Meshless Local Petrov-Galerkin Method Research Articles

OpenMP-based parallel MLPG solver for analysis of heat conduction

PurposeIn the present work, we focus on developing an in-house parallel meshless local Petrov-Galerkin (MLPG) code for the analysis of heat conduction in two-dimensional and three-dimensional regular as well as complex geometries.Design/methodology/approachThe parallel MLPG code has been implemented using open multi-processing (OpenMP) application programming interface (API) on the shared memory multicore CPU architecture. Numerical simulations have been performed to find the critical regions of the serial code, and an OpenMP-based parallel MLPG code is developed, considering the critical regions of the sequential code.FindingsBased on performance parameters such as speed-up and parallel efficiency, the credibility of the parallelization procedure has been established. Maximum speed-up and parallel efficiency are 10.94 and 0.92 for regular three-dimensional geometry (343,000 nodes). Results demonstrate the suitability of parallelization for larger nodes as parallel efficiency and speed-up are more for the larger nodes.Originality/valueFew attempts have been made in parallel implementation of the MLPG method for solving large-scale industrial problems. Although the literature suggests that message-passing interface (MPI) based parallel MLPG codes have been developed, the OpenMP model has rarely been touched. This work is an attempt at the development of OpenMP-based parallel MLPG code for the very first time.

A coupled flow and transport model for simulation of multi-species reactive transport in unconfined aquifer using meshless local Petrov Galerkin (MLPG) method.

An understanding of natural degradation of multiple reactive contaminants in the aquifers is essential before designing the monitoring or remediation programs for polluted aquifers. Since such reactive contaminants are ubiquitous, a number of research works has been performed in the past three decades for the modelling of multi-species reactive transport (MSRT) phenomenon. The widely used finite difference method (FDM) and finite element method (FEM)-based models suffer a drawback of relying on a grid/mesh, which makes the solution unstable. Addressing such difficulties, the latest research on the MSRT models is directed towards the meshless methods. In this study, the meshless local Petrov Galerkin (MLPG) method-based multi-species reactive transport model (MLPG-MSRT) is presented, with an objective to create a robust simulation tool for the prediction of fate of multiple contaminants of the first-order reaction network. The developed model is validated for reversible as well as irreversible reaction networks with the available analytical solutions. Also, the MLPG model for unconfined aquifer flow (UF) is developed, validated, and coupled with the MLPG-MSRT model. The MLPG-UF-MSRT model results are further compared with the established FDM-based MODFLOW-RT3D model solutions for a rectangular and a real field type study. The results showed that the proposed model can simulate MSRT as accurately as the FDM-based models with an additional advantage of simplicity and stability, and thus, is more efficient for complex field problems.

A new meshless local integral equation method

This paper proposes a novel meshless local integral equation (LIE) method for numerical solutions of two and three-dimensional Poisson equations. The proposed method can be regarded as a new variant of the meshless local Petrov-Galerkin (MLPG) method which already has six variants, and so the proposed method can be called MLPG7. A variant of MLPG called local boundary integral equation (LBIE) method, uses a localized fundamental solution of Laplace equation as test function. This test function vanishes on local boundaries and therefore, the derived local equations do not involve the gradient of field variables. The motivation for new formulation is using an average of LBIE equation over radius of local sub-domains instead of a single sub-domain. A new kernel is introduced which is a new modification of fundamental solutions of Laplace equation. It is proved that using the new kernel as test function leads to the same local equations as averaging. Infact, the new formulation is a modification of the LBIE method for which the boundary integral is replaced with a domain integral. It is proved that new local weak formulation is equivalent with strong form. The convergence of the method for the case of regularly spaced nodal points is proved. For trial approximation, this paper uses Gaussian radial basis functions (Gaussian RBF), however, derivative free property of the method allows the use of any non-differentiable basis functions. Stability and convergence of the method are numerically studied by considering both 2D and 3D problems with regular and irregular nodal points. The method is compared with LBIE, finite differences (FD) and RBF-FD methods and the numerical results reveal the significance of the proposed method.

