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ANSYS非线性分析选项:弧长法与预测校正

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"ANSYS软件操作指南" 在ANSYS软件中,弧长法是一种用于处理结构在载荷历史中可能出现物理不稳定性问题的数值求解技术。当结构的载荷-位移曲线出现斜率为0或负值的情况,即结构可能变得不稳定时,弧长法能帮助稳定计算过程。需要注意的是,使用弧长法时,不应与其他特定的分析和载荷步选项结合,例如线搜索(LNSRCH)、时间步长预测(PRED)、自适应下降(NROPT)、自动时间步长(AUTOTS, TIME, DELTIM)或开启时间积分效应(TIMINT)。 5. 时间步长预测(PRED)是一个非线性分析中的选项,可以通过Main Menu | Solution | Load Step Opts | Nonlinear | Predictor路径访问。它可以预测每个子步的第一次平衡迭代,与DOF求解相关,有助于加速收敛。然而,这种方法在包含大转动或粘弹性的分析中效果不理想。 6. 线搜索选项(LNSRCH)则提供了一种替代自适应下降的方法,可以通过Main Menu | Solution | Load Step Opts | Nonlinear | Line Search路径设定。线搜索算法会在硬化响应出现时计算比例因子,用于调整位移增量,其值介于0和1之间。当线搜索选项开启时,自适应下降不会自动激活,因为两者不应同时启用。 在涉及强迫位移的分析中,线搜索值通常在迭代过程中至少使用一次才能达到收敛。这确保了在解决非线性问题时的稳定性和准确性。 ANSYS软件的基本操作涵盖从安装、启动和配置,到模型建立、加载定义、求解过程以及后处理的各个环节。例如,在模型建立阶段,用户需要设置工作目录、作业名和分析标题,定义图形界面过滤参数和单位制,选择适当的单元类型,并设定材料属性。网格划分、耦合和约束也是建模的重要部分。在加载和求解阶段,用户需要定义各种边界条件和求解策略。后处理阶段则涉及通用后处理器、单元表、路径和时间历程后处理器等工具,用于查看和分析计算结果。 通过具体的案例分析,如六方孔螺钉受扳手静力分析、平面问题、轴对称结构、周期对称结构的静力和动力分析,以及预应力结构的模态和谐响应分析,读者可以更深入地理解ANSYS在实际问题中的应用和命令流输入的编写。这些实例详细演示了从问题描述、模型构建、求解到结果解析的全过程,有助于提升用户对ANSYS软件的实际操作能力。

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function points = generate_trajectory(total_points) radius = 2; line_length = 10; % 计算半圆和直线的点数 circle_arc = pi/2 * radius; total_len = circle_arc + line_length + circle_arc n_circle = round(total_points * circle_arc / total_len); n_line = total_points - n_circle - n_circle; % 生成半圆点 (圆心在(0.5,0)) theta = linspace(pi/2, pi, n_circle) - pi/2; x_circle = radius * (1 - cos(theta)); y_circle = radius * sin(theta); % 生成直线点 x_line = linspace(x_circle(end), x_circle(end) + line_length, n_line); y_line = zeros(1, n_line)+y_circle(end); % 生成半圆点 (圆心在(0.5,0)) theta = linspace(0, pi/2, n_circle) - pi/2; x_circle1 = radius * (cos(-theta)) + x_line(end); y_circle1 = radius * sin(-theta) - radius + y_line(end); % 合并点 points = [x_circle', y_circle'; x_line', y_line'; x_circle1', y_circle1']; %% % % 螺旋线参数设置 % a = 0.5; % 螺旋线系数(控制间距) % n_turns = 1; % 螺旋线圈数 % % % % ===== 步骤1:生成高密度采样点 ===== % theta_max = 2 * pi * n_turns; % 最大角度 % density = 5000; % 高密度采样点数量 % theta_dense = linspace(0, theta_max, density); % % % 阿基米德螺旋线参数方程 % x_dense = a * theta_dense .* cos(theta_dense); % y_dense = a * theta_dense .* sin(theta_dense); % % % ===== 步骤2:计算累积弧长 ===== % dx = diff(x_dense); % dy = diff(y_dense); % ds = sqrt(dx.^2 + dy.^2); % 微分弧长 % s_dense = [0, cumsum(ds)]; % 累积弧长 % % % ===== 步骤3:等弧长间隔采样 ===== % s_total = s_dense(end); % 总弧长 % s_eq = linspace(0, s_total, total_points); % 等间距弧长点 % % % 弧长参数插值 % x_eq = interp1(s_dense, x_dense, s_eq, 'linear'); % y_eq = interp1(s_dense, y_dense, s_eq, 'linear'); % % points = [x_eq; y_eq].'; end

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%% 1. 创建三维轨迹点 theta = linspace(0, 6*pi, 30); points = [cos(theta')*2, sin(theta')*2, theta'/5]; %% 2. 弦长参数化(归一化) diffs = diff(points); distances = sqrt(sum(diffs.^2, 2)); t_original = [0; cumsum(distances)]; t_original = t_original / t_original(end); % 归一化到[0,1] %% 3. 创建三次B样条 spline_x = csape(t_original, points(:,1)', 'second'); spline_y = csape(t_original, points(:,2)', 'second'); spline_z = csape(t_original, points(:,3)', 'second'); %% 4. 计算总弧长(关键修改) % 在密集点上采样计算总弧长 t_temp = linspace(0, 1, 1000); pts_temp = [fnval(spline_x, t_temp); fnval(spline_y, t_temp); fnval(spline_z, t_temp)]'; diffs_temp = diff(pts_temp); total_length = sum(sqrt(sum(diffs_temp.^2, 2))); %% 5. 