FEM Isoparametric Concept

Transkript

1 FEM Isoparametric Concept home/lehre/vl-mhs--e/cover_sheet.tex. p./26

2 Table of contents. Interpolation Functions for the Finite Elements 2. Finite Element Types 3. Geometry 4. Interpolation Approach Function 5. Cartesian - Natural Coordinates 6. Lagrange and Hermite Element Family home/lehre/vl-mhs--e/contents.tex. p.2/26

3 Shape Functions for FEM For the finite elements method, the following is valid: The global function of a sought function consists of a sum of local functions: E e= V ω e i dv e Galerkin method: the interpolation function corresponds to weighted function Ritz method: the global variation principle is constructed from the sum of the local variation principles. home/lehre/vl-mhs--e/interpolation.tex. p.3/26

4 Shape functions for FEM Critical step in the FEM: Selection of suitable interpolation functions Interpolation functions are formed through the shape of the finite elements, the approximation order. Finite elements depend on the geometry of the global area, the desired exactness of the area, the simple integration through the area. home/lehre/vl-mhs--e/interpolation2.tex. p.4/26

5 Finite Element Types In order to be able to formulate special physical problems, several elements are often necessary. They are distinguished by the geometry (-D, 2-D or 3-D), the selection of the interpolation function (polynomials; Lagrange or Hermite polynomials), the selection of the element coordinates (Cartesian or natural coordinates), the selection of the variables specified at the nodes (Lagrange group or Hermite group of variables). home/lehre/vl-mhs--e/finite element types.tex. p.5/26

6 Geometry a) Square elements with straight sides b) Square elements with curved sides c) Cubic elements home/lehre/vl-mhs--e/geometry.tex. p.6/26

7 Interpolation Approach Function Polynomials: One dimensional element approximation u = a 0 + a x + a 2 x 2 + a 3 x or u = a 0 + a i x i with i = linear variable i = 2 quadratic variable i = 3 cubic variable -D-Element mit zwei Knoten: 2 For two nodes we need a linear variability. home/lehre/vl-mhs--e/interpolation approach function.tex. p.7/26

8 Interpolation Approach Function Alternative method: r = x l 2 ξ = ξ=0 ξ= u = a 0 + a r for a linear element For each node set up and the constants determined, follows ũ = ω û + ω 2 û 2 = ω N û N home/lehre/vl-mhs--e/interpolation approach function2.tex. p.8/26

9 Interpolation Approach Function with N =, 2 and the interpolation functions ω = 2 ( r) { ω 2 = 2 ( + r) { = a 0 a 0 = a 0 + a 0 = a 0 a = a 0 + a. home/lehre/vl-mhs--e/interpolation approach function2a.tex. p.9/26

10 Interpolation Approach Function x 2 In a quadratic approximation, a 3-node -D element is necessary. Attention: If dimensionless coordinates are used for the derivation of the interpolation functions, one speaks of a natural coordinate system and natural coordinates. home/lehre/vl-mhs--e/interpolation approach function3.tex. p.0/26

11 Interpolation Approach Function Lagrange Elements: Lagrange Elements are necessary to avoid the inversion of the coefficient matrix for the approximation of higher order (see Chung). Hermite Elements: Important in the statistics: The constancy of the derivation of a function at the nodes is secured with the help of Hermite polynomials. home/lehre/vl-mhs--e/interpolation approach function3a.tex. p./26

12 Interpolation Approach Function 2 l 2-node, -D cubic approach ũ(ξ) = H 0 j (ξ)û j + H j (ξ) ( ) û ξ ũ(ξ) = N r w r r =, 2, 3, 4 j =, H H H home/lehre/vl-mhs--e/interpolation approach function4.tex x -0.2 H2. p.2/26

13 Interpolation Approach Function w = ũ N = H 0 = 3ξ 2 + 2ξ 3 w 2 = ũ 2 N 2 = H2 0 = 3ξ 2 2ξ 3 w 3 = ũ 3 N 3 = H = ξ 2ξ 2 + ξ 3 w 4 = ũ 4 N 4 = H2 = ξ 3 ξ 2 w = M also corresponds to a Hermite polynomial of the fifth EI degree! (M ˆ= bending moment) home/lehre/vl-mhs--e/interpolation_approach_function5.tex. p.3/26

14 Cartesian Natural Coordinates Definition: The same parametric function which describes the geometry is used for the interpolation of the variables (shifting, water level, etc.) within an element. Example: determination of the element matrix for a 4-node, 2-D element (see Wollrath). Introduction of a local coordinate system, since in the local coordinate system the basic functions are the same for each element. For example N : s r N = ( r) ( = 4 N 2 = 4 N 3 = 4 N 4 = s) ( + r)( + s) analog: ( r)( + s) ( r)( s) ( + r)( s) home/lehre/vl-mhs--e/cartesian natural coordinates.tex. p.4/26

15 Cartesian Natural Coordinates Further examples for interpolation functions:. Interpolation function for a 2-D element with a 4 to 9 variable node number y s s = Node s = 0 r 3 r = - 7 r = 0 r = + 4 s = - x home/lehre/vl-mhs--e/cartesian_natural_coordinates2.tex. p.5/26

