# Stabilized and variationally consistent nodal integration for meshfree modeling of impact problems

- 983 Downloads
- 24 Citations

## Abstract

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.

## Keywords

Variationally consistent integration Stabilization Integration constraint Meshfree## 1 Introduction

Domain integration in Galerkin meshfree methods has been a topic of interest due to the issues of stability and sub-optimal convergence which arise from the nature of the approximation functions and domain integrations employed. Meshfree approximation functions are in general rational, often with complicated overlapping support structures, and both characteristics contribute to quadrature inaccuracy. In the former case, quadrature schemes such as Gauss cannot provide exact integration of these functions. In the latter case, misalignment of integration cells with supports can cause a great deal of quadrature inaccuracy [12]. These problems are more severe in nodal integration methods, which present an even greater challenge for meshfree methods since they are often employed so that the character of the method is preserved.

Nodal integration methods such as direct nodal integration (DNI) can suffer from stability issues [2, 9] as well as sub-optimal convergence [4, 9] and often require special techniques to alleviate the problems. Beissel and Belytschko [2] showed direct nodal integration for the moving least squares approximation can result in instability due to the fact that the Galerkin equation with nodal integration gives very low energy for saw-tooth modes, resulting from zero or nearly zero derivatives at nodal points. They proposed least-squares stabilization which alleviates the problem, although the technique requires second order derivatives in the Galerkin equation as well as second order consistency of the shape functions. Liu et al. [15] introduced a Taylor series expansion approach to alleviate the instability, but requires third order derivatives. The strain smoothing stabilized conforming nodal integration (SCNI) has been proposed [9, 10] to ease the situation, where derivatives are not directly evaluated at nodes which avoids the instability, but still requires attention because of additional unstable modes which may become excited in certain situations [5, 19].

The sub-optimal convergence of meshfree methods with improper quadrature can be attributed to Strang’s first Lemma [20]. The use of quadrature in the Galerkin equation results in loss of Galerkin orthogonality and subsequently the best approximation property of the solution, and can result in convergence rates much lower than predicted by exact integration, or even solutions which diverge with refinement [4]. Typically background cell integration without higher order quadrature does not provide sufficient accuracy due to the complexity of meshfree shape functions. The approximation functions are often rational with overlapping supports, and it is difficult to provide accurate integration. In early constructions where background grids were adopted for domain integration [3, 18], no particular approach was taken to alleviate quadrature inaccuracy. However in [12] it was recognized that misalignment of integration cells and shape functions supports is a major source of quadrature error, and a scheme was proposed where they align allowing for restoration of convergence rates. Several methods similar in spirit have since been proposed [1, 11, 17], which can also preserve the meshfree character of the Galerkin method but can carry a computational burden.

As an alternative approach, the SCNI method introduced in [9] uses strain smoothing for first order Galerkin exactness and recovers quadratic convergence in the \(L^{2}\) norm for linear basis. SCNI has been applied to other meshfree methods [25] and has also been extended to plates and shells [7, 22, 23]. This technique was later generalized to higher order strain smoothing in [13] giving cubic rates of convergence in the \(L^{2}\) norm. In the recent work in [4], the condition for arbitrary high order Galerkin exactness was derived under the general framework of variational consistency, and several variationally consistent integration (VCI) methods were proposed. Here it was shown that when variational consistency is satisfied, optimal convergence can be attained with far fewer quadrature points than would otherwise be required.

In this work, the variationally consistent integration proposed in [4] is applied to elastodynamics and geometric and material non-linear problems. The VCI methods show more favorable phase and amplitude in transient problems as well as superior performance for large deformation problems compared to their variationally inconsistent counterparts. Stabilization techniques are also employed based on the works in [5, 19], and an eigenvalue analysis of the combined methods is provided.

The outline of the paper is as follows. Section 2 gives a basic overview of domain integration for Galerkin meshfree methods, and demonstrates how variationally inconsistent integration methods can exhibit sub-optimal convergence and how nodal integration can lead to instability. In Sect. 3, several variationally consistent integration methods are introduced along with enhanced stabilization for nodal integration. In Sect. 4, the stabilized VCI methods are applied to several problems demonstrating improved performance over standard methods in the dynamic and large deformation setting. Concluding remarks are then given in Sect. 5.

