Interpolation

An function (

) is defined by a table or file containing the values of the function in discrete points. The interpolation data can be structured (defined on a grid) or unstructured (defined on a generic point cloud).


	The Analytic, Interpolation, and Piecewise functions can also be added to Materials. See Using Functions in Materials.

Definition

Select a — , , , or to define the interpolation function by entering values in a table, by importing interpolation data from a file or from a table under , or using another function respectively.


	The data source for the interpolation must contain real-valued data only.

•

If is selected, enter a and enter coordinates t and function values f(t) into the table cells. A function of one variable can be defined in this way. For functions of two or more variables, such as space-dependent data in 2D and 3D, use a file with the function data. The default is int1. You can move rows up and down in the table, insert a row above the selected rows, remove a row from the table, and clear the table using the buttons underneath the table or by right-clicking in the table.

Optional: Save the parameters to a text file to reuse in other models. Click the button (

) and enter a , including the extension .txt. Click to the text file. The information is saved in space-separated columns in the same order as displayed on screen.

Optional: Import or Load data from a spreadsheet program or other format by saving the file with a .txt extension. Data must be separated by tabs or colons. Click the button (

) and navigate to the text (.txt) file to load and click . If the license includes LiveLink™ for Excel® you can also load interpolation data from a Microsoft Excel Workbook spreadsheet. The data loads to the table. If there is other data already in the table, it adds it after the last row. Move or edit as needed. Trailing columns are removed.

•

If is selected to import interpolation data from a file, select a : , , or . If the license includes LiveLink™ for Excel®, you can also import interpolation data from a Microsoft® Excel Workbook spreadsheet. COMSOL Multiphysics then uses the spreadsheet format and the list is not available.

When the data format is specified, enter the complete network path and name of the interpolation data file in the field, or click to select a text or data file with interpolation data in the dialog box. You can also click the downward arrow beside the button and choose (

) to open the fullscreen window. Click the downward arrow for the menu (

) to choose (

) to move to the row for this file in the window, (

), and (if you have copied a file location) (

). Also choose a decimal separator from the list: (the default) or . Click (

) to import the interpolation data into the model; otherwise, COMSOL Multiphysics references the interpolation data on your file system. When you have imported the interpolation data, the section, under , contains information about the filename, data type, and dimension for the data. Click to save the interpolation data to a file and reference from there instead of including it in the model. Click the button to delete the imported interpolation data from the model. Click the button (

) to ensure that the file is reread when needed.

From the list select , , or . The spreadsheet format is the default format, and that format is the easiest to use for functions defined on an unstructured grid or for general tabulated function values with one or more arguments.

•

If is selected, interpolation data evaluated or imported into a table under or belonging to an Evaluation Group under is treated in the same way as a file using the spreadsheet format.

•

If is selected, enter a . From the list, choose a source function (all applicable functions in the model are available as source functions). The default name is the name of the selected source function. From the list, choose (the default), or choose to use a uniform sampling with the number of sampling points given in the field (default: 1000). The uniform sampling evaluates the function on a uniform grid with a set number of sampling points. The sampling is the same a a uniform sampling with 1000 sampling points. This data source makes it possible to reinterpolate existing functions. This functionality can be useful for creating primitive and inverse functions or for downsampling.

If the interpolation data is given on spreadsheet format and when using a result table, enter a (1–3). For all data formats and table sources, enter information about the functions into the table. Add a and its . The first function in the file has position 1, the following has position 2, and so on. For spreadsheet data, the first columns contain the arguments (typically spatial coordinates); the following columns can contain one or more functions, and the positions entered are the relative position for each function’s data column.

For unstructured interpolation data, COMSOL Multiphysics may internally apply a scaling of the coordinates of the data points to simplify the process of creating an interpolation mesh. Use the list to control this functionality. Select (the default) to apply the scaling if the bounding box of the interpolation points has a bad aspect ratio, to always apply the scaling, or to turn off scaling altogether. Changing the scaling strategy may affect the generated interpolation mesh. The list is only available when unstructured interpolation data can be entered (that is, when or is the data source).