生成等距插值点(核心算法) num_points = 200; % 目标点数 target_spacing = total_length / (num_points-1); % 目标间距 % 初始化 t_equal = zeros(1, num_points); t_equal(1) = 0; % 起点参数 current_length = 0; interpolated_points = zeros(num_points, 3); interpolated_points(1,:) = points(1,:); % 起点 % 迭代计算等距点参数 for i = 2:num_points % 牛顿迭代法求解参数 t_guess = t_equal(i-1) + 0.01; % 初始猜测 for iter = 1:5 % 5次迭代通常足够 % 计算当前位置 pt_current = [fnval(spline_x, t_guess); fnval(spline_y, t_guess); fnval(spline_z, t_guess)]'; % 计算与前一点的距离 segment_vec = pt_current - interpolated_points(i-1,:); current_dist = norm(segment_vec); % 计算导数(速度向量) deriv_x = fnval(fnder(spline_x,1), t_guess); deriv_y = fnval(fnder(spline_y,1), t_guess); deriv_z = fnval(fnder(spline_z,1), t_guess); speed = norm([deriv_x, deriv_y, deriv_z]); % 牛顿法更新参数 t_guess = t_guess - (current_dist - target_spacing)/speed; end t_equal(i) = t_guess; interpolated_points(i,:) = pt_current; end %% 6. 验证等距性(可选) diffs_final = diff(interpolated_points); distances_final = sqrt(sum(diffs_final.^2, 2)); fprintf('平均间距: %.4f\n标准差: %.4f\n最大偏差: %.4f\n', ... mean(distances_final), std(distances_final), max(abs(distances_final - target_spacing))); %% 7. 可视化 figure; plot3(points(:,1), points(:,2), points(:,3), 'ro', 'MarkerSize', 8, 'MarkerFaceColor', 'r'); hold on; plot3(interpolated_points(:,1), interpolated_points(:,2), interpolated_points(:,3), 'b-o', 'LineWidth', 1.5); title('等距B样条插值轨迹'); grid on; axis equal; legend('原始点', '等距插值点', 'Location', 'best'); %% 8. 输出插值点 csvwrite('equidistant_points.csv', interpolated_points); 换成使用PCL实现

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private static double[] latLonToUTM(double lat, double lon, int zoneNumber) { double latRad = lat * DEG2RAD; double lonRad = lon * DEG2RAD; double centralLon = (zoneNumber * 6 - 3) * DEG2RAD; double e = Math.sqrt(1 - Math.pow(SEMI_MINOR_AXIS / SEMI_MAJOR_AXIS, 2)); double e2 = e * e / (1 - e * e); double n = (SEMI_MAJOR_AXIS - SEMI_MINOR_AXIS) / (SEMI_MAJOR_AXIS + SEMI_MINOR_AXIS); double n2 = n * n; double n3 = n * n2; double n4 = n2 * n2; double A = SEMI_MAJOR_AXIS / (1 + n) * (1 + n2/4 + n4/64); double alpha1 = 1.0/2*n - 2.0/3*n2 + 5.0/16*n3 + 41.0/180*n4; double alpha2 = 13.0/48*n2 - 3.0/5*n3 + 557.0/1440*n4; double alpha3 = 61.0/240*n3 - 103.0/140*n4; double alpha4 = 49561.0/161280*n4; double deltaLon = lonRad - centralLon; double s = A * (latRad - alpha1*Math.sin(2*latRad) + alpha2*Math.sin(4*latRad) - alpha3*Math.sin(6*latRad) + alpha4*Math.sin(8*latRad)); double nu = SEMI_MAJOR_AXIS / Math.sqrt(1 - Math.pow(e*Math.sin(latRad), 2)); double t = Math.tan(latRad); double t2 = t * t; double t4 = t2 * t2; double t6 = t4 * t2; double c = e2 * Math.pow(Math.cos(latRad), 2); double c2 = c * c; double cosDeltaLon = Math.cos(deltaLon); double sinDeltaLon = Math.sin(deltaLon); double term1 = s; double term2 = nu/2 * Math.sin(latRad) * Math.cos(latRad) * Math.pow(deltaLon, 2); double term3 = nu/24 * Math.sin(latRad) * Math.pow(Math.cos(latRad), 3) * (5 - t2 + 9*c + 4*c2) * Math.pow(deltaLon, 4); double term4 = nu/720 * Math.sin(latRad) * Math.pow(Math.cos(latRad), 5) * (61 - 58*t2 + t4 + 270*c - 330*t2*c) * Math.pow(deltaLon, 6); double northing = term1 + term2 + term3 + term4; double term5 = nu * Math.cos(latRad) * deltaLon; double term6 = nu/6 * Math.pow(Math.cos(latRad), 3) * (1 - t2 + c) * Math.pow(deltaLon, 3); double term7 = nu/120 * Math.pow(Math.cos(latRad), 5) * (5 - 18*t2 + t4 + 14*c - 58*t2*c) * Math.pow(deltaLon, 5); double easting = 500000 + (term5 + term6 + term7); // 缩放和平移 easting *= 0.9996; northing *= 0.9996; easting += 500000; return new double[]{easting, northing}; } 检查此方法是否有问题

