Solution of the stationary stokes and navier-stokes equations using the modified finite particle method in the framework of a least squares residual method

The present work is concerned with the solution of stationary Stokes and Navier-Stokes flows using the Modified Finite Particle Method for spatial derivative approximations and the Least Square Residual Method for the solution of the linear system deriving from the collocation procedure. The combination of such approaches permits to easily handle the numerical difficulty of the inf-sup conditions, without distinguishing between the discretizations of velocity and pressure fields. The obtained results, both in the cases of linear and non-linear flows, show the robustness of the proposed algorith

On Meshfree GFDM Solvers for the Incompressible Navier-Stokes Equations

Meshfree solution schemes for the incompressible Navier--Stokes equations are usually based on algorithms commonly used in finite volume methods, such as projection methods, SIMPLE and PISO algorithms. However, drawbacks of these algorithms that are specific to meshfree methods have often been overlooked. In this paper, we study the drawbacks of conventionally used meshfree Generalized Finite Difference Method~(GFDM) schemes for Lagrangian incompressible Navier-Stokes equations, both operator splitting schemes and monolithic schemes. The major drawback of most of these schemes is inaccurate local approximations to the mass conservation condition. Further, we propose a new modification of a commonly used monolithic scheme that overcomes these problems and shows a better approximation for the velocity divergence condition. We then perform a numerical comparison which shows the new monolithic scheme to be more accurate than existing schemes

Solution of Heat Transfer and Fluid Flow problems using meshless Radial Basis Function method

In the past, the world of numerical solutions for Heat Transfer and Fluid Flow problems has been dominated by Finite Element Method, Finite Difference Method, Finite Volume Method, and more recently the Boundary Element Method. These methods revolve around using a mesh or grid to solve problems. However, problems with irregular boundaries and domains can be difficult to properly discretize; In this thesis, heat transfer and fluid flow problems are solved using Radial Basis Functions. This method is meshless, easy to understand, and even easier to implement. Radial Basis Functions are used to solve lid-driven cavity flow, natural convection in a square enclosure, flow with forced convection over backward facing step and flow over an airfoil. Codes are developed using MATLAB. The results are compared with COMSOL and FLUENT, two popular commercial codes widely used. COMSOL is a finite element model while FLUENT is a finite volume-based code

Direct solution of Navier-Stokes equations by using an upwind local RBF-DQ method

The differential quadrature (DQ) method is able to obtain quite accurate numerical solutions of differential equations with few grid points and less computational effort. However, the traditional DQ method is convenient only for regular regions and lacks upwind mechanism to characterize the convection of the fluid flow. In this paper, an upwind local radial basis function-based DQ (RBF-DQ) method is applied to solve the Navier-Stokes equations, instead of using an iterative algorithm for the primitive variables. The non-linear collocated equations are solved using the Levenberg-Marquardt method. The irregular regions of 2D channel flow with different obstructions situations are considered. Finally, the approach is validated by comparing the results with those obtained using the well-validated Fluent commercial package

Collocated Discrete Least Squares (CDLS) meshless method for the simulation of power-law fluid flows

AbstractLeast squares is a robust and simple method in function approximation. Collocated Discrete Least Squares (CDLS) is a meshless method based on least squares technique. In this paper CDLS is used with a non-incremental projection method for the solution of incompressible generalized Newtonian fluid flow equations in the simulation of laminar flow of power-law fluids. The scheme is used to solve two benchmark problems named lid-driven cavity flow and flow past a circular cylinder for the power-law fluids

Solution of incompressible fluid flow problems with heat transfer by means of an efficient RBF-FD meshless approach

The localized radial basis function collocation meshless method (LRBFCMM), also known as radial basis function generated finite differences (RBF-FD) meshless method, is employed to solve time-dependent, 2D incompressible fluid flow problems with heat transfer using multiquadric RBFs. A projection approach is employed to decouple the continuity and momentum equations for which a fully implicit scheme is adopted for the time integration. The node distributions are characterized by non-cartesian node arrangements and large sizes, i.e., in the order of $10^5$ nodes, while nodal refinement is employed where large gradients are expected, i.e., near the walls. Particular attention is given to the accurate and efficient solution of unsteady flows at high Reynolds or Rayleigh numbers, in order to assess the capability of this specific meshless approach to deal with practical problems. Three benchmark test cases are considered: a lid-driven cavity, a differentially heated cavity and a flow past a circular cylinder between parallel walls. The obtained numerical results compare very favourably with literature references for each of the considered cases. It is concluded that the presented numerical approach can be employed for the efficient simulation of fluid-flow problems of engineering relevance over complex-shaped domains

Use of Machine Learning for Automated Convergence of Numerical Iterative Schemes

Convergence of a numerical solution scheme occurs when a sequence of increasingly refined iterative solutions approaches a value consistent with the modeled phenomenon. Approximations using iterative schemes need to satisfy convergence criteria, such as reaching a specific error tolerance or number of iterations. The schemes often bypass the criteria or prematurely converge because of oscillations that may be inherent to the solution. Using a Support Vector Machines (SVM) machine learning approach, an algorithm is designed to use the source data to train a model to predict convergence in the solution process and stop unnecessary iterations. The discretization of the Navier Stokes (NS) equations for a transient local hemodynamics case requires determining a pressure correction term from a Poisson-like equation at every time-step. The pressure correction solution must fully converge to avoid introducing a mass imbalance. Considering time, frequency, and time-frequency domain features of its residual’s behavior, the algorithm trains an SVM model to predict the convergence of the Poisson equation iterative solver so that the time-marching process can move forward efficiently and effectively. The fluid flow model integrates peripheral circulation using a lumped-parameter model (LPM) to capture the field pressures and flows across various circulatory compartments. Machine learning opens the doors to an intelligent approach for iterative solutions by replacing prescribed criteria with an algorithm that uses the data set itself to predict convergence

Schwarz alternating domain decomposition approach for the solution of mixed heat convection flow problems based on the method of approximate particular solutions

The incompressible two-dimensional Navier-Stokes equations including thermal energy balance equation are solved by the recently developed Method of Approximate Particular Solutions (MAPS). In a previous authors’ work this method was implemented to solve the two-dimensional Stokes equations by employing the pressure and velocity particular solutions obtained by Oseen’s decomposition with the Multiquadric (MQ) RBF as non-homogeneous term. A pressure-velocity linkage strategy is not required since the pressure particular solutions are obtained from the velocity ones. In the present contribution, the Navier-Stokes equations with Boussinesq approximation are solved by linearizing the convective term in a Picard iterative scheme. With the velocity values obtained at each of the Picard iterations, the energy conservation equation is solved by the MAPS by approximating temperature with the particular solutions of a Poisson problem with the MQ as a forcing term. With the aim of improving the computational efficiency of the global strategy, the two-dimensional domain is split into overlapped rectangular subdomains where the Schwarz Alternating Algorithm is employed to find a solution by using velocity and temperatures values from neighbouring zones as boundary conditions. The mixed convection lid-driven cavity flow problem is solved for moderate Reynolds and low Richardson numbers with the aim of validating the proposed method