Finite difference analysis of shell structures
C. LI (TU Delft - Civil Engineering & Geosciences)
Pierre C.J. Hoogenboom – Mentor (TU Delft - Applied Mechanics)
M. A.N. Hendriks – Mentor (TU Delft - Concrete Structures)
Cor Kasbergen – Mentor (TU Delft - Pavement Engineering)
F.P. van der Meer – Mentor (TU Delft - Applied Mechanics)
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Abstract
A new approach to applying the finite difference method for solving the shell model problem. By solving the general shell differential equations (Sanders-Koiter equations) with first-order finite-difference approximation only, a finite difference algorithm called shell code has been developed.
To this end, a 1200-line Python program has been built. In the process many versions of shell code were considered, including
1) Two programming languages (Python and R)
2) Three interpolations for approximating gradients (three-point and five-point with two end slopes)
3) Determined and over-determined systems of equations (square and rectangular matrices)
4) Four solvers for the systems of equations
The final version of shell code has the following features; five-point interpolation with zero end slope, rectangular matrix, solver lm.fit.sparse (R). Approximately 80% of the shell code results match the finite element results with a deviation less than 5%. It has been proved that the selected version of shell code can solve shell model problems by solving Sanders-Koiter equations with finite difference method. Many previously assumed important factor for affecting shell code results, like number of nodes and interpolation methods, were actually less significant. However, the most vital difference in results occurred under different computation methods. Maybe under further research of developing mathematic tools, finite difference method could be a more promising and practical method for solving model problems.