PFVT quartic velocity class
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classdef ELS4424r < handle % ELS4424r % Container class for 4-node modified quartic Hermite finite % elements on rectangle/quadrilateral (designated by 'S4424'). % The element has 24 degrees-of-freedom and is C1 continuous. % The vector element is the curl of the scalar element and is % C0 continuous. It is used for divergence-free vector fields % such as incompressible velocity and magnetic fields. % % Jonas Holdeman January 2013 properties (Constant) name = 'modified C1 quartic Hermite'; designation = 'S4424r'; shape = 'quadrilateral'; nsides = 4; order = 4; % order of completeness nnodes = 4; % number of nodes nndofs = 6; % max number of nodal dofs tndofe = 24; % total number of dofs for element mxpowr = 5;; % highest power/degree in s 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 (Constant) methods (Static) % Four-node quartic-complete Hermite scalar stream function element % on the reference square [-1,1]x[-1,1]. The designated (4424), the % scalar function (4424) is C1 continuous with C0 curl and with % 24 degrees-of-freedom, 6 dofs per vertex node. % Parameter ni is one of coordinate pairs (-1,-1),(1,-1),(1,1),(-1,1) function s=s(ni,q,r) qi=ni(1); q0=q*ni(1); ri=ni(2); r0=r*ni(2); s=[ 1/64*(1+q0)^2*(1+r0)^2*(3*q0*(1-q0)^2*(2-r0)... +3*r0*(1-r0)^2*(2-q0)+4*(2-q0)*(2-r0)), ... -1/64/ri*(1+q0)^2*(1+r0)^3*(1-r0)*(2-q0)*(5-3*r0), ... 1/64/qi*(1+q0)^3*(1-q0)*(1+r0)^2*(2-r0)*(5-3*q0), ... 1/64/qi^2*(1+q0)^3*(1-q0)^2*(1+r0)^2*(2-r0), ... 1/16/qi/ri*(1+q0)^2*(1-q0)*(1+r0)^2*(1-r0), ... 1/64/ri^2*(1+r0)^3*(1-r0)^2*(1+q0)^2*(2-q0)]; end % s % Four-node cubic-complete Hermite solenoidal vector function element % on the reference square. The vector function is the curl of the % quartic-complete element (4424) with 6 dofs per vertex node. % Parameter ni is one of coordinate pairs (-1,-1),(1,-1),(1,1),(-1,1) function S=S(ni,q,r) qi=ni(1); q0=q*ni(1); ri=ni(2); r0=r*ni(2); S=[3/64*ri*(1+q0)^2*(1-r^2)*((1+q0)*(8-9*q0+3*q^2)+(2-q0)*(1-5*r^2)), ... -1/64*(1+q0)^2*(1+r0)^2*(2-q0)*(7-26*r0+15*r^2), ... 3/64*ri/qi*(1+q0)^3*(1-r^2)*(1-q0)*(5-3*q0), ... 3/64*ri/qi^2*(1+q0)^3*(1-r^2)*(1-q0)^2, ... 1/16/qi*(1+q0)^2*(1+r0)*(1-q0)*(1-3*r0), ... 