On the coefficients of differentiated expansions of ultraspherical polynomials

A formula expressing the coefficients of an expression of ultraspherical polynomials which has been differentiated an arbitrary number of times in terms of the coefficients of the original expansion is proved. The particular examples of Chebyshev and Legendre polynomials are considered

Conforming Chebyshev spectral collocation methods for the solution of laminar flow in a constricted channel

The numerical simulation of steady planar two-dimensional, laminar flow of an incompressible fluid through an abruptly contracting channel using spectral domain decomposition methods is described. The key features of the method are the decomposition of the flow region into a number of rectangular subregions and spectral approximations which are pointwise C(1) continuous across subregion interfaces. Spectral approximations to the solution are obtained for Reynolds numbers in the range 0 to 500. The size of the salient corner vortex decreases as the Reynolds number increases from 0 to around 45. As the Reynolds number is increased further the vortex grows slowly. A vortex is detected downstream of the contraction at a Reynolds number of around 175 that continues to grow as the Reynolds number is increased further

On efficient direct methods for conforming spectral domain decomposition techniques

AbstractA conforming spectral domain decomposition technique is described for the solution of Stokes flow in rectangularly decomposable domains. The matrices arising from such a spectral discretization procedure possess a block tridiagonal structure where these blocks are full submatrices. Efficient direct solution procedures are proposed to take advantage of the matrix structure. A comparison of the methods in terms of computational efficiency is made. Numerical results are presented for the flow through an abruptly contracting channel

Least-squares proper generalized decompositions for weakly coercive elliptic problems

Proper generalised decompositions (PGDs) are a family of methods for efficiently solving high-dimensional PDEs which seek to find a low-rank approximation to the solution of the PDE a priori. Convergence of PGD algorithms can only be proven for problems which are continuous, symmetric and strongly coercive. In the particular case of problems which are only weakly coercive we have the additional issue that weak coercivity estimates are not guaranteed to be inherited by the low-rank PGD approximation. This can cause stability issues when employing a Galerkin PGD approximation of weakly coercive problems. In this paper we propose the use of PGD algorithms based on least-squares formulations which always lead to symmetric and strongly coercive problems and hence provide stable and provably convergent algorithms. Taking the Stokes problem as a prototypical example of a weakly coercive problem, we develop and compare rigorous least-squares PGD algorithms based on continuous least-squares estimates for two different reformulations of the problem. We show that these least-squares PGD provide a much stabler algorithm than an equivalent Galerkin PGD and provide proofs of convergence of the algorithms

A moment-of-fluid method for resolving filamentary structures using a symmetric multi-material approach

Multiphase flows have implications in many areas of engineering. The moment-of-fluid (MOF) method is an interface capturing method using both volume fraction and centroid within a cell for interface reconstruction. A symmetric approach to reconstruct thin structures is presented. Also called filaments, these subcell characteristics involve multi-material reconstruction. In addition, a new optimisation algorithm is presented using a bisection method without any necessary initial condition. Using a Lagrangian approach for dynamic cases, no restrictions are imposed on timestep. The new method is validated using several benchmark cases that have been studied extensively in the literature. A near quadratic order of convergence and high accuracy is achieved while maintaining an acceptable runtime

Magnetic nanodrug delivery in non-Newtonian blood flows

With the goal of determining strategies to maximise drug delivery to a specific site in the body, we developed a mathematical model for the transport of drug nanocarriers (nanoparticles) in the bloodstream under the influence of an external magnetic field. Under the assumption of long (compared to the radius) blood vessels the Navier-Stokes equations are reduced, to a simpler model consistently with lubrication theory. Under these assumptions, analytical results are compared for Newtonian, power-law, Carreau and Ellis fluids, and these clearly demonstrate the importance of shear thinning effects when modelling blood flow. Incorporating nanoparticles and a magnetic field to the model we develop a numerical scheme and study the particle motion for different field strengths. We demonstrate the importance of the non-Newtonian behaviour: for the flow regimes investigated in this work, consistent with those in blood micro vessels, we find that the field strength needed to absorb a certain amount of particles in a non-Newtonian fluid has to be larger than the one needed in a Newtonian fluid. Specifically, for one case examined, a two times larger magnetic force had to be applied in the Ellis fluid than in the Newtonian fluid for the same number of particles to be absorbed through the vessel wall. Consequently, models based on a Newtonian fluid can drastically overestimate the effect of a magnetic field. Finally, we evaluate the particle concentration at the vessel wall and compute the evolution of the particle flux through the wall for different permeability values, as that is important when assessing the efficacy of drug delivery applications. The insights from our work bring us a step closer to successfully transferring magnetic nanoparticle drug delivery to the clinic

