Extension of B-spline Material Point Method for unstructured triangular grids using Powell–Sabin splines

Koster, Pascal de; Tielen, Roel; Wobbes, Elizaveta; Möller, Matthias

doi:10.1007/s40571-020-00328-3

Cited by 16 publications

(5 citation statements)

References 37 publications

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“…The spatial derivative of and can also be computed through the support of the background set of nodes. In this article, N I (x) fall within the class of convex approximation schemes; 6 however, other approximation techniques can be applied [32][33][34][35][36] in the framework of the MPM. Besides, the continuum  is discretized with a finite set of material points §  ⊂ .…”

Section: Spatial Discretization: the Local Max-ent Approximation As Interpolation Techniquementioning

confidence: 99%

“…Spaces

𝒬^{h}

and

𝒲^{h}

might be expressed as:

\begin{align} 𝒬^{h} & : = false{φ \in 𝒬 false| φ = \sum_{I \in 𝒜} φ^{I} N^{I} false(boldx false) false}, \end{align}

\begin{align} 𝒲^{h} & : = false{ψ \in 𝒬 false| ψ = \sum_{I \in 𝒜} ψ^{I} N^{I} false(boldx false) false} . \end{align}

The spatial derivative of

φ

and

ψ

can also be computed through the support of the background set of nodes. In this article,

N^{I} (x)

fall within the class of convex approximation schemes ; 6 however, other approximation techniques can be applied 32‐36 in the framework of the MPM. Besides, the continuum

ℬ

is discretized with a finite set of material points

𝒞 \subset ℬ

.…”

Section: The Discrete Problem: the B‐free Lme‐mpm Frameworkmentioning

confidence: 99%

See 1 more Smart Citation

Toward a local maximum‐entropy material point method at finite strain within a B‐free approach

Molinos

Stickle

Navas

et al. 2021

Numerical Meth Engineering

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The material point method can be regarded as a meshfree extension of the finite element method. This fact has two interesting consequences. On the one hand, this puts a vast literature at our disposal. On the other hand, many of this inheritance has been adopted without questioning it. A clear example of it is the use of the Voigt algebra, which introduces an artificial break point between the formulation of the continuum problem and its discretized counterpart. In the authors' opinion, the use of the Voigt rules leads to a cumbersome formulation where the physical sense of the operations is obscured since the well-known algebra rules are lost. And with them, the intuition about how stresses and strains are related.To illustrate it, we will describe gently and meticulously the whole process to solve the nonlinear governing equations for isothermal finite strain elastodynamics, concluding with a compact set of expressions ready to be implemented effortless. In addition, the classic Newmark-algorithm has been accommodated to the local maximum-entropy material point method framework by means of an incremental formulation. K E Y W O R D SB free, finite strain, material point method, Newmark-, Voigt free * This aspect is controlled at the very beginning of the modelization, specifically during the meshing process.This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

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Section: Spatial Discretization: the Local Max-ent Approximation As Interpolation Techniquementioning

confidence: 99%

“…Spaces

𝒬^{h}

and

𝒲^{h}

might be expressed as:

\begin{align} 𝒬^{h} & : = false{φ \in 𝒬 false| φ = \sum_{I \in 𝒜} φ^{I} N^{I} false(boldx false) false}, \end{align}

\begin{align} 𝒲^{h} & : = false{ψ \in 𝒬 false| ψ = \sum_{I \in 𝒜} ψ^{I} N^{I} false(boldx false) false} . \end{align}

The spatial derivative of

φ

and

ψ

can also be computed through the support of the background set of nodes. In this article,

N^{I} (x)

fall within the class of convex approximation schemes ; 6 however, other approximation techniques can be applied 32‐36 in the framework of the MPM. Besides, the continuum

ℬ

is discretized with a finite set of material points

𝒞 \subset ℬ

.…”

