When the Old Meets the New: Examples of What Established Analytical Methods Look Like in a Modern Computer Environment
This paper was produced for the 2019 NAFEMS World Congress in Quebec Canada
There is a vast number of analytical solutions of Ordinary and Partial Differential Equations (ODE and PDE) available in engineering literature, books, journals and teaching material, starting from the definition of the Partial Differential Equation (PDE) for thin plates formulated by Lagrange in 1811,/1/, see figure 1, the mathematical breakthroughs by Augustin-Louis Cauchy in early 1800s, /2/, and Claude-Louis Navier, /3/, using double Fourier series to solve the problem of a simply supported plate with different types of loads in 1820. The focus in this paper is the analytical solution of rectangular plates.
To solve these PDEs by hand calculations were the norm, limiting the practical use of these mathematical findings significantly. Today’s engineers and designers working in product development have vast computer resources available to them to implement these PDEs for better understanding of the behaviour of rectangular plates. The introduction of Formulation, Validation and Verification in product development has actualised the analytical solutions, as numerical solutions computed using FEA technology must be compared against “exact solutions” for verification.
It is pertinent to ask the question: “What would the forefathers in Classic Solid Mechanics have done if they had our computer resources available to them?” A number of examples are made to show what effective use of state-of-the-art computing can do to revive the classical methods.