# interpolateSolution

Interpolate PDE solution to arbitrary points

## Syntax

## Description

returns the interpolated values at the points in
`uintrp`

= interpolateSolution(`results`

,`querypoints`

)`querypoints`

.

returns the interpolated values of the solution to the time-dependent or
eigenvalue equation or system of such equations at times or modal indices
`uintrp`

= interpolateSolution(___,`iT`

)`iT`

. For a system of time-dependent or eigenvalue
equations, specify both time/modal indices `iT`

and equation
indices `iU`

## Examples

### Interpolate Scalar Stationary Results

Interpolate the solution to a scalar problem along a line and plot the result.

Create the solution to the problem $$-\Delta u=1$$ on the L-shaped membrane with zero Dirichlet boundary conditions.

model = createpde; geometryFromEdges(model,@lshapeg); applyBoundaryCondition(model,"dirichlet", ... "Edge",1:model.Geometry.NumEdges, ... "u",0); specifyCoefficients(model,"m",0,... "d",0,... "c",1,... "a",0,... "f",1); generateMesh(model,"Hmax",0.05); results = solvepde(model);

Interpolate the solution along the straight line from `(x,y) = (-1,-1)`

to `(1,1)`

. Plot the interpolated solution.

xq = linspace(-1,1,101); yq = xq; uintrp = interpolateSolution(results,xq,yq); plot(xq,uintrp) xlabel("x") ylabel("u(x)")

### Interpolate Solution of Poisson's Equation

Calculate the mean exit time of a Brownian particle from a region that contains absorbing (escape) boundaries and reflecting boundaries. Use the Poisson's equation with constant coefficients and 3-D rectangular block geometry to model this problem.

Create the solution for this problem.

model = createpde; importGeometry(model,"Block.stl"); applyBoundaryCondition(model,"dirichlet","Face",[1,2,5],"u",0); specifyCoefficients(model,"m",0,... "d",0,... "c",1,... "a",0,... "f",2); generateMesh(model); results = solvepde(model);

Create a grid and interpolate the solution to the grid.

[X,Y,Z] = meshgrid(0:135,0:35,0:61); uintrp = interpolateSolution(results,X,Y,Z); uintrp = reshape(uintrp,size(X));

Create a contour slice plot for five fixed values of the `y`

coordinate.

contourslice(X,Y,Z,uintrp,[],0:4:16,[]) colormap jet xlabel("x") ylabel("y") zlabel("z") xlim([0,100]) ylim([0,20]) zlim([0,50]) axis equal view(-50,22) colorbar

### Interpolate Scalar Stationary Results Using Query Matrix

Solve a scalar stationary problem and interpolate the solution to a dense grid.

Create the solution to the problem $$-\Delta u=1$$ on the L-shaped membrane with zero Dirichlet boundary conditions.

model = createpde; geometryFromEdges(model,@lshapeg); applyBoundaryCondition(model,"dirichlet", ... "Edge",1:model.Geometry.NumEdges, ... "u",0); specifyCoefficients(model,"m",0,"d",0,"c",1,"a",0,"f",1); generateMesh(model,"Hmax",0.05); results = solvepde(model);

Interpolate the solution on the grid from –1 to 1 in each direction.

v = linspace(-1,1,101); [X,Y] = meshgrid(v); querypoints = [X(:),Y(:)]'; uintrp = interpolateSolution(results,querypoints);

Plot the resulting interpolation on a mesh.

uintrp = reshape(uintrp,size(X)); mesh(X,Y,uintrp) xlabel("x") ylabel("y")

### Interpolate Stationary System

Create the solution to a two-component system and plot the two components along a planar slice through the geometry.

