A RBF partition of unity collocation method based on finite difference for initial-boundary value problems

Meshfree radial basis function (RBF) methods are popular tools used to numerically solve partial differential equations (PDEs). They take advantage of being flexible with respect to geometry, easy to implement in higher dimensions, and can also provide high order convergence. Since one of the main disadvantages of global RBF-based methods is generally the computational cost associated with the solution of large linear systems, in this paper we focus on a localizing RBF partition of unity method (RBF-PUM) based on a finite difference (FD) scheme. Specifically, we propose a new RBF-PUM-FD collocation method, which can successfully be applied to solve time-dependent PDEs. This approach allows to significantly decrease ill-conditioning of traditional RBF-based methods. Moreover, the RBF-PUM-FD scheme results in a sparse matrix system, reducing the computational effort but maintaining at the same time a high level of accuracy. Numerical experiments show performances of our collocation scheme on two benchmark problems, involving unsteady convection-diffusion and pseudo-parabolic equations

Meshless Hemodynamics Modeling And Evolutionary Shape Optimization Of Bypass Grafts Anastomoses

Objectives: The main objective of the current dissertation is to establish a formal shape optimization procedure for a given bypass grafts end-to-side distal anastomosis (ETSDA). The motivation behind this dissertation is that most of the previous ETSDA shape optimization research activities cited in the literature relied on direct optimization approaches that do not guaranty accurate optimization results. Three different ETSDA models are considered herein: The conventional, the Miller cuff, and the hood models. Materials and Methods: The ETSDA shape optimization is driven by three computational objects: a localized collocation meshless method (LCMM) solver, an automated geometry pre-processor, and a genetic-algorithm-based optimizer. The usage of the LCMM solver is very convenient to set an autonomous optimization mechanism for the ETSDA models. The task of the automated pre-processor is to randomly distribute solution points in the ETSDA geometries. The task of the optimized is the adjust the ETSDA geometries based on mitigation of the abnormal hemodynamics parameters. Results: The results reported in this dissertation entail the stabilization and validation of the LCMM solver in addition to the shape optimization of the considered ETSDA models. The LCMM stabilization results consists validating a custom-designed upwinding scheme on different one-dimensional and two-dimensional test cases. The LCMM validation is done for incompressible steady and unsteady flow applications in the ETSDA models. The ETSDA shape optimization include single-objective optimization results in steady flow situations and bi-objective optimization results in pulsatile flow situations. Conclusions: The LCMM solver provides verifiably accurate resolution of hemodynamics and is demonstrated to be third order accurate in a comparison to a benchmark analytical solution of the Navier-Stokes. The genetic-algorithm-based shape optimization approach proved to be very effective for the conventional and Miller cuff ETSDA models. The shape optimization results for those two models definitely suggest that the graft caliber should be maximized whereas the anastomotic angle and the cuff height (in the Miller cuff model) should be chosen following a compromise between the wall shear stress spatial and temporal gradients. The shape optimization of the hood ETSDA model did not prove to be advantageous, however it could be meaningful with the inclusion of the suture line cut length as an optimization parameter

Coupled/combined compact IRBF schemes for fluid flow and FSI problems

The thesis is concerned with the development of compact approximation methods based on Integrated Radial Basis Functions (IRBFs) and their applications in fluid flows and FSI problems. The contributions include (i) new compact IRBF stencils where first- and second-order derivatives are included; (ii) a preconditioning technique where a preconditioner to enhance the stability of the flat IRBF solutions; and, (iii) the incorporation of the proposed stencils into the immersed boundary methods. Numerical experiments show the present schemes generally produce more accurate solutions and better convergence rates than existing methods (e.g. FDM, high-order compact FDM and compact IRBF methods)

Explicit radial basis function collocation method for computing shallow water flows

A simple Explicit Radial Basis Function (RBF) is used to solve the shallow water equations (SWEs) for flows over irregular, frictional topography involving wetting and drying. At first we construct the MQ-RBF interpolation corresponding to space derivative operators. Next, we obtain numerical schemes to solve the SWEs, by using the gradient of the interpolant to approximate the spatial derivative of the differential equation and a third-order explicit Runge-Kutta scheme to approximate the temporal derivative of the differential equation. Then, we verify our scheme against several idealized one-dimensional numerical experiments including dam-break and open channel flows over non-uniform beds (involving shock wave behavior), and moving wet-dry fronts over irregular bed topography. Promising results are obtained

Development of mesh-free methods and their applications in computational fluid dynamics

