A localized radial basis function (RBF) meshless method is developed for coupled viscous fluid flow and convective heat transfer problems. The method is based on new localized radial-basis function (RBF) expansions using Hardy Multiquadrics for the sought-after unknowns. An efficient set of formulae are derived to compute the RBF interpolation in terms of vector products thus providing a substantial computational savings over traditional meshless methods. Moreover, the approach developed in this paper is applicable to explicit or implicit time marching schemes as well as steady-state iterative methods. We apply the method to viscous fluid flow and conjugate heat transfer (CHT) modeling. The incompressible Navier‐Stokes are time marched using a Helmholtz potential decomposition for the velocity field. When CHT is considered, the same RBF expansion is used to solve the heat conduction problem in the solid regions enforcing temperature and heat flux continuity of the solid/fluid interfaces. The computation is accelerated by distributing the load over several processors via a domain decomposition along with an interface interpolation tailored to pass information through each of the domain interfaces to ensure conservation of field variables and derivatives. Numerical results are presented for several cases including channel flow, flow in a channel with a square step obstruction, and a jet flow into a square cavity. Results are compared with commercial computational fluid dynamics code predictions. The proposed localized meshless method approach is shown to produce accurate results while requiring a much-reduced effort in problem preparation in comparison to other traditional numerical methods. DOI: 10.1115/1.2402181

An Efficient Localized Radial Basis Function Meshless Method for Fluid Flow and Conjugate Heat Transfer

A localized radial-basis function (RBF) collocation meshless method is developed for natural-convection heat transfer problems in fully viscous fluid flows. The expansion method is based on the localized collocation of polynomial-augmented Hardy multiquadrics RBF, and it is efficiently formulated to generate derivative fields through simple inner products of small-order vectors. The solution of the Navier-Stokes equations is formulated using a third-order-accurate explicit fractional time-stepping method and a velocity-correction scheme. Several cases are studied and confirmed to attain accurate results when compared to classical benchmark solutions as well as numerical predictions provided by the commercial computational fluid dynamics code Fluent.

Localized Meshless Modeling of Natural-Convective Viscous Flows

The current paper reports on the development and validation of a localized collocation meshless method (LCMM) to model laminar incompressible flows. A high order upwinding scheme was devised to dampen the numerical oscillations arising in convection-dominated flows. Subsequently, the LCMM was analytically validated and demonstrated to yield third-order accurate solutions when compared to a benchmark analytical decaying vortex solution. Numerical validations are provided by comparison with the finite volume commercial (FVM) solver Fluent 6.2. The flow geometry for the numerical validation arises from a biomedical application that consists of modeling blood flow in the interconnection between a bypass graft and an artery. Very good agreement was found between the LCMM and the FVM.

A localized collocation meshless method (LCMM) for incompressible flows CFD modeling with applications to transient hemodynamics

We develop an effective domain decomposition meshless methodology for conjugate heat transfer problems modeled by convecting fully viscous incompressible fluid interacting with conducting solids. The meshless formulation for fluid flow modeling is based on a radial basis function interpolation using Hardy inverse Multiquadrics and a time-progression decoupling of the equations using a Helmholtz potential. The domain decomposition approach effectively reduces the conditioning numbers of the resulting algebraic systems, arising from convective and conduction modeling, while increasing efficiency of the solution process and decreasing memory requirements. Moreover, the domain decomposition approach is ideally suited for parallel computation. Numerical examples are presented to validate the approach by comparing the meshless solutions to finite volume method (FVM) solutions provided by a commercial CFD solver.

Iterative domain decomposition meshless method modeling of incompressible viscous flows and conjugate heat transfer

Improving the blood flow or hemodynamics in the synthetic bypass graft end-to-side distal anastomosis (ETSDA) is an important element for the long-term success of bypass surgeries. An ETSDA is the interconnection between the graft and the operated-on artery. The control of hemodynamic conditions through the ETSDA is mostly dictated by the shape of the ETSDA. Thus, a formal ETSDA shape optimization would serve the goal of improving the ETSDA flowfield. Computational fluid dynamics (CFD) is a convenient tool to quantify hemodynamic parameters; also, the genetic algorithm (GA) is an effective tool to identify the ETSDA optimal shape that modify those hemodynamic quantities such that the optimization objective is met. The present article introduces a unique approach where a meshless CFD solver is coupled to a GA for the purpose of optimizing the ETSDA shape. Three anastomotic models are optimized herein: the conventional ETSDA, the Miller cuff ETSDA and the hood ETSDA. Results demonstrate the effectiveness of...

A meshless CFD approach for evolutionary shape optimization of bypass grafts anastomoses

Mesh reduction methods such as the boundary element methods, method of fundamental solutions or the so-called meshless methods all lead to fully populated matrices. This poses serious challenges for large-scale three-dimensional problems due to storage requirements and iterative solution of a large set of non-symmetric equations. Researchers have developed several approaches to address this issue including the class of fast-multipole techniques, use of wavelet transforms, and matrix decomposition. In this paper, we develop a domain- decomposition, or the artificial sub-sectioning technique, along with a region-by-region iteration algorithm particularly tailored for parallel computation to address the coefficient matrix issue. The meshless method we employ is based on expansions using radial basis functions (RBFs). An efficient physically-based procedure provides an effective initial guess of the temperatures along the sub-domain interfaces. The iteration process converges very efficiently, offers substantial savings in memory, and features superior computational efficiency. The meshless iterative domain decomposition technique is ideally suited for parallel computation. We discuss its implementation under MPI standards on a small Windows XP PC cluster. Numerical results reveal the domain decomposition meshless methods produce accurate temperature predictions while requiring a much- reduced effort in problem preparation in comparison to other traditional numerical methods.

A Parallel Domain Decomposition Technique for Meshless Methods Applications to Large-Scale Heat Transfer Problems

In the current paper we introduce the localized meshless method to resolve the two-dimensional blood flow in the vicinity of a peripheral bypass graft end-to-side distal anastomosis. The goal is to incorporate this new numerical technique in extracting the values of the fluid mechanics wall parameters, such as the wall shear stress and the wall shear stress gradients, which are suggested as contributory factors to the growth of post-operative intimal hyperplasia at the anastomosis. The localized meshless method depends on the Hardy Multiquadrics radial basis function to locally expand the flow variables over a set of nodes distributed in the computational domain. An explicit scheme is adapted for the meshless formulation of the laminar incompressible Navier Stokes equations. Our special interest in the localized meshless method arises from its automated point distribution feature that significantly facilitates the pre-processing of the solution. The blood flow is simulated in three different anastomosis model geometries; the conventional or direct model, the Miller Cuff model, and the Taylor Patch model. The results of the current localized meshless numerical method show a great agreement with the results provided by a well-established finite volume method commercial software.Copyright © 2006 by ASME

Eric A. Mitteff

Papers

A Parallel Domain Decomposition Technique for Meshless Methods Applications to Large-Scale Heat Transfer Problems

Two-Dimensional Meshless Numerical Modeling of the Blood Flow Within Arterial End-to-Side Distal Anastomoses