PFV4 cubic pressure class
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Revision as of 19:26, 15 March 2013 by Jonas Holdeman (Talk | contribs)
classdef ELG3412r < handle % ELG3412r % Container class for 4-node simple cubic Hermite finite elements % on rectangle/quadrilateral (designated by 'G3412r'). % The scalar element is used for scalar fields such as pressure % or temperature. % The vector element is used for irrotational vector fields % such as pressure or thermal gradients or irrotational fluid flow. % % Jonas Holdeman January 2013 properties (Constant) name = 'Simple-cubic Hermite'; designation = 'G3412r'; shape = 'quadrilateral'; nsides = 4; order = 3; % order of completeness nnodes = 4; % number of nodes nndofs = 3; % max number of nodal dofs tndofe = 12; % total number of dofs for element mxpowr = 3; % highest power/degree in g hQuad = @GQuad2; % handle to quadrature rules on rectangles cntr = [0 0]; % reference element centroid nn = [-1 -1; 1 -1; 1 1; -1 1]; % standard nodal order of coords end % properties methods (Static) % Four-node cubic-complete Hermite scalar potential function % element on the reference square. The vector function is the gradient % of this simple-cubic element (3412) with 3 dofs per corner node. % Parameter ni is one of coordinate pairs (-1,-1),(1,-1),(1,1),(-1,1) function g=g(ni,q,r) qi=ni(1); q0=q*ni(1); ri=ni(2); r0=r*ni(2); g=[1/8*(1+q0)*(1+r0)*(2+q0*(1-q0)+r0*(1-r0)), ... -1/8*qi*(1+q0)*(1+r0)*(1-q^2), -1/8*ri*(1+q0)*(1+r0)*(1-r^2)]; end % g % Four-node quadratic-complete Hermite irrotational vector function % element on the reference square. The vector function is the gradient % of the cubic-complete element (3412) with 3 dofs per corner node. % Parameter ni is one of coordinate pairs (-1,-1),(1,-1),(1,1),(-1,1) function gv=G(ni,q,r) qi=ni(1); q0=q*ni(1); ri=ni(2); r0=r*ni(2); gv=[1/8*qi*(1+r0)*(r0*(1-r0)+3*(1-q^2)), ... -1/8*(1+r0)*(1+q0)*(1-3*q0), ... -1/8*qi/ri*(1-r^2)*(1+r0); ... 1/8*ri*(1+q0)*(q0*(1-q0)+3*(1-r^2)), ... -1/8/qi*ri*(1-q^2)*(1+q0), ... -1/8*(1+q0)*(1+r0)*(1-3*r0)]; end % G % First derivatives wrt q & r of four-node quadratic-complete Hermite % gradient vector function element on the reference square. % The vector function is the gradient of the cubic-complete element % (3412) with 3 dofs at each corner node. % Gq = array of q-derivatives of irrotational vectors % Gr = array of r-derivatives of irrotational vectors % Parameter ni is one of coordinate pairs (-1,-1),(1,-1),(1,1),(-1,1) function [Gq,Gr]=DG(ni,q,r) qi=ni(1); q0=q*ni(1); ri=ni(2); r0=r*ni(2); Gq=[-3/4*qi^2*q0*(1+r0), 1/4*qi*(1+r0)*(1+3*q0), 0; ... 1/8*qi*ri*(4-3*q0^2-3*r0^2), -1/8*ri*(1+q0)*(1-3*q0), ... -1/8*qi*(1+r0)*(1-3*r0)]; Gr=[1/8*qi*ri*(4-3*q0^2-3*r0^2), -1/8*ri*(1+q0)*(1-3*q0), ... -1/8*qi*(1+r0)*(1-3*r0); ... -3/4*ri^2*r0*(1+q0), 0, 1/4*ri*(1+q0)*(1+3*r0)]; end % DG % Transpose of the Jacobian matrix at (q,r) function Jt=JacT(Xe,q,r) Jt=Xe*Gm(ELG3412rr.nn(:,:),q,r); end % JacT % Test to see if transformation is affine, returns True or False function isit=isaffine(Xe) isit=sum(abs(Xe(:,1)-Xe(:,2)+Xe(:,3)-Xe(:,4)))<4*eps; end % isaffine % Post-multiplying matrix Ti^-1 function ti=Ti(Xe,m) Jt=Xe*ELG3412r.Gm(ELS32r.nn(:,:),ELG3412r.nn(m,1),ELG3412r.nn(m,2)); JtiD=[Jt(2,2),-Jt(1,2); -Jt(2,1),Jt(1,1)]; % det(J)*inv(J) ti=blkdiag(1,JtiD); end % Ti % Bilinear mapping function from (q,r) in the reference square % [-1.1]x[-1,1] to (x,y) in the straight-sided quadrilateral % finite elements. % The parameter ni can be a vector of coordinate pairs. function g=gm(ni,q,r) g=.25*(1-ni(:,1).*q)*(1+ni(:,2).*r); end % gm % Transposed gradient (derivatives) of scalar bilinear mapping function % The parameter ni can be a vector of coordinate pairs. function G=Gm(ni,q,r) G=[.25*ni(:,1).*(1+ni(:,2).*r), .25*ni(:,2).*(1+ni(:,1).*q)]; end % Gm % Second (cross) derivative of scalar bilinear mapping function % The parameter ni can be a vector of coordinate pairs. function D=DGm(ni,~,~) D=.25*ni(:,1).*ni(:,2); end % DGm end % method (Static) end % classdef