Fields > Table fields

Table field interpolation and extrapolation options

In the Table Field dialog box, on the Table Options page, you can specify the following table field options:

The interpolation method and, if applicable, the extrapolation methodYou can specify an extrapolation method for independent domains that consist of a single variable and are linearly interpolated from a Cartesian, log, or semi-log representation of the tabular data.
When the software creates the interpolator
Whether the interpolator is persistent
Whether to create a default interpolator when the tabular data is incompatible with the selected interpolation method
Whether to shift and scale the independent domainYou can shift and scale independent domains that consist of a single variable and are linearly interpolated from a Cartesian representation of the tabular data.

Interpolation methods

Pre/Post or the solver uses the interpolation method to evaluate the dependent domain at independent domains for which a tabular data point does not exist. The value of the independent domain at which the evaluation occurs is referred to as the lookup point. The value of the dependent domain that the interpolation returns at a lookup point is referred to as the lookup value.

Note:

In this topic, the lookup value is notated as f(x), where x represents the lookup point. The lookup point may represent one or more spatial variables, non-spatial variables, or a combination of spatial variables and a non-spatial variable. The ith tabular data point is notated as (xi,fi), where fi is the value of the dependent domain and xi is the value of the independent domain.

Whether Pre/Post performs the interpolation or the solver performs the interpolation depends on the table field application. The following examples provide insight into which piece of software performs the interpolation.

Suppose that a table field is used to represent a spatially-varying pressure. Pre/Post interpolates the tabular data to obtain the pressure at the applicable nodal locations. These are the values that Pre/Post writes to the input file for the solver.
Suppose that a table field is used to represent how the thickness of a shell mesh varies spatially. Pre/Post interpolates the tabular data to obtain the thickness at the applicable nodal locations. These are the values that Pre/Post writes to the input file for the solver.
Suppose that a table field is used to represent a strain-dependent material property. Pre/Post writes the tabular data directly to the input file for the solver. During the solve, the solver interpolates the tabular data to obtain the value for the material property at the applicable strain.Note: For this case, the interpolation method that the solver uses may not be the same as the interpolation method that you select in Pre/Post.

The interpolation methods that you can select from depend on the number of variables in the independent domain. The following table lists the interpolation methods available for each type of independent domain.

Interpolation method	Single variable independent domain	Two variable independent domain(1)	Three variable independent domain(1)	Four variable independent domain(1)
Linear	X
Akima	X
Akima72	X
Cubic	X
Nearest Neighbor	X	X	X	X
Approximate Nearest Neighbor		X	X	X
Inverse Distance Weighting	X	X	X	X
Delaunay – Fast		X	X
Delaunay – Medium		X	X
Delaunay – Accurate		X	X
Renka's Modified Shepard		X	X	X
(1) When the independent domain of the tabular data contains dissimilar variables such as strain and temperature, or spatial variables and a non-spatial variable, and the tabular data is not lattice data, use a table of fields rather than a table field to minimize interpolation error. For more information, see Table of fields.

Interpolator creation

When you create a table field, you specify the independent and dependent domains, the interpolation method and options, and the tabular data. The software uses this information to create a mathematical representation of the table field that is referred to as the interpolator. Because the creation of an interpolator may be computationally intensive, you can control when the software creates the interpolator.

To create the interpolator when you create the table field, select the Create Interpolator on Apply/OK check box. The software creates the interpolator when you click Apply or OK to close the Table Field dialog box.
To retain the interpolation settings and tabular data to create the interpolator, but not create the interpolator until the table field is plotted, displayed, used in an application such as a load or constraint, or created using the Create Table Interpolator command, clear the Create Interpolator on Apply/OK check box.

Note:

No Interpolator is displayed in the Status column of a table field in the Simulation Navigator when the interpolator for the field has not been created.

Persistent interpolator

When you create a table field, you can select the Persistent Interpolator check box. When you do so, after the software creates the interpolator for the table field, it saves the interpolator with the simulation. Thus, when you reopen the simulation, the software does not need to recreate the interpolator. Depending on the interpolation method and amount of tabular data, this option may improve system performance.

