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ACKNOWLEDGEMENTS First, I would like to express my deepest gratitude to my main supervisor, Prof Ngo Thanh Phong, for his dedicated support, guidance and continuous encouragement during my Ph.D study To me, Prof Ngo Thanh Phong is a very kind mentor with profound knowledge, high patience, generosity and passion of studying science The bright image of Prof Ngo Thanh Phong inspires me not only in my Ph.D study but also in many aspects of my life I would also like to extend a great thank to my co-supervisor, Dr Le Đinh Tuan who provides me the updated and interesting knowledge of the intelligent piezo-electrical material during last many years This inspires me to study profoundly the theory, numerical simulation and experiments related to this new material I would also like to give many thanks to Dr Trinh Anh Ngoc, and Associate Prof Dr Thai Thi Thu Ha for the support and provision of favourable conditions of studying the theory and conducting the experiments of my PhD thesis Highly appreciation is extended to my close friends: Dr Nguyen Thoi Trung, Dr Nguyen Xuan Hung, and Dr Bui Quoc Tinh for the interactive discussion, professional opinions, full cooperation and future objectives Lastly, I greatly appreciate to my extended family, especially to my wife and my lovely children for their eternal love and strong support during my PhD study Without their endless encouragement, understanding and full support, it is impossible for me to finish this thesis Hochiminh city, May 2010 Nguyễn Trần Chân CONTENTS Introduction .8 Chapter : Preliminary of piezoelectricity .18 1.1 Piezoelectric effect 18 1.2 Unidimensional piezoelectricity 18 1.3 Ferroelectric ceramics .20 1.4 PZT piezoelectric ceramic .21 1.5 PVDF piezoelectric polymers 22 Chapter : Mathematical modeling and numerical method .24 2.1 Mathematical Modeling of Piezoelectricity 24 2.1.1 Static electromagnetism 24 2.1.2 Elements of mechanicals 27 2.1.3 The piezoelectric constitutive equations 28 2.1.4 Differential equations (strong form) and weak form .28 2.2 Finite element method .31 2.2.1 Discretization by finite elements 31 2.2.2 An error estimation of FEM .32 2.2.3 Matrix form of the weak form by FEM 33 2.3 Smoothed finite element method for 2D piezoelectricity .35 2.3.1 A smoothing operator on mechanical strains and electric field 36 2.3.2 Smoothed stiffness matrices for piezoelectricity problems 38 Chapter : Numerical results 42 3.1 The traveling wave ultrasonic motor .43 3.2 Bimorph beam of PVDF 57 3.3 Cylinder of PZT 60 3.4 PZT amplifier 67 3.5 Patch test for plane elements 71 3.6 Singer-player piezoelectric strip 74 3.7 Cook’s membrane 78 3.8 MEMS device 80 Chapter : Experiments of piezoelectricity 84 4.1 Measuring equipments 84 4.2 PVDF bimorph beam Experiment 88 4.2.1 Prototyping of PVDF bimorph 88 4.2.2 Measure of PVDF bimorph 89 4.2.3 Remarks on experimental results of PVDF bimorph 93 4.3 PZT amplifier Experiment 94 4.3.1 Prototyping of amplifier 94 4.3.2 Measure of PZT amplifier 96 4.3.3 Remarks on Experimental Results 98 Chapter : Conclusions 100 Publications and accomplishments of thesis 103 References .104 Appendix .116 List of symbols and units for electromechanical quantities Notation iB B , Quantity SI unit Note tensor, matrix magnetic flux Vs/m2 volt-second per density c E square meter elastic coefficients when the electric field is kept constant iD D , tensor, matrix of electric C/m2 displacement field iE E , tensor, matrix of electric square meter N/C