Create a PDE model for two components. Import the geometry of a torus.

model = createpde(2); importGeometry(model,"Torus.stl"); pdegplot(model,"FaceLabels","on");

Set boundary conditions.

gfun = @(region,state)[0,region.z-40]; applyBoundaryCondition(model,"neumann","Face",1,"g",gfun); ufun = @(region,state)[region.x-40,0]; applyBoundaryCondition(model,"dirichlet","Face",1,"u",ufun);

Set the problem coefficients.

specifyCoefficients(model,"m",0,... "d",0,... "c",[1;0;1;0;0;1;0;0;1;0;1; 0;1;0;0;1;0;1;0;0;1],... "a",0,... "f",[1;1]);

Create a mesh and solve the problem.

generateMesh(model); results = solvepde(model);

Interpolate the results on a plane that slices the torus for each of the two components.

[X,Z] = meshgrid(0:100); Y = 15*ones(size(X)); uintrp = interpolateSolution(results,X,Y,Z,[1,2]);

Plot the two components.

```
sol1 = reshape(uintrp(:,1),size(X));
sol2 = reshape(uintrp(:,2),size(X));
figure
surf(X,Z,sol1)
title("Component 1")
```

```
figure
surf(X,Z,sol2)
title("Component 2")
```

### Interpolate Scalar Eigenvalue Results

Solve a scalar eigenvalue problem and interpolate one eigenvector to a grid.

Find the eigenvalues and eigenvectors for the L-shaped membrane.

model = createpde(1); geometryFromEdges(model,@lshapeg); applyBoundaryCondition(model,"dirichlet", ... "Edge",1:model.Geometry.NumEdges, ... "u",0); specifyCoefficients(model,"m",0,... "d",1,... "c",1,... "a",0,... "f",0); r = [0,100]; generateMesh(model,"Hmax",1/50); results = solvepdeeig(model,r);

Interpolate the eigenvector corresponding to the fifth eigenvalue to a coarse grid and plot the result.

[xq,yq] = meshgrid(-1:0.1:1); uintrp = interpolateSolution(results,xq,yq,5); uintrp = reshape(uintrp,size(xq)); surf(xq,yq,uintrp)

### Interpolate Time-Dependent System

Solve a system of time-dependent PDEs and interpolate the solution.

Import slab geometry for a 3-D problem with three solution components. Plot the geometry.

model = createpde(3); importGeometry(model,"Plate10x10x1.stl"); pdegplot(model,"FaceLabels","on","FaceAlpha",0.5)

Set boundary conditions such that face 2 is fixed (zero deflection in any direction) and face 5 has a load of `1e3`

in the positive `z`

-direction. This load causes the slab to bend upward. Set the initial condition that the solution is zero, and its derivative with respect to time is also zero.

applyBoundaryCondition(model,"dirichlet","Face",2,"u",[0,0,0]); applyBoundaryCondition(model,"neumann","Face",5,"g",[0,0,1e3]); setInitialConditions(model,0,0);

Create PDE coefficients for the equations of linear elasticity. Set the material properties to be similar to those of steel. See Linear Elasticity Equations.

E = 200e9; nu = 0.3; specifyCoefficients(model,"m",1,... "d",0,... "c",elasticityC3D(E,nu),... "a",0,... "f",[0;0;0]);

Generate a mesh, setting `Hmax`

to 1.

`generateMesh(model,"Hmax",1);`

Solve the problem for times 0 through `5e-3`

in steps of `1e-4`

.

tlist = 0:1e-4:5e-3; results = solvepde(model,tlist);

Interpolate the solution at fixed `x`

- and `z`

-coordinates in the centers of their ranges, 5 and 0.5 respectively. Interpolate for `y`

from 0 through 10 in steps of 0.2. Obtain just component 3, the `z`

-component of the solution.

```
yy = 0:0.2:10;
zz = 0.5*ones(size(yy));
xx = 10*zz;
component = 3;
uintrp = interpolateSolution(results,xx,yy,zz, ...
component,1:length(tlist));
```

The solution is a 51-by-1-by-51 array. Use `squeeze`

to remove the singleton dimension. Removing the singleton dimension transforms this array to a 51-by-51 matrix which simplifies indexing into it.

uintrp = squeeze(uintrp);

Plot the solution as a function of `y`

and time.

