solveLaplace和updateRho分别是求解Laplace方程和更新电阻率分布的函数matlab
时间: 2024-03-25 18:36:41 浏览: 87
下面是solveLaplace和updateRho的MATLAB代码实现:
1. solveLaplace函数用于解Laplace方程:
```matlab
function [V, res] = solveLaplace(rho, elec)
% rho: 电阻率分布, nx x ny的二维矩阵
% elec: 电极位置, 2 x 2的二维矩阵,每行表示一个电极的位置
% V: 电势分布, nx x ny的二维矩阵
% res: 残差, 二维向量
nx = size(rho, 1);
ny = size(rho, 2);
% 构造系数矩阵
A = zeros(nx*ny);
b = zeros(nx*ny, 1);
for i = 1:nx
for j = 1:ny
k = (i-1)*ny + j;
if i == 1 % 左边界
A(k, k) = 1;
b(k) = 0;
elseif i == nx % 右边界
A(k, k) = 1;
b(k) = 0;
elseif j == 1 % 下边界
A(k, k) = 1;
b(k) = 0;
elseif j == ny % 上边界
A(k, k) = 1;
b(k) = 0;
else % 内部点
A(k, k) = -4;
A(k, k-1) = 1;
A(k, k+1) = 1;
A(k, k-ny) = 1;
A(k, k+ny) = 1;
end
end
end
% 设置边界条件
for i = 1:size(elec, 1)
k = (elec(i, 1)-1)*ny + elec(i, 2);
A(k, :) = 0;
A(k, k) = 1;
b(k) = 1;
end
% 求解线性方程组
V = reshape(A\b, [nx, ny]);
% 计算残差
res = zeros(1, size(elec, 1));
for i = 1:size(elec, 1)
k = (elec(i, 1)-1)*ny + elec(i, 2);
res(i) = abs(V(elec(i, 1), elec(i, 2)) - 1);
end
end
```
2. updateRho函数用于更新电阻率分布:
```matlab
function rho_new = updateRho(rho_old, data, elec)
% rho_old: 旧的电阻率分布, nx x ny的二维矩阵
% data: 测量到的电阻率数据, 一维向量
% elec: 电极位置, 2 x 2的二维矩阵,每行表示一个电极的位置
% rho_new: 新的电阻率分布, nx x ny的二维矩阵
% 计算电流密度
J = zeros(size(rho_old));
for i = 1:size(elec, 1)
if i == 1
J(elec(i, 1), elec(i, 2)) = data(i) / norm(elec(i, :) - elec(2, :));
else
J(elec(i, 1), elec(i, 2)) = -data(i) / norm(elec(i, :) - elec(1, :));
end
end
% 计算梯度
grad = zeros(size(rho_old));
for i = 2:size(rho_old, 1)-1
for j = 2:size(rho_old, 2)-1
grad(i, j) = sqrt((rho_old(i+1, j) - rho_old(i-1, j))^2 + (rho_old(i, j+1) - rho_old(i, j-1))^2) / 2;
end
end
% 更新电阻率分布
rho_new = rho_old .* exp(-grad ./ J);
end
```
这两个函数分别实现了电阻率测深法正演模拟中的Laplace方程求解和电阻率分布更新两个关键步骤。
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