Direct numerical simulations of particle-laden density currents with adaptive, discontinuous finite elements

High-resolution direct numerical simulations (DNSs) are an important tool for the detailed analysis of turbidity current dynamics. Models that resolve the vertical structure and turbulence of the flow are typically based upon the Navier–Stokes equations. Two-dimensional simulations are known to produce unrealistic cohesive vortices that are not representative of the real three-dimensional physics. The effect of this phenomena is particularly apparent in the later stages of flow propagation. The ideal solution to this problem is to run the simulation in three dimensions but this is computationally expensive. <br><br> This paper presents a novel finite-element (FE) DNS turbidity current model that has been built within Fluidity, an open source, general purpose, computational fluid dynamics code. The model is validated through re-creation of a lock release density current at a Grashof number of 5 &times; 10⁶ in two and three dimensions. Validation of the model considers the flow energy budget, sedimentation rate, head speed, wall normal velocity profiles and the final deposit. Conservation of energy in particular is found to be a good metric for measuring model performance in capturing the range of dynamics on a range of meshes. FE models scale well over many thousands of processors and do not impose restrictions on domain shape, but they are computationally expensive. The use of adaptive mesh optimisation is shown to reduce the required element count by approximately two orders of magnitude in comparison with fixed, uniform mesh simulations. This leads to a substantial reduction in computational cost. The computational savings and flexibility afforded by adaptivity along with the flexibility of FE methods make this model well suited to simulating turbidity currents in complex domains

Effiziente Integration und verbesserte Kontaktspannungen für duale Mortar-Formulierungen

Diese Arbeit beschäftigt sich mit Computersimulationen von Kontaktproblemen unter Verwendung der Methode der finiten Elemente. Zur Diskretisierung des Kontakts wird die Mortar-Methode eingesetzt, für welche zwei Modifikationen vorgeschlagen werden. Die erste Modifikation betrifft die numerische Berechnung der sogenannten Kontaktintegrale. Bei der zugehörigen Integration müssen im allgemeinen Fall polynomische Integranden über polygonale Flächen integriert werden. Zur Anwendung gewöhnlicher Quadraturformeln werden die Gebiete üblicherweise in dreieckige Integrationszellen unterteilt. In dieser Arbeit wird eine alternative Unterteilung in viereckige Integrationszellen vorgeschlagen, die dazu führt, dass weniger Integrationspunkte benötigt werden. Durch die in dieser Arbeit beschriebenen numerischen Experimente wird gezeigt, dass dadurch der numerische Aufwand der Integration deutlich reduziert werden kann, ohne die Integrationsgenauigkeit signifikant zu verschlechtern. Die zweite Modifikation dient der Verbesserung der Kontaktspannungen für die duale Mortar-Methode. Bei dieser Methode wird das Lagrange-Multiplikator-Feld mit dualen Formfunktionen approximiert. Daraus resultiert der Vorteil, dass die duale Mortar-Methode im Vergleich zur Standard-Mortar-Methode effizienter ist. Allerdings sind die Kontaktspannungen der dualen Mortar-Methode weniger genau als diejenigen der Standard-Mortar-Methode. In dieser Arbeit wird für die duale Mortar-Methode eine Rückrechnung der Kontaktspannungen basierend auf einer L2-Projektion vorgestellt. Numerische Experimente zeigen, dass durch die vorgeschlagene L2-Projektion die Kontaktspannungsgenauigkeit der dualen Mortar-Methode verbessert wird und vergleichbar zu derjenigen der Standard-Mortar-Methode ist.This work deals with computer simulations of contact problems using the finite element method. Two modifications are proposed for the mortar method, which is the method applied to discretise the contact. The first modification concerns the numerical calculation of so-called contact integrals. For the corresponding integration in general cases polynomial integrands have to be integrated over polygonal areas. In order to use ordinary numerical quadratures the polygonal areas are usually subdivided into triangular integration cells. In this work an alternative subdivision into quadrilateral integration cells is suggested, which yields less integrations points. With the numerical experiments described in this work it is shown that due to this reduction the numerical effort is decreased considerably without deteriorating the accuracy of integration significantly. The second modification improves the contact stresses of the dual mortar method. This method uses dual functions to approximate the Lagrange multiplier field, yielding the advantage that the dual mortar method is more efficient than the standard mortar method. However, the contact stresses of the dual mortar method are less accurate than the contact stresses of the standard mortar method. In this work a modification of the contact stresses based on an L2 projection is presented for the dual mortar method. Numerical experiments show that with the introduced L2 projection the accuracy of the contact stress of the dual mortar method is improved and comparable to the accuracy of the standard mortar method

