Shaofan Li

Meshfree and particle methods and their applications

Recent developments of meshfree and particle methods and their applications in applied mechanics are surveyed. Three major methodologies have been reviewed. First, smoothed particle hydrodynamics ~SPH! is discussed as a representative of a non-local kernel, strong form collocation approach. Second, mesh-free Galerkin methods, which have been an active research area in recent years, are reviewed. Third, some applications of molecular dynamics ~MD! in applied mechanics are discussed. The emphases of this survey are placed on simulations of finite deformations, fracture, strain localization of solids; incompressible as well as compressible flows; and applications of multiscale methods and nano-scale mechanics. This review article includes 397 references. @DOI: 10.1115/1.1431547#

Moving least-square reproducing kernel methods (I) Methodology and convergence

This paper formulates the moving least-square interpolation scheme in a framework of the so-called moving least-square reproducing kernel (MLSRK) representation. In this study, the procedure of constructing moving least square interpolation function is facilitated by using the notion of reproducing kernel formulation, which, as a generalization of the early discrete approach, establishes a continuous basis for a partition of unity. This new formulation possesses the quality of simplicity, and it is easy to implement. Moreover, the reproducing kernel formula proposed is not only able to reproduce any mth order polynomial exactly on an irregular particle distribution, but also serves as a projection operator that can approximate any smooth function globally with an optimal accuracy. In this contribution, a generic m-consistency relation has been found, which is the essential property of the MLSRK approximation. An interpolation error estimate is given to assess the convergence rate of the approximation. It is shown that for sufficiently smooth function the interpolant expansion in terms of sampled values will converge to the original function in the Sobolev norms. As a meshless method, the convergence rate is measured by a new control variable—dilation parameter ρ of the window function, instead of the mesh size h as usually done in the finite element analysis. To illustrate the procedure, convergence has been shown for the numerical solution of the second-order elliptic differential equations in a Galerkin procedure invoked with this interpolant. In the numerical example, a two point boundary problem is solved by using the method, and an optimal convergence rate is observed with respect to various norms.

Finite difference calculus invariant structure of a class of algorithms for the nonlinear Klein-Gordon equation

In a previous work, the authors have presented a formalism for deriving systematically invariant, symmetric finite difference algorithms for nonlinear evolution differential equations that admit conserved quantities. This formalism is herein cast in the context of exact finite difference calculus. The algorithms obtained from the proposed formalism are shown to derive exactly from discrete scalar potential functions using finite difference calculus, in the same sense as that of the corresponding differential equation being derivable from its associated energy function (a conserved quantity). A clear ramification of this result is that the derived algorithms preserve certain discrete invariant quantities, which are the consistent counterpart of the invariant quantities in the continuous case. Results on the nonlinear stability of a class of algorithms that are derived using the proposed formalism, and that preserve energy or linear momentum, are discussed in the context of finite difference calculus. Some ...

Dynamic crack propagation in piezoelectric materials—Part I. Electrode solution

Abstract An analysis is performed for the transient response of a semi-infinite, anti-plane crack propagating in a hexagonal piezoelectric medium. The mixed boundary value problem is solved by transform methods together with the Wiener-Hopf and Cagniard-de Hoop techniques. As a special case, a closed form solution is obtained for constant speed crack propagation under external anti-plane shear loading with the conducting electrode type of electric boundary condition imposed on the crack surface (a second type of boundary condition is considered in Part II of this work). In purely elastic, transversely isotropic elastic solids, there is no antiplane mode surface wave. However, for certain orientations of piezoelectric materials, a surface wave will occur—the BleusteindashGulyaev wave. Since surface wave speeds strongly influence crack propagation, the nature of antiplane dynamic fracture in piezoelectric materials is fundamentally different from that in purely elastic solids, exhibiting many features only associated with the indashplane modes in the elastic case. For a general distribution of crack face tractions, the dynamic stress intensity factor and the dynamic electric displacement intensity factor are derived and discussed in detail for the electrode case. As for inplane elastodynamic fracture, the stress intensity factor and energy release rate go to zero as the crack propagation velocity approaches the surface wave speed. However, the electric displacement intensity does not vanish.

Mesh-free Galerkin simulations of dynamic shear band propagation and failure mode transition

A mesh-free Galerkin simulation of dynamic shear band propagation in an impact-loaded pre-notched plate is carried out in both two and three dimensions.The related experimental work was initially reported by Kalthoff and Winkler (1987), and later re-examined by Zhou et al.(1996a,b), and others. The main contributions of this numerical simulation are as follows: (1) The ductile-to-brittle failure mode transition is observed in numerical simulations for the first time; (2) the experimentally observed dynamic shear band, whose character changes with an increase of impact velocity, propagating along curved paths is replicated; (3) the simulation is able to capture the details of the adiabatic shear band to a point where the periodic temperature profile inside shear band at lm scale can clearly be seen; (4) an intense, high strain rate region is observed in front of the shear band tip, which, we believe, is caused by wave trapping at the shear band tip; it in turn causes damage and stress collapse inside the shear band and provides a key link for self-sustained instability. 2002 Elsevier Science Ltd.All rights reserved.

