Nodal Integration

In most Galerkin mesh-free methods, background integration cells partitioning the problem domain are required to evaluate the weak form. It is therefore worthwhile to consider these methods using the notions of domain decomposition with the integration cells being the subdomains. Presuming that the analytical solution is admissible in the trial solution, domain and boundary integration exactness, which depend on the orders of the employed trial solution and the required solution exactness, are identified for the strict satisfaction of traction reciprocity and natural boundary condition in the weak form. Unfortunately, trial solutions constructed by many mesh-free approximants contain non-polynomial terms which cannot be exactly integrated by Gaussian quadratures. Recently, stabilized conforming (SC) nodal integration for Galerkin mesh-free methods was proposed and illustrated to be linearly exact. This paper will discuss how linear exactness is ensured and how spurious oscillation encountered by direct nodal integration is suppressed in SC nodal integration from a domain decomposition point of view. Moreover, it will be shown that SC nodal integration can be formulated by the Hellinger-Reissner Principle and thus justified in the classical variational sense. Applications of the method to straight beam, plate and curved beam problems are presented.

In most Galerkin mesh-free methods, background integration cells partitioning the problem domain are required to evaluate the weak form. It is therefore worthwhile to consider these methods using the notions of domain decomposition with the integration cells being the subdomains. Presuming that the analytical solution is admissible in the trial solution, domain and boundary integration exactness, which depend on the orders of the employed trial solution and the required solution exactness, are identified for the strict satisfaction of traction reciprocity and natural boundary condition in the weak form. Unfortunately, trial solutions constructed by many mesh-free approximants contain non-polynomial terms which cannot be exactly integrated by Gaussian quadratures. Recently, stabilized conforming (SC) nodal integration for Galerkin mesh-free methods was proposed and illustrated to be linearly exact. This paper will discuss how linear exactness is ensured and how spurious oscillation encountered by direct nodal integration is suppressed in SC nodal integration from a domain decomposition point of view. Moreover, it will be shown that SC nodal integration can be formulated by the Hellinger-Reissner Principle and thus justified in the classical variational sense. Applications of the method to straight beam, plate and curved beam problems are presented.

We present a Lagrangian nodal integration method for the simulation of Newtonian and non-Newtonian free-surface fluid flows. The proposed nodal Lagrangian method uses a finite element mesh to discretize the computational domain and to define the (linear) shape functions for the unknown nodal variables, as in the standard Particle Finite Element Method (PFEM). In this approach, however, the integrals are performed over nodal patches and not over elements, and strains/stresses are defined at nodes and not at Gauss points. This allows to limit the drawbacks associated with the remeshing and leads to a more accurate stress computation than in the classical elemental PFEM. Several numerical tests, in 2D and in 3D, are presented to validate the proposed nodal PFEM. In all cases, the method has shown a very good agreement with analytical solutions and with experimental and numerical results from the literature. A thorough comparison between nodal and elemental PFEMs is also presented, focusing on crucial issues, such as solution accuracy, convergence, mass conservation and sensitivity to mesh distortion. c

We present a Lagrangian nodal integration method for the simulation of Newtonian and non-Newtonian free-surface fluid flows. The proposed nodal Lagrangian method uses a finite element mesh to discretize the computational domain and to define the (linear) shape functions for the unknown nodal variables, as in the standard Particle Finite Element Method (PFEM). In this approach, however, the integrals are performed over nodal patches and not over elements, and strains/stresses are defined at nodes and not at Gauss points. This allows to limit the drawbacks associated with the remeshing and leads to a more accurate stress computation than in the classical elemental PFEM. Several numerical tests, in 2D and in 3D, are presented to validate the proposed nodal PFEM. In all cases, the method has shown a very good agreement with analytical solutions and with experimental and numerical results from the literature. A thorough comparison between nodal and elemental PFEMs is also presented, focusing on crucial issues, such as solution accuracy, convergence, mass conservation and sensitivity to mesh distortion. c

In this paper, we present a novel nodal integration scheme for meshfree Galerkin methods that draws on the mathematical framework of the virtual element method. We adopt linear maximum-entropy basis functions for the discretization of field variables, although the proposed scheme is applicable to any linear meshfree approximant. In our approach, the weak form integrals are nodally integrated using nodal representative cells that carry the nodal displacements and state variables such as strains and stresses. The nodal integration is performed using the virtual element decomposition, wherein the bilinear form is decomposed into a consistency part and a stability part that ensure consistency and stability of the method. The performance of the proposed nodal integration scheme is assessed through benchmark problems in linear and nonlinear analyses of solids for small displacements and small-strain kinematics. Numerical results are presented for linear elastostatics and linear elastodynamics, and viscoelasticity. We demonstrate that the proposed nodally integrated meshfree method is accurate, converges optimally, and is more reliable and robust than a standard cell-based Gauss integrated meshfree method.

