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Matlab 平面波展开法计算二维声子晶体二维声子晶体带结构计算，材料是铅柱在橡胶基体中周期排列，格子为正方形。采用PWE方法计算

完整程序:

%%%%%%%%%%%%%%%%%%%%%%%%%
clear;clc;tic;epssys=1.0e-6; %设定一个最小量，避免系统截断误差或除零错误
 
%%%%%%%%%%%%%%%%%%%%%%%%%%

%定义实际的正空间格子基矢
%%%%%%%%%%%%%%%%%%%%%%%%%%
a=0.02;
a1=a*[1 0];
a2=a*[0 1];
%%%%%%%%%%%%%%%%%%%%%%%%%%

%定义晶格的参数
%%%%%%%%%%%%%%%%%%%%%%%%%%
rho1=11600;E1=4.08e10;mju1=1.49e10;lambda1=mju1*(E1-2*mju1)/(3*mju1-E1); %散射体的材料参数
rho2=1300;E2=1.175e5;mju2=4e4;lambda2=mju2*(E2-2*mju2)/(3*mju2-E2); %基体的材料参数
Rc=0.006; %散射体截面半径
Ac=pi*(Rc)^2; %散射体截面面积
Au=a^2; %二维格子原胞面积
Pf=Ac/Au; %填充率
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%生成倒格基矢
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
b1=2*pi/a*[1 0];
b2=2*pi/a*[0 1];
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%选定参与运算的倒空间格矢量，即参与运算的平面波数量
%设定一个l,m的取值范围，变化l,m即可得出参与运算的平面波集合
NrSquare=10; %选定倒空间的尺度，即l,m（倒格矢G=l*b1+m*b2）的取值范围。
             %NrSquare确定后，使用Bloch波数目可能为(2*NrSquare+1)^2
G=zeros((2*NrSquare+1)^2,2); %初始化可能使用的倒格矢矩阵
i=1;
for l=-NrSquare:NrSquare
    for m=-NrSquare:NrSquare
        G(i,:)=l*b1+m*b2;
        i=i+1;
    end;
end;
NG=i-1; %实际使用的Bloch波数目
G=G(1:NG,:); 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%生成k空间的rho(Gi-Gj),mju(Gi-Gj),lambda(Gi-Gj)值，i,j从1到NG。
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
rho=zeros(NG,NG);mju=zeros(NG,NG);lambda=zeros(NG,NG);
for i=1:NG
    for j=1:NG
        Gij=norm(G(j,:)-G(i,:));
        if (Gij<epssys)
            rho(i,j)=rho1*Pf+rho2*(1-Pf);
            mju(i,j)=mju1*Pf+mju2*(1-Pf);
            lambda(i,j)=lambda1*Pf+lambda2*(1-Pf);
        else
            rho(i,j)=(rho1-rho2)*2*Pf*besselj(1,Gij*Rc)/(Gij*Rc);
            mju(i,j)=(mju1-mju2)*2*Pf*besselj(1,Gij*Rc)/(Gij*Rc);
            lambda(i,j)=(lambda1-lambda2)*2*Pf*besselj(1,Gij*Rc)/(Gij*Rc);
        end;
    end;
end;
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%定义简约布里渊区的各高对称点
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
T=(2*pi/a)*[epssys 0];
M=(2*pi/a)*[1/2 1/2];
X=(2*pi/a)*[1/2 0];
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%对于简约布里渊区边界上的每个k，求解其特征频率
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
THETA_A=zeros(NG,NG); %待解的本征方程A矩阵
THETA_B=zeros(NG,NG); %待解的本征方程B矩阵
Nkpoints=10; %每个方向上取的点数
stepsize=0:1/(Nkpoints-1):1; %每个方向上步长
TX_eig=zeros(Nkpoints,NG); %沿TX方向的波的待解的特征频率矩阵
XM_eig=zeros(Nkpoints,NG); %沿XM方向的波的待解的特征频率矩阵
MT_eig=zeros(Nkpoints,NG); %沿MT方向的波的待解的特征频率矩阵
for n=1:Nkpoints
    fprintf(['\n k-point:',int2str(n),'of',int2str(Nkpoints),'.\n']);
     
