Physical decomposition of thin plate bending problems and their solution by mesh-free methods
We develop and compare a number of mesh-free formulations for solution of plate bending problems using either the Moving Least Square-approximation or Point Interpolation Method-approximation. For solution of the original biharmonic problem, we develop only the local weak formulation to avoid the 4th order derivatives of deflections. The decomposition of the biharmonic operator leads to lower order derivatives of field variables in both the developed strong and local weak formulations which are applicable to arbitrary boundary value problems for thin plate bending. The modified differentiation technique is proposed, in order to increase the accuracy of higher order derivatives of field variables. The accuracy, convergence of accuracy and computational efficiency are studied in simple boundary value problems for circular plate. The discussed methods give reasonable numerical results when applied to decomposed problem, while the methods applied to original biharmonic problem either fail or give unreliable results.
biharmonic operator accuracy computational efficiency poisson equation strong formulation local integral equations approximation of derivatives