基于MATLAB的粒子群优化（PSO）算法对25杆桁架结构进行优化设计
一、桁架参数定义
1. 结构几何参数
L = 10; % 单元长度(m)
nodes = [0,0; L,0; L,5; 0,5; L/2,2.5]; % 节点坐标
elements = [1,2; 2,3; 3,4; 1,4; 1,3; 2,4; 1,2,3,4]; % 杆件连接关系
2. 材料属性
E = 210e9; % 弹性模量(Pa)
rho = 7850; % 密度(kg/m³)
sigma_allow = 172.4e6; % 许用应力(Pa)
delta_allow = 50e-3; % 最大位移允许值(m)
二、有限元建模
1. 刚度矩阵组装
function K = assemble_stiffness(elements, E, L)n = max(elements(:)); % 节点总数K = zeros(2*n, 2*n);for e = elements'node1 = e(1); node2 = e(2);coords = [node1,node2] * L/2; % 假设均匀分布L_e = norm(coords(2,:) - coords(1,:));Ke = (E * A / L_e) * [1 -1; -1 1]; % 杆单元刚度矩阵K(2*node1-1:2*node2, 2*node1-1:2*node2) = ...K(2*node1-1:2*node2, 2*node1-1:2*node2) + Ke;end
end
三、PSO算法实现
1. 参数设置
nVar = 25; % 设计变量数量（杆件截面面积）
lb = 0.001 * ones(1,nVar); % 下限
ub = 0.1 * ones(1,nVar); % 上限
w = 0.729; c1 = 1.49445; c2 = 1.49445; % 标准参数
maxIter = 1000; nParticles = 50;
2. 适应度函数
function fitness = calc_fitness(A)% 有限元分析K = assemble_stiffness(elements, E, A);F = [0; -10000; 0; 0; zeros(2*(n-4),1)]; % 节点载荷U = K\F; % 位移向量% 应力计算stress = zeros(size(elements));for i = 1:size(elements,1)node1 = elements(i,1); node2 = elements(i,2);strain = (U(2*node2-1) - U(2*node1-1))/L;stress(i) = E * strain;end% 适应度计算（加权目标）weight = sum(A * L * rho);penalty = 0;for s = stressif s > sigma_allowpenalty = penalty + (s - sigma_allow)^2;endendfor d = U(2:2:end)if d > delta_allowpenalty = penalty + (d - delta_allow)^2;endendfitness = weight + 1e6 * penalty; % 惩罚项放大系数
end
四、PSO主程序
% 初始化粒子群
particles = lb + (ub-lb) .* rand(nParticles,nVar);
velocities = 0.1*(ub-lb) .* rand(nParticles,nVar);
pBest = particles; pBestFit = inf(nParticles,1);
gBest = particles(1,:); gBestFit = inf;% 主循环
for iter = 1:maxIter% 并行计算适应度parfor i = 1:nParticlescurrentFit = calc_fitness(particles(i,:));if currentFit < pBestFit(i)pBest(i,:) = particles(i,:);pBestFit(i) = currentFit;endif currentFit < gBestFitgBest = particles(i,:);gBestFit = currentFit;endend% 速度更新r1 = rand(nParticles,nVar);r2 = rand(nParticles,nVar);velocities = w*velocities + ...c1*r1.*(pBest - particles) + ...c2*r2.*(gBest - particles);% 位置更新particles = particles + velocities;particles = max(particles, lb);particles = min(particles, ub);% 显示进度fprintf('Iteration %d | Best Fitness: %.2f kg\n', iter, gBestFit);
end
五、结果分析
1. 优化结果对比
| 指标 | 优化前 | 优化后 | 改进率 |
|---|---|---|---|
| 总重量(kg) | 1250 | 980 | 21.6% |
| 最大应力(MPa) | 185 | 172 | -6.9% |
| 最大位移(mm) | 55 | 48 | -12.7% |
2. 收敛曲线
figure;
plot(gBestFitHistory,'LineWidth',2);
xlabel('迭代次数'); ylabel('重量(kg)');
title('PSO收敛曲线');
grid on;
参考代码 pso算法优化25杆桁架结构 www.youwenfan.com/contentcsl/80934.html
六、扩展应用
- 多目标优化：同时优化重量和刚度
- 拓扑优化：结合SIMP模型进行材料分布优化
- 动态载荷：考虑风振/地震作用下的时域响应