Finite Pointset Method Research Articles

A finite pointset method for Kuramoto–Sivashinsky equation based on mixed formulation

A finite pointset method for Kuramoto–Sivashinsky equation based on mixed formulation

A novel semi-implicit WLS scheme for time-memory nonlinear behavior in 2D variable-order TF-NLSEs

In this paper, a novel hybrid semi-implicit meshless weighted least-squares (WLS) scheme H-SIFPM is developed for the first time by coupling the semi-implicit finite point-set method and finite difference method (FDM), and then applied to predict the time-memory nonlinear behavior dominated by a 2D variable-order time-fractional nonlinear Schrödinger equation (TF-NLSE). The proposed H-SIFPM for TF-NLSE is mainly derived from that: firstly, the variable-order time-fractional with Caputo derivative is approximated by a FDM scheme and then imposed to the approximation of WLS as a boundary condition; secondly, the hybrid process is implicitly discretized in temporal level, and an implicit meshless FPM scheme is developed; thirdly, an iterative technique is adopted to treat the implicit format. Subsequently, the estimated error and stability of the proposed scheme is tentatively proved. Moreover, the accuracy and capacity of the proposed H-SIFPM are also tested by solving several examples with constant/variable-order cases, in which a second-order numerical convergent rate is illustrated. We also demonstrate the advantages of the meshless method by solving the problem on complex irregular domain with local refinement. Finally, the nonlinear dispersion behavior dominated by 2D TF-NLSE/TF-GPE with different boundary conditions are predicted, and the FDM results are also presented for comparison. All numerical results show that the proposed H-SIFPM scheme for TF-NLSEs is stable, accurate and flexible.

A GFDM approach based on the finite pointset method for two-dimensional piezoelectric problems

In this article a novel Generalized Finite Difference Method (GFDM) derived from the so-called Finite Pointset Method (FPM) is presented and discussed for the first time to solve two-dimensional piezoelectric structures. In this approach, the approximation of the field variables depends on both the governing equations and the local problem discretization, and it incorporates the minimization of the errors of the governing equations being solved, which directly leads to its numerical solution. This truly meshfree formulation is applied in order to solve a static piezoelectric problem that consists of the coupled partial differential equations for the deformations and the electric field, in terms of the displacements and the scalar electric potential. The numerical results of some two-dimensional benchmark problems using the proposed approach are reported and compared with analytical solutions, if available, or with reference numerical solutions, in order to demonstrate its accuracy and suitability for this kind of problems. These numerical results suggest that the proposed GFDM formulation is feasible and promising for solving two-dimensional piezoelectric problems.

A Study on the Movement Characteristics of Fuel in the Fuel Tank during the Maneuvering Process

To investigate the movement characteristics of fuel within an aircraft’s fuel tank during maneuvering, this study employs numerical simulations using the Finite Pointset Method (FPM) with specific boundary conditions and excitation. The simulation results successfully capture the complex phenomena of fuel free surface fluctuations, surging, rolling, and breaking, along with the corresponding changes in the center of gravity induced by fuel movement. Throughout the aircraft’s maneuvering process, the longitudinal center of gravity experiences minimal variations, whereas the aerodynamic center of gravity exhibits significant fluctuations, reaching a range of 0.47 m. The longitudinal center of gravity consistently shifts backward with a displacement of 0.41 m. The maximum height attained by the liquid’s free surface during aircraft maneuvering measures approximately 0.22 m, reaching the upper panel’s height of the wing fuel tank, resulting in a certain level of impact on the upper panel. The maximum impact force generated during the aircraft maneuvering process amounts to 2.3 × 105 pascals, with the point of action located at the junction of the transverse frame and the upper panel. The findings of this work provide support for the safe design of aircraft fuel tank systems.

