Local Radial Basis Function Collocation Method Research Articles

Elasto-plastic simulation of hot shape rolling of steel by a meshless method

This paper describes the development of meshless simulation of thermomechanics of steel in a continuous hot rolling process using a novel meshless solution procedure. During the process, steel billets at high temperatures are transferred through multiple roll passes, each with a specific groove shape and roll gap. The major outcome of this simulation system is to design the rolling schedule and confirm whether the final shape is inside the acceptable range. For this purpose, a perpendicular 2D slice model approach is used to obtain fast and reasonably accurate simulation results. A return mapping algorithm solves the elasto-plastic material model with linear hardening to obtain displacements, strains and stresses. They are calculated by the local radial basis function collocation method (LRBFCM), which uses the local interpolation of the strong form of partial differential equations. The predefined slice positions are entered into the rolling system, and for each position, the slice contact line and amount of reduction are predicted and used as boundary conditions. Thermal and mechanical models are run sequentially, which is repeated until the final slice position reaches the exit from the last roll. The simulation system was validated with multiple rolling schedules provided by the industrial partner and was successfully used in the steel production plant afterwards. The meshless LRBFCM successfully solves a complex elasto-plastic large deformation problem of hot rolling for the first time.

A new hybrid local radial basis function collocation method for 2.5D thermo-mechanical modelling of continuous casting of steel

This study presents a new strong-form meshless method to solve the thermo-mechanical problem of the solidification process in the continuous casting of steel. A two-dimensional slice that travels in the casting direction is modelled in the Lagrangian system. The newly developed mechanical model is one-way coupled to the thermal model, where the heat flux due to the mould, sprays, rolls and radiation are imposed to solve heat transfer in the strand. The resulting temperature and metallostatic pressure govern the Norton-Hoff visco-plastic model used for computing shrinkage of the solid shell and induced residual stresses. The results are used to estimate critical areas susceptible to hot-tearing formation. The mechanical model uses a generalised plane strain assumption that includes linear strains perpendicular to the slice and enables the computation of the straightening of the strand. The thermo-mechanical model is spatially discretised with a local radial basis function collocation method (LRBFCM). The mechanical part includes a new hybrid method that combines LRBFCM with finite differences for increased stability. The presented work shows how the developed model is used to assess the impact of casting velocity on the solid shell shrinkage and the probability of hot-tearing occurrence in the continuous casting of square billets.

Simulation of temperature field in steel billets moving in the reheating furnace by a meshless method

A simulation of a reheating furnace in a steel production line where the steel billets are heated from room temperature up to 1200 °C is carried out. In this work, governing equations are solved in a strong form by the meshless local radial basis function collocation method (LRBFCM) with explicit time-stepping. The solution of the diffusion equation for the temperature is formulated on a two-dimensional central slice of the billet. The temperature field is solved by considering the positions of multiple billets in the furnace and the radiative and convective heat fluxes on the boundaries of the furnace and the billets. Ray tracing procedure, in which the view factors are computed with a Monte-Carlo method, is employed to determine the radiative heat flux. A sensitivity study is performed on the influence of two different stopping times of the billets in the furnace on the temperature evolution.

Simulation of Temperature Field in Steel Billets during Reheating in Pusher-Type Furnace by Meshless Method

Using a meshless method, a simulation of steel billets in a pusher-type reheating furnace is carried out for the first time. The simulation represents an affordable way to replace the measurements. The heat transfer from the billets with convection and radiation is considered. Inside each of the billets, the heat diffusion equation is solved on a two-dimensional central slice of the billet. The diffusion equation is solved in a strong form by the Local Radial Basis Function Collocation Method (LRBFCM) with explicit time-stepping. The ray tracing procedure solves the radiation, where the view factors are computed with the Monte Carlo method. The changing number of billets in the furnace at the start and the end of the loading and unloading of the furnace is considered. A sensitivity study on billets’ temperature evolution is performed as a function of a different number of rays used in the Monte Carlo method, different stopping times of the billets in the furnace, and different spacing between the billets. The temperature field simulation is also essential for automatically optimizing the furnace’s productivity, energy consumption, and the billet’s quality. For the first time, the LRBFCM is successfully demonstrated for solving such a complex industrial problem.

A stabilized local RBF collocation method for incompressible Navier–Stokes equations

In this work, a stabilized local radial basis function (RBF) collocation method (LRBFCM) is proposed to solve the incompressible Navier–Stokes equations. An improved back ground fictitious grids or meshes technique is proposed to enhance the stability of the LRBFCM. The pressure is computed directly with the coupled velocity field in the Navier–Stokes equations. The explicit time-stepping scheme based on the forward Euler-method is employed for the time discretization, and the free surface of the flow is easy to capture as no numerical integration in the LRBFCM is needed. Numerical results show a high computational speed and a good stability of the proposed LRBFCM.