SUMMARYIn this work, a linear hexahedral element based on an assumed strain finite element technique is presented for the solution of plasticity problems. The element stems from the Nodally Integrated Continuum Element (NICE) formulation and its extensions. Assumed gradient operators are derived via nodal integration from the kinematic‐weighted residual; the degrees of freedom are only the displacements at the nodes. The adopted constitutive model is the classical associative von Mises plasticity model with isotropic and kinematic hardening; in particular, a double‐step midpoint integration algorithm is adopted for the integration and solution of the relevant nonlinear evolution equations. Efficiency of the proposed method is assessed through simple benchmark problems and comparison with reference solutions. Copyright © 2014 John Wiley & Sons, Ltd.

The reproducing kernel particle method (RKPM) is a meshfree method for computational solid mechanics that can be tailored for an arbitrary order of completeness and smoothness. The primary advantage of RKPM relative to standard finiteelement (FE) approaches is its capacity to model large deformations, material damage, and fracture. Additionally, the use of a meshfree approach offers great flexibility in the domain discretization process and reduces the complexity of mesh modifications such as adaptive refinement. We present an overview of the RKPM implementation in the Sierra/SolidMechanics analysis code, with a focus on verification, validation, and software engineering for massively parallel computation. Key details include the processing of meshfree discretizations within a FE code, RKPM solution approximation and domain integration, stress update and calculation of internal force, and contact modeling. The accuracy and performance of RKPM are evaluated using a set of benchmark problems. Solution verification, mesh convergence, and parallel scalability are demonstrated using a simulation of wave propagation along the length of a bar. Initial model validation is achieved through simulation of a Taylor bar impact test. The RKPM approach is shown to be a viable alternative to standard FE techniques that provides additional flexibility to the analyst community.

The numerical modelling of natural disasters such as landslides presents several challenges for conventional mesh-based methods such as the finite element method (FEM) due to the presence of numerically challenging phenomena such as severe material deformation and fragmentation. In contrast, meshfree methods such as the reproducing kernel particle method (RKPM) possess unique features conducive to modelling extreme events such as the absence of a structured mesh and the ease of adaptive refinement, among others. While semi-Lagrangian RKPM (SL RK) where the kernel functions are defined in the current configuration has proven to be effective in extreme event modelling, the computational cost for the re-evaluation of the shape functions every time step is a drawback. A selective employment of SL RK approximation can be a remedy, but the region where the SL RK approximation is needed cannot be pre-determined in most cases. In this work, a space-time evolving coupling of the Lagrangian a...

The explosive welding process is an extreme-deformation problem that involves shock waves, large plastic deformation, and fragmentation around the collision point, which are extremely challenging features to model for the traditional mesh-based methods. In this work, a particle-based Godunov shock algorithm under a semi-Lagrangian reproducing kernel particle method (SL-RKPM) is introduced into the volumetric strain energy to accurately embed the key shock physics in the absence of a mesh or grid, which is shown to also ensure the conservation of linear momentum. For kernel stability, a deformation-dependent anisotropic kernel support update algorithm is proposed, which is shown to capture excessive plastic flow and material separation. A quasi-conforming nodal integration is adopted to avoid the need of updating conforming cells which is tedious in extreme deformations. It is shown that the proposed formulation effectively captures shocks, jet formation, and smooth-to-wavy interface...

This special issue is dedicated to Steve Attaway, who passed away on February 28, 2019. Steve Attaway worked at Sandia National Laboratories in Albuquerque, NM, for over 30 years making significant contributions in highperformance computing, shock physics, meshfree methods, the geosciences, concrete mechanics, and blast effects on structures. Steve's early contributions in meshfree methods included stability analysis of smoothed particle hydrodynamics (SPH) (Swegle, J.

