NURBS-SEM: A hybrid spectral element method on NURBS maps for the solution of elliptic PDEs on surfaces

Non Uniform Rational B-spline (NURBS) patches are a standard way to describe complex geometries in Computer Aided Design tools, and have gained a lot of popularity in recent years also for the approximation of partial differential equations, via the Isogeometric Analysis (IGA) paradigm. However, spectral accuracy in IGA is limited to relatively small NURBS patch degrees (roughly p 648), since local condition numbers grow very rapidly for higher degrees. On the other hand, traditional Spectral Element Methods (SEM) guarantee spectral accuracy but often require complex and expensive meshing techniques, like transfinite mapping, that result anyway in inexact geometries. In this work we propose a hybrid NURBS-SEM approximation method that achieves spectral accuracy and maintains exact geometry representation by combining the advantages of IGA and SEM. As a prototypical problem on non trivial geometries, we consider the Laplace\u2013Beltrami and Allen\u2013Cahn equations on a surface. On these problems, we present a comparison of several instances of NURBS-SEM with the standard Galerkin and Collocation Isogeometric Analysis (IGA)

On the Application of Reduced Basis Methods to Bifurcation Problems in Incompressible Fluid Dynamics

In this paper we apply a reduced basis framework for the computation of flow bifurcation (and stability) problems in fluid dynamics. The proposed method aims at reducing the complexity and the computational time required for the construction of bifurcation and stability diagrams. The method is quite general since it can in principle be specialized to a wide class of nonlinear problems, but in this work we focus on an application in incompressible fluid dynamics at low Reynolds numbers. The validation of the reduced order model with the full order computation for a benchmark cavity flow problem is promising

Alpha-fetoprotein and modified response evaluation criteria in Solid Tumors progression after locoregional therapy as predictors of hepatocellular cancer recurrence and death after transplantation.

Locoregional therapy (LRT) is being increasingly used for the management of hepatocellular cancer (HCC) in patients listed for liver transplantation (LT). Although several selection criteria have been developed, stratifications of survival according to the pathology of explanted livers and pre-LT LRT are lacking. Radiological progression according to the modified Response Evaluation Criteria in Solid Tumors (mRECIST) and alpha-fetoprotein (AFP) behavior was reviewed for 306 patients within the Milan criteria (MC-IN) and 116 patients outside the Milan criteria (MC-OUT) who underwent LRT and LT between January 1999 and March 2010. A prospectively collected database originating from 6 collaborating European centers was used for the study. Sixty-one patients (14.5%) developed HCC recurrence. For both MC-IN and MC-OUT patients, an AFP slope > 15 ng/mL/month and mRECIST progression were unique independent risk factors for HCC recurrence and patient death. When the radiological Milan criteria (MC) status was combined with radiological and biological progression, MC-IN and MC-OUT patients without risk factors had similarly excellent 5-year tumor-free and patient survival rates. MC-IN patients with at least 1 risk factor had worse outcomes, and MC-OUT patients with at least 1 risk factor had the poorest survival (P < 0.001). In conclusion, both radiological and biological modifications permit documentation of the response to LRT in patients waiting for LT. According to these 2 parameters, tumor progression significantly increases the risk of recurrence and patient death not only for MC-OUT patients but also for MC-IN patients. The monitoring of both parameters in combination with the initial radiological MC status is an essential element for further refining the selection criteria for potential liver recipients with HCC. Liver Transpl 19:1108-1118, 2013. © 2013 AASLD