Meshless Local Petrov-Galerkin Approach Research Articles

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.

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.

A meshless local Petrov–Galerkin approach for solving the convection-dominated problems based on the streamline upwind idea and the variational multiscale concept

ABSTRACTA meshless local Petrov–Galerkin (MLPG) approach based on the streamline upwind (SU) idea and the variational multiscale (VMS) concept, called as VMS-SUMLPG method, is herein proposed to solve the convection-dominated problems. In the present VMS-SUMLPG method, the streamline upwind is constructed in the test function to solve the non-self-adjoint matrix. Meanwhile, the VMS concept as a stability term is adopted to alleviate the numerical instability such as spurious oscillations, overshoots, and undershoots. Its numerical accuracy and stability are validated by comparing with the streamline upwind Petrov–Galerkin (SUMLPG) method and the finite volume method with high-order difference schemes for two classical convection-dominated problems at the Peclet number ranging from 106 to 108. It is shown that the numerical solutions of the present VMS-SUMLPG method are accuracy, smoothness, and stability.

Pricing European and American Options Using a Very Fast and Accurate Scheme: The Meshless Local Petrov–Galerkin Method

In this paper, a method for the numerical pricing of American and European options under the Black–Scholes model is introduced. This approach is meshless local Petrov–Galerkin (MLPG) based on local weak form and the moving least squares approximations. The MLPG offers advantages over conventional and strong meshless methods of radial basis function approximations. In this paper, the American option which is a free boundary problem, is reduced to a fixed boundary problem using a Richardson extrapolation technique. Then a time stepping method is employed for the time derivative. Finally numerical results are presented in two test cases. These experiments show that MLPG approach is accurate and fast, and performs significantly better compared to the conventional radial basis functions collocation methods.

2.5D elastic wave propagation in non-homogeneous media coupling the BEM and MLPG methods

This paper describes a 2.5D numerical frequency domain model based on the mutual coupling of the boundary element method (BEM) and the meshless local Petrov–Galerkin (MLPG) method for simulating elastic wave propagation in non-homogeneous media, when the geometry does not change in the z direction. The BEM is used to model the propagation within the unbounded homogeneous domain while the MLPG is used to simulate the confined non-homogeneous domains. The coupling of the two numerical techniques is accomplished directly at the nodal points located at the common interface. Continuity of mechanical displacements and tractions at the interface is imposed through the collocation of continuity equations on the interface with use of the moving least-squares (MLS) scheme. The MLS was also chosen for the approximation of the trial functions for the MLPG formulation.The coupled BEM–MLPG approach is verified against the results provided by an analytical solution developed for a circular multi-layered subdomain, in which the elastic material properties within the circular non-homogeneous region are assumed to vary in the radial direction. Finally, an unbounded medium containing two non-homogeneous inclusions excited by a blast load is used to illustrate the applicability of the proposed model.

The MLPG applied to porous materials with variable stiffness and permeability

Two-dimensional (2-d) and axisymmetric consolidation problems are treated with a meshless local Petrov–Galerkin approach. The porous continuum is modeled with Biot’s theory, where the solid displacements and the pore pressure are chosen as unknowns (u-p-formulation). These unknowns are approximated with independent spatial discretizations using the moving least-squares scheme. The method is validated by a comparison with an 1-d analytical solution. Studies with graded material data show that a variable permeability has a strong influence on the consolidation process. The example of a disturbed zone around a borehole shows as well the importance of graded material data.

Geometrically nonlinear thermomechanical analysis of moderately thick functionally graded plates using a local Petrov–Galerkin approach with moving Kriging interpolation

A meshless local Petrov–Galerkin approach based on the moving Kriging interpolation technique is developed for geometrically nonlinear thermoelastic analysis of functionally graded plates in thermal environments (prescribed a temperature gradient or heat flux). The Kriging interpolation method makes the constructed shape functions possess Kronecker delta function property and thus special techniques for enforcing essential boundary conditions are avoided. In the thermal analysis, the dependency of thermal conductivity of functionally graded materials on temperature is involved, which gives rise to a nonlinear partial differential heat conduction equation. The nonlinear formulation of large deflection of the functionally graded plates is based on the first-order shear deformation plate theory in the von Kármán sense by taking small strains and moderate rotations into account. The incremental form of nonlinear equations is obtained by Taylor series expansion and the tangent stiffness matrix is explicitly developed in two different ways within the framework of the local meshless method. The nonlinear solutions are computed using the Newton–Raphson iteration method. Parametric and convergence studies are conducted to examine the stability of the proposed method and then several selected numerical examples are presented to demonstrate the accuracy and effectiveness of the method for nonlinear bending problems of functionally graded plates in thermal environments.