## 2 Background

In this section we review the issues associated with domain integration in Galerkin meshfree methods. We consider the RK approximation to illustrate the characteristics of the approximations used in meshfree methods. Here it is shown how several integration methods exhibit sub-optimal convergence and instability under certain discretizations.

### 2.1 Reproducing kernel (RK) approximation

### 2.2 Loss of best approximation property and sub-optimal convergence due to inaccurate quadrature rules

The use of inaccurate quadrature in the Galerkin equation results in the loss of the Galerkin orthogonality and could lead to sub-optimal convergence in the Galerkin solution according to Strang’s first Lemma [20]. The concept of variational consistency has been proposed as a means to correct inaccurate quadrature rules to recover Galerkin orthogonality, and consequently achieve optimal convergence in the Galerkin solution [4].

\(L^{2}\) Errors in linear patch test

Method | DNI | \(1\times 1\) GI | \(2\times 2\) GI | \(3\times 3\) GI | \(4\times 4\) GI | \(5\times 5\) GI | SCNI | SNNI |
---|---|---|---|---|---|---|---|---|

\({L}^{2}\hbox { norm}\) | 0.67501 | 1.6402 | 0.0782 | 0.02418 | 0.00668 | 0.00194 | 1.06E\(-\)14 | 0.94592 |

### 2.3 Nodal integration leading to instability

When nodal integration is employed for meshfree methods, instability can arise due to the underestimation of energy associated with small wavelength modes. This is due to the fact that first order derivatives (which appear in the weak form) of the modes are zero or nearly zero at nodal locations. The SCNI method [9, 10] has been introduced which avoids evaluating derivatives at nodal locations, thus circumventing the issue of zero energy modes.

## 3 Variational consistency and stabilization for nodal integration

In this section the concept of variational consistency is reviewed, along with stabilization for nodal integration. It is shown how VCI can restore Galerkin exactness up to the order of completeness in the approximation and provide optimal convergence. Stabilization for nodal integration is introduced, and an eigenvalue analysis for nodal VCI methods with stabilization is also given.

### 3.1 Variationally consistent integration

The concept of variational consistency introduced in [4] can be used as a guideline to construct quadrature schemes and test functions consistent with each other. The variational consistency condition is a generalization of the integration constraint for linear exactness given in [9], where it has been extended to arbitrary high order solutions.

**n**is the unit outward surface normal. The above equation states that the quadrature rule used in the Galerkin equation should be consistent with the test function to achieve Galerkin exactness. If nodal integration is employed, strain smoothing proposed in [9, 10] can be adopted to meet the first order integration constraint:

Simplifications of SCNI for extremely large deformation problems have been proposed such as stabilized non-conforming nodal integration (SNNI) [8, 14]. Here the smoothing zones are simply cells constructed around the nodes with the conforming condition relaxed, as shown in Fig. 4b. As a consequence, the integration constraint is no longer satisfied and sub-optimal convergence is encountered as shown in Fig. 2.

### 3.2 Additional stabilization for nodal integration

Non-zero energy oscillatory modes exist in SCNI when the surface area to volume ratio is small. In the same situation, similar modes also exist in SNNI. Short-wavelength modes associated with only a small amount of energy initiated from the boundary may become excited. When the discretization is fine, or when the volume is comparatively larger than the surface area, these modes remain relatively unchecked [19].

*NS*is the number of stabilization points, \(\tilde{\varepsilon }_L \) is the smoothed strain at node \(L,\varepsilon _K\) is the strain at stabilization point \(K,\mathbf C \) is the matrix of material constants, \(c \) is a stabilization parameter ranging from zero to unity, and \(A_K \) is the cell area associated with point \( K\). Note that for SNNI, the weights for stabilization points are taken as \( A_L/NS \). The distribution of points \(K\) in relation to node \(L\) for SCNI and SNNI is depicted in Fig. 7. In (3.8), the second term is the contribution of the stabilization; in explicit dynamics it leads to an additional internal force term.

