Computes the shape function derivatives and the determinant of the jacobian matrix for quadratic quadrilateral elements. This function is used internally from shapeDerivatives.

Version : 1.0

Author : George Kourakos

email: giorgk@gmail.com

Date 18-Mar-2014

Department of Land Air and Water

University of California Davis

## Input

p: [Np x 3] Coodrinates of nodes [x1 y1 z1; x2 y2 z2;...xn yn zn], where Np is the number of nodes

MSH: [Nel x Np_el] id of elements. Each row correspond to an element. Nel is the number of elements and Np_el is the number of nodes to define the element

n: the integration point where the derivatives will be evaluated.

proj : if proj is true then the elements will be projected on the 2D plane before computing the determinant usign mapElemto2D

## Output

B: Shape function derivatives

Jdet: The determinant of the Jacobian Matrix

## Shape functions

N1=0.25*(xi-1)*(eta-1)*xi*eta; N2=0.25*(xi+1)*(eta-1)*xi*eta;

N3=0.25*(xi+1)*(eta+1)*xi*eta; N4=0.25*(xi-1)*(eta+1)*xi*eta;

N5=-0.5*(xi^2-1)*(eta-1)*eta; N6=-0.5*(xi+1)*(eta^2-1)*xi;

N7=-0.5*(xi^2-1)*(eta+1)*eta; N8=-0.5*(xi-1)*(eta^2-1)*xi;

N9=(xi^2-1)*(eta^2-1);

## Derivatives of shape functions

dN1 = (eta*(2*xi - 1)*(eta - 1))/4; dN2 = (eta*(2*xi + 1)*(eta - 1))/4; dN3 = (eta*(2*xi + 1)*(eta + 1))/4;

dN4 = (eta*(2*xi - 1)*(eta + 1))/4; dN5 = -eta*xi*(eta - 1); dN6 = -((eta^2 - 1)*(2*xi + 1))/2;

dN7 = -eta*xi*(eta + 1); dN8 = -((eta^2 - 1)*(2*xi - 1))/2; dN9 = 2*xi*(eta^2 - 1); (wrt. ksi)

dN10 = (xi*(2*eta - 1)*(xi - 1))/4; dN11 = (xi*(2*eta - 1)*(xi + 1))/4; dN12 = (xi*(2*eta + 1)*(xi + 1))/4;

dN13 = (xi*(2*eta + 1)*(xi - 1))/4; dN14 = -((2*eta - 1)*(xi^2 - 1))/2; dN15 = -eta*xi*(xi + 1);

dN16 = -((2*eta + 1)*(xi^2 - 1))/2; dN17 = -eta*xi*(xi - 1); dN18 = 2*eta*(xi^2 - 1); (wrt. eta)