Smoothing out the bumps in curve fitting

There are many design exploration applications where it is important for performance results to match a certain range of values, whether it be from experimental sources or ideal goals. For example, curves for engine torque vs rpm, bushing deflection vs load, or wing lift vs the angle of attack. Quite often though, the baseline curve data can include fluctuations which makes curve fitting more challenging. There can also be portions of the curve where it is far more important that there be a close fit.

Figure 1. Sample Baseline Curves

Figure 1. Sample Baseline Curves

To tackle these challenges and to also streamline the curve creation, HEEDS 2016.04 contains additional curve tools to ensure better results alignment. There are now added abilities to:

  • Weight curve ranges
  • Normalize RMS values
  • Simplify imported curve data selection

Let’s review these capabilities in detail to show how they can help. 

Weight Factors

A new weighting column has been added to source curve data that allows the user to define importance of different portions of the curve.

The RMS error between the calculated curve and the reference curve is determined by:

CurveFitting-equation

Where:

  • N is the number of points on the curve
  • Yi is the Y value on the curve for point i
  • yi is the Y value on the reference curve corresponding to design point i; the value is calculated using linear interpolation at the x value corresponding to the design point i
  • Fi is the weight factor specified on the reference curve corresponding to the design point i

As indicated in the equations above, the weight factor multiplies the error at each design point. By using a higher weight factor value, the point associated with that weight factor will have a larger contribution to the total RMS value calculated.

The weighting can be uniform, continuously varying, or for specific curve segments. This allows great flexibility in designating the critical portions of the curve data or to reduce the influence of unimportant sections.

Figure 2. Weighting identifies importance of curve regions for RMS calculation.

Figure 2. Weighting identifies importance of curve regions for RMS calculation.

In Figure 2. Above, the weighting is active only for regions of the curve without the spurious experimental data. This improves the curve fitting result as shown below:

Figure3. Improvement in curve fitting through weighting

Figure3. Improvement in curve fitting through weighting

In the first plot of Figure 3, there is no weighting and the non-uniform sections of the reference curve lead to a poor fit. In the second plot the high weighting for just to the smooth sections of the curve lead to an excellent match.

Normalize RMS Error

A new option for curve interpolation is to normalize the RMS error calculation difference.

Figure 4. RMS error normalization setting in HEEDS

Figure 4. RMS error normalization setting in HEEDS

When the range of the reference curve spans orders of magnitude, such as in Figure 5, this option allows errors at low values to be treated with similar importance as errors at high values.

Figure 5.  Logarithmic relationship of Drag on a Sphere to Reynolds number

Figure 5. Logarithmic relationship of drag on a sphere to Reynolds number

Simplify imported curve data selection

When importing reference curves from files, there is a simpler process to map column sources. Each column in the file now has a heading which can be assigned to the X, Y, or WEIGHT components of the reference curve.

Figure 6. Imported Reference Curve Column Assignment

Figure 6. Imported Reference Curve Column Assignment

If the external file has a heading line with X, Y or Weight specified, these are automatically selected by default. This greatly simplifies the import process or the need to modify external files to contain just the necessary curve data.

Summary

HEEDS 2016.04 make tackling curve fitting applications easier than ever before and with greater flexibility to suit your needs. This is just one of many great enhancements in 2016.04. If you have not already done so, I highly recommend contacting your HEEDS account manager to find out about the other ways HEEDS 2016 helps you to discover better designs, faster.

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