In most Galerkin mesh-free methods, background integration cells partitioning the problem domain are required to evaluate the weak form. It is therefore worthwhile to consider these methods using the notions of domain decomposition with the integration cells being the subdomains. Presuming that the analytical solution is admissible in the trial solution, domain and boundary integration exactness, which depend on the orders of the employed trial solution and the required solution exactness, are identified for the strict satisfaction of traction reciprocity and natural boundary condition in the weak form. Unfortunately, trial solutions constructed by many mesh-free approximants contain non-polynomial terms which cannot be exactly integrated by Gaussian quadratures. Recently, stabilized conforming (SC) nodal integration for Galerkin mesh-free methods was proposed and illustrated to be linearly exact. This paper will discuss how linear exactness is ensured and how spurious oscillation encountered by direct nodal integration is suppressed in SC nodal integration from a domain decomposition point of view. Moreover, it will be shown that SC nodal integration can be formulated by the Hellinger-Reissner Principle and thus justified in the classical variational sense. Applications of the method to straight beam, plate and curved beam problems are presented.

This special issue is dedicated to Steve Attaway, who passed away on February 28, 2019. Steve Attaway worked at Sandia National Laboratories in Albuquerque, NM, for over 30 years making significant contributions in highperformance computing, shock physics, meshfree methods, the geosciences, concrete mechanics, and blast effects on structures. Steve's early contributions in meshfree methods included stability analysis of smoothed particle hydrodynamics (SPH) (Swegle, J.

In this work, a linear hexahedral element based on an assumed strain finite element technique is presented for the solution of plasticity problems. The element stems from the Nodally Integrated Continuum Element (NICE) formulation and its extensions. Assumed gradient operators are derived via nodal integration from the kinematic-weighted residual; the degrees of freedom are only the displacements at the nodes. The adopted constitutive model is the classical associative von Mises plasticity model with isotropic and kinematic hardening; in particular, a double-step midpoint integration algorithm is adopted for the integration and solution of the relevant nonlinear evolution equations. Efficiency of the proposed method is assessed through simple benchmark problems and comparison with reference solutions.

In this paper, the linearly conforming radial point interpolation method is extended for geometric nonlinear analysis of plates and cylindrical shells. The Sander's nonlinear shell theory is utilized and the arc-length technique is implemented in conjunction with the modified Newton-Raphson method to solve the nonlinear equilibrium equations. The radial and polynomial basis functions are employed to construct the shape functions with Delta function property using a set of arbitrarily distributed nodes in local support domains. Besides the conventional nodal integration, a stabilized conforming nodal integration is applied to restore the conformability and to improve the accuracy of solutions. Small rotations and deformations, as well as finite strains, are assumed for the present formulation. Comparisons of present solutions are made with the results reported in the literature and good agreements are obtained. The numerical examples have demonstrated that the present approach, combined with arc-length method, is quite effective in tracing the load-deflection paths of snap-through and snap-back phenomena in shell problems.

In this paper, the linearly conforming radial point interpolation method is extended for geometric nonlinear analysis of plates and cylindrical shells. The Sander's nonlinear shell theory is utilized and the arc-length technique is implemented in conjunction with the modified Newton-Raphson method to solve the nonlinear equilibrium equations. The radial and polynomial basis functions are employed to construct the shape functions with Delta function property using a set of arbitrarily distributed nodes in local support domains. Besides the conventional nodal integration, a stabilized conforming nodal integration is applied to restore the conformability and to improve the accuracy of solutions. Small rotations and deformations, as well as finite strains, are assumed for the present formulation. Comparisons of present solutions are made with the results reported in the literature and good agreements are obtained. The numerical examples have demonstrated that the present approach, combined with arc-length method, is quite effective in tracing the load-deflection paths of snap-through and snap-back phenomena in shell problems.

In this paper, the linearly conforming radial point interpolation method is extended for geometric nonlinear analysis of plates and cylindrical shells. The Sander's nonlinear shell theory is utilized and the arc-length technique is implemented in conjunction with the modified Newton-Raphson method to solve the nonlinear equilibrium equations. The radial and polynomial basis functions are employed to construct the shape functions with Delta function property using a set of arbitrarily distributed nodes in local support domains. Besides the conventional nodal integration, a stabilized conforming nodal integration is applied to restore the conformability and to improve the accuracy of solutions. Small rotations and deformations, as well as finite strains, are assumed for the present formulation. Comparisons of present solutions are made with the results reported in the literature and good agreements are obtained. The numerical examples have demonstrated that the present approach, combined with arc-length method, is quite effective in tracing the load-deflection paths of snap-through and snap-back phenomena in shell problems.