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function lspia_fitting_2() clc;clear;close all % 1. 生成轨迹点 (半圆+直线) num_points = 3000; n_control = 50; % 目标控制点数 points = generate_trajectory(num_points); % 3. 抽取50个控制点 control_idx = select_control_points(points, n_control); control_points = points(control_idx, :); n_control = size(control_points, 1); % 4. LSPIA拟合参数设置 p_degree = 3; % 三次B样条 max_iter = 1000; % 最大迭代次数 tolerance = 0.001; % 收敛阈值 mu = 0.002; % 松弛因子 % 5. 定义起点和终点切向量约束 start_tangent = [0, 1]; % 起点切向量 (水平向右) end_tangent = [0, -1]; % 终点切向量 (垂直向下) % 6. 计算原始点参数化 [u_values] = parameterize_points(points); % 7. 计算节点向量 knots = compute_knot_vector(control_points, p_degree, n_control); % 8. 预计算基函数矩阵 N_matrix = precompute_basis(u_values, knots, p_degree, n_control, num_points); sss = tic; % 9. LSPIA迭代(包含切向量约束) [final_control, iter] = constrained_lspia_iteration(... points, control_points, N_matrix, ... mu, max_iter, tolerance, ... points(1,:), points(end,:), ... knots... ); toc(sss) fprintf('LSPIA converged after %d iterations\n', iter); sss = tic; total_length = bspline_length(final_control, knots); toc(sss) % total_length1 = fast_bspline_length(final_control, knots); % fprintf('LSPIA BSpline length %f %f\n', total_length, total_length1); fprintf('LSPIA BSpline length %f\n', total_length); % sss = tic; % [usol ppp] = bspline_arc_length_param(final_control, knots, total_length, 3*total_length/4) % toc(sss) % 10. 计算并验证切向量 [actual_start_tangent, actual_end_tangent] = calculate_curve_tangents(... final_control, knots); fprintf('起点切向量设定: (%.2f, %.2f), 实际: (%.2f, %.2f)\n', ... start_tangent(1), start_tangent(2), ... actual_start_tangent(1), actual_start_tangent(2)); fprintf('终点切向量设定: (%.2f, %.2f), 实际: (%.2f, %.2f)\n', ... end_tangent(1), end_tangent(2), ... actual_end_tangent(1), actual_end_tangent(2)); % 11. 可视化结果 plot_fitting_results(points, control_points, final_control, knots, p_degree, n_control, N_matrix, ... iter); end %% 新增函数:带切向量约束的LSPIA迭代 function [P, iter] = constrained_lspia_iteration(... points, P0, N_matrix, ... mu, max_iter, tolerance, ... start_point, end_point, ... U) P = P0; iter = 0; err = inf; prev_err = inf; while iter < max_iter && err > tolerance % 计算当前拟合点 Y = N_matrix * P; % 计算残差 delta = points - Y; % 计算控制点增量 deltaP = mu * (N_matrix' * delta); % 更新控制点 P_new = P + deltaP; % 固定起点和终点位置 P_new(1,:) = start_point; P_new(end,:) = end_point; % 计算误差 err = sum(sum(delta.^2))/size(points, 1); % 自适应步长调整 if err > 10*prev_err mu = mu * 0.5; end prev_err = err; % 更新迭代变量 P = P_new; iter = iter + 1; end iter end %% 新增函数:计算曲线切向量 function [start_tangent, end_tangent] = calculate_curve_tangents(P, U) start_tangent = bspline_tangent(0, P, U); end_tangent = bspline_tangent(1, P, U); end function tangent = bspline_tangent(u, control_points, knots) % 计算三次B样条曲线在参数u处的切向量 % 输入: % u - 参数值 (标量) % control_points - 控制点矩阵 (n x dim, 每行一个控制点) % knots - 节点向量 (1 x (n+4)) % 输出: % tangent - 切向量 (1 x dim) % 参数检查 if nargin < 3 error('需要3个输入参数: u, control_points, knots'); end % 获取控制点数和维度 n = size(control_points, 1); % 控制点数量 dim = size(control_points, 2); % 曲线维度 % 验证节点向量长度 if length(knots) ~= n + 4 error('节点向量长度应为控制点数+4'); end % 1. 参数u的合法性检查与截断 u_min = knots(4); u_max = knots(end-3); u = max(u, u_min); u = min(u, u_max); % 2. 查找u所在的节点区间 k = find(u >= knots, 1, 'last'); % 找到满足knots(k) <= u的最大索引k k = min(k, n); % 确保k不超过控制点最大索引 k = max(k, 4); % 确保k不小于4 % 3. 