The local radial basis function collocation method for elastic wave propagation analysis in 2D composite plate

In this paper, the local radial basis function collocation method (LRBFCM) is applied to simulate the in-plane elastic wave propagation in composite plate. In order to reduce the influence caused by large aspect ratio, an improved LRBFCM based on direct method is developed so that the governing equations can be numerically decoupled by using bilinear nodes in different directions. Several numerical examples are given to demonstrate the accuracy and efficiency of the proposed technique in simulating the elastic wave propagation with large aspect ratio.

Fractional Dual-Phase-Lag Model for Nonlinear Viscoelastic Soft Tissues

The primary goal of this paper is to create a new fractional boundary element method (BEM) model for bio-thermomechanical problems in functionally graded anisotropic (FGA) nonlinear viscoelastic soft tissues. The governing equations of bio-thermomechanical problems are briefly presented, including the fractional dual-phase-lag (DPL) bioheat model and Biot’s model. The more complex shapes of nonlinear viscoelastic soft tissues can be handled by the boundary element method, which also avoids the need for the interior domain to be discretized. The fractional dual-phase-lag bioheat equation was solved using the general boundary element method (GBEM) based on the local radial basis function collocation method (LRBFCM). The poroelastic fields are then calculated using the convolution quadrature boundary element method (CQBEM) The numerical findings show that our proposed numerical model is valid, efficient, and accurate.

Adaptive selection strategy of shape parameters for LRBF for solving partial differential equations

Radial basis function (RBF) is a basis function suitable for scattered data interpolation and high dimensional function interpolation, wherein the independent shape parameters have a direct impact on the accuracy of calculation results. Research in this field is mostly concerned with the shape parameter selection strategy based on the premise of global distribution or block regional distribution. In this paper, a shape parameter selection strategy is proposed, which is used for the local RBF collocation method (LRBF) for solving partial differential equations. It overcomes many limitations of the traditional methods applied to LRBF. In this strategy, a set of twin matrices similar to the interpolation matrix are constructed to evaluate the error of the model. In addition, the penalty term contained in the twin matrix is used to relax the influence of the far end region on the target node. Since the objective problem is nonlinear, a particle swarm optimization algorithm (PSO) is employed to minimize the training objective and adjust the shape parameters of the basis function at each iteration. Extensive numerical results showed the effectiveness of the error estimation strategy, by providing a good shape parameter and better solution accuracy. At the end of the paper, the generality of the shape parameter optimization framework based on this strategy is discussed through three examples.

A Computational Model for Nonlinear Biomechanics Problems of FGA Biological Soft Tissues

The principal objective of this work was to develop a semi-implicit hybrid boundary element method (HBEM) to describe the nonlinear fractional biomechanical interactions in functionally graded anisotropic (FGA) soft tissues. The local radial basis function collocation method (LRBFCM) and general boundary element method (GBEM) were used to solve the nonlinear fractional dual-phase-lag bioheat governing equation. The boundary element method (BEM) was then used to solve the poroelastic governing equation. To solve equations arising from boundary element discretization, an efficient partitioned semi-implicit coupling algorithm was implemented with the generalized modified shift-splitting (GMSS) preconditioners. The computational findings are presented graphically to display the influences of the graded parameter, fractional parameter, and anisotropic property on the bio-thermal stress. Different bioheat transfer models are presented to show the significant differences between the nonlinear bio-thermal stress distributions in functionally graded anisotropic biological tissues. Numerical findings verified the validity, accuracy, and efficiency of the proposed method.

A novel local radial basis function collocation method for multi-dimensional piezoelectric problems

The local radial basis function collocation method (LRBFCM), a strong-form formulation of the meshless numerical method, is proposed for solving piezoelectric medium problems. The proposed numerical algorithm is based on the local Kansa method using variable shape parameter. We introduce a novel technique for the determination of shape parameter in the LRBFCM, which leads to greater accuracy, and simplicity. The implemented algorithm is first verified with a 2D Poisson equation. Then, we employed LRBFCM in a numerical simulation for 2D and 3D piezoelectric problems involving mutual coupling of the electric field and elastodynamic equations for mechanical field. The presented meshless method is verified using corresponding results obtained from the finite element method and moving least squares meshless local Petrov–Galerkin method. In particular, the 2D piezoelectric problem is verified with an exact solution.