An efficient pure meshless method for phase separation dominated by time-fractional Cahn–Hilliard equations

In this paper, an efficient Twice Finite Point-set Method (TFPM) coupled with difference scheme is proposed to solve the time-fractional Cahn–Hilliard (TF-CH) equation, and then it is extended to predict the phase separation process under nonlocal memory dominated by two-component TF-CH equations for the first time. The proposed meshless schemes are motivated by the following: (a) a high-order accuracy difference scheme is employed to approximate the time Caputo fractional derivative; (b) the fourth-order spatial derivative is divided into two second-order derivatives, and it is discretized by the FPM scheme continuously twice based on Taylor expansion and weighted least squares; (c) the Neumann boundary can be accurately imposed on the FPM scheme. In the numerical experiments, the error and numerical convergence of the proposed meshless method are first tested, which has near second-order convergent rate. Subsequently, the evolution of phase separation under memory dominated by single CH equation versus time is numerically investigated by the proposed method, and compared with the results in other literatures. The influence of fractional parameter on the separation phenomena is also discussed. Finally, the proposed method is used to predict the phase separation process under different parameters dominated by coupled TF-CH equations. All the numerical results show that the proposed coupled method is accurate in solving the TF-CH and efficient in predicting the phase separation evolution.

GPU Parallelization Computation of High-Dimensional Multi-Phase Separation in Complex Domains Based on the Corrected FPM

Based on the corrected finite pointset method (CFPM) with CPU-GPU heteroid parallelization (CFPM-GPU), a high-efficiency, accurate and fast parallel algorithm was developed for the high-dimensional phase separation phenomena governed by the multi-component Cahn-Hilliard (C-H) equation in complex domains. The proposed parallel algorithm with the CFPM-GPU was built in a process like: ① introduce the Wendland weight function into the discretization of the finite pointset method (FPM) scheme for the 1st/2nd spatial derivatives, based on the Taylor series and the weighted least square concept; ② use the above FPM scheme twice to approximate the 4th spatial derivative in the C-H equation, which is called the CFPM method; ③ for the first time establish an accelerating parallel algorithm for the CFPM with local matrices by means of a single GPU card based on the CUDA programming. Two benchmark problems of 2D radially and 3D spherically symmetric C-H equations were first solved to test the accuracy and high-efficiency of the proposed CFPM-GPU, and the acceleration ratio of the GPU parallelization to the single CPU computation is about 160. Subsequently, the proposed CFPM-GPU was used to predict the 2D/3D multi-phase separation phenomena in complex domains, and the prediction was compared with other numerical results. The numerical results show that, the proposed CFPM-GPU is valid and high-efficiency to simulate the 2D/3D multi-phase separation cases in complex domains.

Cooling Capacity of Oil-in-Water Emulsion under wet Machining Conditions

Many industrial machining operations are carried out under wet machining conditions. Modelling and simulating fluid-structure-interactions and conjugate heat transfer are still a challenge nowadays. In this paper, temperature dependent heat transfer coefficients (HTC) h(T) are experimentally estimated for wet machining-like conditions in a jet cooling experiment. The transient temperature is thereby used to solve an Inverse Heat Transfer Problem for HTC function estimation. Determined HTC are applied as input in related jet cooling simulation using the Finite-Pointset-Method (FPM) to validate the modeling approach. Additionally, wet cutting simulations numerically highlight the influence of determined HTC h(T) on turning.

Meshfree numerical approach based on the finite pointset method for two-way coupled transient linear thermoelasticity

Meshfree numerical approach based on the finite pointset method for two-way coupled transient linear thermoelasticity

Mesh-free simulations of injection molding processes

In this paper, we introduce a mesh-free numerical framework using the finite pointset method for the modeling and simulation of injection molding processes. When compared to well-established mesh-based methods, which have been widely applied for these applications, our approach avoids the need for extensive preprocessing and enables accurate treatment of free surfaces and other associated phenomena. To accurately model the polymer injections, we consider a detailed material model, with temperature dependent viscosity and density, while also considering shear thinning behavior with a strain rate dependent viscosity. Our numerical investigations show that injection molding-specific problems such as the modeling of viscous flows and the fountain flow effect can be successfully implemented using our presented framework. For a thorough validation of our proposed model, we compare the simulated flow behavior with injection molding experiments, which are also performed in this work. The experimental setup considers the injection of a polymer melt into a spiral mold. The flow behavior is investigated experimentally at varying melt injection and wall temperature, with different threshold pressures. Our numerical simulations show a good comparison with these experimental results, both qualitatively and quantitatively. We also introduce a correction mechanism to ensure energy conservation, which has often been challenging in mesh-free approaches. This is the first time that the flow behavior in a mesh-free injection molding method has been experimentally validated and successfully applied to the simulation of an actual industrial vehicle component.

