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Meshfree methods
About: Meshfree methods is a research topic. Over the lifetime, 2216 publications have been published within this topic receiving 69596 citations.
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TL;DR: A novel trans- fer operator is introduced, which interacts in the transition zone, where both methods coexist, and this coupling between the two simulation techniques can be considered as saddle point problem and the stability of this coupling is shown.
Abstract: In contrast to the well known meshbased methods like the finite element method, meshfree methods do not rely on a mesh. The advantage of meshfree methods lies in the fact, that they need no mesh generation and can thus better cope with geometric changes and high dimensional problems. However besides their great applicability, meshfree methods are rather time con- suming. Thus, it seems favorable to combine both methods, by using meshfree methods only in a small part of the domain, where a mesh is disadvantageous, and a meshbased method for the rest of the domain. We motivate, that this coupling between the two simulation techniques can be considered as saddle point problem and show the stability of this coupling. Thereby a novel trans- fer operator is introduced, which interacts in the transition zone, where both methods coexist.
25 Mar 2013
TL;DR: One of the most widely used meshless methods, the element-free Galerkin method (EFGM), is used in this research and Maximum entropy shape functions are used instead of the conventional moving least squares based formulations'.
Abstract: Three-dimensional problems with both material and geometrical nonlinearities are of practical importance in many engineering applications, e.g. geomechanics, metal forming and biomechanics. Traditionally, these problems are simulated using an adaptive finite element method (FEM). However, the FEM faces many challenges in modeling these problems, such as mesh distortion and selection of a robust refinement algorithm. Adaptive meshless methods are a more recent technique for modeling these problems and can overcome the inherent mesh based drawbacks of the FEM but are computationally expensive. To take advantage of the good features of both methods, in the method proposed in this paper, initially the whole of the problem domain is modeled using the FEM. During an analysis those elements which violate a predefined error measure are automatically converted to a meshless zone. This zone can be further refined by adding nodes, overcoming computationally expensive FE remeshing. Therefore an appropriate coupling between the FE and the meshless zone is vital for the proposed formulation. One of the most widely used meshless methods, the element-free Galerkin method (EFGM), is used in this research. Maximum entropy shape functions are used instead of the conventional moving least squares based formulations'. These shape functions posses a weak Kronecker delta property at the boundaries of the problem domain, which allows the essential boundary conditions to be imposed directly and also helps to avoid the use of a transition region in the coupling between the FE and the EFG regions. Total Lagrangian formulation is preferred over the updated Lagrangian formulation for modeling finite deformation due to its computational efficiency. The well-established error estimation procedure of Zienkiewicz-Zhu is used in the FE region to determine the elements requiring conversion to the EFGM. The Chung and Belytschko error estimator is used in the EFG region for further adaptive refinement. Numerical examples are presented to demonstrate the performance of the current approach in three
26 Mar 2015
TL;DR: Chen et al. as mentioned in this paper proposed the use of a pressure projection scheme based on a similar approach used in fluid mechanics called the polynomial pressure projection (PPP), which is applied to the Galerkin weak form to penalise the error between the pressure field and the projected pressure field.