Section: The Discrete Problem: the B‐free Lme‐mpm Frameworkmentioning

confidence: 99%

Toward a local maximum‐entropy material point method at finite strain within a B‐free approach

Molinos

Stickle

Navas

et al. 2021

Numerical Meth Engineering

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show abstract

“…Huang et al, 2015;Bandara et al, 2016;Wang et al, 2016c, b). This formulation provides an objective (invariant by rotation or frameindifferent) stress rate measure (de Souza Neto et al, 2011) and is simple to implement. The Jaumann rate of the Cauchy stress is defined as…”

Section: Rate Formulation and Elasto-plasticitymentioning

confidence: 99%

A fast and efficient MATLAB-based MPM solver: fMPMM-solver v1.1

et al. 2020

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Abstract. We present an efficient MATLAB-based implementation of the material point method (MPM) and its most recent variants. MPM has gained popularity over the last decade, especially for problems in solid mechanics in which large deformations are involved, such as cantilever beam problems, granular collapses and even large-scale snow avalanches. Although its numerical accuracy is lower than that of the widely accepted finite element method (FEM), MPM has proven useful for overcoming some of the limitations of FEM, such as excessive mesh distortions. We demonstrate that MATLAB is an efficient high-level language for MPM implementations that solve elasto-dynamic and elasto-plastic problems. We accelerate the MATLAB-based implementation of the MPM method by using the numerical techniques recently developed for FEM optimization in MATLAB. These techniques include vectorization, the use of native MATLAB functions and the maintenance of optimal RAM-to-cache communication, among others. We validate our in-house code with classical MPM benchmarks including (i) the elastic collapse of a column under its own weight; (ii) the elastic cantilever beam problem; and (iii) existing experimental and numerical results, i.e. granular collapses and slumping mechanics respectively. We report an improvement in performance by a factor of 28 for a vectorized code compared with a classical iterative version. The computational performance of the solver is at least 2.8 times greater than those of previously reported MPM implementations in Julia under a similar computational architecture.

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“…Usually, a fixed regular Cartesian grid is used for MPM throughout the simulation, and unstructured grids can also be employed. Koster et al 5 use of high-order, C1-continuous Powell-Sabin B-splines on unstructured grids. The MPM has been successfully applied to many challenging problems involving large deformation where conventional Lagrangian methods often fail owing to mesh distortion or element entanglements, such as high-speed impact and penetration, [6][7][8] landslide, 9,10 fluid-structure interaction, 11,12 and explosive-related simulations.…”

Section: Introductionmentioning

confidence: 99%

Generalized particle domain method: An extension of material point method generates particles from the CAD files

Wang,

Dong,

Zhang

et al. 2024

Numerical Meth Engineering

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SummaryIn this paper, a generalized particle domain method (GPDM) is proposed and developed within the framework of the convected particle domain interpolation method. This new method generates particles directly from non‐uniform rational B‐spline (NURBS)‐based CAD file of a continuum body. The particle domain corresponds to a NURBS element even for trimmed elements of solids with complex geometries. The shape functions and the gradient of shape functions are evaluated using NURBS basis functions to map material properties between particles and grid nodes. It approves that this proposed GPDM can track the domain of particles accurately and avoid the issue of cell‐crossing instability. Several numerical examples are presented to demonstrate the high performance of this proposed new particle domain method. It is shown that the results obtained using the proposed GPDM are consistent with the experimental data reported in the literature. Further development of the generalized particle domain method can provide a link to the material point method and isogeometric analysis.

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Extension of B-spline Material Point Method for unstructured triangular grids using Powell–Sabin splines

Cited by 16 publications

References 37 publications

Toward a local maximum‐entropy material point method at finite strain within a B‐free approach

Toward a local maximum‐entropy material point method at finite strain within a B‐free approach

A fast and efficient MATLAB-based MPM solver: fMPMM-solver v1.1

Generalized particle domain method: An extension of material point method generates particles from the CAD files

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