Development of a mesh generator and implementation of the discontinuous Galerkin Finite Element Method in 3D to avoid geometric stiffening effects.

1 February 2019

Abstract

In this bachelor’s thesis, the discontinuous Galerkin Finite Element Method for three-dimensional mechanical problems is considered. The objective of this work is to extend an existing two-dimensional element code to 3D for linear 8-node volume elements, in order to investigate the avoidance of geometric stiffening effects.

The shear locking effect leads to inaccurate deformation representation in bending-dominant problems when lower-order Lagrange polynomials are used to solve the mechanical problem. The theory of the discontinuous Galerkin Finite Element Method (dG-FEM) aims to remedy this phenomenon.

To describe the formulation of the dG Finite Element Method, first, the stiffening effect phenomenon is introduced. Section 2 defines locking and its different types, with a detailed explanation of shear locking in subsequent subsections.

To address the shear stiffening problem, algorithms must be developed within this bachelor’s thesis that can create and solve such problems. The first step is achieved through the development of a mesh generator tasked with the automated creation of 3D discontinuous meshes. The functionality and various processes necessary for creating a discontinuous mesh are detailed in Chapter 3, along with a Program Flowchart (PAP) illustrating the operation of the automated mesh generator.

Section 4 focuses on formulating the discontinuous Galerkin Finite Element Method. This chapter presents the strong and weak forms of the dG formulation, along with describing the components that expand the weak form and their influence on the equation.

The final section of this bachelor’s thesis presents the results of simulations demonstrating the quality of the dG Finite Element method compared to the stiffening effect.

In conclusion, a summary and outlook are provided at the end of the bachelor’s thesis.

Literature

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