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

Li, Shaofan; Liu, Wing Kam

doi:10.1002/(sici)1097-0207(19990530)45:3<251::aid-nme583>3.0.co;2-i

Cited by 169 publications

(46 citation statements)

References 25 publications

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“…Furthermore, the non-locality of mesh-free interpolation not only allows smooth discontinuous strain ÿeld, but also allows us to use 'the special lumping technique' to produce high-quality, detailed resolution shear bands. In addition, the results presented here show that meshless discretization provides a favorable environment for h-adaptive reÿnement and by using the meshless hierarchical partition of unity proposed in Reference [18], a spectral (wavelet) adaptive procedure has been successfully implemented to reÿne shear-band solutions.…”

Section: Discussionmentioning

confidence: 89%

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

Liu

2000

Int. J. Numer. Meth. Engng.

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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 © 2000 John Wiley & Sons, Ltd.

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Section: Discussionmentioning

confidence: 89%

“…The particular mesh-free method used in this numerical simulation is the latest version of reproducing kernel particle method (RKPM). A detailed presentation of the methodology can be found in Li and Liu [18]. For clarity, some of the technical ingredients of the method are outlined below.…”

Section: Reproducing Kernel Hierarchical Partition Of Unitymentioning

confidence: 99%

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

Liu

2000

Int. J. Numer. Meth. Engng.

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“…Strong form collocation for second-order differential equations requires taking second-order differentiation on the RK shape functions of (20.75), which is time consuming, especially in calculating higher order derivatives of M −1 (x) at every evaluation point x. As such, we introduce the approximation of u ,β [31][32][33] as…”

Section: Reproducing Kernel and Gradient Reproducing Kernel Approximamentioning

confidence: 99%

Image‐Based Multiscale Modeling of Porous Bone Materials

Yang

Wei

Chen

2013

Multiscale Simulations and Mechanics of Biological Materials

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Bones comprise about one-fifth of an individual body weight, with the main functions in supporting and protecting organs, performing movements, and producing the blood cells, among others. Based on the microstructural composition, bones can be classified as cortical bone (compact bone) and trabecular bone (cancellous bone or spongy bone). With reference to Figure 20.1a for a femur long bone, the cortical bone constitutes about 80% of the human skeleton mass and forms an outer layer of bones, while the trabecular bone fills the interior with a porous and cancellous structure; see the microstructure of the femur long bone shown in Figure 20.1b. In general, the cortical bone is stiffer, harder, and denser than the trabecular bone so that it has the ability to protect organs and support the body movement, in addition to its ability to transmit chemical components. In contrast, the trabecular bone has a larger surface area, lower density and stiffness, and higher porosity than those of the cortical bone. Owing to the porous nature of the trabecular bone, there exists space for the blood vessels and bone marrow to flow inside the spongy structure, as can be seen from the computed tomography (CT) scan of the trabecular bone microstructure in Figure 20.2. As such, the bone materials can be characterized as porous media of solid skeleton with the pores filled with fluid. Consequently, the fluid-saturated porous material has been introduced to describe the constitutive behavior of bone materials.Homogenization methods have been introduced to provide a multiscale paradigm for analysis of poroelastic materials, and yield macroscopic balance laws that resemble the Biot's theory [1] while embedding microscopic features [2][3][4]. This work is motivated by the fact that as high-resolution digital imaging techniques emerge, such as the advancement of micro-CT and micro-magnetic resonance imaging (micro-MRI), the investigation of mechanical properties of cortical and trabecular bones can Multiscale Simulations and Mechanics of Biological Materials, First Edition. Edited by Shaofan Li and Dong Qian.

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“…As one of meshfree methods, the reproducing kernel particle method (RKPM) is systematically formulated in Liu et al [23,20,24], Li and Liu [16,17]. The 3-D tri-linear RKPM meshfree shape function used in computations is constructed by using following polynomial basis,…”

Section: -D Reproducing Kernel Particle Shape Functionmentioning

confidence: 99%

Numerical simulations of large deformation of thin shell structures using meshfree methods

Wei

Liu

2000

Computational Mechanics

140

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In this paper, meshfree simulations of large deformation of thin shell structures is presented. It has been shown that the window function based meshfree interpolants can be used to construct highly smoothed (high order``manifold'') shape functions for three-dimensional (3-D) meshfree discretization/interpolation, which can be used to simulate large deformation of thin shell structures while avoiding ill-conditioning as well as stiffening in numerical computations.The main advantage of such 3-D meshfree continuum approach is its simplicity in both formulation and implementation as compared to shell theory approach, or degenerated continuum approach. Moreover, it is believed that the accuracy of the computation may increase because of using 3-D exact formulation. Possible mechanism to relieve shear/volumetric locking due to the meshfree interpolation is discussed. Several examples have been computed by using a meshfree, explicit, total Lagrangian formulation. Towards to developing a self-contact algorithm, a novel meshfree contact algorithm is proposed in the end.

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

Cited by 169 publications

References 25 publications

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

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

Image‐Based Multiscale Modeling of Porous Bone Materials

Numerical simulations of large deformation of thin shell structures using meshfree methods

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