that no gaps or ovrlaps occur in th ara. That mans that paramtric coordinats should b usd for intrpolations instad of physical coordinats, bcaus paramtric coordinats dgnrat to lowr dgr polynomials on thir boundaris and can uniquly provid continuity across curvd lmnt intrfacs. To illustrat th problms of trying to us physical coordinat intrpolation consi a gnral quailatral lmnt. Th four cornrs provid th data ndd to dscrib th gomtric shap of th lmnt, and how a solution would vary ovr th lmnt. Try an incomplt quaatic as u(x, y) = c + c 2 x + c 3 y + c 4 x y It is similar to th unit coordinat quailatral givn in th prvious chaptr. Th four constants could b valuatd from th cornr valus. Howvr, a problm thn dvlops along any dg of th lmnt. On an dg th quation for y along th straight lin in slop-intrcpt form is y(x) = mx + b Thn th physical intrpolation along th dg bcoms u b (x) = c + c 2 x + c 3 (mx + b) + c 4 x (mx + b) u b (x) = (c + bc 3 ) + (c 2 + c 3 m + c 4 b)x + c 4 mx 2 u b (x) = d + d 2 x + d 3 x 2 which rquirs thr pics of data on th dg, but only two vrtx valus ar availabl. Thrfor, and unknown solution and/or th shap of th dg is not uniqu basd on th two vrtx valus shard with a nighboring lmnt. Ths contrasts with th unit coordinat quailatral with u(r, s) = c + c 2 r + c 3 s + c 4 r s whr on an dg, such as s=, rducs from th incomplt quaatic to a complt linar intrpolation u b (r) = u(r, s = ) = c + c 2 r + + That can b uniquly dfind by th two vrtx valus that ar shard with anothr lmnt adjacnt to that dg. Thrfor, to valuat gomtric proprtis in two- and thr-dimnsions paramtric intrpolation must b usd, along with its associatd Jacobian matrix. Th x- coordinat data, x, and y-coordinat data, y, can b intrpolatd in a non-dimnsional spac,, such as on a unit paramtric coordinat quailatral, r, s, n n x(r) = H k (r, s) x k = H(r, s) x = x T H(r, s) T k= n n y(r, s) = H k (r, s) k= y k So th Jacobian of an arbitrarily shapd (within rason) lmnt is Pag 5 of

6 J x y = [ x s y s ] H(r, s) = [ H(r, s) s ] [x y ] n p n s = (n p n n ) (n n n s ) is a squar matrix numrically valuatd and any local point of intrst by forming th product of th local intrpolation ivativs with th rctangular input matrix of nodal coordinats. Th invrs Jacobian matrix and th dtrminant of th Jacobian ar asily valuatd numrically. To b a rasonably shapd lmnt th Jacobian dtrminant must b positiv vrywhr in th lmnt. Th ara calculation is A = da = n = da A n = J d = and a typical lmnt ara is valuatd by numrical intgration as A = J d n q = J (r q, s q ) w q q= n = A = which shows that a distortd lmnt with a ngativ dtrminant, J (r q, s q ) <, can lad to nonphysical ngativ aras (and momnts of inrtia, and analysis matrics in latr chaptrs). Gnrally, to avoid lmnt distortion a mid-sid nod should not b offst prpndicular to th straight lin btwn vrtics by mor than 2% of th straight lin distanc. Th spcific mathmatical rstrictions on lmnt shaps ar wll known and ar accountd for in automatic msh gnrators. Th supplid Matlab script chcks that th dtrminant is positiv du to manually gnratd mshs. Likwis, othr gomtric proprtis ar valuatd by numrical intgration. Th momnt of inrtia of th arbitrarily shapd rgion is I xx = y 2 da = n I xx = n = y 2 da = y 2 (r, s) J d = A n = whr again n q = y 2 (r q, s q ) J (r q, s q ) I xx q= n y(r q, s q ) = n k= H k (r q, s q ) y k. w q A msh of straight sidd triangls can rasonably approximat any planar shap. If th Jacobian is constant (as in straight sidd triangls and rctangls) thn xact intgration is possibl: I xx = y T m y m H(r, s) T H(r, s) J d (In th following chaptrs this intgral oftn occurs and is rfrrd to as th gnralizd mass matrix.) For a straight sidd linar intrpolation triangl Pag 6 of

