Two algorithms for fast 2D node generation: application to RBF meshless discretization of diffusion problems and image halftoning

Mesh generation techniques for traditional mesh based numerical approaches such as FEM and FVM have now reached a good degree of maturity. There is no such an acknowledged background when dealing with node generation techniques for meshless numerical approaches, despite their theoretical simplicity and efficiency; furthermore node generation can be put in connection with some well-known image approximation techniques. Two node generation algorithms are here proposed and employed in the numerical solution of 2D steady state diffusion problems by means of a local Radial Basis Function (RBF) meshless method. Finally, such algorithms are also tested for greyscale image approximation through stippling

Accurate Stabilization Techniques for RBF-FD Meshless Discretizations with Neumann Boundary Conditions

A major obstacle to the application of the standard Radial Basis Function-generated Finite Difference (RBF-FD) meshless method is constituted by its inability to accurately and consistently solve boundary value problems involving Neumann boundary conditions (BCs). This is also due to ill-conditioning issues affecting the interpolation matrix when boundary derivatives are imposed in strong form. In this paper these ill-conditioning issues and subsequent instabilities affecting the application of the RBF-FD method in presence of Neumann BCs are analyzed both theoretically and numerically. The theoretical motivations for the onset of such issues are derived by highlighting the dependence of the determinant of the local interpolation matrix upon the boundary normals. Qualitative investigations are also carried out numerically by studying a reference stencil and looking for correlations between its geometry and the properties of the associated interpolation matrix. Based on the previous analyses, two approaches are derived to overcome the initial problem. The corresponding stabilization properties are finally assessed by succesfully applying such approaches to the stabilization of the Helmholtz-Hodge decomposition

Distributed Lagrange Multiplier Functions for Fictitious Domain Method with Spectral/hp Element Discretization

A fictitious domain approach for the solution of second-order linear differential problems is proposed; spectral/hp elements have been used for the discretization of the domain. The peculiarity of our approach is that the Lagrange multipliers are particular distributed functions, instead of classical \u3b4 Dirac (impulsive)multipliers. In this paper we present the formulation and the application of this approach to 1D and 2D Poisson problems and 2D Stokes flow (biharmonic equation)

