Meshless method – Review on recent developments

Patel, Vivek; Rachchh, Nikunj V.

doi:10.1016/j.matpr.2020.02.328

Cited by 19 publications

(5 citation statements)

References 44 publications

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“…However, this approach requires performing tracking on a regular mesh, constraining the location of the tracking points to the nodes of this mesh. To remove the need for a regular grid for computation, meshless methods, and specifically radial basis function (RBF)-based methods, can be employed to solve relatively simple mechanical problems with complex geometries from sparse source points ( Patel and Rachchh, 2020 ).…”

Section: Introductionmentioning

confidence: 99%

Fast strain mapping in abdominal aortic aneurysm wall reveals heterogeneous patterns

Bracco,

Broda,

Lorenzen

et al. 2023

Front. Physiol.

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Abdominal aortic aneurysm patients are regularly monitored to assess aneurysm development and risk of rupture. A preventive surgical procedure is recommended when the maximum aortic antero-posterior diameter, periodically assessed on two-dimensional abdominal ultrasound scans, reaches 5.5 mm. Although the maximum diameter criterion has limited ability to predict aneurysm rupture, no clinically relevant tool that could complement the current guidelines has emerged so far. In vivo cyclic strains in the aneurysm wall are related to the wall response to blood pressure pulse, and therefore, they can be linked to wall mechanical properties, which in turn contribute to determining the risk of rupture. This work aimed to enable biomechanical estimations in the aneurysm wall by providing a fast and semi-automatic method to post-process dynamic clinical ultrasound sequences and by mapping the cross-sectional strains on the B-mode image. Specifically, the Sparse Demons algorithm was employed to track the wall motion throughout multiple cardiac cycles. Then, the cyclic strains were mapped by means of radial basis function interpolation and differentiation. We applied our method to two-dimensional sequences from eight patients. The automatic part of the analysis took under 1.5 min per cardiac cycle. The tracking method was validated against simulated ultrasound sequences, and a maximum root mean square error of 0.22 mm was found. The strain was calculated both with our method and with the established finite-element method, and a very good agreement was found, with mean differences of one order of magnitude smaller than the image spatial resolution. Most patients exhibited a strain pattern that suggests interaction with the spine. To conclude, our method is a promising tool for investigating abdominal aortic aneurysm wall biomechanics as it can provide a fast and accurate measurement of the cyclic wall strains from clinical ultrasound sequences.

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Section: Introductionmentioning

confidence: 99%

Fast strain mapping in abdominal aortic aneurysm wall reveals heterogeneous patterns

Bracco,

Broda,

Lorenzen

et al. 2023

Front. Physiol.

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show abstract

“…For the last few decades, there has been growing interest into meshless methods. Smoothed particle hydrodynamics [1][2][3], generalized finite difference method [4][5][6], reproducing kernel particle method [7][8][9], element-free Galerkin method [10][11][12], hp-clouds [13][14][15], partition of unity [16][17][18], finite point method [19][20][21] and radial basis function based finite difference (RBF-FD) method are some of the popular meshless methods. Meshless methods utilize only point clouds as a form of discretization for any domain.…”

Section: Introductionmentioning

confidence: 99%

A High-Order Accurate Meshless Method for Solution of Incompressible Fluid Flow Problems

Shahane,

Radhakrishnan,

Vanka

2020

Preprint

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Meshless solution to differential equations using radial basis functions (RBF) is an alternative to grid based methods commonly used. Since it does not need an underlying connectivity in the form of control volumes or elements, issues such as grid skewness that adversely impact accuracy are eliminated.Gaussian, Multiquadrics and inverse Multiquadrics are some of the most popular RBFs used for the solutions of fluid flow and heat transfer problems. But they have an additional shape parameter that has to be fine tuned for accuracy and stability. Moreover, they also face stagnation error when the point density is increased for accuracy. Recently, Polyharmonic splines (PHS) with appended polynomials have been shown to solve the above issues and give rapid convergence of discretization errors with the degree of appended polynomials. In this research, we extend the PHS-RBF method for the solution of incompressible Navier-Stokes equations. A fractional step method with explicit convection and explicit diffusion terms is combined with a pressure Poisson equation to satisfy momentum and continuity equations. Systematic

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“…This flexibility allows for better accuracy and reliability in numerical simulations and the ability to adjust the computational domain to fit specific solution requirements. Compared to the finite element method, commonly used in solid mechanics, the meshless method solves the limitations of the finite element method in terms of tricky mesh generation for complex shapes, low accuracy of stress calculation, and difficulty of adaptive analysis [40][41][42].…”

Section: Meshless Methodsmentioning

confidence: 99%

Progress and Challenges of Integrated Machine Learning and Traditional Numerical Algorithms: Taking Reservoir Numerical Simulation as an Example

Chen,

Zhang,

et al. 2023

Mathematics

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Machine learning techniques have garnered significant attention in various engineering disciplines due to their potential and benefits. Specifically, in reservoir numerical simulations, the core process revolves around solving the partial differential equations delineating oil, gas, and water flow dynamics in porous media. Discretizing these partial differential equations via numerical methods is one cornerstone of this simulation process. The synergy between traditional numerical methods and machine learning can enhance the precision of partial differential equation discretization. Moreover, machine learning algorithms can be employed to solve partial differential equations directly, yielding rapid convergence, heightened computational efficiency, and accuracies surpassing 95%. This manuscript offers an overview of the predominant numerical methods in reservoir simulations, focusing on integrating machine learning methodologies. The innovations in fusing deep learning techniques to solve reservoir partial differential equations are illuminated, coupled with a concise discussion of their inherent advantages and constraints. As machine learning continues to evolve, its conjunction with numerical methods is poised to be pivotal in addressing complex reservoir engineering challenges.

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Meshless method – Review on recent developments

Cited by 19 publications

References 44 publications

Fast strain mapping in abdominal aortic aneurysm wall reveals heterogeneous patterns

Fast strain mapping in abdominal aortic aneurysm wall reveals heterogeneous patterns

A High-Order Accurate Meshless Method for Solution of Incompressible Fluid Flow Problems

Progress and Challenges of Integrated Machine Learning and Traditional Numerical Algorithms: Taking Reservoir Numerical Simulation as an Example

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