field e coupled constant-coefficient coulombs per newtons per coulomb C/m2 coulombs per square metre fVV i ,f vector of volume force N/m3 Newton per cubic metre fSS i ,f vector of surface force N/m2 newtons per square metre fP vector of point force N newtons iH H , tensor, matrix magnetic A/m field iJ J , density of electric current ampere per meter A/m2 ampere per meter P P i, tensor, matrix of electric C/m2 polarisation qS electric surface charge coulombs per square metre C/m2 coulombs per square metre qP ijS S , point charge C coulomb tensor, matrix of strain t time s second , stress N/m2 newtons per T ij T square metre iu u , U vector of displacement m meter density of internal energy x i the cartesian coordinates electric scalar potential V volt permittivity of the dielectric F/m farads per meter permittivity of vacuum F/m farads per meter permittivity when strain is F/m farads per meter C/m3 coulombs per S kept constant e free charge density cubic metre density of surface charge C/m2 coulombs per square meter v ijk Mass of volume m3 cubic metre the permulation tensor List of abbreviations Notation Signification FE Finite element FEM Finite element method SFEM Smoothing finite element method ES-FEM edge–based smoothed finite element method PZT Lead Zirconate titanate (PbZryTi1-yO3) PVDF Polyvinylidene Fluoride (-CH2-CF2-)n MEMs MicroElectroMechanical systems T3 Triangle element in FEM Q4 Quadrilateral element in FEM ES-T3 Triangle element in ES-FEM PCM Meshless point collocation method DCSELAB National key lab of digital control & system engineering Introduction Introduction General description of the problem The phenomena of transforming directly the different forms of energy of intelligent materials leads to many impressive and interesting applications This has opened many new directions of research which have a great potential in the future This also helps to create high increased values for the products Therefore, researches and applications of intelligent materials are currently the hot topics These materials exist in many forms such as: piezoelectric materials, shape memory alloys , photoelastic materials, electrostrictive materials and relate to many type of energy convertions In particular, the domain of electromechanical energy conversion lives a strong growing in recently relating mainly to the progress of piezoelectric materials The piezoelectric materials form a complex system, which utilizes two physical phenomena, that is an interaction between an electric field and a field of mechanical displacement It results in two effects The first one, called direct piezoelectric effect, corresponds with the appearance of an electric tension when a mechanical force is applied: the material then plays the role of a piezoelectric sensor Conversely, the reverse piezoelectric effect corresponds to the dilation or the contraction of piezoelectric material when one applies an electric tension to it The material then behaves like a piezoelectric actuator The comprehensive study of these phenomena, generally coupled between them, is a difficult task but necessary because of its vast technological applications Subjected to electric fields, these materials are capable to produce high forces in a reduced volumes Inversely, subjected to external forces conducting to Introduction displacement and stress fields, the piezoelectric structure produces an elastic field Due to this special property, the piezoelectric material has an important role and has been applied widely in many fields