# evaluate

Interpolate data to selected locations

**This function supports the legacy workflow. Using the [p,e,t]
representation of FEMesh data is not recommended. Use interpolateSolution and evaluateGradient to interpolate a PDE solution and its gradient to
arbitrary points without switching to a [p,e,t]
representation.**

## Description

## Examples

### Interpolate to Matrix of Values

This example shows how to interpolate a solution to a scalar problem using a `pOut`

matrix of values.

Solve the equation $$-\Delta u=1$$ on the unit disk with zero Dirichlet conditions.

g0 = [1;0;0;1]; % circle centered at (0,0) with radius 1 sf = 'C1'; g = decsg(g0,sf,sf'); % decomposed geometry matrix model = createpde; gm = geometryFromEdges(model,g); % Zero Dirichlet conditions applyBoundaryCondition(model,"dirichlet", ... "Edge",(1:gm.NumEdges), ... "u",0); [p,e,t] = initmesh(gm); c = 1; a = 0; f = 1; u = assempde(model,p,e,t,c,a,f); % solve the PDE

Construct an interpolator for the solution.

F = pdeInterpolant(p,t,u);

Generate a random set of coordinates in the unit square. Evaluate the interpolated solution at the random points.

rng default % for reproducibility pOut = rand(2,25); % 25 numbers between 0 and 1 uOut = evaluate(F,pOut); numNaN = sum(isnan(uOut))

numNaN = 9

`uOut`

contains some `NaN`

entries because some points in `pOut`

are outside of the unit disk.

### Interpolate to x, y Values

This example shows how to interpolate a solution to a scalar problem using `x`

, `y`

values.

Solve the equation $$-\Delta u=1$$ on the unit disk with zero Dirichlet conditions.

g0 = [1;0;0;1]; % circle centered at (0,0) with radius 1 sf = 'C1'; g = decsg(g0,sf,sf'); % decomposed geometry matrix model = createpde; gm = geometryFromEdges(model,g); % Zero Dirichlet conditions applyBoundaryCondition(model,"dirichlet", ... "Edge",(1:gm.NumEdges), ... "u",0); [p,e,t] = initmesh(gm); c = 1; a = 0; f = 1; u = assempde(model,p,e,t,c,a,f); % solve the PDE

Construct an interpolator for the solution.

`F = pdeInterpolant(p,t,u); % create the interpolant`

Evaluate the interpolated solution at grid points in the unit square with spacing `0.2`

.

[x,y] = meshgrid(0:0.2:1); uOut = evaluate(F,x,y); numNaN = sum(isnan(uOut))

numNaN = 12

`uOut`

contains some `NaN`

entries because some points in the unit square are outside of the unit disk.

### Interpolate a Solution with Multiple Components

This example shows how to interpolate the solution to a system of `N`

= 3 equations.

Solve the system of equations $$-\Delta u=f$$ with Dirichlet boundary conditions on the unit disk, where

$$f={[\mathrm{sin}(x)+\mathrm{cos}(y),\mathrm{cosh}(xy),\frac{xy}{1+{x}^{2}+{y}^{2}}]}^{T}.$$

g0 = [1;0;0;1]; % circle centered at (0,0) with radius 1 sf = 'C1'; g = decsg(g0,sf,sf'); % decomposed geometry matrix model = createpde(3); gm = geometryFromEdges(model,g); applyBoundaryCondition(model,"dirichlet", ... "Edge",(1:gm.NumEdges), ... "u",zeros(3,1)); [p,e,t] = initmesh(g); c = 1; a = 0; f = char('sin(x) + cos(y)','cosh(x.*y)','x.*y./(1+x.^2+y.^2)'); u = assempde(model,p,e,t,c,a,f); % solve the PDE

Construct an interpolant for the solution.

`F = pdeInterpolant(p,t,u); % create the interpolant`

Interpolate the solution at a circle.

s = linspace(0,2*pi); x = 0.5 + 0.4*cos(s); y = 0.4*sin(s); uOut = evaluate(F,x,y);

Plot the three solution components.

npts = length(x); plot3(x,y,uOut(1:npts),"b") hold on plot3(x,y,uOut(npts+1:2*npts),"k") plot3(x,y,uOut(2*npts+1:end),"r") hold off view(35,35)

### Interpolate a Time-Varying Solution

This example shows how to interpolate a solution that depends on time.

Solve the equation

$$\frac{\partial u}{\partial t}-\Delta u=1$$

on the unit disk with zero Dirichlet conditions and zero initial conditions. Solve at five times from 0 to 1.

g0 = [1;0;0;1]; % circle centered at (0,0) with radius 1 sf = 'C1'; g = decsg(g0,sf,sf'); % decomposed geometry matrix model = createpde; gm = geometryFromEdges(model,g); % Zero Dirichlet conditions applyBoundaryCondition(model,"dirichlet", ... "Edge",(1:gm.NumEdges), ... "u",0); [p,e,t] = initmesh(gm); c = 1; a = 0; f = 1; d = 1; tlist = 0:1/4:1; u = parabolic(0,tlist,model,p,e,t,c,a,f,d);

52 successful steps 0 failed attempts 106 function evaluations 1 partial derivatives 13 LU decompositions 105 solutions of linear systems

Construct an interpolant for the solution.

