Documents published in Scipedia

• Under-resolved Direct Numerical Simulation of NACA0012 at Stall

  M. Lahooti, G. Vivarelli, F. Montomoli, S. Sherwin

  eccomas2022.

  
  

  Abstract
  In this work a high-order spectral-h/p element solver is employed to efficiently but accurately resolve the flow field around the NACA0012 aerofoil.

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• A PINN computational study for a scalar 2D inviscid Burgers model with Riemann data

  R. Carniello, J. Florindo, E. Abreu

  eccomas2022.

  
  

  Abstract

  In this work, we present an application of modern deep learning methodologies to the numerical solution of two dimensional hyperbolic partial differential equations in transport models. More specifically, we employ a supervised deep neural network that takes into account of initial-boundary value problems for a scalar, 2D inviscid Burgers model including the case with Riemann data, whose solutions develop discontinuity, containing both shock wave and rarefaction wave. We also apply the proposed PINN approach to the linear advection equation with periodic sinusoidal initial condition. Our results suggest that a relatively simple deep learning model was capable of achieving promising results in the linear advection and inviscid Burgers equation with rarefaction, providing numerical evidence of good approximation of weakentropy solutions to the case of nonlinear 2D inviscid Burgers model. For the Riemann problems, the neural network performed better when rarefaction wave is predominant. The premises underlying these preliminary results as an integrated physics-informed deep learning approach are promising. However, there are hints of evidence suggesting specific fine tuning on the PINN methodology for solving hyperbolic-transport problems in the presence of shock formation in the solutions.

  Abstract
  In this work, we present an application of modern deep learning methodologies to the numerical solution of two dimensional hyperbolic partial differential equations in transport [...]

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• Using Graph Neural Network for gas-liquid interface reconstruction in Volume Of Fluid methods

  M. BUCCI, J. GRATIEN, T. FANEY, T. NAKANO, G. CHARPIAT

  eccomas2022.

  
  

  Abstract

  The volume of fluid (VoF) method is widely used in multi-phase flow simulations to track and locate the interface between two immiscible fluids. A major bottleneck of the VoF method is the interface reconstruction step due to its high computational cost and low accuracy on unstructured grids. We propose a machine learning enhanced VoF method based on Graph Neural Networks (GNN) to accelerate the interface reconstruction on general unstructured meshes. We first develop a methodology to generate a synthetic dataset based on paraboloid surfaces discretized on unstructured meshes. We then train a GNN based model and perform generalization tests. Our results demonstrate the efficiency of a GNN based approach for interface reconstruction in multi-phase flow simulations in the industrial context.

  Abstract
  The volume of fluid (VoF) method is widely used in multi-phase flow simulations to track and locate the interface between two immiscible fluids. A major bottleneck of the [...]

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• Parametric Compressible Flow Predictions using Physics-Informed Neural Networks

  S. Wassing, S. Langer, P. Bekemeyer

  eccomas2022.

  
  

  Abstract

  The numerical approximation of solutions to the compressible Euler and Navierstokes equations is a crucial but challenging task with relevance in various fields of science and engineering. Recently, methods from deep learning have been successfully employed for solving partial differential equations by incorporating the equations into a loss function that is minimized during the training of a neural network. This approach yields a so-called physics-informed neural network which does not rely on a classical discretization and can address parametric problems in a straightforward manner. Therefore, it avoids characteristic difficulties of traditional approaches, such as finite volume methods. This has raised the question, whether physics-informed neural networks may be a viable alternative to conventional methods for computational fluid dynamics. Here, we show a new physics-informed neural network training procedure to approximately solve the two-dimensional compressible Euler equations, which makes use of artificial dissipation during the training process. We demonstrate how additional dissipative terms help to avoid unphysical results and how the additional numerical viscosity can be reduced during training while iterating towards a solution. Furthermore, we showcase how this approach can be combined with parametric boundary conditions. Our results highlight the appearance of unphysical results when solving compressible flows with physics-informed neural networks and offer a new approach to overcome this problem. We therefore expect that the presented methods enable the application of physics-informed neural networks for previously difficult to solve problems.

  Abstract
  The numerical approximation of solutions to the compressible Euler and Navierstokes equations is a crucial but challenging task with relevance in various fields of science [...]

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• Feedback reconstruction techniques for optimal control problems on a tree structure

  A. Alla, L. Saluzzi

  eccomas2022.

  
  

  Abstract

  The computation of feedback control using Dynamic Programming equation is a difficult task due the curse of dimensionality. The tree structure algorithm is one the methods introduced recently that mitigate this problem. The method computes the value function avoiding the construction of a space grid and the need for interpolation techniques using a discrete set of controls. However, the computation of the control is strictly linked to control set chosen in the computation of the tree. Here, we extend and complete the method selecting a finer control set in the computation of the feedback. This requires to use an interpolation method for scattered data which allows us to reconstruct the value function for nodes not belonging to the tree. The effectiveness of the method is shown via a numerical example.

  Abstract
  The computation of feedback control using Dynamic Programming equation is a difficult task due the curse of dimensionality. The tree structure algorithm is one the methods [...]

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• Structure-Preserving Neural Networks for the N-body Problem

  P. Horn, B. Koren, S. Portegies Zwart

  eccomas2022.

