An Improved Implementation and Inalysis of the Diaz and O'Rourke Algorithm for Finding the Simpson Point of a Convex Polygon

This paper focuses on the well-known Diaz and O'Rourke [M. Diaz and J. O'Rourke, Algorithms for computing the center of area of a convex polygon, Visual Comput. 10 (1994), 432–442.] iterative search algorithm to find the Simpson Point of a market, described by a convex polygon. In their paper, they observed that their algorithm did not appear to converge pointwise, and therefore, modified it to do so. We first present an enhancement of their algorithm that improves its time complexity from O(log2?) to O(n log 1/?). This is then followed by a proof of pointwise convergence and derivation of explicit bounds on convergence rates of our algorithm. It is also shown that with an appropriate interpretation, our convergence results extend to all similar iterative search algorithms to find the Simpson Point – a class that includes the original unmodified Diaz–O'Rourke algorithm. Finally, we explore how our algorithm and its convergence guarantees might be modified to find the Simpson Point when the demand distribution is non-uniform

Computational Geometry Column 45

The algorithm of Edelsbrunner for surface reconstruction by wrapping a set of points in R3 is described

Outflow boundary conditions for 3D simulations of non-periodic blood flow and pressure fields in deformable arteries

The simulation of blood flow and pressure in arteries requires outflow boundary conditions that incorporate models of downstream domains. We previously described a coupled multidomain method to couple analytical models of the downstream domains with 3D numerical models of the upstream vasculature. This prior work either included pure resistance boundary conditions or impedance boundary conditions based on assumed periodicity of the solution. However, flow and pressure in arteries are not necessarily periodic in time due to heart rate variability, respiration, complex transitional flow or acute physiological changes. We present herein an approach for prescribing lumped parameter outflow boundary conditions that accommodate transient phenomena. We have applied this method to compute haemodynamic quantities in different physiologically relevant cardiovascular models, including patient-specific examples, to study non-periodic flow phenomena often observed in normal subjects and in patients with acquired or congenital cardiovascular disease. The relevance of using boundary conditions that accommodate transient phenomena compared with boundary conditions that assume periodicity of the solution is discussed

Computational Geometry Column 37

Open problems from the 15th Annual ACM Symposium on Computational Geometry

Computational fluid dynamics combustion analysis evaluation

This study involves the development of numerical modelling in spray combustion. These modelling efforts are mainly motivated to improve the computational efficiency in the stochastic particle tracking method as well as to incorporate the physical submodels of turbulence, combustion, vaporization, and dense spray effects. The present mathematical formulation and numerical methodologies can be casted in any time-marching pressure correction methodologies (PCM) such as FDNS code and MAST code. A sequence of validation cases involving steady burning sprays and transient evaporating sprays will be included

Aortic Coarctation: Recent Developments in Experimental and Computational Methods to Assess Treatments for this Simple Condition

Coarctation of the aorta (CoA) is often considered a relatively simple disease, but long-term outcomes suggest otherwise as life expectancies are decades less than in the average population and substantial morbidity often exists. What follows is an expanded version of collective work conducted by the authors\u27 and numerous collaborators that was presented at the 1st International Conference on Computational Simulation in Congenital Heart Disease pertaining to recent advances for CoA. The work begins by focusing on what is known about blood flow, pressure and indices of wall shear stress (WSS) in patients with normal vascular anatomy from both clinical imaging and the use of computational fluid dynamics (CFD) techniques. Hemodynamic alterations observed in CFD studies from untreated CoA patients and those undergoing surgical or interventional treatment are subsequently discussed. The impact of surgical approach, stent design and valve morphology are also presented for these patient populations. Finally, recent work from a representative experimental animal model of CoA that may offer insight into proposed mechanisms of long-term morbidity in CoA is presented

Towards a theory of automated elliptic mesh generation

The theory of elliptic mesh generation is reviewed and the fundamental problem of constructing computational space is discussed. It is argued that the construction of computational space is an NP-Complete problem and therefore requires a nonstandard approach for its solution. This leads to the development of graph-theoretic, combinatorial optimization and integer programming algorithms. Methods for the construction of two dimensional computational space are presented

Lower Bounds for Pinning Lines by Balls

A line L is a transversal to a family F of convex objects in R^d if it intersects every member of F. In this paper we show that for every integer d>2 there exists a family of 2d-1 pairwise disjoint unit balls in R^d with the property that every subfamily of size 2d-2 admits a transversal, yet any line misses at least one member of the family. This answers a question of Danzer from 1957

Incorporating the Aortic Valve into Computational Fluid Dynamics Models using Phase-Contrast MRI and Valve Tracking

The American Heart Association states about 2% of the general population have a bicuspid aortic valve (BAV). BAVs exist in 80% of patients with aortic coarctation (CoA) and likely influences flow patterns that contribute to long-term morbidity post-surgically. BAV patients tend to have larger ascending aortic diameters, increased risk of aneurysm formation, and require surgical intervention earlier than patients with a normal aortic valve. Magnetic resonance imaging (MRI) has been used clinically to assess aortic arch morphology and blood flow in these patients. These MRI data have been used in computational fluid dynamics (CFD) studies to investigate potential adverse hemodynamics in these patients, yet few studies have attempted to characterize the impact of the aortic valve on ascending aortic hemodynamics. To address this issue, this research sought to identify the impact of aortic valve morphology on hemodynamics in the ascending aorta and determine the location where the influence is negligible. Novel tools were developed to implement aortic valve morphology into CFD models and compensate for heart motion in MRI flow measurements acquired through the aortic valve. Hemodynamic metrics such as blood flow velocity, time-averaged wall shear stress (TAWSS), and turbulent kinetic energy (TKE) induced by the valve were compared to values obtained using the current plug inflow approach. The influence of heart motion on these metrics was also investigated, resulting in the underestimation of TAWSS and TKE when heart motion was neglected. CFD simulations of CoA patients exhibiting bicuspid and tricuspid aortic valves were performed in models including the aortic sinuses and patient-specific valves. Results indicated the aortic valve impacted hemodynamics primarily in the ascending aorta, with the BAV having the greatest influence along the outer right wall of the vessel. A marked increase in TKE is present in aortic valve simulations, particularly in BAV patients. These findings suggest that future CFD studies investigating altered hemodynamics in the ascending aorta should accurately replicate aortic valve morphology. Further, aortic valve disease impacts hemodynamics in the ascending aorta that may be a predictor of the development or progression of ascending aortic dilation and possible aneurysm formation in this region