3D elastic dental analysis by a local RBF collocation method

In this paper, the local radial basis function collocation method (LRBFCM) is presented to analyze 3D elastic problems of complex geometries. In particular, a real tooth domain built by cone beam computer tomography (CBCT) data is considered. The reconstruction process of a tooth shape, the choice of the local influence domain in the LRBFCM, as well as the treatments of the boundary conditions are detailed described in this paper. Linear elastic analyses of five different examples are carried out to validate the effectiveness of the proposed approach. The stress difference between a tooth with the dental caries and a healthy tooth is investigated and discussed by the 3D elastic deformation and stress analysis.

An improved local radial basis function collocation method based on the domain decomposition for composite wall

In this study, the idea of the domain decomposition is applied to further enhance the computational efficiency of the local radial basis function collocation method (LRBFCM). Unlike the traditional LRBFCM, the computational domain is decomposed into small unit square domains. Then the LRBFCM is further applied to each unit square domain, and the whole computational domain is later analyzed by considering interface conditions. The direct method is proposed to deal with the instability caused by the derivative calculation at the interface. Then the proposed LRBFCM is further extended to the composite walls. Different cases are carried out to test the efficiency of the proposed method. The accuracy of the numerical results is validated by comparing with the finite element method (FEM).

Application of the local RBF collocation method to natural convection in a 3D cavity influenced by a magnetic field

This paper explores, for the first time, the application of the novel mesh-free local radial basis function collocation method (LRBFCM) to the solution of a multi-physics problem in three dimensions. A related benchmark problem is solved by considering the natural convection of an incompressible Newtonian fluid in a differentially heated cubic cavity with and without the application of a magnetic field. The research is limited to typical magnetic fields used in the magnetohydrodynamic processing of liquid metals. For this purpose the assumption of small magnetic Reynolds numbers Rem ≪ 1 is made. Spatial discretization is performed by local non-uniform collocation with scaled multiquadrics radial basis functions (RBFs) with the shape parameter set to a constant value and the explicit Euler formula used to perform the time stepping. The involved temperature, velocity and pressure fields are represented on overlapping seven-nodded sub-domains. The pressure-velocity coupling is resolved by the fractional step method. The originality of the contribution represents LRBFCM solution of the classic three-dimensional steady natural convection benchmark for Rayleigh numbers from 105 to 107 and Prandtl number 0.71, and its extension to Prandtl number 0.1, and Hartman numbers 0, 10, 50 and 100. The accuracy of the LRBFCM is found to be comparable with the published benchmark results obtained using established numerical methods.

A meshless collocation method for band structure simulation of nanoscale phononic crystals based on nonlocal elasticity theory

In this paper, the band structures of nanoscale phononic crystals based on the nonlocal elasticity theory are calculated by using a meshfree local radial basis function collocation method (LRBFCM). The direct method is applied to enhance the stability of the derivative calculations in the LRBFCM. A simple summation in the LRBFCM is proposed to deal with the integration related to the nonlocal stresses or tractions. The LRBFCM for the band structure calculations is validated by the results obtained with the first-principle and the transfer matrix (TM) method for one-dimensional (1D) phononic crystals, as well as the comparison of the frequency responses of the two-dimensional (2D) periodic structures.

Response of a Porous Seabed around an Immersed Tunnel under Wave Loading: Meshfree Model

Seabed instability surrounding an immersed tunnel is a vital engineering issue regarding the design and maintenance for submarine tunnel projects. In this study, a numerical model based on the local radial basis function collocation method (LRBFCM) is developed to evaluate the seabed behaviour in a marine environment, in which the seabed is treated as the porous medium and governed by Biot’s “ u − p ” approximation. As for the flow field above the seabed, the VARANS equations are used to simulate the fluid motion and properties. The present model is validated with analytical solutions and experimental data which show a good capacity of the integrated model. Both wave and current loading are considered in this study. Parametric studies are carried out to investigate the effects of wave characteristics and soil properties. Based on the numerical results, the maximum liquefaction depth around the immersed tunnel could be deeper under the wave loading with long wave period (T) and large wave height (H). Moreover, a seabed with lower permeability ( K s ) and degree of saturation ( S r ) is more likely to be liquefied.

A local radial basis function collocation method for band structure computation of 3D phononic crystals

In this paper, we further extend the local radial basis function collocation method (LRBFCM) for efficient computation of band structures of phononic crystals from 2D to 3D. The proposed LRBFCM uses one fictitious node to tackle instability problems caused by calculation of derivatives of the wave pressure. A few examples of sound pressure wave propagation are modelled to validate the developed method. Comparisons with finite element modeling demonstrate the high stability and efficiency of the new method in computation of band structures of 3D phononic crystals.

