Toward transient finite element simulation of thermal deformation of machine tools in real-time

Abstract

Finite element models without simplifying assumptions can accurately describe the spatial and temporal distribution of heat in machine tools as well as the resulting deformation. In principle, this allows to correct for displacements of the Tool Centre Point and enables high precision manufacturing. However, the computational cost of FEM models and restriction to generic algorithms in commercial tools like ANSYS prevents their operational use since simulations have to run faster than real-time. For the case where heat diffusion is slow compared to machine movement, we introduce a tailored implicit-explicit multi-rate time stepping method of higher order based on spectral deferred corrections. Using the open-source FEM library DUNE, we show that fully coupled simulations of the temperature field are possible in real-time for a machine consisting of a stock sliding up and down on rails attached to a stand. 15 though accurate specification of boundary conditions can be a challenge [4]. Accurate transient finite element models are very useful as they can provide spatially and temporally resolved temperature fields for machines with complex designs and geometries [3,5]. In contrast to empirical approaches [6, 7], the parameters in FEM are physical quantities that can, at least theoretically, be 20 measured. Since reduced models are typically machine-specific, their derivation also comes with a high cost in terms of person hours. In contrast, the mesh for FEM models can be generated automatically, e.g. from CAD files, even for machines with complex geometries.The disadvantage of FEM models is their high computational cost, which 25 is why often reduced models are employed, sacrificing accuracy or generality for speed. Running full time-dependent FEM models is considered too computationally expensive to be possible in real-time: "application of the original FE-models without any simplifications [...] for model-based control-integrated correction is very time-consuming and thus impractical" [8]. Despite only re-30 solving one machine part and employing a time-averaged heat source instead of a full coupling, Galant et al. report a computation time of around 5 hours to simulate a milling machine with 16,626 degrees-of-freedom over 16 hours using ANSYS (corresponding to η = 3.2). To the best of the authors' knowledge, there are no reports of simulations solving in real-time the fully coupled tran-35 sient FEM problem for a machine with moving parts without simplifications.Recent review papers also make no mention of such efforts [1,3,9]. A combination of finite differences and FEM, called FDEM, has been proposed that reduces computational effort but still relies on the use of macro elements to reduce the size of the solved system [2]. With respect to FDM and FEM, in a 40 review from 2017, Cao et al. state that "[...], due to the low efficiency, the computational models were rarely used in online thermal error compensation" [10], mentioning only approaches that rely on steady-state FEM models [11,12].A key reason is probably...

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