Property preserving reformulation of constitutive laws for the conformation tensor

The challenge for computational rheologists is to develop efficient and stable numerical schemes in order to obtain accurate numerical solutions for the governing equations at values of practical interest of the Weissenberg numbers. This study presents a new approach to preserve the symmetric positive definiteness of the conformation tensor and to bound the magnitude of its eigenvalues. The idea behind this transformation is lies with the matrix logarithm formulation. Under the logarithmic transformation, the eigenvalue spectrum of the new conformation tensor varies from infinite positive to infinite negative. But, reconstruction the classical formulation from unbounded eigenvalues doesn't achieve meaningful results. This enhanced formulation, hyperbolic tangent, prevails the previous numerical failure by bounding the magnitude of eigenvalues in a manner that positive definite is always satisfied. In order to evaluate the capability of the hyperbolic tangent formulation we performed a numerical simulation of FENE-P fluids in a rectangular channel in the context of the finite element method. Under this new transformation, the maximum attainable Weissenberg number increases 21.4% and 112.5% comparing the standard log-conformation and classical constitutive equation respectively

A quadtree-based adaptive moment-of-fluid method for interface reconstruction with filaments

Implementation of quadtree adaptive mesh refinement (AMR) to the moment-of-fluid (MOF) method is presented in the context of an interface capturing method. Filaments, thinner than a cell size, are resolved using a computationally efficient technique on an unconstrained quadtree structure. The centroid defect relative to its cell size is used as the refinement criterion, together with an enhanced refinement calculation and subsequently its volume conservation. In addition, different approaches are proposed to ensure mass conservation during the computation. This MOF-AMR framework is validated for a range of benchmark problems which are studied widely in the literature. There is no restriction on the choice of CFL number for the purely Lagrangian advection method considered here and this has advantages when combined with AMR. The current quadtree MOF-AMR method leads to much improved computational efficiency and accuracy relative to its grid size compared with a uniform grid. Higher levels of refinement can be costly, therefore the efficiency of mesh resolution is further discussed in relation to the choice of time step and number of AMR levels

Reduced Efficacy of Anti-A\u3cem\u3eβ\u3c/em\u3e Immunotherapy in a Mouse Model of Amyloid Deposition and Vascular Cognitive Impairment Comorbidity

Vascular cognitive impairment and dementia (VCID) is the second most common form of dementia behind Alzheimer\u27s disease (AD). It is estimated that 40% of AD patients also have some form of VCID. One promising therapeutic for AD is anti-Aβ immunotherapy, which uses antibodies against Aβ to clear it from the brain. While successful in clearing Aβ and improving cognition in mice, anti-Aβ immunotherapy failed to reach primary cognitive outcomes in several different clinical trials. We hypothesized that one potential reason the anti-Aβ immunotherapy clinical trials were unsuccessful was due to this high percentage of VCID comorbidity in the AD population. We used our unique model of VCID-amyloid comorbidity to test this hypothesis. We placed 9-month-old wild-type and APP/PS1 mice on either a control diet or a diet that induces hyperhomocysteinemia (HHcy). After being placed on the diet for 3 months, the mice then received intraperotineal injections of either IgG2a control or 3D6 for another 3 months. While we found that treatment of our comorbidity model with 3D6 resulted in decreased total Aβ levels, there was no cognitive benefit of the anti-Aβ immunotherapy in our AD/VCID mice. Further, microhemorrhages were increased by 3D6 in the APP/PS1/control but further increased in an additive fashion when 3D6 was administered to the APP/PS1/HHcy mice. This suggests that the use of anti-Aβ immunotherapy in patients with both AD and VCID would be ineffective on cognitive outcomes