Precision of meshfree methods and application to forward modeling of two-dimensional electromagnetic sources

Meshfree method offers high accuracy and computational capability and constructs the shape function without relying on predefined elements. We comparatively analyze the global weak form meshfree methods, such as element-free Galerkin method （EFGM）, the point interpolation method （PIM）, and the radial point interpolation method （RPIM）. Taking two dimensional Poisson equation as an example, we discuss the support-domain dimensionless size, the field nodes, and background element settings with respect to their effect on calculation accuracy of the meshfree method. RPIM and EFGM are applied to controlled- source two-dimensional electromagnetic modeling with fixed shape parameters. The accuracy of boundary conditions imposed directly and by a penalty function are discussed in the case of forward modeling of two-dimensional magnetotellurics in a homogeneous medium model. The coupling algorithm of EFG-PIM and EFG-RPIM are generated by integrating the PIM or RPIM and EFGM. The results of the numerical modeling suggest the following. First, the proposed meshfree method and corresponding coupled methods are well-suited for electromagnetic numerical modeling. The accuracy of the algorithm is the highest when the support-domain dimensionless size is 1.0 and the distribution of field nodes is consistent with the nodes of background elements. Second, the accuracy of PIM and RPIM are lower than that of EFGM for the Poisson equation but higher than EFGM for the homogeneous medium MT response. Third, RPIM overcomes the matrix inversion problem of PIM and has a wider selection of support-domain dimensionless sizes as compared to RPIM.

A Size-Dependent Functionally Graded Higher Order Plate Analysis Based on Modified Couple Stress Theory and Moving Kriging Meshfree Method

A size-dependent computational approach for bending,free vibration and buckling analyses of isotropic and sandwich functionally graded(FG)microplates is in this study presented.We consider both shear deformation and small scale effects through the generalized higher order shear deformation theory and modified couple stress theory(MCST).The present model only retains a single material length scale parameter for capturing properly size effects.A rule of mixture is used to model material properties varying through the thickness of plates.The principle of virtual work is used to derive the discrete system equations which are approximated by moving Kriging interpolation(MKI)meshfree method.Numerical examples consider the inclusions of geometrical parameters,volume fraction,boundary conditions and material length scale parameter.Reliability and effectiveness of the present method are confirmed through numerical results.

Computational Study of Collective Cell Migration By Meshfree Method

The collective cell migration behavior on a substrate was studied using RKPM meshfree method.The cells were modeled as nematic liquid crystal with hyperelastic cell nucleus.The cell-substrate and cell-cell interactions were modeled by coarse-grained potential forces.Through this study,the pulling and pushing phenomenon during collective cell migration process was observed and it was found that the individual cell mobility significantly influenced the collective cell migratory behavior.More self-propelled cells are in the system along the same direction,the faster the collective group migrates toward coordinated direction.The parametric study on cell-cell adhesion strength indicated that as the adhesion strength increases,the collective cell migration speed increases.It also showed that the mechanical stress in leader cell is higher than stress in follower cells.

A meshfree method and its applications to elasto-plastic problems

Standard finite element approaches are still ineffective in handling extreme material deformation, such as cases of large deformations and moving discontinuities due to severe mesh distortion. Among meshfree methods developed to overcome the ineffectiveness, Reproducing Kernel Particle Method (RKPM) has demonstrated its great suitability for structural analysis.This paper presents applications of RKPM in elasto-plastic problems after a review of meshfree methods and an introduction to RKPM. A slope stability problem in geotechnical engineering is analyzed as an illustrative case. The corresponding numerical simulations are carried out on an SGI Onyx3900 supercomputer. Comparison of the RKPM and the FEM under identical conditions showed that the RKPM is more suitable for problems where there exists extremely large strain such as in the case of slope sliding.

MESHFREE METHOD BASED ON POINT COLLOCATION FOR METAL FORMING SIMULATION

A meshfree method based on reproducing kernel approximation and point collocation is presented for analysis of metal ring compression. The point collocation method is a true meshfree method without the employment of a background mesh. It is shown that, in a point collocation approach, the remesh problem because of the mesh distortion in FEM （finite element method） and the low efficiency in Galerkin-based meshfree method are avoided. The corrected kernel functions are introduced to the stabilization of free-surface boundary conditions. The solution of symmetric ring compression problem is compared with a conventional finite element solution, and reasonable results have been obtained.

