Automatic symbolic computation for discontinuous Galerkin finite element methods

The implementation of discontinuous Galerkin finite element methods (DGFEMs) represents a very challenging computational task, particularly for systems of coupled nonlinear PDEs, including multiphysics problems, whose parameters may consist of power series or functionals of the solution variables. Thereby, the exploitation of symbolic algebra to express a given DGFEM approximation of a PDE problem within a high level language, whose syntax closely resembles the mathematical definition, is an invaluable tool. Indeed, this then facilitates the automatic assembly of the resulting system of (nonlinear) equations, as well as the computation of Frechet derivative(s) of the DGFEM scheme, needed, for example, within a Newton-type solver. However, even exploiting symbolic algebra, the discretisation of coupled systems of PDEs can still be extremely verbose and hard to debug. Thereby, in this article we develop a further layer of abstraction by designing a class structure for the automatic computation of DGFEM formulations. This work has been implemented within the FEniCS package, based on exploiting the Unified Form Language. Numerical examples are presented which highlight the simplicity of implementation of DGFEMs for the numerical approximation of a range of PDE problems

Numerical modelling of chemical vapour deposition reactors

In this thesis we study the chemical reactions and transport phenomena which occur in a microwave power assisted chemical vapour deposition (MPA-CVD) reactor which facilitates diamond growth. First we introduce a model of an underlying binary gas flow and its chemistry for a hydrogen gas mixture. This system is heated by incorporating a microwave frequency electric field, operating in a resonant mode in the CVD chamber. This heating facilitates the dissociation of hydrogen and the generation of a gas discharge plasma, a key component of carbon deposition in industrial diamond manufacture. We then proceed to summarise the discontinuous Galerkin (DG) finite element discretisation of the standard hyperbolic and elliptic partial differential operators which typically occur in conservation laws of continuum models. Additionally, we summarise the non-stabilised discontinuous Galerkin formulation of the time harmonic Maxwell operator. These schemes are then used as the basis for the discretisation method employed for the numerical approximation of the MPA-CVD model equations. The practical implementation of the resulting DG MPA-CVD model is an extremely challenging task, which is prone to human error. Thereby, we introduce a mathematical approach for the symbolic formulation and computation of the underlying finite element method, based on automatic code generation. We extend this idea further such that the DG finite element formulation is automatically computed following the user's specification of the convective and viscous flux terms of the underlying PDE system in this symbolic framework. We then devise a method for writing a library of automatically generated DG finite element formulations for a hierarchy of partial differential equations with automatic treatment of prescribed boundary conditions. This toolbox for automatically computing DG finite element solutions is then applied to the DG MPA-CVD model. In particular, we consider reactor designs inspired by the AIXTRON and LIMHP reactors which are analysed extensively in the literature

Thermal modeling of subduction zones with prescribed and evolving 2D and 3D slab geometries

The determination of the temperature in and above the slab in subduction zones, using models where the top of the slab is precisely known, is important to test hypotheses regarding the causes of arc volcanism and intermediate-depth seismicity. While 2D and 3D models can predict the thermal structure with high precision for fixed slab geometries, a number of regions are characterized by relatively large geometrical changes. Examples include the flat slab segments in South America that evolved from more steeply dipping geometries to the present day flat slab geometry. We devise, implement, and test a numerical approach to model the thermal evolution of a subduction zone with prescribed changes in slab geometry over time. Our numerical model approximates the subduction zone geometry by employing time dependent deformation of a B\'ezier spline which is used as the slab interface in a finite element discretization of the Stokes and heat equations. We implement the numerical model using the FEniCS open source finite element suite and describe the means by which we compute approximations of the subduction zone velocity, temperature, and pressure fields. We compute and compare the 3D time evolving numerical model with its 2D analogy at cross-sections for slabs that evolve to the present-day structure of a flat segment of the subducting Nazca plate

A divergence free C0C^0-RIPG stream function formulation of the incompressible Stokes system with variable viscosity

Pointwise divergence free velocity field approximations of the Stokes system are gaining popularity due to their necessity in precise modelling of physical flow phenomena. Several methods have been designed to satisfy this requirement; however, these typically come at a greater cost when compared with standard conforming methods, for example, because of the complex implementation and development of specialized finite element bases. Motivated by the desire to mitigate these issues for 2D simulations, we present a $C^0$-interior penalty Galerkin (IPG) discretization of the Stokes system in the stream function formulation. In order to preserve a spatially varying viscosity this approach does not yield the standard and well known biharmonic problem. We further employ the so-called robust interior penalty Galerkin (RIPG) method; stability and convergence analysis of the proposed scheme is undertaken. The former, which involves deriving a bound on the interior penalty parameter is particularly useful to address the $\mathcal{O}(h^{-4})$ growth in the condition number of the discretized operator. Numerical experiments confirming the optimal convergence of the proposed method are undertaken. Comparisons with thermally driven buoyancy mantle convection model benchmarks are presented

