q4plate.cpp

Finite element program for mechanical problem. It can solve various problem in solid problem

#include "q4plate.h"
#include "global.h"
#include "globmat.h"
#include "genfile.h"
#include "node.h"
#include "element.h"
#include "intpoints.h"
#include <stdlib.h>
#include <math.h>


// w,fx,fy
q4plate::q4plate (void)
{
  long i,j;

  nne=4;  ndofe=12;  tncomp=5;  napfun=2;  ned=4;  nned=2;
  intordmm=3; intordb=2;  ssst=plate;
  
  nb=2;

  ncomp = new long [nb];
  ncomp[0]=3;
  ncomp[1]=2;

  cncomp = new long [nb];
  cncomp[0]=0;
  cncomp[1]=3;

  nip = new long* [nb];
  intordsm = new long* [nb];
  for (i=0;i<nb;i++){
    nip[i] = new long [nb];
    intordsm[i] = new long [nb];
  }
  nip[0][0]=9;  nip[0][1]=0;
  nip[1][0]=0;  nip[1][1]=4;

  tnip=0;
  for (i=0;i<nb;i++){
    for (j=0;j<nb;j++){
      tnip+=nip[i][j];
    }
  }

  intordsm[0][0]=3;  intordsm[0][1]=0;
  intordsm[1][0]=0;  intordsm[1][1]=2;
}



q4plate::~q4plate (void)
{
  long i;
  
  for (i=0;i<nb;i++){
    delete [] nip[i];
    delete [] intordsm[i];
  }
  delete [] nip;
  delete [] intordsm;

  delete [] cncomp;
  delete [] ncomp;
}


void q4plate::eleminit (long eid)
{
  long ii,jj;
  Mt->elements[eid].nb=nb;
  Mt->elements[eid].intordsm = new long* [nb];
  Mt->elements[eid].nip = new long* [nb];

  for (ii=0;ii<nb;ii++){
    Mt->elements[eid].intordsm[ii] = new long [nb];
    Mt->elements[eid].nip[ii] = new long [nb];
    for (jj=0;jj<nb;jj++){
      Mt->elements[eid].intordsm[ii][jj]=intordsm[ii][jj];
      Mt->elements[eid].nip[ii][jj]=nip[ii][jj];
    }
  }
}

/**
   function approximates function defined by nodal values

   @param xi,eta - coordinates on element
   @param nodval - nodal values
*/
double q4plate::approx (double xi,double eta,vector &nodval)
{
  double f;
  vector bf(nne);
  
  bf_lin_4_2d (bf.a,xi,eta);
  
  scprd (bf,nodval,f);
  
  return f;
}

/**
   function assembles %matrix of transformation for Q4plate
   %matrix reduces 16 DOFs to 12 DOFs
   
   20.3.2002
*/
void q4plate::atd_matrix (matrix &atd,vector &x,vector &y,long eid)
{
  double sx,sy;
  matrix p(16,16),p1(16,16),ct(4,2);
  vector dl(4);
  long i,j,i1,i2,i3,i4,ii;

  dl[0]=sqrt((x[1]-x[0])*(x[1]-x[0])+(y[1]-y[0])*(y[1]-y[0]));
  dl[1]=sqrt((x[2]-x[1])*(x[2]-x[1])+(y[2]-y[1])*(y[2]-y[1]));
  dl[2]=sqrt((x[3]-x[2])*(x[3]-x[2])+(y[3]-y[2])*(y[3]-y[2]));
  dl[3]=sqrt((x[0]-x[3])*(x[0]-x[3])+(y[0]-y[3])*(y[0]-y[3]));
  // matrix vector of sides {cos,sin} 
  ct[0][0]=(x[1]-x[0])/dl[0];
  ct[0][1]=(y[1]-y[0])/dl[0];
  ct[1][0]=(x[2]-x[1])/dl[1];
  ct[1][1]=(y[2]-y[1])/dl[1];
  ct[2][0]=(x[3]-x[2])/dl[2];
  ct[2][1]=(y[3]-y[2])/dl[2];
  ct[3][0]=(x[0]-x[3])/dl[3];
  ct[3][1]=(y[0]-y[3])/dl[3];
  fillm (0.0,p);
  fillm (0.0,p1);
  fillm (0.0,atd);

