Construction of Serendipity Shape Functions by Geometrical Probability

Transcription

1 J. Basc. Appl. Sc. Res., ()56-56, 0 0, TextRoad Publcaton ISS 00-0 Journal of Basc and Appled Scentfc Research Constructon of Serendpty Shape Functons by Geometrcal Probablty Kamal Al-Dawoud Department of mathematcs & Statstcs Qassm Unversty\ On sabbatcal leave from Mutah Unversty Jordan ABSTRACT In ths paper the rectangular fnte element (FE) s consdered as a serendpty famly (O.Zenkewcz, R.Taylor, 000): lnear ( nodes), quadratc ( nodes), cubc ( nodes). A cubc FE s receved wth standard ( parameters) and modfed ( parameters) are functons of such a form. The paper shows a way of correctng the defcences of shape functons forms FESF usng a weghted average of varous models. KEYWORDS: Geometrcal probablty, fnte element method, serendpty famly, parametrc nterpolaton.. ITRODUCTIO Fnte elements of serendpty famly (FESF) form are a useful class of dscrete models. For the frst tme these elements were obtaned n 6 wth the help of resourceful recrutng because they dd not lend any formalzaton. FESF are the result of modfcaton of FE Lagrangan famles by elmnatng nternal nodes. The focus of ths paper s on cubc FE serendpty famly. In the standard model parameters, an nterpolaton polynomal concdes wth the number of nodes on the boundary (). There are several ways of solvng the problem of constructng the standard functons of the form: nverse matrx method, systematc generaton of shape functons (Taylor s procedure), and the probablstc-geometrcal method (PGM). The man advantage of PGM over other methods s that t allows dscoverng the hdden parameters n the nterpolaton procedure. For example, the cubc FE PGM provdes a -parameter nterpolaton. The correspondng shape functons of here are gven n explct form. These results ndcate that at FESF hgher orders, there are many sutable bases. Ths means that the PGM has opened new opportuntes for solvng urgent problems of optmzng functons of such a form. The paper shows a way of correctng the defcences of functons forms FESF usng a weghted average of varous models. In ths paper prove the cubc FE serendpty famly; there s a - parametrc nterpolaton polynomal that satsfes the condtons of Lagrange.. RESULTS AD DISCUSSIO In FEM ntally appled only fnte elements to the famly of Lagrangan [-, 6, ]. Optons of Lagrangan form of famly (Fg., a) are the natural generalzaton of the one-dmensonal Lagrangan nterpolaton. These functons are obtaned by multplyng the one-dmensonal polynomals (separaton of varables). Interpolaton and the computatonal drawbacks of Lagrangan famly are well known [, -]. If the FE famles of Lagrangan exclude parastc nternal nodes, we get FE serendpty famly (Fg., b). (a) 6 5 *Correspondng Author: Kamal Al-Dawoud, Department of mathematcs & Statstcs, Qassm Unversty\ On sabbatcal leave from Mutah Unversty Jordan. Emal: aldawoud55@yahoo.com 56

2 Al-Dawoud, 0 Lnear FE Quadratc FE Cubc FE (b) 6 5 Fg. Square elements: (a) Lagrang famly; (b) serendpty famly The key dea of PGM and the procedure for applyng the method n problems of constructng shape functons of Lagrangan elements are descrbed n [, ]. In ths paper the effectveness of geometrc probablty n FE serendpty famly s consdered. Frst, the quadratc FE serendpty famly s consdered. As usual [-] s a square х (, ), nodes are regularly located on ts border (Fg., a). (a) (b) (c) M M 5 Fg. Quadratc FE: (а) general vew; (b) composton of smple elements for, elements for, Baselne data contan numbers: the coordnates of nodes 5 5 ; (c) composton of smple, and the nodal values of the functon f,,. The challenge s to construct a polynomal nterpolaton P,, f () The functon form must possess the followng characterstcs (condtons of Lagrange type): where k k k k, () - s the Kronecker delta, s the number of functons and k s the number of nodes; 50

