Question

我在这里找到了一个使用三角形元素的例子。然后我继续修改它生成网格的方式，用矩形元素替换三角形元素，但我不知道如何整合它们。这是我的版本：

print(Decimal(string:"0.23"))

我开始编辑用于整合三角形元素的代码部分，但我不确定如何继续，或者甚至以类似的方式对矩形元素进行编码。

更新

因此，我尝试将Dohyun所建议的内容纳入我有限的理解中，以及我现在所拥有的内容：

%3.6  femcode.m

% [p,t,b] = squaregrid(m,n) % create grid of N=mn nodes to be listed in p
% generate mesh of T=2(m-1)(n-1) right triangles in unit square
m=6; n=5; % includes boundary nodes, mesh spacing 1/(m-1) and 1/(n-1)
[x,y]=ndgrid((0:m-1)/(m-1),(0:n-1)/(n-1)); % matlab forms x and y lists
p=[x(:),y(:)]; % N by 2 matrix listing x,y coordinates of all N=mn nodes
nelems=(m-1)*(n-1);
t=zeros(nelems,3);
for e=1:nelems
    t(e) = e + floor(e/6);
    t(e, 2) = e + floor(e/6) + 1;
    t(e, 3) = e + floor(e/6) + 6;
    t(e, 4) = e + floor(e/6) + 7;
end
% final t lists 4 node numbers of all rectangles in T by 4 matrix 
b=[1:m,m+1:m:m*n,2*m:m:m*n,m*n-m+2:m*n-1]; % bottom, left, right, top 
% b = numbers of all 2m+2n **boundary nodes** preparing for U(b)=0

% [K,F] = assemble(p,t) % K and F for any mesh of rectangles: linear phi's
N=size(p,1);T=nelems; % number of nodes, number of rectangles
% p lists x,y coordinates of N nodes, t lists rectangles by 4 node numbers
K=sparse(N,N); % zero matrix in sparse format: zeros(N) would be "dense"
F=zeros(N,1); % load vector F to hold integrals of phi's times load f(x,y)

for e=1:T  % integration over one rectangular element at a time
  nodes=t(e,:); % row of t = node numbers of the 3 corners of triangle e
  Pe=[ones(4,1),p(nodes,:),p(nodes,1).*p(nodes,2)]; % 4 by 4 matrix with rows=[1 x y xy] 
  Area=abs(det(Pe)); % area of triangle e = half of parallelogram area
  C=inv(Pe); % columns of C are coeffs in a+bx+cy to give phi=1,0,0 at nodes
  % now compute 3 by 3 Ke and 3 by 1 Fe for element e
  grad=C(2:3,:);Ke=Area*grad'*grad; % element matrix from slopes b,c in grad
  Fe=Area/3; % integral of phi over triangle is volume of pyramid: f(x,y)=1
  % multiply Fe by f at centroid for load f(x,y): one-point quadrature!
  % centroid would be mean(p(nodes,:)) = average of 3 node coordinates
  K(nodes,nodes)=K(nodes,nodes)+Ke; % add Ke to 9 entries of global K
  F(nodes)=F(nodes)+Fe; % add Fe to 3 components of load vector F
end   % all T element matrices and vectors now assembled into K and F

% [Kb,Fb] = dirichlet(K,F,b) % assembled K was singular! K*ones(N,1)=0
% Implement Dirichlet boundary conditions U(b)=0 at nodes in list b
K(b,:)=0; K(:,b)=0; F(b)=0; % put zeros in boundary rows/columns of K and F 
K(b,b)=speye(length(b),length(b)); % put I into boundary submatrix of K
Kb=K; Fb=F; % Stiffness matrix Kb (sparse format) and load vector Fb

% Solving for the vector U will produce U(b)=0 at boundary nodes
U=Kb\Fb  % The FEM approximation is U_1 phi_1 + ... + U_N phi_N

% Plot the FEM approximation U(x,y) with values U_1 to U_N at the nodes 
% surf(p(:,1),p(:,2),0*p(:,1),U,'edgecolor','k','facecolor','interp');
% view(2),axis equal,colorbar

我认为我可以使用定积分而不是数值积分，但结果与原始程序的积分不匹配。

Result of my program

Result of original program

Answer 1

对于P1三角形单元的情况，渐变恰好是常数。因此，整合只是area*grad'*grad。

然而，对于双线性情况，梯度的内积是二阶多项式。因此，您需要使用数值积分。

因此，在简单乘法中，你需要另一个循环来计算正交点的基值。

另外，你的面积公式是错误的。搜索仿射映射。

我在github上有一个存储库，它使用任意顺序多项式实现从1D到3D的泊松方程。如果您有兴趣，请访问https://github.com/dohyun64/fem_dohyun

用MATLAB中矩形元素的有限元方法逼近泊松方程

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