## 4 Numerical examples

Nomenclature for domain integration

Standard | Stabilized | VC corrected | Stabilized and VC corrected |
---|---|---|---|

SCNI | MSCNI | – | – |

SNNI | MSNNI | VC-SNNI | VC-MSNNI |

DNI | – | VC-DNI | – |

### 4.1 Tube problem

### 4.2 Wave propagation in an elastic bar

**-**VC methods as shown in Fig. 15. Here it is seen that the VCI methods can provide much higher accuracy in both phase and amplitude compared to the variationally inconsistent methods.

### 4.3 Taylor bar impact

Properties of aluminum bar

Young’s modulus, \(E\) | 78.2 GPa |

Poisson’s ratio, \(\nu \) | 0.3 |

Density, \(\rho \) | 2,700 \(\hbox {kg}/\hbox {m}^{3}\) |

Yield stress, \(\sigma _{Y}\) | 0.29 GPa |

The added stabilization for SNNI shows a large improvement in the pattern of deformation, whereas SCNI and VC-SNNI show less of an improvement with stabilization. These results also agree with the results in Sect. 3, where only a marginal improvement is provided by stabilization when the solution by VCI methods is already stable. Comparing all the integration methods, it can be seen that MSCNI, MSNNI and VC-MSNNI provide the best solutions, although VC-DNI and VC-SNNI also perform well.

## 5 Conclusions

In this work it has been demonstrated that several commonly used domain integration methods can exhibit both instability and sub-optimal convergence due to inaccurate quadrature. Our particular interest is the improvement of the SNNI nodal integration method which provides greater simplicity for domain integration in fragment-impact problems, but suffers from low accuracy and instability.

To address both accuracy and stability, VCI methods with additional stabilization have been introduced. The VCI method recovers Galerkin exactness to an order consistent with the order of completeness in the approximation functions. The VCI methods are formulated under the assumed strain framework and can be conveniently enriched with strain energy stabilization for enhanced stability.

An eigenvalue analysis has been provided to show the enhanced stability of the proposed methods. Several numerical examples have been given to examine the performance of VCI methods with stabilization. For wave propagation problems, standard methods show large errors in phase and amplitude, while their variationally consistent counterparts do not. For large deformation impact problems, solutions for the VCI and VC corrected methods were also superior to their uncorrected counterparts, with stabilized variationally consistent methods (MSCNI and VC-MSNNI) showing the best performance.