A new LRBFCM-GBEM modeling algorithm for general solution of time fractional-order dual phase lag bioheat transfer problems in functionally graded tissues

The main objective of this article is to propose a new hybrid modeling algorithm based on combining local radial basis function collocation method (LRBFCM) and general boundary element method (GBEM) for solving time fractional-order dual phase lag bioheat transfer problems in functionally graded tissues. The LRBFCM was developed and implemented using an implicit time-stepping technique and Caputo time fractional derivative for solving the fractional-order governing equation without dual phase lags. Due to suitability of the GBEM for modeling of bioheat transfer in functionally graded tissues. Therefore, GBEM is applied for solving the dual phase lags governing equation without fractional-order derivative. The numerical results are depicted graphical forms to show the effects of functionally graded parameter, fractional-order parameter and anisotropy on the nonlinear temperature distribution. Also, these numerical results demonstrate the validity and accuracy of the proposed algorithm and technique.

Solution of three-dimensional temperature and turbulent velocity field in continuously cast steel billets with electromagnetic stirring by a meshless method

This paper demonstrates the development of meshless Local Radial Basis Function Collocation Method (LRBFCM) for solution of three dimensional (3D) turbulent molten steel flow and solidification under the influence of electromagnetic stirring (EMS) and its application to continuous casting process of steel billets. A mixture continuum approach is used to formulate the coupled set of macroscopic equations for mass, momentum, energy, turbulent kinetic energy, and dissipation rate in Cartesian coordinates. The mushy zone is treated as a Darcy porous media. The explicit Chorin fractional step method is used to resolve the pressure-velocity coupling. The LRBFCM uses explicit time discretization and spatial discretization with shape-adaptive multiquadrics radial basis functions collocation on non-uniform seven-noded influence domains, and displacement-adaptive upwind scheme. The Lorentz force due to EMS is provided by a one-way coupling by Elmer software. The advantages of the meshless method are trivial adaptation of the nodal densities, simple upgrade to 3D from previous two-dimensional models, and high accuracy. A study on the influence of changing the EMS parameters on the calculated temperature and velocity fields is performed. The paper confirms the ability of LRBFCM meshless solution of a realistic 3D multiphysics industrial problem with complicated swirling flow pattern.

A local radial basis function collocation method for band structure computation of phononic crystals with scatterers of arbitrary geometry

• A new numerical algorithm based on the LRBFCM is proposed. • The stability of the LRBFCM is greatly increased. • The boundary or interface condition of complex geometry can be easily treated. • The improved LRBFCM is applied to the PCs with a scatterer of complex geometry. A numerical algorithm based on the local radial basis function collocation method (LRBFCM) is developed to efficiently compute the derivatives of primary field quantities. Instead of a direct calculation of the derivatives by partial differentiation of the shape functions as in traditional numerical approaches, the derivative calculation in the present work is performed using a simple finite difference scheme with an introduced fictitious node. The developed algorithm is geometrically very flexible and can be easily applied to the continuity and boundary conditions of arbitrary geometries, which require an accurate derivative computation of the primary field quantities. The developed LRBFCM are applied to phononic crystals with scatterers of arbitrary geometry, which has not yet been reported before to the authors’ knowledge. A few examples for anti-plane elastic wave propagation are modelled to validate the developed LRBFCM. A comparison with finite element modelling shows that the present method is efficient and flexible.

A Rolling Simulation System for Reversing Rolling Mills

In this work a rolling simulation system has been developed for rolling schedules which consists of multiple reversing rolling mills. A slice model approach is applied where the position of a slice can only be determined by considering total deformation. Each slice is parallel to each other and perpendicular to the rolling direction. The solution of coupled thermal and mechanical models over each slice, at a given time and position, are achieved by a novel meshless Local Radial Basis Function Collocation Method (LRBFCM). Mechanical material model obeys ideal plastic flow rule defined by Von Mises. Unknown fields over the slices are interpolated by a certain number of collocation points distributed over the physical domain and its boundary. A system of equations is solved for each collocation point considering its local neighbouring points in the range between 5 and 7. A non-linear system of equations is solved by direct iteration. Groove geometries of each roll are implemented in a compatible way with the slice model and every roll has a horizontal orientation. In between each rolling pass the billet is rotated either 90 or 45 degrees clockwise or counter clockwise. Reduction at each of the passes can be very high, and in such cases, the material completely fills up the groove. This requires a special attention regarding the contact boundary conditions and the collocation node distribution due to numerical instability issues. Coulomb model of friction or sticking boundary conditions are used at the contact boundaries and Gauss-Seidel iterative elliptic node generation algorithm is used for redistributing collocation nodes over the physical domain, when necessary. The simulation results for arbitrary initial position of the slice in the billet are shown in terms of temperature, displacement, strain and stress fields as well as roll forces and torques. A user friendly computer application is created for industrial use based on C# and .NET.