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The paper presents meshless methods based on the mixed Meshless Local Petrov-Galerkin approach used for solving linear fourth-order differential equations. In all the methods presented here, the primary variable and its derivatives up to the third order are approximated separately. Three different mixed meshless methods are derived by different choices of test and trial functions and are verified using available analytical and reference solutions. The numerical performance of the presented algorithms is demonstrated by several representative numerical examples.

Meshless approaches enable discretizations of a computational model only by a set of nodes, which do not need to be connected to elements. This paper presents the meshless local Petrov-Galerkin method, which belongs to truly meshless approaches, as it does not require any kind of mesh or background cells for either interpolation or integration. Full displacement and mixed formulations are presented. The full displacement approach is used for the solution of a three-dimensional elasto-static problem, while the mixed approach is applied for the modeling of deformation responses of shell-like structures. The modeling of material discontinuities is performed by the mixed meshless local Petrov-Galerkin approach by employing the collocation method. The efficiency and accuracy of all the presented methods are tested and compared with finite element formulations in numerical examples. It is demonstrated that the meshless approaches may be considered an alternative to the well-known finite element method regarding certain problems.

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