Radial Basis Function-Finite Difference Meshless Methods for CFD Problems

The objective of this dissertation is to investigate the numerical properties of the RBF-FD meshless approach when it is employed for the solution of CFD problems, with particular reference to fluid-flow problems defined over complex-shaped domains. This objective has been accomplished by developing a MATLAB code which is composed by several elements characterizing the meshless approach to a CFD problem. The work presented in this thesis has focused on the analysis and development of these characterizing elements which are essential in developing an innovative, robust, accurate and flexible numerical method. The node generation is the first problem that has been tackled and a significant portion of this work is dedicated to this element since it is the foundation of every meshless approach. Different algorithms have been proposed for the generation of node distributions on 2D and 3D complex-shaped domains. Such node distributions then proved to be very suitable for the use with RBF-FD discretizations. The developed node generation algorithms are extremely efficient and are based on very simple principles: this is an important insight since the possibility to easily deal with complex geometries represents the main theoretical advantage of meshless methods over mesh-based methods. The RBF interpolation, which is the key element of the RBF-FD method, is then thoroughly studied by exploring all the variables which influence the construction of an accurate and robust interpolant over scattered nodes. The multiquadric RBF has been chosen and extensive numerical tests are conducted for 2D and 3D cases. On the base of these analysis, a RBF-FD code has been developed for the local approximation of the partial derivatives of an unknown function which is defined only at scattered nodes in 2D/3D. The coupling of a node generation algorithm to a RBF-FD scheme leads to an actual meshless approach which can be used to discretize a given PDE over a domain with possible arbitrary shape. Such meshless approach has been employed to perform several test cases for different 2D/3D model problems which have fundamental importance in CFD applications. The solution phase then follows the RBF-FD discretization, and its role in the simulation chain is just as important as the previous aspects. For this purpose, novel multicloud techniques have been proposed for the acceleration of the convergence in the solution of the system of equations arising from RBF-FD discretizations in the case of a 2D Poisson equation. Such multicloud techniques have proven to bring substantial improvements over the traditional solvers employed with the RBF-FD discretizations. Finally, the RBF-FD approach is employed to solve actual 2D/3D fluid-flow problems in the case of the time-dependent, incompressible Navier-Stokes equations. Important problems are addressed, e.g., the development of stable discretizations when dealing with the pressure-velocity coupling using primitive variables, leading to an efficient and stable RBF-FD approach which can be used for the accurate solution of time-dependent fluid-flow problems over arbitrarily shaped domains in 2D and in 3D.The objective of this dissertation is to investigate the numerical properties of the RBF-FD meshless approach when it is employed for the solution of CFD problems, with particular reference to fluid-flow problems defined over complex-shaped domains. This objective has been accomplished by developing a MATLAB code which is composed by several elements characterizing the meshless approach to a CFD problem. The work presented in this thesis has focused on the analysis and development of these characterizing elements which are essential in developing an innovative, robust, accurate and flexible numerical method. The node generation is the first problem that has been tackled and a significant portion of this work is dedicated to this element since it is the foundation of every meshless approach. Different algorithms have been proposed for the generation of node distributions on 2D and 3D complex-shaped domains. Such node distributions then proved to be very suitable for the use with RBF-FD discretizations. The developed node generation algorithms are extremely efficient and are based on very simple principles: this is an important insight since the possibility to easily deal with complex geometries represents the main theoretical advantage of meshless methods over mesh-based methods. The RBF interpolation, which is the key element of the RBF-FD method, is then thoroughly studied by exploring all the variables which influence the construction of an accurate and robust interpolant over scattered nodes. The multiquadric RBF has been chosen and extensive numerical tests are conducted for 2D and 3D cases. On the base of these analysis, a RBF-FD code has been developed for the local approximation of the partial derivatives of an unknown function which is defined only at scattered nodes in 2D/3D. The coupling of a node generation algorithm to a RBF-FD scheme leads to an actual meshless approach which can be used to discretize a given PDE over a domain with possible arbitrary shape. Such meshless approach has been employed to perform several test cases for different 2D/3D model problems which have fundamental importance in CFD applications. The solution phase then follows the RBF-FD discretization, and its role in the simulation chain is just as important as the previous aspects. For this purpose, novel multicloud techniques have been proposed for the acceleration of the convergence in the solution of the system of equations arising from RBF-FD discretizations in the case of a 2D Poisson equation. Such multicloud techniques have proven to bring substantial improvements over the traditional solvers employed with the RBF-FD discretizations. Finally, the RBF-FD approach is employed to solve actual 2D/3D fluid-flow problems in the case of the time-dependent, incompressible Navier-Stokes equations. Important problems are addressed, e.g., the development of stable discretizations when dealing with the pressure-velocity coupling using primitive variables, leading to an efficient and stable RBF-FD approach which can be used for the accurate solution of time-dependent fluid-flow problems over arbitrarily shaped domains in 2D and in 3D

Node generation in complex 3D domains for heat conduction problems solved by RBF-FD meshless method

A novel algorithm is presented and employed for the fast generation of meshless node distributions over arbitrary 3D domains defined by using the stereolithography (STL) file format. The algorithm is based on the node-repel approach where nodes move according to the mutual repulsion of the neighboring nodes. The iterative node-repel approach is coupled with an octree-based technique for the efficient projection of the nodes on the external surface in order to constrain the node distribution inside the domain. Several tests are carried out on three different mechanical components of practical engineering interest and characterized by complexity of their geometry. The generated node distributions are then employed to solve a steady-state heat conduction test problem by using the Radial Basis Function-generated Finite Differences (RBF-FD) meshless method. Excellent results are obtained in terms of both quality of the generated node distributions and accuracy of the numerical solutions

A Fully Meshless Approach to the Numerical Simulation of Heat Conduction Problems over Arbitrary 3D Geometries

One of the goals of new CAE (Computer Aided Engineering) software is to reduce both time and costs of the design process without compromising accuracy. This result can be achieved, for instance, by promoting a \u201cplug and play\u201d philosophy, based on the adoption of automatic mesh generation algorithms. This in turn brings about some drawbacks, among others an unavoidable loss of accuracy due to the lack of specificity of the produced discretization. Alternatively it is possible to rely on the so called \u201cmeshless\u201d methods, which skip the mesh generation process altogether. The purpose of this paper is to present a fully meshless approach, based on Radial Basis Function generated Finite Differences (RBF-FD), for the numerical solution of generic elliptic PDEs, with particular reference to time-dependent and steady 3D heat conduction problems. The absence of connectivity information, which is a peculiar feature of this meshless approach, is leveraged in order to develop an efficient procedure that accepts as input any given geometry defined by a stereolithography surface (.stl file format). In order to assess its performance, the aforementioned strategy is tested over multiple geometries, selected for their complexity and engineering relevance, highlighting excellent results both in terms of accuracy and computational efficiency. In order to account for future extensibility and performance, both node generation and domain discretization routines are entirely developed using Julia, an emerging programming language that is rapidly establishing itself as the new standard for scientific computing

