Multigrid finite element method in stress analysis of three-dimensional elastic bodies of heterogeneous structure

11th International Conference on "Mesh methods for boundary-value problems and applications"

Matveev, A.D.^1

Institute of Computational Modeling SB RAS, Krasnoyarsk, Akademgorodok

660036, Russia^1

关键词: Composite solids;    Discrete modeling;    Discrete models;    Heterogeneous structures;    Inhomogeneous structure;    Laminated plate;    Numerical solution;    Rectangular parallelepipeds;   

Others  :  https://iopscience.iop.org/article/10.1088/1757-899X/158/1/012067/pdf
DOI  :  10.1088/1757-899X/158/1/012067

来源: IOP

【 摘 要 】

To calculate the three-dimensional elastic body of heterogeneous structure under static loading, a method of multigrid finite element is provided, when implemented on the basis of algorithms of finite element method (FEM), using homogeneous and composite threedimensional multigrid finite elements (MFE). Peculiarities and differences of MFE from the currently available finite elements (FE) are to develop composite MFE (without increasing their dimensions), arbitrarily small basic partition of composite solids consisting of single-grid homogeneous FE of the first order can be used, i.e. in fact, to use micro approach in finite element form. These small partitions allow one to take into account in MFE, i.e. in the basic discrete models of composite solids, complex heterogeneous and microscopically inhomogeneous structure, shape, the complex nature of the loading and fixation and describe arbitrarily closely the stress and stain state by the equations of three-dimensional elastic theory without any additional simplifying hypotheses. When building the m grid FE, m of nested grids is used. The fine grid is generated by a basic partition of MFE, the other m - 1 large grids are applied to reduce MFE dimensionality, when m is increased, MFE dimensionality becomes smaller. The procedures of developing MFE of rectangular parallelepiped, irregular shape, plate and beam types are given. MFE generate the small dimensional discrete models and numerical solutions with a high accuracy. An example of calculating the laminated plate, using three-dimensional 3-grid FE and the reference discrete model is given, with that having 2.2 milliards of FEM nodal unknowns.