Abstract: The pressure projection method [1] has been used successfully in meshfree mixed formulations to model large deformations of hyperelastic materials because it stabilises pressure oscillations and overcomes volumetric locking near the incompressibility limit. However, the current pressure projection method is limited to the solutions of nearly-incompressible problems, due to the use of a finite bulk modulus as a penalty factor to enforce incompressibility [1]. In order to solve ideally incompressible hyperelasticity, we propose the use of a pressure projection scheme based on a similar approach used in fluid mechanics called the polynomial pressure projection (PPP) [2]. A stabilisation term, similar to the PPP, is appended to the Galerkin weak form to penalise the error between the pressure field and the projected pressure field. In this case, the projected pressure field cannot be eliminated through projection onto the displacement field, because the stabilisation term vanishes for pressures within the penalisation operation [2]. We have developed a total Lagrangian meshfree framework to solve the modified Galerkin weak form with numerical differencing. This has not been possible until now because numerical differencing perturbs the fictitious field approximated from the moving least squares (MLS) shape functions. To overcome this, we performed a full transformation [3] between nodal and fictitious values, enabling the perturbation of material points for numerical differencing. Using our method, essential boundary conditions can be imposed easily. We validated the accuracy of our framework against an analytic solution for uniaxial beam extension, and a finite element solution for a 2D annulus under inflation and radial torsion. The predicted pressure fields agreed well with the benchmark solutions. Our goal is to apply this framework to simulate large hyperelastic deformations of soft tissues. In particular, we hope to simulate breast tissue reorientation under different gravity loading conditions, as well as during mammographic compression. In conclusion, we have developed and validated a new meshfree framework that reliably and robustly predicts large deformations of ideally incompressible soft tissues. Keywords : Meshfree methods, Hyperelasticity, Incompressibility, Stabilisation method References: Chen, J-S., Pan C. (1998). A Pressure Projection Method for Nearly Incompressible Rubber Hyperelasticity, Part I: Theory. Journal of Applied Mechanics. 63(4), 862-868. Dohrmann, C.R. and Bochev, P.B. (2004). A stabilized finite element method for stokes problem based on polynomial pressure projections. International Journal of Numerical Methods in Fluids, 46: 183–201. Chen, J-S., Pan C., Wu, C-T., Liu W.K. (1996). Reproducing Kernel Particle Methods for large deformations analysis of non-linear structures. Computational Methods in Applied Mechanical Engineering. 139: 195-227.
TL;DR: In this paper, a set of mesh-free numerical schemes for derivative terms is developed, which are compatible with the coordinates of Cartesian, cylindrical, and spherical coordinates, based on the comparisons between numerical and theoretical solutions, errors and convergences are assessed by a posteriori method.
Abstract: In order to improve the approximation of spatial derivatives without meshes, a set of meshfree numerical schemes for derivative terms is developed, which is compatible with the coordinates of Cartesian, cylindrical, and spherical. Based on the comparisons between numerical and theoretical solutions, errors and convergences are assessed by a posteriori method, which shows that the approximations for functions and derivatives are of the second accuracy order, and the scale of the support domain has some influences on numerical errors but not on accuracy orders. With a discrete scale h=0.01, the relative errors of the numerical simulation for the selected functions and their derivatives are within 0.65%.
26 Oct 2011
TL;DR: In this paper, the capacity of particle methods to compute pressure fields and hydrodynamic loads subsequently is analyzed for wave fluid flows, where the authors show that the particle scattering at the stages, following the water impact, is larger than the classic SPH.
Abstract: The study of wave fluid flows is now under special consideration in view of serious effects, caused by dams breaking and consequent formation of moving waves, their interaction with solids and structures, uprush on shore, etc. Thereby solving the problem of hydrodynamic loads estimation is important for designing the shape and stiffness of the structures, interacting with oncoming waves. Such problems, due to large deformations of free surfaces, are very complex, and meshless methods proved to be the most suitable for numerical simulation of them. Particle methods form the special class of meshless methods, which mainly based on the strong form of governing equations of gas dynamics and fluid dynamics. The peculiar representatives of particle methods are Smoothed Particle Hydrodynamics (SPH) (Lucy, 1977; Gingold & Monaghan, 1977) and Incompressible SPH (ISPH) (Cummins & Rudman, 1999; Shao & Lo, 2003; Lee et al., 2008). Large amount of papers, devoted to numerical simulations of free surface flows using SPH or ISPH, demonstrated a high degree of efficiency of both methods in obtaining the kinematic characteristics of flows, though it has been revealed, that ISPH shows a larger particle scattering at the stages, following the water impact, in comparison with the classic SPH, where particles are more ordered. However, dynamic characteristics of flows are still hard to compute, especially it concerns the classic SPH. The objective of the chapter is to analyze the capacity of the methods to compute pressure fields and hydrodynamic loads subsequently.