Introduction
In this article, the finite element method (FEM) is discussed, which has been commonly used to solve the problems of geometrically nonlinear mechanics. This approach is primarily used to solve elastic-plastic problems, stress-strain analysis, as well as contact and friction analysis.
Discussion
The finite element method is a discreteapproach for solving variational problems. This method involves the division of the domain into a finite number of elements, each of which is described in terms of its geometry, material properties and loading conditions. The governing equations of the problem are then discretized to linear systems of equations. These equations are then solved for the unknowns of interest.
The most common type of finite element method is the displacement-based FEM. In this approach, a finite element model is defined by its node points, which can be seen as a collection of variables, each representing the displacement of a point. The values of the displacement vector can then be used to approximate the solution of the underlying problem. This makes the method suitable for the approximation of nonlinear problems, as the displacement vector can be updated at each iteration of the solution algorithm.
In order to solve a geometrically nonlinear problem, the finite element method must incorporate an iterative algorithm that is capable of taking into account the stiffness of the elements. The displacement-based FEM utilizes an incremental-iterative approach, in which the stiffness of the elements is continually updated as the load increases. The load is then increased incrementally and the solution procedure is repeated until the solution converges.
The finite element method can be used to solve a variety of engineering problems, including but not limited to those involving nonlinearities, large deformations, contact and friction, plasticity, and fracture mechanics. In addition, the method can also be used for geometrical nonlinear optimization problems.
In order to solve a given problem, the finite element method must be properly formulated. This requires the selection of an appropriate finite element formulation, as well as an appropriate numerical solution strategy. In addition, the finite element model requires careful consideration of boundary conditions, material properties, and loading conditions.
Conclusion
The finite element method has proven to be an effective method for solving a variety of geometrically nonlinear mechanics problems. The method is capable of accurately approximating the solution of the underlying problem, and can be used to solve a variety of engineering problems. In order to properly utilize the method, a formulation must first be chosen and proper consideration of the boundary conditions, material properties, and loading conditions must be considered.
