Quadratic (or Nonlinear) 9-node isoparametric quadrilateral element in Finite element analysis (FEA)

Previously, 4-node linear isoparametric elements have been covered in the following videos:
   • FEA Isoparametric Quadrilaterals Part 1: J...  
   • FEA Isoparametric Quadrilaterals Part 2: T...  
   • FEA Isoparametric Quadrilaterals Part 3: G...  
   • FEA Isoparametric Quadrilaterals Part 4: E...  
While this is a big step up from the 3-node constant strain triangle (CST) element (   • Stiffness matrix for constant strain trian...  ), there are still some limitations - especially with bending. This video explores how the addition of mid-side nodes can provide higher order shape functions which address these limitations. This video covers the nine-node version of a 2D isoparametric quadrilateral element which includes a center node in addition to the mid-side nodes. Note that there is also an eight-node version that is commonly used which does not include the center node.

0:00 Introduction
0:39 Limitations of the 4-node linear isoparametric element
1:36 Definition of "shear locking"
2:58 How quadratic elements address limitations linear isoparametric elements
4:06 Shape functions for the 9-node quadratic isoparametric element
8:17 Position functions and Jacobian matrix for 9-node quadratic isoparametric element
10:10 Stiffness matrix equation for 9-node quadratic isoparametric element
11:28 Displacement functions and the strain-displacement matrix [B] for 9-node quadratic isoparametric element
13:26 Gauss integration for 9-node quadratic isoparametric element
17:34 Reflection questions

Suggested answers to reflection questions
1.) Quadratic (also called "Nonlinear") elements are especially useful in bending because they counter the "shear locking" effect present in linear elements. They also can map to curved edges such as fillets easier.
2.) More Gauss integration points (or sampling points) are required to evaluate the stiffness matrix for the 9-node quadratic isoparametric element because the shape functions are now quadratic instead of linear (as for the 4-node isoparametric element). This means that the polynomials within the integrand of the stiffness matrix have a higher order and thus require more Gauss integration points. According to Gauss quadrature, four Gauss integration points should be used in each direction but FEA typically uses three in each direction.
3.) The increased computational cost comes in two forms:
First the setup time increases because the stiffness matrix is now 18x18 instead of 8x8. Also, there are now 9 Gauss integration points instead of 4. This means that each element would require 2916 separate computations instead of 256 (not to mention that each computation is more involved).
Second, the number of total degrees of freedom in the model, n, would increase by ~9/4. Since solving {q} = inv([K])*{F} is roughly proportional to n^2, the solve process would roughly increase by a factor of 5. Since most of the solve time typically corresponds to solving {q} = inv([K])*{F}, this would have the most drastic effect on solution time.