Measures of Success: A Snapshot of the Montana Wolf Program in 2009

Montana’s gray wolf (Canis lupus) population continues to be secure, while the political and legal environments remain dynamic. Wolf delisting is a two-step process. Biological recovery criteria must be met and clearly demonstrated, along with an adequate regulatory framework. Secondly, the delisting decision must be upheld during inevitable legal challenges. The northern Rockies wolf population has met or exceeded numeric and connectivity requirements for many years. The northern Rockies gray wolf population was initially delisted in 2008, but a legal challenge reinstated federal legal protections under the Endangered Species Act mid-summer. By the end of 2008, Montana Fish, Wildlife and Parks estimated a minimum of 497 wolves in 84 verified packs, 34 of which met the definition of breeding pair. Federal delisting efforts resumed early in 2009 and took effect throughout Montana on 4 May. The second delisting decision was challenged again in Federal Court, although a preliminary injunction request to reinstate federal protections was denied in September. With delisting in Montana, the wolf was automatically reclassified as a species in need of management. Montana’s laws, administrative rules, and management plan also took effect. Montana Tribes lead wolf management activities on their respective reservations. The first fair chase wolf hunting season in Montana occurred in 2009. Seventy-two wolves were harvested through a quota-based framework. Wolves and their management continue to be controversial to a diversity of publics for a wide variety of reasons. Nonetheless, Montana’s wolf program has a solid regulatory foundation and the population is biologically sound. This presentation will provide an update on a variety of topics

Numerical modelling of MPA-CVD reactors with the discontinuous Galerkin finite element method

In this article we develop a fully self consistent mathematical model describing the formation of a hydrogen plasma in a microwave power assisted chemical vapour deposition (MPA-CVD) reactor employed for the manufacture of synthetic diamond. The underlying multi-physics model includes constituent equations for the background gas mass average velocity, gas temperature, electromagnetic field energy and plasma density. The proposed mathematical model is numerically approximated based on exploiting the discontinuous Galerkin finite element method. We demonstrate the practical performance of this computational approach on a variety of CVD reactor geometries for a range of operating conditions

Adaptive Wolf Management: The Regulated Public Harvest Component

Montana’s wolf (Canis lupus) conservation and management plan is based on adaptive management principles and includes regulated public harvest as a population management tool. The need and opportunity to implement public harvest in 2008, 2009, and 2010 required Montana Fish, Wildlife and Parks (FWP) to develop a stepped down adaptive management framework specific to harvest. For 2008 and 2009, FWP set modest objectives: implement a harvest, maintain a recovered population, and begin the learning process to inform development of future hunting regulations and quotas. In 2010, FWP used a formal Structured Decision Making Process to more clearly define priorities and challenges of setting a wolf season, outline objectives of a successful season, and evaluate consequences and trade-offs between alternative management actions. For all years, FWP used a modeling process to simulate a wide range of harvest rates across three harvest units and to predict harvest effects on the minimum number of wolves, packs and breeding pairs. Model inputs were derived from minimum wolf numbers observed in the field. Modeling allowed consideration of a range of harvest quotas, predicted outcomes, and risk that harvest could drive the population below federally-required minimums. It also facilitated explicit consideration of how well a particular quota achieved objectives and how to adapt future regulations and quotas. Legal challenges to federal delisting restricted implementation of the first fair chase hunting season to 2009. Montana’s wolf population is securely recovered, despite the dynamic political and legal environments. Regardless, FWP remains committed to a scientific, data-driven approach to adaptive management

Soluble TNF Mediates the Transition from Pulmonary Inflammation to Fibrosis

BACKGROUND: Fibrosis, the replacement of functional tissue with excessive fibrous tissue, can occur in all the main tissues and organ systems, resulting in various pathological disorders. Idiopathic Pulmonary Fibrosis is a prototype fibrotic disease involving abnormal wound healing in response to multiple sites of ongoing alveolar epithelial injury. METHODOLOGY/PRINCIPAL FINDINGS: To decipher the role of TNF and TNF-mediated inflammation in the development of fibrosis, we have utilized the bleomycin-induced animal model of Pulmonary Fibrosis and a series of genetically modified mice lacking components of TNF signaling. Transmembrane TNF expression is shown to be sufficient to elicit an inflammatory response, but inadequate for the transition to the fibrotic phase of the disease. Soluble TNF expression is shown to be crucial for lymphocyte recruitment, a prerequisite for TGF-b1 expression and the development of fibrotic lesions. Moreover, through a series of bone marrow transfers, the necessary TNF expression is shown to originate from the non-hematopoietic compartment further localized in apoptosing epithelial cells. CONCLUSIONS: These results suggest a primary detrimental role of soluble TNF in the pathologic cascade, separating it from the beneficial role of transmembrane TNF, and indicate the importance of assessing the efficacy of soluble TNF antagonists in the treatment of Idiopathic Pulmonary Fibrosis