Reproducing kernel hierarchical partition of unity, Part I—formulation and theory

This work is concerned with developing the hierarchical basis for meshless methods. A reproducing kernel hierarchical partition of unity is proposed in the framework of continuous representation as well as its discretized counterpart. To form such hierarchical partition, a class of basic wavelet functions are introduced. Based upon the built-in consistency conditions, the differential consistency conditions for the hierarchical kernel functions are derived. It serves as an indispensable instrument in establishing the interpolation error estimate, which is theoretically proven and numerically validated. For a special interpolant with different combinations of the hierarchical kernels, a synchronized convergence effect may be observed. Being different from the conventional Legendre function based p-type hierarchical basis, the new hierarchical basis is an intrinsic pseudo-spectral basis, which can remain as a partition of unity in a local region, because the discrete wavelet kernels form a ‘partition of nullity’. These newly developed kernels can be used as the multi-scale basis to solve partial differential equations in numerical computation as a p-type refinement. Copyright

Dynamic crack propagation in piezoelectric materials-Part II. Vacuum solution

Abstract In Part I of this work, antiplane dynamic crack propagation in piezoelectric materials was studied under the condition that crack surfaces behaved as though covered with a conducting electrode. Piezoelectric surface wave phenomena were clearly seen to be critical to the behavior of the moving crack. Closed form results were obtained for stress and electric displacement intensities at the crack tip in the subsonic crack speed range; the major result is that the energy release rate vanishes as the crack speed approaches the surface (Bleustein-Gulyaev) wave speed. In this paper, an alternative assumption is made that between the growing crack surfaces there is a permeable vacuum free space, in which the electrostatic potential is nonzero. By coupling the piezoelectric equations of the solid phase with the electric charge equation in the vacuum region, a closed form solution is again obtained. In contrast to the electrode case of Part I, this case allows both applied charge and applied traction loading. In addition, the work of Part I is extended to examine piezoelectric crack propagation over the full velocity range of subsonic, transonic and supersonic speeds. Several aspects of the results are explored. The energy release rate in this case does not go to zero when the crack propagating velocity approaches the surface wave speed, even if there is only applied traction loading. When the crack exceeds the Bleustein-Gulyaev wave speed, the character of the crack-tip singularities of the physical fields depends on both speed regime and type of loading. At the other extreme, the quasi-static limit of the dynamic solution furnishes a set of new static solutions. The general permeability assumptions made here allow for fully coupled conditions that are ruled out by the a priori interfacial assumptions made in previously published solutions.

Moving least-square reproducing kernel method Part II: Fourier analysis

In Part I of this work, the moving least-square reproducing kernel (MLSRK) method is formulated and implemented. Based on its generic construction, an m-consistency structure is discovered and the convergence theorems are established. In this part of the work, a systematic Fourier analysis is employed to evaluate and further establish the method. The preliminary Fourier analysis reveals that the MLSRK method is stable for sufficiently dense, non-degenerated particle distribution, in the sense that the kernel function family satisfies the Riesz bound. One of the novelties of the current approach is to treat the MLSRK method as a variant of the ‘standard’ finite element method and depart from there to make a connection with the multiresolution approximation. In the spirits of multiresolution analysis, we propose the following MLSRK transformation, Fm,kϱ,hu=∑i=1npu,K,ϱiKϱh(x−xi,x)w1 The highlight of this paper is to embrace the MLSRK formulation with the notion of the controlled fp-approximation. Based on its characterization, the Strang-Fix condition for example, a systematic procedure is proposed to design new window functions so they can enhance the computational performance of the MLSRK algorithm. The main effort here is to obtain a constant correction function in the interior region of a general domain, i.e. Cρh = 1. This can create a leap in the approximation order of the MLSRK algorithm significantly, if a highly smooth window function is embedded within the kernel. One consequence of this development is the synchronized convergence phenomenon—a unique convergence mechanism for the MLSRK method, i.e. by properly tuning the dilation parameter, the convergence rate of higher-order error norms will approach the same order convergence rate of the L2 error norm—they are synchronized.

Numerical simulations of strain localization in inelastic solids using mesh-free methods

In this paper, a comprehensive account on using mesh-free methods to simulate strain localization in inelastic solids is presented. Using an explicit displacement-based formulation in mesh-free computations, high-resolution shear-band formations are obtained in both two-dimensional (2-D) and three-dimensional (3-D) simulations without recourse to any mixed formulation, discontinuous/incompatible element or special mesh design. The numerical solutions obtained here are insensitive to the orientation of the particle distributions if the local particle distribution is quasi-uniform, which, to a large extent, relieves the mesh alignment sensitivity that finite element methods suffer. Moreover, a simple h-adaptivity procedure is implemented in the explicit calculation, and by utilizing a mesh-free hierarchical partition of unity a spectral (wavelet) adaptivity procedure is developed to seek high-resolution shear-band formations. Moreover, the phenomenon of multiple shear band and mode switching are observed in numerical computations with a relatively coarse particle distribution in contrast to the costly fine-scale finite element simulations. Copyright

Mesh-free simulations of shear banding in large deformation

Abstract Mesh-free approximation is used in numerical simulations of strain localization under large deformation. An explicit displacement based mesh-free formulation is used in both two-dimensional and three-dimensional computations. The spatial isotropy of mesh-free interpolant is demonstrated through the numerical example to show that mesh-free methods possess certain “mesh objectivity” that alleviates the notorious mesh-alignment sensitivity associated with numerical simulation of strain localization. It is also demonstrated in the paper that mesh-free interpolants can accurately capture finite shear deformation under large mesh distortion without recourse to special mesh design and remeshing. Moreover, curved shear band surface and multiple shear band interactions are captured in numerical simulations.