计算相关的基函数值 % 3.1 计算零次基函数(N0) N0 = zeros(1, 7); % 存储k-3到k+3的基函数 for i = 1:7 idx = k - 4 + i; % 当前节点索引 if knots(idx) <= u && u <= knots(idx+1) N0(i) = 1; else N0(i) = 0; end end % 3.2 计算一次基函数(N1) N1 = zeros(1, 6); % 存储k-3到k+2的基函数 for i = 1:6 idx = k - 4 + i; % 当前节点索引 % 计算左侧部分 denom_left = knots(idx+1) - knots(idx); if denom_left ~= 0 left = (u - knots(idx)) / denom_left; else left = 0; end % 计算右侧部分 denom_right = knots(idx+2) - knots(idx+1); if denom_right ~= 0 right = (knots(idx+2) - u) / denom_right; else right = 0; end N1(i) = left * N0(i) + right * N0(i+1); end % 3.3 计算二次基函数(N2) N2 = zeros(1, 5); % 存储k-3到k+1的基函数 for i = 1:5 idx = k - 4 + i; % 当前节点索引 % 计算左侧部分 denom_left = knots(idx+2) - knots(idx); if denom_left ~= 0 left = (u - knots(idx)) / denom_left; else left = 0; end % 计算右侧部分 denom_right = knots(idx+3) - knots(idx+1); if denom_right ~= 0 right = (knots(idx+3) - u) / denom_right; else right = 0; end N2(i) = left * N1(i) + right * N1(i+1); end % 4. 计算基函数的一阶导数 dN3 = zeros(1, 4); % 存储k-3到k的基函数导数 for i = 1:4 idx = k - 4 + i; % 当前节点索引 % 计算第一部分 denom1 = knots(idx+3) - knots(idx); if denom1 ~= 0 term1 = 3 * N2(i) / denom1; else term1 = 0; end % 计算第二部分 denom2 = knots(idx+4) - knots(idx+1); if denom2 ~= 0 term2 = 3 * N2(i+1) / denom2; else term2 = 0; end dN3(i) = term1 - term2; end % 5. 计算切向量 tangent = zeros(1, dim); for i = 1:4 ctrl_idx = k - 4 + i; % 控制点索引(k-3, k-2, k-1, k) tangent = tangent + dN3(i) * control_points(ctrl_idx, :); end % 归一化切向量(可选) if norm(tangent) > eps tangent = tangent / norm(tangent); end end function point = curve_point(u, ctrl_point, n_control, knots, p_degree) basis = compute_basis(u, knots, p_degree, n_control); point = basis * ctrl_point; end %% 修改可视化函数:显示切向量 function plot_fitting_results(points, init_control, final_control, knots, p_degree, n, N_matrix, iter) figure; % 绘制原始轨迹 subplot(2,2,1); plot(points(:,1), points(:,2), 'b-', 'LineWidth', 1.5); hold on; plot(init_control(:,1), init_control(:,2), 'ro', 'MarkerSize', 8, 'LineWidth', 2); title('原始轨迹和初始控制点'); legend('原始轨迹', '控制点'); axis equal; grid on % 生成高密度拟合曲线 u_fine = linspace(0, 1, 5000); curve = zeros(length(u_fine), 2); sss = tic; for i = 1:length(u_fine) curve(i,:) = curve_point(u_fine(i), final_control, n, knots, p_degree); end [total_length] = chord_points(curve) toc(sss) % 绘制最终拟合曲线 subplot(2,2,2); plot(points(:,1), points(:,2), 'b-', 'LineWidth', 1); hold on; plot(curve(:,1), curve(:,2), 'r-', 'LineWidth', 1); plot(final_control(:,1), final_control(:,2), 'go-', 'MarkerSize', 8, 'LineWidth', 1.5); curve1 = zeros(length(u_fine), 2); for i = 1:length(u_fine) curve1(i,:) = curve_point(u_fine(i), init_control, n, knots, p_degree); end plot(curve1(:,1), curve1(:,2), 'k-', 'LineWidth', 1); knots = compute_knot_vector(final_control, p_degree, n); curve2 = zeros(length(u_fine), 2); for i = 1:length(u_fine) curve2(i,:) = curve_point(u_fine(i), final_control, n, knots, p_degree); end plot(curve2(:,1), curve2(:,2), 'm-', 'LineWidth', 1); [total_length] = chord_points(curve2) title('拟合曲线和切向量约束'); legend('原始轨迹', '拟合曲线', '控制点'); axis equal; grid on % 绘制控制点变化 subplot(2,2,3); plot(init_control(:,1), init_control(:,2), 'ro-', 'LineWidth', 1.5); hold on; plot(final_control(:,1), final_control(:,2), 'go-', 'LineWidth', 1.5); title('控制点迭代', iter); legend('初始控制点', '最终控制点'); axis equal; grid on % 计算点到曲线的最小距离 subplot(2,2,4); hold on; % 计算当前拟合点 Y = N_matrix * final_control; % 计算残差 err = sqrt(sum((points - Y).