Local Radial Basis Function Collocation Research Articles

Fourth-order phase field modelling of brittle fracture with strong form meshless method

This study aims to find a solution for crack propagation in 2D brittle elastic material using the local radial basis function collocation method. The staggered solution of the fourth-order phase field and mechanical model is structured with polyharmonic spline shape functions augmented with polynomials. Two benchmark tests are carried out to assess the performance of the method. First, a non-cracked square plate problem is solved under tensile loading to validate the implementation by comparing the numerical and analytical solutions. The analysis shows that the iterative process converges even with a large loading step, whereas the non-iterative process requires smaller steps for convergence to the analytical solution. In the second case, a single-edge cracked square plate subjected to tensile loading is solved, and the results show a good agreement with the reference solution. The effects of the incremental loading, length scale parameter, and mesh convergence for regular and scattered nodes are demonstrated. This study presents a pioneering attempt to solve the phase field crack propagation using a strong-form meshless method. The results underline the essential role of the represented method for an accurate and efficient solution to crack propagation. It also provides valuable insights for future research towards more sophisticated material models.

The localized radial basis function collocation method for dendritic solidification, solid phase sintering and wetting phenomenon based on phase field

Phase-field has been effectively applied to many complex problems according to the mesh based method. However, the computational speed of the numerical method based on phase-field still needs improved. In this paper, an improved localized radial basis function collocation method (LRBFCM) based on the adaptive support domain is employed to the phase-field methods. The proposed adaptive support domain can increase the stability of the LRBFCM, and the improved LRBFCM is much more efficient than the traditional finite element method (FEM) in coupling with phase-field methods. The proposed approach is further applied to the single-phase dendrite solidification, two-phase sintering, and three-phase wetting phenomena. We compare the efficiency of the proposed LRBFCM with different numerical methods, which show that the LRBFCM combined with the Fourier spectral method can deal with the three-phase model with more than ten million nodes easily.

A POD based reduced-order local RBF collocation approach for time-dependent nonlocal diffusion problems

A fast algorithm based on reduced-order model (ROM) is proposed for unsteady nonlocal diffusion models. It combines proper orthogonal decomposition (POD) approach and collocation method with local radial basis functions (RBFs), which makes it possible for using ROM to solve nonlocal models. Several numerical experiments showed that this approach significantly reduce the computational cost of nonlocal models while keep the similar convergent behavior compared with the RBF collocation methods.

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.

Fourth-order phase-field simulation of cracks using strong form meshless method

The prediction of crack propagation in fracture mechanics is still a fundamental challenge. In recent decades, phase-field formulation emerged as a powerful tool for the simulation of crack evolution, which offers a continuous representation of cracks from intact to broken material. For the first time, the present study combines a fourth-order phase field and a strong-form meshless method to investigate crack propagation in brittle elastic material. The local radial basis function collocation method, structured with the augmented polyharmonic splines, is used to solve the coupled mechanical and phase-field models. Due to the smooth nature of the fourth-order phase-field model, the sharp transition at the crack surface can be avoided. It is found that the presented method successfully predicts the crack propagation for a material subjected to tensile loading. The obtained results agree well with the benchmark case. Further study will include elastoplastic material behaviour and applications to crack formation in different steel processing steps.

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.

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.

Assessment of Local Radial Basis Function Collocation Method for Diffusion Problems Structured with Multiquadrics and Polyharmonic Splines

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.

An adaptive support domain for the in-compressible fluid flow based on the localized radial basis function collocation method

In this paper, an adaptive support domain scheme is proposed to the in-compressible fluid flow based on the localized radial basis function collocation method (LRBFCM). The idea of the adaptive support domain avoids complex mathematical derivations of the finite difference method (FDM). Unlike other upwind schemes in the LRBFCM, the support domain uses the sampling nodes similar to different finite difference schemes, and the improved LRBFCM is further applied to the in-compressible fluid flows. Considering the Kronecker tensor product and Cholesky decomposition techniques, the improved LRBFCM can be easily used to the lid-driven problems with Reynolds numbers up to 100,000 by a simple mathematical derivation. The proposed LRBFCM has an adaptive support domain similar to the finite difference schemes with much less CPU time costs.

