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ex_piezoelectric1.m File Reference

Description

EX_PIEZOELECTRIC1 Piezoelectric bending of a beam example.

[ FEA, OUT ] = EX_PIEZOELECTRIC1( VARARGIN ) Example for piezoelectric bending of a beam. Reference: Hwang, W. S., Park, H. C. Finite Element Modeling of Piezoelectric Sensors and Actuators. AIAA Journal, 31(5), pp 930-937, 1993.

Accepts the following property/value pairs.

Input       Value/{Default}        Description
-----------------------------------------------------------------------------------
l           scalar {0.12}          Length of beam
h           scalar {2e-3}          Height (thickness) of beam
delV        scalar {200}           Potential difference
igrid       scalar 1/{0}           Cell type (0=quadrilaterals, 1=triangles)
hmax        scalar {20}            Cell resolution
sfund       string {sflag1}        Shape function for displacements
sfunp       string {sflag1}        Shape function for potential
iplot       scalar 0/{1}           Plot solution (=1)
                                                                                  .
Output      Value/(Size)           Description
-----------------------------------------------------------------------------------
fea         struct                 Problem definition struct
out         struct                 Output struct

Code listing

 cOptDef = { ...
   'l',        0.12; ...
   'h',        2e-3; ...
   'delV',     100; ...
   'igrid',    0; ...
   'hmax',     20; ...
   'sfund',    'sflag2'; ...
   'sfunp',    'sflag1'; ...
   'iplot',    1; ...
   'fid',      1 };
 [got,opt] = parseopt( cOptDef, varargin{:} );
 fid       = opt.fid;
 i_impl    = 1;

% Geometry definition.
 fea.sdim = { 'x' 'y' };   % Names of space coordinates.
 gobj1    = gobj_rectangle( 0, opt.l,  0,       opt.h/2, 'R1' );
 gobj2    = gobj_rectangle( 0, opt.l, -opt.h/2, 0,       'R2' );
 fea.geom.objects = { gobj1 gobj2 };


% Grid generation.
 switch opt.igrid
   case -1
     fea.grid = rectgrid( opt.hmax, opt.hmax, [ 0 opt.l; -opt.h/2 opt.h/2] );
     fea.grid = quad2tri( fea.grid );
   case 0
     fea.grid = rectgrid( opt.hmax, opt.hmax, [ 0 opt.l; -opt.h/2 opt.h/2] );
     fea.grid.s( selcells( fea, 'y<=0' ) ) = 2;
   case 1
     fea.grid = gridgen( fea, 'hmax', opt.h/(opt.hmax/4.6), 'fid', fid );
     fea.grid.s(:) = 1;
 end
 fea.grid.s( selcells( fea, 'y<=0' ) ) = 2;


% Equation coefficients.
 Emod =  2e9;       % Modulus of elasticity
 nu   =  0.29;      % Poissons ratio
 Gmod =  0.775e9;   % Shear modulus
 d31  =  0.22e-10;  % Piezoelectric sn coefficient
 d33  = -0.3e-10;   % Piezoelectric sn coefficient
 prel = 12;         % Relative electrical permittivity
 pvac = 0.885418781762-11;   % Electrical permittivity of vacuum

% Constitutive relations.
 constrel   = [ Emod/(1-nu^2)    nu*Emod/(1-nu^2) 0    ;
                nu*Emod/(1-nu^2) Emod/(1-nu^2)    0    ;
                0                0                Gmod ];
 piezoel_st = [ 0 d31 ;
                0 d33 ;
                0 0   ];
 piezoel_sn = constrel*piezoel_st;
 dielmat_st = [ prel 0    ;
                0    prel ]*pvac ;
 dielmat_sn = [ dielmat_st - piezoel_st'*piezoel_sn ];

% Populate coefficient matrices (negative sign due to fem partial integration).
 c{1,1} = { constrel(1,1)   constrel(1,3) ;
            constrel(1,3)   constrel(3,3) };
 c{1,2} = { constrel(1,3)   constrel(1,2) ;
            constrel(3,3)   constrel(2,3) };
 c{2,1} = c{1,2}';
 c{2,2} = { constrel(3,3)   constrel(1,3) ;
            constrel(1,3)   constrel(2,2) };
 c{1,3} = { piezoel_sn(1,1) piezoel_sn(1,2) ;
            piezoel_sn(3,1) piezoel_sn(3,2) };
 if( i_impl )
   for i=1:4
     c{1,3}{i} = [num2str(c{1,3}{i}),'*(2*(y<0)-1)'];
   end
 else
   for i=1:4
     fea.expr{i,1} = ['piezoel_sn1',num2str(i)];
     fea.expr{i,2} = { -c{1,3}{i} c{1,3}{i} };
     c{1,3}{i} = ['piezoel_sn1',num2str(i)];
   end
 end
 c{3,1} = c{1,3}';
 c{2,3} = { piezoel_sn(3,1) piezoel_sn(3,2);
            piezoel_sn(2,1) piezoel_sn(2,2) };
 if( i_impl )
   for i=1:4
     c{2,3}{i} = [num2str(c{2,3}{i}),'*(2*(y<0)-1)'];
   end
 else
   for i=1:4
     fea.expr{i+4,1} = ['piezoel_sn2',num2str(i)];
     fea.expr{i+4,2} = { -c{2,3}{i} c{2,3}{i} };
     c{2,3}{i} = ['piezoel_sn2',num2str(i)];
   end
 end
 c{3,2} = c{2,3}';
 c{3,3} = { dielmat_sn(1,1) dielmat_sn(2,1) ;
            dielmat_sn(2,1) dielmat_sn(2,2) };


