Meshfree method for large deformation in dynamic problems

We present a new meshless method, Taylor-SPH, for the numerical analysis of large deformation dynamic problems. This method is based on the previous work developed by the authors to solve solid dynamics problems within the framework of small deformation theory. The governing equations are given in terms of stress and velocity using the updated Lagrangian approach. The Jaumann rate of the Cauchy stress is used to get an objective stress rate tensor. The Taylor-SPH method is based on two sets of particles resulting on avoiding the classical tensile instability. In order to assess the accuracy of the proposed method, numerical examples based on elastic material involving large deformation are solved.


Introduction
The Finite Element Method (FEM) has been successfully used for solving partial differential equations (PDEs) in solid and fluid mechanics. However, the FEM presents a few drawbacks that have to be improved for modelling large deformations and failure problems. One of the main drawbacks is the remeshing procedure. The presence of large deformations is accompanied by severe mesh distortion. The remeshing technique is generally used to avoid the mesh distortions. However, the remeshing is too much time consuming and reduces the accuracy of the numerical solutions. To overcome difficulties related to the mesh, various meshfree methods have been developed in the last two decades and have been used in many areas with considerable success. In the meshfree methods, the approximation of variables is constructed based on scattered points without mesh connectivity. Therefore, meshfree methods can deal in a straightforward manner with large deformation and failure problems without the difficulties encountered in mesh-based methods. A wide variety of meshfree methods have been proposed over the past decades and has been successfully applied to many problems in computational mechanics, for more details see e.g. [1,2]. In this paper, we present a new meshfree method, Taylor-SPH (TSPH), for the numerical analysis of large deformation problems under dynamic conditions. This method is based on the previous work developed by  to solve solid dynamics problems within the framework of small deformation theory. The equations are written in the form of a system of first order hyperbolic PDEs. The principal variables are stress and velocity. The proposed algorithm is based on two sets of particles resulting on avoiding the classical tensile instability. To illustrate the performance of the proposed method, numerical examples involving large deformation are solved using the Taylor-SPH method.

Governing equations
Consider a problem defined in domain Ω bounded by Γ and Γ such that Γ = Γ ∪ Γ . Using the updated Lagrangian approach and the Jaumann rate of the Cauchy stress, the governing equations with boundary conditions are written in terms of stress and velocity as follows Here represents the velocity vector, is the Cauchy stress tensor, is the body forces per unit mass; is the gradient operator and is the density, is the rate of deformation tensor, is the spin tensor and is the elastic tensor. ̅ and ̅ are prescribed velocity and stress respectively; n is the outward normal to the domain. The above system (1a, 1b) can alternatively be written in a concise manner as

Numerical methods: Taylor-SPH
The Taylor-SPH method is used to discretize the above equation (2). The TSPH is a collocation meshfree method which consists of applying first the time discretization using a Taylor series expansion in two steps and the space discretization using the corrected SPH method. The application of the TSPH on the equation (2) leads to is the set of real particles J such that ‖ − ‖ ≤ 2ℎ. ̃ and ∇ are the normalized kernel and the corrected gradient respectively. The parameter h represents the smoothing length that defines the size of the kernel support.


: is the set of virtual particles J such that ‖ − ‖ ≤ 2ℎ.

1D elastic bar: Test of stability
The aim of this example is to verify the stability of the new algorithm on the smoothing length parameter. The problem consists of a shock wave that propagates in 1D elastic bar. It is well known that the smoothing length h is an important parameter in the SPH method. First, we can write the smoothing length as: ℎ = . Δ ; where is a factor that defines the radius of the kernel function. To investigate the influence of h, three values of have been used in the numerical analysis. The error is computed using the L2 norm of the velocity. The results are summarized in Table1. Table 1 Sensitivity of the solution on smoothing length It can be observed that when = ℎ/Δ is within the range of 0.6-1.2, the numerical solution is in very good agreement with the analytical solution. No dispersion no diffusion appears in the numerical result. As is increased (≥1.6), the number of particles increases and the oscillations appear in the numerical solution. The error increases and the solution loses its accuracy. This example shows that the Taylor-SPH method avoids the SPH tension instability and it can be used for the propagation of shock wave in elastic media provided to take 0.6 ≤ ≤ 1.2.

Cantilever beam
In this example, the Taylor-SPH method is used to solve a 2D bending problem. The problem consists of a cantilever beam subjected to a vertical load P at its free end. The beam has dimensions L x D and a unit thickness. The exact solution of this problem is given by: -Vertical displacement

Conclusion
The Taylor-SPH meshfree method for large deformation in dynamic problems has been presented. The main advantage of the proposed method is the best accuracy because of (i) using stress and velocity as main variables in the PDEs written with the updated Lagrangian approach (ii) and using two types of particles in the time discretization at time steps ( +   1   2 ) and ( + 1) resulting on avoiding the classical tensile instability. From the numerical examples presented above, we can conclude that the Taylor-SPH method: avoids numerical instabilities, achieves an excellent convergence with small number of particles, provides accurate results for bending problems and shock wave propagation.