For the common case where the data source contains function values that are functions of the spatial coordinates, select the check box. Then select the frame to which the spatial coordinates are attached from the list (the default is for the spatial frame). Then the function can be called without arguments when used in the model; the spatial coordinates are added as function arguments automatically. If you choose from the list, you can select the and, if selected, enter a shape function order (default: 1) in the field. Doing so makes it possible to evaluate variables using Lagrange interpolation with cached values evaluated at Lagrange points on the computational mesh when using an interpolation function to define spatially dependent variables through spatial coordinates as arguments to the interpolation function. The check box is available for nodes in a branch when the is or when using a in 1D models.


	The Interpolation functions support 1, 2, or 3 arguments. You cannot define functions with more than three (3) arguments because the algorithm creates a mesh for the point cloud, which is not possible in four dimensions or higher.

An Example of Importing a File Data Source into a Parameter Table

The file named temp.txt contains temperature measurements in nine points in the plane:

The data columns contain x-coordinates, y-coordinates, and temperature values, respectively. To use this file as an interpolation function called tempfun, perform the following steps.

Select from the list.

Enter a (the complete network path) or to locate a file to import.

From the list select .

Enter a . In this example, enter 2.

Enter the name tempfun.

Enter its as 1. The first function in the file has position 1, the following has position 2, and so on. The position in file for a function is the column after the spatial coordinates (or other function arguments) where it is defined. In this example with two arguments (spatial coordinates), the third column is Position 1 in the file.

If desired, adjust the interpolation and extrapolation settings in the section (see below).

Use the function tempfun with x and y as input arguments in a 2D model to get the interpolated value for the temperature at any position. If the check box is selected, use tempfun without adding the input arguments.

Examples of Spreadsheet, Sectionwise, and Grid File Formats

In all file formats, use a % (percent) sign to indicate a comment line. Values and indices can be separated by space, comma, semicolon, or tab characters.

Spreadsheet File

A spreadsheet file contains function data for space-dependent functions or general input variables and function values for functions of one or more variables (that is, the functions do not need to be functions of space coordinates but can also be functions of something else, such as temperature).

%Header (optional)

Columns containing x, y (optional), and z (optional), or any other arguments, followed by function data columns.

The number of input variable data columns can be 1, 2, or 3 with no limit on the number of subsequent function data columns.

Sectionwise File

A sectionwise file has coordinates and function values. The sectionwise format can be used for exporting results and for importing numerical data defined on an unstructured first-order tetrahedral mesh (triangular mesh in 2D) as interpolation table data. It is also possible to use this format in 1D and can then, contrary to the spreadsheet format, be used for 1D geometries with disconnected intervals. It cannot be used directly for defining a geometry or mesh. A sectionwise file contains unstructured mesh and function data for space-dependent functions. The functions need not be functions of space coordinates (x, y, and z) but can also be functions of some other quantities, such as deformation or temperature. The row indices in the format below are 1-based indices.

%Coordinates

One to three columns containing x, y (optional), and z (optional) coordinate values

or other input variable values

%Elements

Triangulation where each row contains the row indices of the points in the Coordinates

section for one element(edge, triangular, or tetrahedral)
%Data (funname)

Column of data values for each point

It is possible to include more than one function in the file as long as a %Data header separates them one from the other.

Grid File

A grid file contains data values on a grid. You supply the grid points as x coordinate values (in 1D), x and y coordinate values in 2D, and x, y, and z coordinate values in 3D.

%Grid

x grid points separated by spaces, commas, semicolons, or tab characters

y grid points separated by spaces,commas, semicolons, or tab characters (optional)

z grid points separated by spaces,commas, semicolons, or tab characters (optional)

%Data

Data values separated by spaces, commas, semicolons, or tab characters

The grid coordinate section lists the unique coordinate values. Each data row contains values for different x grid points for fixed values of y and z (in 3D). The rows first increase the y grid value and then the z grid value. The grid points can also represent another independent variable that the data values depend on. For example, the “grid points” can be temperature values and the data values the thermal conductivity at these temperatures.

It is possible to include more than one function in the file as long as a %Data header separates them one from the other. For grid points that are outside of the computational mesh, the data values will be NaN.


	It is important to use a comment line starting with % to separate the grid points or other interpolation points and the data values that are associated with these coordinates or interpolation points.

