Local integral formulations for thin plate bending problems

Abstract eng:
It is well known that high order derivatives of field variables in the governing equations give rise to difficulties in solution of boundary value problems because of worse accuracy of numerically evaluated high order derivatives. The order of the differential operator can be decreased by decomposing this operator into two lower order differential operators with introducing new field variable. In two recent decades, solution of many engineering problems as well as problems of mathematical physics have been reformulated by using various mesh free formulations with meshless approximations. In this paper, we present that the decomposition of the biharmonic equation into two Poisson equations is applicable to general case of boundary conditions and any shape of the boundary edge of the plate, if we use the Local Integral Equation (LIE) formulation and a meshless approximation for primary field variables. Besides the standard advantages of mesh free formulations remember the new advantage consisting in decreasing the order of the derivatives of field variables. Instead of the third order derivatives of the deflection field in the formulation for the biharmonic equation the highest order of the derivatives in the present formulation does not exceed the first order in cases of clamped and/or simply supported edges while the second order derivatives are required in the case of free edges on plates. Several illustrative examples will be presented for comparison of accuracy, convergence and computational efficiency achieved by using various approaches.

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