電磁場有限元方法EX2.2-里茲法求解泊松方程控制的邊值問題

簡單學習一下有限元法的基礎理論，書本為電磁場有限元經典教材：
THE FINITE ELEMENT METHOD IN ELECTROMAGNETICS, JIAN-MING JIN

1、問題定義

1.1、英文描述

1.2、中文描述

給定由方程定義的邊值問題：

$$\frac{{\mathrm{d}}^2?}{\mathrm{d}{x}^2}+2?=0(0<x<1)$$

邊界條件：

$${?|}_{x=0}=0\text{和}{?|}_{x=1}=1$$

證明該問題的泛函$F$為：

$$F(\widetilde{?})=\frac{1}{2}{\int }_0^1\left[{\left(\frac{\mathrm{d}\widetilde{?}}{\mathrm{d}x}\right)}^2?2{\widetilde{?}}^2\right]\mathrm{d}x$$

應用里茨方法，使用試探函數：

$$\widetilde{?}(x)=x+\sum _{n=1}^N{c}_n\sin (n\pi x)(\text{取?}N=2)$$

求近似解，并將結果與精確解：

$$?=\frac{\sin (\sqrt{2}x)}{\sin \sqrt{2}}$$

進行比較。

2、解題

2.1、解題思路

這題比較簡單，使用經典里茲方法，將試探函數$\widetilde{?}(x)$帶入泛函$F$中，然后對泛函$F$求c1和c2的偏導即可（$F$對c1，c2的偏導為0，以此求得極小值）。變分公式的證明見最后。

下面使用matlab實現代碼思路

2.2、Matlab代碼實現

2.2.1 定義試探函數和變分問題

定義試探函數和變分問題，并將試探函數帶入變分：

%% Solve
syms x c1 c2
% define the trail function and variational functional
phi = x+c1*sin(pi*x) +c2*sin(2*pi*x);
F=0.5*int((diff(phi, x))^2-2*(phi^2),x,0,1);
disp('F=');
pretty(simplify(F))

2.2.2 變分F對c1、c2變量求偏導

% Find the extreme value of the functional. calculate the partial derivative of c
diff_c1 = diff(F, c1);  % ?F/?c1
diff_c2 = diff(F, c2);  % ?F/?c2

2.2.3 F對c1、c2變量偏導為0，建立方程并求解

% Solve the system of equations for variables [c1, c2]
equations = [diff_c1 == 0, diff_c2 == 0];
solutions = solve(equations, [c1, c2]);
disp('Solution:');
% Extract all solutions for c1 and c2
c1_val = double(solutions.c1);
c2_val = double(solutions.c2);
% Display results (both symbolic and numeric forms)
disp(['c1 = ', num2str(c1_val)]);
disp(['c2 = ', num2str(c2_val)]);

2.2.4 對比理論解和數值解，并繪圖

phi_curve=subs(phi, {c1, c2}, {c1_val, c2_val});
phi_curve_exact=sin(sqrt(2)*x)/sin(sqrt(2));%% Draw figure
figure(1)
fplot(phi_curve, [0, 1], ...'Color', [0, 0.4470, 0.7410], ... % MATLAB經典藍'LineStyle', '--', ...             % 虛線'LineWidth', 2.5, ...'Marker', 'o', ...                 % 圓形標記'MarkerSize', 7, ...'MarkerFaceColor', 'auto', ...     % 自動填充'MarkerEdgeColor', 'k', ...        % 黑色邊框'DisplayName', sprintf('Numerical: $\\phi(x) = x + %.2f\\sin(\\pi x) + %.2f\\sin(2\\pi x)$', c1_val, c2_val));hold on
fplot(phi_curve_exact, [0, 1], ...'Color', [0.8500, 0.3250, 0.0980], ... % MATLAB經典紅'LineStyle', '-', ...                   % 實線'LineWidth', 2.5, ...'Marker', 's', ...                      % 正方形標記'MarkerSize', 8, ...'MarkerFaceColor', 'auto', ...           % 自動填充'DisplayName', 'Exact: $\frac{\sin(\sqrt{2} x)}{\sin(\sqrt{2})}$');xlabel('Position (x)', 'FontSize', 14);
ylabel('Function Value \phi(x)', 'FontSize', 14);
title('Comparison of Numerical and Exact Solutions', 'FontSize', 16);
xlim([0, 1]);legend('Interpreter', 'latex', 'Location', 'best', 'FontSize', 12);set(gca, 'FontSize', 12, 'GridAlpha', 0.3, 'Box', 'on');
grid minor;

2.3、結果對比

可以看到數值解和解析解是幾乎一致的：

3、全部代碼

% Exercise 2.2 THE FINITE ELEMENT METHOD IN ELECTROMAGNETICS JIN JIAN MINGclc
clear
%% Solve
syms x c1 c2
% define the trail function and variational functional
phi = x+c1*sin(pi*x) +c2*sin(2*pi*x);
F=0.5*int((diff(phi, x))^2-2*(phi^2),x,0,1);
disp('F=');
pretty(simplify(F))% Find the extreme value of the functional. calculate the partial derivative of c
diff_c1 = diff(F, c1);  % ?F/?c1
diff_c2 = diff(F, c2);  % ?F/?c2% Solve the system of equations for variables [c1, c2]
equations = [diff_c1 == 0, diff_c2 == 0];
solutions = solve(equations, [c1, c2]);
disp('Solution:');
% Extract all solutions for c1 and c2
c1_val = double(solutions.c1);
c2_val = double(solutions.c2);
% Display results (both symbolic and numeric forms)
disp(['c1 = ', num2str(c1_val)]);
disp(['c2 = ', num2str(c2_val)]);phi_curve=subs(phi, {c1, c2}, {c1_val, c2_val});
phi_curve_exact=sin(sqrt(2)*x)/sin(sqrt(2));%% Draw figure
figure(1)
fplot(phi_curve, [0, 1], ...'Color', [0, 0.4470, 0.7410], ... % MATLAB經典藍'LineStyle', '--', ...             % 虛線'LineWidth', 2.5, ...'Marker', 'o', ...                 % 圓形標記'MarkerSize', 7, ...'MarkerFaceColor', 'auto', ...     % 自動填充'MarkerEdgeColor', 'k', ...        % 黑色邊框'DisplayName', sprintf('Numerical: $\\phi(x) = x + %.2f\\sin(\\pi x) + %.2f\\sin(2\\pi x)$', c1_val, c2_val));hold on
fplot(phi_curve_exact, [0, 1], ...'Color', [0.8500, 0.3250, 0.0980], ... % MATLAB經典紅'LineStyle', '-', ...                   % 實線'LineWidth', 2.5, ...'Marker', 's', ...                      % 正方形標記'MarkerSize', 8, ...'MarkerFaceColor', 'auto', ...           % 自動填充'DisplayName', 'Exact: $\frac{\sin(\sqrt{2} x)}{\sin(\sqrt{2})}$');xlabel('Position (x)', 'FontSize', 14);
ylabel('Function Value \phi(x)', 'FontSize', 14);
title('Comparison of Numerical and Exact Solutions', 'FontSize', 16);
xlim([0, 1]);legend('Interpreter', 'latex', 'Location', 'best', 'FontSize', 12);set(gca, 'FontSize', 12, 'GridAlpha', 0.3, 'Box', 'on');
grid minor;

4、變分公式的證明