^2, 2)); min_dists = compute_min_distances(points, curve); min_dists1 = compute_min_distances(points, curve1); plot(err, 'y-', 'LineWidth', 1.0); plot(min_dists, 'b-', 'LineWidth', 1.0); plot(min_dists1, 'k-', 'LineWidth', 1.0); % 标记最大误差点 [max_err, max_idx] = max(min_dists); [max_err1, max_idx1] = max(min_dists1); plot(max_idx, max_err, 'ro', 'MarkerSize', 10, 'LineWidth', 2); text(max_idx, max_err, sprintf(' 最大误差: %.4f', max_err)); plot(max_idx1, max_err1, 'ro', 'MarkerSize', 10, 'LineWidth', 2); text(max_idx1, max_err1, sprintf(' 最大误差: %.4f', max_err1)); title('点到曲线最小距离误差'); xlabel('点索引'); ylabel('最小距离误差'); grid on; fprintf('平均最小距离误差: %.6f\n', mean(min_dists)); fprintf('最大最小距离误差: %.6f\n', max_err); end %% 以下函数保持不变(同之前的实现) %% 轨迹生成函数 function points = generate_trajectory(total_points) radius = 2; line_length = 10; % 计算半圆和直线的点数 circle_arc = pi/2 * radius; total_len = circle_arc + line_length + circle_arc n_circle = round(total_points * circle_arc / total_len); n_line = total_points - n_circle - n_circle; % 生成半圆点 (圆心在(0.5,0)) theta = linspace(pi/2, pi, n_circle) - pi/2; x_circle = radius * (1 - cos(theta)); y_circle = radius * sin(theta); % 生成直线点 x_line = linspace(x_circle(end), x_circle(end) + line_length, n_line); y_line = zeros(1, n_line)+y_circle(end); % 生成半圆点 (圆心在(0.5,0)) theta = linspace(0, pi/2, n_circle) - pi/2; x_circle1 = radius * (cos(-theta)) + x_line(end); y_circle1 = radius * sin(-theta) - radius + y_line(end); % 合并点 points = [x_circle', y_circle'; x_line', y_line'; x_circle1', y_circle1']; %% % % 螺旋线参数设置 % a = 0.5; % 螺旋线系数(控制间距) % n_turns = 1; % 螺旋线圈数 % % % % ===== 步骤1:生成高密度采样点 ===== % theta_max = 2 * pi * n_turns; % 最大角度 % density = 5000; % 高密度采样点数量 % theta_dense = linspace(0, theta_max, density); % % % 阿基米德螺旋线参数方程 % x_dense = a * theta_dense .* cos(theta_dense); % y_dense = a * theta_dense .* sin(theta_dense); % % % ===== 步骤2:计算累积弧长 ===== % dx = diff(x_dense); % dy = diff(y_dense); % ds = sqrt(dx.^2 + dy.^2); % 微分弧长 % s_dense = [0, cumsum(ds)]; % 累积弧长 % % % ===== 步骤3:等弧长间隔采样 ===== % s_total = s_dense(end); % 总弧长 % s_eq = linspace(0, s_total, total_points); % 等间距弧长点 % % % 弧长参数插值 % x_eq = interp1(s_dense, x_dense, s_eq, 'linear'); % y_eq = interp1(s_dense, y_dense, s_eq, 'linear'); % % points = [x_eq; y_eq].'; end %% 控制点选择函数 function control_indices = select_control_points(points, n_control) n = size(points, 1); % 1. 计算每个点的曲率 curvatures = compute_discrete_curvature(points); % 2. 识别关键区域 [extreme_regions, constant_regions] = identify_key_regions(curvatures, n); % 3. 分配控制点到关键区域 control_indices = []; % 起点区域:分配5个控制点 start_region = 1:floor(0.05*n); start_indices = round(linspace(1, length(start_region), 5)); control_indices = [control_indices; start_region(start_indices)']; % 终点区域:分配5个控制点 end_region = (n - floor(0.05*n)+1:n); end_indices = round(linspace(1, length(end_region), 5)); control_indices = [control_indices; end_region(end_indices)']; % 曲率极值区间:每个区域分配至少5个控制点 for i = 1:size(extreme_regions, 1) region_start = extreme_regions(i, 1); region_end = extreme_regions(i, 2); region_size = region_end - region_start + 1; % 计算需要分配的控制点数(至少5个) num_pts = max(5, min(10, round(5 * region_size / n))); region_indices = round(linspace(region_start, region_end, num_pts)); control_indices = [control_indices; region_indices']; end % 曲率稳定区间:每个区域分配至少5个控制点 for i = 1:size(constant_regions, 1) region_start = constant_regions(i, 1); region_end = constant_regions(i, 2); region_size = region_end - region_start + 1; % 计算需要分配的控制点数(至少5个) num_pts = max(5, min(10, round(5 * region_size / n))); region_indices = round(linspace(region_start, region_end, num_pts)); control_indices = [control_indices; region_indices']; end % 4. 