Convergent and stable domain integration that is also computationally efficient remains a challenge for Galerkin meshfree methods. High order quadrature can achieve stability and optimal convergence, but it is prohibitively expensive for practical use. On the other hand, low order quadrature consumes much less CPU but can yield non-convergent, unstable solutions. In this work, an accelerated, convergent, and stable nodal integration is developed for the reproducing kernel particle method. A stabilization scheme for nodal integration is proposed based on implicit gradients of the strains at the nodes that offers a computational cost similar to direct nodal integration. The method is also formulated in a variationally consistent manner, so that optimal convergence is achieved. A significant efficiency enhancement over a comparable stable and convergent nodal integration scheme is demonstrated in a complexity analysis and in CPU time studies. A stability analysis is also given, and several examples are provided to demonstrate the effectiveness of the proposed method for both linear and nonlinear problems.

Galerkin meshfree methods can suffer from instability and suboptimal convergence if the issue of quadrature is not properly addressed. The instability due to quadrature is further magnified in high strain rate events when nodal integration is used. In this paper, several stable and convergent nodal integration methods are presented and applied to transient and large deformation impact problems, and an eigenvalue analysis of the methods is also provided. Optimal convergence is attained using variationally consistent integration, and stability is achieved by employing strain smoothing and strain energy stabilization. The proposed integration methods show superior performance over standard nodal integration in the wave propagation and Taylor bar impact problems tested.

Meshfree methods have been formulated based on Galerkin type weak formulation and collocation type strong formulation. The approximation functions commonly used in the Galerkin based meshfree methods are the moving least-squares (MLS) and reproducing kernel (RK) approximations, while the radial basis functions (RBFs) are usually employed in the strong form collocation method. Galerkin type formulation in conjunction with approximation functions with polynomial reproducibility yields algebraic convergence. Alternatively, strong form collocation method with RBF approximation offers exponential convergence, however the method is suffered from ill-conditioning due to its "nonlocal" approximation. In this work, we discuss stability issues related to nodal integration of Galerkin type meshfree method and ill-conditioning of the radial basis collocation method. We show how to combine the advantages of RBF and RK approximations to yield a local approximation that is better conditioned than that of the radial basis collocation method, while at the same time offers a higher rate of convergence that that of Galerkin type reproducing kernel method.

A Hermite reproducing kernel (RK) approximation and a sub-domain stabilized conforming integration (SSCI) are proposed for solving thin-plate problems in which second-order differentiation is involved in the weak form. Although the standard RK approximation can be constructed with an arbitrary order of continuity, the proposed approximation based on both deflection and rotation variables is shown to be more effective in solving plate problems. By imposing the Kirchhoff mode reproducing conditions on deflectional and rotational degrees of freedom simultaneously, it is demonstrated that the minimum normalized support size (coverage) of kernel functions can be significantly reduced. With this proposed approximation, the Galerkin meshfree framework for thin plates is then formulated and the integration constraint for bending exactness is also derived. Subsequently, an SSCI method is developed to achieve the exact pure bending solution as well as to maintain spatial stability. Numerical examples demonstrate that the proposed formulation offers superior convergence rates, accuracy and efficiency, compared with those based on higher-order Gauss quadrature rule.

Galerkin meshfree methods can suffer from instability and suboptimal convergence if the issue of quadrature is not properly addressed. The instability due to quadrature is further magnified in high strain rate events when nodal integration is used. In this paper, several stable and convergent nodal integration methods are presented and applied to transient and large deformation impact problems, and an eigenvalue analysis of the methods is also provided. Optimal convergence is attained using variationally consistent integration, and stability is achieved by employing strain smoothing and strain energy stabilization. The proposed integration methods show superior performance over standard nodal integration in the wave propagation and Taylor bar impact problems tested.

Galerkin meshfree methods can suffer from instability and suboptimal convergence if the issue of quadrature is not properly addressed. The instability due to quadrature is further magnified in high strain rate events when nodal integration is used. In this paper, several stable and convergent nodal integration methods are presented and applied to transient and large deformation impact problems, and an eigenvalue analysis of the methods is also provided. Optimal convergence is attained using variationally consistent integration, and stability is achieved by employing strain smoothing and strain energy stabilization. The proposed integration methods show superior performance over standard nodal integration in the wave propagation and Taylor bar impact problems tested.