Transient Non-linear Heat Conduction Solution by a Dual Reciprocity Boundary Element Method with an Effective Posteriori Error Estimator

A Dual Reciprocity Boundary Element Method is formulated to solve non-linear heat conduction problems. The approach is based on using the Kirchhoff transform along with lagging of the effective non-linear thermal diffusivity. A posteriori error estimate is used to provide effective estimates of the temporal and spatial error. A numerical example is used to demonstrate the approach

Singular Superposition/Boundary Element Method for Reconstruction of Multi-dimensional Heat Flux Distributions with Application to Film Cooling Holes

A hybrid singularity superposition/boundary element-based inverse problem method for the reconstruction of multi-dimensional heat flux distributions is developed. Cauchy conditions are imposed at exposed surfaces that are readily reached for measurements while convective boundary conditions are unknown at surfaces that are not amenable to measurements such as the walls of the cooling holes. The purpose of the inverse analysis is to determine the heat flux distribution along cooling hole surfaces. This is accomplished in an iterative process by distributing a set of singularities (sinks) inside the physical boundaries of the cooling hole (usually along cooling hole centerline) with a given initial strength distribution. A forward steady-state heat conduction problem is solved using the boundary element method (BEM), and an objective function is defined to measure the difference between the heat flux measured at the exposed surfaces and the heat flux predicted by the BEM under the current strength distribution of the singularities. A Genetic Algorithm (GA) iteratively alters the strength distribution of the singularities until the measuring surfaces heat fluxes are matched, thus satisfying Cauchy conditions. The distribution of the heat flux at the walls of the cooling hole is determined in a post-processing stage after the inverse problem is solved. The advantage of this technique is to eliminate the need of meshing the surfaces of the cooling holes, which requires a large amount of effort to achieve a high quality mesh. Moreover, the use of singularity distributions significantly reduces the number of parameters sought in the inverse problem, which constitutes a tremendous advantage in solving the inverse problem, particularly in the application of film cooling holes

An RBF Interpolation Blending Scheme for Effective Shock-Capturing

In recent years significant focus has been given to the study of Radial basis functions (RBF), especially in their use on solving partial differential equations (PDE). RBF have an impressive capability of inter- polating scattered data, even when this data presents localized discontinuities. However, for infinitely smooth RBF such as the Multiquadrics, inverse Multiquadrics, and Gaussian, the shape parameter must be chosen properly to obtain accurate approximations while avoiding ill-conditioning of the interpolating matrices. The optimum shape parameter can vary significantly depending on the field, particularly in locations of steep gradients, shocks, or discontinuities. Typically, the shape parameter is chosen to be high value to render flatter RBF therefore yielding a high condition number for the ensuing interpola- tion matrix. However, this optimization strategy fails for problems that present steep gradients, shocks or discontinuities. Instead, in such cases, the optimal interpolation occurs when the shape parameter is chosen to be low in order to render steeper RBF therefore yielding low condition number for the interpolation matrix. The focus of this work is to demonstrate the use of RBF interpolation to capture the behaviour of steep gradients and shocks by implementing a blending scheme that combines high and low shape parameters. A formulation of the RBF blending interpolation scheme along with test- ing and validation through its implementation in the solution of the Burger’s linear advection equation and compressible Euler equations using a Localized RBF Collocation Meshless Method (LRC-MM) is presented in this paper

A Coupled Localized RBF Meshless/DRBEM Formulation for Accurate Modeling of Incompressible Fluid Flows

Velocity-pressure coupling schemes for the solution of incompressible fluid flow problems in Computational Fluid Dynamics (CFD) rely on the formulation of Poisson-like equations through projection methods. The solution of these Poisson-like equations represent the pressure correction and the velocity correction to ensure proper satisfaction of the conservation of mass equation at each step of a time-marching scheme or at each level of an iteration process. Inaccurate solutions of these Poisson-like equations result in meaningless instantaneous or intermediate approximations that do not represent the proper time-accurate behavior of the flow. The fact that these equations must be solved to convergence at every step of the overall solution process introduces a major bottleneck for the efficiency of the method. We present a formulation that achieves high levels of accuracy and efficiency by properly solving the Poisson equations at each step of the solution process by formulating a Localized RBF Collocation Meshless Method (LRC-MM) solution approach for the approximation of the diffusive and convective derivatives while employing the same framework to implement a Dual-Reciprocity Boundary Element Method (DR-BEM) for the solution of the ensuing Poisson equations. The same boundary discretization and point distribution employed in the LRC-MM is used for the DR-BEM. The methodology is implemented and tested in the solution of a backward-facing step problem

Application of an RBF Blending Interpolation Method To Problems With Shocks

Radial basis functions (RBF) have become an area of research in recent years, especially in the use of solving partial differential equations (PDE). Radial basis functions have an impressive capability in interpolating scattered data, even for data with discontinuities. Although, for infinitely smooth radial basis functions such as the multi-quadrics and inverse multi-quadrics, the shape parameter must be chosen properly to obtain accurate approximations while avoiding ill-conditioning of the interpolating matrices. The optimum shape parameter can vary depending on the field, such as in locations of sharp gradients or shocks. Typically, the shape parameter is chosen to maintain a high conditioning number for the interpolation matrix, rendering the RBF smooth [1–10]. However, this strategy fails for a problem with a shock or sharp discontinuity. Instead, in such cases the conditioning number must be kept small. The focus of this work is then to demonstrate the use of RBF interpolation in the approximation of sharp gradients or shocks by use of a RBF blending interpolation approach. This RBF blending interpolation approach is used to maintain the optimum shape parameter depending on the field. The approach is able to sense gradients or shocks in the field and adjust the shape parameter accordingly to keep excellent accuracy. Presented in this work, is an explanation of the RBF blending interpolation methodology and testing of the RBF blending interpolation approach by solving the Burger’s equation using the virtual finite difference method