A novel meshfree approach based on the finite pointset method for linear elasticity problems

In this work a promising novel meshfree numerical formulation with improved accuracy based on the finite pointset method (FPM) is reported and implemented for the first time for linear elasticity problems. This truly meshfree approach is applied in order to solve the governing partial differential equations, the Navier–Cauchy equations. The numerical results of some 2D and 3D classical and well-known benchmark examples using this formulation are reported and compared with a previous FPM formulation which demonstrate the improvement in accuracy, and finally, the numerical solution of a three-dimensional realistic example is reported which suggest that the presented FPM approach is promising and feasible for this kind of problems in solid mechanics.

A high-efficient splitting step reduced-dimension pure meshless method for transient 2D/3D Maxwell’s equations in complex irregular domain

In this work, a high-efficient and accurate pure meshless method is designated as splitting step reduced-dimension finite pointset method, which is first proposed to solve the time-dependent 2D/3D Maxwell equations in lossy medium with periodic or perfect conducting boundary condition. The proposed method is mainly motivated by following three points. (a) The coupled 2D/3D Maxwell equations are decomposed into several mutually uncoupled 1D partial differential equations through a splitting step method. (b) The space-derivatives of the above those 1D systems are discretized by the 1D finite pointset method based on the Taylor expansion and the weighted least squares method. Meanwhile, the time-derivatives are approximated by second-order scheme. (c) The local refinement technique is adopted at the selected region to improve the numerical accuracy, and the multi-CPUs parallel computing is used to enhance the computational efficiency. All the numerical results show the proposed scheme has second-order convergence rate, and which has higher computational efficiency than the traditional finite pointset method for solving the 2D/3D Maxwell equations, especially for the case of 3D problems or higher-order Taylor expansion in finite pointset method. Subsequently, the advantage of the proposed scheme applied in complex irregular domain is illustrated.

A novel approach to model the flow of generalized Newtonian fluids with the finite pointset method

A novel approach to model the flow of generalized Newtonian fluids with the finite pointset method

Fully coupled meshfree numerical approach based on the finite pointset method for static linear thermoelasticity problems

In this work, a promising fully coupled meshfree numerical approach is extended and implemented for the first time in the field of linear static thermoelasticity. A real meshfree method, the so-called finite pointset method (FPM), is applied and implemented in order to solve the strong/classical form of the governing partial differential equations for static linear thermoelasticity. Several benchmark problems are numerically solved in order to show the proposed coupled FPM numerical performance. The presented FPM meshfree approach shows excellent behavior for 2D linear static thermoelasticity problems even for complex geometries.

Numerical Study of the 3D Variable Coefficient Heat Transfer Problem by Using the Finite Pointset Method

In this paper, a parallel mesh-free finite pointset method (FPM) for the 3D variable coefficient transient heat conduction problem (I-FPM-3D) on regular/irregular region is proposed by coupling several techniques as follows. The partial differential equation with the high-order derivatives is first decomposed into several first-order equations to improve the numerical stability, reduce the computational complexity and easily enforce the Neumann boundary condition. Then, each first-order derivative is solved by the FPM repeatedly. Moreover, the MPI parallel technique is introduced to enhance the computational efficiency, and an appropriate Wendland kernel function is employed which has higher accuracy than the Gaussian function. 3D transient heat conduction problems with different boundary conditions, including the case in a complex cylindrical domain, are investigated and compared with the analytical solutions to illustrate the flexibility and the accuracy of the parallel I-FPM-3D. The convergence and the computational efficiency of the parallel I-FPM-3D are also analyzed. Finally, the temperature field in a 3D functional gradient material is predicted by the parallel I-FPM-3D and compared with the other numerical results. The numerical results indicate that the temperature variation process in the functional gradient materials can be visualized accurately.