of advanced technology such as: ultrasound technology, traveling wave ultrasonic motors, nano-positioning systems, microcontroller systems [25], actuators [9][12], sensors, biochip [13] From a mathematic point of view, the piezoelectric domain is governed by the Maxwell’s equations [73] in the material and the general equation of the elasticity The coupling between two equilibrium of forces on the surface of interaction At low frequencies (up to hundreds of KHz) analytical solutions of this type of problems exits only for certain simple configurations For problems where geometry is complex, we must use methods of discretisation of fields, in particular the finite element method (FEM) The researchs in this area Nowadays, there are seldom research projects about this material in Vietnam While the inevitable exploitation of the piezoelectric materials boosts the study on piezolectric phenomenon in worldwide The most popular are piezoelectric ceramics (PZT) and piezopolymers (PVDF) It is used for a large number of applications Almost the research reports and applications of the piezoelectric material have been derived from the long-term and large-scale research projects with the cooperation of many leading experts and Ph D students, and these research reports have been considered as the technological secrets The research of piezolectric field relating to a wide range of many sciences In scope of thesis, we only mention the development in mathematical modeling, numerical methods and experiment of some well known groups such as: -Tzou, H.S Dept of Mech Eng., Kentucky Univ Robotics and Automation Proceedings, 1989 IEEE International Conference In which the development of a light-weight robot end-effector using polymeric piezoelectric bimorph The 10 C.33.44.55.54.78.65.5.43.22.2.4 22.Tai lieu Luan 66.55.77.99 van Luan an.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.33.44.55.54.78.655.43.22.2.4.55.22 Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an Appendix-Patch test for plane elements 74.3*10^3 113*10^3 -13.84*10^6 -6.98*10^6 13.84*10^6 5.47*10^9]; patch_tes=inv(prop_mater); s11=patch_tes(1,1); s13=patch_tes(1,2); g31=patch_tes(1,3); fb1=[1 11]; ff1=[s11*sigma_zeros*gcoord(1,1) s13*sigma_zeros*gcoord(1,2) g31*sigma_zeros*gcoord(1,2)]; fb2=[3 12]; ff2=[s11*sigma_zeros*gcoord(2,1) s13*sigma_zeros*gcoord(2,2) g31*sigma_zeros*gcoord(2,2)]; fb3=[5 13]; ff3=[s11*sigma_zeros*gcoord(3,1) s13*sigma_zeros*gcoord(3,2) g31*sigma_zeros*gcoord(3,2)]; fb4=[7 14]; ff4=[s11*sigma_zeros*gcoord(4,1) s13*sigma_zeros*gcoord(4,2) g31*sigma_zeros*gcoord(4,2)]; bc1=[fb1 fb2 fb3 fb4]; bc2=[ff1 ff2 ff3 ff4]; ff=zeros(sdof+sdof/2,1); [kk,ff]=feaplyc2(K,ff,bc1,bc2); u=kk\ff; sigma_element=zeros(3,4); D_electric=zeros(2,4); for iel=1:nel % loop for the total number of elements index =[ele_nods(iel,1)*2-1 ele_nods(iel,1)*2 ele_nods(iel,2)*2-1 ele_nods(iel,2)*2 ele_nods(iel,3)*2-1 ele_nods(iel,3)*2]; % extract system dofs for (ied)-th SD index1=[index [sdof+ele_nods(iel,:)]]; q=[u(index1(1)) u(index1(2)) u(index1(3)) u(index1(4)) u(index1(5)) u(index1(6))]; E=[u(index1(7)) u(index1(8)) u(index1(9))]; x1=gcoord(iel,1); y1=gcoord(iel,2); x2=gcoord(iel,1); y2=gcoord(iel,2); x3=gcoord(iel,1); y3=gcoord(iel,2); xcoord=[x1 x2 x3]; 178 Stt.010.Mssv.BKD002ac.email.ninhd 77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77t@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn C.33.44.55.54.78.65.5.43.22.2.4 22.Tai lieu Luan 66.55.77.99 van Luan an.