Isoparametric Quadrilaterals

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Isoparametric Quadrilaterals

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Chapter 17: ISOPARAMETRIC QUADRILATERALS

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TABLE OF CONTENTS
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§17.1. Introduction §17.2. Partial Derivative Computation §17.2.1. The Jacobian . . . . . . . §17.2.2. Shape Function Derivatives . . §17.2.3. Computing the Jacobian Matrix . §17.2.4. The Strain-Displacement Matrix §17.2.5. *A Shape Function Implementation §17.3.Numerical Integration by Gauss Rules §17.3.1. One Dimensional Rules . . . §17.3.2. Implementation of 1D Rules . . §17.3.3. Two Dimensional Rules . . . §17.3.4. Implementation of 2D Gauss Rules §17.4. The Stiffness Matrix §17.5. *Integration Variants §17.5.1. *Weighted Integration . . . . §17.5.2. *Selective Integration . . . . . . . . . . §17. Notes and Bibliography . . . . . . . §17. References . . .. . . . . . . . . §17. Exercises . . . . . . . . . . . .

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17–3 17–3 17–317–4 17–4 17–5 17–5 17–6 17–6 17–8 17–8 17–9 17–10 17–11 17–11 17–12 17–12 17–12 17–13

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17–3 §17.1. Introduction

§17.2

PARTIAL DERIVATIVE COMPUTATION

In this Chapter the isoparametric representation of element geometry and shape functions discussed in the previous Chapter is used to construct quadrilateral elements for the plane stress problem. Formulas given in Chapter 14 for thestiffness matrix and consistent load vector of general plane stress elements are of course applicable to these elements. For a practical implementation, however, we must go through more specific steps: 1. 2. 3. Construction of shape functions. Computations of shape function derivatives to form the strain-displacement matrix. Numerical integration over the element by Gauss quadrature rules.

Thefirst topic was dealt in the previous Chapter in recipe form, and is systematically covered in the next one. Assuming the shape functions have been constructed (or readily found in the FEM literature) the second and third items are combined in an algorithm suitable for programming any isoparametric quadrilateral. The implementation of the algorithm in the form of element modules is partly explained inthe Exercises of this Chapter, and covered more systematically in Chapter 23. We shall not deal with isoparametric triangles here to keep the exposition focused. Triangular coordinates, being linked by a constraint, require “special handling” techniques that would complicate and confuse the exposition. Chapter 24 discusses isoparametric triangular elements in detail. §17.2. Partial DerivativeComputation Partial derivatives of shape functions with respect to the Cartesian coordinates x and y are required for the strain and stress calculations. Because shape functions are not directly functions of x and y but of the natural coordinates ξ and η, the determination of Cartesian partial derivatives is not trivial. The derivative calculation procedure is presented below for the case of anarbitrary isoparametric quadrilateral element with n nodes. §17.2.1. The Jacobian In quadrilateral element derivations we will need the Jacobian of two-dimensional transformations that connect the differentials of {x, y} to those of {ξ, η} and vice-versa. Using the chain rule:  ∂x ∂x   ∂ξ ∂ξ  dx dξ dξ dx  ∂ξ ∂η  dξ  ∂x ∂y  dx = = JT = J−T , = . ∂ y ∂ y  dη ∂η ∂η  dy dy dη dη dy ∂ξ ∂η ∂x ∂y(17.1) Here J denotes the Jacobian matrix of (x, y) with respect to (ξ, η), whereas J−1 is the Jacobian matrix of (ξ, η) with respect to (x, y):  ∂x ∂y   ∂ξ ∂η  J= ∂(x, y)  ∂ξ ∂ξ  = = ∂x ∂y  ∂(ξ, η) ∂η ∂η 1 ∂(ξ, η)  ∂ x ∂ x  J11 J12 =  ∂ξ ∂η  = , J−1 = J21 J22 ∂(x, y) J ∂y ∂y J22 −J12 , −J21 J11

(17.2) where J = |J| = det(J) = J11 J22 − J12 J21 . In FEM work J and J−1 are...