A convected particle least square interpolation material point method

Applying the convected particle domain interpolation (CPDI) to the material point method has many advantages over the original material point method, including significantly improved accuracy. However, in the large deformation regime, the CPDI still may not retain the expected convergence rate. The paper proposes an enhanced CPDI formulation based on least square reconstruction technique. The convected particle least square interpolation (CPLS) material point method assumes the velocity field inside the material point domain as nonconstant. This velocity field in the material point domain is mapped to the background grid nodes with a moving least squares reconstruction. In this paper, we apply the improved moving least squares method to avoid the instability of the conventional moving least squares method due to a singular matrix. The proposed algorithm can improve convergence rate, as illustrated by numerical examples using the method of manufactured solutions.     

Enhancement of the material point method using B‐spline basis functions

The material point method (MPM) enhanced with B‐spline basis functions, referred to as B‐spline MPM (BSMPM), is developed and demonstrated using representative quasi‐static and dynamic example problems. Smooth B‐spline basis functions could significantly reduce the cell‐crossing error as known for the original MPM. A Gauss quadrature scheme is designed and shown to be able to diminish the quadrature error in the BSMPM analysis of large‐deformation problems for the improved accuracy and convergence, especially with the quadratic B‐splines. Moreover, the increase in the order of the B‐spline basis function is also found to be an effective way to reduce the quadrature error and to improve accuracy and convergence. For plate impact examples, it is demonstrated that the BSMPM outperforms the generalized interpolation material point (GIMP) and convected particle domain interpolation (CPDI) methods in term of the accuracy of representing stress waves. Thus, the BSMPM could become a promising alternative to the MPM, GIMP, and CPDI in solving certain types of transient problems.     

Controlling the onset of numerical fracture in parallelized implementations of the material point method (MPM) with convective particle domain interpolation (CPDI) domain scaling

The material point method is well suited for large‐deformation problems in solid mechanics but requires modification to avoid cell‐crossing errors as well as extension instabilities that lead to numerical (nonphysical) fracture. A promising solution is convected particle domain interpolation (CPDI), in which the integration domain used to map data between particles and the background grid deforms with the particle, based on the material deformation gradient. While eliminating the extension instability can be a benefit, it is often desirable to allow material separation to avoid nonphysical stretching. Additionally, large stretches in material points can complicate parallel implementation of CPDI if a single particle domain spans multiple computational patches. A straightforward modification to the CPDI algorithm allows a user‐specified scaling of the particle integration domain to control the numerical fracture response, which facilitates parallelization. Combined with particle splitting, the method can accommodate materials with arbitrarily large failure strains. Used with a smeared damage/softening model, this approach will prevent nonphysical numerical fracture in situations where the material should remain intact, but the effect of a single velocity field on localization may still produce errors in the post‐failure response. Details are given for both 2‐D and 3‐D implementations of the scaling algorithm. Copyright © 2015 John Wiley & Sons, Ltd.     

Coupling lattice Boltzmann and material point method for fluid-solid interaction problems involving massive deformation

This paper presents a coupling lattice Boltzmann and material point method (LBMPM) for fluid-solid interaction problems involving massive deformation. The convected particle domain interpolation-based material point method is adopted to solve the structure responses due to the particularly advantage on dynamic massive deformation simulations and the lattice Boltzmann method is utilized for its reliability and simplicity to simulate the complex fluid flow. The coupling strategy for these two methods is based on the consistent conditions with respect to displacement, velocity, and force, respectively, on the interface between fluid and solid parts, including the unified interpolation bounce-back scheme for curved boundaries, the Galilean invariant momentum exchange method for hydrodynamic forces, the force imposing strategy particularly for massive deformation, and the refilling algorithm for moving boundaries. There is no remeshing operation needed in the proposed LBMPM for both solid and fluid parts even when solid massive deformation and fluid complex flow are considered. Three representative numerical examples are carried out and the simulation results demonstrate that the proposed LBMPM is capable of simulate complex bidirectional fluid-solid interaction processes with the superiority for the problems involving solid dynamic large deformation behaviors and complex fluid flow.     

