Arbitrary-Lagrangian-Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes

We present a new family of high order accurate fully discrete one-step Discontinuous Galerkin (DG) finite element schemes on moving unstructured meshes for the solution of nonlinear hyperbolic PDE in multiple space dimensions, which may also include parabolic terms in order to model dissipative transport processes. High order piecewise polynomials are adopted to represent the discrete solution at each time level and within each spatial control volume of the computational grid, while high order of accuracy in time is achieved by the ADER approach. In our algorithm the spatial mesh configuration can be defined in two different ways: either by an isoparametric approach that generates curved control volumes, or by a piecewise linear decomposition of each spatial control volume into simplex sub-elements. Our numerical method belongs to the category of direct Arbitrary-Lagrangian-Eulerian (ALE) schemes, where a space-time conservation formulation of the governing PDE system is considered and which already takes into account the new grid geometry directly during the computation of the numerical fluxes. Our new Lagrangian-type DG scheme adopts the novel a posteriori sub-cell finite volume limiter method, in which the validity of the candidate solution produced in each cell by an unlimited ADER-DG scheme is verified against a set of physical and numerical detection criteria. Those cells which do not satisfy all of the above criteria are flagged as troubled cells and are recomputed with a second order TVD finite volume scheme. The numerical convergence rates of the new ALE ADER-DG schemes are studied up to fourth order in space and time and several test problems are simulated. Finally, an application inspired by Inertial Confinement Fusion (ICF) type flows is considered by solving the Euler equations and the PDE of viscous and resistive magnetohydrodynamics (VRMHD).Comment: 39 pages, 21 figure

An Efficient Approach for Solving Mesh Optimization Problems Using Newton’s Method

We present an efficient approach for solving various mesh optimization problems. Our approach is based on Newton’s method, which uses both first-order (gradient) and second-order (Hessian) derivatives of the nonlinear objective function. The volume and surface mesh optimization algorithms are developed such that mesh validity and surface constraints are satisfied. We also propose several Hessian modification methods when the Hessian matrix is not positive definite. We demonstrate our approach by comparing our method with nonlinear conjugate gradient and steepest descent methods in terms of both efficiency and mesh quality

Moving mesh Virtual Element Methods

This thesis explores the development and analysis of moving mesh Virtual Element Methods for partial differential equations on time-dependent domains. This thesis presents the first moving mesh method to purely use the Virtual Element Method, an isoparametric Virtual Element Method for approximating partial differential equations on curved domains and a high-order Arbitrary Lagrangian-Eulerian Virtual Element Method for problems on time-dependent domains with moving boundaries. Each contribution successfully demonstrates the applicability and accuracy of Virtual Element Methods in existing moving mesh algorithms, achieving similar orders of accuracy compared to classical Finite Element Method approaches. The results suggest that the flexibility of moving mesh methods can be greatly improved by incorporating more general mesh structures, including polygons and curved-edged polygons, proving the Virtual Element Method offers an effective extension to classical approaches. This work provides a foundation for future research in Virtual Element Methods for more complex problems on time-dependent domains and developing the analysis to support proposed moving mesh methods

Moving mesh Virtual Element Methods

This thesis explores the development and analysis of moving mesh Virtual Element Methods for partial differential equations on time-dependent domains. This thesis presents the first moving mesh method to purely use the Virtual Element Method, an isoparametric Virtual Element Method for approximating partial differential equations on curved domains and a high-order Arbitrary Lagrangian-Eulerian Virtual Element Method for problems on time-dependent domains with moving boundaries. Each contribution successfully demonstrates the applicability and accuracy of Virtual Element Methods in existing moving mesh algorithms, achieving similar orders of accuracy compared to classical Finite Element Method approaches. The results suggest that the flexibility of moving mesh methods can be greatly improved by incorporating more general mesh structures, including polygons and curved-edged polygons, proving the Virtual Element Method offers an effective extension to classical approaches. This work provides a foundation for future research in Virtual Element Methods for more complex problems on time-dependent domains and developing the analysis to support proposed moving mesh methods