Application extension of the meshless local Petrov-Galerkin method: Non-Newtonian fluid flow implementations

A computer code is developed to enhance the meshless local Petrov-Galerkin method capability to solve non-Newtonian fluid flow problems. In particular, the mesh-free method here, is to be empowered to simulate and solve the two-dimensional laminar incompressible power-law cases for the first time. The newly developed computer code expresses the appropriate governing equations in terms of the vorticity-stream function formulation and sustains a weighting function of unity. The local quadrature domain integrated by parts over the appropriate control volumes provides the local weak form of the considered governing equations. The extended scheme implements the moving least square interpolation technique to approximate the obtained field variables. The ability of the developed code to handle the Ostwald-de Waele (also known as the power-law) fluid flow occurrences were assessed through comparing the obtained results of the extended method to those of the conventional mesh-based methods for some benchmarking cases available in the paper. Based on this comparison, the extended code demonstrates a good capability to address the power-law fluid flow problems for different indices for one of the most popular non-Newtonian fluid flow models.

Petrov–Galerkin zonal free element method for 2D and 3D mechanical problems

AbstractIn this article, a novel weak‐form zonal Petrov–Galerkin free element method is proposed for two‐ and three‐dimensional linear mechanical problems. By absorbing the advantages of finite block method and strong‐form finite element method, the block mapping technique is used in the free element method. Combining the characteristics of the meshless local Petrov–Galerkin method, the local Petrov–Galerkin formulation based on the zonal free element method is formed at last. Besides, the local integral domain selected in the local collocation element is circular or spherical to simplify programming. The transformation of the local integral domain between the physical and normalized spaces is given for two‐ and three‐dimensional problems. The comparison of accuracy and convergence between the new proposed Petrov–Galerkin method and the conventional methods is carried out. Some challenging examples including fracture mechanics problems and a complex 3D problem are given to validate the convergence and accuracy of the proposed method.

A Multi-Scale Continuum Study for Boron Nitride Nanotubes Based on the Meshless Local Petrov–Galerkin Method

A Multi-Scale Continuum Study for Boron Nitride Nanotubes Based on the Meshless Local Petrov–Galerkin Method

Simulation of predator–prey system with two-species, two chemicals and an additional chemotactic influence via direct meshless local Petrov–Galerkin method

PurposeThis paper aims to introduce a new numerical approach based on the local weak form and the Petrov–Galerkin idea to numerically simulation of a predator–prey system with two-species, two chemicals and an additional chemotactic influence.Design/methodology/approachIn the first proceeding, the space derivatives are discretized by using the direct meshless local Petrov–Galerkin method. This generates a nonlinear algebraic system of equations. The mentioned system is solved by using the Broyden’s method which this technique is not related to compute the Jacobian matrix.FindingsThis current work tries to bring forward a trustworthy and flexible numerical algorithm to simulate the system of predator–prey on the nonrectangular geometries.Originality/valueThe proposed numerical results confirm that the numerical procedure has acceptable results for the system of partial differential equations.

Numerical study of the variable-order time-fractional mobile/immobile advection-diffusion equation using direct meshless local Petrov-Galerkin methods

In this paper, we use direct meshless local Petrov-Galerkin (DMLPG) methods for solving the variable-order time-fractional mobile/immobile advection-diffusion equation in two dimensions. The basis of the DMLPG methods is on the generalized moving least-square (GMLS) approximation and local weak form of the equation. Since the GMLS approximation uses basic polynomials as shape functions, there is no need to construct complex shape functions in the moving least-square (MLS) approximation. In addition, the calculation of numerical integrals in the local weak form of the equation is simple. We also give proof of the convergence and stability of the time-discrete scheme. The analytical results show that the semi-discrete time scheme is unconditionally stable and the convergence order is O(Δt2). Moreover, to show the accuracy and efficiency of the methods, we use the regular and irregular domains with distributions of uniform and scattered nodes in numerical examples.