For example, when you use the Delaunay interpolation methods, the software creates a Delaunay triangulation/tetrahedralization of the independent domain from the tabular data. If the Persistent Interpolator check box is selected, the software does not need to recreate the Delaunay triangulation/tetrahedralization during subsequent sessions. It only recreates the Delaunay triangulation/tetrahedralization when you make changes to the tabular data.

Default interpolator

When you create a table field, you can select the Fallback to Default Interpolator check box. When you do so, the software overrides the interpolation method when tabular data is incompatible with the chosen method and instead uses the nearest neighbor interpolation method. Incompatible tabular data can result from too few tabular data points or issues regarding the quality of the tabular data.

For example, because a Delaunay triangularization requires at least three tabular data points, if you select a Delaunay method for a 2D independent domain and specify only two tabular data points, the software cannot create a Delaunay triangularization. If the Fallback to Default Interpolator check box is selected, the software detects this incompatibility and overrides the Delaunay method with the nearest neighbor method.

For the same situation, if you clear the Fallback to Default Interpolator check box, the software does not create an interpolator and displays an error message in the Information window.

Interpolation accuracy

As a general rule, the finer the resolution of the tabular data, the more accurate the interpolated result. To check the accuracy of the interpolation, consider using one of the following methods:

If the table field has a spatial independent domain, you can display it by selecting the check box for the field in the Simulation Navigator. You can then use the commands contained in the Field Display Properties dialog box to refine the display of the field.For information on displaying fields, see Displaying fields.
If the table field defines how a boundary condition varies spatially, you can create a contour plot of the boundary condition and use the Identify command to check the value of the boundary condition at a variety of locations.For information on creating a contour plot of a boundary condition, see Contour plots of boundary conditions.
You can plot the field.For information on plotting fields, see Graphing fields.
You can create an expression from the table field. You can then evaluate the expression at specific values of the independent domain.For information on creating an expression from a field, see Fields in expressions, Field function, and Mapped field function.For information on evaluating an expression at specified values of the independent domain, see Test an expression.

Linear interpolation

Note:

Simcenter Nastran supports the linear interpolation and extrapolation options that are discussed in this section. Other solvers may not.

When you select Linear from the Interpolation list, from the Algorithm list, you can optionally select that the abscissa and ordinate be logarithmic. The following table lists the formula that the software uses to calculate the lookup value for each option.

Algorithm option	Abscissa (X-axis)	Ordinate (Y-axis)	Table lookup value, f(x)	Applicability
Linear Linear	Linear	Linear	\left( {\frac{{{x_j} - x}}{{{x_j} - {x_i}}}} \right){f_i} + \left( {\frac{{x - {x_i}}}{{{x_j} - {x_i}}}} \right){f_j}	The best selection for arbitrarily shaped curves that you represent with many tabular data points or curves of the form f(x) = mx + b that plot as a straight line on a Cartesian plot and can be represented with as few as two tabular data points.
Log Log	Logarithmic	Logarithmic	{e^{\left( {\frac{{\ln {x_j} - \ln x}}{{\ln {x_j} - \ln {x_i}}}\ln {f_i} + \frac{{\ln x - \ln {x_i}}}{{\ln {x_j} - \ln {x_i}}}\ln {f_j}} \right)}}	The best selection for curves of the form f(x) = bxm that plot as a straight line on a log-log plot and can be represented with as few as two tabular data points.
Log Linear	Logarithmic	Linear	\left( {\frac{{\ln {x_j} - \ln x}}{{\ln {x_j} - \ln {x_i}}}} \right){f_i} + \left( {\frac{{\ln x - \ln {x_i}}}{{\ln {x_j} - \ln {x_i}}}} \right){f_j}
Linear Log	Linear	Logarithmic	{e^{\left( {\frac{{{x_j} - x}}{{{x_j} - {x_i}}}\ln {f_i} + \frac{{x - {x_i}}}{{{x_j} - {x_i}}}\ln {f_j}} \right)}}	The best selection for curves of the form f(x) = bemx that plot as a straight line on a semi-log plot (where x is plotted linearly and f(x) is plotted logarithmically) and can be represented with as few as two tabular data points.
For interpolation, (xi,fi) and (xj,fj) are the two bounding tabular data points nearest to x. For extrapolation, (xi,fi) and (xj,fj) are either the two starting tabular data points or the two ending tabular data points nearest to x.

