Nodal and Mixed Bases  Modal basis  Triangle Basis:

Quadrilateral Bases: ${\bf QuHH}$ and ${\bf QuHN}$ 

For quadrilateral elements Q¹ is a bijection so we do not need to worry about the coordinate singularity as in the triangle case. We are free to choose a set of modes that are C⁰ compatible with the triangle expansion. An obvious choice that guarantees a high degree of orthogonality is the ${\bf QuHH}$ basis:
 
 

Vertex Modes:

Edge Modes $(2 \leq n, m < N)$:

 Interior Modes $( 2 \leq m,n < N)$:

 We represent this basis for N=5 in figure 3.4.
 
 

 

Figure 3.4: Mode shapes for the quadrilateral modal basis (${\bf QuHH}$) with N=5.

 
 
 

Alternatively, we could also choose a second set of modes with an even better orthogonality relationship. To this end, we use the Legendre interpolant functions, which we will investigate more thoroughly in the next section. These nodes are used in a collocation manner to maximize the discrete orthogonality of the modes in the interior region. More specifically, the Legendre quadrature points are used as nodal points as well. We also modify the vertex modes and replace the edge modes with tensor products of the one-dimensional modal basis and the Legendre basis. Similarly, we replace the interior modes with tensor products of only the Legendre basis. This means that edge modes from one edge are orthogonal to edge modes from another edge and all the interior modes. The interior modes are mutually orthogonal and hence we can perform inner products and backwards transforms in O(N²) operations. Before we write the basis we need to define an appropriate Lagrange interpolant in terms of the Legendre polynomial $P_n(x) \equiv P_n^{0,0}$ as:

$$h_n(r) = - \frac{(1-r^2)P_N^\prime(r)}{N(N+1)P_N(r_n)(r-r_n)}$$    where r_i denote location of roots of $(1-r^2)P_N^\prime(r)=0$ in the interval [-1,1]. Also, by definition we have that $h_n(r_m) =\delta_{mn}$. This construction destroys the hierarchy of the basis in the interior of the element. More specifically, this QuHN basis is defined as follows:
 
 

Vertex Modes:

Edge Modes $(2 \leq n, m < N)$:

 Interior Modes $( 2 \leq m,n < N)$:

 

 

Figure 3.5: Mode shapes for the quadrilateral mixed modal basis ${\bf QuHN}$ with N=5.

 

Note that according to our convention we have that h₁(-1)=1 and h_N(1)=1, etc. In figure 3.5 we see that the edge modes now have behavior localized to the edge, and the same is true for the vertex modes. In figure 3.6 we present graphically the inner product $(\phi_{mn}, \phi_{pq})$ which is the mass matrix for this basis. Here, we order the vertices first (around the origin), followed by the edges, and the interior contributions. We verify that indeed there exists great sparsity in this matrix indicative of the strong discrete orthogonality between modes.
 
 

 

Figure 3.6: Mass matrix for the quadrilateral mixed modal basis (${\bf QuHN}$) with N=15.

 

10/24/1998