Numerical investigation of wave interaction with two closely spaced floating boxes using particle method

This paper applies Improved Meshless Local Petrov-Galerkin method based on Rankine source function (IMLPG_R) to simulate two-dimensional wave interaction with two closely spaced floating boxes. The present study considers four different box configurations based on the degree of freedom, namely FF (both boxes fixed), HH (only heave allowed), RR (only roll allowed), and MM (both heave and roll allowed). The influences of the different degrees of freedom of the boxes on the wave elevation in the gap, and forces acting on the boxes, are investigated. Regular waves with a wave period ranging from 0.6s to 1.6s and two focused wave groups based on the JONSWAP spectrum with a peak frequency of 0.869Hz and 1.081Hz, respectively, are employed for the analysis. The piston mode resonance frequencies of different configurations were observed to vary slightly depending on the motion restrained. Compared to the FF configuration, the MM configuration showed a 28.3% increase in wave amplification in the gap. Some unique characteristics observed in the response amplitude operator (RAO) and flow patterns around the boxes are explained using instantaneous velocity and vorticity diagrams. The response of various configurations under a nonlinear transient-focused wave is also examined. The first-order and second-order time series of wave elevation in the gap is obtained by using the four-phase combination method.

Simulation of reactive transport in porous media using Meshless Local Petrov Galerkin (MLPG) and combination of Meshless Weak and Strong (MWS) form methods

In this paper, two meshless methods, namely, a weak form Meshless Local Petrov Galerkin (MLPG) method, and Meshless Weak Strong (MWS) form method, obtained by combining MLPG with a strong form Radial Point Collocation Method (RPCM), are presented for simulation of advection-dispersion-reaction phenomena of the contaminants in the porous media. The first-order decay and sorption reactions are considered in this study. The Crank Nicolson scheme is applied for the time discretization. The weak form MLPG is a truly meshless and robust numerical technique, that can be applied to complex aquifer systems with derivative boundaries. However, in this method, the computational time is increased due to the integration, which is not essential for simple problems. Thus, the MLPG method is further coupled with a strong form RPCM with an aim of decreasing the background integration, by modelling only the nodes around the derivative boundaries using MLPG method and the other nodes by a direct RPCM which do not require integration. The proposed MWS model automatically converts into a complete RPCM model if there are no derivative boundaries. Thus, this model being both accurate and computationally efficient is suitable for simple and moderately complex aquifer systems and MLPG is the most stable and reliable method for modelling the most complex aquifer problems. Both the developed models are tested with available analytical solutions and applied for hypothetical case studies. The results prove the efficiency of the models and the applicability of each model is described in detail.

Meshless local Petrov-Galerkin method for 2D fractional Fokker-Planck equation involved with the ABC fractional derivative

This paper examines a time fractional version of the 2D Fokker-Planck equation involved with the Atangana-Baleanu-Caputo fractional derivative, under the Dirichlet boundary conditions. A fully discretization approach based on the meshless local Petrov-Galerkin method and ( 3 − β ) -order approximation is proposed for this equation. More precisely, we apply the meshless local Petrov-Galerkin method based on the Moving Kriging interpolation to discretize the space domain, and utilize the ( 3 − β ) -order approximation together with the θ -weighted finite difference method to discretize the temporal domain. By implementing this method, we get a solution for the problem by solving a system of algebraic equations. The validity of the method is investigated by solving four examples.

On the dynamics and stability of size-dependent symmetric FGM plates with electro-elastic coupling using meshless local Petrov-Galerkin method

In present study, the equations of motion of sandwich symmetric functionally graded size-dependent plates integrated with piezoelectric layers are derived using Hamilton’s variational principle (with modified Lagrangian) based on a higher-order nonlocal strain gradient theory in conjunction with Reddy’s third-order shear deformation plate kinematics. For the first time, a set of sixth-order partial differential equations are derived and solved by meshless local Petrov-Galerkin method to investigate the bifurcation buckling and free vibration of plates under electromechanical in-plane forces with considered nanostructure and various boundary conditions. The shifted and scaled moving least-squares approach is utilized for approximation of unknown variables, and the Gauss weight function is used as test function for deriving local weak form of governing equations. The Gauss-Legendre quadrature is employed for numerical evaluation of weak forms. A power-law distribution and a half cosine variation are applied to model the variation of materials properties and electric potentials, respectively. The obtained results are verified with well-known solutions in the literature. The effect of nonlocal parameter, material length scale parameter, power-law index, predefined electric field, axial compressive and tensile forces, volume ratio of FG core and piezoelectric layers, and stiffness of Pasternak foundation on the critical buckling loads and natural frequencies are presented and discussed comprehensively. The results obtained can be used in the analysis, design and control of composite nanostructures that are widely used in MEMS and NEMS devices.