  
  

  Abstract

  In order to understand when it is useful to build physics constraints into neural networks, we investigate different neural network topologies to solve the N -body problem. Solving the chaotic N -body problem with high accuracy is a challenging task, requiring special numerical integrators that are able to approximate the trajectories with extreme precision. In [1] it is shown that a neural network can be a viable alternative, offering solutions many orders of magnitude faster. Specialized neural network topologies for applications in scientific computing are still rare compared to specialized neural networks for more classical machine learning applications. However, the number of specialized neural networks for Hamiltonian systems has been growing significantly during the last years [3, 5]. We analyze the performance of SympNets introduced in [5], preserving the symplectic structure of the phase space flow map, for the prediction of trajectories in N -body systems. In particular, we compare the accuracy of SympNets against standard multilayer perceptrons, both inside and outside the range of training data. We analyze our findings using a novel view on the topology of SympNets. Additionally, we also compare SympNets against classical symplectic numerical integrators. While the benefits of symplectic integrators for Hamiltonian systems are well understood, this is not the case for SympNets.

  Abstract
  In order to understand when it is useful to build physics constraints into neural networks, we investigate different neural network topologies to solve the N -body problem. [...]

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• An accelerated deflation preconditioner for parametric systems based on subspace recycling

  D. Panagiotopoulos, W. Desmet, E. Deckers

  eccomas2022.

  
  

  Abstract

  Krylov subspace recycling [1] is often deployed to accelerate the iterative solution of sequences of linear systems. Such approaches reuse a continuously updated deflation subspace to reach a converged solution within a low number of iterations. This procedure is justified for problems that describe gradually evolving phenomena, such as crack propagation, and thus involve a sequence of systems that are not simultaneously available. However considering parametric systems, these techniques might induce an unnecessary overhead cost. Specifically, by constantly updating the recycled subspace a new projection on the newly constructed subspace needs to be operated for each new system, inducing a cost that scales with O( × N2) for dense systems, where N is the size of the system and is the size of the employed recycled basis. In that context, this work proposes an accelerated recycling procedure for parametric systems that is inspired by the Galerkin Model Order Reduction strategy and employs an offline ­ online operation splitting. In the offline part, the subspace to be recycled is constructed via an Automatic Krylov subspaces Recycling algorithm (AKR) [2] and the parametric system is projected on the subspace to yield a Reduced Order Model (ROM). Then, in the online part the construction of the deflation preconditioner only requires employing the ROM and as a result the cost of constructing the preconditioner is reduced to O(2). The proposed procedure is tested on a randomly parametrized linear system and is compared the non-deflated GMRES algorithm and a conventional recycling strategy presented in [11].

  Abstract
  Krylov subspace recycling [1] is often deployed to accelerate the iterative solution of sequences of linear systems. Such approaches reuse a continuously updated deflation [...]

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• Mixed finite element formulations and energy-momentum time integrators for thermo-viscoelastic gradient-based fiber-reinforced continua

  J. Dietzsch, M. Groß, I. Kalaimani

  eccomas2022.

  
  

  Abstract

  Nowadays, fibre-reinforced materials and their accurate dynamic simulation play a significant role in the construction of lightweight structures. On the one hand, we are dealing with locking of the matrix material as well as the fibres, thermal expansion, the directed heat conduction through the fibres and viscoelastic behaviour in such materials. The material reinforcement is performed by fiber rovings with a separate bending stiffness, which can be modelled by second order gradients. On the other hand, we also want to perform accurate long-term simulations. In this presentation, we focus on numerically stable dynamic long-time simulations with locking free meshes, and thus use higherorder accurate energy-momentum schemes emanating from mixed finite element methods. We adapt the variational-based space-time finite element method in Reference [1] to the material formulation, and additionally include independent fields to obtain well-known mixed finite elements [2, 3].

  Abstract
  Nowadays, fibre-reinforced materials and their accurate dynamic simulation play a significant role in the construction of lightweight structures. On the one hand, we are dealing [...]

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• A tensor-product finite element cochain complex with arbitrary continuity

  F. Bonizzoni, G. Kanschat

  eccomas2022.

  
  

  Abstract

  We develop tensor product finite element cochain complexes of arbitrary smoothness on Cartesian meshes of arbitrary dimension. The first step is the construction of a onedimensional Cm-conforming finite element cochain complex based on a modified Hermite interpolation operator, which is proved to commute with the exterior derivative by means of a general commutation lemma. Adhering to a strict tensor product construction we then derive finite element complexes in higher dimensions.

  Abstract
  We develop tensor product finite element cochain complexes of arbitrary smoothness on Cartesian meshes of arbitrary dimension. The first step is the construction of a onedimensional [...]

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• Least-Squares and DPG approximation of eigenvalue associated to coupled problems

  F. Bertrand

  eccomas2022.

  
  

  Abstract

  The Arts and Science Contest of the ECCOMAS Young Investigators Committee (EYIC) aims to show science in all its beauty and elegance, by visualizing scientific works with an artistic point of view. The competition was open to every participant with an accepted the visuals were uploaded in 4 groups on the ECCOMAS Facebook page, one group per week. The visual with the most likes of each group was put in the finalists short-list, and a jury chose the final winners of the contest. The starry night of reaction diffusion won this competition and the present contribution presents the numerical method behind the picture.

  Abstract
  The Arts and Science Contest of the ECCOMAS Young Investigators Committee (EYIC) aims to show science in all its beauty and elegance, by visualizing scientific works with [...]

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