The fully-implicit log-conformation formulation and its application to three-dimensional flows

The stable and efficient numerical simulation of viscoelastic flows has been a constant struggle due to the High Weissenberg Number Problem. While the stability for macroscopic descriptions could be greatly enhanced by the log-conformation method as proposed by Fattal and Kupferman, the application of the efficient Newton-Raphson algorithm to the full monolithic system of governing equations, consisting of the log-conformation equations and the Navier-Stokes equations, has always posed a problem. In particular, it is the formulation of the constitutive equations by means of the spectral decomposition that hinders the application of further analytical tools. Therefore, up to now, a fully monolithic approach could only be achieved in two dimensions, as, e.g., recently shown in [P. Knechtges, M. Behr, S. Elgeti, Fully-implicit log-conformation formulation of constitutive laws, J. Non-Newtonian Fluid Mech. 214 (2014) 78-87]. The aim of this paper is to find a generalization of the previously made considerations to three dimensions, such that a monolithic Newton-Raphson solver based on the log-conformation formulation can be implemented also in this case. The underlying idea is analogous to the two-dimensional case, to replace the eigenvalue decomposition in the constitutive equation by an analytically more "well-behaved" term and to rely on the eigenvalue decomposition only for the actual computation. Furthermore, in order to demonstrate the practicality of the proposed method, numerical results of the newly derived formulation are presented in the case of the sedimenting sphere and ellipsoid benchmarks for the Oldroyd-B and Giesekus models. It is found that the expected quadratic convergence of Newton's method can be achieved.Comment: 21 pages, 9 figure

Fully-implicit log-conformation formulation of constitutive laws

Subject of this paper is the derivation of a new constitutive law in terms of the logarithm of the conformation tensor that can be used as a full substitute for the 2D governing equations of the Oldroyd-B, Giesekus and other models. One of the key features of these new equations is that - in contrast to the original log-conf equations given by Fattal and Kupferman (2004) - these constitutive equations combined with the Navier-Stokes equations constitute a self-contained, non-iterative system of partial differential equations. In addition to its potential as a fruitful source for understanding the mathematical subtleties of the models from a new perspective, this analytical description also allows us to fully utilize the Newton-Raphson algorithm in numerical simulations, which by design should lead to reduced computational effort. By means of the confined cylinder benchmark we will show that a finite element discretization of these new equations delivers results of comparable accuracy to known methods.Comment: 21 pages, 5 figure

Automatic implementation of material laws: Jacobian calculation in a finite element code with TAPENADE

In an effort to increase the versatility of finite element codes, we explore the possibility of automatically creating the Jacobian matrix necessary for the gradient-based solution of nonlinear systems of equations. Particularly, we aim to assess the feasibility of employing the automatic differentiation tool TAPENADE for this purpose on a large Fortran codebase that is the result of many years of continuous development. As a starting point we will describe the special structure of finite element codes and the implications that this code design carries for an efficient calculation of the Jacobian matrix. We will also propose a first approach towards improving the efficiency of such a method. Finally, we will present a functioning method for the automatic implementation of the Jacobian calculation in a finite element software, but will also point out important shortcomings that will have to be addressed in the future.Comment: 17 pages, 9 figure

Implementing the Singular Value Decomposition in the Helmholtz Analytics Toolkit HeAT

Singular value decomposition (SVD) is a fundamental tool in data science and often used as, e.g., a preprocessing step. When dealing with very large data sets as they often arise at the DLR, performing the analysis in a scalable way on HPC systems can be necessary; this also includes the computation of the SVD. In this talk we present work in progress regarding our implementation of a parallel SVD within the PyTorch- and mpi4py-based HPC-data analytics software HEAT (Helmholtz Analytics Toolkit) developed at DLR, JSC, and KIT (Götz et al., 2020 IEEE International Conference on Big Data, pp. 276-287)

An eigenvalue-free implementation of the log-conformation formulation

The log-conformation formulation, although highly successful, was from the beginning formulated as a partial differential equation that contains an, for PDEs unusual, eigenvalue decomposition of the unknown field. To this day, most numerical implementations have been based on this or a similar eigenvalue decomposition, with Knechtges et al. (2014) being the only notable exception for two-dimensional flows. In this paper, we present an eigenvalue-free algorithm to compute the constitutive equation of the log-conformation formulation that works for two- and three-dimensional flows. Therefore, we first prove that the challenging terms in the constitutive equations are representable as a matrix function of a slightly modified matrix of the log-conformation field. We give a proof of equivalence of this term to the more common log-conformation formulations. Based on this formulation, we develop an eigenvalue-free algorithm to evaluate this matrix function. The resulting full formulation is first discretized using a finite volume method, and then tested on the confined cylinder and sedimenting sphere benchmarks

HeAT -- a Distributed and GPU-accelerated Tensor Framework for Data Analytics

To cope with the rapid growth in available data, the efficiency of data analysis and machine learning libraries has recently received increased attention. Although great advancements have been made in traditional array-based computations, most are limited by the resources available on a single computation node. Consequently, novel approaches must be made to exploit distributed resources, e.g. distributed memory architectures. To this end, we introduce HeAT, an array-based numerical programming framework for large-scale parallel processing with an easy-to-use NumPy-like API. HeAT utilizes PyTorch as a node-local eager execution engine and distributes the workload on arbitrarily large high-performance computing systems via MPI. It provides both low-level array computations, as well as assorted higher-level algorithms. With HeAT, it is possible for a NumPy user to take full advantage of their available resources, significantly lowering the barrier to distributed data analysis. When compared to similar frameworks, HeAT achieves speedups of up to two orders of magnitude.Comment: 10 pages, 8 figures, 5 listings, 1 tabl

HeAT – a Distributed and GPU-accelerated Tensor Framework for Data Analytics

In order to cope with the exponential growth in available data, the efficiency of data analysis and machine learning libraries have recently received increased attention. Although corresponding array-based numerical kernels have been significantly improved, most are limited by the resources available on a single computational node. Consequently, kernels must exploit distributed resources, e.g., distributed memory architectures. To this end, we introduce HeAT, an array-based numerical programming framework for large-scale parallel processing with an easy-to-use NumPy-like API. HeAT utilizes PyTorch as a node-local eager execution engine and distributes the workload via MPI on arbitrarily large high-performance computing systems. It provides both low-level array-based computations, as well as assorted higher-level algorithms. With HeAT, it is possible for a NumPy user to take advantage of their available resources, significantly lowering the barrier to distributed data analysis. Compared with applications written in similar frameworks, HeAT achieves speedups of up to two orders of magnitude