Hutton - Fundamentals of Finite Element Analysis (523155), страница 36
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Therefore,all the first partial derivatives of the field variable are constant. In structuralapplications, the tetrahedral element is a constant strain element; in general, theelement exhibits constant gradients of the field variable in the coordinatedirections.Other elements of the tetrahedral family are depicted in Figure 6.18. Theinterpolation functions for the depicted elements are readily written in volumecoordinates, as for higher-order two-dimensional triangular elements. Note particularly that the second element of the family has 10 nodes and a cubic form forthe field variable and interpolation functions. A quadratic tetrahedral elementcannot be constructed to exhibit geometric isotropy even if internal nodes areincluded.(a)(b)Figure 6.18 Higher-order tetrahedral elements:(a) 10 node.
(b) 20 node.Hutton: Fundamentals ofFinite Element Analysis6. Interpolation Functionsfor General ElementFormulation© The McGraw−HillCompanies, 2004Text6.7 Three-Dimensional Elements872c4(⫺1, 1, ⫺1)65(⫺1, 1, 1)yr(1, ⫺1, ⫺1)txz(1, 1, ⫺1)s2b3191122a(a)(⫺1, ⫺1, 1)(1, ⫺1, 1)(b)Figure 6.19 Eight-node brick element: (a) Global Cartesian coordinates. (b) Naturalcoordinates with an origin at the centroid.6.7.2 Eight-Node Brick ElementThe so-called eight node brick element (rectangular parellopiped) is shown inFigure 6.19a in reference to a global Cartesian coordinate system.
Here, weutilize the natural coordinates r, s, t of Figure 6.19b, defined asr =x − x̄as=y − ȳbt=z − z̄c(6.67)where 2a, 2b, 2c are the dimensions of the element in the x, y, z coordinates,respectively, and the coordinates of the element centroid arex̄ =x2 − x12ȳ =y3 − y22z̄ =z5 − z12(6.68)The natural coordinates are defined such that the coordinate values vary between−1 and +1 over the domain of the element. As with the plane rectangularelement, the natural coordinates provide for a straightforward development ofthe interpolation functions by using the appropriate monomial terms to satisfynodal conditions.
As we illustrated the procedure in several previous developments, we do not repeat the details here. Instead, we simply write the interpolation functions in terms of the natural coordinates and request that the readerHutton: Fundamentals ofFinite Element Analysis1926. Interpolation Functionsfor General ElementFormulationCHAPTER 6© The McGraw−HillCompanies, 2004TextInterpolation Functions for General Element Formulationverify satisfaction of all nodal conditions. The interpolation functions areN1 =1(1 − r)(1 − s)(1 + t)8N2 =1(1 + r)(1 − s)(1 + t)8N3 =1(1 + r)(1 + s)(1 + t)8N4 =1(1 − r)(1 + s)(1 + t)81N5 = (1 − r)(1 − s)(1 − t)8N6 =1(1 + r)(1 − s)(1 − t)8N7 =1(1 + r)(1 + s)(1 − t)8N8 =1(1 − r)(1 + s)(1 − t)8(6.69)and the field variable is described as(x , y, z) =8N i (r, s, t )i(6.70)i=1If Equation 6.70 is expressed in terms of the global Cartesian coordinates, itis found to be of the form(x , y, z) = a0 + a1 x + a2 y + a3 z + a4 x y + a5 x z + a6 yz + a7 x yz(6.71)showing that the field variable is expressed as an incomplete, symmetric polynomial.
Geometric isotropy is therefore assured. The compatibility requirement issatisfied, as is the completeness condition. Recall that completeness requires thatthe first partial derivatives must be capable of assuming constant values (for C 0problems). If, for example, we take the first partial derivative of Equation 6.71with respect to x, we obtain∂= a1 + a4 y + a5 z + a7 yz∂x(6.72)which certainly does not appear to be constant at first glance.