Factors affecting accuracy of radial point interpolation meshfree method for 3-D solid mechanics

Recently,the radial point interpolation meshfree method has gained popularity owing to its advantages in large deformation and discontinuity problems,however,the accuracy of this method depends on many factors and their influences are not fully investigated yet.In this work,three main factors,i.e.,the shape parameters,the influence domain size,and the nodal distribution,on the accuracy of the radial point interpolation method(RPIM)are systematically studied and conclusive results are obtained.First,the effect of shape parameters(R,q)of the multi-quadric basis function on the accuracy of RPIM is examined via global search.A new interpolation error index,closely related to the accuracy of RPIM,is proposed.The distribution of various error indexes on the R q plane shows that shape parameters q[1.2,1.8]and R[0,1.5]can give good results for general 3-D analysis.This recommended range of shape parameters is examined by multiple benchmark examples in 3D solid mechanics.Second,through numerical experiments,an average of 30 40 nodes in the influence domain of a Gauss point is recommended for 3-D solid mechanics.Third,it is observed that the distribution of nodes has significant effect on the accuracy of RPIM although it has little effect on the accuracy of interpolation.Nodal distributions with better uniformity give better results.Furthermore,how the influence domain size and nodal distribution affect the selection of shape parameters and how the nodal distribution affects the choice of influence domain size are also discussed.

A meshfree-based local Galerkin method with condensation of degree of freedom for elastic dynamic analysis

Condensation technique of degree of freedom is first proposed to improve the computational efficiency of meshfree method with Galerkin weak form for elastic dy- namic analysis. In the present method, scattered nodes with- out connectivity are divided into several subsets by cells with arbitrary shape. Local discrete equation is established over each cell by using moving Kriging interpolation, in which the nodes that located in the cell are used for approxima- tion. Then local discrete equations can be simplified by con- densation of degree of freedom, which transfers equations of inner nodes to equations of boundary nodes based on cells. The global dynamic system equations are obtained by as- sembling all local discrete equations and are solved by using the standard implicit Newmark＇s time integration scheme. In the scheme of present method, the calculation of each cell is carried out by meshfree method, and local search is imple- mented in interpolation. Numerical examples show that the present method has high computational efficiency and good accuracy in solving elastic dynamic problems.

Simulation of Seawater Intrusion in Coastal Confined Aquifer Using a Point Collocation Method Based Meshfree Model

Seawater intrusion caused by groundwater over-exploitation from coastal aquifers poses a severe problem in many regions. Formulation of proper pumping strategy using a simulation model can assure sustainable supply of fresh water from the coastal aquifers. The focus of the present study is on the development of a numerical model based on Meshfree (MFree) method to study the seawater intrusion problem. For the simulation of seawater intrusion problem, widely used models are based on Finite Difference (FDM) and Finite Element (FEM) Methods, which demand well defined grids/meshes and considerable pre-processing efforts. Here, MFree Point Collocation Method (PCM) based on the Radial Basis Function (RBF) is proposed for the simulation. Diffusive interface approach with density-dependent dispersion and solution of flow and solute transport is adopted. These equations are solved using PCM with appropriate boundary conditions. The developed model has been verified with Henry’s problem, and found to be satisfactory. Further the model has been applied to another established problem and an attempt is made to examine the influence of important system parameters including pumping and recharge on the seawater intrusion. The PCM based MFree model is found computationally efficient as preprocessing is avoided when compared to other numerical methods.

ICM Method Combined with Meshfree Approximation for Continuum Structure

The independent continuous mapping（ICM） method is integrated into element free Galerkin method and a new implementation of topology optimization for continuum structure is presented.To facilitate the enforcement of the essential boundary condition and derivative of various sensitivities,a singular weight function in element free Galerkin method is introduced.Material point variable is defined to illustrate the condition of material point and its vicinity instead of element or node.The topological variables field is constructed by moving least square approximation which inherits the continuity and smoothness of the weight function.Due to reciprocal relationships between the topological variables and design variables,various structural responses sensitivities are derived according to the method for calculating the partial derivatives of compound functions.Numerical examples indicate that checkerboard pattern and mesh-dependence phenomena are overcome without additional restriction methods.

Meshfree Finite Volume Element Method for Constrained Optimal Control Problem Governed by Random Convection Diffusion Equations

In this paper,we investigate a stochastic meshfree finite volume element method for an optimal control problem governed by the convection diffusion equations with random coefficients.There are two contributions of this paper.Firstly,we establish a scheme to approximate the optimality system by using the finite volume element method in the physical space and the meshfree method in the probability space,which is competitive for high-dimensional random inputs.Secondly,the a priori error estimates are derived for the state,the co-state and the control variables.Some numerical tests are carried out to confirm the theoretical results and demonstrate the efficiency of the proposed method.

