Abstract
Abstract The goal of this paper is to present various types of iterative solvers: gradient iteration, Newton’s method and a quasi-Newton method, for the finite element solution of elliptic problems arising in Gao type beam models (a geometrical type of nonlinearity, with respect to the Euler–Bernoulli hypothesis). Robust behaviour, i.e., convergence independently of the mesh parameters, is proved for these methods, and they are also tested with numerical experiments.
Highlights
The numerical study of the deformation of thin beams and plates is a widespread problem in elasticity theory and engineering practice since such elastic structures regularly appear in several real applications; see, e.g., [2, 12, 16,17,18]
The goal of this paper is to present various types of iterative solvers: gradient iteration, Newton’s method and a quasi-Newton method, for the finite element solution of elliptic problems arising in Gao type beam models
The goal of this paper is to present various types of such iterative solvers in the setting of the finite element method (FEM) and, in particular, to show the robust behaviour of these methods, i.e., convergence independently of the mesh parameters
Summary
The numerical study of the deformation of thin beams and plates is a widespread problem in elasticity theory and engineering practice since such elastic structures regularly appear in several real applications; see, e.g., [2, 12, 16,17,18] These models generally lead to fourth-order equations. The goal of this paper is to present various types of such iterative solvers in the setting of the finite element method (FEM) and, in particular, to show the robust behaviour of these methods, i.e., convergence independently of the mesh parameters The presentation of these methods, based on a Hilbert space framework, includes the proper forms of the Sobolev gradient iteration and of Newton’s method adapted to the given beam problem.
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