A Study on the LATIN-PGD Method: Analysis of Some Variants in the Light of the Latest Developments
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ORIGINAL PAPER
A Study on the LATIN‑PGD Method: Analysis of Some Variants in the Light of the Latest Developments R. Scanff1,2 · S. Nachar1 · P. ‑A. Boucard1 · D. Néron1 Received: 27 March 2020 / Accepted: 22 October 2020 © CIMNE, Barcelona, Spain 2020
Abstract The LATIN-PGD method is a powerful alternative to the Newton–Raphson scheme for solving non-linear time-dependent problems in combination with reduced-order modeling methods. Many developments have been carried out over the last few decades and have led to some variants of the LATIN-PGD method. However, only few comparisons have been made between these variants and none using the digital resources now available. In this article, a comparison between the two major variants of the LATIN-PGD method, as well as with the Newton–Raphson one, is performed using a unified software which highlights the assets of the LATIN-PGD. Various test cases dealing with elasto-visco-plastic problems are undertaken, including comparison with commercial solvers, which reveals the interesting time saving in favor of the LATIN-PGD method.
1 Introduction Simulations are becoming widespread in a lot of areas within companies and in all phases of the product’s life cycle: from design to virtual prototyping through validation and maintenance. The key to these simulations lies in the physical model chosen, whose an arbitration is commonly made between the accuracy of the model and the calculation time, especially in design phases requiring numerous simulations in a very short time. On the one hand, the constant increase in computing capacity as well as the advent of parallel and GPU architecture has enabled more complex problems to be dealt with while maintaining acceptable computational times. But, on the other hand, these evolutions have not been followed from a software viewpoint by the development of
* R. Scanff ronan.scanff@ens‑paris‑saclay.fr; [email protected] S. Nachar stephane.nachar@ens‑paris‑saclay.fr P. ‑A. Boucard pierre‑alain.boucard@ens‑paris‑saclay.fr D. Néron david.neron@ens‑paris‑saclay.fr 1
Université Paris-Saclay, ENS Paris-Saclay, CNRS, LMT - Laboratoire de Mécanique et Technologie, 91190 Gif‑sur‑Yvette, France
Siemens Industry Software SAS, 92320 Châtillon, France
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new non-linear resolution engines better adapted to current computing architectures. The vast majority of the problems encountered in the mechanics of non-linear structures are evolutionary problems that is to say the behavior of the structure depends on its history: plasticity, visco-plasticity, damage or fatigue are part of this framework. Classical resolution algorithms such as the Newton–Raphson method use this vision and a time discretization to accurately determine at each time step the behavior of the structure. An alternative approach was proposed by Ladevèze [62, 56] under the name of the Large Time Increment (LATIN) method where the solution is no longer looked for sequentially but sought by successive corrections over the complete spatio-temporal space via an algori
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