An Updated Lagrangian Particle Hydrodynamics (ULPH)-NOSBPD Coupling Approach forModeling Fluid-Structure Interaction Problem

A fluid-structure interaction approach is proposed in this paper based onNon-Ordinary State-Based Peridynamics(NOSB-PD)and Updated Lagrangian Particle Hydrodynamics(ULPH)to simulate the fluid-structure interaction problem with large geometric deformation and material failure and solve the fluid-structure interaction problem of Newtonian fluid.In the coupled framework,the NOSB-PD theory describes the deformation and fracture of the solid material structure.ULPH is applied to describe the flow of Newtonian fluids due to its advantages in computational accuracy.The framework utilizes the advantages of NOSB-PD theory for solving discontinuous problems and ULPH theory for solving fluid problems,with good computational stability and robustness.A fluidstructure coupling algorithm using pressure as the transmission medium is established to deal with the fluidstructure interface.The dynamic model of solid structure and the PD-ULPH fluid-structure interaction model involving large deformation are verified by numerical simulations.The results agree with the analytical solution,the available experimental data,and other numerical results.Thus,the accuracy and effectiveness of the proposed method in solving the fluid-structure interaction problem are demonstrated.The fluid-structure interactionmodel based on ULPH and NOSB-PD established in this paper provides a new idea for the numerical solution of fluidstructure interaction and a promising approach for engineering design and experimental prediction.

h-ADAPTIVITY ANALYSIS BASED ON MULTIPLE SCALE REPRODUCING KERNEL PARTICLE METHOD

An h-adaptivity analysis scheme based on multiple scale reproducing kernel particle method was proposed, and two node refinement strategies were constructed using searching-neighbor-nodes（SNN） and local-Delaunay-triangulation（LDT） techniques, which were suitable and effective for h-adaptivity analysis on 2-D problems with the regular or irregular distribution of the nodes. The results of multiresolution and h- adaptivity analyses on 2-D linear elastostatics and bending plate problems demonstrate that the improper high-gradient indicator will reduce the convergence property of the h- adaptivity analysis, and that the efficiency of the LDT node refinement strategy is better than SNN, and that the presented h-adaptivity analysis scheme is provided with the validity, stability and good convergence property.

A moving Kriging interpolation-based boundary node method for two-dimensional potential problems

In this paper, a meshfree boundary integral equation （BIE） method, called the moving Kriging interpolation- based boundary node method （MKIBNM）, is developed for solving two-dimensional potential problems. This study combines the DIE method with the moving Kriging interpolation to present a boundary-type meshfree method, and the corresponding formulae of the MKIBNM are derived. In the present method, the moving Kriging interpolation is applied instead of the traditional moving least-square approximation to overcome Kronecker＇s delta property, then the boundary conditions can be imposed directly and easily. To verify the accuracy and stability of the present formulation, three selected numerical examples are presented to demonstrate the efficiency of MKIBNM numerically.

A node-based smoothed point interpolation method for dynamic analysis of rotating flexible beams

We proposed a mesh-free method, the called node-based smoothed point interpolation method(NS-PIM),for dynamic analysis of rotating beams. A gradient smoothing technique is used, and the requirements on the consistence of the displacement functions are further weakened. In static problems, the beams with three types of boundary conditions are analyzed, and the results are compared with the exact solution, which shows the effectiveness of this method and can provide an upper bound solution for the deflection.This means that the NS-PIM makes the system soften. The NS-PIM is then further extended for solving a rigid-flexible coupled system dynamics problem, considering a rotating flexible cantilever beam. In this case, the rotating flexible cantilever beam considers not only the transverse deformations,but also the longitudinal deformations. The rigid-flexible coupled dynamic equations of the system are derived via employing Lagrange’s equations of the second type. Simulation results of the NS-PIM are compared with those obtained using finite element method(FEM) and assumed mode method. It is found that compared with FEM, the NS-PIM has anti-ill solving ability under the same calculation conditions.