Fuzzy Meshless Local Petrov-Galerkin Method for Analysis of Imprecisely Defined Systems

Many engineering systems contain uncertainties that cannot be described by either deterministic or probabilistic approaches. For example, the geometry, material properties, external actions (loads), and boundary conditions may be imprecise in a practical mechanical or structural system. The uncertainties present may be associated with parameters that are vague, imprecise, or linguistic. A fuzzy meshless local Petrov-Galerkin approach is developed in this work for the analysis of imprecisely defined systems. Starting from the basic concepts of fuzzy sets, fuzzy arithmetic, and fuzzy calculus (differentiation and integration) the various steps of the meshless local Petrov-Galerkin approach involved in the derivation of the equations are redefined using fuzzy concepts. The resulting fuzzy equations are derived using a fuzzy version of Gaussian elimination procedure coupled with truncation. A one-dimensional heat-transfer problem involving conduction, convection, and radiation with vaguely defined thermal properties of the material and boundary conditions and two stress-analysis problems (a beam and a plate with a hole) with vaguely defined material properties and external loads are considered to demonstrate the methodology. A truncation scheme is used to overcome the difficulty of overestimating the width of the response characteristics of the system due to repeated fuzzy arithmetic operations in the solution of the fuzzy equations. A simple sequential linearization scheme is used to handle the nonlinear boundary-value problem associated with the radiation boundary condition in the heat-transfer problem. The present approach represents a unique methodology that enables the handling of certain types of imprecisely known data more realistically compared to the existing procedures.

A TRULY-MESHLESS GALERKIN METHOD, THROUGH THE MLPG "MIXED" APPROACH

A truly meshless Galerkin method is formulated in the present study, as a special case of the general Meshless Local Petrov-Galerkin (MLPG) ”Mixed” approach. The Galerkin method is implemented as a truly meshless method, for solving elasto-static problems. In the present Galerkin method, the test function is chosen to be the same as the trial function, as a special case of the MLPG approach. However, the MLPG local weak form is written over a local sub-domain which is completely independent from the trial or test functions. Even though in the present Galerkin approach, the trial and test functions are the same, the present MLPG approach (wherein the support sizes of the nodal trial and test function domains, as well as the size of the local subdomain over which the local weak-form is considered, can be arbitrary) may lead to either symmetric or unsymmetric ”stiffness” matrices. These matrices are sparse and are well-conditioned. The present MLPG Galerkin Mixed Method does not require any background meshes (or cells) for performing the numerical integration of the local weak-forms, and makes the present method to be truly meshless. In addition, the mixed approach is also used to interpolate the nodal values of strains independently from the nodal values of displacements. The present mixed approach eliminates the expensive process of directly differentiating the interpolations for displacements in the entire domain, to find the derivatives, such as strains and stresses. The present MLPG Galerkin Mixed Method is not plagued by the so-called LBB conditions, which are common in the Galerkin Mixed Finite Element Method. Numerical examples are included to demonstrate the advantages of the present method: i) the truly meshless implementation; ii) the simplicity of the mixed approach wherein lower-order polynomial basis and smaller support sizes can be used; and iii) higher accuracies and computational efficiencies, and iv) no LBB conditions.

Meshless Local Petrov-Galerkin Approach in Solving Microwave Guide Problems

This paper describes a meshless approach to obtain solutions for propagating microwave problems. The Meshless Local Petrov-Galerkin (MLPG) method is used, based on a local weak form tested with the Heaviside step function. The field is approximated by the Point Interpolation method using radial basis function with additional polynomial basis (RPIMp). TEAM workshop problem 18 is solved and its solution is compared with references from the literature. The results are in good agreement, showing that MLPG can be used as a good alternative to finite elements for this kind of problems.

