T.
Belytschko, Y. Guo, W. K. Liu and S. P. Xiao
International Journal for Numerical Methods in Engineering, Vol 48, 2000
A unified stability analysis of meshless methods with Eulerian and Lagrangian kernels is presented. Three types of instabilities were identified in one dimension: an instability due to rank deficiency, a tensile instability and an instability which is also found in continua. The stability of particle methods with Eulerian and Lagrangian kernels are markedly different: Lagrangian kernels do not exhibit the tensile instability. In both kernels, the instability due to rank deficiency can be suppressed by stress points. In two dimensions the stabilizing effect of stress points is dependent on their locations. It was found that the best approach to stable particle discretizations is to use Lagrangian kernels with stress points. The stability of the least square stabilization was also studied.
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T.
Belytschko and S. P. Xiao
Compuers and Mathematics with Applications, Vol 43, 2002
The stability of discretizations by particle methods with corrected derivatives is analyzed. It is shown that the standard particle method (which is equivalent to the element-free Galerkin method with an Eulerian kernel and nodal quadrature) has two sources of instability: 1. rank deficiency of the discrete equations; 2. distortion of the material instability. The latter leads to the so-called tensile instability. It is shown that a Lagrangian kernel with addition of stress points eliminates both instabilities. Examples that verify the stability of the new formulation are given.
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T.
Belytschko, S. P. Xiao and C. Parimi
International Journal for Numerical Methods in Engineering, Vol 57, 2003
Topology optimization is formulated in terms of the nodal variables that control an implicit function description of the shape. The implicit function is constrained by upper and lower bounds, so that only a band of nodal variables needs to be considered in each step of the optimization. The weak form of the equilibrium equation is expressed as a Heaviside function of the implicit function; the Heaviside function is regularized to permit the evaluation of sensitivities. We show that the method is a dual of the Bendsoe-Kikuchi method. The method is applied both to problems of optimizing single material and multi-material configurations; the latter is made possible by enrichment functions based on the extended finite element method that enable discontinuous derivatives to be accurately treated within an element. The method is remarkably robust and we found no instances of checkerboarding. The method handles toplogical merging and separation without any apparent difficulties.
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S.
P. Xiao
Wave Motion, 2004, Vol 40, 263-276, 2004
A finite element flux-corrected transport (FE-FCT) method is proposed for the study of shock wave propagation in solids. The FCT algorithm contains two stages: transport and antidiffusion. The total Lagrangian finite element method is used here and the FCT algorithm is only applied to correct the nodal velocities along the lines of the structured mesh. An implicit function is implemented into the finite element method so that the objective with arbitrary boundries can be modeled with structured mesh. Both one-dimensional and two-dimensional examples show that the FE-FCT method can efficiently eliminate the oscillations behind the shock wave fronts.
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T.
Rabczuk, T. Belytschko and S. P. Xiao
Computer Methods in Applied Mechanics and Engineering, Vol 193, 2004
A large deformation particle method based on the Krongauz-Belytschko corrected gradient meshfree method with Lagrangian kernels is developed. In this form, the gradient is corrected by a linear transformation so that linear completeness is satisfied. For the test functions, Shepard functions are used; this guarantees that the patch test is met. Lagrangian kernels are introduced to eliminate spurious distortions of the domain of material stability. A mass allocation scheme is developed that captures correct reflection of waves without any explicit application of traction boundary conditions. In addition, the Lagrangian kernel versions of various forms of smooth particle methods (SPH), including the standard forms and the Randles-Libersky modification are presented and studied. Results are obtained for a variety of problems that compare this method to standard form of SPH, the Randles-Libersky correction and large deformation versions of the element-free Galerkin method (EFG).
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S. P. Xiao and T. Belytschko
Advances in Computational Mathematics, Vol 23: 171-190, 2005
Material instabilities are precursors to phenomena such as shear bands and fracture. Therefore, numerical methods that are intended for failure simulation need to reproduce the onset of material instabilities with resonable fidelity. Here the effectiveness of particle discretizations in reproducing of the onset of material instabilities is analyzed in two dimensions. For this purpose, a simplified hyperelastic law and a Blatz-Ko material are used. It is shown that the Eulerian kernels used in smooth particle hydrodynamics severely distort the domain of material stability. In particular, for the uniaxial case, material instabilities occur at much lower stresses, which is often called the tensile instability. On the other hand, for Lagrangian kernels, the domain of material instability is reproduced very well. We also show that particle methods without stress points exhibit instabilities due to rank deficiency of the discrete equations.
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S. P. Xiao
International Journal for Numerical Methods in Engineering, Vol 66: 364-380,
2006
This paper introduces a non-oscillatory method, finite
element flux-corrected transport (FE-FCT) method, for spallation studies. This
method includes the implementation of the FCT algorithm into the total
Lagrangian finite element method. Therefore, the FE-FCT method can efficiently
eliminate fluctuations behind shock wave fronts. In multi-dimensional
simulations, the FCT algorithm can correct each component of nodal velocities
separately along the lines of structured meshes. The requirement of structured
meshes is satisfied by using an implicit function that can describe arbitrary
boundaries of an object under structured meshes. In this paper, I apply such
non-oscillatory method in spallation studies by using different damage models.
The one-dimensional and two-dimensional examples show that the FE-FCT method
can predict the spallation and the spall thickness more accurately than the
conventional FE method.
This paper is available in PDF form .
S. P. Xiao
Communications for Numerical Methods in Engineering, Vol 23: 71-84, 2007
This paper proposes a new lattice Boltzmann method for the
study of shock wave propagation in elastic solids. The method, which implements
a flux-corrected transport (FCT) algorithm, contains three stages: collision,
streaming, and correction. In the collision stage, distribution functions are
updated. In the streaming stage, the distribution functions are shifted between
lattice points. Generally, a partial differential equation (PDE) is solved in
the streaming stage, and three methods can be used: the finite difference
method, the finite element method, and the meshfree particle method. The latter
two methods support the use of unstructured meshes in the LB method. The FCT
algorithm is used in the correction stage to revise the distribution functions
at lattice points, so oscillations behind shock wave fronts can be eliminated
efficiently. In this method, schemes for shock wave reflections at fixed and
free boundaries are developed based on the bounce-back technique. A similar
technique is used to treat wave reflections and transmissions at material
interfaces of composites. Several one-dimensional examples show that this
LB-FCT method can provide ideal depictions of shock wave propagation in
structures, especially composite structures. Therefore, the method is expected
to study the failure mechanism of structures under dynamic loads.
This paper is available in PDF form .
T. Rabczuk, S. P. Xiao and M. Sauer
Communications for Numerical Methods in Engineering, Vol 22:
1031-1065, 2007
This paper reviews several novel and older methods for
coupling meshfree particle methods, particularly the elementfree Galerkin (EFG)
method and the Smooth Particle Hydrodynamics (SPH), with finite elements.We
study master slave couplings where particles are fixed across the finite
element boundary, coupling via interface shape functions such that consistency
conditions are satisfied, bridging domain coupling, compatibility coupling with
Lagrange multipliers and hybrid coupling methods where forces from the
particles are applied via their shape functions on the FE nodes and vice versa.
The hybrid coupling methods are well suited for large deformations and
adaptivity and the coupling procedure is independent from the particle distance
and nodal arrangement.We will study the methods for several static and dynamic
applications, compare the results to analytical and experimental data and show
advantages and drawbacks of the methods.
This paper is available in PDF form .