% Dependent variable names.
 fea.dvar  = { 'u' 'v' 'V' };
 n_dvar    = length(fea.dvar);

% Finite element shape functions.
 fea.sfun  = { opt.sfund opt.sfund opt.sfunp };

% Define equations.
 bilinear_form = [ 2 2 3 3 ;
                   2 3 2 3 ];
 for i=1:n_dvar
   for j=1:n_dvar
     fea.eqn.a.form{i,j} = bilinear_form;
     fea.eqn.a.coef{i,j} = c{i,j}(:)';
   end
 end

% Source terms (set to zero).
 fea.eqn.f.form = { 1 1 1 };
 fea.eqn.f.coef = { 0 0 0 };


% Boundary conditions.
 n_bdr          = max(fea.grid.b(3,:));   % Number of boundaries.
 fea.bdr.d      = cell(n_dvar,n_bdr);
 fea.bdr.n      = cell(n_dvar,n_bdr);
 i_top          = findbdr( fea, ['y>=',num2str(opt.h/2-sqrt(eps))] );
 i_bottom       = findbdr( fea, ['y<=',num2str(-opt.h/2+sqrt(eps))] );
 i_left         = findbdr( fea, ['x<=',num2str(sqrt(eps))] );
 [fea.bdr.d{3,i_top}]    = deal( opt.delV );   % Set potential to dV on top boundary.
 [fea.bdr.d{3,i_bottom}] = deal( 0 );          % Set potential to 0V on bottom boundary.
 [fea.bdr.d{1:2,i_left}] = deal( 0 );          % Set displacements to 0 on left boundary.


% Check and parse problem struct.
 fea = parseprob( fea );

% Call to stationary solver.
 fea.sol.u = solvestat( fea, 'fid', fid );


% Postprocessing.
 if( opt.iplot )
   YSCALE = 3;
   axlim  = [0, opt.l, -YSCALE*opt.h/2, YSCALE*opt.h/2];
   DSCALE = 20;

   subplot(2,2,1)
   postplot( fea, 'surfexpr', 'u', 'isoexpr', 'u', 'setaxes', 'off' )
   axis(axlim);
   title('x-displacement')

   subplot(2,2,2)
   postplot( fea, 'surfexpr', 'v', 'isoexpr', 'v', 'setaxes', 'off' )
   axis(axlim);
   title('y-displacement')

   subplot(2,2,3)
   postplot( fea, 'surfexpr', 'V', 'isoexpr', 'V', 'setaxes', 'off' )
   axis(axlim);
   title('Electric potential')

   subplot(2,2,4)
   plotsubd( fea, 'labels', 'off', 'setaxes', 'off' )
   axis(axlim);
   title(['Displacement plot (at ',num2str(DSCALE),' times scale)'])

   ind1 = sub2ind( size(fea.grid.c), fea.grid.b(2,:), fea.grid.b(1,:) );
   p1b  = fea.grid.p( :, fea.grid.c(ind1) );
   ind2 = sub2ind( size(fea.grid.c), mod(fea.grid.b(2,:),size(fea.grid.c,1))+1, fea.grid.b(1,:) );
   p2b  = fea.grid.p( :, fea.grid.c(ind2) );
   up1  = DSCALE*evalexpr( 'u', p1b, fea );
   vp1  = DSCALE*evalexpr( 'v', p1b, fea );
   up2  = DSCALE*evalexpr( 'u', p2b, fea );
   vp2  = DSCALE*evalexpr( 'v', p2b, fea );
   hold on
   for i=1:size(p1b,2)
     plot( [p1b(1,i)+up1(i) p2b(1,i)+up2(i)], [p1b(2,i)+vp1(i) p2b(2,i)+vp2(i)], '-b', 'linewidth', 2 )
   end
   set( gca, 'ytick', [] )
 end


% Error checking.
 ind_dof_v  = fea.eqn.dofm{2}(:) + fea.eqn.ndof(1);
 out.v_max  = max(abs( fea.sol.u(ind_dof_v) ));
 v_max_ref  = abs(-3/2*d31*opt.delV*(opt.l/opt.h)^2);
 out.err    = abs(v_max_ref - out.v_max)/v_max_ref;
 out.pass   = out.err <= 0.1;

 if ( nargout==0 )
   clear fea out
 end