A nodally integrated thermo-mechanical meshfree formulation with application to fused deposition modeling
Open Access
- Author:
- Lin, Kuan Chung
- Graduate Program:
- Civil Engineering
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- July 06, 2020
- Committee Members:
- Michael Charles Hillman, Dissertation Advisor/Co-Advisor
Michael Charles Hillman, Committee Chair/Co-Chair
Gordon Patrick Warn, Committee Member
Pinlei Chen, Committee Member
Charles E Bakis, Outside Member
Kostas Papakonstantinou, Committee Member
Patrick Joseph Fox, Program Head/Chair - Keywords:
- Meshfree methods
Fused deposition modeling
Nodal integration
Coupled thermal mechanical formulation
Naturally stablized nodal integration
Variationally consistent integration - Abstract:
- The basic physical processes behind fused deposition modeling (FDM) are understood; however, the quality of the product is sensitive to how exactly the process is carried out. There are many parameters in FDM that will impact the product quality. Currently, experimental trial-and-error is the main method employed to understand how the product quality is affected by various control parameters, which is physically and financially inefficient. In order to gain an understanding of melt flow behavior in a more effective way, capabilities for validated simulation of the processes in FDM is desirable. The deposition process in FDM involves extrusion of the material, layer-by-layer. Thus self-contact and fusion of the material occur, resulting in evolving topology of the problem domain through time. Meshfree methods are promising candidates to reliably simulate topological domain changes since connectivity can be constructed on-the-fly, unlike mesh-based methods where connectivity is fixed based on the element discretization in the material (undeformed) configuration. However, two key issues need addressing in these methods. First, meshfree approximations are non-interpelotory, and essential (kinematic) boundary conditions cannot be imposed directly, causing issues with attaining high-order accuracy. Another key issue that needs careful consideration is the issue of how to perform quadrature of the weak form in Galerkin meshfree methods. Nodal integration is desirable for domain integration; meaning that the meshfree particles are used as the integration points themselves. This approach is highly efficient and keeps the desirable meshfree characteristics of meshfree methods on the discrete level with quadrature, but can yield non-convergent results due to the inherent low-order integration accuracy, and solution instability caused by severely underestimating the energy of saw-tooth modes. To remedy the issue with enforcing boundary conditions, two consistent weak forms are developed for meshfree methods. These weak forms remedy the fact that most meshfree approximations cannot satisfy the requirements of the traditional weak formulation. Numerical examples demonstrate that using these techniques, p-refinement, and h-refinement with p-th order rates can be achieved, previously unavailable in meshfree methods with strong enforcement of boundary conditions. A new concept of a-refinement is further introduced, also previously unavailable. In order to remedy the instability inherent in nodal integration, and to provide improved solution accuracy over direct nodal integration, variationally consistent naturally stabilized nodal integration (VC-NSNI) is proposed for the fully coupled thermo-mechanical problem. A naturally stabilized nodal integration is first developed, in which the Taylor series expansion applied to strains in solid mechanics problems is extended to include the an expansion on the gradient of temperature in the thermoelastic problem, which results in a stabilized nodal integration for the two-field problem. For accuracy, the variational consistency conditions are derived for thermo-elasticity, where the two independent fields are considered. A variationally consistent integration (VCI) technique is then proposed, in order to meet the constraints. Several benchmark problems are tested using the combined VC-NSNI approach, and the improved accuracy and stability of the proposed method is demonstrated. Next, generalized thermoelasticity theories with finite propagation speeds, namely the Lord and Shulman, and a Green and Lindsay theories are investigated. These theories result in hyperbolic equations which are more amenable to explicit dynamic simulations than the parabolic type. A meshfree approach for solving the associated fully coupled governing equations is then developed. Next, finite-strain thermoplasticity is developed and validated. Finally, these techniques are extended to finite-strain, thermo-mechanical problems with viscosity. Free-surface conditions are also considered, namely, surface tension and heat flux boundary conditions. The approach is verified by comparing simulation results to analytical solutions for melt-flow profiles in the liquefier, and simulations of FDM using other numerical methods.