Derivation, simulation and validation of poroelastic models in dental biomechanics

Poroelasticity and mechanics of growth are playing an increasingly relevant role in biomechanics. This work is a self- contained and holistic presentation of the modeling and simulation of non-linear poroelasticity with and without growth inhomogeneities. Balance laws of poroelasticity are derived in Cartesian coordinates. These allow to write the governing equations in a form that is general but also readily implementable. Closure relations are formally derived from the study of dissipation. We propose an approximation scheme for the poroelasticity problem based on an implicit Euler method for the time discretization and a finite element method for the spatial discretization. The non-linear system is solved by means of Newton's method. Time integration of the growth tensor is discussed for the specific case in which the rate of inelastic deformations is prescribed. We discuss the stability of the mixed finite element discretization of the arising saddle-point problem. We show that a linear finite element approximation of both the unknowns, that is not LBB compliant for the elasticity problem, is nevertheless stable when applied to the linearized poroelasticity problem. This choice enables a fast assembling phase. The discretization of the poroelastic system may present unphysical oscillations if the spatial and temporal step-sizes are not properly chosen. We study the source of these wiggles by comparing the pressure Schur complement to a reaction- diffusion problem. From our analysis, we define a novel Péclet number for the poroelastic system and we show how it depends on the shear and bulk moduli of the solid phase. This number allows to introduce a stability condition that ensures that the solution is free of unphysical oscillations. If this condition on the Péclet number is not met, we introduce a fluid pressure Laplacian stabilization in order to remove the wiggles. This stabilization technique depends on a numerical parameter, whose optimal value is given by the derived Péclet number. Finally, we propose a coupled elastic-poroelastic model for the simulation of a tooth-periodontal ligament system. Because of the high resolution required by this system, we develop an efficient multigrid Newton's method for the non-linear poroelasticity system. The stability condition has again a significant influence on the performances of this solver. If the condition on the Péclet number is not satisfied on all levels of the multigrid algorithm, poor convergence rates or even divergence of the solver can be observed. The stabilization of the coarse grid operators with the optimal fluid pressure Laplacian method is a simple and efficient method to improve the convergence rate of the multigrid solver applied to this saddle-point system. We validate our coupled model against experimental measurements realized by the group of Prof. Bourauel at the University of Bonn

Development of novel Reynolds-averaged Navier-Stokes turbulence models based on Lie symmetry constraints

In the present work, the problem of RANS (Reynolds-Averaged Navier–Stokes) turbulence modeling is investigated from a novel angle by considering recently discovered constraints arising from Lie symmetry analysis. In this context, symmetries are defined as variable transformations that leave invariant a given equation. For equations describing physical phenomena,it is usually observed that their symmetries correspond to physical principles encoded in the equations. The key idea behind using symmetry methods for modeling tasks is that the physical principles encoded in an exact equation should also be present in a model for these equations. Lie symmetry theory establishes a mathematical framework to formalize this notion. The symmetries that govern turbulence fall into two main categories: Classical symmetries,which are present in the Navier–Stokes equations as well as in all statistical descriptions of turbulence, and statistical symmetries, which are found exclusively in statistical descriptions of turbulence and have no counterpart in the unaveraged Navier–Stokes equations. Even though the explicit use of symmetry methods in turbulence modeling is not yet prevalent, many well-established constraints imposed on turbulence models to prevent physically unreasonable behavior actually stem from symmetry arguments. This has led to a situation where the constraints implied by classical symmetries, which correspond to fundamental principles found throughout classical mechanics, have generally been taken into account when constructing turbulence models since the 1970s. Roughly speaking, two-equation eddy viscosity models are the simplest class of models to fulfill all of them. Statistical symmetries, on the other hand, are connected to special properties of turbulent statistics, and are, therefore, not as intuitive as the classical symmetries. As a result, they have so far been overlooked in turbulence modeling. The main goal of the present work is to devise a turbulence model while taking these statistical symmetries into account. This task turns out to be challenging because the combined set of classical and statistical symmetries imposes considerable restrictions on the possible form of the model equations. To overcome this challenge, a formal modeling algorithm is adapted and applied to turbulence modeling. Its results hint at the necessity for auxiliary velocity-like and pressure-like variables. With these model variables, possible model skeletons, both for an eddy-viscosity type model and for a Reynolds stress model, are developed. Subsequently, these simple base models are evolved into full turbulence models by applying them to canonical flows. Due to the complexity of the resulting Reynolds stress model, the emphasis is placed on developing a modified version of the k-ε-model that fulfills the statistical symmetries. This new model is calibrated against a wide range of canonical flows, where it performs at least equally well or better than the standard k-ε-model. Furthermore, the implementation of the standard k-ω-model in the in-house DG (Discontinuous Galerkin) solver BoSSS (Bounded Support Spectral Solver) is presented. Additionally, a special-purpose solver is developed that allows efficient numerical calculations with the modified k-ε-model for simple flows. The obtained results match well with experimental data

