Documents published in Scipedia

• A Mixed Finite Element Formulation for Arbitrary Element Geometries and Nearly-Incompressible Finite Elasticity

  B. Sauren, S. Klarmann, S. Klinkel

  eccomas2022.

  
  

  Abstract

  In this contribution, the displacement-based scaled-boundary finite element method (SBFEM) is extended to a mixed displacement-pressure formulation for the geometrically and materially nonlinear analysis of nearly-incompressible solids. The displacements and pressures are both parameterized by a scaled-boundary approach, for which an interpolation in the scaling direction is used. Here, higher-order interpolation functions may be employed. It is shown that by introducing the pressure as a field variable, volumetric locking is no longer present. The approach is valid for arbitrary scaling center locations, which can be either inside or outside of the element domain. Other than that, the formulation is valid for non-star-convex element geometries. Numerical examples show that the method is capable of alleviating volumetric locking for arbitrary polygonal meshes and is beneficial in comparison to the displacement-based method when it comes to modeling nearly-incompressible materials.

  Abstract
  In this contribution, the displacement-based scaled-boundary finite element method (SBFEM) is extended to a mixed displacement-pressure formulation for the geometrically and [...]

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• Multiscale kinetic transport models for the spread of epidemics with uncertain data

  G. Bertaglia

  eccomas2022.

  
  

  Abstract

  Most epidemiological models are rooted in the pioneering work proposed by Kermack and McKendrick and are based on systems of deterministic ODEs, which describe the temporal evolution of the spread of an infectious disease assuming population and territorial homogeneity. Generally, the concept of the average behavior of a population is sufficient to have a first reliable description of an epidemic development, but the inclusion of the spatial component becomes crucial when it is necessary to consider spatially heterogeneous interventions, as in the case of the COVID-19 pandemic. Moreover, any realistic data-driven model must take into account the large uncertainty in the values reported by official sources such as the amount of infectious individuals. In this work, drawing inspiration from kinetic theory, recent advances on the development of stochastic multiscale kinetic transport models for the spread of epidemics under uncertain data are presented. The propagation of the infectious disease is described by the spatial movement and interactions of individuals divided into commuters moving in the territory on a wide scale and non-commuters acting only on urban scales. The resulting models are solved numerically through a suitable stochastic Asymptotic-Preserving IMEX Runge-Kutta Finite Volume Collocation Method, which ensures a consistent treatment of the system of equations, without loss of accuracy when entering in the stiff, diffusive regime. Application studies concerning the spread of the COVID-19 pandemic in Italy assess the validity of the proposed methodology.

  Abstract
  Most epidemiological models are rooted in the pioneering work proposed by Kermack and McKendrick and are based on systems of deterministic ODEs, which describe the temporal [...]

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• Modelling the COVID-19 pandemic: variants and vaccines

  A. Kubik, A. Ramos, B. Ivorra, M. Vela-Pérez, M. Ferrández

  eccomas2022.

  
  

  Abstract

  In December 2019, a new coronavirus, the SARS-CoV-2, was detected in the Chinese city of Wuhan. Since then, many mathematical models have been developed to study the possible evolution of the COVID-19 disease and shed some light on the different biological processes of concern. On 14 December 2020, the United Kingdom reported a potentially more contagious and lethal variant of the virus, at the same time that different vaccines were being tested in order to prevent severe forms of the disease. In the following lines, we revisit a model proposed by our team, which took into account these two determining facts, showing its performance with real Italian data.

  Abstract
  In December 2019, a new coronavirus, the SARS-CoV-2, was detected in the Chinese city of Wuhan. Since then, many mathematical models have been developed to study the possible [...]

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• Extended virtual element method for two-dimensional fracture modeling in linear elasticity

  A. Chiozzi, E. Benvenuti, G. Manzini, N. Sukumar

  eccomas2022.

  
  

  Abstract

  The virtual element method (VEM), is a stabilized Galerkin scheme deriving from mimetic finite differences, which allows for very general polygonal meshes, and does not require the explicit knowledge of the shape functions within the problem domain. In the VEM, the discrete counterpart of the continuum formulation of the problem is defined by means of a suitable projection of the virtual shape functions onto a polynomial space, which allows the decomposition of the bilinear form into a consistent part, reproducing the polynomial space, and a correction term ensuring stability. In the present contribution, we outline an extended virtual element method (X-VEM) for two-dimensional elastic fracture problems where, drawing inspiration from the extended finite element method (X-FEM), we extend the standard virtual element space with the product of vector-valued virtual nodal shape functions and suitable enrichment fields, which reproduce the singularities of the exact solution. We define an extended projection operator that maps functions in the extended virtual element space onto a set spanned by the space of linear polynomials augmented with the enrichment fields. Numerical examples in 2D elastic fracture are worked out to assess convergence and accuracy of the proposed method for both quadrilateral and general polygonal meshes.

  Abstract
  The virtual element method (VEM), is a stabilized Galerkin scheme deriving from mimetic finite differences, which allows for very general polygonal meshes, and does not require [...]

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• Numerical treatment of the vectorial equations of solar oscillations

  T. Alemán, M. Halla, C. Lehrenfeld, P. Stocker

  eccomas2022.

  
  

  Abstract

  Driven by the challenging task of finding robust discretization methods for Galbrun's equation, we investigate conditions for stability and different aspects of robustness for different finite element schemes on a simplified version of the equations. The considered PDE is a second order indefinite vector-PDE which remains if only the highest order terms of Galbrun's equation are taken into account. A key property for stability is a Helmholtz-type decomposition which results in a strong connection between stable discretizations for Galbrun's equation and Stokes and nearly incompressible linear elasticity problems.

  Abstract
  Driven by the challenging task of finding robust discretization methods for Galbrun's equation, we investigate conditions for stability and different aspects of robustness [...]

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• Hybrid Finite-Volume/Discontinuous Galerkin framework for the solution of Multiphysics problems using unstructured meshes

  V. Maltsev, P. Tsoutsanis, M. Skote

  eccomas2022.

  
  

  Abstract

  A hybrid FV/DG framework is developed for the simulation of compressible multispecies flows on unstructured meshes with a five-equation Diffuse-Interface Model [1]. The high order DG method is employed for the purpose of limiting the material interface smearing typical of the diffuse-interface models resulting from excessive numerical dissipation [2, 3]. In order to ensure high-order accuracy in smooth flow regions and non-oscillatory behaviour near shocks or material interfaces, the hybrid scheme resorts to the underlying FV method when invalid cells are detected by a troubled cell indicator checking the unlimited DG solution, and enables a highorder non-linear CWENOZ reconstruction [4,5] if the solution present oscillations. The CWENO and CWENOZ type reconstruction uses a high-order polynomial for the central stencil and a lower-order polynomial for the directional stencils enhancing robustness and efficiency of classic WENO schemes. To achieve consistency in advecting material interfaces at constant pressure and velocity, the source term from the non-conservative equation is discretised compatibly with the Riemann solver, following the work of Johnsen and Colonius [6].

  Abstract
  A hybrid FV/DG framework is developed for the simulation of compressible multispecies flows on unstructured meshes with a five-equation Diffuse-Interface Model [1]. The high [...]

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• 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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