Tutorial on Hybridizable Discontinous Galerkin (HDG) for second-order elliptic problems

The HDG is a new class of discontinuous Galerkin (DG) methods that shares favorable properties with classical mixed methods such as the well known Raviart-Thomas methods. In particular, HDG provides optimal convergence of both the primal and the dual variables of the mixed formulation. This property enables the construction of superconvergent solutions, contrary to other popular DG methods. In addition, its reduced computational cost, compared to other DG methods, has made HDG an attractive alternative for solving problems governed by partial differential equations. A tutorial on HDG for the numerical solution of second-order elliptic problems is presented. Particular emphasis is placed on providing all the necessary details for the implementation of HDG methods.Peer ReviewedPreprin

Allogeneic blood stem cell transplantation after a reduced-intensity, preparative regimen : A pilot study in patients with refractory malignancies

BACKGROUND. The immune-mediated graft-versus-tumor (GVT) effect plays a therapeutic role in the treatment of patients with hematologic malignancies who undergo allogeneic hematopoietic stem cell transplantation (HSCT). More recently, it was reported that a GVT effect also occurred in patients who underwent transplantation for metastatic renal carcinoma. The authors carried out a pilot trial of allogeneic transplantation after a reduced-intensity, preparative regimen in patients with refractory malignancies, including solid tumors. The objectives of the current study were to evaluate the feasibility of this approach in terms of toxicity and engraftment and to document evidence of GVT effects. METHODS. Seventeen patients with Stage IV malignancies (7 patients with renal cell carcinoma, 3 patients with sarcoma, 2 patients with breast carcinoma, 2 patients with Hodgkin disease, 1 patient with ovarian carcinoma, 1 patient with melanoma, and 1 patient with both melanoma and renal cell carcinoma) that were not amenable to further conventional treatment were enrolled. The median patient age was 43 years (range, 10-60 years). The Eastern Cooperative Oncology Group performance status (PS) was 0-1 in 11 patients and 2-3 in 6 patients. Preparative treatment consisted of reduced-intensity chemotherapy with fludarabine (30 mg/m2 per day for 4 consecutive days) and cyclophosphamide (30 mg/Kg per day for 2 consecutive days) prior to allogeneic HSCT from a human leukocyte antigen-identical sibling. The median number of CD34+ cells infused was 6.06 X 106/kg (range, 1.5-14.0 X 106/kg). Graft-versus-host disease (GVHD) prophylaxis consisted of cyclosporin-A and short-term methotrexate. RESULTS. Patients who had a PS of 2-3 prior to undergoing HSCT experienced Grade 4 hematologic toxicities and Grade 65 3 organ toxicities and died of either treatment-related complications or disease progression within 100 days from transplantation. By contrast, 10 of 11 patients who had a PS of 0-1 prior to undergoing HSCT experienced only short-lasting, Grade 64 3 neutropenia and thrombocytopenia and no organ toxicity; 1 of 10 patients died of graft failure on Day +29 after undergoing HSCT. By Day +90, 100% donor chimerism was documented in all patients with a past history of heavy chemotherapy, whereas mixed donor chimerism was observed in the 4 patients with a past history of only 1 line of chemotherapy and/or immunotherapy prior to entering the HSCT program. Grade 2-3 acute GVHD occurred in 5 patients. Among patients with a follow-up > 100 days, 2 complete responses and 3 transitory partial responses were recorded. CONCLUSIONS. With this conditioning regimen, full donor chimerism was achieved rapidly only in patients who had received previous intensive chemotherapy. In a proportion of patients with refractory malignancies, allogeneic transplantation resulted in tumor regression. This novel therapeutic strategy may provide little benefit in patients with poor PS and rapidly progressing disease

HDG-NEFEM with Degree Adaptivity for Stokes Flows

This paper presents the first degree adaptive procedure able to directly use the geometry given by a CAD model. The technique uses a hybridisable discontinuous Galerkin discretisation combined with a NURBS-enhanced rationale, completely removing the uncertainty induced by a polynomial approximation of curved boundaries that is common within an isoparametric approach. The technique is compared against two strategies to perform degree adaptivity currently in use. This paper demonstrates, for the first time, that the most extended technique for degree adaptivity can easily lead to a non-reliable error estimator if no communication with CAD software is introduced whereas if the communication with the CAD is done, it results in a substantial computing time. The proposed technique encapsulates the CAD model in the simulation and is able to produce reliable error estimators irrespectively of the initial mesh used to start the adaptive process. Several numerical examples confirm the findings and demonstrate the superiority of the proposed technique. The paper also proposes a novel idea to test the implementation of high-order solvers where different degrees of approximation are used in different elements

A high-order non field-aligned approach for the discretization of strongly anistropic diffusion operators in magnetic fusion

In this work we present a hybrid discontinuous Galerkin scheme for the solution of extremely anisotropic diffusion problems arising in magnetized plasmas for fusion applications. Unstructured meshes, non-aligned with respect to the dominant diffusion direction, allow an unequalled flexibility in discretizing geometries of any shape, but may lead to spurious numerical diffusion. Curved triangles or quadrangles are used to discretize the poloidal plane of the machine, while a structured discretization is used in the toroidal direction. The proper design of the numerical fluxes guarantees the correct convergence order at any anisotropy level. Computations performed on well-designed 2D and 3D numerical tests show that non-aligned discretizations are able to provide spurious diffusion free solutions as long as high-order interpolations are used. Introducing an explicit measure of the numerical diffusion, a careful investigation is carried out showing an exponential increase of this latest with respect to the non-alignment of the mesh with the diffusion direction, as well as an exponential decrease with the polynomial degree of interpolation. A brief assessment of the method with respect to two finite-difference schemes using non-aligned discretization, but classically used in fusion modeling, is also presented

Numerical Study of 2D Vertical Axis Wind and Tidal Turbines with a Degree-Adaptive Hybridizable Discontinuous Galerkin Method

The book encompasses novel CFD techniques to compute offshore wind and tidal applications. Computational fluid dynamics (CFD) techniques are regarded as the main design tool to explore the new engineering challenges presented by offshore wind and tidal turbines for energy generation. The difficulty and costs of undertaking experimental tests in offshore environments have increased the interest in the field of CFD which is used to design appropriate turbines and blades, understand fluid flow physical phenomena associated with offshore environments, predict power production or characterise offshore environments, amongst other topics.Peer ReviewedPostprint (author's final draft

A high-order non field-aligned approach for the discretization of strongly anistropic diffusion operators in magnetic fusion

In this work we present a hybrid discontinuous Galerkin scheme for the solution of extremely anisotropic diffusion problems arising in magnetized plasmas for fusion applications. Unstructured meshes, non-aligned with respect to the dominant diffusion direction, allow an unequalled flexibility in discretizing geometries of any shape, but may lead to spurious numerical diffusion. Curved triangles or quadrangles are used to discretize the poloidal plane of the machine, while a structured discretization is used in the toroidal direction. The proper design of the numerical fluxes guarantees the correct convergence order at any anisotropy level. Computations performed on well-designed 2D and 3D numerical tests show that non-aligned discretizations are able to provide spurious diffusion free solutions as long as high-order interpolations are used. Introducing an explicit measure of the numerical diffusion, a careful investigation is carried out showing an exponential increase of this latest with respect to the non-alignment of the mesh with the diffusion direction, as well as an exponential decrease with the polynomial degree of interpolation. A brief assessment of the method with respect to two finite-difference schemes using non-aligned discretization, but classically used in fusion modeling, is also presented