Meshless Local Petrov-Galerkin Research Articles

Numerical modelling of forced heaving of mono hull and twin hull in particle method

In this paper, the particle method for handling oscillation of floating body using variable particle spacing is reported. Heave oscillation of floating mono hull and twin hull on the free surface in a viscous fluid is performed using Improved Meshless Local Petrov-Galerkin method based on Rankine Source function (IMLPG_R). The IMLPG_R method is a particle-based method that will solve the interaction between fluid and floating body using Navier-Stokes equation. The pressure at each time step is estimated from Pressure Poisson equation, in order to incorporate the heaving of the floating bodies, some improvements like free surface identification, gradient calculations has been introduced in the present paper to handle the variable resolutions in modelling. Simulations are performed for different frequency and amplitude of oscillations for mono and twin hull forced motions to show the robustness of the developed model to handle both small and steep oscillations.

Interpolating meshless local Petrov-Galerkin method for steady state heat conduction problem

In many meshfree methods, moving least squares scheme (MLS) has been used to generate meshfree shape functions. Imposition of Dirichlet boundary conditions is difficult task in these methods as the MLS approximation is devoid of Kronecker delta property. A new variant of the MLS approximation scheme, namely interpolating moving least squares scheme, possesses Kronecker delta property. In the current work, a novel interpolating meshless local Petrov-Galerkin (IMLPG) method has been developed based on the interpolating MLS approximation for two and three dimensional steady state heat conduction in regular and complex domain. The interpolating MLPG method shows two advantages over standard meshless local Petrov-Galerkin (MLPG) method i.e. higher computational efficiency and ease to impose EBCs at similar accuracy level. Performance of three different test functions in-conjunction with interpolating MLPG method has been shown.

Node-to-Node Realization of Meshless Local Petrov Galerkin (MLPG) Fully in GPU

This paper presents an end-to-end massively parallelized procedure for the solution of boundary value problems on Graphics Processing Units (GPU). The proposal is an integrated strategy that not only entails the calculation of nodal contributions, and the stiffness matrix assembly using the Meshless Local Petrov Galerkin Method (MLPG) but also the iterative solution of the system of algebraic equations in combination with methods from the Conjugate Gradient (CG) family. This end-to-end solution is fully developed using the Compute Unified Device Architecture (CUDA) platform without the need for extra data movement between the device and host after the matrix assembly. This is possible thanks to the parallel nature of the MLPG; each node designates a thread on the device. The introduced solution is wholly executed in the GPU, with minimal auxiliary structures or global synchronization points. The proposed approach was applied to the solution of a simple electromagnetic problem, and a sevenfold speedup was observed.

Study of a linear viscoelastic band by the meshless local Petrov–Galerkin method (MLPG)

Study of a linear viscoelastic band by the meshless local Petrov–Galerkin method (MLPG)

The Slope Stability Solution Using Meshless Local Petrov-Galerkin Method

Abstract The paper deals with use of the meshless method for slope stability analysis. There are many formulations of the meshless methods. The article presents the Meshless Local Petrov-Galerkin method (MLPG) – local weak formulation of the equilibrium equations. The main difference between meshless methods and the conventional finite element method (FEM) is that meshless shape functions are constructed using randomly scattered set of points without any relation between points. The shape function construction is the crucial part of the meshless numerical analysis. The numerical example of the slope stability was calculated using meshless computer code and compared with FEM and limit equilibrium (LEM) results.

Meshless fracture analysis of 3D planar cracks with generalized thermo-mechanical stress intensity factors

This paper implements three dimensional meshless local Petrov–Galerkin method in the linear elastic fracture mechanics analyses of isotropic functionally graded and homogeneous materials under different thermo-mechanical loads. Energy equation based on Lord–Shulman non-Fourier heat conduction law coupled with equation of motion is applied to evaluate the effect of theoretical and realistic relaxation times on transient thermal stress intensity factors along 3D thorough or semielliptical cracks under thermo-mechanical shock. A new coordinate transform method with linear weight function is introduced to evaluate stress intensity factors by generalized interaction integral method based on incompatibility formulation. Simple linear test function is introduced, which is approximated via compatible radial basis functions and leads in eliminating internal boundary integrals of meshless method and reducing computational time. Shape functions are constructed in every Gauss point by determining the closest domain point method in preprocessing. Parametric studies are carried out in order to determine suitable radial basis functions shape parameters and penalty method parameter. Reasonable accuracy and the small number of used points are the main advantages of this method over finite element method.