  for (i=0;i<nne;i++){
    i1=3*i;
    i2=i1+1;
    i3=i1+2;
    i4=12+i;
    p[i1][0]=1.;
    p[i1][1]=x[i];
    p[i1][2]=y[i];
    p[i1][3]=x[i]*y[i];
    p[i1][4]=x[i]*x[i];
    p[i1][5]=y[i]*y[i];
    p[i1][6]=x[i]*x[i]*y[i];
    p[i1][7]=y[i]*y[i]*x[i];
    p[i1][8]=x[i]*x[i]*x[i];
    p[i1][9]=y[i]*y[i]*y[i];
    p[i1][10]=x[i]*x[i]*x[i]*y[i];
    p[i1][11]=y[i]*y[i]*y[i]*x[i];
    p[i2][5]=y[i]*2.;
    p[i2][6]=x[i]*x[i];
    p[i2][7]=y[i]*x[i]*2.;
    p[i2][9]=y[i]*y[i]*3.;
    p[i2][10]=x[i]*x[i]*x[i];
    p[i2][11]=y[i]*y[i]*x[i]*3.;
    p[i2][14]=1.;
    p[i2][15]=x[i];
    p[i3][4]=-x[i]*2.;
    p[i3][6]=-x[i]*y[i]*2.;
    p[i3][7]=-y[i]*y[i];
    p[i3][8]=-x[i]*x[i]*3.;
    p[i3][10]=-x[i]*x[i]*y[i]*3.;
    p[i3][11]=-y[i]*y[i]*y[i];
    p[i3][12]=-1.;
    p[i3][13]=-y[i];

    ii=i+1;

    if (ii > 3) ii=ii-4;
    
    sy=(y[i]+y[ii])/2.;
    sx=(x[i]+x[ii])/2.;
    p[i4][1]= ct[i][0];
    p[i4][2]= ct[i][1];
    p[i4][3]= sy*ct[i][0]+sx*ct[i][1];
    p[i4][12]=-ct[i][0];
    p[i4][13]=-sy*ct[i][0];
    p[i4][14]=-ct[i][1];
    p[i4][15]=-sx*ct[i][1];
  }

  double *ee;
  ee = new double [16*16];
  for (i=0;i<16;i++){
    for (j=0;j<16;j++){
      ee[i*16+j]=0.0;
    }
    ee[i*16+i]=1.0;
  }
  
  gemp (p.a,p1.a,ee,16,16,10.e-5,1);
//  invm (p,p1);  without move zerodiagonal
  delete [] ee;

/*  
  if(eid==122){
    fprintf (Out,"\n\n p1");
    for (i=0;i<16;i++){
      fprintf (Out,"\n");
      for (j=0;j<16;j++){
      fprintf (Out," %le",p1[i][j]);
	  }
    }
*/

  for (i=0;i<p1.n;i++){
    atd[i][0] =p1[i][0]  -p1[i][12]/dl[0]+p1[i][15]/dl[3];
    atd[i][1] =p1[i][1]  -p1[i][12]*ct[0][1]/2.-p1[i][15]*ct[3][1]/2.;
    atd[i][2] =p1[i][2]  +p1[i][12]*ct[0][0]/2.+p1[i][15]*ct[3][0]/2.;
    atd[i][3] =p1[i][3]  -p1[i][13]/dl[1]+p1[i][12]/dl[0];
    atd[i][4] =p1[i][4]  -p1[i][13]*ct[1][1]/2.-p1[i][12]*ct[0][1]/2.;
    atd[i][5] =p1[i][5]  +p1[i][13]*ct[1][0]/2.+p1[i][12]*ct[0][0]/2.;
    atd[i][6] =p1[i][6]  -p1[i][14]/dl[2]+p1[i][13]/dl[1];
    atd[i][7] =p1[i][7]  -p1[i][14]*ct[2][1]/2.-p1[i][13]*ct[1][1]/2.;
    atd[i][8] =p1[i][8]  +p1[i][14]*ct[2][0]/2.+p1[i][13]*ct[1][0]/2.;
    atd[i][9] =p1[i][9]  -p1[i][15]/dl[3]+p1[i][14]/dl[2];
    atd[i][10]=p1[i][10] -p1[i][15]*ct[3][1]/2.-p1[i][14]*ct[2][1]/2.;
    atd[i][11]=p1[i][11] +p1[i][15]*ct[3][0]/2.+p1[i][14]*ct[2][0]/2.;
  }
  
