Contemporary Meshfree Methods for Three Dimensional Heat Conduction Problems
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ORIGINAL PAPER
Contemporary Meshfree Methods for Three Dimensional Heat Conduction Problems M. Afrasiabi1 · M. Roethlin1 · K. Wegener1 Received: 6 February 2019 / Accepted: 16 July 2019 © CIMNE, Barcelona, Spain 2019
Abstract This work aims to be a fairly comprehensive study on the respective performance of several meshfree schemes selected for 3D heat conduction problems. A wide array of such methods is implemented in this paper, two of which are employed in 3D for the first time. These methods are compared in a systematic fashion: First, their ability to approximate the Laplacian operator, the key ingredient of the heat equation, is examined. Synthetic benchmarks as well as a real-world engineering problem where experimental data is available follow. In the interest of reproducibility and knowledge dissemination, the complete source code is made public and can be downloaded from: https://github.com/mroethli/thermal_iwf.
1 Introduction Particle-based Lagrangian methods have emerged as viable numerical techniques in scientific and engineering simulations. The salient feature of such methods is that they discretize a continuum by only a set of nodal points, or particles, without mesh constraints. This appealing attribute of meshfree methods makes them geometrically flexible yet computationally efficient for some applications. Introduced by Gingold and Monaghan [1] in 1977, Smoothed Particle Hydrodynamics(SPH) is one of the earliest meshfree methods, and perhaps the most popular one. The invention of SPH was triggered by the application in solving astrophysical problems, such as the formation and evolution of protostars or galaxies [1, 2]. Soon after its debut, and due to the distinct privilege of functioning with no underlying grid, SPH has been widely adopted by the computational science community. A broad array of solid and fluid mechanics applications ranging from metal cutting simulations [3–9], laser drilling [10], and brittle dynamic crack growth [11] to CFD [12–14], droplet spreading and solidification [15–17] has been successfully carried out using SPH, to name a few. It is understood that the avoidance of cumbersome remeshing procedure * M. Afrasiabi [email protected] 1
Department of Mechanical and Process Engineering, Institute of Machine Tools and Manufacturing (IWF), ETH Zürich, Leonhardstrasse 21, 8092 Zurich, Switzerland
required by mesh-based methods (like FEM), especially for highly dynamic and large deformations problems, serves as a very interesting feature when considering a Lagrangian grid undergoing severe distortions. This situation could be exemplified by metal forming and to a much larger extent in cutting simulations. There are, however, some shortcomings associated with the standard SPH method, among which the well-known problem of particle deficiency near/on boundaries is a sore point often subject to criticism [18]. This issue may come to play an even more rampant role when the approximation of higher order derivatives is of interest. To remedy this boundary deficiency, mesh
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