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Isoparametric Elements

Isoparametric Elements. Structural Mechanics Displacement-based Formulations. Fundamental Dilemma. A primary reason engineers go to FEA is complex geometry But elements give the most accurate results when they have regular shapes (isosceles triangles, squares)

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Isoparametric Elements

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  1. Isoparametric Elements Structural Mechanics Displacement-based Formulations

  2. Fundamental Dilemma • A primary reason engineers go to FEA is complex geometry • But elements give the most accurate results when they have regular shapes (isosceles triangles, squares) • You should always minimize element distortion when you create a mesh (more on this later …) • It is also important to understand how element shape is managed …interpolation (shape) functions

  3. Reduced Accuracy • These elements work, but not well …

  4. Isoparametric Elements • There are two roles of interpolation in FEA: • Defining the location of interior points within an element in terms of nodal values (geometry interpolation) • Defining the displacement of interior points within an element in terms of nodal values (result interpolation) • There is no fundamental reason why both types of interpolation must be conducted in the same way • But a common class of highly versatile elements does just that • Iso = same; the same basis for geometry and result interpolation

  5. Bilinear Quadrilateral (Q4) • Interpolation involves the summation of nodal values multiplied by corresponding shapes functions geometry interpolation field variable interpolation - where - nodal coordinates nodal displacements shape functions

  6. Shape Functions • Shape functions have a value of 1.0 at the “corresponding” node and a value of 0.0 at all others (the function “belongs” to a node) • They span a normalized domain, typically [-1,1] over each spatial dimension

  7. Element Geometry Interpolation • Edges of adjacent elements match (no overlaps, gaps) as long as common nodes are shared • There are consistent interior point locations defined by the interpolation functions (e.g. you can define the “center” of an element)

  8. Example 2D Element N1 = (3,2) N2 = (11,3) N3 = (10,10) N4 = (4,9) N3 The shape functions establish a geometric equivalence between elements with different node locations… N4 Y N2 N1 X

  9. Field Quantity Interpolation • We assume the field quantity (e.g. x-component of displacement) varies within the element as a sum of node-weighted shape functions (visualized here as height above the plane) • If the field quantity actually does vary in this way (or close to it) then the element choice (size, order) is justified nodal values u1 = 2 u2 = 3 u3 = 4 u4 = 5 X Y At the element “center” …

  10. Multiple Elements element 2 N5 = (0,-5) N6 = (9,-3) N3 = (11,3) N4 = (3,2) element 1 … … N1 = (3,2) N2 = (11,3) N3 = (10,10) N4 = (4,9) u1 = 2 u2 = 3 u3 = 4 u4 = 5 u5 = 1 u6 = 2.5 X Y

  11. C0 Continuous • A key feature of these elements is their continuity • The value (C0) of an interpolated quantity is continuous across element boundaries • No geometric “gaps” or “overlaps” along the edges of shared elements • Field quantities transition without any “steps” in value • But the slope (C1) (derivative) of a field quantity is not continuous along element edges • As element density increases the transitions become less abrupt • It is possible to construct C1 shape functions, but they require “slope nodes” and are not as computationally efficient as mesh refinement

  12. Higher-Order Elements • The overall scheme stays exactly the same • But there are more nodes and different shape function definitions

  13. Multiple Elements

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