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Povzetek: This paper aims to systematically assess the local radial basis function collocation method, structured with multiquadrics (MQs) and polyharmonic splines (PHSs), for solving steady and transient diffusion problems. The boundary value test involves a rectangle with Dirichlet, Neuman, and Robin boundary conditions, and the initial value test is associated with the Dirichlet jump problem on a square. The spectra of the free parameters of the method, i.e., node density, timestep, shape parameter, etc., are analyzed in terms of the average error. It is found that the use of MQs is less stable compared to PHSs for irregular node arrangements. For MQs, the most suitable shape parameter is determined for multiple cases. The relationship of the shape parameter with the total number of nodes, average error, node scattering factor, and the number of nodes in the local subdomain is also provided. For regular node arrangements, MQs produce slightly more accurate results, while for irregular node arrangements, PHSs provide higher accuracy than MQs. PHSs are recommended for use in diffusion problems that require irregular node spacing.
Ključne besede: meshless method, polyharmonic splines, multiquadrics, augmentation, heat diffusion equation
Objavljeno v DiRROS: 04.04.2024; Ogledov: 106; Prenosov: 44
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Povzetek: We have extended the existing two-dimensional rigid solid phase benchmark for binary substance with the solid phase motion in the present paper. Incompressible laminar Newtonian flow is assumed, and a standard mixture formulation is used for the mass, momentum, energy, and solute transport. A coherency solid motion model accounts for the free-floating grains, assuming that the solid velocity is proportional to the mixture velocity and the liquid fraction. The lever rule is used to describe the mass fractions of the phases. A two-dimensional benchmark is solved using the semi-implicit meshless diffuse approximate method with an adaptive subdomain upwinding strategy. The results of the meshless method are compared to the finite volume method results with a reasonable agreement. The new benchmark results show that the solid motion has an essential effect on the macrosegregation pattern.
Ključne besede: solidification, meshless methods, grain motion, benchmarks
Objavljeno v DiRROS: 21.03.2024; Ogledov: 138; Prenosov: 67
Celotno besedilo (1,11 MB)
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Povzetek: After the continuous hot-rolling process, steel bars are immediately placed on the cooling bed. At the beginning of the cooling, the material is at high temperatures, and the yield strength is low. Due to thermal load, yield strength can be exceeded, and permanent plastic strains start accumulating, resulting in possible unwanted shape changes and residual stresses. The present paper aims to develop a thermo-mechanical model for studying and eliminating undesirable phenomena before the products leave the cooling bed. The governing equations are solved for the two-dimensional slice in a strong form, and a modified version of the radial basis function generated finite difference (RBF-FD) method [1]. The initial bar geometry is obtained from the existing meshless hot-rolling simulation system [2]. The thermal and mechanical models are one-way coupled, i.e. the temperature solution represents a driving force for the stress and strain solution. The temperature field is obtained with explicit propagation in time. The convective and radiative heat fluxes on the boundary are updated at each time step using the ray tracing procedure to determine the radiative heat flux. The mechanical part is solved by considering the small strain elasto-plasticity, where the isotropic von Mises temperature-dependent hardening is employed. The global system of nonlinear equations of the mechanical part is solved by the Newton-Raphson method. The closest point projection method is used to solve the constitutive relations. A sensitivity study is performed on the influence of cooling intensity on a rectangular steel bar’s temperature, stress and strain field. We defined the most influential factors for defect formation. For the first time, a novel meshless RBF-FD method is successfully used for solving such a complex industrial problem. The model will be perspectively upgraded from the slice to the three-dimensional model to enable also bending.
Ključne besede: cooling bed, steel bars, thermo-mechanics, strong form, meshless method
Objavljeno v DiRROS: 21.03.2024; Ogledov: 90; Prenosov: 52
Celotno besedilo (1,62 MB)
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Povzetek: This paper represents Part 2 of the parallel paper Part 1, where the strong form hybrid RBF-FD method was developed for solving thermo-elasto-plastic problems. It addresses the industrial application of this novel meshless method to steel bars cooling on a cooling bed (CB) where the formation of residual stress is of primary interest. The study investigates the impact of the distance between the bars and the distance to the heat shield above the CB on radiative heat fluxes and, consequently, on thermo-mechanical response. The thermal model is solved on bars cross-section with a RBF-FD method where augmented polyharmonic splines are used for the local approximation. View factors, computed with a Monte-Carlo method, are included in radiative heat fluxes. The thermal solution is incrementally applied on a mechanical model that assumes a generalised plane strain state and captures bars bending. The study employs a hybrid RBF-FD method to resolve a nonlinear discontinuous mechanical problem successfully. The simulation of the process shows how different process parameters influence the thermo-mechanical response of the bars.
Ključne besede: steel bars, cooling bed, thermo-mechanical modelling, hybrid radial basis function, generated finite differences, residual stresses
Objavljeno v DiRROS: 28.02.2024; Ogledov: 127; Prenosov: 100
Celotno besedilo (8,55 MB)
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Povzetek: The method of fundamental solutions with a subdomain technique is used for the solution of the free boundary problem associated with a two-phase Stokes flow in a 2D geometry. The solution procedure is based on the collocation of the boundary conditions with the Stokeslets. It is formulated for the flow of unmixing fluids in contact, where the velocity, pressure field, and position of the free boundary between the fluids must be determined. The standard formulation of the method of fundamental solutions is, for the first time, upgraded for the case with mixed velocity and pressure boundary conditions and verified on a T-splitter single-phase flow with unsymmetric pressure boundary conditions. The standard control volume method is used for the reference solution. The accurate evaluation of the velocity derivatives, which are required to calculate the balance of forces at the free boundary between the fluids, is achieved in a closed form in contrast to previous numerical attempts. An algorithm for iteratively calculating the position of the free boundary that involves displacement, smoothing and repositioning of the nodes is elaborated. The procedure is verified for a concurrent flow of two fluids in a channel. The velocity and velocity derivatives show fast convergence to the analytical solution. The developed boundary meshless method is easy to code, accurate and computationally efficient since only collocation at the fixed and free boundaries is needed.
Ključne besede: Stokes equations, two-phase flow, free boundary problems, method of fundamental solutions, subdomain technique
Objavljeno v DiRROS: 28.02.2024; Ogledov: 139; Prenosov: 96
Celotno besedilo (3,40 MB)
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