Meshfree Linear Solver Selection and Variational Geometric Formulations in Computational Mechanics
Open Access
- Author:
- Wang, Yanran
- Graduate Program:
- Civil Engineering (MS)
- Degree:
- Master of Science
- Document Type:
- Master Thesis
- Date of Defense:
- June 17, 2022
- Committee Members:
- Mike Hillman, Thesis Advisor/Co-Advisor
Kostas Papakonstantinou, Committee Member
Jay Regan, Professor in Charge/Director of Graduate Studies
Pinlei Chen, Committee Member
Christian Peco, Committee Member - Keywords:
- meshfree
Complexity Analyses
Linear Solver Selection
Geometric Interpretations
Variational formulation - Abstract:
- The development of the use of meshfree approximations in PDEs has significantly alleviated longstanding challenges for the finite element method (FEM), such as mesh dependence and mesh distortion issues. Moreover, the discretization, with uncoupled completeness, continuity, and support (nodal influence), provides excellent flexibility that can be leveraged for difficult problems such as a complex fracture. However, the construction of the meshfree shape functions is generally computationally more intensive than the FEM shape functions, and the stiffness matrix has a non-trivial increase in bandwidth (as well as the CPU scaling of the internal force vector in explicit calculations). In addition, while many attempts have been made to utilize the versatility of meshfree approximations for fracture, a robust formulation remains illusive. To examine the overall effectiveness of meshfree methods for the boundary value problems at hand, comprehensive complexity and performance comparisons between various domain and essential boundary treatment methods are first presented. Without loss of generality, the Galerkin class is considered, which in and of itself has many permutations: the globally and rational, non-interpolatory nature of the meshfree shape functions complicate the numerical domain integration and essential boundary reinforcements when solving the boundary value problems. Many domain integration methods and kinematic constraint reinforcement techniques have been developed, each with its own complexity. In addition, these integration and essential boundary reinforcement methods will result in linear systems of equations with different characteristics when solving the same PDE (e.g., symmetric or non-symmetric, positive definite or semi-definite, and so on). Furthermore, it must also be kept in mind that solving the linear system of equations to obtain the unknown field variables contributes to computational time. Therefore, various linear system solver algorithms are investigated in this thesis to aid the solver selection for different meshfree domain and essential boundary treatment methods. Apart from the complexity and performance comparisons, this work also focuses on developing a general framework for meshfree particle methods that is compatible with the discrete flavor of numerical fracture mechanics, such as the Cohesive Zone Method (CZM). While finite-volume and finite element type techniques can explicitly define potential surfaces, a Galerkin particle discretization does not. As such, the geometric information between the particles in the Galerkin form meshfree formulations is investigated in this work in order to identify potential \emph{virtual} cohesive crack surfaces and outward normals without explicitly defining the individual nodal domain, i.e., partitioning the domain with nodal Voronoi cells. A remaining missing piece, then, is to define the constitutive laws at these virtual surfaces. A gap-traction relation is defined for CZM, where the work is directly related to the fracture energy. However, for the general Galerkin framework, continuous fields exist on any given interior surface, virtual or not, so it is not clear how to define a constitutive law for fracture in this case since there is no gap. The microplane model idea is first explored in this work to this end, wherein the microplanes are defined according to the orientation of integration points within a spherical domain, and the nodal strain tensors are projected onto each microplane to form in-plane strain vectors in the original model. This leveraged to develop a microplane model using actual nodal Voronoi surfaces as an initial test, inspired by the traditional microplane model but instead selects edges of a nodal Voronoi cell as "microplanes" to project the strain tensor at a node. A variational formulation of the proposed 2D Voronoi edge microplane model is then presented. Although the current formulation can qualitatively represent various solution fields for elliptic PDEs, it does not converge with $h$ and $p$ refinement. Therefore, while progress is made in this area, the direct use of a variational microplane model seems unsuitable if high accuracy is demanded, and further investigation is warranted. The reproducing kernel finite volume method (RKFM), a newly proposed finite volume-based method with RK approximation, has achieved some successful results for dynamic crack simulations utilizing CZM with Voronoi surfaces. Nevertheless, RKFM results in an asymmetrical stiffness matrix, which is the main culprit for its temporal instability when employing nonuniform nodal distributions. This work puts some effort into developing reformulations of RKFM to address its temporal instability. In the end, a discontinuous Roth-type reformulation of RKFM, which computes the volumetric stress under the RKFM framework while the deviatoric stress is formulated using RKPM with nodal integration, achieves stable results for the wave propagation benchmark problem.