This paper explores the application of the meshless local radial basis function collocation method for the solution of coupled heat transfer and fluid flow problems with a free surface. The method employs the representation of temperature, velocity and pressure fields on overlapping five noded sub-domains through collocation by using radial basis functions (RBFs). This simple representation is then used to compute the first and the second derivatives of the fields from the respective derivatives of the RBFs. The energy and momentum equations are solved through explicit time integration scheme. For numerical efficiency, the artificial compressibility method with characteristic based split technique is firstly adopted to solve the pressure–velocity coupled equations. The performance of the method is assessed based on solving the classical two-dimensional De Vahl Davis steady natural convection benchmark problem with an upper free surface for Rayleigh number ranged from 10**3 to 10**5 and Prandtl number 0.71.
COBISS.SI-ID: 3715835
Purpose: The purpose of this work is to upgrade our previous developments of Local Radial Basis Function Collocation Method (LRBFCM) for heat transfer, fluid flow, and electromagnetic problems to thermoelastic problems and to study its numerical performance with the aim to build a multiphysics meshless computing environment based on LRBFCM. Design/methodology/approach: Linear thermoelastic problems for homogenous isotropic body in two dimensions are considered. The stationary stress equilibrium equation is written in terms of deformation field. The domain and boundary can be discretized with arbitrary positioned nodes where the solution is sought. Each of the nodes has its influence domain, encompassing at least six neighboring nodes. The unknown displacement field is collocated on local influence domain nodes with shape functions that consist of a linear combination of multiquadric radial basis functions and monomials. The boundary conditions are analytically satisfied on the influence domains which contain boundary points. The action of the stationary stress equilibrium equation on the constructed interpolation results in a sparse system of linear equations for solution of the displacement field. Findings: The performance of the method is demonstrated on three numerical examples: bending of a square, thermal expansion of a square and thermal expansion of a thick cylinder. Error is observed to be composed of two contributions, one proportional to a power of internodal spacing and the other to a power of the shape parameter. The latter term is the reason for the observed accuracy saturation, while the former term describes the order of convergence. The explanation of the observed error is given for the smallest number of collocation points (six) used in local domain of influence. The observed error behavior is explained by considering the Taylor series expansion of the interpolant. The method can achieve high accuracy and performs well for the examples considered. Originality: LRBFCM has been developed for thermoelasticity and its error behaviour studied. A robust way of controlling the error was devised from consideration of the condition number. The performance of the method has been demonstrated for a large number of the nodes and on uniform and non-uniform node arrangements Research Limitations: The method can at the present cope with linear thermoelasticity. Other, more complicated material behavior (visco-plasticity for example), will be tackled in one of our future publications.
COBISS.SI-ID: 3976187
The solution of Stokes flow problems with Dirichlet and Neumann boundary conditions is performed by a non-singular Method of Fundamental Solutions which does not require artificial boundary, i.e. source points of fundamental solution coincide with the collocation points on the geometrical boundary. The fundamental solution of the Stokes pressure and velocity is obtained from the analytical solution due to the action of the Dirac delta type of force. Instead of Dirac delta force, a non-singular function, called blob, with a free parameter epsilon is employed, which limits to Dirac delta function when epsilon limits to zero. The analytical expressions for related Stokes flow pressure and velocity around such regularized sources have been derived for rational and exponential blobs in an ordered way. The solution of the problem is sought as a linear combination of the fields due to the regularized sources that coincide with the boundary and with their intensities chosen in such a way that the solution complies with the boundary conditions. A two dimensional driven cavity numerical example and a flow between parallel plates is chosen to assess the properties of the method. The results of the posed Method of Regularized Sources (MRS) have been compared with the results obtained by the fine-grid second-order classical Finite Difference Method and analytical solution. The results converge with finer discretisation, however they depend on the value of epsilon. The method gives reasonably accurate results for the range of epsilon between 0.1 and 0.5 of the typical nodal distance on the boundary. Exponential blobs give slightly better results than the rational blobs, however they require slightly more computing time. A robust and efficient strategy to find the optimal value of epsilon is needed in the perspective.
COBISS.SI-ID: 4115195
This paper represents a continuation of numerical results regarding the recently proposed industrial numerical benchmark test, obtained by a meshless method. A part of the benchmark test, involving turbulent fluid flow with solidification in two dimensions, was recently published. A preliminary macrosegregation upgrade and a three dimensional test were published too. Previous tests were bound to calculations in mold and sub-mold regions only. In the present paper, reference calculations in two dimensions are presented for the entire strand. The physical model is established on a set of macroscopic equations for mass, energy, momentum, species, turbulent kinetic energy, and dissipation rate. The mixture continuum model is used to treat the solidification system. The mushy zone is modeled as a Darcy porous media with Kozeny-Carman permeability relation, where the morphology of the porous media is modeled by a constant value. The incompressible turbulent flow of the molten steel is described by the Low-Reynolds-Number k-ε turbulence model, closed by the Abe-Kondoh-Nagano closure coefficients and damping functions. Lever microsegregation model is used. The numerical method is established on explicit time-stepping, collocation with scaled multiquadrics radial basis functions with adaptive selection of its shape on non-uniform five-nodded influence domains. The velocity–pressure coupling of the incompressible flow is resolved by the explicit Chorin’s fractional step method. The advantages of the method are its simplicity and efficiency, since no polygonisation is involved, easy adaptation of the nodal points in areas with high gradients, almost the same formulation in two and three dimensions, high accuracy and low numerical diffusion.
COBISS.SI-ID: 3925499
Purpose: In this study we upgrade our previous developments of the Local Radial Basis Function Collocation Method (LRBFCM) for heat transfer, fluid flow, electromagnetic problems and linear thermoelasticity to dynamic coupled thermoelasticity problems. Design/methodology/approach: We solve a thermoelastic benchmark by considering a linear thermoelastic plate under thermal and pressure shock. Spatial discretization is performed by a local collocation with multiquadrics augmented by monomials. The implicit Euler formula is used to perform the time stepping. The system of equations obtained from the formula is solved using a Newton-Raphson algorithm with GMRES to iteratively obtain the solution. The LRBFCM solution is compared with the reference FEM solution and, in one case, with a solution obtained using the meshless local Petrov-Galerkin method. Findings: The performance of the LRBFCM is found to be comparable to the FEM, with some differences near the tip of the shock front. The LRBFCM appears to converge to the mesh-converged solution more smoothly than the FEM. Also, the LRBFCM seems to perform better than the MLPG in the studied case. Originality: For the first time, the LRBFCM has been applied to problems of coupled thermoelasticity. Research Limitations: The performance of the LRBFCM near the tip of the shock front appears to be suboptimal, since it does not capture the shock front as well as the FEM. With the exception of a solution obtained using the meshless local Petrov-Galerkin method, there is no other high-quality reference solution for the considered problem in the literature yet. In most cases, therefore, we are able to compare only two mesh-converged solutions obtained by the authors using two different discretization methods. The shock-capturing capabilities of the method should be studied in more detail. The presentation was considered outstanding and suggested to be published in extended form in International Journal of Numerical Methods in Heat & Fluid Flow.
COBISS.SI-ID: 4114939