Interpolation and Extrapolation

The interpolation and extrapolation settings control how the program evaluates the function between the discrete points where it is defined by the table or file, and the behavior of the function outside the domain where it is defined by the table or file.

Select an method:

•

For functions of one variable select , (the default interpolation method), , or .

interpolation selects the value of the nearest point where the function is defined.

uses a linear polynomial to interpolate the function between the points where it is defined.

interpolation uses a piecewise-cubic Hermite polynomial with continuous first derivatives. It preserves the shape of the data and respects monotonicity.

The method also performs interpolation with a piecewise cubic polynomial. Here, even second derivatives are continuous; however, the method does not necessarily respect monotonicity.

•

For functions of more than one variable, select or . The other options are not supported.

Select an method to specify how to treat arguments outside the grid or mesh of points.

•

. No extrapolation outside the defined interpolation range.

•

. Uses the value from the closest point inside the grid (for structured interpolation) or the value from the closest mesh element (for unstructured interpolation). The function evaluates the polynomial from the closest grid point at the actual point where a value is requested. This is the default extrapolation method.


	For interpolation functions with three arguments, the Constant method will not reliably find the closest mesh element. It will find some point on the boundary of the interpolation mesh.

•

. The function is linear outside the grid with a continuous function value and continuous first derivative at the boundary of the grid. or must be selected from the list.

•

. Evaluates the polynomial from the closest grid point at the actual point where a value is requested.

•

. Uses a single value, such as zero or NaN (Not-a-Number), everywhere outside the grid or mesh. Enter the value in the field.


	Unstructured extrapolation supports a constant or a specific value only.

Units

In this section you define the units of the function (the output) and the arguments (the inputs) in the and table, respectively.

Plot Parameters

If you have selected to define the inverse function (see below), this section becomes available. Select the check box to plot the inverse function instead of the interpolation function itself when you click the button (

) to generate a preview plot.

A plot group for an interpolation function contains three connected line graphs: a line graph of the piecewise function and two line graphs using dashed red lines, Left Extrapolation and Right Extrapolation, which show how the selected extrapolation extends the interpolation function on both sides. In addition, a point graph shows the interpolation points (interpolation nodes).

Related Functions

In this section, you can define the following related functions:

Inverse Function

The inverse function is available for interpolation functions defined by a local table only. If you want to define the inverse function f −1 for the interpolation function f, select the check box and enter a function name for the inverse function in the field (the default is the name of the interpolation function with a suffix _inv). If you want to plot the inverse function instead of the interpolation function itself, first select the check box in the sections, which is only available when you have chosen to define the inverse function.


	The inverse function only exists if the function is strictly monotonic.

Primitive Function

If you want to define the primitive function F for the interpolation function f, select the check box and enter a function name for the primitive function in the field (the default is the name of the interpolation function with a suffix _prim). You can use the primitive function to sample from the interpolation function. The primitive function is the integral of the interpolation function. The integration constant is such that the value of the primitive function is zero at the first data point. In other words, if the interpolation function is f(x), then the primitive function is

where x0 is the x-coordinate of the first data point.

Random Function

To define a random function, select the check box with a name that you specify in the field (the default for an interpolation function int1 is rn_int1). You can use this function to sample from the interpolation function. Random functions can be useful in particle tracing, for example. Suppose, for example, that you want to specify an arbitrary energy distribution function for the initial energy of a bunch of particles. In order to do this, you need to randomly sample from an interpolation function, not merely call it. The random function makes this possible. To use a random function for this purpose, select the check box, and then call it in an Inlet feature, for example, using rn_int1(cpt.pidx), where cpt.pidx is a variable that is uniquely valued for each particle.

The default number of arguments is 1. If you want to use additional arguments, type a number in the field.

You specify the range of the random function using the list. The default is . Choose to provide limits in the and fields.


	See Common Settings for the Function Nodes for information about the Units section.

If you have the:

•

Acoustics Module, see Muffler with Perforates: Application Library path .

•

CFD Module, see Transonic Flow in a Sajben Diffuser: Application Library path .

•

Corrosion Module, see Cathodic Protection of Steel in Reinforced Concrete: Application Library path .

•

Heat Transfer Module, see Temperature Field in a Cooling Flange: Application Library path .

•

Pipe Flow Module, see Geothermal Heating from a Pond Loop: Application Library path .