Generalized multiple scale reproducing kernel particle methods

Liu, Wing Kam; Chen, Yi-Jung; Uras, R. A.; Chang, C. T.

doi:10.1016/s0045-7825(96)01081-x

Cited by 193 publications

(105 citation statements)

References 22 publications

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“…These include, but are not limited to, the Diffuse Element Method of Nayroles et al [65], the Element Free Galerkin method of Belytschko et al [10], the Reproducing Kernel Particle Method of Liu and Chen [56], and the Meshless Local Petrov-Galerkin method of Atluri and Zhu [4]. A number of overviews and reviews have appeared regarding these methods [9,59,58,60,61], though the interested reader will find no shortage of new developments.…”

Section: Application Of Meshfree Methods To Modeling Fracturementioning

confidence: 99%

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Physics-based modeling of brittle fracture: cohesive formulations and the application of meshfree methods

Klein

Foulk

Chen

et al. 2001

Theoretical and Applied Fracture Mechanics

149

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Simulation of generalized fracture and fragmentation remains an ongoing challenge in computational fracture mechanics. There are difficulties associated not only with the formulation of physically-based models of material failure, but also with the numerical methods required to treat geometries that change in time. The issue of fracture criteria is addressed in this work through a cohesive view of material, meaning that a finite material strength and work to fracture are included in the material description. In this study, we present both surface and bulk cohesive formulations for modeling brittle fracture, detailing the derivation of the formulations, fitting relations, and providing a critical assessment of their capabilities in numerical simulations of fracture. Due to their inherent adaptivity and robustness under severe deformation, meshfree methods are especially well-suited to modeling fracture behavior. We describe the application of meshfree methods to both bulk and surface approaches to cohesive modeling. We present numerical examples highlighting the capabilities and shortcomings of the methods in order to identify which approaches are best-suited to modeling different types of fracture phenomena.

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Section: Application Of Meshfree Methods To Modeling Fracturementioning

confidence: 99%

“…The Reproducing Kernel Particle Method (RKPM) is formulated within the framework of wavelet theory. As a result, the method can make use of the multi-resolution character of wavelets to produce a hierarchical, multiscale solution [57,59]. We have not taken advantage of these feature of RKPM.…”

Section: Introductionmentioning

confidence: 99%

Physics-based modeling of brittle fracture: cohesive formulations and the application of meshfree methods

Klein

Foulk

Chen

et al. 2001

Theoretical and Applied Fracture Mechanics

149

View full text Add to dashboard Cite

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“…Hence, a typical cell-cell interaction force can be denoted as To incorporate IFEM with protein molecular dynamics, the cell-cell interaction force is applied on the surfaces of cells: (22) 3.6.3. Blood viscoelasticity-Blood plasma can be accurately modeled with a Newtonian fluid model, yet blood flows do exhibit non-Newtonian or viscoelastic behaviors, in particular under low Reynolds numbers.…”

Section: Rbc Aggregation and Blood Rheologymentioning

confidence: 99%

“…Unlike the Dirac delta functions in the IB method which yield C 1 continuity [19,20], the discretized delta function in IFEM is the C n shape function often employed in the meshfree reproducing Kernel particle method (RKPM) [21][22][23][24]. Because of the higher order smoothness in the RKPM delta function, the accuracy is increased in the coupling procedures between fluid and solid domains [17].…”

Section: Introductionmentioning

confidence: 99%

Immersed finite element method and its applications to biological systems

Liu

Farrell

et al. 2006

Computer Methods in Applied Mechanics and Engineering

Self Cite

251

190

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This paper summarizes the newly developed immersed finite element method (IFEM) and its applications to the modeling of biological systems. This work was inspired by the pioneering work of Professor T.J.R. Hughes in solving fluid-structure interaction problems. In IFEM, a Lagrangian solid mesh moves on top of a background Eulerian fluid mesh which spans the entire computational domain. Hence, mesh generation is greatly simplified. Moreover, both fluid and solid domains are modeled with the finite element method and the continuity between the fluid and solid subdomains is enforced via the interpolation of the velocities and the distribution of the forces with the reproducing Kernel particle method (RKPM) delta function. The proposed method is used to study the fluid-structure interaction problems encountered in human cardiovascular systems. Currently, the heart modeling is being constructed and the deployment process of an angioplasty stent has been simulated. Some preliminary results on monocyte and platelet deposition are presented. Blood rheology, in particular, the shear-rate dependent de-aggregation of red blood cell (RBC) clusters and the transport of deformable cells, are modeled. Furthermore, IFEM is combined with electrokinetics to study the mechanisms of nano/bio filament assembly for the understanding of cell motility.

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“…Their main advantage is avoiding the use of a mesh due to the development of new shape functions that allow the interpolation of field variables to be accomplished at a global level or a local level, which makes them very adequate for large strain problems. Some of them are Smooth Particle Hydrodynamics (SPH) method [6], Diffuse Element Method (DEM) [7], Element Free Galerkin (EFG) method [8], hpcloud Method [9], Reproducing Kernel Particle Method (RKPM) [10], Natural Element Method (NEM) [11], Meshless Galerkin Method using Radial Basis Function (RBF) 2 ISRN Mechanical Engineering [12], and Meshless Local Petrov-Glerkin Method (MLPG) [13]. Most of these methods with the exception of MLPG are not really meshless methods, since they use the background mesh for the numerical integration of the weak form.…”

Section: Introductionmentioning

confidence: 99%

The Meshless Local Petrov-Galerkin Method for Large Deformation Analysis of Hyperelastic Materials

Sun

2011

ISRN Mechanical Engineering

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Nonlinear formulations of the meshless local Petrov-Galerkin method (MLPG) are presented for the large deformation analysis of hyperelastic materials which are considered to be incompressible or nearly incompressible. The MLPG method requires no explicit mesh in computation and therefore avoids mesh distortion difficulties. In this paper, a simple Heaviside test function is chosen for reducing the computational effort by simplifying domain integrals for hyperelasticity problems. Trial functions are constructed using the radial basis function (RBF) coupled with a polynomial basis function. The plane stress hypothesis and a pressure projection method are employed to overcome the incompressibility or nearly incompressibility in the plane stress and plane strain problems, respectively. Effects of the sizes of local subdomain and interpolation domain on the performance of the present MLPG method are investigated. The behaviour of shape parameters of multiquadrics (MQ) function has been studied. Numerical results for several examples show that the present method is effective in dealing with large deformation hyperelastic materials problems.

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Generalized multiple scale reproducing kernel particle methods

Cited by 193 publications

References 22 publications

Physics-based modeling of brittle fracture: cohesive formulations and the application of meshfree methods

Physics-based modeling of brittle fracture: cohesive formulations and the application of meshfree methods

Immersed finite element method and its applications to biological systems

The Meshless Local Petrov-Galerkin Method for Large Deformation Analysis of Hyperelastic Materials

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