1/64/ri*(1+q0)^2*(1+r0)^2*(1-r0)*(1-5*r0)*(2-q0); ... -3/64*qi*(1+r0)^2*(1-q^2)*((1+r0)*(8-9*r0+3*r^2)+(2-r0)*(1-5*q^2)), ... 3/64*qi/ri*(1+r0)^3*(1-q^2)*(1-r0)*(5-3*r0), ... -1/64*(1+r0)^2*(1+q0)^2*(2-r0)*(7-26*q0+15*q^2), ... -1/64/qi*(1+r0)^2*(1+q0)^2*(1-q0)*(1-5*q0)*(2-r0), ... -1/16/ri*(1+r0)^2*(1+q0)*(1-r0)*(1-3*q0), ... -3/64*qi/ri^2*(1+r0)^3*(1-q^2)*(1-r0)^2]; end % S % First derivatives wrt q & r of four-node cubic-complete Hermite % solenoidal vector function element on the reference square. % The vector function is the curl of the quartic-complete element % (4424) with 6 dofs at each corner node. % SQ = array of q-derivatives of solenoidal vectors % SR = array of r-derivatives of solenoidal vectors % Parameter ni is one of coordinate pairs (-1,-1),(1,-1),(1,1),(-1,1) function [SQ,SR]=DS(ni,q,r) qi=ni(1); q0=q*ni(1); ri=ni(2); r0=r*ni(2); SQ=[9/64*qi*ri*(1-q^2)*(1-r^2)*(6-5*q^2-5*r^2), ... -3/64*qi*(1-q^2)*(1+r0)^2*(1-3*r0)*(7-5*r0), ... 3/64*ri*(1+q0)^2*(1-r^2)*(1-3*q0)*(7-5*q0), ... 3/64*ri/qi*(1+q0)^2*(1-r^2)*(1-q0)*(1-5*q0), ... 1/16*(1+q0)*(1+r0)*(1-3*q0)*(1-3*r0), ... 3/64*qi/ri*(1-q^2)*(1+r0)^2*(1-r0)*(1-5*r0); ... 3/32*qi^2*q0*(1+r0)^2*(r0*(10*q^2+3*r^2-7)+20-20*q^2-6*r^2), ... -3/32*qi^2/ri*q0*(1+r0)^3*(1-r0)*(5-3*r0),... 3/16*qi*(1-q^2)*(1+r0)^2*(2-r0)*(1+5*q0), ... 1/16*(1+q0)*(1+r0)^2*(2-r0)*(1+2*q0-5*q^2), ... 1/8*qi/ri*(1+r0)^2*(1-r0)*(1+3*q0), ... 3/32*qi^2/ri^2*q0*(1+r0)^3*(1-r0)^2]; SR=[-3/32*ri^2*r0*(1+q0)^2*(q0*(10*r^2+3*q^2-7)+20-20*r^2-6*q^2), ... 3/16*ri*(1+q0)^2*(1-r^2)*(2-q0)*(1+5*r0),... -3/32*ri^2*r0/qi*(1+q0)^2*(1-q^2)*(5-3*q0), ... -3/32*ri^2/qi^2*r0*(1+q0)*(1-q^2)^2, ... -1/8*ri/qi*(1+q0)*(1-q^2)*(1+3*r0), ... -1/16*(1+q0)^2*(1+r0)*(2-q0)*(1+2*r0-5*r^2); ... -9/64*qi*ri*(1-q^2)*(1-r^2)*(6-5*q^2-5*r^2), ... 3/64*qi*(1+r0)^2*(1-q^2)*(1-3*r0)*(7-5*r0), ... -3/64*ri*(1+q0)^2*(1-r^2)*(1-3*q0)*(7-5*q0), ... -3/64*ri/qi*(1+q0)*(1-q^2)*(1-r^2)*(1-5*q0), ... -1/16*(1+q0)*(1+r0)*(1-3*q0)*(1-3*r0), ... -3/64*qi/ri*(1-q^2)*(1-r^2)*(1+r0)*(1-5*r0)]; end % DS % Second derivatives wrt {qq,qr,rr} of four-node quadratic-complete % Hermite solenoidal vector function element on the reference square. function [Sqq,Sqr,Srr]=D2S(ni,q,r) qi=ni(1); q0=q*ni(1); ri=ni(2); r0=r*ni(2); Sqq=[-9/32*qi^2*ri*q0*(1-r^2)*(11-10*q^2-5*r^2), ... 