1. Assessment of local radial basis function collocation method for diffusion problems structured with multiquadrics and polyharmonic splinesIzaz Ali, Umut Hanoglu, Robert Vertnik, Božidar Šarler, 2024, original scientific article Abstract: 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.
Keywords: meshless method, polyharmonic splines, multiquadrics, augmentation, heat diffusion equation
Published in DiRROS: 04.04.2024; Views: 72; Downloads: 32
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2. Application of a meshless space-time adaptive approach to phase-field modelling of polycrystalline solidificationTadej Dobravec, Boštjan Mavrič, Božidar Šarler, 2023, published scientific conference contribution Abstract: We have developed a 2-D numerical meshless adaptive approach for phase-field modelling of dendritic solidification. The quadtree-based approach decomposes the computational domain into quadtree sub-domains of different sizes. The algorithm generates uniformly-distributed computational nodes in each quadtree sub-domain. We apply the meshless radial basis function generated finite difference method and the forward Euler scheme to discretise governing equations in each computational node. The fixed ratio between the characteristic size and the node spacing of a quadtree sub-domain ensures space adaptivity. The adaptive time-stepping accelerates the calculations further. In the framework of previous research studies, we used the approach to solve quantitative phase-field models for single dendrite growth in pure melts and dilute binary alloys. In the present study, we upgrade the solution procedure for the modelling growth of multiple differently oriented dendrites. Along with the space-time adaptive approach, we apply non-linear preconditioning of the phase-field equation to increase computational efficiency. We investigate a novel numerical approach's accuracy and computational efficiency by simulating the equiaxed dendrite growth from a dilute binary alloy.
Keywords: dendritic growth, phase-field method, meshless methods, polycrystalline solidification
Published in DiRROS: 21.03.2024; Views: 101; Downloads: 58
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3. A meshless numerical solution of thermo-mechanics of hot-rolled steel bars on a cooling bedGašper Vuga, Boštjan Mavrič, Umut Hanoglu, Božidar Šarler, 2023, published scientific conference contribution Abstract: 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.
Keywords: cooling bed, steel bars, thermo-mechanics, strong form, meshless method
Published in DiRROS: 21.03.2024; Views: 83; Downloads: 47
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4. A coupled domain–boundary type meshless method for phase-field modelling of dendritic solidification with the fluid flowTadej Dobravec, Boštjan Mavrič, Rizwan Zahoor, Božidar Šarler, 2023, original scientific article Abstract: Purpose - This study aims to simulate the dendritic growth in Stokes flow by iteratively coupling a domain and boundary type meshless method. Design/methodology/approach - A preconditioned phase-field model for dendritic solidification of a pure supercooled melt is solved by the strong-form space-time adaptive approach based on dynamic quadtree domain decomposition. The domain-type space discretisation relies on monomial augmented polyharmonic splines interpolation. The forward Euler scheme is used for time evolution. The boundary-type meshless method solves the Stokes flow around the dendrite based on the collocation of the moving and fixed flow boundaries with the regularised Stokes flow fundamental solution. Both approaches are iteratively coupled at the moving solid–liquid interface. The solution procedure ensures computationally efficient and accurate calculations. The novel approach is numerically implemented for a 2D case. Findings - The solution procedure reflects the advantages of both meshless methods. Domain one is not sensitive to the dendrite orientation and boundary one reduces the dimensionality of the flow field solution. The procedure results agree well with the reference results obtained by the classical numerical methods. Directions for selecting the appropriate free parameters which yield the highest accuracy and computational efficiency are presented. Originality/value - A combination of boundary- and domain-type meshless methods is used to simulate dendritic solidification with the influence of fluid flow efficiently.
Keywords: dendritic solidification, Stokes flow, phase-field method, space-time adaptivity, meshless methods, RBF-FD, modified method of regularised sources
Published in DiRROS: 07.02.2024; Views: 147; Downloads: 64
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