* Objective: convex quadratic
* Constraints: convex quadratic

***********************************************************************
* Continuum elements.
* Iterative round up/down procedure to force a 0/1 solution.
***********************************************************************
$Set N  15
$Set M  15
$Set N1 14
$Set M1 14

Set i  / i0*i1 / ;
Alias(i,j);
Alias(i,k);
Alias(i,l);
Set b  / b1*b2 / ;

Set X  / x0*x%N% / ;
Set Y  / y0*y%M% / ;
Set X0(X) / x1*x%N1% / ;
Set Y0(Y) / y1*y%M1% / ;
Set X01(X) / x1*x%N1% / ;
Set Y01(Y) / y1*y%M1% / ;
Set d / d1,d2 / ;


Parameter   lambda ;  lambda = 1.0 ;
Parameter       mu ;      mu = 1.0 ;
Parameter porosity ; porosity = 0.70 ;
Parameter Vol ; Vol = 1 ; Vol = porosity*%n%*%m%;
Parameter reducedVol; reducedVol := Vol;

Set ANCHORS(x,y) ; ANCHORS(x,y) = no;


Parameter load(x,y,d) ; load(x,y,d) = 0 ;
load[x,y,'d1']${((ord(x)-1) eq 0  ) and ((ord(y)-1) gt 0) and ((ord(y)-1) lt %m%) } = -1;
load[x,y,'d1']${((ord(x)-1) eq %n%) and ((ord(y)-1) gt 0) and ((ord(y)-1) lt %m%) } =  1;
load[x,y,'d2']${((ord(y)-1) eq 0  ) and ((ord(x)-1) gt 0) and ((ord(x)-1) lt %n%) } = -1;
load[x,y,'d2']${((ord(y)-1) eq %m%) and ((ord(x)-1) gt 0) and ((ord(x)-1) lt %n%) } =  1;

load['x0','y0','d1']     = -1/2 ;  load['x0','y0','d2']     = -1/2 ;
load['x%n%','y0','d1']   =  1/2 ;  load['x%n%','y0','d2']   = -1/2 ;
load['x%n%','y%m%','d1'] =  1/2 ;  load['x%n%','y%m%','d2'] =  1/2 ;
load['x0','y%m%','d1']   = -1/2 ;  load['x0','y%m%','d2']   =  1/2 ;


    set ELEMENTS := X0 cross Y0;

Parameter xi(i) ; xi('i0') = -1 ;  xi('i1') = 1 ;

Parameter axx(i,j,k,l) ;   axx(i,j,k,l) = xi[i]*xi[k]*(1 + xi[j]*xi[l]/3);
Parameter ayy(i,j,k,l) ;   ayy(i,j,k,l) = xi[j]*xi[l]*(1 + xi[i]*xi[k]/3);
Parameter axy(i,j,k,l) ;   axy(i,j,k,l) = xi[i]*xi[l];
Parameter ayx(i,j,k,l) ;   ayx(i,j,k,l) = xi[j]*xi[k];

Parameter K_big(i,j,k,l,d,b) ;
    K_big[i,j,k,l,'d1','b1'] = (lambda + 2*mu)*axx[i,j,k,l] + mu*ayy[i,j,k,l];
    K_big[i,j,k,l,'d2','b2'] = (lambda + 2*mu)*ayy[i,j,k,l] + mu*axx[i,j,k,l];
    K_big[i,j,k,l,'d1','b2'] = lambda*axy[i,j,k,l] + mu*ayx[i,j,k,l];
    K_big[i,j,k,l,'d2','b1'] = lambda*ayx[i,j,k,l] + mu*axy[i,j,k,l];

Parameter u(x,y,d);
Parameter rho(x,y); rho[x,y] := 0.5;

***********************************************************************
* nonlinear model for w, theta  (w is u and theta is xi in my notes)
***********************************************************************
Variable  w(x,y,d) , theta , work ;

Equation
*To_anchor(x,y,d) ,
element_eqs(x,y) , Def_obj_1 ;

element_eqs(x,y)$((X0(X))and(Y0(Y))and(rho[x,y] ne 1))..
        sum{(i,j,k,l),
        (
        w[x+ord(i),y+(ord(j)-1),'d1']*K_big[i,j,k,l,'d1','b1']*w[x+ord(k),y+(ord(l)-1),'d1'] +
        w[x+ord(i),y+(ord(j)-1),'d2']*K_big[i,j,k,l,'d2','b1']*w[x+ord(k),y+(ord(l)-1),'d1'] +
        w[x+ord(i),y+(ord(j)-1),'d1']*K_big[i,j,k,l,'d1','b2']*w[x+ord(k),y+(ord(l)-1),'d2'] +
        w[x+ord(i),y+(ord(j)-1),'d2']*K_big[i,j,k,l,'d2','b2']*w[x+ord(k),y+(ord(l)-1),'d2']
         ) }
        =l= theta/reducedVol;

*To_anchor(x,y,d)$ANCHORS(x,y)..    w[x,y,d] =e= 0 ;

Def_obj_1.. work =e=   { 2 * sum{(x,y,d),  load[x,y,d]*w[x,y,d] } -
                sum{(x,y)$((X0(X))and(Y0(Y))and(rho[x,y] eq 1)),
                sum{(i,j,k,l),
                (
        w[x+ord(i),y+(ord(j)-1),'d1']*K_big[i,j,k,l,'d1','b1']*w[x+ord(k),y+(ord(l)-1),'d1'] +
        w[x+ord(i),y+(ord(j)-1),'d2']*K_big[i,j,k,l,'d2','b1']*w[x+ord(k),y+(ord(l)-1),'d1'] +
        w[x+ord(i),y+(ord(j)-1),'d1']*K_big[i,j,k,l,'d1','b2']*w[x+ord(k),y+(ord(l)-1),'d2'] +
        w[x+ord(i),y+(ord(j)-1),'d2']*K_big[i,j,k,l,'d2','b2']*w[x+ord(k),y+(ord(l)-1),'d2']
                 )}}
        - theta  }/100000000
        ;

model nlmodel /  element_eqs,
*                 To_anchor ,
                 Def_obj_1 /;

w.lo[x,y,d] = 0.00;
theta.lo = 0.01 ;

solve nlmodel using nlp maximazing work;
u[x,y,d] := w.l[x,y,d];

work.l = 100000000*work.l;
***********************************************************************
* linear model for sigma
***********************************************************************
Variable sigma(x,y) , zero  ;
Equation compat(x,y,d), tot_vol , Def_obj_2 ;

compat(x,y,d)$( (X0(X)) and (Y0(Y)) and (ord(X) gt 1) and (ord(Y) gt 1) )..

     sum{(i,j,k,l)$( (X0(X-(ord(i)-1) ) ) and (Y0(Y-(ord(j)-1))) and (ord(X)-(ord(i)-1) gt 1 )  and (ord(Y)-(ord(j)-1) gt 1 ) ),
     (
         K_big[i,j,k,l,d,'b1']*u[x+(ord(k)-ord(i)),y+(ord(l)-ord(j)),'d1'] +
         K_big[i,j,k,l,d,'b2']*u[x+(ord(k)-ord(i)),y+(ord(l)-ord(j)),'d2']
      )*sigma[x-(ord(i)-1),y-(ord(j)-1)]}
      - load[x,y,d] =e= 0 ;


tot_vol..
    sum {(x,y)$((X0(X))and(Y0(Y))), sigma[x,y]} =e= Vol ;

Def_obj_2.. zero =e= 0 ;


model linmodel / compat, tot_vol , Def_obj_2 /;

solve linmodel  using nlp minimazing zero;