However, if weapply the derivative operation to Equation 6.70 while noting that∂1 ∂=∂xa ∂r(6.73)Hutton: Fundamentals ofFinite Element Analysis6. Interpolation Functionsfor General ElementFormulation© The McGraw−HillCompanies, 2004Text6.8 Isoparametric Formulationthe result is11∂=(1 − s)(1 + t )(2 − 1 ) +(1 + s)(1 + t )(3 − 4 )∂x8a8a+11(1 − s)(1 − t )(6 − 5 ) +(1 + s)(1 − t )(7 − 8 )8a8a(6.74)Referring to Figure 6.19, observe that, if the gradient of the field variable in thex direction is constant, ∂ /∂ x = C, the nodal values are related by∂2 = 1 +dx = 1 + C (2a)∂x∂3 = 4 +dx = 4 + C (2a)∂x(6.75)∂ 6 = 5 +dx = 5 + C (2a)∂x∂7 = 8 +dx = 8 + C (2a)∂xSubstituting these relations into Equation 6.74, we find∂1=[(1 − s)(1 + t )(2aC ) + (1 + s)(1 + t )(2aC )∂x8a+ (1 − s)(1 + t )(2aC ) + (1 + s)(1 − t )(2aC )](6.76a)which, on expansion and simplification, results in∂≡C∂x(6.76b)Observing that this result is valid at any point (r, s, t) within the element, itfollows that the specified interpolation functions indeed allow for a constant gradient in the x direction.
Following similar procedures shows that the other partialderivatives also satisfy the completeness condition.6.8 ISOPARAMETRIC FORMULATIONThe finite element method is a powerful technique for analyzing engineeringproblems involving complex, irregular geometries. However, the two- and threedimensional elements discussed so far in this chapter (triangle, rectangle, tetrahedron, brick) cannot always be efficiently used for irregular geometries.Consider the plane area shown in Figure 6.20a, which is to be discretized via amesh of finite elements. A possible mesh using triangular elements is shown inFigure 6.20b. 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 increasingly193Hutton: Fundamentals ofFinite Element Analysis1946.
Interpolation Functionsfor General ElementFormulationCHAPTER 6© The McGraw−HillCompanies, 2004TextInterpolation Functions for General Element Formulation(a)(b)(c)(d)Figure 6.20(a) A domain to be modeled. (b) Triangular elements.(c) Rectangular elements. (d) Rectangular and quadrilateralelements.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 smallestcharacteristic dimension is known as the aspect ratio. Large aspect ratios increasethe inaccuracy of the finite element representation and have a detrimental effecton convergence of finite element solutions [8].
An aspect ratio of 1 is ideal butcannot always be maintained. (Commercial finite element software packages provide warnings when an element’s aspect ratio exceeds some predetermined limit.)In Figure 6.20b, to maintain a reasonable aspect ratio for the inner elements, itwould be necessary to reduce the height of each row of elements as the center ofthe sector is approached.
This observation is also in keeping with the convergencerequirements of the h-refinement method. Although the triangular element can beused to closely approximate a curved boundary, other considerations dictate arelatively large number of elements and associated computation time.If we consider rectangular elements as in Figure 6.20c (an intentionallycrude mesh for illustrative purposes), the problems are apparent. Unless theelements are very small, the area of the domain excluded from the model (theHutton: Fundamentals ofFinite Element Analysis6. Interpolation Functionsfor General ElementFormulation© The McGraw−HillCompanies, 2004Text6.8 Isoparametric Formulationshaded 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, the astute reader 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 themodel. The shaded areas of Figure 6.20c could be modeled by three-node triangular elements. Such combination of element types may not be the best interms of solution accuracy since the rectangular element and the triangular element have, by necessity, different order polynomial representations of the fieldvariable.
The field variable is continuous across such element boundaries; this isguaranteed by the finite element formulation. However, conditions on derivativesof the field variable for the two element types are quite different. On a curvedboundary such as that shown, the triangular element used to fill the “gaps” left bythe rectangular elements may also have adverse aspect ratio characteristics.Now examine Figure 6.20d, which shows the same area meshed with rectangular elements and a new element applied near the periphery of the domain. Thenew element has four nodes, straight sides, but is not rectangular.
(Please notethat the mesh shown is intentionally coarse for purposes of illustration.) The newelement is known as a general two-dimensional quadrilateral element and is seento mesh ideally with the rectangular element as well as approximate the curvedboundary, just like the triangular element. The four-node quadrilateral element isderived from the four-node rectangular element (known as the parent element)element via a mapping process. Figure 6.21 shows the parent element and itsnatural (r, s) coordinates and the quadrilateral element in a global Cartesiancoordinate system.
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