}

/**
  function assembles %matrix of base functions for w
  (approximation of deflection)

  20.3.2002
*/
void q4plate::bf_matrix (matrix &n,matrix &atd,vector &x,vector &y,vector &l)
{
  double sx,sy;
  int i;
  
  // first point is in I Q
  sx=(x[0]*(1+l[0])*(1+l[1])+x[1]*(1-l[0])*(1+l[1])+x[2]*(1-l[0])*(1-l[1])+x[3]*(1+l[0])*(1-l[1]))/4.;
  sy=(y[0]*(1+l[0])*(1+l[1])+y[1]*(1-l[0])*(1+l[1])+y[2]*(1-l[0])*(1-l[1])+y[3]*(1+l[0])*(1-l[1]))/4.;
  // base w
  for (i=0;i<atd.n;i++){
    n[0][i]=atd[0][i]+sx*atd[1][i]+sy*atd[2][i]+sx*sy*atd[3][i]+sx*sx*atd[4][i]
      +sy*sy*atd[5][i]+sx*sx*sy*atd[6][i]+sy*sy*sx*atd[7][i]+sx*sx*sx*atd[8][i]
      +sy*sy*sy*atd[9][i]+sx*sx*sx*sy*atd[10][i]+sx*sy*sy*sy*atd[11][i];
  }
  
}

/**
   function assembles bending part of the geometrical %matrix of base functions Q4plate
   20.3.2002
*/
void q4plate::geom_matrix_bending (matrix &gm,matrix &atd,vector &x,vector &y,vector &l)
{
  double sx,sy;
  int i;
  // first point is in I Q
  sx=(x[0]*(1+l[0])*(1+l[1])+x[1]*(1-l[0])*(1+l[1])+x[2]*(1-l[0])*(1-l[1])+x[3]*(1+l[0])*(1-l[1]))/4.;
  sy=(y[0]*(1+l[0])*(1+l[1])+y[1]*(1-l[0])*(1+l[1])+y[2]*(1-l[0])*(1-l[1])+y[3]*(1+l[0])*(1-l[1]))/4.;
  // original gm matrix, which would have to multiply by matrix atd
  //  gm[0][4]=-2.;  gm[0][6]=-y*2.;  gm[0][8]=-x*6.;  gm[0][10]=-x*y*6.;
  //  gm[1][5]=-2.;  gm[1][7]=-x*2.;  gm[1][9]=-y*6.;  gm[1][11]=-x*y*6.;
  //  gm[2][6]=-x*4.; gm[2][7]=-y*4.; gm[2][10]=-x*x*6.; gm[2][11]=-y*y*6.; gm[2][13]=-1.; gm[2][15]=-1.;
  
  for (i=0;i<atd.n;i++){
    gm[0][i]=-2.*atd[4][i]-2.*sy*atd[6][i]-6.*sx*atd[8][i]-6.*sx*sy*atd[10][i];
    gm[1][i]=-2.*atd[5][i]-2.*sx*atd[7][i]-6.*sy*atd[9][i]-6.*sx*sy*atd[11][i];
    gm[2][i]=-4.*sx*atd[6][i]-4.*sy*atd[7][i]-6.*sx*sx*atd[10][i]-6.*sy*sy*atd[11][i]-atd[13][i]-atd[15][i];
  }
  
}

/**
   function assembles shear part of the geometrical %matrix of base functions Q4plate
   20.3.2002
*/
void q4plate::geom_matrix_shear (matrix &gm,matrix &atd,vector &x,vector &y,vector &l)
{
  long i;
  double sx,sy;
  
