Simple and robust element-free Galerkin method with almost interpolating shape functions for finite deformation elasticity

Bourantas, George C.; Zwick, Benjamin F.; Joldes, Grand Roman; Wittek, Adam; Miller, Karol

doi:10.1016/j.apm.2021.03.007

Cited by 18 publications

(8 citation statements)

References 43 publications

(38 reference statements)

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“…This fact shows us that the share of nodes in the vicinity of the node of interest is lower. The nodes in the immediate vicinity of the node of interest have the largest share [6], [7]. Such a situation is preferable for the description of phenomena with a strong local character, as it is the case of impact problems in general and high speed impact in particular [4], [5].…”

Section: Silvia Marzavanmentioning

confidence: 99%

On the Choice of the Weight Function and its Parameters in the Element Free Galerkin Method

Marzavan¹

2022

J.Mil.Technol.

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Our paper is focused on a subject having a great importance when Element Free Galerkin (EFG) method is used. Namely, it is about the choosing of some charateristic parameters of the kernel function. Practically, the EFG method user has to decide upon some aspects like which kernel function is going to be used, then which is the most fitted dimension of the domain support, which is the influence of the internodal distance for a better accuracy and others. This paper is based on the theoretical fundamentals of the EFG method and on our practical experience in using of the method. Our research presents some comments and recommendations accompanied by examples and processing of some results coming from practice. The author considers that despite all the progress made in the development of the EFG method, its use in the analysis of mechanical structures needs a good experience regarding some aspects on which the indications are few, or vague, or missing. The numerical examples in this work, are performing by using original programs in Matlab for solving simple problems. We consider that our conclusions are very useful not only for the beginners in using of EFGM but for those who want to know better and especially why it is better to use an one or other option.

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Section: Silvia Marzavanmentioning

confidence: 99%

On the Choice of the Weight Function and its Parameters in the Element Free Galerkin Method

Marzavan¹

2022

J.Mil.Technol.

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“…For any point

x \in normalΩ

, IMMLS function

u^{h} ((), x)

is used to approximate the unknown field variable

u ((), x) .

u^{h} ((), x) goodbreak= \sum_{j = 1}^{m} p_{j} ((), x) a_{j} ((), x) goodbreak= P^{T} ((), x) a ((), x),

where

P^{T} ((), x) = ([],,,, p_{1} ((), x), p_{2} ((), x), \dots, p_{m} ((), x))

(‘ T ’ means ‘matrix transpose’) are basis functions and

a ((), x)

is a vector containing the coefficients

a_{j} ((), x)

((), j = 1, 2, \dots, m)

, with

m

being the number of terms in the vector of basis function

P^{T} ((), x)

. These coefficients are calculated from the minimisation of an error functional

J

, which is defined based on the weighted least squares errors and including additional constraints on the coefficient

α

corresponding to the second‐degree monomials in the basis 28,44

\begin{matrix} true \\ J (x) = \sum_{i = 1}^{n} [{((), u^{h} ((), x_{i}) goodbreak- u ((), x_{i}))}^{2} W (d_{i})] + μ_{x^{2}} α_{x^{2}}^{2} + μ_{y^{2}} a_{y^{2}}^{2} + μ_{z^{2}} α_{z^{2}}^{2} ... \end{matrix}

…”

Section: Numerical Modelmentioning

confidence: 99%

“…The components of the vector

μ = ([], μ_{x^{2}} μ_{y^{2}} μ_{z^{2}} μ_{xy} μ_{xz} μ_{yz})

are positive weights 46 (in this study, weights are small,

μ_{x^{2}} = μ_{y^{2}} = μ_{z^{2}} = μ_{xy} = μ_{xz} = μ_{yz} = 10^{- 7}

) for the additional constraints. The weight function is defined as 44,45 :

W ((), d_{i}) goodbreak= \frac{{({((), \frac{d_{i}}{r})}^{2} + ε)}^{- 2} - {(1 + ε)}^{- 2}}{ε^{- 2} - {(1 + ε)}^{- 2}} .

where

r

is the influence domain radius and

ε = 10^{- 5}

is a regularisation parameter.…”

Section: Numerical Modelmentioning

confidence: 99%

“…These coefficients are calculated from the minimisation of an error functional J, which is defined based on the weighted least squares errors and including additional constraints on the coefficient α corresponding to the second-degree monomials in the basis. 28,44 J…”

Section: Interpolating Modified Moving Least-squares Shape Functionsmentioning

confidence: 99%

“…We use the interpolating modified moving least-squares shape (IMMLS) functions. 44 Because of their interpolating properties, essential boundary conditions can be imposed with very high accuracy without any additional effort. 44,45 For any point x Ω, IMMLS function u h x ð Þ is used to approximate the unknown field variable u x ð Þ:…”

Section: Interpolating Modified Moving Least-squares Shape Functionsmentioning

confidence: 99%

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Computer simulation of tumour resection‐induced brain deformation by a meshless approach

Bourantas

Zwick

et al. 2021

Numer Methods Biomed Eng

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Tumour resection requires precise planning and navigation to maximise tumour removal while simultaneously protecting nearby healthy tissues. Neurosurgeons need to know the location of the remaining tumour after partial tumour removal before continuing with the resection. Our approach to the problem uses biomechanical modelling and computer simulation to compute the brain deformations after the tumour is resected. In this study, we use meshless Total Lagrangian explicit dynamics as the solver. The problem geometry is extracted from the patient-specific magnetic resonance imaging (MRI) data and includes the parenchyma, tumour, cerebrospinal fluid and skull. The appropriate non-linear material formulation is used. Loading is performed by imposing intra-operative conditions of gravity and reaction forces between the tumour and surrounding healthy parenchyma tissues. A finite frictionless sliding contact is enforced between the skull (rigid) and parenchyma. The meshless simulation results are compared to intra-operative MRI sections. We also calculate Hausdorff distances between the computed deformed surfaces (ventricles and tumour cavities) and surfaces observed intra-operatively. Over 80% of points on the ventricle surface and 95% of points on the tumour cavity surface were successfully registered (results within the limits of two times the original in-plane resolution of the intra-operative image). Computed results demonstrate the potential for our method in estimating the tissue deformation and tumour boundary during the resection.

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Consistent $$\overline {\boldsymbol {C}}$$ Element-Free Galerkin Method for Finite Strain Analysis

Areias¹,

Carapau

Lopes

et al. 2022

Recent Advances in Mechanics and Fluid-Structure Interaction With Applications

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Simple and robust element-free Galerkin method with almost interpolating shape functions for finite deformation elasticity

Cited by 18 publications

References 43 publications

On the Choice of the Weight Function and its Parameters in the Element Free Galerkin Method

On the Choice of the Weight Function and its Parameters in the Element Free Galerkin Method

Computer simulation of tumour resection‐induced brain deformation by a meshless approach

Consistent $$\overline {\boldsymbol {C}}$$ Element-Free Galerkin Method for Finite Strain Analysis

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