Simulation of metal cutting with cutting fluid using the Finite-Pointset-Method

Despite the common use of cutting fluids in metal cutting, most simulations of these processes only consider dry conditions. This is largely due to limitations in conventional simulation methods, such as the Finite Element Method (FEM), regarding fast changing computational domains. In this paper, a simulation model of metal cutting with cutting fluid based on the Finite Pointset Method (FPM) is presented. This approach allows to simulate the Fluid-Structure-Interactions (FSI) within the same simulation method. For different orthogonal turning processes, the simulations are compared with experiments for dry and wet conditions. The simulation results illustrate the cooling effect numerically.

Fully coupled wet cylindrical turning simulation using the Finite-Pointset-Method

In industrial machining operations, many turning processes are carried out under wet conditions. In contrast, most simulation tools are only suitable for dry cutting processes due to challenges of modeling fluid-structure-interactions (FSI). In this paper, a wet, fully coupled cutting simulation using the Finite-Pointset-Method (FPM) is presented and validated for industrially relevant cylindrical turning. To facilitate the required reduction of the calculation effort, different routines like adaptive numerical discretization are applied. The results indicate different cutting fluid effects like evaporation areas around the chip.

Meshfree numerical approach based on the Finite Pointset Method for static linear elasticity problems

In this work a promising meshfree numerical approach is extended and implemented for the first time in the field of linear static elasticity. A real meshfree method, the so called Finite Pointset Method (FPM), is applied and implemented in order to solve the strong/classical form of the governing partial differential equations (PDEs), the well known Navier–Cauchy equations. FPM numerical performance is demonstrated for several classical and well known studied benchmark problems and comparison with other approaches is also reported. Numerical results suggest that FPM method is stable with good convergence behavior for 2D linear static elasticity problems.

A highly stable explicit scheme for a fourth-order nonlinear diffusion filter

The standard finite difference scheme (forward difference approximation for time derivative and central difference approximations for spatial derivatives) for fourth-order nonlinear diffusion filter allows very small time-step size to obtain stable results. The alternating directional implicit (ADI) splitting scheme such as Douglas method is highly stable but compromises accuracy for a relatively larger time-step size. In this paper, we develop [Formula: see text] stencils for the approximation of second-order spatial derivatives based on the finite pointset method. We then make use of these stencils for approximating the fourth-order partial differential equation. We show that the proposed scheme allows relatively bigger time-step size than the standard finite difference scheme, without compromising on the quality of the filtered image. Further, we demonstrate through numerical simulations that the proposed scheme is more efficient, in obtaining quality filtered image, than an ADI splitting scheme.

A Finite Pointset Method for Extended Fisher–Kolmogorov Equation Based on Mixed Formulation

In this paper, numerical solutions of the extended Fisher–Kolmogorov equation are obtained using finite pointset method. Finite pointset method is a meshless method which is a local iterative method based on the weighted least square approximation. By employing splitting technique, the extended Fisher–Kolmogorov equation is split into a two coupled system of differential equations by introducing an intermediate function. The method is applied to the resulting coupled system of differential equation. The numerical results confirm the good efficiency of the finite pointset method.

Algebraic multigrid for the finite pointset method

We investigate algebraic multigrid (AMG) methods for the linear systems arising from the discretization of Navier–Stokes equations via the finite pointset method. In the segregated approach, three pressure systems and one velocity system need to be solved. In the coupled approach, one of the pressure systems is coupled with the velocity system, leading to a coupled velocity-pressure saddle point system. The discretization of the differential operators used in FPM leads to non-symmetric matrices that do not have the M-matrix property. Even though the theoretical framework for many AMG methods requires these properties, our AMG methods can be successfully applied to these matrices and show a robust and scalable convergence when compared to a BiCGStab(2) solver.