Numerical Integration Over Polygonal Domains using Convex ... - Ijecs

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and ( x 1, y1) ( x N 1, y N 1) . We have assumed in the above eqn. (32 ) that PN can be discretised into a set of M triangles, T n ( n 1, 2,3,...., M ). In the numerical integration formula of section 6.1, we have xy Tn Tpqr PQR, an arbitrary triangle with vertices ( x , y ), p, q, r in Cartesian space ( x , y). The numerical integration formula for ( ) xy is already explained in section 6.1. We can get T f n T PQRn pqr higher accuracy for the integral ( f ) by using the refined triangular mesh of the polygonal domain. We P N have written a computer code in MATLAB for this purpose. We first decompose the given polygonal domain into a coarse mesh of M triangles T (say), as expressed in eqn.(32). We then refine the mesh containing M triangles into a new mesh with n n 2 M triangles, satisfying the relation T . This n 2 n t p p1 division can be carried by using the linear transformations connecting the Cartesian space ( x, y) and the local 2 space ( u, v). We divide the standard triangle (right isosceles) in the ( u, v) space into n right isosceles triangles and then use the linear transformation to obtain the corresponding Cartesian nodal coordinates using the linear transformation. The nodal connectivity data in ( u, v) space is also determined for the subdivisions. This is prepared for each subdivision and incorporated into the computer code. The computer code can obtain integral values II ( f i ( x, y)), PN by dividing the P N into meshes with refinements. The first mesh is the coarse mesh with M triangles, the subsequent meshes will have n 2 M ( 2 2 2 2 2 2 n 2 2 , 3 , 4 , 5 , 6 , 7 ,...... ) triangles. We know that the numerical algorithm of section 6.1 is for an arbitrary triangle which is further divided into three special quadrilaterals. Thus the polygonal domain is divided into3 n 2 M special quadrilaterals. This computer code written in MATLAB is appended for reference. 7 Boundary Integration over the Plane Polygon In this section, we develop a numerical scheme to determine the integrals over a plane polygon. This will help us in verifying the computed values of some typical integrals considered in the next section. We have also developed a computer code based on MATLAB for this purpose. This will verify the computed values of integrals obtained by use of the numerical scheme developed in the previous section. It is well known that one of the remarkable theorems of multivariate calculus, Green’s theorem for a plane region is of immense value for the computation of multiple integrals over the plane polygon. We know that f ( x, y) dxdy F( x, y) dy, F( x, y) f ( x, y) dx , where f is continuous on a domain with piecewise smooth boundary described counterclockwise) gives in principle an appealing tool for numerical cubature, since it transforms a two-dimensional integration problem into a one-dimensional integration problem. Its practical use, however, requires the knowledge of a primitive of the integrand. However, this restriction can be overcome by proper application of the available theoretical knowledge on integration. We now show that by the application of Green’s theorem, the integration over the polygonal domain with n- edges can be obtained as a sum of n triangles expanded with respect to the origin which does not require the knowledge of a primitive of the integrand. We now state and prove the following lemma 4 and theorems 1-2 which will help us in developing the computer codes. Theorem 1. The integral