1 / 13

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)

goro
Télécharger la présentation

Isoparametric Elements

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related