7 2 2 m = [ 2 ], I 2 xx = [ y y 2 y 3 ] [ 2 ] { y 2 } y 3 Ths calculations for planar aras ar asily xtndd to solids of rvolution by noting th diffrntial volum is dv = 2π R da = 2π R(r, s) J d Whn th y-axis is th axis of rvolution thn th radius is R(r, s) = x(r, s). Gomtric proprtis of rgular solids ar obtaind in a similar fashion. 3.3 Paramtric surfacs Many applications will latr rquir th intgration of matrics ovr arbitrary surfacs. For xampl, if a sgmnt of a sphr is subjctd to a normal prssur thn to dtrmin th applid forc componnts you must intgrat th product of th valu of th prssur and th surfac normal vctor ovr th surfac sgmnt. Th xtnsion of dtrmining th abov paramtric ara proprtis to computing th gomtric proprtis of a gnral surfac is straight forward by using us paramtric gomtry. Th gomtric proprtis of a gnral surfac ar dfind by th scalar intgrals; Ara A = ds First momnts M x = y ds, M y = x ds, M z = z ds Scond momnts I xx = (y 2 + z 2 ) ds, I yy = (x 2 + z 2 ) ds, I zz = (x 2 + y 2 ) ds Product momnts I xy = x y ds, I xz = x z ds, I yz = y z ds X-cntroids x = x ds /A, y = y ds /A, z = z ds /A y whr ds is th diffrntial ara of th curvd surfac. Th gnrally curvd surfac is scribd by th two paramtric curvs. In th paramtric domain th two paramtric coordinats ar orthogonal, but whn thy gt mappd onto a thr-dimnsional surfac thy gnrally ar no longr mutually prpndicular. In chaptr, th two tangnt vctors at a point on th surfac ar T r = R = x y z i + j + k and T s = R s, and T r = R Likwis, T s ds = R is th physical diffrntial lngth along th paramtric r-curv on th surfac. s ds is th physical diffrntial lngth along th paramtric s-curv on th surfac. Thrfor, thir vctor cross-product is a vctor normal to th surfac having th units of (diffrntial) surfac ara. In othr words, thir cross product is T r T s ds = n ds whr n is th unit vctor normal to th surfac. In othr words, th diffrntial surfac ara is th magnitud of th cross-product tims th paramtric diffrntial masur: Pag 7 of

8 ds = T r T s ds = (T r T r )(T s T s ) (T r T s ) 2 ds. This can b usd to find gomtric proprtis. Latr it will b usd to gt th rsultant forc from a prssur distribution: F x F = { F y } = p(r, s) { F S x n x n y n x } ds = p(r, s) n ds S Summing ovr th surfac boundary sgmnts and numrically intgrating ach numrically givs n_b n_b n_q F = b= p(r, s) n ds = p(r q, s q ) (T r T r )(T s T s ) (T r T s ) 2 b= w q. S b q= Rcall from Chaptr that th two tangnt vctors at a point, in matrix notation, ar found by intrpolation as [ T r x y z ] = [ ] = [ Hb (r, s) T s x s y s z s H b ] [x y z ]. (r, s) s Pag 8 of

9 4 Summary Planar Curv Trminology Lngth L = ds First momnt M x = y ds, First momnt M y = x ds X-cntroid x = M y /L, Y-cntroid y = M x /L Scond momnt I xx = y 2 ds, Scond momnt I yy = x 2 ds Product momnt I xy = x y ds Position on curv Tangnt x(r) = G(r) x = x T G(r) T, T (r) = x(r) y(r) G(r) i + j = Straight lin linar lmnt y(r) = G(r) y = y T G(r) T x i + G(r) y j L = ( x ) 2 + ( y ) 2 = (x 2 x ) 2 + (y 2 y ) 2 M y = L [ 2 2 ] {x x} = R x, M x = L [ ] {y y} = R y 2 Gnral curvd lmnt Quaaturs I yy = x T m x = L 6 [ x x 2 ] [ 2 2 ] {x x 2 } I xy = L 6 [ x x 2 ] [ 2 2 ] {y y }, Ixx = L 6 [ y y 2 ] [ 2 2 ] {y y } 2 L ( x(r) 2 ) R G(r) ( x(r) 2 ) m G(r) T G(r) ( x(r) 2 ) + ( y(r) ) 2 + ( y(r) ) 2 + ( y(r) ) 2 n q L = x T G(r q ) T G(r q ) x + y T G(r q ) T G(r q )y q= w q Pag 9 of

10 n q R = G(r) x T G(r q ) T G(r q ) x + y T G(r q ) T G(r q )y q= w q n q m = G(r) T G(r) x T G(r q ) T G(r q ) x + y T G(r q ) T G(r q )y q= w q Physical diffrntial masurs: Planar curv Planar ara Volum of rvolution dl = ( x(r q) da = J d dv = 2π R J d )2 + ( y(r q) )2 Surfac ara Solid volum ds = ( R R ) ( R s R s ) ( R R s ) 2 d dv = J d Pag of

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