Enriched reproducing kernel particle method for fractional advection–diffusion equation

The reproducing kernel particle method （RKPM） has been efficiently applied to problems with large deformations, high gradients and high modal density. In this paper, it is extended to solve a nonlocal problem modeled by a fractional advectiondiffusion equation （FADE）, which exhibits a boundary layer with low regularity. We formulate this method on a moving least-square approach. Via the enrichment of fractional-order power functions to the traditional integer-order basis for RKPM, leading terms of the solution to the FADE can be exactly reproduced, which guarantees a good approximation to the boundary layer. Numerical tests are performed to verify the proposed approach.

An efficient Galerkin meshfree formulation for shear deformable beam under finite deformation

This paper proposes a geometrically nonlinear total Lagrangian Galerkin meshfree formulation based on the stabilized conforming nodal integration for efficient analysis of shear deformable beam.The present nonlinear analysis encompasses the fully geometric nonlinearities due to large deflection,large deformation as well as finite rotation.The incremental equilibrium equation is obtained by the consistent linearization of the nonlinear variational equation.The Lagrangian meshfree shape function is utilized to discretize the variational equation.Subsequently to resolve the shear and membrane locking issues and accelerate the computation,the method of stabilized conforming nodal integration is systematically implemented through the Lagrangian gradient smoothing operation.Numerical results reveal that the present formulation is very effective.

Analysis of piezoelectric ceramic multilayer actuators based on an electro-mechanical coupled meshless method

This paper presents an efficient meshless method for analyzing cracked piezoelectric structures subjected to mechanical and electrical loading. In this method, an element free Galerkin （EFG） formulation, an enriched basic function and some special shape functions that contain discontinuous derivatives are employed. Based on the moving least squares （MLS） interpolation approach, the EFG method is one of the promising methods for dealing with problems involving progressive crack growth. Since the method is meshless and no element connectivity data are needed, the burdensome remeshing procedure required in the conventional finite element method （FEM） is avoided. The numerical results show that the proposed method can yield an accurate near-tip stress field in an infinite piezoelectric plate containing an interior hole. In another example studying a ceramic multilayer actuator, the proposed model was found to be accurate in the simulation of stress and electric field concentrations arround the abrupt end of an internal electrode.

Explicit formulations and performance of LSFD method on Cartesian mesh

Performance of the LSFD method is compared with conventional FD schemes. Generally, 9-point stencils for 2D cases and 27-point stencils for 3D cases are used for the approximation of the first and second order derivatives obtained with conventional central difference schemes. When the same stencils are used, explicit LSFD formulations for approximation of the first and second order derivatives are presented. The LSFD formulations are actually a combination of conventional central difference schemes along relevant mesh lines. It has been found that LSFD formulations need much less iteration steps than the conventional FD schemes to converge, and the ratio of mesh spacing in the x and y directions is an important parameter in the LSFD application, with a great impact on stability of LSFD computation.

A Novel Meshfree Analysis of Transient Heat Conduction Problems Using RRKPM

By introducing the radial basis functions(RBFs)into the reproducing kernel particle method(RKPM),the calculating accuracy and stability of the RKPM can be improved,and a novel meshfree method of the radial basis RKPM(meshfree RRKPM)is proposed.Meanwhile,the meshfree RRKPM is applied to transient heat conduction problems(THCP),and the corresponding equations of the meshfree RRKPM for the THCP are derived.The twopoint time difference scheme is selected to discretize the time of the THCP.Finally,the numerical results illustrate the effectiveness of the meshfree RRKPM for the THCP.

The Method of Fundamental Solutions for Two-Dimensional Elastostatic Problems with Stress Concentration and Highly Anisotropic Materials

The method of fundamental solutions(MFS)is a boundary-type and truly meshfree method,which is recognized as an efficient numerical tool for solving boundary value problems.The geometrical shape,boundary conditions,and applied loads can be easily modeled in the MFS.This capability makes the MFS particularly suitable for shape optimization,moving load,and inverse problems.However,it is observed that the standard MFS lead to inaccurate solutions for some elastostatic problems with stress concentration and/or highly anisotropic materials.In thiswork,by a numerical study,the important parameters,which have significant influence on the accuracy of the MFS for the analysis of two-dimensional anisotropic elastostatic problems,are investigated.The studied parameters are the degree of anisotropy of the problem,the ratio of the number of collocation points to the number of source points,and the distance between main and pseudo boundaries.It is observed that as the anisotropy of the material increases,there will be more errors in the results.It is also observed that for simple problems,increasing the distance between main and pseudo boundaries enhances the accuracy of the results;however,it is not the case for complicated problems.Moreover,it is concluded that more collocation points than source points can significantly improve the accuracy of the results.