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.33.44.55.54.78.655.43.22.2.4.55.22 Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an Appendix-Patch test for plane elements ycoord=[y1 y2 y3]; sigma=zeros(3,1); D=zeros(2,1); for i=1:nnel % loop for nodes of (iel)-th element nod(i)=ele_nods(iel,i); % extract nodes for (iel)-th element x(i)=gcoord(nod(i),1); % extract x value of the node y(i)=gcoord(nod(i),2); % extract y value of the node end % % find the derivatives of shape functions % dNdx=(1/(2*area_T3(iel)))*[(y(2)-y(3)) (y(3)-y(1)) (y(1)-y(2))]; %derivatives w.r.t x dNdy=(1/(2*area_T3(iel)))*[(x(3)-x(2)) (x(1)-x(3)) (x(2)-x(1))]; %derivatives w.r.t y Be=get_Bmat_2D(nnel,dNdx,dNdy); % strain-displacement matrix of elements % save into matrix containing strain-displacement m Bfi=[dNdx(1) dNdx(2) dNdx(3); dNdy(1) dNdy(2) dNdy(3)]; sigma = sigma + D_piezo*Be*q + e'*Bfi*E; D = D + e*Be*q - epxilon*Bfi*E; sigma_element(:,iel) = sigma; D_electric(:,iel) = D; end ketqua_EST3=[u(9);u(10);u(15);sigma_element(:,4);D_electric(:,4)] 179 Stt.010.Mssv.BKD002ac.email.ninhd 77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77t@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn C.33.44.55.54.78.65.5.43.22.2.4 22.Tai lieu Luan 66.55.77.99 van Luan an.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.33.44.55.54.78.655.43.22.2.4.55.22 Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an Appendix- Singer-player piezoelectric strip Code_singerplayer_trip clear all;clc format long global nel sdof nnel global gcoord ele_nods D_piezo ndof e epxilon % -element - node -L=1; % length x (mm) H=1; % length y (mm) V=1*10^-6; % electric V Sigmazero=-5; % stress (N/m2) % -% input data for control parameters % -lx= 8; %lx= 4; ly= 8; %lx= 4; % number element on x axis % number element on y axis % -dx=L/lx; % Length side of an element on x axis dy=H/ly; % Length side of an element on y axis gcoord=[]; % Nodes coordinate for j=-ly/2:1:ly/2 for i=1:lx+1 gcoord=[gcoord; (i-1)*dx j*dy;]; end end % air=0.0; for i=1:size(gcoord,1) if (gcoord(i,1)~=0)&(gcoord(i,1)~=1) if (gcoord(i,2)~=-0.5)&(gcoord(i,2)~=0.5)&(gcoord(i,2)~=0) r=random('beta',1,1); r=air*(2*r-1); gcoord(i,1) = gcoord(i,1)+ dx*r; gcoord(i,2) = gcoord(i,2)+ dy*r; end end if (gcoord(i,2)==0)&(gcoord(i,1)>0)&(gcoord(i,1)node no and j->x or y % -for i=1:nx+1 for j=1:ny+1 h=L2-(j-1)*(L2-L3)/ny; gcoord((nx+1)*(j-1)+i,1)=(i-1)*L1/nx; gcoord((nx+1)*(j-1)+i,2)=(i-1)*L1*(h/L1)/nx+(j-1)*L2/ny; end end % % input data for nodal connectivity for each element % nodes=[];lx=nx;ly=ny; for i=1:nx 185 Stt.010.Mssv.BKD002ac.email.ninhd 77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77t@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn C.33.44.55.54.78.65.5.43.22.2.4 22.Tai lieu Luan 66.55.77.99 van Luan an.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.33.44.55.54.78.655.43.22.2.4.55.22 Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an Appendix-Cook’s membrane for j=1:ny nodes=[nodes;(lx+1)*(j-1)+i (lx+1)*(j-1)+i+1 (lx+1)*j+i+1;(lx+1)*(j-1)+i (lx+1)*j+i+1 (lx+1)*j+i]; end end % -number_nodes=size(gcoord,1); ele_nods=nodes; m=size(nodes,1); eldof1=[1:1:m]'; eldof=[eldof1 nodes(:,1)*2-1 nodes(:,1)*2 nodes(:,2)*2-1 nodes(:,2)*2 nodes(:,3)*2-1 nodes(:,3)*2]; % -calculating K matrix nnel=3;% nnel = number of nodes per element ndof=2;% ndof = number of degrees of freedom (dofs) per node D_piezo=[139 74.3 ; %PZT4 74.3 113 0; 0 25.6]*10^3; matmtx=D_piezo; e=[0 13.44; %PZT4 -6.98 13.84 0]*10^6; %coulomb/m2 epxilon=[6 0; %PZT4 5.47]*10^9; %F/m nel=length(ele_nods);%% nel = total number of elements in system sdof=length(gcoord)*2;% sdof = total dofs in system [edge_data,ele_edgs]=get_edge_data;%ma tran canh gom nhung ptu nao [area_edg,area_T3] = cal_area_edge_T3(edge_data); [K]=cal_K_ESFEM_T3_aver_a(edge_data,area_edg,area_T3); % [r,c]=size(K); fc=zeros(c,1);%tao vec to tai co kich thuoc tuong ung voi K ff=zeros(1,c);%tao vec to zero de gan dieu kien bien vao bm=[]; f=[]; % duoi load=[]; for inn=1:number_nodes %[number-nodes, ]=size(nodes) if (gcoord(inn,1)==0) bm=[bm inn]; end 186 Stt.010.Mssv.BKD002ac.email.ninhd 77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77t@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn C.33.44.55.54.78.65.5.43.22.2.4 22.Tai lieu Luan 66.55.77.99 van Luan an.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.33.44.55.54.78.655.43.22.2.4.55.22 Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an Appendix-Cook’s membrane if (gcoord(inn,1)==48) load=[load inn]; end end f=[1:1:(nx+1)]; % - boundary conditions -siz=size(load,2); for i_dof=load fc(i_dof*2,1)=f_distribute/(siz-1); end fc(load(1)*2)=f_distribute/(siz-1)/2; %f=1/2 for first and last node fc(load(siz)*2)=f_distribute/(siz-1)/2; %f=1/2 for first and last node AAA=[]; % chuyen vi = for i_dof=bm AAA=[AAA; (2*i_dof-1);2*i_dof]; end fbm=AAA'; ffbm(1,fbm)=0; AAA=[]; % dien the =0V (duoi) for i_dof=f AAA=[AAA; number_nodes*2+i_dof]; end fbe0=AAA'; ff1(1,fbe0)=0; bc1=[fbm fbe0 ]; bc2=[ffbm(1,fbm) ff1(1,fbe0)]; bc=[bc1' bc2']; % xuat ket qua u_32=solveq(K,fc,bc); ucentral=u_32((nx+1)*(ny/2+1)*2); phicentral=u_32(number_nodes*2+(nx+1)*(ny/2+1)); chuyenvi=[chuyenvi;ucentral];dien=[dien;phicentral]; end u_exac=[ 2.109e-4 2.109e-4 2.109e-4 2.109e-4]; %nghiem tham khao dien_exac=[ 1.732e-8 1.732e-8 1.732e-8 1.732e-8];%nghiem tham khao chuyenvi=[0.166169351354934 0.201785991518924 0.208969664907645 0.210340079588422]*1e-3; %nghiem so da tinh o tren 187 Stt.010.Mssv.BKD002ac.email.ninhd 77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77t@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn C.33.44.55.54.78.65.5.43.22.2.4 22.Tai lieu Luan 66.55.77.99 van Luan an.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.33.44.55.54.78.655.43.22.2.4.55.22 Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an Appendix-Cook’s membrane dien=[0.083482882993942 0.145550415460886 0.166080114080411 0.170367953019360]*1e-7;%nghiem so da tinh o tren ss_cv=(chuyenvi)./u_exac(1); ss_dien=(dien)./dien_exac(1); ref=[1 1 1]; subplot(2,1,1) hold on plot(luoi(1:4),ss_cv,'-*') plot(luoi(1:4),ref,'k') legend('ES','Ref sol') xlabel('Number of element') ylabel('Normalized vertical displacement v') subplot(2,1,2) hold on plot(luoi(1:4),ss_dien,'-h') plot(luoi(1:4),ref,'k') legend('ES','Ref sol') xlabel('Number of element') ylabel('Normalized electric potiental') 188 Stt.010.Mssv.BKD002ac.email.ninhd 77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77t@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn C.33.44.55.54.78.65.5.43.22.2.4 22.Tai lieu Luan 66.55.77.99 van Luan an.