Parallel Multiphase Navier-Stokes Solver

We study and implement methods to solve the variable density Navier-Stokes equations. More specifically, we study the transport equation with the level set method and the momentum equation using two methods: the projection method and the artificial compressibility method. This is done with the aim of numerically simulating multiphase fluid flow in gravity oil-water-gas separator vessels. The result of the implementation is the parallel Aspen software framework based on the massively parallel deal.II . For the transport equation, we briefly discuss the theory behind it and several techniques to stabilize it, especially the graph laplacian artificial viscosity with higher order elements. Also, we introduce the level set method to model the multiphase flow and study ways to maintain a sharp surface in between phases. For the momentum equation, we give an overview of the two methods and discuss a new projection method with variable time stepping that is second order in time. Then we discuss the new third order in time artificial compressiblity method and present variable density version of it. We also provide a stability proof for the discrete implicit variable density artificial compressibility method. For all the methods we introduce, we conduct numerical experiments for verification, convergence rates, as well as realistic models

Advances in computational modelling of turbidity currents using the finite-element method

Turbidity currents are one of the main processes by which sediment is moved from the continental shelf to the deep ocean. They are a potential environmental hazard and they form a significant component of the stratigraphic record. Computational modelling is an important tool for understanding turbidity current dynamics, for augmenting experimental analyses, and for interpreting data that is collected in the field. This work begins by presenting a depth-averaged turbidity current model that is differentiated to facilitate the use of gradient-based optimisation algorithms. These optimisation algorithms are applied in selecting model parameters to best fit model output with data obtained in the field. To the best of the author's knowledge this is the first published work where optimisation of input parameters is applied to turbidity current modelling. The work also presents the first high resolution three-dimensional simulation of a turbidity current using the finite element method. One of the key benefits of the finite element method is the ability to easily accommodate complex domain geometries. As such this model is uniquely capable of producing high resolution simulations of turbidity currents in unconstrained complex domains. Methods of reducing the computational cost of these very expensive simulations are explored. The use of Large Eddy Simulation is shown to provide some improvements at moderate simulation resolutions. Unstructured mesh optimisation is shown to reduce the cost of these simulations by approximately two orders of magnitude when compared to a fixed mesh simulation. The savings afforded by the use of these techniques make the problem tractable using finite elements and will enable simulation of turbidity currents in complex and expansive domains where DNS modelling was previously unachievable.Open Acces

Model Adaptivity and Numerical Solutions Using Sensitivity Analysis

In this work we consider the dependence of solutions to a partial differential equations system on its data. The problem of interest is a coupled model of nonlinear flow and transport in porous media, with applications, e.g. to environmental modeling. The model of flow we consider is known as the non-Darcy model, and its solutions: the velocity, and pressure unknowns, depend on the coefficients of permeability and inertia, and other data such as boundary conditions. In turn, the transport solutions depend on the velocity of the fluid, and on boundary and initial conditions. Furthermore, one can be interested in a particular quantity computable from the flow and transport solutions, and represented by a functional. In this work we evaluate rigorously the sensitivity, i.e., the derivative, of the solutions, or of the quantity of interest, upon the data. Due to its delicate nature, the sensitivity is evaluated either in a direct way, called Forward Sensitivity, or via an adjoint method, which only uses a variational form. Our first contribution is that we find a way to find the sensitivity for the coupled flow and transport model without having to solve multiple flow problems. Second, we prove the well-posedness of the flow problem, and set up the numerical approximation using the framework similar to that of expanded mixed finite element methods. Next, the numerical approximation of the problem leads to a nonlinear system of discrete equations, which is difficult to solve. To aid in solving this system, we propose to take advantage of sensitivity analysis, which is used in a novel way within a homotopy framework. The theoretical results in this thesis are illustrated with numerical simulations. The code, in Python, for the examples is provided.Keywords: non-Darcy, Numerical solutions, Model adaptivity, Sensitivity analysis, Continuatio