Second‐order convected particle domain interpolation (CPDI2) with enrichment for weak discontinuities at material interfaces

Convected particle domain interpolation (CPDI) is a recently developed extension of the material point method, in which the shape functions on the overlay grid are replaced with alternative shape functions, which (by coupling with the underlying particle topology) facilitate efficient and algorithmically straightforward evaluation of grid node integrals in the weak formulation of the governing equations. In the original CPDI algorithm, herein called CPDI1, particle domains are tracked as parallelograms in 2‐D (or parallelepipeds in 3‐D). In this paper, the CPDI method is enhanced to more accurately track particle domains as quadrilaterals in 2‐D (hexahedra in 3‐D). This enhancement will be referred to as CPDI2. Not only does this minor revision remove overlaps or gaps between particle domains, it also provides flexibility in choosing particle domain shape in the initial configuration and sets a convenient conceptual framework for enrichment of the fields to accurately solve weak discontinuities in the displacement field across a material interface that passes through the interior of a grid cell. The new CPDI2 method is demonstrated, with and without enrichment, using one‐dimensional and two‐dimensional examples. Copyright © 2013 John Wiley & Sons, Ltd.     

A weighted least squares particle‐in‐cell method for solid mechanics

A novel meshfree method is proposed that incorporates features of the material point (MPM) and generalized interpolation material point (GIMP) methods and can be used within an existing MPM/GIMP implementation. Weighted least squares kernel functions are centered at stationary grid nodes and used to approximate field values and gradients. Integration is performed over cells of the background grid and material boundaries are approximated with an implicit surface. The proposed method avoids nearest‐neighbor searches while significantly improving accuracy over MPM and GIMP. Implementation is discussed in detail and several example problems are solved, including one manufactured solution which allows measurement of dynamic, non‐linear, large deformation performance. Advantages and disadvantages of the method are discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

A fast and robust hybrid method for block‐structured mesh deformation with emphasis on FSI‐LES applications

The present work introduces an efficient technique for the deformation of block‐structured grids occurring in simulations of fluid–structure interaction (FSI) problems relying on large‐eddy simulation (LES). The proposed hybrid approach combines the advantages of the inverse distance weighting (IDW) interpolation with the simplicity and low computational effort of transfinite interpolation (TFI), while preserving the mesh quality in boundary layers. It is an improvement over the state‐of‐the‐art currently in use. To reach this objective, in a first step, three elementary mesh deformation methods (TFI, IDW, and radial basis functions) are investigated based on several test cases of different complexities analyzing not only their capabilities but also their computational costs. That not only allows to point out the advantages of each method but also demonstrates their drawbacks. Based on these specific properties of the different methods, a hybrid methodology is suggested that splits the entire grid deformation into two steps: first, the movement of the block‐boundaries of the block‐structured grid and second, the deformation of each block of the grid. Both steps rely on different methodologies, which allows to work out the most appropriate method for each step leading to a reasonable compromise between the grid quality achieved and the computational effort required. Finally, a hybrid IDW‐TFI methodology is suggested that best fits to the specific requirements of coupled FSI‐LES applications. This hybrid procedure is then applied to a real‐life FSI‐LES case. Copyright © 2016 John Wiley & Sons, Ltd.     

A local grid refinement scheme for B-spline material point method

A local grid refinement scheme for the material point method with B-spline basis functions (BSMPM) is developed based on the concept of bridging domain approach. The fine grid is defined in the local large-deformation regions to accurately capture the complex material responses, whereas other spatial domains are discritized by coarse grids. In the overlapping domain between the fine and coarse grids, the constraint of particle displacements obtained with different grids is enforced using the Lagrange multiplier method to eliminate the spurious stress reflection at the fine/coarse grid interface. Representative numerical examples have shown that the BSMPM simulations with the proposed local grid refinement scheme could provide the solutions in good agreement with those obtained with the uniformly fine grid, and that no significant spurious stress reflection is induced at the fine/coarse grid interface, even for the bridging domain size as small as the cell size of the fine grid. It is also found that the proposed local grid refinement method can significantly reduce the BSMPM computational time compared with the cases for uniformly fine grids. A multitime-step algorithm is presented and shown to considerably enhance the efficiency of the present local grid refinement scheme with no compromise in accuracy.     