The option you select from the Values outside Table list determines how the software calculates the table lookup value for the table field when x lies outside the range of the tabular data.

If you select Undefined, the table field is not assigned a value.
If you select Extrapolate, the value of the table field at values of x less than that defined by the tabular data is extrapolated from the two starting data points. The value of the table field at values of x greater than that defined by the tabular data is extrapolated from the two ending data points.The formula that the software uses to extrapolate the tabular data is the same as the formula the software uses to interpolate the tabular data for the option you select from the Algorithm list, except that the (xi,fi) and (xj,fj) are either the two starting tabular data points or the two ending tabular data points.
If you select Constant, the software uses the values of the table field at the starting data point and the ending data point for the table lookup.For example, suppose (100,5) and (200,15) are the starting and ending data points, respectively. For values of x less than 100, the table lookup is 5. For values of x greater than 200, the table lookup is 15.
If you select User Defined, the software uses the value that you enter in the Independent Value < Minimum box for values of x less than that defined by the tabular data, and the value that you enter in the Independent Value > Maximum box for values of x greater than that defined by the tabular data.

When you select Linear Linear from the Algorithm list, you can optionally shift and scale the tabular data.

To use the tabular data directly, clear the Independent Value Shift X1 and Independent Value Divisor X2 check boxes. The returned lookup value depends on the table field application.For defining temperature-dependent structural material properties like elastic modulus, Poisson's ratio, and so on, the lookup returns f(x).For defining temperature-dependent thermal material properties like thermal conductivity, heat capacity, and so on, the lookup returns zf(x), where z is the nominal value for the material property as listed on the corresponding material entry in the solver input file.For all other applications, the lookup returns f(x).
To shift the tabular data, select the Independent Value Shift X1 check box and enter a value for the X1 parameter in the Independent Value Shift X1 box. When the solver needs a value at x, the table lookup is at (x – X1). The returned lookup value depends on the table field application.For defining temperature-dependent structural and thermal material properties, the lookup returns zf(x – X1*), where z is the nominal value for the material property as listed on the corresponding material entry in the solver input file.For all other applications, the lookup returns f(x – X1)*.
To scale the tabular data, select the Independent Value Divisor X2 check box and enter a value for the X2 parameter in the Independent Value Divisor X2 box. When the solver needs a value at x, the table lookup is at (x/X2).For defining temperature-dependent structural and thermal material properties, the lookup returns zf(x/X2*), where z is the nominal value for the material property as listed on the corresponding material entry in the solver input file.For all other applications, the lookup returns f(x/X2)*.
To both shift and scale the tabular data, select both check boxes and enter values for the X1 and X2 parameters in the Independent Value Shift X1 and Independent Value Divisor X2 boxes. When the solver needs a value at x, the table lookup is at ((x – X1)/X2).For defining temperature-dependent structural and thermal material properties, the lookup returns zf((x – X1*)/X2), where z is the nominal value for the material property as listed on the corresponding material entry in the solver input file.For all other applications, the lookup returns f((x – X1)/X2)*.

Akima interpolation

The software supports the original Akima method (Akima 72) and a newer version of the original Akima method. For many applications, both methods produce nearly identical results.

When you select an Akima interpolation method from the Interpolation list, the software creates a smooth curve that has the following properties:

The curve passes through each tabular data point.
The curve is a piecewise-cubic polynomial. That is, a cubic polynomial is defined for each interval between a pair of adjacent data points.
The curve and the slope of the curve are continuous over the entire range of tabular data.
The curve minimizes overshoot. That is, the maximum or minimum value predicted by the curve is close to the maximum or minimum value of the dependent variable in the tabular data.
Localized perturbations to the tabular data are not propagated throughout the curve. That is, changes in the tabular data alter the shape of the curve only near the locations of the changes.

To obtain the value at the lookup point, the software evaluates the cubic polynomial that corresponds to the two tabular data points that bound the lookup point.

When you select an Akima interpolation method, table lookup does not occur for values of x that lie outside the range of the tabular data.