  // first point is in I Q
  sx=(x[0]*(1+l[0])*(1+l[1])+x[1]*(1-l[0])*(1+l[1])+x[2]*(1-l[0])*(1-l[1])+x[3]*(1+l[0])*(1-l[1]))/4.;
  sy=(y[0]*(1+l[0])*(1+l[1])+y[1]*(1-l[0])*(1+l[1])+y[2]*(1-l[0])*(1-l[1])+y[3]*(1+l[0])*(1-l[1]))/4.;
  
  for (i=0;i<atd.n;i++){
    gm[0][i]=atd[1][i]+sy*atd[3][i]-atd[12][i]-sy*atd[13][i];
    gm[1][i]=atd[2][i]+sx*atd[3][i]-atd[14][i]-sx*atd[15][i];
  }
}

/**
   function assembles stiffness matrices for bending and shear

   function extracts components of the bending
   part of stiffness %matrix of the material to the matrix db

   20.3.2002
*/
void q4plate::dmatblock (matrix &dd,matrix &d,long ri, long ci, double t)
{
  double c;
  
  fillm (0.0,dd);

  if (ri==0 && ci==0){
  c=t*t*t;
  dd[0][0] = c*d[0][0];  dd[0][1] = c*d[0][1];  dd[0][2] = c*d[0][2];
  dd[1][0] = c*d[1][0];  dd[1][1] = c*d[1][1];  dd[1][2] = c*d[1][2];
  dd[2][0] = c*d[2][0];  dd[2][1] = c*d[2][1];  dd[2][2] = c*d[2][2];
  }
  if (ri==1 && ci==1){
    dd[0][0] = t*d[3][3]*5.0/6.0;  dd[0][1] = 0.0;
    dd[1][0] = 0.0;                dd[1][1] = t*d[4][4]*5.0/6.0;
  }
}


/**
   function assembles transformation %matrix from local nodal coordinate
   system to the global coordinate system x_g = T x_l
   
   base vectors e1 and e2 have to contain 2 components
   the components are in the midplane
   
   @param nodes - element nodes
   @param tmat - transformation %matrix
   
   JK,
*/
void q4plate::transfmat (ivector &nodes,matrix &tmat)
{
  long i,n,m;
  n=nodes.n;
  m=tmat.m;
  for (i=0;i<m;i++){
    tmat[i][i]=1.0;
  }
  
  for (i=0;i<n;i++){
    if (Mt->nodes[nodes[i]].transf>0){
      tmat[i*3+1][i*3+1]=Mt->nodes[nodes[i]].e1[0];  tmat[i*3+1][i*3+2]=Mt->nodes[nodes[i]].e2[0];
      tmat[i*3+2][i*3+1]=Mt->nodes[nodes[i]].e1[1];  tmat[i*3+2][i*3+2]=Mt->nodes[nodes[i]].e2[1];
    }
  }
}

/**
   function computes stiffness %matrix of Q4plate element
   
   this function is used in plate elements (function is called
   by function res_stiffness_matrix) and shell elements
   
   @param eid - element id
   @param ri,ci - row and column indeces
   @param sm - stiffness %matrix
   @param x,y - nodal coordinates
   
   20.3.2002
*/
void q4plate::stiffness_matrix (long eid,long ri, long ci, matrix &sm,vector &x, vector &y)
{
  long i,j,ii,jj,ipp;
  double jac,a,a0,a1,thick;
  ivector nodes(nne);
  vector w,gp,l(2),t(nne);
  matrix gm,dd,atd(16,12),d(5,5);

  Mt->give_elemnodes (eid,nodes);
  Mc->give_thickness (eid,nodes,t);

  // translation to center it is necesary for w which in local coord. and no in relation
  a=(x[0]+x[1]+x[2]+x[3])/4.;
  x[0]=x[0]-a;
  x[1]=x[1]-a;
  x[2]=x[2]-a;
  x[3]=x[3]-a;
  a=(y[0]+y[1]+y[2]+y[3])/4.;
  y[0]=y[0]-a;
  y[1]=y[1]-a;
  y[2]=y[2]-a;
  y[3]=y[3]-a;

  //     jakobian
  a = (x[2]-x[0])*(y[3]-y[1])-(y[2]-y[0])*(x[3]-x[1]);
  a0=-(x[3]-x[2])*(y[1]-y[0])+(y[3]-y[2])*(x[1]-x[0]);
  a1=-(x[2]-x[1])*(y[3]-y[0])+(y[2]-y[1])*(x[3]-x[0]);

  atd_matrix (atd,x,y,eid);

  fillm (0.0,sm);