“Mixed” Meshless Time-Domain Adaptive Algorithm for Solving Elasto-Dynamics Equations

A time-domain adaptive algorithm was developed for solving elasto-dynamics problems through a mixed meshless local Petrov-Galerkin finite volume method (MLPG5). In this time-adaptive algorithm, each time-dependent variable is interpolated by a time series function of n-order, which is determined by a criterion in each step. The high-order series of expanded variables bring high accuracy in the time domain, especially for the elasto-dynamic equations, which are second-order PDE in the time domain. In the present mixed MLPG5 dynamic formulation, the strains are interpolated independently, as are displacements in the local weak form, which eliminates the expensive differential of the shape function. In the traditional MLPG5, both shape function and its derivative for each node need to be calculated. By taking the Heaviside function as the test function, the local domain integration of stiffness matrix is avoided. Several numerical examples, including the comparison of our method, the MLPG5–Newmark method and FEM (ANSYS) are given to demonstrate the advantages of the presented method: (1) a large time step can be used in solving a elasto-dynamics problem; (2) computational efficiency and accuracy are improved in both space and time; (3) smaller support sizes can be used in the mixed MLPG5.

Monitoring network design with MLPG-TLBO hybrid model (case study Birjand, Iran)

As the groundwater quantitative monitoring aimed to determine the factors affecting the aquifer behavior plays an important role in its regional management, studying the temporal and spatial groundwater level variations requires a comprehensive monitoring network. Effort has been made in this study to introduce a new linked simulation–optimization method, named MLPG-TLBO to quantitatively monitor the Birjand aquifer and determine the optimal points for piezometers. This model uses meshless local Petrov Galerkin (MLPF) method in the simulation part and teaching–learning-based optimization (TLBO) method in the optimization part. The objective function, in this study, is to minimize the difference between the groundwater level observed in piezometers and obtained computationally by the model. Since this coupled model is independent from the meshing, it eliminates the mesh-dependent shortcomings and, hence, yields more accurate results. It has been calibrated and validated in previous Birjand area studies and has led to acceptable results. By implementing the model in Birjand aquifer, the optimal positions of ten piezometers were determined mostly in areas where the density of the extraction wells was lower. Finally, the RMSE of both MLPG-TLBO and FDM was obtained to be 0.334 m and 1.483 m for 10 optimal piezometers. The RMSE value for MLPG-TLBO has shown a 0.423 m reduction compared to its previous value. This difference is quite meaningful as it shows good performance of this method in designing an optimal network for the aquifer.

An improved impermeable solid boundary scheme for Meshless Local Petrov–Galerkin method

Meshless methods have become an essential numerical tool for simulating a wide range of flow–structure interaction problems. However, the way by which the impermeable solid boundary condition is implemented can significantly affect the accuracy of the results and computational cost. This paper develops an improved boundary scheme through a weak formulation for the boundary particles based on Pressure Poisson’s Equation (PPE). In this scheme, the wall boundary particles simultaneously satisfy the PPE in the local integration domain by adopting the Meshless Local Petrov–Galerkin method with the Rankine source solution (MLPG_R) integration scheme (Ma, 2005b) and the Neumann boundary condition, i.e., normal pressure gradient condition, on the wall boundary which truncates the local integration domain. The new weak formulation vanishes the derivatives of the unknown pressure at wall particles and is discretized in the truncated support domain without extra artificial treatment. This improved boundary scheme is validated by analytical solutions, numerical benchmarks, and experimental data in the cases of patch tests, lid-driven cavity, flow over a cylinder and monochromic wave generation. Second-order convergent rate is achieved even for disordered particle distributions. The results show higher accuracy in pressure and velocity, especially near the boundary, compared to the existing boundary treatment methods that directly discretize the pressure Neumann boundary condition.