Axisymmetric form of the generalized interpolation material point method

This paper reformulates the axisymmetric form of the material point method (MPM) using generalized interpolation material point (GIMP) methods. The reformulation led to a need for new shape functions and gradients specific for axisymmetry that were not available before. The new shape functions differ most from planar shape functions near the origin where r = 0. A second purpose for this paper was to evaluate the consequences of axisymmetry on a variety MPM extensions that have been developed since the original work on axisymmetric MPM. These extensions included convected particle domain integration (CPDI), traction boundary conditions, explicit cracks, multimaterial mode MPM for contact, thermal conduction, and solvent diffusion. Some examples show that the axisymmetric shape functions work well and are especially crucial near the origin. One real‐world example is given for modeling a cylinder‐penetration problem. Finally, a check list for software development describes all tasks needed to convert 2D planar or 3D codes to include an option for axisymmetric MPM. Copyright © 2014 John Wiley & Sons, Ltd.     

A novel singular node‐based smoothed finite element method (NS‐FEM) for upper bound solutions of fracture problems

It is well known that the lower bound to exact solutions in linear fracture problems can be easily obtained by the displacement compatible finite element method (FEM) together with the singular crack tip elements. It is, however, much more difficult to obtain the upper bound solutions for these problems. This paper aims to formulate a novel singular node‐based smoothed finite element method (NS‐FEM) to obtain the upper bound solutions for fracture problems. In the present singular NS‐FEM, the calculation of the system stiffness matrix is performed using the strain smoothing technique over the smoothing domains (SDs) associated with nodes, which leads to the line integrations using only the shape function values along the boundaries of the SDs. A five‐node singular crack tip element is used within the framework of NS‐FEM to construct singular shape functions via direct point interpolation with proper order of fractional basis. The mix‐mode stress intensity factors are evaluated using the domain forms of the interaction integrals. The upper bound solutions of the present singular NS‐FEM are demonstrated via benchmark examples for a wide range of material combinations and boundary conditions. Copyright © 2010 John Wiley & Sons, Ltd.     

Multi‐scale methods

In this paper four multiple scale methods are proposed. The meshless hierarchical partition of unity is used as a multiple scale basis. The multiple scale analysis with the introduction of a dilation parameter to perform multiresolution analysis is discussed. The multiple field based on a 1‐D gradient plasticity theory with material length scale is also proposed to remove the mesh dependency difficulty in softening/localization problems. A non‐local (smoothing) particle integration procedure with its multiple scale analysis are then developed. These techniques are described in the context of the reproducing kernel particle method. Results are presented for elastic‐plastic one‐dimensional problems and 2‐D large deformation strain localization problems to illustrate the effectiveness of these methods. Copyright © 2000 John Wiley & Sons, Ltd.     

Analysis and reduction of quadrature errors in the material point method (MPM)

The material point method (MPM) has demonstrated itself as a computationally effective particle method for solving solid mechanics problems involving large deformations and/or fragmentation of structures, which are sometimes problematic for finite element methods (FEMs). However, similar to most methods that employ mixed Lagrangian (particle) and Eulerian strategies, analysis of the method is not straightforward. The lack of an analysis framework for MPM, as is found in FEMs, makes it challenging to explain anomalies found in its employment and makes it difficult to propose methodology improvements with predictable outcomes. In this paper we present an analysis of the quadrature errors found in the computation of (material) internal force in MPM and use this analysis to direct proposed improvements. In particular, we demonstrate that lack of regularity in the grid functions used for representing the solution to the equations of motion can hamper spatial convergence of the method. We propose the use of a quadratic B‐spline basis for representing solutions on the grid, and we demonstrate computationally and explain theoretically why such a small change can have a significant impact on the reduction in the internal force quadrature error (and corresponding ‘grid crossing error’) often experienced when using MPM. Copyright © 2008 John Wiley & Sons, Ltd.     

