shapeDerivPrism_quad

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

Version : 1.0

Author : George Kourakos

email: giorgk@gmail.com

web : http://groundwater.ucdavis.edu/msim

Date 18-Mar-2014

Department of Land Air and Water

University of California Davis

Contents

Usage

[B Jdet]=shapeDerivPrism_quad(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

N(1) = (xi*zta*(2*xi - 1)*(zta - 1))/2;

N(2) = (eta*zta*(2*eta - 1)*(zta - 1))/2;

N(3) = (zta*(zta - 1)*(eta + xi - 1)*(2*eta + 2*xi - 1))/2;

N(4) = (xi*zta*(2*xi - 1)*(zta + 1))/2;

N(5) = (eta*zta*(2*eta - 1)*(zta + 1))/2;

N(6) = (zta*(zta + 1)*(eta + xi - 1)*(2*eta + 2*xi - 1))/2;

N(7) = 2*eta*xi*zta*(zta - 1);

N(8) = -2*eta*zta*(zta - 1)*(eta + xi - 1);

N(9) = -(xi*zta*(zta - 1)*(4*eta + 4*xi - 4))/2;

N(10) = 2*eta*xi*zta*(zta + 1);

N(11) = -2*eta*zta*(zta + 1)*(eta + xi - 1);

N(12) = -(xi*zta*(zta + 1)*(4*eta + 4*xi - 4))/2;

N(13) = -xi*(2*xi - 1)*(zta^2 - 1);

N(14) = -eta*(2*eta - 1)*(zta^2 - 1);

N(15) = -(zta^2 - 1)*(eta + xi - 1)*(2*eta + 2*xi - 1);

N(16) = -4*eta*xi*(zta^2 - 1);

N(17) = 4*eta*(zta^2 - 1)*(eta + xi - 1);

N(18) = xi*(zta^2 - 1)*(4*eta + 4*xi - 4);

Derivatives of shape functions

wrt. ksi:

dN1 = xi*zta*(zta - 1) + (zta*(2*xi - 1)*(zta - 1))/2;

dN2 = 0;

dN3 = (zta*(zta - 1)*(2*eta + 2*xi - 1))/2 + zta*(zta - 1)*(eta + xi - 1);

dN4 = xi*zta*(zta + 1) + (zta*(2*xi - 1)*(zta + 1))/2;

dN5 = 0;

dN6 = (zta*(zta + 1)*(2*eta + 2*xi - 1))/2 + zta*(zta + 1)*(eta + xi - 1);

dN7 = 2*eta*zta*(zta - 1);

dN8 = -2*eta*zta*(zta - 1);

dN9 = - 2*xi*zta*(zta - 1) - (zta*(zta - 1)*(4*eta + 4*xi - 4))/2;

dN10 = 2*eta*zta*(zta + 1);

dN11 = -2*eta*zta*(zta + 1);

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

dN13 = - 2*xi*(zta^2 - 1) - (2*xi - 1)*(zta^2 - 1);

dN14 = 0;

dN15 = - (zta^2 - 1)*(2*eta + 2*xi - 1) - 2*(zta^2 - 1)*(eta + xi - 1);

dN16 = -4*eta*(zta^2 - 1);

dN17 = 4*eta*(zta^2 - 1);

dN18 = 4*xi*(zta^2 - 1) + (zta^2 - 1)*(4*eta + 4*xi - 4);

wrt. eta:

dN19 = 0;

dN20 = eta*zta*(zta - 1) + (zta*(2*eta - 1)*(zta - 1))/2;

dN21 = (zta*(zta - 1)*(2*eta + 2*xi - 1))/2 + zta*(zta - 1)*(eta + xi - 1);

dN22 = 0;

dN23 = eta*zta*(zta + 1) + (zta*(2*eta - 1)*(zta + 1))/2;

dN24 = (zta*(zta + 1)*(2*eta + 2*xi - 1))/2 + zta*(zta + 1)*(eta + xi - 1);

dN25 = 2*xi*zta*(zta - 1);

dN26 = - 2*eta*zta*(zta - 1) - 2*zta*(zta - 1)*(eta + xi - 1);

dN27 = -2*xi*zta*(zta - 1);

dN28 = 2*xi*zta*(zta + 1);

dN29 = - 2*eta*zta*(zta + 1) - 2*zta*(zta + 1)*(eta + xi - 1);

dN30 = -2*xi*zta*(zta + 1);

dN31 = 0;

dN32 = - (2*eta - 1)*(zta^2 - 1) - 2*eta*(zta^2 - 1);

dN33 = - (zta^2 - 1)*(2*eta + 2*xi - 1) - 2*(zta^2 - 1)*(eta + xi - 1);

dN34 = -4*xi*(zta^2 - 1);

dN35 = 4*(zta^2 - 1)*(eta + xi - 1) + 4*eta*(zta^2 - 1);

dN36 = 4*xi*(zta^2 - 1);

wrt. zeta:

dN37 = (xi*zta*(2*xi - 1))/2 + (xi*(2*xi - 1)*(zta - 1))/2;

dN38 = (eta*zta*(2*eta - 1))/2 + (eta*(2*eta - 1)*(zta - 1))/2;

dN39 = (zta*(eta + xi - 1)*(2*eta + 2*xi - 1))/2 + ((zta - 1)*(eta + xi - 1)*(2*eta + 2*xi - 1))/2;

dN40 = (xi*zta*(2*xi - 1))/2 + (xi*(2*xi - 1)*(zta + 1))/2;

dN41 = (eta*zta*(2*eta - 1))/2 + (eta*(2*eta - 1)*(zta + 1))/2;

dN42 = (zta*(eta + xi - 1)*(2*eta + 2*xi - 1))/2 + ((zta + 1)*(eta + xi - 1)*(2*eta + 2*xi - 1))/2;

dN43 = 2*eta*xi*(zta - 1) + 2*eta*xi*zta;

dN44 = - 2*eta*zta*(eta + xi - 1) - 2*eta*(zta - 1)*(eta + xi - 1);

dN45 = - (xi*zta*(4*eta + 4*xi - 4))/2 - (xi*(zta - 1)*(4*eta + 4*xi - 4))/2;

dN46 = 2*eta*xi*(zta + 1) + 2*eta*xi*zta;

dN47 = - 2*eta*zta*(eta + xi - 1) - 2*eta*(zta + 1)*(eta + xi - 1);

dN48 = - (xi*zta*(4*eta + 4*xi - 4))/2 - (xi*(zta + 1)*(4*eta + 4*xi - 4))/2;

dN49 = -2*xi*zta*(2*xi - 1);

dN50 = -2*eta*zta*(2*eta - 1);

dN51 = -2*zta*(eta + xi - 1)*(2*eta + 2*xi - 1);

dN52 = -8*eta*xi*zta;

dN53 = 8*eta*zta*(eta + xi - 1);

dN54 = 2*xi*zta*(4*eta + 4*xi - 4);

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