A meshless Reissner plate bending procedure using local radial point interpolation with an efficient integration scheme

This paper presents a numerical analysis of bending plates considering Reissner’s hypothesis. A truly meshless method designated Meshless Local Petrov-Galerkin (MLPG) method is used to obtain a linear system of equation. A simplified Radial Point Interpolation Method (RPIM) approximation scheme, by centering both quadrature and local interpolation subdomains in the same field point, is proposed to increase the computational efficiency of the MLPG. Moreover, the shear locking effect is also analyzed. Results obtained by the application of the presented formulation are discussed in this work, considering plates with different geometries and boundary conditions, and compared, in terms of precision and efficiency, with solutions obtained via Finite Element Method.

Finite and infinite block Petrov–Galerkin method for cracks in functionally graded materials

• A novel finite and infinite block Petrov–Galerkin method is applied for crack problems in functionally graded materials . • The problem domain is discretized by a few blocks in the computation which saves tedious mesh generation . • The displacements and stresses are continuous at interfaces of neighboring blocks. • Due to the usage of Petrov–Galerkin formulation, the BPGM is a weak-form method. • Numerical results show that the proposed approach is accurate and efficient. This paper presents a novel weak-form block Petrov–Galerkin method (BPGM) for linear elastic and crack problems in functionally graded materials with bounded and unbounded problem domains. The main idea of this approach is to combine the meshless local Petrov–Galerkin method with block method. Once the problem domain is discretized into several sub-regions, named blocks, which can be mapped into normalized square domains. The weak-form Petrov–Galerkin method and polynomial series of interpolations are employed in each block. The computational efficiency is rigorously examined against the strong-form finite block method, the finite element technique and meshless approaches. Numerical results demonstrate that the BPGM possesses the following important properties: (1) only a few blocks are required for calculating problems in unbounded regions which saves tedious work of meshing; (2) the displacements and stresses are continuous at the interfaces of neighboring blocks; (3) due to the use of weak formulation, the continuity requirements of the approximation functions are reduced and numerical results are stable; (4) because of using Lagrange polynomial interpolation, highly accurate solutions can be obtained with a small amount of nodes in each block.

Extending the Meshless Local Petrov–Galerkin Method to Solve Stabilized Turbulent Fluid Flow Problems

The aim of this paper is to extend the meshless local Petrov–Galerkin method to solve stabilized turbulent fluid flow problems. For the unsteady incompressible turbulent fluid flow problems, the Spalart–Allmaras model is used to stabilize the governing equations, and the meshless local Petrov–Galerkin method is extended based on the vorticity-stream function to solve the turbulent flow problems. In this study, the moving least squares scheme interpolates the field variables. The proposed method solves three standard test cases of the turbulent flow over a flat plate, turbulent flow through a channel, and turbulent flow over a backward-facing step for evaluation of the method’s capability, accuracy, and validity purposes. Based on the comparison of the three test cases results with those of the experimental and conventional numerical works available in the literature, the proposed method shows to be accurate and quite implemental. The new extended method in this study together with the previously published works of the authors (on extending the meshless local Petrov–Galerkin method to solve laminar flow problems) now, for the first time, empower the meshless method to solve both laminar and turbulent flow problems.

A solution to the singularity problem in the meshless method for neutron diffusion equation

In recent years, the moving least square (MLS) approximation has been widely used for numerical analysis of scattered data. Many numerical methods for solving partial differential equations are developed based on the MLS approximation.The MLS method is ideal for problems where the geometry of the domain is complex. An example of this is the neutron diffusion calculation. The collocation method is known that it has fast convergence speed, however, it is unstable and non-robust in many cases. In particular, it is the reason resulting in the fluctuation of the solution at the position that is on overlapping boundary region. This drawback can be dealt with the meshless local Petrov-Galerkin weak form for nodes on the interface boundary. In this paper, a combination of the weak form of the meshless local Petrov-Galerkin (MLPG) and the collocation method are applied to solve the neutron diffusion equation. Trial functions employed in the weak form of MLPG are constructed via the MLS method. In most MLPG methods numerical integration is required. The exception to this is the case of the collocation method in which the test functions are Dirac delta functions. Therefore, if the shape function is not constructed precisely enough, within an acceptable error range, the second order derivative of the shape function will lack accuracy. One of the reasons for the inaccuracy of the shape function is singularity problems occurring in the process of constructing the shape function. In this study, a solution is introduced to eliminate the singularity problem, allowing us to obtain the derivative of the shape function with sufficient precision. Finally, neutron diffusion problems are implemented in a combination of the weak-form of the meshless local Petrov-Galerkin (MLPG) and the collocation method is used to evaluate the efficiency and accuracy.

