有限元分析基本流程：幾何創建->邊界條件設置->物性參數設置->模型求解->結果后處理（可視化）。
問題描述：

matlab代碼（非PDE TOOLBOX實現）：

%% Create an electromagnetic model for electrostatic analysis. close all; clear all; emagmodel = createpde('electromagnetic','electrostatic');%% Create the geometry % Rectangle is code 3, 4 sides, followed by x-coordinates and then y-coordinates R1 = [3,4,0,12,12,0,0,0,2,2]'; % air R2 = [3,4,0,12,12,0,2,2,4,4]'; % dielectric slab R3 = [3,4,0,12,12,0,4,4,8,8]'; % air h=0.1;w=1; R4 = [3,4,4.5,5.5,5.5,4.5,4,4,4.1,4.1]'; R5 = [3,4,6.5,7.5,7.5,6.5,4,4,4.1,4.1]'; % geometry description matrix geom = [R1,R2,R3,R4,R5]; % Names for the two geometric objects ns = (char('R1','R2','R3','R4','R5'))'; % Set formula sf = '(R1+R2+R3)-R4-R5'; % Create geometry gd = decsg(geom,sf,ns); % geometry description geometryFromEdges(emagmodel,gd);% View geometry figure(1); pdegplot(emagmodel,'EdgeLabels','on','SubdomainLabels','on') axis equal%% Solve problem mu = 3.0; % relative permittivity of dielectric slab emagmodel.VacuumPermittivity = 8.8541878128E-12; electromagneticProperties(emagmodel,'RelativePermittivity',1.0, 'RelativePermeability',1.0,...,'Face',1); electromagneticProperties(emagmodel,'RelativePermittivity',1.0, 'RelativePermeability',1.0,...'Face',2); electromagneticProperties(emagmodel,'RelativePermittivity',mu, 'RelativePermeability',1.0, ...'Face',3); % Apply the voltage boundary conditions on the edges of the square. electromagneticBC(emagmodel,'Voltage',0.0,'Edge',[1 2 8 9 10 11 12 13]); electromagneticBC(emagmodel,'Voltage',5.0,'Edge',[3 4 15 19]); electromagneticBC(emagmodel,'Voltage',-5.0,'Edge',[5 6 17 20]);% Generate the mesh. mesh = generateMesh(emagmodel,'Hmax',0.15); figure(2);pdeplot(mesh);results = solve(emagmodel); % plot electric field figure(3); pdeplot(emagmodel,'FlowData',[results.ElectricField.Ex ...results.ElectricField.Ey]) axis equal % plot electric potential figure(4); pdeplot(emagmodel,'XYData',results.ElectricPotential,'ZData',results.ElectricPotential,...'ColorMap','hot','Contour','on');% 1.calculate the potential at points P(8,5),Q(4,4),R(6,1),S(6,3),T(2,3) P_V = interpolateElectricPotential(results,8,5); P_E = interpolateElectricField(results,8,5); P_D = interpolateElectricFlux(results,8,5); disp('Potential, electric field |E| and flux density |D| at point P(8,5) is: '); disp(P_V);disp(sqrt(P_E.Ex^2+P_E.Ey^2));disp(sqrt(P_D.Dx^2+P_D.Dy^2));Q_V = interpolateElectricPotential(results,4,4); Q_E = interpolateElectricField(results,4,4); Q_D = interpolateElectricFlux(results,4,4); disp('Potential, electric field |E| and flux density |D| at point Q(4,4) is: '); disp(Q_V);disp(sqrt(Q_E.Ex^2+Q_E.Ey^2));disp(sqrt(Q_D.Dx^2+Q_D.Dy^2));R_V = interpolateElectricPotential(results,6,1); R_E = interpolateElectricField(results,6,1); R_D = interpolateElectricFlux(results,6,1); disp('Potential, electric field |E| and flux density |D| at point R(6,1) is: '); disp(R_V);disp(sqrt(R_E.Ex^2+R_E.Ey^2));disp(sqrt(R_D.Dx^2+R_D.Dy^2));S_V = interpolateElectricPotential(results,6,3); S_E = interpolateElectricField(results,6,3); S_D = interpolateElectricFlux(results,6,3); disp('Potential, electric field |E| and flux density |D| at point S(6,3) is: '); disp(S_V);disp(sqrt(S_E.Ex^2+S_E.Ey^2));disp(sqrt(S_D.Dx^2+S_D.Dy^2));T_V = interpolateElectricPotential(results,2,3); T_E = interpolateElectricField(results,2,3); T_D = interpolateElectricFlux(results,2,3); disp('Potential, electric field |E| and flux density |D| at point T(2,3) is: '); disp(T_V);disp(sqrt(T_E.Ex^2+T_E.Ey^2));disp(sqrt(T_D.Dx^2+T_D.Dy^2));% 2.potential cotour plot figure(5); phi = results.ElectricPotential; pdeplot(emagmodel,'XYData',phi,'Contour','on','ColorMap','hot');%% 3. calculate cap Dx = results.ElectricFluxDensity.Dx; Dy = results.ElectricFluxDensity.Dy;Q = sum(sqrt(Dx.*Dx+Dy.*Dy)); dU = 10; % V cap = Q/dU;

結果如下：

總結

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