剩余控制点按曲率分配 allocated = numel(control_indices); remaining = n_control - allocated; if remaining > 0 % 计算曲率权重(曲率高的区域权重高) weights = curvatures / sum(curvatures); weights(control_indices) = 0; % 避免重复分配 % 累积权重 cumulative_weights = cumsum(weights); total_weight = cumulative_weights(end); % 目标权重间隔 target_weights = linspace(0, total_weight, remaining + 1); target_weights = target_weights(2:end-1); % 基于权重选择控制点 for i = 1:length(target_weights) idx = find(cumulative_weights >= target_weights(i), 1); if ~isempty(idx) control_indices = [control_indices; idx]; end end end control_indices = [linspace(1,5,5).';control_indices;linspace(n-5+1,n,5).']; % 确保索引唯一且有序 control_indices = unique(control_indices); % 5. 如果控制点不足,补充随机点 if numel(control_indices) < n_control missing = n_control - numel(control_indices); candidate_indices = setdiff(1:n, control_indices); rand_indices = randperm(numel(candidate_indices), missing); control_indices = [control_indices; candidate_indices(rand_indices)']; control_indices = sort(control_indices); end % 6. 如果控制点过多,随机移除 if numel(control_indices) > n_control excess = numel(control_indices) - n_control; remove_indices = randperm(numel(control_indices), excess); control_indices(remove_indices) = []; end end function [extreme_regions, constant_regions] = identify_key_regions(curvatures, n) % 识别曲率极值区域(极大值和极小值) extreme_regions = []; % 设置阈值 max_curv = max(curvatures); threshold_high = 0.7 * max_curv; % 高曲率阈值 threshold_low = 0.3 * max_curv; % 低曲率阈值 % 识别高曲率区域 high_curv = curvatures > threshold_high; [region_start, region_end] = find_continuous_regions(high_curv); % 合并相邻区域 merged_regions = []; if ~isempty(region_start) current_start = region_start(1); current_end = region_end(1); for i = 2:length(region_start) if region_start(i) - current_end < 20 % 相邻区域阈值 current_end = region_end(i); else merged_regions = [merged_regions; current_start, current_end]; current_start = region_start(i); current_end = region_end(i); end end merged_regions = [merged_regions; current_start, current_end]; % 只保留显著的高曲率区域 min_region_size = 0.02 * n; % 最小区域大小 for i = 1:size(merged_regions, 1) region_size = merged_regions(i, 2) - merged_regions(i, 1) + 1; if region_size >= min_region_size extreme_regions = [extreme_regions; merged_regions(i, :)]; end end end % 识别低曲率区域(曲率稳定区域) constant_regions = []; low_curv = curvatures < threshold_low; [region_start, region_end] = find_continuous_regions(low_curv); % 合并相邻区域 merged_regions = []; if ~isempty(region_start) current_start = region_start(1); current_end = region_end(1); for i = 2:length(region_start) if region_start(i) - current_end < 50 % 相邻区域阈值 current_end = region_end(i); else merged_regions = [merged_regions; current_start, current_end]; current_start = region_start(i); current_end = region_end(i); end end merged_regions = [merged_regions; current_start, current_end]; % 只保留显著的低曲率区域 min_region_size = 0.05 * n; % 最小区域大小 for i = 1:size(merged_regions, 1) region_size = merged_regions(i, 2) - merged_regions(i, 1) + 1; if region_size >= min_region_size constant_regions = [constant_regions; merged_regions(i, :)]; end end end end function curvature = compute_discrete_curvature(points) % 计算离散点曲率(基于三点定圆法) % 输入:points - n×2矩阵,表示点坐标 [x1,y1; x2,y2; ...] % 输出:curvature - n×1向量,每个点对应的曲率值 n = size(points, 1); curvature = zeros(n, 1); if n < 3 error('至少需要3个点才能计算曲率'); end % 处理第一个点(使用前三个点计算) A = points(1, :); B = points(2, :); C = points(3, :); curvature(1) = three_point_curvature(A, B, C); % 处理中间点(使用相邻三点计算) for i = 2:n-1 A = points(i-1, :); B = points(i, :); C = points(i+1, :); curvature(i) = three_point_curvature(A, B, C); end % 处理最后一个点(使用最后三个点计算) A = points(n-2, :); B = points(n-1, :); C = points(n, :); curvature(n) = three_point_curvature(A, B, C); end function k = three_point_curvature(A, B, C) % 通过三点计算中间点B的曲率 % 基于三点定圆法:曲率 k = 1/R,R为外接圆半径 % 计算各边向量 AB = B - A; BC = C - B; AC = C - A; % 计算各边长度 a = norm(BC); % 边BC(对顶点A) b = norm(AC); % 边AC(对顶点B) c = norm(AB); % 边AB(对顶点C) % 计算三角形面积(使用叉积模) cross_val = (B(1)-A(1))*(C(2)-A(2)) - (C(1)-A(1))*(B(2)-A(2)); area = 0.5 * abs(cross_val); % 计算外接圆半径 R = abc/(4S) % 当三点共线时(面积为0),曲率为0 if area < eps || any([a, b, c] < eps) k = 0; else R = (a*b*c) / (4*area); k = 1 / R; end end function [region_start, region_end] = find_continuous_regions(logical_vec) % 找到连续的真值区域 diff_vec = diff([0; logical_vec; 0]); region_start = find(diff_vec == 1); region_end = find(diff_vec == -1) - 1; end %% 参数化函数 function [u_values] = parameterize_points(points) n = size(points, 1); chords = zeros(n-1, 1); for i = 1:n-1 chords(i) = norm(points(i+1,:) - points(i,:)); end total_chord = sum(chords); u_values = [0; cumsum(chords)/total_chord]; end function [total_chord] = chord_points(points) n = size(points, 1); chords = zeros(n-1, 1); for i = 1:n-1 chords(i) = norm(points(i+1,:) - points(i,:)); end total_chord = sum(chords); end %% 节点向量计算 function U = compute_knot_vector(ctrl_pts, p, n) % 计算控制点弦长 chords = zeros(n-1,1); for i = 1:n-1 chords(i) = norm(ctrl_pts(i+1,:) - ctrl_pts(i,:)); end % 控制点参数化 t = [0; cumsum(chords)/sum(chords)]; % 构建节点向量 U = zeros(1, n+p+1); U(1:p+1) = 0; U(end-p:end) = 1; % 内部节点计算 for j = p+2:n sum_val = 0; for i = j-p:j-1 sum_val = sum_val + t(i); end U(j) = sum_val / p; end end %% 基函数计算 (修复数值稳定性) function N = compute_basis(u, U, p, n) % 找到参数u所在的节点区间 k = find_span(u, U, n, p); % 计算非零基函数 N_vec = zeros(1, n); basis = get_basis_functions(u, p, U, k); % 归一化处理防止数值不稳定 basis_sum = sum(basis); if basis_sum > 0 basis = basis / basis_sum; % 确保基函数和为1 end % 确定有效索引范围 start_idx = max(1, k-p); end_idx = min(n, k); valid_length = end_idx - start_idx + 1; % 仅赋值有效范围内的基函数 N_vec(start_idx:end_idx) = basis(1:valid_length); N = N_vec; end function k = find_span(u, U, n, p) % 修正边界处理 if u >= U(n+1) k = n; % 最后一个有效区间 return; end low = p+1; high = n+1; k = floor((low+high)/2); while u < U(k) || u >= U(k+1) if u < U(k) high = k; else low = k; end k = floor((low+high)/2); end end function basis = get_basis_functions(u, p, U, k) basis = zeros(1, p+1); left = zeros(1, p+1); right = zeros(1, p+1); basis(1) = 1; for j = 1:p left(j) = u - U(k+1-j); right(j) = U(k+j) - u; saved = 0; for r = 0:j-1 denom = right(r+1) + left(j-r); if abs(denom) < 1e-10 % 防止除零错误 temp = 0; else temp = basis(r+1) / denom; end basis(r+1) = saved + right(r+1) * temp; saved = left(j-r) * temp; end basis(j+1) = saved; end end %% 预计算基函数矩阵 function N_matrix = precompute_basis(u_values, U, p, n, num_points) N_matrix = zeros(num_points, n); for i = 1:num_points N_matrix(i,:) = compute_basis(u_values(i), U, p, n); end % 添加数值稳定性检查 if any(any(isnan(N_matrix))) || any(any(isinf(N_matrix))) error('基函数矩阵包含NaN或Inf值,请检查节点向量和参数化'); end end %% 计算点到曲线的最小距离(新增函数) function min_dists = compute_min_distances(points, curve) num_points = size(points, 1); min_dists = zeros(num_points, 1); % 使用KD树加速最近邻搜索 kdtree = KDTreeSearcher(curve); for i = 1:num_points % 查找曲线上的最近点 [idx, dist] = knnsearch(kdtree, points(i, :)); min_dists(i) = dist; % 可视化调试:显示最近点连线 % plot([points(i,1), curve(idx,1)], [points(i,2), curve(idx,2)], 'm:'); end end function total_length = bspline_length(ctrl_pts, knots) % 输入: % ctrl_pts - 控制点矩阵 (n+1) x dim (dim为2或3) % knots - 节点向量 (1 x (n+5) 向量) % 输出: 三次B样条曲线的总长度 % ===== 1. 