Time‐domain vector potential technique for the meshless radial point interpolation method

A time‐domain meshless algorithm based on vector potentials is introduced for the analysis of transient electromagnetic fields. The proposed numerical algorithm is a modification of the radial point interpolation method, where radial basis functions are used for local interpolation of the vector potentials and their derivatives. In the proposed implementation, solving the second‐order vector potential wave equation intrinsically enforces the divergence‐free property of the electric and magnetic fields. Furthermore, the computational effort associated with the generation of a dual node distribution (as required for solving the first‐order Maxwell's equations) is avoided. The proposed method is validated with several examples of 2D waveguides and filters, and the convergence is empirically demonstrated in terms of node density or size of local support domains. It is further shown that inhomogeneous node distributions can provide increased convergence rates, that is, the same accuracy with smaller number of nodes compared with a solution for homogeneous node distribution. A comparison of the magnetic vector potential technique with conventional radial point interpolation method is performed, highlighting the superiority of the divergence‐free formulation. Copyright © 2015 John Wiley & Sons, Ltd.     

The natural neighbour Petrov–Galerkin method for elasto‐statics

In this paper, an efficient and accurate meshless natural neighbour Petrov–Galerkin method (NNPG) is proposed to solve elasto‐static problems in two‐dimensional space. This method is derived from the generalized meshless local Petrov–Galerkin method (MLPG) as a special case. In the NNPG, the local supported trial functions are constructed based on the non‐Sibsonian interpolation and test functions are taken as the three‐node triangular FEM shape functions. The local weak forms of the equilibrium equation and the boundary conditions are satisfied in local polygonal sub‐domains. These sub‐domains are constructed with Delaunay tessellations and domain integrals are evaluated over included Delaunay triangles by using Gaussian quadrature scheme. As this method combines the advantages of natural neighbour interpolation with Petrov–Galerkin method together, no stiffness matrix assembly is required and no special treatment is needed to impose the essential boundary conditions. Several numerical examples are presented and the results show the presented method is easy to implement and very accurate for these problems. Copyright © 2005 John Wiley & Sons, Ltd.     

A dynamic mesh strategy applied to the simulation of flapping wings

A robust and efficient dynamic grid strategy based on an overset grid coupled with mesh deformation technique is proposed for simulating unsteady flow of flapping wings undergoing large geometrical displacement. The dynamic grid method was implemented using a hierarchical unstructured overset grid locally coupled with a fast radial basis function (RBF)‐based mapping approach. The hierarchically organized overset grid allows transferring the grid resolution for multiple blocks and overlapping/embedding the meshes. The RBF‐based mapping approach is particularly highlighted in this paper in view of its considerable computational efficiency compared with conventional RBF evaluation. The performance of the proposed dynamic mesh strategy is demonstrated by three typical unsteady cases, including a rotating rectangular block in a fixed domain, a relative movement between self‐propelled fishes and the X‐wing type flapping‐wing micro air vehicle DelFly, which displays the clap‐and‐fling wing‐interaction phenomenon on both sides of the fuselage. Results show that the proposed method can be applied to the simulation of flapping wings with satisfactory efficiency and robustness. Copyright © 2016 John Wiley & Sons, Ltd.     