A direct meshless local collocation method for solving stochastic Cahn–Hilliard–Cook and stochastic Swift–Hohenberg equations

In this study, the direct meshless local Petrov–Galerkin (DMLPG) method has been employed to solve the stochastic Cahn–Hilliard–Cook and Swift–Hohenberg equations. First of all, we discretize the temporal direction by a finite difference scheme. In order to obtain a fully discrete scheme the direct meshless local collocation method is used to discretize the spatial variable and the Euler–Maruyama method is used for time discretization. The used method is a truly meshless technique. In order to illustrate the efficiency and accuracy of the explained numerical technique, we study two stochastic models with their applications in biology and engineering, i.e., the stochastic Cahn–Hilliard–Cook equation and a stochastic Swift–Hohenberg model.

Numerical Analysis of a Semi-Infinite Solid with Temperature Dependent Thermal Conductivity using Truly Meshfree Method

This article presents Meshless Local Petrov-Galerkin (MLPG) method to obtain the numerical solution of linear and non-linear heat conduction in a semi-infinite solid object with specific heat flux. Moving least square approximants are used to approximate the unknown function of temperature T(x) with Th(x). These approximants are constructed by using a linear basis, a weight function and a set of non-constant coefficients. Essential boundary condition is imposed by the penalty function method. A predictor-corrector scheme based on direct substitution iteration has been applied to address the non-linearity and two-level  method for temporal discretization. The accuracy of MLPG method is verified by comparing the results for the simplified versions of the present model with the exact solutions. Once the accuracy of MLPG method is established, the method is further extended to investigate the effects of temperature-dependent properties.

DMLPG method for numerical simulation of soliton collisions in multi-dimensional coupled damped nonlinear Schrödinger system which arises from Bose–Einstein condensates

In this paper, the direct meshless local Petrov–Galerkin (DMLPG) method is applied to find the numerical solution of coupled damped nonlinear Schrödinger system in one, two and three-dimensional spaces. The propagation properties of single soliton, double and triple solitons of coupled damped nonlinear Schrödinger system are simulated and the interactions between these solitons are studied numerically. The efficient time differencing Runge–Kutta method is utilized for the time discretization. DMLPG shifts the numerical integrations over low-degree polynomials rather than over complicated shape functions and this significantly increases the computational efficiency of DMLPG in comparison with the other meshless local weak form methods especially in two and three dimensions. The main aim of this paper is to show that the DMLPG method can be simply used for solving high-dimensional system of non-linear partial differential equations especially coupled damped nonlinear Schrödinger system. The numerical results confirm the good efficiency of the proposed method for solving our model.

Meshless local Petrov–Galerkin solution of the neutron transport equation with streamline-upwind Petrov–Galerkin stabilization

The meshless local Petrov–Galerkin (MLPG) method is applied to the steady-state and k-eigenvalue neutron transport equations, which are discretized in energy using the multigroup approximation and in angle using the discrete ordinates approximation. To prevent oscillations in the neutron flux, the MLPG transport equation is stabilized by the streamline upwind Petrov–Galerkin (SUPG) method. Global neutron conservation is enforced by using moving least squares basis and weight functions and appropriate SUPG parameters. The cross sections in the transport equation are approximated in accordance with global particle balance and without constraint on their spatial dependence or the location of the basis and weight functions. The equations for the strong-form meshless collocation approach are derived for comparison to the MLPG equations. The method of manufactured solutions is used to verify the resulting MLPG method in one, two and three dimensions. Results for realistic problems, including two-dimensional pincells, a reflected ellipsoid and a three-dimensional problem with voids, are verified by comparison to Monte Carlo simulations.

An improved pseudospectral meshless radial point interpolation (PSMRPI) method for 3D wave equation with variable coefficients

In this paper, a pseudospectral meshless radial point interpolation (PSMRPI) technique is applied to the three-dimensional wave equation with variable coefficients subject to given appropriate initial and Dirichlet boundary conditions. The present method is a kind of combination of meshless methods and spectral collocation techniques. The point interpolation method along with the radial basis functions is used to construct the shape functions as the basis functions in the frame of the spectral collocation methods. These basis functions will have Kronecker delta function property, as well as unitary possession. In the proposed method, operational matrices of higher order derivatives are constructed and then applied. The merit of this innovative method is that, it does not require any kind of integration locally or globally over sub-domains, as it is essential in meshless methods based on Galerkin weak forms, such as element-free Galerkin and meshless local Petrov–Galerkin methods. Therefore, computational cost of PSMRPI method is low. Further, it is proved that the procedure is stable with respect to the time variable over some conditions on the 3D wave model, and the convergence of the technique is revealed. These latest claims are also shown in the numerical examples, which demonstrate that PSMRPI provides excellent rate of convergence.