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.33.44.55.54.78.655.43.22.2.4.55.22 Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an Appendix-MEMs Code_bimorphmems clear all;clc format long global nel sdof nnel global gcoord ele_nods matmtx ndof e epxilon L=10; %length of x Micrometrer H=1; %length of y % - nhap dien the mesh_vol=6; if mesh_vol ==1 V=1; elseif mesh_vol==2 V=2; elseif mesh_vol==3 V=5; elseif mesh_vol==5 V=10; elseif mesh_vol==6 V=15; elseif mesh_vol==7 V=20; elseif mesh_vol==8 V=25; else V=50; end mesh_grid=1; % neu mesh_grid = la luoi tinh % neu mesh_grid = la luoi tho if mesh_grid ==1 lx=60;ly=6;%lx=80;ly=10; else lx=40;ly=2; end % -dx=L/lx; % the length of side of element in x-axis dy=H/ly; % the length of side of element in y-axis gcoord=[]; % toa cua cac nodes 189 Stt.010.Mssv.BKD002ac.email.ninhd 77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77t@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn C.33.44.55.54.78.65.5.43.22.2.4 22.Tai lieu Luan 66.55.77.99 van Luan an.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.33.44.55.54.78.655.43.22.2.4.55.22 Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an Appendix-MEMs for j=0:1:ly for i=1:lx+1 gcoord=[gcoord; (i-1)*dx j*dy;]; end end % number of dofs per node; % % input data for nodal connectivity for each element % nodes=[]; for i=1:lx for j=1:ly nodes=[nodes;(lx+1)*(j-1)+i (lx+1)*(j-1)+i+1 (lx+1)*j+i+1;(lx+1)*(j-1)+i (lx+1)*j+i+1 (lx+1)*j+i]; end end % number_nodes=size(gcoord,1); ele_nods=nodes; m=size(nodes,1); eldof1=[1:1:m]'; eldof=[eldof1 nodes(:,1)*2-1 nodes(:,1)*2 nodes(:,2)*2-1 nodes(:,2)*2 nodes(:,3)*2-1 nodes(:,3)*2]; % -calculating K matrix nnel=3;% nnel = number of nodes per element ndof=2;% ndof = number of degrees of freedom (dofs) per node % PVDF D_piezo=[2.18 0.633 0; 0.633 2.18 0; 0 0.775]*10^-3; % hooke matrix N/Mm2 e=[0 0; 4.6 4.6 0]*10^-8; %N/VMm epxilon=[1.062 0; 1.062]*10^-10; %N/V2 matmtx=D_piezo; nel=length(ele_nods);%% nel = total number of elements in system sdof=length(gcoord)*2;% sdof = total dofs in system [edge_data,ele_edgs]=get_edge_data;%ma tran canh gom nhung ptu nao [area_edg,area_T3] = cal_area_edge_T3(edge_data); [K]=cal_K_ESFEM_T3_aver_a(edge_data,area_edg,area_T3); 190 Stt.010.Mssv.BKD002ac.email.ninhd 77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77t@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn C.33.44.55.54.78.65.5.43.22.2.4 22.Tai lieu Luan 66.55.77.99 van Luan an.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.33.44.55.54.78.655.43.22.2.4.55.22 Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an Appendix-MEMs % [r,c]=size(K); fc=zeros(c,1); %tao vec to tai co kich thuoc tuong ung voi K ff=zeros(1,c); %tao vec to zero de gan dieu kien bien vao bm=[]; f=[]; %tren w=[]; %duoi mid=[]; order_con=[]; for inn=1:number_nodes %number_nodes=size(gcoord,1) if (gcoord(inn,1)==0) bm=[bm inn]; order_con=[order_con inn]; end if (gcoord(inn,1)==10) order_con=[order_con inn]; end if (gcoord(inn,2)==1)&(gcoord(inn,1)>0) f=[f inn]; end if (gcoord(inn,2)==0)&(gcoord(inn,1)>0) w=[w inn]; end if (gcoord(inn,2)==0.5)&(gcoord(inn,1)>0) mid=[mid inn]; end end % - boundary conditions -tam=[]; for i_dof=bm tam=[tam; 2*i_dof-1; 2*i_dof]; end fbm=tam'; ffbm(1,fbm)=0; % chuyen vi = tam=[]; for i_dof=f tam=[tam; number_nodes*2+i_dof]; end fbef=tam'; ff1(1,fbef)=V; % dien the =1V (tren) unit V 191 Stt.010.Mssv.BKD002ac.email.ninhd 77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77t@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn C.33.44.55.54.78.65.5.43.22.2.4 22.Tai lieu Luan 66.55.77.99 van Luan an.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.C.33.44.55.54.78.655.43.22.2.4.55.22 Do an.Tai lieu Luan van Luan an Do an.Tai lieu Luan van Luan an Do an Stt.010.Mssv.BKD002ac.email.ninhd 77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77.77.99.44.45.67.22.55.77.C.37.99.44.45.67.22.55.77t@edu.gmail.com.vn.bkc19134.hmu.edu.vn.Stt.010.Mssv.BKD002ac.email.ninhddtt@edu.gmail.com.vn.bkc19134.hmu.edu.vn