To calculate the stress state of elastic three-dimensional plates and beams under static loading a multigrid finite element method implemented on the basis of algorithms of finite element method (FEM), using three-dimensional multigrid finite ele-ments (MFE) of heterogeneous structure has been provided. The differences of MFE from currently available finite elements (FE) are as follows. When building - grid FE of nested grids is used. The fine grid generates a partition taking into ac-count inhomogeneous structure and shape of MFE, the other large grids are applied to reduce MFE dimensionality, with MFE dimension decreas-ing when is increasing. The peculiarities and advantages of MFE are to develop MFE, arbitrarily small basic partitions of composite plates and beams containing the 1st order single-grid FE can be used, i.e. in fact, the finite element micro ap-proach is applied. These partitions allow one to take into account in MFE the complex heterogene-ous and microscopically inhomogeneous structure, shape and complex loading and fixing nature and to describe the stress and stain state by the equa-tions of three-dimensional elastic theory without any additional simplifying hypotheses. The essence of MFE is as follows. At a basic partition (on the fine grid) of - grid FE, , the total potential energy as a function of many variables depend-ing on the fine grid nodal displacements has been determined. On the other coarse grids (en-closed in the fine one), the displacement functions used to reduce the dimension of the function that allows one developing MFE of small dimension are found by FEM. The procedures of developing MFE of rectangular parallelepiped of plate and beam types are given. The advantages of MFE are: they produce small dimensional discrete models and high accuracy numerical solutions. An example of calculating the laminated plate, using three-dimensional 3-grid FE and the reference discrete model are given, with that having 623 millions of FEM nodal unknowns.
elasticity, composites, plates, beams, multigrid finite elements, micro-approach, high accuracy
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