Tracer methods are widespread in computational geodynamics for modeling the advection of chemical data. However, they present certain numerical challenges, especially when used over long periods of simulation time. We address two of these in this work: the necessity for mass conservation of chemical composition fields and the need for the velocity field to be pointwise divergence free to avoid gaps in tracer coverage. We do this by implementing the hybrid discontinuous Galerkin (HDG) finite element (FE) method combined with a mass conserving constrained projection of the tracer data. To demonstrate the efficacy of this system, we compare it to other common FE formulations of the Stokes system and projections of the chemical composition. We provide a reference of the numerical properties and error convergence rates which should be observed by using these various discretization schemes. This serves as a tool for verification of existing or new implementations. We summarize these data in a reproduction of a published Rayleigh-Taylor instability benchmark, demonstrating the importance of careful choices of appropriate and compatible discretization methods for all aspects of geodynamics simulations.@en

This thesis presents a numerical framework for simulating advection-dominated flows which reconciles the advantages of Eulerian mesh-based schemes with those of a Lagrangian particle-based discretization strategy. Particularly, the strategy proposed in this thesis inherits the diffusion-free properties as in Lagrangian particle-based advection, while simultaneously possessing high-order accuracy and local conservation properties as in state-of-the-art Eulerian mesh-based discretization strategies. These properties render the scheme particularly apt for simulating flow- and transport processes in which the physical diffusion is low, such as turbulent flows, or simulating flow problems with sharp and complex-shaped interfaces, such as the air-water interface in breaking ocean waves.@en

This paper introduces LEOPART, an add-on for the open-source finite element software library FENICS to seamlessly integrate Lagrangian particle functionality with (Eulerian) mesh-based finite element (FE) approaches. LEOPART- which is so much as to say: ‘Lagrangian–Eulerian on Particles’ - contains tools for efficient, accurate and scalable advection of Lagrangian particles on simplicial meshes. In addition, LEOPART comes with several projection operators for exchanging information between the scattered particles and the mesh and vice versa. These projection operators are based on a variational framework, which allows extension to high-order accuracy. In particular, by implementing a dedicated PDE-constrained particle–mesh projection operator, LEOPART provides all the tools for diffusion-free advection, while simultaneously achieving optimal convergence and ensuring conservation of the projected particle quantities on the underlying mesh. A range of numerical examples that are prototypical to passive and active tracer methods highlight the properties and the parallel performance of the different tools in LEOPART. Lastly, future developments are identified. The source code for LEOPART is actively maintained and available under an open-source license at https://bitbucket.org/jakob_maljaars/leopart.@en

A generic particle–mesh method using a hybridized discontinuous Galerkin (HDG) framework is presented and validated for the solution of the incompressible Navier–Stokes equations. Building upon particle-in-cell concepts, the method is formulated in terms of an operator splitting technique in which Lagrangian particles are used to discretize an advection operator, and an Eulerian mesh-based HDG method is employed for the constitutive modeling to account for the inter-particle interactions. Key to the method is the variational framework provided by the HDG method. This allows to formulate the projections between the Lagrangian particle space and the Eulerian finite element space in terms of local (i.e. cellwise) ℓ2-projections efficiently. Furthermore, exploiting the HDG framework for solving the constitutive equations results in velocity fields which excellently approach the incompressibility constraint in a local sense. By advecting the particles through these velocity fields, the particle distribution remains uniform over time, obviating the need for additional quality control. The presented methodology allows for a straightforward extension to arbitrary-order spatial accuracy on general meshes. A range of numerical examples shows that optimal convergence rates are obtained in space and, given the particular time stepping strategy, second-order accuracy is obtained in time. The model capabilities are further demonstrated by presenting results for the flow over a backward facing step and for the flow around a cylinder.@en

By combining concepts from particle-in-cell (PIC) and hybridized discontinuous Galerkin (HDG) methods, we present a particle–mesh scheme for flow and transport problems which allows for diffusion-free advection while satisfying mass and momentum conservation – locally and globally – and extending to high-order spatial accuracy. This is achieved via the introduction of a novel particle–mesh projection operator which casts the particle–mesh data transfer as a PDE-constrained optimization problem, permitting advective flux functionals at cell boundaries to be inferred from particle trajectories. This optimization problem seamlessly fits in a HDG framework, whereby the control variables in the optimization problem serve as advective fluxes in the HDG scheme. The resulting algebraic problem can be solved efficiently using static condensation. The performance of the scheme is demonstrated by means of numerical examples for the linear advection–diffusion equation and the incompressible Navier–Stokes equations. The results demonstrate optimal spatial accuracy, and when combined with a θ time integration scheme, second-order temporal accuracy is shown.@en