Akima72 interpolation

To use the original Akima method, select Akima72 from the Interpolation list. When you do so, the software fits a cubic polynomial of the form:

f(x) = a + bx + c{x^2} + d{x^3}

through adjacent tabular data points so that continuity of slope is maintained.

For example, suppose that you use the original Akima interpolator to fit a piecewise-cubic curve through five data points: (x1,f1), (x2,f2), (x3,f3), (x4,f4), and (x5,f5), where x1 < x2 < x3 < x4 < x5.

The software must determine values for the constants in the following four cubic polynomials:

{f_{12}}(x) = {a_{12}} + {b_{12}}x + {c_{12}}{x^2} + {d_{12}}{x^3}

{f_{23}}(x) = {a_{23}} + {b_{23}}x + {c_{23}}{x^2} + {d_{23}}{x^3}

{f_{34}}(x) = {a_{34}} + {b_{34}}x + {c_{34}}{x^2} + {d_{34}}{x^3}

{f_{45}}(x) = {a_{45}} + {b_{45}}x + {c_{45}}{x^2} + {d_{45}}{x^3}

where the subscripts indicate the interval over which the polynomial is valid.

For this example, 16 constants must be determined. Thus, 16 independent conditions are required.

Eight conditions are obtained by requiring the polynomials to pass through the data points.

{f_{12}}({x_1}) = {f_1}	{f_{34}}({x_3}) = {f_3}
{f_{12}}({x_2}) = {f_2}	{f_{34}}({x_4}) = {f_4}
{f_{23}}({x_2}) = {f_2}	{f_{45}}({x_4}) = {f_4}
{f_{23}}({x_3}) = {f_3}	{f_{45}}({x_5}) = {f_5}

The remaining eight conditions are obtained by fitting the cubic polynomials to estimates of the slope at each data point.

{f'_{12}}({x_1}) = {s_1}	{f'_{34}}({x_3}) = {s_3}
{f'_{12}}({x_2}) = {s_2}	{f'_{34}}({x_4}) = {s_4}
{f'_{23}}({x_2}) = {s_2}	{f'_{45}}({x_4}) = {s_4}
{f'_{23}}({x_3}) = {s_3}	{f'_{45}}({x_5}) = {s_5}

where s1, s2, and so on are the estimated slopes at data points 1, 2, and so on.

The slope at point 3 is estimated as follows:

{s_3} = \frac{{\left| {{m_{45}} - {m_{34}}} \right|{m_{23}} + \left| {{m_{23}} - {m_{12}}} \right|{m_{34}}}}{{\left| {{m_{45}} - {m_{34}}} \right| + \left| {{m_{23}} - {m_{12}}} \right|}}

where m12 is the straight-line slope between points 1 and 2, m23 is the straight-line slope between points 2 and 3, and so on.

Note:

If the example included more data points, the slope estimates at the other interior points would be calculated similarly.

The slopes at the beginning two and ending two data points are estimated differently.

For information on how Akima72 estimates the slope at these points, see Reference 1.

Akima interpolation

To use the newer Akima method, select Akima from the Interpolation list. When you do so, the software fits a cubic polynomial through adjacent tabular data points, similar to Akima72. However, the procedure the software uses to estimate the slopes is different.

For more information on Akima, see Reference 2.

Cubic interpolation

When you select Cubic from the Interpolation list, the software creates a smooth curve that has the following properties:

The curve passes through each tabular data point.
The curve is a piecewise-cubic polynomial. That is, a cubic polynomial is defined for each interval between a pair of adjacent data points.
The curve, the slope of the curve, and the curvature of the curve are continuous over the entire range of tabular data.

However, unlike the Akima methods, which are also piecewise-cubic representations of the tabular data, the curve the software produces when you select the Cubic option is more prone to:

Overshooting.
Propagating localized perturbations throughout the entire range of tabular data.

When you select Cubic from the Interpolation list, the software fits a cubic polynomial of the form:

f(x) = a + bx + c{x^2} + d{x^3}

through adjacent tabular data points so that continuity of slope and curvature are maintained.

For example, suppose that you use the cubic interpolator to fit a piecewise-cubic curve through five data points: (x1,f1), (x2,f2), (x3,f3), (x4,f4), and (x5,f5), where x1 < x2 < x3 < x4 < x5.