II over a simple polygon with N - oriented edges l ik ( k i 1), i 1,2,3 ,..., N xy each with end points x , y ) and x , y ) in the xy – plane is expressible as N i1 T ( i i ( k k II ( f ) f ( x, y) dxdy , ( f is a smooth function on the domain xy) (33) xy xy i 0k where, x x y , and Proof: Let N 1 1, N 1 y1 xy T refers to the triangle with vertices ( x , y ), (0, 0), ( x , y ), ( k i 1 ). i 0 k i i k k ( x , y) f ( x, y) dx and consider the following expression and apply the Green’s theorem: H. T. Rathod a IJECS Volume 2 Issue 8 August, 2013 Page No.2576-2610 Page 2583
x y f x y dxdy ( , ) dxdy ( , ) xy xy xy xy xy xy x Tj 0i Tk 0 j Ti 0k Tj 0i Tk 0 j Ti 0k = ( x, y) dy xy xy xy T j 0i Tk 0 j T i 0k = ( x, y) dy ( x, y) dy ( x, y) dy l j 0 l0i lij lk 0 l0 j l jk li 0 l j 0k lki = ( x, y) dy = ( x , y) dy = f ( x, y) dxdy (34) xy xy lij l jk lki T ijk T ijk xy Thus, we have proved that the triangle Tijk expands into three new triangles with respect to the origin. We may note that in deriving the above result of eqn.(34), the following fact is used. ( x, y) dy ( x, y) dy 0 l0i li 0 ( x, y) dy ( x, y) dy l j 0 0 j l ( x , y) dy ( x, y) dy l k 0 0k l 0 0 The general result of eqn.(33) can be readily proved on similar lines. Theorem 2. Let us denote the triangle spanned by vertices ( xi , yi ), (0,0), ( xk , yk ), ( k i 1) as the integral of a smooth function f over the region II xy i 0k T ( f ) ( x k y x y ) i i k 1 1 0 0 rf ( r( x Proof: Let us consider the integral: i x ki s), r( y i xy Ti 0 k y ki is expressible as s)) drds xy Ti 0 k . Then H. T. Rathod a IJECS Volume 2 Issue 8 August, 2013 Page No.2576-2610 Page 2584 (35) II xy ( f ) f ( x, y) dxdy (36) Tijk xy Tijk The parametric equations of the oriented triangle in the xy - plane with vertices spanned by x , y ), ( x , y ) ( i i j j and ( xk , yk ), ( k i 1) which map this arbitrary triangle into a unit right isosceles triangle in the uv – plane are x x x u x v, y y y u y v (37) i where, ji ki i ji ji j ki 0 u, v 1, u v 1, x x x , x x x , y y y , y y y (38) We have then ( x, y) dxdy dudv ( x ji yki xkiy ji ) dudv ( u, v) xy = ( 2 dudv ijk ) i ki k = ( 2 area of the triangle T xy ijk ) dudv (39) and thus, we define xy ( x y x y ) (40) 2 ijk ji ki ki ji Using the above eqns.(37) –(39) in eqn.(36) gives us II xy i 0k T 2 1 1u xy ijk 0 0 f (( x i x ji u x ki v),( y i y ji u y ki i ji v)) dudv Using the transformations u 1 r, v rs (42) we can map the above integral in eqn.(36) into an equivalent integral over the rectangle ( r , s) / 0 r, s 1 . This gives us j i ki k i (41)
Page 1 and 2: www.ijecs.in International Journal
Page 3 and 4: In this section, we describe a spec
Page 5 and 6: 4 Explicit forms of the Jacobians W
Page 7 and 8: II ( f ) can be computed as finite
Page 9: x y x y x y (1) (1) (2) (2) (3) (3)
Page 13 and 14: 8 Numerical Examples In our earlier
Page 15 and 16: 8.3 Some Integrals over the Polygon
Page 17 and 18: 30 0.40000000059051 0.4000000001043
Page 19 and 20: 20 1.00000000000000 1.0000000000000
Page 21 and 22: 30 0.71828182542692 0.7182818260030
Page 23 and 24: Gauss Order NUMBER OF SPECIAL QUADR
Page 25 and 26: 35 0.00842118094149 0.0084211809414
Page 27 and 28: 35 0.03141452863239 0.0314145286323
Page 29 and 30: 20 0.54541595958491 0.5453925639380
Page 31 and 32: ___________________________________
Page 33 and 34: ___________________________________
Page 35 and 36: RULE) (GLQ=GAUSS LEGENDRE QUADRATUR
Page 37: [2] K. J. Bathe, Finite Element Pro

Numerical Integration Over Polygonal Domains using Convex ... - Ijecs

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