TimeDependentResults

Time-dependent PDE solution and derived quantities

Description

A TimeDependentResults object contains the solution of a PDE and its gradients in a form convenient for plotting and postprocessing.

• A TimeDependentResults object contains the solution and its gradient calculated at the nodes of the triangular or tetrahedral mesh, generated by generateMesh.

• Solution values at the nodes appear in the NodalSolution property.

• The solution times appear in the SolutionTimes property.

• The three components of the gradient of the solution values at the nodes appear in the XGradients, YGradients, and ZGradients properties.

• The array dimensions of NodalSolution, XGradients, YGradients, and ZGradients enable you to extract solution and gradient values for specified time indices, and for the equation indices in a PDE system.

To interpolate the solution or its gradient to a custom grid (for example, specified by meshgrid), use interpolateSolution or evaluateGradient.

Creation

There are several ways to create a TimeDependentResults object:

• Solve a time-dependent problem using the solvepde function. This function returns a PDE solution as a TimeDependentResults object. This is the recommended approach.

• Solve a time-dependent problem using the parabolic or hyperbolic function. Then use the createPDEResults function to obtain a TimeDependentResults object from a PDE solution returned by parabolic or hyperbolic. Note that parabolic and hyperbolic are legacy functions. They are not recommended for solving PDE problems.

Properties

Mesh — Finite element mesh

FEMesh object

Finite element mesh, returned as a FEMesh Properties object.

NodalSolution — Solution values at the nodes

vector | array

Solution values at the nodes, returned as a vector or array. For details about the dimensions of NodalSolution, see Dimensions of Solutions, Gradients, and Fluxes.

Data Types: double
Complex Number Support: Yes

SolutionTimes — Solution times

real vector

Solution times, returned as a real vector. SolutionTimes is the same as the tlist input to solvepde, or the tlist input to the legacy parabolic or hyperbolic solvers.

Data Types: double

XGradients — x-component of gradient at the nodes

vector | array

x-component of the gradient at the nodes, returned as a vector or array. For details about the dimensions of XGradients, see Dimensions of Solutions, Gradients, and Fluxes.

Data Types: double
Complex Number Support: Yes

YGradients — y-component of gradient at the nodes

vector | array

y-component of the gradient at the nodes, returned as a vector or array. For details about the dimensions of YGradients, see Dimensions of Solutions, Gradients, and Fluxes.

Data Types: double
Complex Number Support: Yes

ZGradients — z-component of gradient at the nodes

vector | array

z-component of the gradient at the nodes, returned as a vector or array. For details about the dimensions of ZGradients, see Dimensions of Solutions, Gradients, and Fluxes.

Data Types: double

Object Functions

evaluateCGradient	Evaluate flux of PDE solution
evaluateGradient	Evaluate gradients of PDE solutions at arbitrary points
interpolateSolution	Interpolate PDE solution to arbitrary points

Examples

Solution of a Parabolic Problem

Solve a parabolic problem with 2-D geometry.

Create and view the geometry: a square with a circular subdomain.

% Square centered at (1,1)
rect1 = [3;4;0;2;2;0;0;0;2;2];
% Circle centered at (1.5,0.5)
circ1 = [1;1.5;.75;0.25];
% Append extra zeros to the circle
circ1 = [circ1;zeros(length(rect1)-length(circ1),1)];
gd = [rect1,circ1];
ns = char('rect1','circ1');
ns = ns';
sf = 'rect1+circ1';
[dl,bt] = decsg(gd,sf,ns);
pdegplot(dl,'EdgeLabels','on','FaceLabels','on')
axis equal
ylim([-0.1,2.1])

Include the geometry in a PDE model.

model = createpde();
geometryFromEdges(model,dl);

Set boundary conditions that the upper and left edges are at temperature 10.

applyBoundaryCondition(model,'dirichlet','Edge',[2,3],'u',10);

Set initial conditions that the square region is at temperature 0, and the circle is at temperature 100.

setInitialConditions(model,0);
setInitialConditions(model,100,'Face',2);

Define the model coefficients.

specifyCoefficients(model,'m',0,'d',1,'c',1,'a',0,'f',0);

Solve the problem for times 0 through 1/2 in steps of 0.01.

generateMesh(model,'Hmax',0.05);
tlist = 0:0.01:0.5;
results = solvepde(model,tlist);

Plot the solution for times 0.02, 0.04, 0.1, and 0.5.

sol = results.NodalSolution;
subplot(2,2,1)
pdeplot(model,'XYData',sol(:,3))
title('Time 0.02')
subplot(2,2,2)
pdeplot(model,'XYData',sol(:,5))
title('Time 0.04')
subplot(2,2,3)
pdeplot(model,'XYData',sol(:,11))
title('Time 0.1')
subplot(2,2,4)
pdeplot(model,'XYData',sol(:,51))
title('Time 0.5')