A new LBIE method for solving elastodynamic problems

A new local boundary integral equation (LBIE) method for the solution of elastodynamic problems in both frequency and time domain is proposed. Non-uniformly distributed points covering the analyzed domain are used for the interpolation of the involved fields. The key-point of the proposed methodology is that the support domain of each point is divided into parts with the aid of cells formed by connecting the point of interest with the nearby points. Then an efficient radial basis functions (RBF) interpolation scheme is exploited for the representation of displacements in each cell, while on the intersections between the local domains and the global boundary, tractions are treated as independent variables via conventional boundary elements. For each point the corresponding LBIE is written in terms of displacements only, since on the boundary of support domains tractions are eliminated with the aid of the elastostatic companion solution. The integration in support domains is performed easily and with high accuracy, while due to cells the extension of the method to three dimensions is straightforward. Transient solutions are obtained after inversion of frequency domain results with the inverse fast Fourier transform (FFT). Two representative numerical examples that demonstrate the accuracy of the proposed methodology are provided.     

A Three-dimensional Adaptive Strategy with Uniform Background Grid in Element-free Galerkin Method for Extremely Large Deformation Problems

A novel three-dimensional adaptive element-free Galerkin method (EFGM) based on a uniform background grid is proposed to cope with the problems with extremely large deformation. On the basis of this uniform background grid, an interior adaptive strategy through an error estimation within the analysis domain is developed. By this interior adaptive scheme, additional adaptive nodes are inserted in those regions where the solution accuracy needs to be improved. As opposed to the fixed uniform background grid, these inserted nodes can move along with deformation to describe the particular local deformation of the structure. In addition, a triangular surface technique is adopted to depict the geometry of the three-dimensional structure and a new surface adaptive strategy on the surface of the structure is also proposed. The complicated geometry of the three-dimensional structure can be thus analyzed precisely even under extremely large deformation. Besides, the contact regions of the structure can be determined accurately when the contact behavior occurs. Therefore, the present EFGM adaptive strategy not only retains the advantage of the uniform background grid for solving the extremely deformed problems, but also enhances the solution accuracy in the interior and surface of the structure.     

On the convergence analysis,stability, and implementation of meshless local radial point interpolation on a class of three‐dimensional wave equations

In this article, the meshless local radial point interpolation method is applied to analyze three space dimensional wave equations of the form subject to given initial and Dirichlet boundary conditions. The main difficulty of the great number of methods in full 3‐D problems is the large computational costs. In meshless local radial point interpolation method, it does not require any background integration cells, so that all integrations are carried out locally over small quadrature domains of regular shapes such as circles or squares in two dimensions and spheres or cubes in three dimensions. The point interpolation method with the help of radial basis functions is proposed to construct shape functions that have Kronecker delta function property. A weak formulation with the Heaviside step function converts the set of governing equations into local integral equations on local subdomains. A two‐step time discretization method is employed to evaluate the time derivatives. This suggests Crank‐Nicolson technique to be applied on the right hand side of the equation. The convergence analysis and stability of the method are fully discussed. Three illustrative examples are presented, and satisfactory agreements are achieved. It is shown theoretically that the proposed method is unconditionally stable for the second example whereas it is not for the first and third ones. Copyright © 2015 John Wiley & Sons, Ltd.     

Local transmitting boundaries for the generalized interpolation material point method

The boundary‐value problems of mechanics can be solved using the material point method with explicit solver formulations. In explicit formulations, even quasi‐static problems are solved as if dynamic, which means that waves are reflected at computational boundaries, generating spurious oscillations in the solution to the boundary‐value problem. Such oscillations can be reduced to a level such that they are barely noticeable with the use of transmitting boundaries. Current implementations of transmitting boundaries in the material point method are limited to the standard viscous boundary. The absence of any stiffness component in the standard viscous boundary may lead to an undesirable finite rigid‐body motion over time. This motion can be minimized through the adoption of the transmitting cone boundary that approximates the stiffness of the unbounded domain. This paper lays out the implementation of the transmitting cone boundary for the generalized interpolation material point method. The cone boundary reflection‐canceling tractions can be applied to either the edges or the centroids of material points; this paper discusses the implications of both approaches.