3 J. Basc. Appl. Sc. Res., ()56-56, 0, o Besdes, functon of the form, s to ensure belongs to a partcular sde of the square, then the functon, the law of quadratc parabola ( knots). () C contnuty at the boundary: f the node along ths part of changng Frst, two well-known method of soluton of the problem are brefly descrbed... Algebrac (matrx) method [, 5]. Intally nterpolaton polynomal s wrtten n a general terms (by means of Pascal's trangle): P, 5 6, () uncertan functon., where ow usng the nput data, a system of lnear algebrac equatons x s represented: 5 6 f,, (5) The determnant of a matrx system (5) s not equal to zero, whch guarantees the unqueness of the soluton. Solvng system (5), we get only one set of coeffcents, whch are expressed n terms of the number,, f. The substtuton of nto () and rearrangement of terms on the polynomal f leads to a type (). In dong so, we get the functon form n an explct form:,,, ;,,,, 5, ;, 6, (6) () ;.. Taylor procedure [, 6, ]. Intally the shape functons of nodes n the mddle of the partes s desgned: 5, 6,,. To ths end, the correspondng Lagrange polynomal of second degree n one drecton s multpled by the lnear functon n other areas. Ths allows the partes to obtan the desred degree of the polynomal. Ths automatcally s condton (). For an angular node ths method does not gve the desred result, as along as one sde shape functon changes lnearly, so n the mddle t gves 0.5 nstead of zero. However, a lnear combnaton of blnear functons, and functons of the form secondary nodes gve the desred result. () For example, 5, 5

4 Al-Dawoud, 0,. make a lnear combnaton of blnear functon,, and, and the forms functons: ow, to get,, 5,, 5,, () As t can be seen, the procedure of Taylor leads to the same result. Ths should be expected, snce the determnant of a matrx system (5) s non zero...applcaton of PGM [, ]. ote that the FEs of hgher order can always be represented as compostons of smple FE. The number of smple elements s determned by the order of the polynomal of nterpolaton. In ths case, each node s assocated wth two smple FE. For example, n the desgn,, two smple FEs are consdered:---, and -5- (Fg., b). In each of these FEs a random pont s throwng. In Fg., b, c the area of "successful" outcomes s shaded. Calculatons of the probabltes for the smple geometrc elements are descrbed n detal n [, ]. In ths case we get (Fg., b):, ;,. ow, accordng to the formula of multplyng the probabltes of two ndependent events we get (). ote. In desgnng the angular shape functons,, n some models of hgher order t s convenent to use a trangle wth a curved hypotenuse. When buldng 5, the formula of multplcaton gves the probablty 5, quadratc form FE buld, and,, two smple FE are consdered (Fg., c): 5--- and In ths case,. It s clear that functons of the 5. All functons are gven n [5]. The results are summerzed n the followng theorem. Theorem. On a quadratc FE serendpty famly, there s only an -parametrc polynomal, satsfyng the condtons (), ().. A cubc fnte element serendpty famly. Theorem. At the cubc FE serendpty famly a unque -parametrc nterpolaton polynomal that satsfes the condtons of type (), (). Proof: ths element s shown n Fg., a, b. 5

5 J. Basc. Appl. Sc. Res., ()56-56, 0 0 0, (a) (b) Fg. Cubc FE: (а) - sub-element composton to A cubc functon changes,, (b) - sub-element composton to 5, along the borders of - and -, and can be obtaned by usng two FEs: --- lnear and the quadratc -5- (Fg., a). The rght trangle wth a curved hypotenuse s ndependent []. Here t s used for the constructon of, on the cubc FE. As the hypotenuse of -5- s a crcle passng through the nodes Frst of all, apply the sub-element --- and determne the lkelhood of a random pont n the rectangle opposte node (shaded). Then In the p. sub-element -5-, the probablty of an accdental fall nto the strp regon 0 and s p 0 0 ow, accordng to the formula of multplyng probabltes we get: Generally, for,,,, 0 (0), 0,, To construct, 0 () 5 (Fg., b) the three-fe wth the general node 5: 5---0, and s consdered. The area of "successful" outcomes s shaded. It s clear that we had thrown three random ponts (one n each FE).The probablty of fallng nto a jont hatched rectangles s gven by 5