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    %对于TX（正方格子）方向上的每个k值，求解其特征频率
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    TX_step=stepsize(n)*(X-T)+T;
     
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    %n 求本征矩阵的元素
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    for i=1:NG
        for j=1:NG
            kGi=TX_step+G(i,:);
            kGj=TX_step+G(j,:);
            THETA_A(i,j)=mju(i,j)*dot(kGi,kGj);
            THETA_B(i,j)=rho(i,j); 
        end;
    end;
     
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    %求解TX（正方格子）方向上的k矩阵的特征频率
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    TX_eig(n,:)=sort(sqrt(eig(THETA_A,THETA_B))).';
    
    
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    %对于XM（正方格子）方向上的每个k值，求解其特征频率
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    XM_step=stepsize(n)*(M-X)+X;
     
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    %n 求本征矩阵的元素
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    for i=1:NG
        for j=1:NG
            kGi=XM_step+G(i,:);
            kGj=XM_step+G(j,:);
            THETA_A(i,j)=mju(i,j)*dot(kGi,kGj);
            THETA_B(i,j)=rho(i,j); 
        end;
    end;
     
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    %求解XM（正方格子）方向上的k矩阵的特征频率
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    XM_eig(n,:)=sort(sqrt(eig(THETA_A,THETA_B))).';
    
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    %对于MT（正方格子）方向上的每个k值，求解其特征频率
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    MT_step=stepsize(n)*(T-M)+M;
     
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    %n 求本征矩阵的元素
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    for i=1:NG
        for j=1:NG
            kGi=MT_step+G(i,:);
            kGj=MT_step+G(j,:);
            THETA_A(i,j)=mju(i,j)*dot(kGi,kGj);      
            THETA_B(i,j)=rho(i,j); 
        end;
    end;
     
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    %求解MT（正方格子）方向上的k矩阵的特征频率
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    MT_eig(n,:)=sort(sqrt(eig(THETA_A,THETA_B))).';  
end;
fprintf('\n Calculation Time:%d sec',toc);
save pbs2D
     
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%绘制声子晶体能带结构图
%首先将特定方向（正方格子：TX,XM,MT）离散化
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
kaxis=0;
TXaxis=kaxis:norm(T-X)/(Nkpoints-1):(kaxis+norm(T-X));
kaxis=kaxis+norm(T-X);
XMaxis=kaxis:norm(M-X)/(Nkpoints-1):(kaxis+norm(X-M));
kaxis=kaxis+norm(X-M);
MTaxis=kaxis:norm(T-M)/(Nkpoints-1):(kaxis+norm(T-M));
kaxis=kaxis+norm(T-M);
 
Ntraject=3; %所需绘制的特定方向的数目
EigFreq=zeros(Ntraject*Nkpoints,1);
figure(1)
hold on;
Nk=Nkpoints;
 
 
for k=1:NG 
    for i=1:Nkpoints 
        EigFreq(i+0*Nk)=TX_eig(i,k)/(2*pi); 
        EigFreq(i+1*Nk)=XM_eig(i,k)/(2*pi); 
        EigFreq(i+2*Nk)=MT_eig(i,k)/(2*pi); 
    end; 
    plot(TXaxis(1:Nk),EigFreq(1+0*Nk:1*Nk),'b',... 
         XMaxis(1:Nk),EigFreq(1+1*Nk:2*Nk),'b',... 
         MTaxis(1:Nk),EigFreq(1+2*Nk:3*Nk),'b'); 
end;
grid on;
hold off;
titlestr='传统平面波展开法计算得到的二维声子晶体能带结构图';
title(titlestr);
xlabel('波矢k');
ylabel('频率f/Hz');
 
axis([0 MTaxis(Nkpoints) 0 800]);
set(gca,'XTick',[TXaxis(1) TXaxis(Nkpoints) XMaxis(Nkpoints) MTaxis(Nkpoints)]);
xtixlabel=char('T','X','M','T');
set(gca,'XTickLabel',xtixlabel);