F = pdeInterpolant(p,t,u);

Interpolate the solution at `x = 0.1`

, `y = -0.1`

, and all available times.

x = 0.1; y = -0.1; uOut = evaluate(F,x,y)

`uOut = `*1×5*
0 0.1809 0.2278 0.2388 0.2413

The solution starts at 0 at time 0, as it should. It grows to about 1/4 at time 1.

### Interpolate to a Grid

This example shows how to interpolate an elliptic solution to a grid.

**Define and Solve the Problem**

Use the built-in geometry functions to create an L-shaped region with zero Dirichlet boundary conditions. Solve an elliptic PDE with coefficients $$c=1$$, $$a=0$$, $$f=1$$, with zero Dirichlet boundary conditions.

[p,e,t] = initmesh("lshapeg"); % Predefined geometry u = assempde("lshapeb",p,e,t,1,0,1); % Predefined boundary condition

**Create an Interpolant**

Create an interpolant for the solution.

F = pdeInterpolant(p,t,u);

**Create a Grid for the Solution**

xgrid = -1:0.1:1; ygrid = -1:0.2:1; [X,Y] = meshgrid(xgrid,ygrid);

The resulting grid has some points that are outside the L-shaped region.

**Evaluate the Solution On the Grid**

uout = evaluate(F,X,Y);

The interpolated solution `uout`

is a column vector. You can reshape it to match the size of `X`

or `Y`

. This gives a matrix, like the output of the `tri2grid`

function.

Z = reshape(uout,size(X));

## Input Arguments

`F`

— Interpolant

output of `pdeInterpolant`

Interpolant, specified as the output of `pdeInterpolant`

.

**Example: **`F = pdeInterpolant(p,t,u)`

`pOut`

— Query points

matrix with two or three rows

Query points, specified as a matrix with two or three rows. The first row
represents the `x`

component of the query points, the
second row represents the `y`

component, and, for 3-D
geometry, the third row represents the `z`

component.
`evaluate`

computes the interpolant at each column of
`pOut`

. In other words, `evaluate`

interpolates at the points `pOut(:,k)`

.

**Example: **`pOut = [-1.5,0,1;`

1,1,2.2]

**Data Types: **`double`

`x`

— Query point component

vector or array

Query point component, specified as a vector or array. `evaluate`

interpolates at either 2-D points
`[x(k),y(k)]`

or at 3-D points
`[x(k),y(k),z(k)]`

. The `x`

and
`y`

, and `z`

arrays must contain the
same number of entries.

`evaluate`

transforms query point components to the
linear index representation, such as `x(:)`

.

**Example: **`x = -1:0.2:3`

**Data Types: **`double`

`y`

— Query point component

vector or array

Query point component, specified as a vector or array. `evaluate`

interpolates at either 2-D points
`[x(k),y(k)]`

or at 3-D points
`[x(k),y(k),z(k)]`

. The `x`

and
`y`

, and `z`

arrays must contain the
same number of entries.

`evaluate`

transforms query point components to the
linear index representation, such as `y(:)`

.

**Example: **`y = -1:0.2:3`

**Data Types: **`double`

`z`

— Query point component

vector or array

Query point component, specified as a vector or array. `evaluate`

interpolates at either 2-D points
`[x(k),y(k)]`

or at 3-D points
`[x(k),y(k),z(k)]`

. The `x`

and
`y`

, and `z`

arrays must contain the
same number of entries.

`evaluate`

transforms query point components to the
linear index representation, such as `z(:)`

.

**Example: **`z = -1:0.2:3`

**Data Types: **`double`

## Output Arguments

`uOut`

— Interpolated values

array

Interpolated values, returned as an array. `uOut`

has the
same number of columns as the data `u`

used in creating
`F`

. If `u`

depends on time,
`uOut`

contains a column for each time step. For
time-independent `u`

, `uOut`

has one
column.

The number of rows in `uOut`

is the number of equations
in the PDE system, `N`

, times the number of query points,
`pOut`

. The first `pOut`

rows
correspond to equation 1, the next `pOut`

rows correspond
to equation 2, and so on.

If a query point is outside the mesh, `evaluate`

returns `NaN`

for that
point.

## More About

### Element

An *element* is a basic
unit in the finite-element method.

For 2-D problems, an element is a triangle in the `model.Mesh.Element`

property. If the triangle represents a linear element, it has nodes only at the triangle
corners. If the triangle represents a quadratic element, then it has nodes at the triangle
corners and edge centers.

For 3-D problems, an element is a tetrahedron with either four or ten points. A four-point (linear) tetrahedron has nodes only at its corners. A ten-point (quadratic) tetrahedron has nodes at its corners and at the center point of each edge.

For details, see Mesh Data.

## Algorithms

For each point where a solution is requested (`pOut`

), there are two
steps in the interpolation process. First, the *element* containing
the point must be located and second, interpolation within that element must be
performed using the element shape functions and the values of the solution at the
element’s node points.

## Version History

**Introduced in R2014b**

## MATLAB 명령

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