Analysis of the band structure of transient in-plane elastic waves based on the localized radial basis function collocation method

In this paper, the localized radial basis function collocation method is applied to investigate the in-plane elastic wave propagation of 2D phononic crystals. The direct method with local support domain in one line is used to enhance the stability of the localized radial basis function collocation method when dealing with the derivative calculations in the boundary and interface conditions. The Houbolt method is employed to discretize the time, and then the Fast Fourier Transform is adopted to obtain the band structure according to the wave vectors in the first Brillouin Zone. Different acoustic impedance ratios, filling rates, and lattice forms are studied. The numerical results can be used to validate the band structures obtained directly from the frequency domain.

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.

Local radial basis function collocation method preserving maximum and monotonicity principles for nonlinear differential equations

AbstractIn this paper, a hybrid numerical scheme based on combining exponential time differencing (ETD) and local radial basis function collocation method was constructed. Model problems with different boundary conditions were considered, and the resulting linear system was carefully analyzed. The relation between the number of points employed in the local radial basis function collocation method and the condition number of the coefficient matrix was given. For application, three typical differential equations were investigated, that is, the Allen–Cahn equation for checking the maximum‐preserving property, the combustion equation for checking the monotonicity‐preserving property, and the Gray–Scott system for checking the robustness of the proposed scheme. Numerical examples show the effectiveness of the proposed method.

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.

A meshless multiscale method for simulating hemodynamics

Simulating hemodynamic quantities such as pressure and velocity are of great interest to clinicians to aid in surgical planning. To accurately simulate modifications to a region of vasculature, the entire system must be modeled. To facilitate this, a localized Radial-Basis Function (RBF) collocation Meshless flow solver is developed and tightly coupled to a 0D Lumped-Parameter Model (LPM) for solution of the peripheral circulation. The Meshless solver uses localized RBF collocations at data points that are automatically generated according to the geometry. The incompressible flow equations are updated by an explicit time-marching scheme based on a pressure-velocity correction algorithm. The inlets and outlets of the domain are tightly coupled with the LPM which contains elements that draw from a fluid-electrical analogy such as resistors, capacitors, and inductors that represent the viscous resistance, vessel compliance, and flow inertia, respectively. The localized RBF Meshless approach is well-suited for modeling complicated non-Newtonian hemodynamics due to ease of spatial discretization, ease of addition of multi-physics interactions such as fluid-structure interaction of the vessel wall, and ease of parallelization for fast computations. This work introduces the tight coupling of Meshless methods and LPMs for fast and accurate hemodynamic simulations.

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.

A Novel Localized Meshless Method for Solving Transient Heat Conduction Problems in Complicated Domains

This paper first attempts to solve the transient heat conduction problem by combining the recently proposed local knot method (LKM) with the dual reciprocity method (DRM). Firstly, the temporal derivative is discretized by a finite difference scheme, and thus the governing equation of transient heat transfer is transformed into a non-homogeneous modified Helmholtz equation. Secondly, the solution of the non-homogeneous modified Helmholtz equation is decomposed into a particular solution and a homogeneous solution. And then, the DRM and LKM are used to solve the particular solution of the non-homogeneous equation and the homogeneous solution of the modified Helmholtz equation, respectively. The LKM is a recently proposed local radial basis function collocation method with the merits of being simple, accurate, and free of mesh and integration. Compared with the traditional domain-type and boundary-type schemes, the present coupling algorithm could be treated as a really good alternative for the analysis of transient heat conduction on high-dimensional and complicated domains. Numerical experiments, including two- and three-dimensional heat transfer models, demonstrated the effectiveness and accuracy of the new methodology.

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.

Numerical simulation of 3D double-nozzles printing by considering a stabilized localized radial basis function collocation method

In this work, a stabilized local radial basis function collocation method (LRBFCM) is proposed to simulate the electrochemical 3D double-nozzles printing. A forward Euler-method is proposed to discretize time in the fluid flow described by Navier-Stokes (NS) equations. The background fictitious grids or meshes are employed to enhance the stability of the LRBFCM in solving the fluid flow. The details of the LRBFCM for simulating the pressure and velocity fields of the electrolyte during the electrochemical 3D printing process are introduced. The accuracy of the proposed method is validated by corresponding experiments, and good agreement can be observed between the numerical results and the experimental measurements.

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.