Multi-Scale Cardiovascular Flow Analysis by an Integrated Meshless-Lumped Parameter Model

A computational tool that integrates a Radial basis function (RBF)-based Meshless solver with a Lumped Parameter model (LPM) is developed to analyze the multi-scale and multi-physics interaction between the cardiovascular flow hemodynamics, the cardiac function, and the peripheral circulation. The Meshless solver is based on localized RBF collocations at scattered data points which allows for automation of the model generation via CAD integration. The time-accurate incompressible flow hemodynamics are addressed via a pressure-velocity correction scheme where the ensuing Poisson equations are accurately and efficiently solved at each time step by a Dual-Reciprocity Boundary Element method (DRBEM) formulation that takes advantage of the integrated surface discretization and automated point distribution used for the Meshless collocation. The local hemodynamics are integrated with the peripheral circulation via compartments that account for branch viscous resistance (R), flow inertia (L), and vessel compliance (C), namely RLC electric circuit analogies. The cardiac function is modeled via time-varying capacitors simulating the ventricles and constant capacitors simulating the atria, connected by diodes and resistors simulating the atrioventricular and ventricular-arterial valves. This multi-scale integration in an in-house developed computational tool opens the possibility for model automation of patient-specific anatomies from medical imaging, elastodynamics analysis of vessel wall deformation for fluid-structure interaction, automated model refinement, and inverse analysis for parameter estimation

Meshless 2D direct numerical simulation and heat transfer in a backward-facing step with heat conduction in the step

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. With this new procedure, preliminary calculations with 2D steady state flows show that viscous effects become negligible faster that ever predicted numerically. The fundamental idea of this method lays on several important inconsistencies found in three of the most popular techniques used in CFD, segregated procedures, as well as in other formulations. The inconsistencies found become important in elliptic flows and they might lead to some wrong solutions. Preliminary calculations done in 2D laminar flows, suggest that the numerical diffusion and interpolation error are much important at low speeds, mainly when both, viscous and inertia forces are present. With this competitive and efficient procedure, the solution of the 2D Direct Numerical Simulation of turbulent flow with heat transfer on a backward-facing step is presented. The thermal energy is going to be transferred to the fluid through conduction on the step, with both constant temperature and heat flux conditions in the back wall of the step. The variation of the local Nusselt Number through the wall will be studied and its corresponding effect in the energy transfer to the fluid

Meshless Modeling of Flow Dispersion and Progressive Piping in Poroelastic Levees

Performance data on earth dams and levees continue to indicate that piping is one of the major causes of failure. Current criteria for prevention of piping in earth dams and levees have remained largely empirical. This paper aims at developing a mechanistic understanding of the conditions necessary to prevent piping and to enhance the likelihood of self-healing of cracks in levees subjected to hydrodynamic loading from astronomical and meteorological (including hurricane storm surge-induced) forces. Systematic experimental investigations are performed to evaluate erosion in finite-length cracks as a result of transient hydrodynamic loading. Here, a novel application of the localized collocation meshless method (LCMM) to the hydrodynamic and poroelastic problem is introduced to arrive at high-fidelity field solutions. Results from the LCMM numerical simulations are designed to be used as an input, along with the soil and erosion parameters obtained experimentally, to characterize progressive piping

Computational Fluid Dynamics in Congenital Heart Disease

Computational fluid dynamics has been applied to the design, refinement, and assessment of surgical procedures and medical devices. This tool calculates flow patterns and pressure changes within a virtual model of the cardiovascular system. In the field of paediatric cardiac surgery, computational fluid dynamics is being used to elucidate the optimal approach to staged reconstruction of specific defects and study the haemodynamics of the resulting anatomical configurations after reconstructive or palliative surgery. In this paper, we review the techniques and principal findings of computational fluid dynamics studies as applied to a few representative forms of congenital heart disease

Evaluation of Stent and Baffle Deformation in Hybrid Comprehensive Stage II Procedure

Introduction: Hypoplastic Left Heart Syndrome (HLHS) is a Congenital Heart Disease (CHD) that leads to a single ventricle circulation (SV). The existing three-stage palliative operation leads to 50% survival rates. To reduce the morbidity and mortality rate associated with the procedure, an alternative technique called Hybrid Comprehensive Stage II (HCSII) featuring the inclusion of a stent and baffle in the left and right pulmonary arteries shown is proposed. The included stent included has the potential to become fractured as a result of oscillatory asymmetric external loads. Materials and Methods: A dynamically-scaled mock flow loop (MFL) study of HCSII shows the effects of fluid pressure on the stent and baffle to infer long term complications validated with numerical simulations. The MFL includes a patient-specific 3D printed model of the reconstructed anatomy, incorporating an intra-pulmonary baffle graft and a stent. Through the inclusion of the digital video otoscope DE500, videos of the stent and baffle are captured and post-processed to determine baffle displacement during the systolic and diastolic phases. Stent deformation is quantified using Scanning Electron Microscope (SEM).Experimental results are cross-validated, using finite element analysis done in Abaqus. Results and Discussion: The displacement of the baffle is tracked in three different locations throughout the cycles. Between peak systole to peak diastole, the computed baffle displacement for each tracked location, based on the processed image data, is 38, 4 and 6 pixels respectively. Conclusions: For 10 cycles, the stent and the baffle deformations are small. Results indicate the left and right pulmonary flow remain unobstructed despite cyclic deformation of the baffle, hence the likelihood of patient death due to total pulmonary obstruction following stent collapse is low