PSBFEM-Abaqus: Development of User Element Subroutine (UEL) for Polygonal Scaled Boundary Finite Element Method in Abaqus

The polygonal scaled boundary finite element method (PSBFEM) is a novel method integrating the standard scaled boundary finite element method (SBFEM) and the polygonal mesh technique. This work discusses developing a PSBFEM framework within the commercial finite element software Abaqus. The PSBFEM is implemented by the User Element Subroutine (UEL) feature of the software. The details on the main procedures to interact with Abaqus, defining the UEL element, and solving the stiffness matrix by the eigenvalue decomposition are present. Moreover, we also develop the preprocessing module and the postprocessing module using the Python script to generate meshes automatically and visualize results. Several benchmark problems from two-dimensional linear elastostatics are solved to validate the proposed implementation. The results show that PSBFEM-UEL has significantly better than FEM convergence and accuracy rate with mesh refinement. The implementation of PSBFEM-UEL can conveniently use arbitrary polygon elements by the polygon/quadtree discretizations in the Abaqus. The developed UEL and the associated input files can be downloaded from https://github.com/hhupde/PSBFEM-Abaqus.

Linear elastostatics

In this chapter the basic field equations in terms of displacement, strain, and stress, and typical boundary conditions which are necessary to formulate a complete three-dimensional boundary-value-problem for linear elasticity in the static case (i.e., neglect of inertial terms) are stated. The uniqueness of a solution to such a boundary value problem is discussed. There are a number of alternate ways that one can approach the statement of an elastostatic boundary-value-problem. The first major approach is obtained by reducing the set of field equations by expressing them solely in terms of the displacement field --- the Navier equations --- while in the second major approach the general system of equations may be reformulated by eliminating the displacement and strain fields and casting the system solely in terms of the stress field --- the Beltrami-Michell equations. The special formulation of idealized two-dimensional boundary value problems is presented the two basic theories of plane strain and plane stress for isotropic materials are discussed.

Introduction to the finite element method for linear elastostatics

This chapter introduces the widely-used finite element method applied to solving two-dimensional boundary value problems in linear elastostatics under plane strain or plane stress conditions. While the chapter illustrates the main structure of the finite element method using the equations of linear elasticity, the method can also be applied to a wide variety of other problems in science and engineering.

Solutions to some classical problems in linear elastostatics

This chapter presents and discusses the solution of several classical problems in linear elastostatics, including thick-walled spheres and cylinders under external and internal pressure; bending and torsion of prismatic bars of arbitrary cross section; and the use of Airy’s stress function method to solve several two-dimensional plane strain and plane stress traction boundary value problems, including a demonstration of the extent of the Saint-Venant effect. The discussion also includes an analysis of the asymptotic stress and deformation fields near the tips of sharp cracks, and a discussion of stress intensity factors which are of importance in linear elastic fracture mechanics.

On the Asymptotic Behavior of D-Solutions to the Displacement Problem of Linear Elastostatics in Exterior Domains

We study the asymptotic behavior of solutions with finite energy to the displacement problem of linear elastostatics in a three-dimensional exterior Lipschitz domain.

Examples for an extended Barabási-Albert model with random initial degrees

We develop a Papkovich-Neuber type representation formula for the solutions of the Navier-Lamé equation of linear elastostatics for spatial star-shaped domains. This representation is compared to the existing ones.

Development of User Element Routine (UEL) for Cell-Based Smoothed Finite Element Method (CSFEM) in Abaqus

In this paper, we discuss the implementation of a cell-based smoothed finite element method (CSFEM) within the commercial finite element software Abaqus. The salient feature of the CSFEM is that it does not require an explicit form of the derivative of the shape functions and there is no need for isoparametric mapping. This implementation is accomplished by employing the user element subroutine (UEL) feature in Abaqus. The details on the input data format together with the proposed user element subroutine, which forms the core of the finite element analysis are given. A few benchmark problems from linear elastostatics in both two and three dimensions are solved to validate the proposed implementation. The developed UELs and the associated input files can be downloaded from https://github.com/nsundar/SFEM_in_Abaqus .

A Note on the Displacement Problem of Elastostatics with Singular Boundary Values

The displacement problem of linear elastostatics in bounded and exterior domains with a non-regular boundary datum a is considered. Precisely, if the elastic body is represented by a domain of class C k ( k ≥ 2 ) of R 3 and a ∈ W 2 − k − 1 / q , q ( ∂ Ω ) , q ∈ ( 1 , + ∞ ) , then it is proved that there exists a solution which is of class C ∞ in the interior and takes the boundary value in a well-defined sense. Moreover, it is unique in a natural function class.