A local Petrov-Galerkin approach with moving Kriging interpolation for solving transient heat conduction problems

A meshless Local Petrov-Galerkin approach based on the moving Kriging interpolation (Local Kriging method; LoKriging hereafter) is employed for solving partial different equations that govern the heat flow in two- and three-dimensional spaces. The method is developed based on the moving Kriging interpolation for constructing shape functions at scattered points, and the Heaviside step function is used as a test function in each sub-domain to avoid the need for domain integral in symmetric weak form. As the shape functions possess the Kronecker delta function property, essential boundary conditions can be implemented without any difficulties. The traditional two-point difference method is selected for the time discretization scheme. For computation of 3D problems, a novel local sub-domain from the polyhedrons is used for evaluating the integrals. Several selected numerical examples are presented to illustrate the performance of the LoKriging method.

The Nonlinear Meshless Local Petrov-Galerkin (MLPG) Method from the Nonlinear Regular Local Boundary Integral Equation

The meshless local Petrov-Galerkin approach based on a regular local boundary integral equation is successfully extended to solve nonlinear boundary value problems. The present method is truly meshless, as no mesh connectivity is needed for interpolating the solution variables and for integrating the weak form. Compared to the original MLPG method, the present method does not need the derivatives of the shape functions in constructing the stiffness matrix for those nodes with no displacement specified on their local boundaries. Numerical examples show that high rates of convergence with mesh refinement are achievable, and the computational results are quite accurate.

Meshless local Petrov–Galerkin approach for coupled radiative and conductive heat transfer

A meshless local Petrov–Galerkin approach is employed for solving the coupled radiative and conductive heat transfer in absorbing, emitting and scattering media. The meshless local Petrov–Galerkin approach with upwind scheme for radiative transfer is based on the discrete ordinate equations. The moving least square approximation is used to construct the shape function. Three particular test cases for coupled radiative and conductive heat transfer are examined to verify this new approximate method. The dimensionless temperatures and the dimensionless heat fluxes are obtained. The results are compared with the other benchmark approximate solutions. By comparison, the results show that the meshless local Petrov–Galerkin approach has a good accuracy in solving the coupled radiative and conductive heat transfer in absorbing, emitting and scattering media.

The Applications of Meshless Local Petrov-Galerkin (MLPG) Approaches in High-Speed Impact, Penetration and Perforation Problems

This paper presents the implementation of a three-dimensional dynamic code, for contact, impact, and penetration mechanics, based on the Meshless Local Petrov-Galerkin (MLPG) approach. In the current im- plementation, both velocities and velocity-gradients are interpolated independently,and their compatibility is en- forced only at nodal points. As a result, the time con- suming differentiations of the shape functions at all in- tegration points is avoided, and therefore, the numerical process becomes more stable and efficient. The ability of the MLPG code for solving high-speed contact, im- pact and penetration problems with large deformations and rotations is demonstrated through several compu- tational simulations, including the Taylor impact prob- lem, and some ballisticimpact and perforation problems. The computational times for the above simulations are recorded, and are compared with those of the popular finite element code (Dyna3D), to demonstrate the effi- ciency of the present MLPG approach.

Meshless approach for coupled radiative and conductive heat transfer in one-dimensional graded index medium

A meshless local Petrov–Galerkin (MLPG) approach is employed for solving the coupled radiative and conductive heat transfer in a one-dimensional slab with graded index media. The angular distribution term in discrete ordinate equation of radiative transfer within a one-dimensional graded index slab is discretized by a step scheme, and the meshless approach for radiative transfer is based on the discrete ordinate equation. A moving least-squares approximation is used to construct the shape function. Two particular test cases for coupled radiative and conductive heat transfer within a one-dimensional graded index slab are examined to verify this new approximate method. The temperatures and the radiative heat fluxes are obtained. The results are compared with the other benchmark approximate solutions. By comparison, the results show that the MLPG approach has a good accuracy in solving the coupled radiative and conductive heat transfer in one-dimensional graded index media.

Meshless method for radiation heat transfer in graded index medium

Because ray goes along a curved path determined by the Fermat principle, curved ray tracing is very difficult and complex in graded index media. To avoid the difficult and complex computation of curved ray trajectories, a meshless local Petrov–Galerkin approach based on discrete-ordinate equations is developed to solve the radiative transfer problem in multi-dimensional absorbing–emitting–scattering semitransparent graded index media. A moving least square approximation is used to construct the shape function. Two particular test problems in radiative transfer are taken as examples to verify this meshless approach. The predicted temperature distributions and the dimensionless radiative heat fluxes are determined by the proposed method and compared with the other benchmark approximate solutions. The results show that the meshless local Petrov–Galerkin approach based on discrete-ordinate equations has a good accuracy in solving the radiative transfer problems in absorbing–emitting–scattering semitransparent graded index media.