6 Al-Dawoud, 0 Generally, for unts 5, 6,, 0:, 5,, ; () For nodes,,, :,, ; () These functons form are known to be n the FEM [, -].The method of the nverse matrx, the procedure of Taylor, PGM provde a convergence of results. ote. In ths case, to the polynomal (), the components, 0,, are added. Ths polynomal s called the standard polynomal [6]. The man dsadvantages of the standard model FEST-: the exstence of multple zeros at the nodes, as well the unnatural unform dstrbuton of mass n unts of force (negatve pressure n the corner nodes). These shortcomngs can be overcome by PGM. It s a new method can that detect the hdden parameters of the nterpolaton, whch can alter the functons of the form of quanttatve and qualtatve. From Pascal's trangle, t s easy to establsh that the pattern of behavor, on the approprate sde FESF- allows for varaton n the number of parameters of a polynomal of to 6. Theorem. At the cubc FE serendpty famly, there s a -parametrc nterpolaton polynomal that satsfes the condtons of type (), (). Proof : we wll proceed wth a constructve manner. To do ths, consder a dfferent composton of smple sub-element to FESF- (Fg., a, b ). (a) 0 (b) Fg. Cubc FE: (а) - sub-element composton to buld;, (b) - sub-element composton to buld, 5 To construct,, three smple sub-element (Fg., a) wth a common node : ---, -5-, -6- are consdered. In each sub-element, thrown random pont subdoman "successful" outcomes of 5

7 J. Basc. Appl. Sc. Res., ()56-56, 0 the tests are shaded. The probablty of fallng wthn the shaded area s expressed n terms of the relatve area of sub-"successful" outcomes: p ; p ; Accordng to the rule of multplyng the probabltes we get: Generally, for,,, p.,.,,, To construct, () 5, three sub-elements (Fg., b) wth a common node 5: 5---0, 5-0--,and are consdered. Subdoman "successful" outcomes are shaded. The product of the probablty of fallng nto the subdoman "successful" outcomes provdes: Generally, for 5 5, 6,,0,,, ; (5) For nodes,,,,, ; (6) Testng shows that the condtons k, k k,, nterpolaton polynomal P,, f are met, but the has the th parameter. The modfed model () - (6) dffers from the standard model () - () so that the number of multple zeros s decreased, and the negatve pressure n the unform dstrbuton of mass forces on the nodes dsappeared. Corollary. The emergence of alternatve bases () - (6) allows to generate a myrad of bases on a formula weghted averagng: S M,,, () 55

8 Al-Dawoud, 0 where S - the standard bass; M - Modfed base; 0 - Weghtng factor. Concluson. The emergence of alternatve bases for the elements of serendpty famly opens up new possbltes for solvng the problem of optmzng the propertes of serendpty models. The proof of the exstence of FESF- nterpolaton polynomals, 5 and 6 parameters s of great nterest. In addton, PGM offers great promse for generatng functons of the form of serendpty elements n D. REFERECES. Gallager R.H. 5, Fnte Element Analyss: Fundamentals. Prentce Hall, ew Jersey.. Kamal Al-Dawoud, Khomchenko A.. Constructon of Lagrange Interpolatonal Polynomals Usng Geometrcal Probablty // Umm Al-Qura Unversty Journal of Scence Medcne Engneerng. Vol., o P orre D.H., de Vres G., An Introducton to Fnte Element Analyss. Academc Press, ew York.. Segerlnd L.J. 5, Appled Fnte Element Analyss. John Wley, ew York. 5. Zenkewcz O.C., Taylor R.L. 000, The Fnte Element Method. Volume : The Bass. Butterworth Henemann. 6. Zenkewcz O.C., Morgan K., Fnte Elements and Approxmaton. John Wley, ew York.. Mtchell A.R., Wat R., The Fnte Element Method n Partal Dfferental Equatons. John Wley, ew York. 56