同济大学：《有限元方法》课程教学资源（教案讲义）Chapter 7 Isoparametric Element

2014/5/15 线■ Isoparametric Element ⊙ Two-dimensional element calculations Two-dimensional element calculations =) Note that the boundary segment1 1

2014/5/15 1 Isoparametric Element ISOPARAMETRIC (ELEMENT) FORMULATION (a) A domain to be modeled. (b) Triangular elements.(c) Rectangular elements. (d) Rectangular and quadrilateral elements. The finite element method is a powerful technique for analyzing engineering problems involving complex, irregular geometries. However, the two- and three dimensional elements discussed so far (triangle, rectangle, tetrahedron, brick) cannot always be efficiently used for irregular geometries. Consider the plane area shown in Figure a, which is to be discretized via a mesh of finite elements. A possible mesh using triangular elements is shown in Figure b. Note that the outermost “row” of elements provides a chordal approximation to the circular boundary, and as the size of the elements is decreased (and the number of elements increased), the approximation becomes increasingly closer to the actual geometry. However, also note that the elements in the inner “rows” become increasingly slender (i.e., the height to base ratio is large). In general, the ratio of the largest characteristic dimension of an element to the smallest characteristic dimension is known as the aspect ratio. Large aspect ratios increase the inaccuracy of the finite element representation and have a detrimental effect on convergence of finite element solutions. An aspect ratio of 1 is ideal but cannot always be maintained. In Figure b, to maintain a reasonable aspect ratio for the inner elements, it would be necessary to reduce the height of each row of elements as the center of the sector is approached. This observation is also in keeping with the convergence requirements of the h-refinement method. Although the triangular element can be used to closely approximate a curved boundary, other considerations dictate a relatively large number of elements and associated computation time. If we consider rectangular elements as in Figure c (an intentionally crude mesh for illustrative purposes), the problems are apparent. Unless the elements are very small, the area of the domain excluded from the model (the shaded area in the figure) may be significant. For the case depicted, a large number of very small square elements best approximates the geometry. At this point, you may think, Why not use triangular and rectangular elements in the same mesh to improve the model? Indeed, a combination of the element types can be used to improve the geometric accuracy of the model. The shaded areas of Figure c could be modeled by three-node triangular elements. Such combination of element types may not be the best in terms of solution accuracy since the rectangular element and the triangular element have, by necessity, different order polynomial representations of the field variable. The field variable is continuous across such element boundaries; this is guaranteed by the finite element formulation. However, conditions on derivatives of the field variable for the two element types are quite different. On a curved boundary such as that shown, the triangular element used to fill the “gaps” left by the rectangular elements may also have adverse aspect ratio characteristics. Now examine Figure d, which shows the same area meshed with rectangular elements and a new element applied near the periphery of the domain. The new element has four nodes, straight sides, but is not rectangular. (Please note that the mesh shown is intentionally coarse for purposes of illustration.) The new element is known as a general two-dimensional quadrilateral element and is seen to mesh ideally with the rectangular element as well as approximate the curved boundary, just like the triangular element. The four-node quadrilateral element is derived from the four-node rectangular element (known as the parent element) element via a mapping process. For complex geometries © 2002 Brooks/Cole Publishing / Thomson Learning™ General quadrilateral elements Elements with curved sides

2014/5/15 Two-dimensional element calculations Transformation equations .We assume that the mappings smy-n mae he .We need to figure out how we tomake all these sdefined in terms of Transformation equations Transformation equations From the equations above we can conclude that xnae=u上 Constructing the transformation T emie.N(st) =∑N6网 where M is the number of nodes in the master element 2

2014/5/15 2 In isoparametric formulation 1. Shape functions first expressed in (s,t) coordinate system i.e., Ni (s,t) 2. The isoparamtric mapping relates the (s,t) coordinates with the global coordinates (x,y) , i.e., if you are given a point (s,t) in isoparametric coordinates, then you can compute the coordinates of the point in the (x,y) coordinate system using the equations 3. The inverse map will never be explicitly computed. It is laborious to find the inverse map s(x,y) and t(x,y) and we do not do that. Instead we compute the integrals in the domain of the parent element. i i i i i i y N s t y x N s t x ( , ) ( , )    

2014/5/15 Constructing the transformation T Geometric interpretation of the Jacobian The finite element basis r=xis.n).y=y(s.n)and (.n da-Jda 9 tumed inside ou Transformation equations Example problem Let us return to the definition of Consider the following master element. Applying the mapping: 一一兰m leads to: m-a.ga-m ,,m,5,9 等等兽将 V=0-0+ Example problem Example problem .Examine if the following mapping is invertable. Exanhmappinis inverable. =是，G=0g+4gn=+ y-y8m-emRm-ta y兰ygn-gm-g-+ Note thatthe map T,is invertable. is the ratio of the areaof to area of .Clockwise numbeing of nodes is prohibited 3

