Serendipity và chức năng hình dạng Lagrange được so sánh trong Hình 3,11 Lagrange yếu tố bong bóng "chế độ bổ sung và, do đó, có thể để mô tả chính xác hình dạng phức tạp. Các yếu tố hình tam giác có thể được hình thành từ các thành phần tứ giác, | 40 The Boundary Element Method with Programming If the element shape functions for the quadratic element are derived from Lagrange polynomials then there is an additional node at the centre of the element Figure . The shape functions are given by Ln i j 41 Ai2 Bj1 Bj 2 Bj3 o o-------o 9------- -------9 r 9 Y T o-----o-------o o-------o-------o T----- ---1 T T o-----o---o Figure Serendipity and Lagrange shape functions DISCRETISATION AND INTERPOLATION 41 Aụ is defined in equation and Bjm m fj m Uj -Um Bjm 1 if j m where i and j are the column and row numbers of the nodes. This numbering is defined in Figure . The nodes are given by n 1 1 1 n 2 1 2 n 1 2 4 n 2 2 3 n 1 3 8 n 2 3 6 n 3 1 5 n 3 2 7 n 3 3 9 The Serendipity and Lagrange shape functions are compared in Figure The Lagrange element has an additional bubble mode and is therefore able to describe complicated shapes more accurately. Triangular elements can be formed from quadrilateral elements by assigning the same global node number to two or three corner nodes. Such degenerate elements are shown in Figure . Figure Linear and quadratic degenerate elements Alternatively triangular elements may be defined using the iso-parametric concept. In Figure we show a triangular element in the global and local coordinate system. The shape functions for the transformation are defined as4 1 7 1 N3 7 7 42 The Boundary Element Method with Programming Figure Triangular linear element in global and local coordinate system As can be seen in Figure the shape functions are represented by planes. 1 7 Figure Shape functions of linear triangular boundary element 3 7 A It is also possible to define a triangular element with a quadratic shape function. The shape functions for the mid-side nodes are given by N4 4f 1 -f-7 N5 4 77 n6 M 1