//  ii=Mt->elements[eid].ipp[sip][0];
  for (ii=0;ii<nb;ii++){
    for (jj=0;jj<nb;jj++){
      if (intordsm[ii][jj]==0)  continue;
      allocv (intordsm[ii][jj],w); allocv (intordsm[ii][jj],gp);
      allocm (ncomp[jj],ndofe,gm);
      allocm (ncomp[ii],ncomp[jj],dd);

      ipp=Mt->elements[eid].ipp[ri+ii][ci+jj];
      gauss_points (gp.a,w.a,intordsm[ii][jj]);

	  for (i=0;i<intordsm[ii][jj];i++){
		 l[0]=gp[i];  
		for (j=0;j<intordsm[ii][jj];j++){
		  l[1]=gp[j];
		  jac=(a+a0*l[0]+a1*l[1])/8.*w[i]*w[j];

		  if(ii==0) geom_matrix_bending (gm,atd,x,y,l);
		  else if(ii==1) geom_matrix_shear (gm,atd,x,y,l);
		  Mm->matstiff (d,ipp);
	      thick=approx (l[0],l[1],t);
          dmatblock (dd, d,ii,jj,thick);
          bdbjac (sm,gm,dd,gm,jac);
		  ipp++;
		}
	  }
      destrm (dd);  destrm (gm);  destrv (gp);  destrv (w);
	}
  }


/*
  fprintf (Out,"\n\n kontrola matice tuhosti");
  for (i=0;i<ndofe;i++){
    fprintf (Out,"\n");
    for (j=0;j<ndofe;j++){fprintf (Out," %le",sm[i][j]);}
  }
  if(eid<3)
  {
    fprintf (Out,"\n\n diagonalni prvky desky, %ld,\n",eid);
    for (i=0;i<ndofe;i++){fprintf (Out," %le",sm[i][i]);}
  }
*/  

}


/**
   function assembles stiffness %matrix of plane stress rectangular
   finite element with biquadratic approximation functions
   
   @param eid - element id
   @param sm - stiffness %matrix
   
*/
void q4plate::res_stiffness_matrix (long eid,matrix &sm)
{
  long transf;
  ivector nodes(nne);
  vector x(nne),y(nne);
  Mt->give_node_coord2d (x,y,eid);
  Mt->give_elemnodes (eid,nodes);
  
  stiffness_matrix (eid,0,0,sm,x,y);

  //  transformation of stiffness matrix
  transf = Mt->locsystems (nodes);
  if (transf>0){
    matrix tmat (ndofe,ndofe);
    transfmat (nodes,tmat);
    glmatrixtransf (sm,tmat);
  }
}


/**
   function computes initial stress stiffness %matrix of Q4 element
   
   @param eid - element id
   @param ri,ci - row and column indices
   @param ism - i-s matrix
   
   15.11.2003
*/
void q4plate::initstr_matrix (long eid,long ri,long ci,matrix &ism)
{
  long i,i1,i2,j,j1,j2,k,ii,jj;
  double jac,a,a0,a1,thick,xx,yy;
  ivector nodes(nne);
  vector w,gp,l(2),x(nne),y(nne),z(nne),sig(3),sx(3),sy(3),t(nne),dwx(12),dwy(12);
  matrix tran(3,3),atd(16,12);

// sig....... Axial stress

  Mt->give_elemnodes (eid,nodes);
  Mc->give_thickness (eid,nodes,t);

// translation to center it is necesary for w which in local coord. and no in relation
  a=(x[0]+x[1]+x[2]+x[3])/4.;
  x[0]=x[0]-a;
  x[1]=x[1]-a;
  x[2]=x[2]-a;
  x[3]=x[3]-a;
  a=(y[0]+y[1]+y[2]+y[3])/4.;
  y[0]=y[0]-a;
  y[1]=y[1]-a;
  y[2]=y[2]-a;
  y[3]=y[3]-a;
  