Numerical solution of an inverse source problem for a time-fractional PDE via direct meshless local Petrov–Galerkin method

In this paper, we employ an improved Meshless Local Petrov–Galerkin (MLPG), namely Direct Meshless Local Petrov–Galerkin (DMLPG) method, for solving an inverse time-dependent source problem for two-dimensional fractal mobile/immobile solute transport equation on regular and irregular domains. In the weak form DMLPG method, numerical integrations apply over low-degree polynomial basis functions instead of complicated moving least squares (MLS) shape functions of MLPG method. Therefore, the computational costs often reduce in the DMLPG method. In space domain, we employ the DMLPG method and the time derivatives of problem are discretized based on finite difference schemes. Numerical results demonstrate efficiency and accuracy of the proposed algorithm for solving the inverse source problem.

A meshless weak strong form method for the groundwater flow simulation in an unconfined aquifer

A reliable flow simulation is essential for better groundwater management. However, the complex flow behavior cannot be simulated by analytical methods and numerical models are often required. The Finite Difference and Finite Element Methods (FDM, FEM) are the most common and reliable numerical methods for groundwater studies. These methods require a background grid/mesh, which makes them complex. Thus, the meshless methods, which do not use a pre-defined grid/mesh, were developed. The meshless methods can be classified as strong and weak form methods. The strong form methods are simple and computationally efficient. The weak form methods are complex and require higher computational time, but are most suitable to derivative boundary problems. In this study, for the first time, groundwater flow behavior is studied by combining weak and strong form methods, with high accuracy in less computational time. The meshless weak strong form (MWS) model is developed by coupling the weak-form Meshless Local Petrov Galerkin (MLPG) method with the strong form Radial Point Collocation method (RPCM). The efficiency of the MWS model is verified using 1D and 2D hypothetical problems. Further, the MWS model is applied for a field problem and the results show the computational efficiency and accuracy of the model.

Application of Meshless local Petrov-Galerkin approach for steady state groundwater flow modeling

Abstract The complicated behavior of groundwater system in an arid aquifer is generally studied by solving the governing equations using either analytical or numerical methods. In this regard, analytical methods are just for some aquifers with regular boundaries. Numerical methods used for this aim are finite difference (FDM) and finite element methods (FEM) which are engaged for some simple aquifers. Using them in the complex cases with irregular boundaries has some shortcomings, dependent on meshes. In this study, meshless local Petrov-Galerkin (MLPG) method based on the moving kriging (MK) approximation function is used to simulate groundwater flow in steady state over three aquifers, two standard and a real field aquifer. Moving kriging function known as new function which reduces the uncertain parameter. For the first aquifer, a simple rectangular aquifer, MLPG-MK indicates good agreement with analytical solutions. In the second one, aquifer conditions get more complicated. However, MLPG-MK reveals results more accurate than FDM. RMSE for MLPG-MK and FDM is 0.066 and 0.322 m respectively. In the third aquifer, Birjand unconfined aquifer located in Iran is investigated. In this aquifer, there are 190 extraction wells. The geometry of the aquifer is irregular as well. With this challenging issues, MLPG-MK again shows satisfactory accuracy. As the RMSE for MLPG-MK and FDM are 0.483 m and 0.566 m. therefore, planning for this aquifer based on the MLPG-MK is closer to reality.

Buckling Analysis of Non-Prismatic Columns Subjected to Non-Uniform Loading Using the Meshless Local Petrov-Galerkin Method

Buckling Analysis of Non-Prismatic Columns Subjected to Non-Uniform Loading Using the Meshless Local Petrov-Galerkin Method

Numerical analysis of Annular fin of different profiles with a meshfree method

This article deals with heat transfer from the annular fin of different profiles. Three different materials: aluminum, copper, and steel are considered in numerical simulation of the problem. The paper uses the meshless local Petrov-Galerkin (MLPG) method on the ordinary differential equation derived from energy balance on a differential element of variable cross-sectional radial fin in conductive-convective environment. The weighted residual method is used to develop discrete equations. The moving least square approximants are used for function approximation and the penalty method for enforcement of the essential boundary condition. A new term, Fin Heat Transfer Ratio (fHTR) is introduced to understand the relative volume-adjusted performance of any profile annular fin compare to the rectangular profile annular fin under similar conditions. The triangular profile is found to be best among all based the volume-adjusted performance.