## References

- 1.Atluri SN, Shen S (1998) A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics. Comput Mech 22:117–127zbMATHMathSciNetCrossRefGoogle Scholar
- 2.Beissel S, Belytschko T (1996) Nodal integration of the element-free Galerkin method. Comput Methods Appl Mech Eng 139:49–74zbMATHMathSciNetCrossRefGoogle Scholar
- 3.Belytschko T, Lu YY, Gu L (1994) Element-free Galerkin methods. Int J Numer Methods Eng 37:229–256zbMATHMathSciNetCrossRefGoogle Scholar
- 4.Chen JS, Hillman M, Rüter M (2013) An arbitrary order variationally consistent integration method for Galerkin meshfree methods. Int J Numer Methods Eng 95:387–418CrossRefGoogle Scholar
- 5.Chen JS, Hu W, Puso M, Wu Y, Zhang X (2006) Strain smoothing for stabilization and regularization of Galerkin meshfree methods, vol. 57. In: Lecture notes computational science and engineering. pp. 57–76Google Scholar
- 6.Chen JS, Pan C, Wu CT, Liu WK (1996) Reproducing kernel particle methods for large deformation analysis of non-linear structures. Comput Methods Appl Mech Eng 139:195–227zbMATHMathSciNetCrossRefGoogle Scholar
- 7.Chen JS, Wang D (2006) A constrained reproducing kernel particle formulation for shear deformable shell in Cartesian coordinate. Int J Numer Methods Eng 68:151–172zbMATHCrossRefGoogle Scholar
- 8.Chen JS, Wu Y, Guan PC, Teng H, Gaidos J, Hofstetter K, Alsaleh M (2009) A semi-Lagrangian reproducing kernel formulation for modeling earth moving operations. Mech Mater 41:670–683CrossRefGoogle Scholar
- 9.Chen JS, Wu CT, Yoon S, You Y (2001) A stabilized conforming nodal integration for Galerkin mesh-free methods. Int J Numer Methods Eng 50:435–466zbMATHCrossRefGoogle Scholar
- 10.Chen JS, Yoon S, Wu CT (2002) Nonlinear version of stabilized conforming nodal integration for Galerkin meshfree methods. Int J Numer Methods Eng 53:2587–2615zbMATHCrossRefGoogle Scholar
- 11.De S, Bathe KJ (2000) The method of finite spheres. Comput Mech 24:329–345zbMATHMathSciNetCrossRefGoogle Scholar
- 12.Dolbow J, Belytschko T (1999) Numerical integration of the Galerkin weak form in meshfree methods. Comput Mech 23:219–230zbMATHMathSciNetCrossRefGoogle Scholar
- 13.Duan Q, Li X, Zhang H, Belytschko T (2012) Second-order accurate derivatives and integration schemes for meshfree methods. Int J Numer Methods Eng 92:399–424MathSciNetCrossRefGoogle Scholar
- 14.Guan PC, Chi SW, Chen JS, Slawson TR, Roth MJ (2011) Semi-Lagrangian reproducing kernel particle method for fragment-impact problems. Int J Impact Eng 38:1033–1047 Google Scholar
- 15.Liu GR, Zhang GY, Wang YY, Zhong ZH, Li GY, Han X (2007) A nodal integration technique for meshfree radial point interpolation method (NI-RPIM). Int J Solids Struct 44:3840–3890Google Scholar
- 16.Liu WK, Jun S, Zhang YF (1995) Reproducing kernel particle methods. Int J Numer Methods Fluids 20:1081–1106zbMATHMathSciNetCrossRefGoogle Scholar
- 17.Liu Y, Belytschko T (2010) A new support integration scheme for the weakform in mesh-free methods. Int J Numer Methods Eng 82:699–715zbMATHMathSciNetGoogle Scholar
- 18.Nayroles B, Touzot G, Villon P (1992) Generalizing the finite element method: diffuse approximation and diffuse elements. Comput Mech 10:307–318zbMATHCrossRefGoogle Scholar
- 19.Puso MA, Chen JS, Zywicz E, Elmer W (2008) Meshfree and finite element nodal integration methods. Int J Numer Methods Eng 74:416–446zbMATHMathSciNetCrossRefGoogle Scholar
- 20.Strang G, Fix G (2008) An analysis of the finite element method, 2nd edn. Wellesley-Cambridge Press, MassachusettsGoogle Scholar
- 21.Taylor G (1948) The use of flat-ended projectiles for determining dynamic yield stress, Part I. Proc R Soc Lond Ser A 194:289–299CrossRefGoogle Scholar
- 22.Wang D, Chen JS (2004) Locking free stabilized conforming nodal integration for meshfree Mindlin–Reissner plate formulation. Comput Methods Appl Mech Eng 193:1065–1083zbMATHCrossRefGoogle Scholar
- 23.Wang D, Chen JS (2008) A Hermite reproducing kernel approximation for thin plate analysis with sub-domain stabilized conforming integration. Int J Numer Methods Eng 74:368–390zbMATHCrossRefGoogle Scholar
- 24.Wilkins ML, Guinan MW (1973) Impact of cylinders on rigid boundary. J Appl Phys 44:1200–1206CrossRefGoogle Scholar
- 25.Yoo JW, Moran B, Chen JS (2004) Stabilized conforming nodal integration in the natural-element method. Int J Numer Methods Eng 60:861–890zbMATHCrossRefGoogle Scholar