Large eddy simulation of primary liquid-sheet breakup

This research project aims at providing the aeronautical industry with a modelling capability to simulate the fuel injection in gas turbine combustion chambers. The path to this objective started with the review of state-of-the-art numerical techniques to model the primary breakup of liquid fuel into droplets. Based on this and keeping in mind the requirements of the industry, our modelling strategy led to the generation of a mass-conservative method for efficient atomisation modelling on unstructured meshes. This goal has been reached with the creation of high-order numerical schemes for unstructured grids, the development of an efficient numerical method that transports the liquid-vapour interface accurately while conserving mass and the implementation of an algorithm that outputs the droplet boundary conditions to separate combustion codes. Both high-order linear and WENO schemes have been created for general polyhedral meshes. The notorious complexity of high-order schemes on 3D mixed-element meshes has been handled by the creation of a series of algorithms. These include the tetrahedralisation of the mesh, which allows generality of the approach while remaining efficient and affordable, together with a novel approach to stencil generation and a faster interpolation of the solution. The performance of the scheme has been demonstrated on typical two-dimensional and three-dimensional test cases for both linear and non-linear hyperbolic partial differential equations. The conservative level set method has been extended to unstructured meshes and its performance has been improved in terms of robustness and accuracy. This was achieved by solving the equations for the transport of the liquid volume fraction with our novel WENO scheme for polyhedral meshes and by adding a flux-limiter algorithm. The resulting method, named robust conservative level set, conserves mass to machine accuracy and its ability to capture the physics of the atomisation is demonstrated in this thesis. To be readily applicable to the simulation of atomisation, the novel interfacecapturing technique has been embedded in a framework — within the open source CFD code OpenFOAM — that solves the velocity and pressure fields, outputs droplet characteristics and runs in parallel. In particular, the production of droplet boundary conditions involves a set of routines handling the selection of drops in the level set field, the calculation of relevant droplet characteristics and their storage into data files. An n-halo parallelisation method has been implemented in OpenFOAM to perform the computations at the expected order of accuracy. Finally, the modelling capability has been demonstrated on the simulation of primary liquid-sheet breakup with relevance to fuel injection in aero-engine combustors. The computation has demonstrated the ability of the code to capture the physics accurately and further illustrates the potential of the numerical approach

Interface Tracking and Solid-Fluid Coupling Techniques with Coastal Engineering Applications

Multi-material physics arise in an innumerable amount of engineering problems. A broadly scoped numerical model is developed and described in this thesis to simulate the dynamic interaction of multi-fluid and solid systems. It is particularly aimed at modelling the interaction of two immiscible fluids with solid structures in a coastal engineering context; however it can be extended to other similar areas of research. The Navier Stokes equations governing the fluids are solved using a combination of finite element (FEM) and control volume finite element (CVFE) discretisations. The sharp interface between the fluids is obtained through the compressive transport of material properties (e.g. material concentration). This behaviour is achieved through the CVFE method and a conveniently limited flux calculation scheme based on the Hyper-C method by Leonard (1991). Analytical and validation test cases are provided, consisting of steady and unsteady flows. To further enhance the method, improve accuracy, and exploit Lagrangian benefits, a novel moving mesh method is also introduced and tested. It is essentially an Arbitrary Lagrangian Eulerian method in which the grid velocity is defined by semi-explicitly solving an iterative functional minimisation problem. A multi-phase approach is used to introduce solid structure modelling. In this approach, solution of the velocity field for the fluid phase is obtained using Model B as explained by Gidaspow (1994, page 151). Interaction between the fluid phase and the solids is achieved through the means of a source term included in the fluid momentum equations. The interacting force is calculated through integration of this source term and adding a buoyancy contribution. The resulting force is passed to an external solid-dynamics model such as the Discrete Element Method (DEM), or the combined Finite Discrete Element Method (FEMDEM). The versatility and novelty of this combined modelling approach stems from its ability to capture the fluid interaction with particles of random size and shape. Each of the three main components of this thesis: the advection scheme, the moving mesh method, and the solid interaction are individually validated, and examples of randomly shaped and sized particles are shown. To conclude the work, the methods are combined together in the context of coastal engineering applications, where the complex coupled problem of waves impacting on breakwater amour units is chosen to demonstrate the simulation possibilities. The three components developed in this thesis significantly extend the application range of already powerful tools, such as Fluidity, for fluids-modelling and finite discrete element solids-modelling tools by bringing them together for the first time

7th International Meshing Roundtable '98

The goal of the 7th International Meshing Roundtable is to bring together researchers and developers from industry, academia, and government labs in a stimulating, open environment for the exchange of technical information related to the meshing process. In the past, the Roundtable has enjoyed significant participation from each of these groups from a wide variety of countries