3/32*qi^2*q0*(1+r0)^2*(1-3*r0)*(7-5*r0) , ... -9/16*qi*ri*(1-r^2)*(1-q^2)*(1+5*q0), ... -3/16*ri*(1+q0)*(1-r^2)*(1+2*q0-5*q^2), ... -1/8*qi*(1+r0)*(1+3*q0)*(1-3*r0), ... -3/32*qi^2/ri*q0*(1+r0)^2*(1-r0)*(1-5*r0); ... 3/32*qi^3*(1+r0)^2*(6+(r0-2)*(30*q^2+3*r^2-7)), ... -3/32*qi^3/ri*(1+r0)^3*(1-r0)*(5-3*r0), ... 3/16*qi^2*(1+r0)^2*(2-r0)*(5-2*q0-15*q^2), ... 3/16*qi*(1+r0)^2*(2-r0)*(1-2*q0-5*q^2), ... 3/8*qi^2/ri*(1+r0)^2*(1-r0), ... 3/32*qi^3/ri^2*(1+r0)^3*(1-r0)^2]; Sqr=[9/32*qi*ri^2*r0*(1-q^2)*(5*q^2-11+10*r^2), ... 9/16*qi*ri*(1-q^2)*(1-r^2)*(5*r0+1), ... -3/32*ri^2*(q0+1)^2*r0*(-1+3*q0)*(-7+5*q0), ... 3/32*ri^2/qi*(1-q^2)*(1+q0)*r0*(-1+5*q0), ... 1/8*ri*(q0+1)*(-1+3*q0)*(1+3*r0), ... 3/16*qi*(1-q^2)*(1+r0)*(5*r^2-2*r0-1); ... -9/32*qi^2*ri*q0*(1-r^2)*(5*r^2-11+10*q^2), ... -3/32*qi^2*q0*(3*r0-1)*(5*r0-7)*(1+r0)^2, ... 9/16*qi*ri*(1-q^2)*(1-r^2)*(1+5*q0), ... -3/16*ri*(q0+1)*(1-r^2)*(5*q^2-2*q0-1), ... -1/8*qi*(1+r0)*(3*r0-1)*(1+3*q0), ... -3/32*qi^2/ri*q0*(5*r0-1)*(1-r^2)*(1+r0)]; Srr=[-3/32*ri^3*(1+q0)^2*(6+(q0-2)*(30*r^2+3*q^2-7)), ... 3/16*ri^2*(1+q0)^2*(2-q0)*(5-2*r0-15*r^2), ... -3/32*ri^3/qi*(1+q0)^2*(1-q^2)*(5-3*q0), ... -3/32*ri^3/qi^2*(1+q0)*(1-q^2)^2, ... -3/8*ri^2/qi*(1+q0)*(1-q^2), ... -3/16*ri*(1+q0)^2*(2-q0)*(1-2*r0-5*r^2); ... 9/32*qi*ri^2*r0*(1-q^2)*(11-5*q^2-10*r^2), ... -9/16*qi*ri*(1-q^2)*(1-r^2)*(1+5*r0), ... 3/32*ri^2*r0*(1+q0)^2*(1-3*q0)*(7-5*q0), ... 3/32*ri^2/qi*(1+q0)*(1-q^2)*r0*(1-5*q0), ... 1/8*ri*(1+q0)*(1-3*q0)*(1+3*r0), ... 3/16*qi*(1-q^2)*(1+r0)*(1+2*r0-5*r^2)]; end % D2S % Transpose of the Jacobian matrix at (q,r) function Jt=JacT(Xe,q,r) Jt=Xe*Gm(ELS4424r.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*ELS4424r.Gm(ELS4424r.nn(:,:),ELS4424r.nn(m,1),ELS4424r.nn(m,2)); JtiD=[Jt(2,2),-Jt(1,2); -Jt(2,1),Jt(1,1)]; % det(J)*inv(J) T2=[Jt(1,1)^2, 2*Jt(1,1)*Jt(1,2), Jt(1,2)^2; ... Jt(1,1)*Jt(2,1), Jt(1,1)*Jt(2,2)+Jt(1,2)*Jt(2,1), Jt(1,2)*Jt(2,2);... Jt(2,1)^2, 2*Jt(2,1)*Jt(2,2), Jt(2,2)^2]; ti=blkdiag(1,JtiD,T2); Bxy=Xe*ELS4424r.DGm(ELS4424r.nn(:,:),ELS4424r.nn(m,1), ... ELS4424r.nn(m,2)); % 2nd cross-derivative ti(5,2:3)=Bxy([2,1])'; 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 % Transposed second cross-derivative of scalar bilinear mapping function % The parameter ni can be a vector of coordinate pairs. function D=DGm(ni,q,r) D=.25*ni(:,1).*ni(:,2); end % DGm end % method (Static) end % classdef