One Line of Development of the Galerkin Projection Method in Problems of Stationary Solid Mechanics (Review)

• Published in: International applied mechanics Vol. 59; no. 1; pp. 1 - 58
• Main Authors: Grigorenko, Ya. M., Bespalova, O. I., Grigorenko, O. Ya
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.01.2023; Springer; Springer Nature B.V
• Subjects: Analysis; Applications of Mathematics; Boundary conditions; Continuity (mathematics); Differential equations; Elastic media; Elastic properties; Engineering; Galerkin method; Independent variables; Mathematical analysis; Mechanical properties; Mechanics; Ordinary differential equations; Physical properties; Real numbers; Solid mechanics; Stationary processes; stationary deformation; spatial bodies; linear and nonlinear boundary-value problems; Galerkin method; eigenvalue problems; three stages of development; shell elements
• ISSN: 1063-7095; 1573-8582
• DOI: 10.1007/s10778-023-01198-x

One of the line of development of the projective Galerkin method intended for efficient solution of problems with a large number of independent variables and the application of its modifications to the study of stationary processes in solid mechanics are considered. The subject of study is three successive stages of development of the method characterized by an expansion of the class of unknowns in the approximate expression: from the class of real numbers to the class of continuous functions (differentiable several times) of one of the variables (one-dimensional functions), and, finally, to a set of one-dimensional functions that include all variables. The structure of the projection of the initial multidimensional problem changes to fit, respectively, a system of algebraic equations, a system of ordinary differential equations, and, finally, a system of one-dimensional problems for various domain variables. According to the development of the Galerkin method, the classes of problems of stationary solid mechanics are also expanded in terms of eliminating certain constraints on the shape of the object, boundary conditions, physical and mechanical properties of the elastic medium, and others. The application of Galerkin methods at different stages of its development is demonstrated by solving two-dimensional and three-dimensional problems of statics and vibrations of a wide class of elements of shells and solids.