The shifted interface method for solid mechanics: An embedded domain approach
Abstract
Historically, embedded domain discretizations have been confined to the realm of fluid mechanics applications. They have shown considerable impact, especially when the complex geometrical shapes pose severe challenges to conformal mesh generation. The imposition of boundary conditions is more challenging in the case of embedded/immersed methods that in the case of conformal method (i.e., methods in which the computational grid conforms to the domain boundary). In recent work, the PI and his research group have developed a new embedded finite element method, named Shifted Boundary Method (or SB method). This new approach overcomes the difficulty on matrix conditioning and algorithmic stability that the occurrence of small cut cells produced in standard embedded methods. This without creating complicated data structures and limiting in general computational complexity to a minimum. The key feature of the SB method is the idea of shifting the location where boundary conditions are applied from the true to the surrogate boundary, and to appropriately modify the shifted boundary conditions, enforced weakly, in order to preserve optimal convergence rates of the numerical solution. This process yields a method that is simple, robust, and efficient. In previous work of the PIÕs research team, the SB method was successfully applied to the Poisson, Stokes, linear advection-diffusion, and Navier-Stokes equations. In the case of Poisson and linear advection-diffusion equations, proofs of stability and convergence were also derived. Typically, embedded methods are of finite volume type if fluid problems are the target. However, the use of the finite element framework is more powerful, because it has a much more developed and self-contained mathematical structure for the analysis of stability and accuracy. For example, in the SB method, with strongly leverage Nitsche-type techniques to impose the boundary condition weakly, which are in general superior to Lagrange multiplier or penalty-type approaches, currently the only two choices in the context of finite volume discretizations. Although embedded methods of finite element seem naturally applicable to solid mechanics, the existing literature on the subject is scarce, if non-existing. From a certain perspective, it may seem strange, at first sight, to consider an embedded method for solid mechanics, but there can be many benefits. If we look at modern manufacturing techniques, like additive manufacturing (e.g., 3D-printing and related techniques), the geometric complexity of the final shapes is considerably higher than in more standard forging, milling, drilling processes. This geometric complexity is intrinsic in additive manufacturing processes, in which materials are added layer upon layer rather than removed with very specific tools from a bulk block. As a consequence, existing mesh generators may be severely challenged. Additionally, if one would like to simulate the actual process of manufacturing, with the purpose of computing residual stresses and improving/optimizing the overall production cycle, then thermo-chemo- mechanical simulations of solid with advancing interfaces are required, a task for which embedded methods seem a natural choice. These topics are the scope and goals of this proposal. With this proposal, we would also like to pursue embedded solid mechanics computations of structures, with emphasis on growth/erosion processes and advancing/receding interfaces, respectively. Erosion and growth process are relevant in many applications of importance for the Army Research Office, such as problems associated with melting, phase transition, structure damage, structural and soil erosion.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Feb 14, 2019
- Source ID
- W911NF1810308
Entities
People
- Guglielmo Scovazzi
Organizations
- Army Contracting Command
- Duke University
- United States Army