**Case Study: Rapid Conversion of a Technical Paper to a Mathematical Model**

**Introduction**

A technical paper, in PDF form, containing an algorithm with over sixty linear, nonlinear and integral equations is converted to a mathematical model in TK Solver and subsequently linked to Excel and a web application. The math model was completed in one day and the links and refinements to the interface were completed on day two.

**The Model Building Process**

The technical paper was ESDU Data Item 85007, entitled “Contact Phenomena III: Calculation of Individual Stress Components in Concentrated Elastic Contacts Under Combined Normal and Tangential Loading.” It was made available in PDF form.

The limits of the integrals were determined from the solution of a sixth order polynomial. It was necessary to find all the roots and then use the largest real root in the upper limit of integration. TK Solver includes a root finder that was perfect for the task, based on Bairstow’s algorithm for finding real and complex roots of nth-order polynomials. A loop was added to select the best one for this application. TK’s unique capability to mix rule-based equation solving with calls to procedural subroutines was a time-saver.

**Interfacing with Excel**

The TK file was linked to an Excel spreadsheet, as shown below. There are several advantages to this. The underlying formulas can be hidden. The interface can be customized. The TK model can be used as a function within a larger spreadsheet application. The spreadsheet and model can be packaged together and shared with anyone.

**Interfacing with the Web**

The TK Solver model was also processed using a UTS tool called Galaxy, which automatically builds a browser interface for a web-based application. Again, the variables displayed in the browser interface are dynamically linked to the model.

**Odds and Ends**

There are a number of additional features of TK Solver that make it an excellent tool for such applications.

*Units management*is one TK’s strengths. In this application, unit conversions are available for all the inputs and outputs. You can globally swap between US and SI units or mix and match as you like. This feature carries through to the Excel and Web interfaces as well.

*Table and plot generation*is another asset. Any mathematical model built in TK Solver can be enhanced without any additional programming required. As shown, the effects of varying the friction angle are plotted and automatically updated as other variables change. Even the locations of the plot annotations dynamically change.

*Backsolving*is perhaps TK Solver’s most valuable feature. Backsolving allows users to input a desired value for one or more outputs and solve for the required inputs. One example for this model would be to input a stress value and have TK determine the required friction angle or contact dimension. Here is one such example. Note that fy, the component stress with the maximum absolute value, has been input and the contact length, a, has been solved.