* Objective: convex quadratic
* Constraints: convex quadratic

*****************************************************************
* Continuum elements.
* Iterative procedure to impose upper bounds of 1 on sigmas.
*****************************************************************
$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) / x0*x%N1% / ;
Set Y0(Y) / y0*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;
Parameter beta; beta := 2 ;
Set ANCHORS(x,y) ; ANCHORS(x,y) = no;

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

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

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/exp(log(reducedVol)*beta);

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

Def_obj_1.. -work =e=    2 * sum{(x,y,d)$((X0(X))and(Y0(Y))),  load[x,y,d]*w[x,y,d] } -
                            sum{(x,y)$((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
        ;

model nlmodel /  element_eqs,
*                 To_anchor ,
                 Def_obj_1 /;
w.lo[x,y,d] = 0.001;
theta.lo = 0 ;


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


*****************************************************************
* convex 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 ;

Def_obj_2.. zero =e= 0 ;


model linmodel / compat,  Def_obj_2 /;

solve linmodel  using nlp minimazing zero;