# shapeDerivHex_Lin

Computes the shape function derivatives and the determinant of the jacobian matrix for hexahedral linear 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

## Usage

[B Jdet]=shapeDerivHex_Lin(p, MSH, n)

## 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.

## Output

B: Shape function derivatives

Jdet: The determinant of the Jacobian Matrix

%% Shape functions N1 = 0.125*(1-ksi)*(1-eta)*(1-zta);

N2=0.125*(1+ksi)*(1-eta)*(1-zta);

N3 = 0.125*(1+ksi)*(1+eta)*(1-zta);

N4 = 0.125*(1-ksi)*(1+eta)*(1-zta);

N5 = 0.125*(1-ksi)*(1-eta)*(1+zta);

N6 = 0.125*(1+ksi)*(1-eta)*(1+zta);

N7 = 0.125*(1+ksi)*(1+eta)*(1+zta);

N8 = 0.125*(1-ksi)*(1+eta)*(1+zta);

## Derivatives of shape functions

wrt. ksi:

dN1 = -((eta - 1)*(zta - 1))/8;

dN2 = ((eta - 1)*(zta - 1))/8;

dN3 = -((eta + 1)*(zta - 1))/8;

dN4 = ((eta + 1)*(zta - 1))/8;

dN5 = ((eta - 1)*(zta + 1))/8;

dN6 = -((eta - 1)*(zta + 1))/8;

dN7 = ((eta + 1)*(zta + 1))/8;

dN8 = -((eta + 1)*(zta + 1))/8;

wrt. eta:

dN9 = -((ksi - 1)*(zta - 1))/8;

dN10 = ((ksi + 1)*(zta - 1))/8;

dN11 = -((ksi + 1)*(zta - 1))/8;

dN12 = ((ksi - 1)*(zta - 1))/8;

dN13 = ((ksi - 1)*(zta + 1))/8;

dN14 = -((ksi + 1)*(zta + 1))/8;

dN15 = ((ksi + 1)*(zta + 1))/8;

dN16 = -((ksi - 1)*(zta + 1))/8;

wrt. zeta:

dN17 = -((eta - 1)*(ksi - 1))/8;

dN18 = ((eta - 1)*(ksi + 1))/8;

dN19 = -((eta + 1)*(ksi + 1))/8;

dN20 = ((eta + 1)*(ksi - 1))/8;

dN21 = ((eta - 1)*(ksi - 1))/8;

dN22 = -((eta - 1)*(ksi + 1))/8;

dN23 = ((eta + 1)*(ksi + 1))/8;

dN24 = -((eta + 1)*(ksi - 1))/8;