Ph.DDOCTOR OF PHILOSOPH

Meshless Direct Numerical Simulation of Turbulent Incompressible Flows

A meshless direct pressure-velocity coupling procedure is presented to perform Direct Numerical Simulations (DNS) and Large Eddy Simulations (LES) of turbulent incompressible flows in regular and irregular geometries. The proposed method is a combination of several efficient techniques found in different Computational Fluid Dynamic (CFD) procedures and it is a major improvement of the algorithm published in 2007 by this author. This new procedure has very low numerical diffusion and some preliminary calculations with 2D steady state flows show that viscous effects become negligible faster that ever predicted numerically. The fundamental idea of this proposal lays on several important inconsistencies found in three of the most popular techniques used in CFD, segregated procedures, streamline-vorticity formulation for 2D viscous flows and the fractional-step method, very popular in DNS/LES. The inconsistencies found become important in elliptic flows and they might lead to some wrong solutions if coarse grids are used. In all methods studied, the mathematical basement was found to be correct in most cases, but inconsistencies were found when writing the boundary conditions. In all methods analyzed, it was found that it is basically impossible to satisfy the exact set of boundary conditions and all formulations use a reduced set, valid for parabolic flows only. For example, for segregated methods, boundary condition of normal derivative for pressure zero is valid only in parabolic flows. Additionally, the complete proposal for mass balance correction is right exclusively for parabolic flows. In the streamline-vorticity formulation, the boundary conditions normally used for the streamline function, violates the no-slip condition for viscous flow. Finally, in the fractional-step method, the boundary condition for pseudo-velocity implies a zero normal derivative for pressure in the wall (correct in parabolic flows only) and, when the flows reaches steady state, the procedure does not guarantee mass balance. The proposed procedure is validated in two cases of 2D flow in steady state, backward-facing step and lid-driven cavity. Comparisons are performed with experiments and excellent agreement was obtained in the solutions that were free from numerical instabilities. A study on grid usage is done. It was found that if the discretized equations are written in terms of a local Reynolds number, a strong criterion can be developed to determine, in advance, the grid requirements for any fluid flow calculation. The 2D-DNS on parallel plates is presented to study the basic features present in the simulation of any turbulent flow. Calculations were performed on a short geometry, using a uniform and very fine grid to avoid any numerical instability. Inflow conditions were white noise and high frequency oscillations. Results suggest that, if no numerical instability is present, inflow conditions alone are not enough to sustain permanently the turbulent regime. Finally, the 2D-DNS on a backward-facing step is studied. Expansion ratios of 1.14 and 1.40 are used and calculations are performed in the transitional regime. Inflow conditions were white noise and high frequency oscillations. In general, good agreement is found on most variables when comparing with experimental data

Development of Reduced-Order Meshless Solutions of Three-Dimensional Navier Stokes Transport Phenomena

Emerging meshless technologies are very promising for numerically solving Euler and Navier-Stokes transport systems in one-, two-, and three-dimensions (3-D). The Reduced-Order Meshless (ROM) technique developed in this work is applicable to a wide array of transport physics systems (i.e., fluid flow, heat transfer, gas dynamics, internal combustion flow and chemical reactions, and solid-liquid mixture flow) with various types of boundary and initial conditions. Such applications to be benchmarked in this work include one- and two-dimensional advection, and two- and three-dimensional convection-diffusion problems (Burgers’ equation). Computational solutions to these boundary-value problems will be demonstrated using the ROM approach and the predicted solutions will be posted against the Meshless Local Petrov-Galerkin (MLPG) method and exact solutions to these problems when they exist. Extensions to 3-D phenomenology will be attempted based on the conclusions obtained from computational studies to establish the existence, smoothness, and boundedness of 3-D Navier-Stokes transport systems. An approximated benchmark solution of the Navier-Stokes equations is also developed in this work using a linearized perturbation analysis. The classical paper on gas turbine throughflow, Three Dimensional Flows in Turbomachines (Marble, 1964), outlines this procedure for approximation, and produces solutions for a class of axisymmetric problems. An investigation into the behavior of these solutions uncovered a series of inconsistencies in the paper, which are outlined in detail and corrected when known to be in error.This research was supported by The Ohio State University College of Engineering

Finite Element Method for Analysis of Conjugate Heat Transfer between Solid and Unsteady Viscous Flow

A fractional four-step finite element method for analyzing conjugate heat transfer between solid and unsteady viscous flow is presented. The second-order semi-implicit Crank-Nicolson scheme is used for time integration and the resulting nonlinear equations are linearized without losing the overall time accuracy. The streamline upwind Petrov-Galerkin method (SUPG) is applied for the weighted formulation of the Navier-Stokes equations. The method uses a three-node triangular element with equal-order interpolation functions for all the variables of the velocity components, the pressure and the temperature. The main advantage of the method presented is to consistently couple heat transfer along the fluid-solid interface. Four test cases, which are the lid-driven cavity flow, natural convection in a square cavity, transient flow over a heated circular cylinder and forced convection cooling across rectangular blocks, are selected to evaluate the efficiency of the method presented

Entropically damped artificial compressibility for the discretization corrected particle strength exchange method in incompressible fluid mechanics

peer reviewedWe present a consistent mesh-free numerical scheme for solving the incompressible Navier–Stokes equations. Our method is based on entropically damped artificial compressibility for imposing the incompressibility con- straint explicitly, and the Discretization-Corrected Particle Strength Exchange (DC-PSE) method to consistently discretize the differential operators on mesh-free particles. We further couple our scheme with Brinkman penalization to solve the Navier–Stokes equations in complex geometries. The method is validated using the 3D Taylor–Green vortex flow and the lid-driven cavity flow problem in 2D and 3D, where we also compare our method with hr-SPH and report better accuracy for DC-PSE. In order to validate DC-PSE Brinkman penalization, we study flow past obstacles, such as a cylinder, and report excellent agreement with previous studies.R-AGR-3952 - C20/MS/14610324-PorSol (01/02/2021 - 31/01/2024) - OBEIDAT Ana