A particle-mesh strategy is presented for scalar transport problems which provides diffusion-free advection, conserves mass locally (i.e. cellwise) and exhibits optimal convergence on arbitrary polyhedral meshes. This is achieved by expressing the convective field naturally located on the Lagrangian particles as a mesh quantity by formulating a dedicated particle-mesh projection based via a PDE-constrained optimization problem. Optimal convergence and local conservation are demonstrated for a benchmark test, and the application of the scheme to mass conservative density tracking is illustrated for the Rayleigh–Taylor instability.@en

In this work the feasibility of a numerical wave tank using a hybrid particle-mesh method is investigated. Based on the Fluid Implicit Particle Method (FLIP) a generic formulation for the hybrid method is presented for incompressible multi-phase flows involving large density jumps and wave generating boundaries. The performance of the method is assessed for a standing wave, and the generation and propagation of a solitary wave over a flat and a sloping bed. These benchmark tests demonstrate that the method is a promising and attractive tool for simulating the nearshore propagation of waves.@en

The last decade has seen a growing interest in cohesive zone models for fatigue applications. These cohesive zone models often suffer from a lack of generality and applying them typically requires calibrating a large number of model-specific parameters. To improve on these issues a new method has been proposed in this paper based on the Thick Level Set approach. In this concept, material degradation due to cyclic loading is the result of interaction between damage evolution and fracture mechanics. The Thick Level Set formulation has been extended to interface elements, in order to allow for separation of strain energy in the bulk and energy required for surface creation. Global fracture parameters, derived from a free energy description governing the interface elements, are used as input for the empirical crack growth rate relation (Paris' equation). It must be emphasized that in contrast to existing fatigue models, the Thick Level Set approach does not require the definition of a damage evolution law. Instead, damage is updated automatically by a continuously moving damage front. It is shown that applicability is not limited to fatigue behavior of linear elastic materials; elastic-plastic materials such as steels can be analysed as well. The sensitivity of model parameters is investigated and discussed and the practical relevance is explored for standard test configurations.@en

The last decade has seen a growing interest in cohesive zone models for fatigue applications. These cohesive zone models often suffer from a lack of generality and applying them typically requires calibrating a large number of model-specific parameters. To improve on these issues a new method has been proposed in this paper based on the Thick Level Set approach. In this concept, material degradation due to cyclic loading is the result of interaction between damage evolution and fracture mechanics. The Thick Level Set formulation has been extended to interface elements, in order to allow for separation of strain energy in the bulk and energy required for surface creation. Global fracture parameters, derived from a free energy description governing the interface elements, are used as input for the empirical crack growth rate relation (Paris' equation). It must be emphasized that in contrast to existing fatigue models, the Thick Level Set approach does not require the definition of a damage evolution law. Instead, damage is updated automatically by a continuously moving damage front. It is shown that applicability is not limited to fatigue behavior of linear elastic materials; elastic-plastic materials such as steels can be analysed as well. The sensitivity of model parameters is investigated and discussed and the practical relevance is explored for standard test configurations.@en

In this work the feasibility of a numerical wave tank using a hybrid particle-mesh method is investigated. Based on the fluid implicit particle method (FLIP) a formulation for the hybrid method is presented for incompressible multiphase flows involving large density jumps and wave generating boundaries. The performance of the method is assessed for a standing wave and for the generation and propagation of a solitary wave over a flat and a sloping bed. A comparison is made with results obtained with a well-established SPH package. The tests demonstrate that the method is a promising and attractive tool for simulating the nearshore propagation of waves.@en

In this work the feasibility of a numerical wave tank using a hybrid particle-mesh method is investigated. Based on the fluid implicit particle method (FLIP) a formulation for the hybrid method is presented for incompressible multiphase flows involving large density jumps and wave generating boundaries. The performance of the method is assessed for a standing wave and for the generation and propagation of a solitary wave over a flat and a sloping bed. A comparison is made with results obtained with a well-established SPH package. The tests demonstrate that the method is a promising and attractive tool for simulating the nearshore propagation of waves.@en