A Fat boundary-type method for localized nonhomogeneous material problems

Abstract

Problems with localized nonhomogeneous material properties arise frequently in many applications and are a well-known source of difficulty in numerical simulations. In certain applications (including additive manufacturing), the physics of the problem may be considerably more complicated in relatively small portions of the domain, requiring a significantly finer local mesh compared to elsewhere in the domain. This can make the use of a uniform mesh numerically unfeasible. While nonuniform meshes can be employed, they may be challenging to generate (particularly for regions with complex boundaries) and more difficult to precondition. The problem becomes even more prohibitive when the region requiring a finer-level mesh changes in time, requiring the introduction of refinement and derefinement techniques. To address the aforementioned challenges, we employ a technique related to the Fat boundary method [1,2,20] as a possible alternative. We analyze the proposed methodology from a mathematical point of view and validate our findings on two-dimensional numerical tests.

Two-level Method Part-scale Thermal Analysis of Laser Powder Bed Fusion Additive Manufacturing

“…which holds for the formulation shown in [30][31][32], no longer holds. In practice, we find this error is small, provided that the jump across γ is not overly large (following the theory for the related Fat-boundary method given in [39]).…”

“…The two-level method is based on a re-formulation of the heat transfer equation (1) as two coupled problems referred in the following as the local and the global problem. Let us consider a global domain Ω + and a local domain Ω − ⊂ Ω + , such that, following the notation introduced in [30], T + and T − are the local and the global solution respectively. The thermal conductivity κ is defined as:…”

“…In order to achieve a more efficient numerical solution, which is necessary to deal with the LPBF thermal problem at part-scale, the original formulations of the global and local problem as presented by Viguerie et al [30,31], Viguerie and Auricchio [32] are slightly changed such as:…”

“…When compared to the two-level method as implemented in [30,32], this formulation greatly simplifies and reduces the cost of the assembly and evaluation process. More specifically, this novel formulation reduces the number of necessary integral evaluations, no longer needs the storage and evaluation of the derivatives of κ, and avoids the necessity of evaluating volumetric integrals non-conformal to the mesh discretization, which may ultimately require the application of an adaptive integration scheme, such as the ones adopted in other immersed boundary approaches (e.g., the Finite Cell Method [41] or the Cut Finite Element Method [42]).…”

“…In particular, a part-scale thermal model is developed and experimentally validated using the two-level method first presented by Viguerie et al [30] and analyzed in detail in [31]. Such a numerical approach is a local-global methodology which allows one to couple the results of a local, finer discretization with a coarser, global representation of the problem solutions.…”