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2014/5/15 Example problem Example problem .Examine if the following mapping is invertable. Examine th following mapping is invertable T =兰n=g+g+- ，生yg2g家gg0 y=兰ygm=g+2gm-0-500 is smaller near node 1.largest near node 3. .All angles of quadrilateral elements need to be<t The .知+ +6,-0+6,-0-1,+0- ，wm moune the nto an is 到- -m []m-号 4

2014/5/15 4 s t 1 1 2 1 3 4 1 1 Problem: Consider the following isoparamteric map y 1 2 3 4 (3,1) (6,1) (6,6) (3,6) x ISOPARAMETRIC COORDINATES y GLOBAL COORDINATES 1 1 2 2 3 3 4 4 1 1 2 2 3 3 4 4 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) x N s t x N s t x N s t x N s t x y N s t y N s t y N s t y N s t y         The isoparamtric map (1 )(1 ) (1 )(1 ) (1 )(1 ) (1 )(1 ) 6 3 3 6 4 4 4 4 3(1 ) 2 (1 )(1 ) (1 )(1 ) (1 )(1 ) (1 )(1 ) 6 6 1 1 4 4 4 4 7 5 2 s t s t s t s t x s s t s t s t s t y t                              In this case, we may compute the inverse map, but we will NOT do that! The Jacobian matrix NOTE: The diagonal terms are due to stretching of the sides along the x-and y￾directions. The off-diagonal terms are zero because the element does not shear. 3 0 2 5 0 2 x y s s J x y t t                             3(1 ) 2 7 5 2 s x t y     since 1 2/ 3 0 15 and det( ) 0 2/ 5 4 J J          Figure above shows the parent element and its natural (r, s) coordinates and the quadrilateral element in a global Cartesian coordinate system. The geometry of the quadrilateral element is described by Mapping of a parent element into an isoparametric element. where the Gi (x , y) can be considered as geometric interpolation functions, and each such function is associated with a particular node of the quadrilateral element. Given the geometry and the form of Equations above, each function Gi (x , y) must evaluate to unity at its associated node and to zero at each of the other three nodes

2014/5/15 蜜 --立冰 多 )m- c→ h2-D case -∑。5-∑x 4-∑N%∑N%ΣN 2城gwg =N+空+EN g=∑M,+每+O 与 Note that N.N..0 4.g being 2 locnl coordinate

2014/5/15 5 These conditions are exactly the same as those imposed on the interpolation functions of the parent element. Consequently, the interpolation functions for the parent element can be used for the geometric functions, if we map the coordinates so that Consequently, the geometric expressions become Clearly, we can also express the field variable variation in the quadrilateral element as Since the same interpolation functions are used for both the field variable and description of element geometry, the procedure is known as isoparametric (constant parameter) mapping. The element defined by such a procedure is known as an isoparametric element

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2014/5/15 6 Good and bad isoparametric mappings of 4-node quadrilateral from the {ξ, η} reference plane onto the {x, y} physical plane. In formulating element characteristic matrices, various derivatives of the interpolation functions with respect to the global coordinates are required. In isoparametric elements, both element geometry and variation of the interpolation functions are expressed in terms of the natural coordinates of the parent element, so some additional mathematical complication arises. Specifically, we must compute ∂ Ni /∂ x and ∂ Ni /∂ y (and, possibly, higher-order derivatives). Since the interpolation functions are expressed in (r, s) coordinates, we can formally write these derivatives as Jacobian matrix The isoparametric formulation is by no means limited to linear parent elements. Many higher￾order isoparametric elements have been formulated and used successfully. Figure above depicts the isoparametric elements corresponding to the six-node triangle and the eight-node rectangle. Owing to the mapping being described by quadratic functions of the parent elements, the resulting elements have curved boundaries, which are also described by quadratic functions of the global coordinates. Such elements can be used to closely approximate irregular boundaries. However, note that curved elements do not, in general, exactly match a specified boundary curve. •Same shape functions are used to interpolate nodal coordinates and displacements •Shape functions are defined for an idealized mapped element (e.g. square for any quadrilateral element) •Advantages include more flexible shapes and compatibility •We pay the price in complexity and require numerical integration to calculate stiffness matrices and equivalent loads The developments presented in this chapter show how interpolation functions for one-, two-, and three-dimensional elements can be obtained via a systematic procedure. Also, the algebraically tedious procedure can often be bypassed using intuition and logic when natural coordinates are used. The interpolation functions discussed are standard polynomial forms but by no means exhaustive of the interpolation functions that have been developed for use in finite element analysis. As the objective of this text is to present the fundamentals of finite element analysis, the material of this chapter is intended to cover the basic concepts of interpolation functions without proposing to be comprehensive. The treatment here is intended to form a basis for formulation of finite element models of various physical problems in following chapters. In general, every element and the associated interpolation functions discussed here can be applied to specific problems, as is illustrated in the remainder of the text. CONCLUSIONS