Semi-Lagrangian reproducing kernel formulation and application to modeling earth moving operations

Guan, Pai-Chen; Chen, J.S.; Wu, Yi-Chien; Teng, H. Henry; Gaidos, Joseph G.; Hofstetter, Karin; Alsaleh, Mustafa I.

doi:10.1016/j.mechmat.2009.01.030

Cited by 62 publications

(20 citation statements)

References 16 publications

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“…The formation of conforming strain smoothing domains in SCNI can be cumbersome in problems subjected to topological change in geometry, and stabilized nonconforming nodal integration (SNNI) has been introduced as a simplification of SCNI. Figure (b) depicts a typical smoothing scheme for SNNI where the smoothing zones are nonconforming.…”

Section: Introductionmentioning

confidence: 99%

An arbitrary order variationally consistent integration for Galerkin meshfree methods

Chen

Hillman

Rüter

2013

Numerical Meth Engineering

161

110

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SUMMARYBecause most approximation functions employed in meshfree methods are rational functions with overlapping supports, sufficiently accurate domain integration becomes costly, whereas insufficient accuracy in the domain integration leads to suboptimal convergence. In this paper, we show that it is possible to achieve optimal convergence by enforcing variational consistency between the domain integration and the test functions, and optimal convergence can be achieved with much less computational cost than using higher‐order quadrature rules. In fact, stabilized conforming nodal integration is variationally consistent, whereas Gauss integration and nodal integration are not. In this work the consistency conditions for arbitrary order exactness in the Galerkin approximation are set forth explicitly. The test functions are then constructed to be variationally consistent with the integration scheme up to a desired order. Attempts are also made to correct methods that are variationally inconsistent via modification of test functions, and several variationally consistent methods are derived under a unified framework. It is demonstrated that the solution errors of PDEs due to quadrature inaccuracy can be significantly reduced when the variationally inconsistent methods are corrected with the proposed method, and consequently the optimal convergence rate can be either partially or fully restored. Copyright © 2013 John Wiley & Sons, Ltd.

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Section: Introductionmentioning

confidence: 99%

An arbitrary order variationally consistent integration for Galerkin meshfree methods

Chen

Hillman

Rüter

2013

Numerical Meth Engineering

161

110

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“…In the semi-Lagrangian RK approximation (Chen and Wu 2007a), the reproducing conditions are constructed in the current configuration, where the nodal neighbor list is updated by redefining the kernel support coverage. In this way, the nodal points follow the motion of material points under a Lagrangian description, while the mesh distortion issues associated with conventional meshbased methods are effectively avoided (Guan et al 2009;Chi et al 2015;Sherburn et al 2015;Wei et al 2019)…”

Section: Semi-lagrangian Rk Approximationmentioning

confidence: 99%

Accelerated and Stabilized Meshfree Method for Impact-Blast Modeling

Chen

Baek

Huang

et al. 2020

Structures Congress 2020

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Meshfree methods such as the reproducing kernel particle method (RKPM) are well suited for modeling materials and solids undergoing fracture and damage processes, and nodal integration is a natural choice for modeling this class of problems. However, nodal integration suffers from spatial instability, and the excessive material deformation and damage process could also lead to kernel instability in RKPM. This paper reviews the recent advances in nodal integration for meshfree methods that are stable, accurate, and with optimal convergence. A variationally consistent integration (VCI) is introduced to allow correction of low order quadrature rules to achieve optimal convergence, and several stabilization techniques for nodal integration are employed. The application of the stabilized RKPM with nodal integration for shock modeling, fracture to damage multiscale mechanics, and materials modeling in extreme events, are demonstrated. These include the modeling of man-made disasters such as fragmentimpact processes, penetration, shock, and blast events will be presented to demonstrate the effectiveness of the new developments.

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“…It yields the total (damaged) effective stress

{\overset{true¯}{σ}}_{ij}

{\overset{true¯}{σ}}_{ij} = {\overset{σ}{true}}_{ij}^{dev} ((), 1 - d) + ((), ((), 1 - d) {\overset{σ}{true}}_{kk}^{+} + {\overset{σ}{true}}_{kk}^{-}) δ_{ij}

where superscript dev, +, and − indicate deviatoric, tensile, and compressive parts of the corresponding terms, respectively. Following Guan et al, the damage parameter d is defined as

d = \frac{c_{1} ((), η - c_{2})}{η ((), c_{1} - c_{2})}, η \geq c_{2}

where η is the norm of the deviatoric strain (ie,

η = \sqrt{ε_{ij}^{dev} ε_{ji}^{dev}}

) used as a means to identify material damage. The damage parameter c 2 specifies the initiation point, when material starts to damage (ie, d = 0).…”

Section: Saturated Deformable Porous Mediamentioning

confidence: 99%

“…This mapping between the two configurations breaks down when extreme deformation occurs. To circumvent such issue, the shape functions of semi‐Lagrangian RKPM are constructed in the current configuration, thus avoiding the severe distortion or even separation of the support domain. The discretization (ie, nodal points) of semi‐Lagrangian RK formulation, however, still follows a Lagrangian description to track internal variables of the same material points at each time step, while the support of the RK shape function maintains a fixed shape and size.…”

Section: Semi‐lagrangian Rkpmmentioning

confidence: 99%

“…Besides the capability to analyze slope stability as effectively as FEMs, the present method can also naturally model run‐out simulation. This is due to the construction of the approximation functions in the current configuration, which readily allows extreme deformation and material separation. Because of such principle, the method can also naturally detect contacting bodies with contact forces directly calculated from a constitutive model, which can be aided by a level‐set algorithm to enhance accuracy of the contacting surfaces.…”

Section: Introductionmentioning

confidence: 99%

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u‐p semi‐Lagrangian reproducing kernel formulation for landslide modeling

Siriaksorn

Chi

Foster

et al. 2017

Num Anal Meth Geomechanics

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This paper presents a u-p (displacement-pressure) semi-Lagrangian reproducing kernel (RK) formulation to effectively analyze landslide processes. The semi-Lagrangian RK approximation is constructed based on Lagrangian discretization points with fixed kernel supports in the current configuration. As a result, it tracks state variables at discretization points while allowing extreme deformation and material separation that is beyond the capability of Lagrangian formulations. The u-p formulation following Biot theory is incorporated into the formulation to describe poromechanics of saturated geomaterials.In addition, a stabilized nodal integration method to ensure stability of the domain integration and kernel contact algorithms to model contact between bodies are introduced in the u-p semi-Lagrangian RK formulation. The proposed method is verified with several numerical examples and validated with an experimental result and the field data of an actual landslide.

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Semi-Lagrangian reproducing kernel formulation and application to modeling earth moving operations

Cited by 62 publications

References 16 publications

An arbitrary order variationally consistent integration for Galerkin meshfree methods

An arbitrary order variationally consistent integration for Galerkin meshfree methods

Accelerated and Stabilized Meshfree Method for Impact-Blast Modeling

u‐p semi‐Lagrangian reproducing kernel formulation for landslide modeling

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