Weakly-Singular Traction and Displacement Boundary Integral Equations and Their Meshless Local Petrov-Galerkin Approaches

The general meshless local Petrov-Galerkin (MLPG) weak forms of the displacement and traction boundary integral equations (BIEs) are presented for solids undergoing small deformations. Using the directly derived non-hyper-singular integral equations for displacement gradients, simple and straight-forward derivations of weakly singular traction BIEs for solids undergoing small deformations are also presented. As a framework for meshless approaches, the MLPG weak forms provide the most general basis for the numerical solution of the non-hyper-singular displacement and traction BIEs. By employing the various types of test functions, several types of MLPG/BIEs are formulated. Numerical examples show that the present methods are very promising, especially for solving the elastic problems in which the singularities in displacements, strains, and stresses are of primary concern.

A New Implementation of the Meshless Finite Volume Method, Through the MLPG "Mixed" Approach

The Meshless Finite Volume Method (MFVM) is developed for solving elasto-static problems, through a new Meshless Local Petrov-Galerkin (MLPG) approach. In this MLPG mixed approach, both the strains as well as displacements are interpolated, at randomly distributed points in the domain, through lo- cal meshless interpolation schemes such as the mov- ing least squares(MLS) or radial basis functions(RBF). The nodal values of strains are expressed in terms of the independently interpolated nodal values of displace- ments, by simply enforcing the strain-displacement rela- tionships directly by collocation at the nodal points. The MLPG local weak form is then written for the equilib- rium equations over the local sub-domains, by using the nodal strains as the independent variables. By taking the Heaviside function as the test function, the local domain integration is avoided; this leads to a Meshless Finite Vol- ume Method, which is a counterpart to the mesh-based finite volume method that is popular in computational fluid dynamics. The present approach eliminates the ex- pensive process of directly differentiating the MLS inter- polations for displacements in the entire domain, to find the strains, especially in 3D cases. Numerical examples are included to demonstrate the advantages of the present methods: (i) lower-order polynomial basis can be used in the MLS interpolations; (ii) smaller support sizes can be used in the MLPG approach; and (iii) higher accuracies and computational efficiencies are obtained. keyword: Meshless Local Petrov-Galerkin approach (MLPG), Finite Volume Methods, Mixed Methods, Ra- dial Basis Functions (RBF), and Moving Least Squares (MLS).

An accurate solution to the meshless local petrov‐galerkin formulation in elastodynamics

A meshless local Petrov‐Galerkin (MLPG) method for solving elastodynamic problems is developed and numerically implemented. The proposed MLPG approach is based on a locally symmetric weak form and shape functions from the moving least squares (MLS) approximation. This approach is truly meshless, as it does not involve a finite element mesh, either to interpolate the solution variables, or to integrate the energy. However, complex vibrating‐modes or ‐frequencies may arise from asymmetric mass and stiffness matrices formulated by the MLPG method. Unlike the commonly used finite difference methods such as the Newmark method, the accurate approach in this study is the time‐discontinuous Galerkin (TDG) method for solving second‐order ordinary differential equations in the time domain. Numerical results indicate that the TDG method provides very stable and accurate results in the sense that the crucial modes are accurately integrated and the spurious modes are successfully filtered out.

The meshless local Petrov-Galerkin (MLPG) approach for solving problems in elasto-statics

The meshless local Petrov-Galerkin (MLPG) approach is an effective method for solving boundary value problems, using a local symmetric weak form and shape functions from the moving least squares approximation. In the present paper, the MLPG method for solving problems in elasto-statics is developed and numerically implemented. The present method is a truly meshless method, as it does not need a “finite element mesh”, either for purposes of interpolation of the solution variables, or for the integration of the energy. All integrals in the formulation can be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. The essential boundary conditions in the present formulation are imposed by a penalty method, as the essential boundary conditions can not be enforced directly when the non-interpolative moving least squares approximation is used. Several numerical examples are presented to illustrate the implementation and performance of the present MLPG method. The numerical examples show that the present MLPG approach does not exhibit any volumetric locking for nearly incompressible materials, and that high rates of convergence with mesh refinement for the displacement and energy norms are achievable. No post-processing procedure is required to compute the strain and stress, since the original solution from the present method, using the moving least squares approximation, is already smooth enough.