16 Cartesian Natural Coordinates i = 5 i = 6 i = 7 i = 8 i = 9 h = ( + r)( + s) 4 h h 2 8 h 4 9 h 2 = ( r)( + s) 4 h 2 5 h 2 6 h 4 9 h 3 = ( r)( s) h 2 6 h 2 7 h 4 9 h 4 = ( + r)( s) h 2 7 h 2 8 h 4 9 h 5 = ( 2 r2 )( + s) h 2 9 h 6 = ( 2 s2 )( r) h 2 9 h 7 = ( 2 r2 )( s) h 2 9 h 8 = ( 2 s2 )( + r) h 2 9 h 9 = ( r 2 )( s 2 ) home/lehre/vl-mhs--e/cartesian natural coordinates3.tex. p.6/26

17 Cartesian Natural Coordinates 2. Interpolation function of a 3-D element with an 8 to 20 variable node number t z r s y x home/lehre/vl-mhs--e/cartesian_natural_coordinates4b.tex. p.7/26

18 Cartesian Natural Coordinates h = g (g 9 + g 2 + g 7 )/2 h 6 = g 6 (g 3 + g 4 + g 8 )/2 h 2 = g 2 (g 9 + g 0 + g 8 )/2 h 7 = g 7 (g 4 + g 5 + g 9 )/2 h 3 = g 3 (g 0 + g + g 9 )/2 h 8 = g 8 (g 5 + g 6 + g 20 )/2 h 4 = g 4 (g + g 2 + g 20 )/2 h i = g i for i = 9,..., 20 h 5 = g 5 (g 3 + g 6 + g 7 )/2 home/lehre/vl-mhs--e/cartesian natural coordinates4.tex. p.8/26

19 Cartesian Natural Coordinates g i = 0, when node i is not contained; g i = G(r, r i ) G(s, s i ) G(t, t i ) otherwise G(β, β i ) = 2 ( + β iβ) für β i = ± G(β, β i ) = ( β 2 ) für β i = 0 β = r, s, t In technical applications (e.g. ground water grad W ĥ), terms often are to differentiate or integrate according to the Cartesian coordinates. home/lehre/vl-mhs--e/cartesian natural coordinates4a.tex. p.9/26

20 Cartesian Natural Coordinates Since the function is depicted through isoparametric coordinates, one looks for a transformation relation between both coordinate systems. e.g.: With chain rule: x = r y = r grad N i = ( r ) x + s ( ) r y + s ( s ) x ( ) s y x y N i = r x r y s x s y r s home/lehre/vl-mhs--e/cartesian natural coordinates4a.tex. p.20/26

21 Cartesian Natural Coordinates The calculation of r, etc., is not easily possible. That is why the x following path is taken: for the 2-D 4-node element, the inverse relation is: r s = (J ˆ= Jacobi Matrix) x r x s y r y s [ ] x y = J x y home/lehre/vl-mhs--e/cartesian natural coordinates5.tex. p.2/26

22 Cartesian Natural Coordinates The Jacobi matrix can easily be determined by making use of the relation: x = n e i= ω i x i x = [ω, ω 2, ω 3, ω 4 ] (linear interpolation of the coordinates between the nodes) x x 2 x 3 y = [ω, ω 2, ω 3, ω 4 ] x 4 y y 2 y 3 y 4 home/lehre/vl-mhs--e/cartesian natural coordinates6.tex. p.22/26

23 Cartesian Natural Coordinates r s = = 4 N r N s N 2 r N 2 s N 3 r N 3 s N 4 r N 4 s x y x 2 y 2 x 3 y 3 x 4 y 4 } {{ } J [ ] x y ( + s) ( + s) ( s) ( s) x 2 y 2 ( + r) ( r) ( r) ( + r) x 3 y 3 } {{ x 4 y 4 } J x y x y. home/lehre/vl-mhs--e/cartesian natural coordinates7.tex. p.23/26

24 Z.B.: Cartesian Natural Coordinates y 2 x cm cm 4 cm 2 cm cm 3 2 cm y x 0.75 cm 4 x = 3r ; y = 2s J = x= 4 (+r)(+s)() + (-r)(+s)(-) + (-r)(-s)(-) + (+r)(-s)(+) y = 4 (+r)(+s)(5/4) + (-r)(+s)(/4) + (-r)(-s)(-3/4) + (+r)(-s)(-3/4) J = 4 (+s) 0 (3+r) home/lehre/vl-mhs--e/cartesian natural coordinates8.tex. p.24/26

25 Cartesian Natural Coordinates It is therefore: x y z = J r s t. home/lehre/vl-mhs--e/cartesian natural coordinates9.tex. p.25/26

26 Lagrange and Hermite Element Family Division of the elements into two catagories (independent of the geometric shape): The Lagrange family consists of finite elements for which the values are specified at the nodes, while for the Hermite family the function, as well the derivations, are determined at the nodes. The Lagrange family and the Hermite family can both be depicted through polynomials, which can be derived from the Pascal triangle/tetrahedron. E.g.: cubic approach for a 2-D problem: c 0 + c x + c 2 y + c 3 x 2 + c 4 xy + c 5 y 2 + c 6 x 3 + c 7 x 2 y + c 8 xy 2 + c 9 y 3 = û home/lehre/vl-mhs--e/lagrange and hermite.tex. p.26/26