参数验证和初始化 ===== p = 3; % 三次B样条 n = size(ctrl_pts, 1) - 1; % 控制点索引最大值 dim = size(ctrl_pts, 2); % 曲线维度 (2D/3D) % 验证节点向量长度: 应满足 m = n + p + 1 if length(knots) ~= n + p + 2 error('节点向量长度错误: 应为 n+p+2 = %d, 实际为 %d', n+p+2, length(knots)); end % ===== 2. 计算导数曲线 (二次B样条) ===== deriv_ctrl = zeros(n, dim); % 导数控制点 for i = 1:n denom = knots(i+p+1) - knots(i+1); if abs(denom) < 1e-10 deriv_ctrl(i, :) = zeros(1, dim); else deriv_ctrl(i, :) = p * (ctrl_pts(i+1, :) - ctrl_pts(i, :)) / denom; end end deriv_knots = knots(2:end-1); % 导数节点向量 % ===== 3. 定义速度函数 ||C'(u)|| ===== function s = speed(u) % 计算二次B样条曲线在u处的导数向量 deriv_vec = zeros(1, dim); for j = 1:n N = basis_function(j-1, 2, u, deriv_knots); % 二次基函数 deriv_vec = deriv_vec + N * deriv_ctrl(j, :); end s = norm(deriv_vec); % 欧几里得范数 end % % ===== 4. B样条基函数计算 (递归实现) ===== % function N = basis_function(i, p, u, knots_vec) % % i: 基函数索引 (0-based) % % p: 次数 % % u: 参数值 % % knots_vec: 节点向量 % % % 递归终止条件 % if p == 0 % N = (knots_vec(i+1) <= u) && (u < knots_vec(i+2)); % if u == knots_vec(end) && i == length(knots_vec)-p-2 % N = 1; % 处理右端点 % end % return; % end % % % 递归计算 % N = 0; % denom1 = knots_vec(i+p+1) - knots_vec(i+1); % if denom1 > 1e-10 % N1 = basis_function(i, p-1, u, knots_vec); % N = N + (u - knots_vec(i+1)) / denom1 * N1; % end % % denom2 = knots_vec(i+p+2) - knots_vec(i+2); % if denom2 > 1e-10 % N2 = basis_function(i+1, p-1, u, knots_vec); % N = N + (knots_vec(i+p+2) - u) / denom2 * N2; % end % end % ===== 5. 自适应辛普森积分 ===== function I = adaptive_simpson(f, a, b, tol) % 递归自适应辛普森积分 c = (a + b) / 2; fa = f(a); fc = f(c); fb = f(b); % 基本辛普森近似 S = (b - a) * (fa + 4*fc + fb) / 6; % 递归计算左右区间 S_left = (c - a) * (fa + 4*f((a+c)/2) + fc) / 6; S_right = (b - c) * (fc + 4*f((c+b)/2) + fb) / 6; S2 = S_left + S_right; % 误差估计 err = abs(S2 - S); if err < 15 * tol I = S2 + (S2 - S) / 15; else I = adaptive_simpson(f, a, c, tol/2) + ... adaptive_simpson(f, c, b, tol/2); end end % ===== 6. 计算曲线长度 ===== u_min = knots(p+1); % 有效参数起点 (u_3) u_max = knots(end-p); % 有效参数终点 (u_{m-3}) total_length = adaptive_simpson(@speed, u_min, u_max, 1e-6); end function N = basis_function(i, p, u, knots) % 计算B样条基函数N_{i,p}(u) % 输入: % i: 基函数索引 (从0开始) % p: 样条阶数 % u: 参数值 % knots: 节点向量(行向量) % 输出: % N: 基函数值 % ===== 特殊情况处理 ===== if u < knots(1) || u > knots(end) N = 0; return; end % ===== 递归计算基函数 (Cox-de Boor算法) ===== if p == 0 % 零阶基函数 if u >= knots(i+1) && u < knots(i+2) N = 1; else N = 0; end else % 计算递归系数 denom1 = knots(i+p+1) - knots(i+1); denom2 = knots(i+p+2) - knots(i+2); term1 = 0; term2 = 0; if denom1 > eps term1 = (u - knots(i+1)) / denom1 * basis_function(i, p-1, u, knots); end if denom2 > eps term2 = (knots(i+p+2) - u) / denom2 * basis_function(i+1, p-1, u, knots); end N = term1 + term2; end end 显示完整的C代码

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基鑫阁
2025.05.16
本文详述了如何在ANSYS中使用弧长法稳定数值求解,特别适用于结构潜在的物理不稳定情况。
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尹子先生
2025.05.08
文档强调了与弧长法不兼容的分析选项,帮助用户避免常见的模拟错误。
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乔木Leo
2025.02.26
对于需要精确模拟大型转动和粘弹性行为的专业人士来说,本资源是一大助力。🍜
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我只匆匆而过
2025.02.02
介绍了弧长法的使用场景和注意事项,为工程分析提供了宝贵指导。
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邢小鹏
2025.01.04
详细解释了时间步长预测和线搜索选项的功能及其在特定分析中的效用,值得深入研究。
Yu-Demon321
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