NUMERICAL SOLUTION OF HEAT CONDUCTION PROBLEMS BY MEANS OF A MESHLESS METHOD WITH PROPER POINT DISTRIBUTIONS

A meshless method based on Radial Basis Function (RBF) interpolation and direct collocation is used to systematically solve steady state heat conduction problems on several 2D domains with different boundary conditions. The point sets, or distributions, needed by the method have been generated using two techniques: a relaxed quadtree technique and a high quality, geometry dependent, technique. The resulting solutions are then compared to the corresponding analytical solutions; FEM (Finite Method) solutions have also been computed as reference. These tests showed good convergence properties (II or IV order) for many of the considered cases, with an error of the same order of magnitude of classical FEM solutions, while a limited number of cases showed fluctuating convergence curves. These features confirm that this numerical approach could be an effective technique in the numerical simulation of practical heat conduction problems

Analysis of geometric uncertainties in CFD problems solved by RBF-FD meshless method

In order to analyze incompressible and laminar fluid flows in presence of geometric uncertainties on the boundaries, the Non-Intrusive Polynomial Chaos method is employed, which allows the use of a deterministic fluid dynamic solver. The quantification of the fluid flow uncertainties is based on a set of deterministic response evaluations, which are obtained through a Radial Basis Function-generated Finite Differences meshless method. The use of such deterministic solver represents the key point of the analysis, thanks to the computational efficiency and similar accuracy over the traditional mesh-based numerical methods. The validation of the proposed approach is carried out through the solution of the flow past a 2D spinning cylinder near a moving wall and the flow over a backward-facing step, in presence of stochastic geometries. The applicability to practical problems is demonstrated through the investigation of geometric uncertainty effects on the forced convection of Al2O3-water nanofluid laminar flow in a grooved microchannel

A Gold Nanoparticle Nanonuclease Relying on a Zn(II) Mononuclear Complex

Similarly to enzymes, functionalized gold nanoparticles efficiently catalyze chemical reactions, hence the term nanozymes. Herein, we present our results showing how surface-passivated gold nanoparticles behave as synthetic nanonucleases, able to cleave pBR322 plasmid DNA with the highest efficiency reported so far for catalysts based on a single metal ion mechanism. Experimental and computational data indicate that we have been successful in creating a catalytic site precisely mimicking that suggested for natural metallonucleases relying on a single metal ion for their activity. It comprises one Zn(II) ion to which a phosphate diester of DNA is coordinated. Importantly, as in nucleic acids-processing enzymes, a positively charged arginine plays a key role by assisting with transition state stabilization and by reducing the pK(a) of the nucleophilic alcohol of a serine. Our results also show how designing a catalyst for a model substrate (bis-p-nitrophenylphosphate) may provide wrong indications as for its efficiency when it is tested against the real target (plasmid DNA)

A Gold Nanoparticle Nanonuclease Relying on a Zn(II) Mononuclear Complex

Similarly to enzymes, functionalized gold nanoparticles efficiently catalyze chemical reactions, hence the term nanozymes. Herein, we present our results showing how surface-passivated gold nanoparticles behave as synthetic nanonucleases, able to cleave pBR322 plasmid DNA with the highest efficiency reported so far for catalysts based on a single metal ion mechanism. Experimental and computational data indicate that we have been successful in creating a catalytic site precisely mimicking that suggested for natural metallonucleases relying on a single metal ion for their activity. It comprises one Zn(II) ion to which a phosphate diester of DNA is coordinated. Importantly, as in nucleic acids-processing enzymes, a positively charged arginine plays a key role by assisting with transition state stabilization and by reducing the pK(a) of the nucleophilic alcohol of a serine. Our results also show how designing a catalyst for a model substrate (bis-p-nitrophenylphosphate) may provide wrong indications as for its efficiency when it is tested against the real target (plasmid DNA)