  //     jakobian
  a = (x[2]-x[0])*(y[3]-y[1])-(y[2]-y[0])*(x[3]-x[1]);
  a0=-(x[3]-x[2])*(y[1]-y[0])+(y[3]-y[2])*(x[1]-x[0]);
  a1=-(x[2]-x[1])*(y[3]-y[0])+(y[2]-y[1])*(x[3]-x[0]);

  atd_matrix (atd,x,y,eid);

  fillm (0.0,ism);

//  ii=Mt->elements[eid].ipp[sip][0];
  for (ii=0;ii<nb;ii++){
    for (jj=0;jj<nb;jj++){
      if (intordsm[ii][jj]==0)  continue;
      allocv (intordsm[ii][jj],w); allocv (intordsm[ii][jj],gp);

	  for (i=0;i<intordsm[ii][jj];i++){
		 l[0]=gp[i];  
		for (j=0;j<intordsm[ii][jj];j++){
		  l[1]=gp[j];
  	      xx=(x[1]*(1-l[0])*(1+l[1])+x[2]*(1-l[0])*(1-l[1])+x[3]*(1+l[0])*(1-l[1]))/4.;
	      yy=(y[1]*(1-l[0])*(1+l[1])+y[2]*(1-l[0])*(1-l[1])+y[3]*(1+l[0])*(1-l[1]))/4.;
          for (k=0;k<dwx.n;k++){
           dwx[i]=atd[1][k]+yy*atd[3][k]+2.*xx*atd[4][k]+2.*xx*yy*atd[6][k]+
                  yy*yy*atd[7][k]+3.*xx*xx*atd[8][k]+3.*xx*xx*yy*atd[10][k]+yy*yy*yy*atd[11][k];
           dwy[i]=atd[2][k]+xx*atd[3][k]+2.*yy*atd[5][k]+xx*xx*atd[6][k]+2.*yy*xx*     
                  atd[7][k]+3.*yy*yy*atd[9][k]+xx*xx*xx*atd[10][k]+3.*xx*yy*yy*atd[11][k];
		  }  
  
	      thick=approx (l[0],l[1],t);
		  jac=(a+a0*l[0]+a1*l[1])/8.*w[i]*w[j]*thick/24.;
          for (i2=0;i2<sig.n;i2++){
            i1=3*i2-3;
            for (j2=0;j2<nne;j2++){
             j1=3*j2-3;
             ism[i1][j1]=ism[i1][j1]+( dwx[i]*dwx[j]*sig[0]
                +(dwx[i]*dwy[j]+dwx[j]*dwy[i])*sig[2]+dwy[i]*dwy[j]*sig[1] )*jac;
			}
		  }
		}
	  }
      destrv (gp);  destrv (w);
	}
  }
       

	
	
}



/**
   function computes load %matrix of the plate rectangular finite element
   load vector is obtained after premultiplying load %matrix
   by nodal load values
   
   @param eid - number of element
   @param lm - load %matrix
   
   JK, 3.10.2006
*/
void q4plate::load_matrix (long eid,matrix &lm)
{
  long i,j;
  double jac,xi,eta,w1,w2,a,a0,a1,thick;
  ivector nodes(nne);
  vector x(nne),y(nne),w(intordmm),gp(intordmm),t(nne),l(2);
  matrix atd(16,12),n(napfun,ndofe);
  
  Mt->give_elemnodes (eid,nodes);
  Mc->give_thickness (eid,nodes,t);
  
  Mt->give_node_coord2d (x,y,eid);
  gauss_points (gp.a,w.a,intordmm);


  a=(x[0]+x[1]+x[2]+x[3])/4.;
  x[0]=x[0]-a;
  x[1]=x[1]-a;
  x[2]=x[2]-a;
  x[3]=x[3]-a;
  a=(y[0]+y[1]+y[2]+y[3])/4.;
  y[0]=y[0]-a;
  y[1]=y[1]-a;
  y[2]=y[2]-a;
  y[3]=y[3]-a;


  a = (x[2]-x[0])*(y[3]-y[1])-(y[2]-y[0])*(x[3]-x[1]);
  a0=-(x[3]-x[2])*(y[1]-y[0])+(y[3]-y[2])*(x[1]-x[0]);
  a1=-(x[2]-x[1])*(y[3]-y[0])+(y[2]-y[1])*(x[3]-x[0]);