[X,Y] = ndgrid(yy,tlist); figure surf(X,Y,uintrp) xlabel("Y") ylabel("Time") title("Deflection at x = 5, z = 0.5") zlim([0,14e-5])

## Input Arguments

`results`

— PDE solution

`StationaryResults`

object (default) | `TimeDependentResults`

object | `EigenResults`

object

PDE solution, specified as a `StationaryResults`

object,
a `TimeDependentResults`

object,
or an `EigenResults`

object.
Create `results`

using `solvepde`

, `solvepdeeig`

,
or `createPDEResults`

.

**Example: **`results = solvepde(model)`

`xq`

— *x*-coordinate query points

real array

*x*-coordinate query points, specified as a real array.
`interpolateSolution`

evaluates the solution at the
2-D coordinate points `[xq(i),yq(i)]`

or at the 3-D
coordinate points `[xq(i),yq(i),zq(i)]`

. So
`xq`

, `yq`

, and (if present)
`zq`

must have the same number of entries.

`interpolateSolution`

converts query points to column
vectors `xq(:)`

, `yq(:)`

, and (if present)
`zq(:)`

. The returned solution is a column vector of
the same size. To ensure that the dimensions of the returned solution is
consistent with the dimensions of the original query points, use
`reshape`

. For example, use ```
uintrp =
reshape(gradxuintrp,size(xq))
```

.

**Data Types: **`double`

`yq`

— *y*-coordinate query points

real array

*y*-coordinate query points, specified as a real array.
`interpolateSolution`

evaluates the solution at the
2-D coordinate points `[xq(i),yq(i)]`

or at the 3-D
coordinate points `[xq(i),yq(i),zq(i)]`

. So
`xq`

, `yq`

, and (if present)
`zq`

must have the same number of entries.
Internally, `interpolateSolution`

converts query points
to the column vector `yq(:)`

.

**Data Types: **`double`

`zq`

— *z*-coordinate query points

real array

*z*-coordinate query points, specified as a real array.
`interpolateSolution`

evaluates the solution at the
3-D coordinate points `[xq(i),yq(i),zq(i)]`

. So
`xq`

, `yq`

, and
`zq`

must have the same number of entries. Internally,
`interpolateSolution`

converts query points to the
column vector `zq(:)`

.

**Data Types: **`double`

`querypoints`

— Query points

real matrix

Query points, specified as a real matrix with either two rows for 2-D
geometry, or three rows for 3-D geometry. `interpolateSolution`

evaluates the solution at the coordinate
points `querypoints(:,i)`

, so each column of
`querypoints`

contains exactly one 2-D or 3-D query
point.

**Example: **For 2-D geometry, ```
querypoints = [0.5,0.5,0.75,0.75;
1,2,0,0.5]
```

**Data Types: **`double`

`iU`

— Equation indices

vector of positive integers

Equation indices, specified as a vector of positive integers.
Each entry in `iU`

specifies an equation index.

**Example: **`iU = [1,5]`

specifies the indices
for the first and fifth equations.

**Data Types: **`double`

`iT`

— Time or mode indices

vector of positive integers

Time or mode indices, specified as a vector of positive integers.
Each entry in `iT`

specifies a time index for time-dependent
solutions, or a mode index for eigenvalue solutions.

**Example: **`iT = 1:5:21`

specifies the time or
mode for every fifth solution up to 21.

**Data Types: **`double`

## Output Arguments

`uintrp`

— Solution at query points

array

Solution at query points, returned as an array. For query points that are
outside the geometry,
`uintrp`

= `NaN`

. For
details about dimensions of the solution, see Dimensions of Solutions, Gradients, and Fluxes.

## Version History

**Introduced in R2015b**

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