RBF-based meshless local Petrov Galerkin method for the multi-dimensional convection–diffusion-reaction equation

In this paper, the meshless local Petrov Galerkin (MLPG) method is employed to analyze convection–diffusion-reaction equation based on radial basis function (RBF) collocation method. Compared with traditional Petrov Galerkin method, MLPG method is not limited to the particular interpolation domain and integration domain. In the MLPG method, interpolation domain is chosen through a special method which ensures that the number of neighboring nodes around the interpolation point (or collocation point) is constant. And all integrations are carried out locally over small quadrature domains of regular shapes such as two-dimensional (2D) squares and three-dimensional (3D) cubes. Crank–Nicolson scheme is applied to the time discretization to guarantee the unconditional stability of present method. Thanks to the Kronecker delta function property of the shape functions for RBF interpolation, we can easily obtain the error estimates that MLPG method has second order convergent rate in time and space simultaneously. Finally, numerical examples are presented to show the accuracy and efficiency of the MLPG method.

A meshless local Petrov–Galerkin method for solving the neutron diffusion equation

The goal of this study is to solve the neutron diffusion equation by using a meshless method and evaluate its performance compared to traditional methods. This paper proposes a novel method based on coupling the meshless local Petrov–Galerkin approach and the moving least squares approximation. This computational procedure consists of two main steps. The first involved applying the moving least squares approximation to construct the shape function based on the problem domain. Then, the obtained shape function was used in the meshless local Petrov–Galerkin method to solve the neutron diffusion equation. Because the meshless method is based on eliminating the mesh-based topologies, the problem domain was represented by a set of arbitrarily distributed nodes. There is no need to use meshes or elements for field variable interpolation. The process of node generation is simply and fully automated, which can save time. As this method is a local weak form, it does not require any background integration cells and all integrations are performed locally over small quadrature domains. To evaluate the proposed method, several problems were considered. The results were compared with those obtained from the analytical solution and a Galerkin finite element method. In addition, the proposed method was used to solve neutronic calculations in the small modular reactor. The results were compared with those of the citation code and reference values. The accuracy and precision of the proposed method were acceptable. Additionally, adding the number of nodes and selecting an appropriate weight function improved the performance of the meshless local Petrov–Galerkin method. Therefore, the proposed method represents an accurate and alternative method for calculating core neutronic parameters.

Two-level meshless local Petrov Galerkin method for multi-dimensional nonlinear convection–diffusion equation based on radial basis function

In this article, a two-level meshless local Petrov Galerkin method (MLPG) is proposed to analyze nonlinear convection–diffusion equation based on the radial basis function (RBF) collocation method. Two-level method is employed to save the computation time, solving one small linearized convection–diffusion equation in a coarse nodal distribution by Newton iterative scheme and then processing one linear equation on a fine nodal distribution. The convergence analysis for the MLPG method and the two-level method is proven by applying the RBF interpolation method. Finally, the numerical examples provide a sufficient support and show that the proposed method is efficient for solving nonlinear convection–diffusion equation.

Elastic analysis of arbitrary shape plates using Meshless local Petrov–Galerkin method

In this study the stress analysis of orthotropic thin plate with arbitrary shapes for different boundary conditions is investigated. Meshfree method is applied to static analysis of thin plates with various geometries based on the Kirchhoff‎ classical plate theory. According to the meshfree method the domain of the plates are expressed through a set of nodes without using mesh. In this method, a set of nodes are defined in a standard rectangular domain, then via a third order map, these nodes are transferred to the main domain of the original geometry; therefore the analysis of the plates can be done. Herein, Meshless local Petrov–Galerkin (MLPG) as a meshfree numerical method is utilized. The MLS function in MLPG does not satisfy essential boundary conditions using Delta Kronecker. In the MLPG method, direct interpolation of the boundary conditions can be applied due to constructing node by node of the system equations. The detailed parametric study is conducted, focusing on the arbitrary geometries of the thin plates. Results show that the meshfree method provides better accuracy rather than finite element method. Also, it is found that trend of the figures have good agreement with relevant published papers.

Anisotropic transient thermoelasticity analysis in a two-dimensional decagonal quasicrystal using meshless local Petrov–Galerkin (MLPG) method

The meshless local Petrov–Galerkin (MLPG) method is employed for anisotropic transient thermoelasticity analysis of 2D decagonal quasicrystals (QCs) subjected to transient thermal and mechanical shock loadings. The wave type model and the elasto-hydrodynamic model are applied to derive the phonon and phason governing equations, respectively. The temperature affects only the phonon field. To find the temperature distributions on the assumed 2D domain, the anisotropic heat conduction problem is solved using the MLPG method. Also, the MLPG method is successfully employed to obtain the transient behaviors of both phonon and phason displacements by solving the governing equations in local integral equations (LIEs) forms. Making use a unit step function as the test functions in the local weak-form of governing equations, we derived the local integral equations (LIEs) considered on small subdomains identical with support domains of test functions around each node. The radial basis functions are used for approximation of the spatial variation of field variables. The Laplace-transform technique is utilized to discretize the time variations.