Figure 4 - uploaded by Zhidong Han
Content may be subject to copyright.
Source publication
This paper presents a new method for the computational mechanics of large strain deformations of solids, as a fundamental departure from the currently popular finite element methods (FEM). The currently widely popular primal FEM: (1) uses element-based interpolations for displacements as the trial functions, and element-based interpolations of disp...
Similar publications
We study a time explicit nite volume method for a rst order conservation law with a multiplica-tive source term involving a Q-Wiener process. After having presented the denition of a measure-valued weak entropy solution of the stochastic conservation law, we apply a nite volume method together with with Godunov scheme for the space discretization,...
In this work, we consider the discretization of nonlinear hyperbolic systems in nonconservative form with the high-order discontinuous Galerkin spectral element method (DGSEM) based on collocation of quadrature and interpolation points (Kopriva and Gassner (2010) [39]). We present a general framework for the design of such schemes that satisfy a se...
Citations
... Points. Alternatively, one could postulate criteria for rupture between two subdomains, based on the discontinuity of the Eshelby tractions [13] between the two segments. ...
The Fragile Points Method (FPM) is a stable and elementarily simple, meshless Galerkin weak-form method, employing simple, local, polynomial, Point-based, discontinuous and identical trial and test functions which are derived from the Generalized Finite Difference method. Numerical Flux Corrections are introduced in the FPM to resolve the inconsistency caused by the discontinuous trial functions. Given the simple polynomial characteristic of trial and test functions, integrals in the Galerkin weak form can be calculated in the FPM without much effort. With the global matrix being sparse, symmetric and positive definitive, the FPM is suitable for large-scale simulations. Additionally, because of the inherent discontinuity of trial and test functions, we can easily cut off the interaction between Points and introduce cracks, rupture, fragmentation based on physical criteria. In this paper, we have studied the applications of the FPM to linear elastic mechanics and several numerical examples of 2D linear elasticity are computed. The results suggest the FPM is accurate, robust, consistent and convergent. Volume locking does not occur in the FPM for nearly incompressible materials. Besides, a new, simple and efficient approach to tackle pre-existing cracks in the FPM is also illustrated in this paper and applied to mode-I crack problems.
... In meshless methods, the physical domain of interest is entirely presented by scattered field points without considering meshes and elements, so the overall analyses are totally free from the issues of mesh entanglement and distortion. This attractive feature has motivated the development of this group of methods in the past 20 years, such as smooth particle hydrodynamics [217,218], the element-free Galerkin method [219], the meshless local Petrov-Galerkin method [220], the reproducing kernel particle method [221], the meshless max-ent method [222,223], Galerkin-based methods with max-ent basis functions [109] and the isogeometric meshless method [27,214,[224][225][226][227][228][229]. ...
Conventional mesh-based methods for solid mechanics problems su�er from issues resulting
from the use of a mesh, therefore, various meshless methods that can be grouped into those based on weak or strong forms of the underlying problem have been proposed to address these problems by using only points for discretisation. Compared to weak form meshless methods, strong form meshless methods have some attractive features because of the absence of any background mesh and avoidance of the need for numerical integration, making the implementation straightforward. The objective of this thesis is to develop a novel numerical method based on strong form point collocation methods for solving problems with geometric non-linearity including membrane problems. To address some issues in existing strong form meshless methods, the local maximum entropy point collocation method is developed, where the basis functions possess some advantages such as the weak Kronecker-Delta property on boundaries. r- and h-adaptive strategies are investigated in the proposed method and are further combined into a novel rh-adaptive approach, achieving the prescribed accuracy with the optimised locations and limited number of points. The proposed meshless method with h-adaptivity is then extended to solve geometrically non-linear problems described in a Total Lagrangian formulation, where h-adaptivity is again employed after the initial calculation to improve the accuracy of the solution e�ciently. This geometrically non-linear method is �nally developed to analyse membrane problems, in which the out-of-plane deformation for membranes complicates the governing PDEs and the use of hyperelastic materials makes the computational modelling of membrane problems challenging. The Newton-Raphson arc-length method is adopted here to capture the snap-through behaviour in hyperelastic membrane problems. Several numerical examples are presented for each proposed algorithm to validate the proposed methodology and suggestions are made for future work leading on from the �ndings of this thesis.
... MLPGEM combines the Meshless Local Petrov-Galerkin (MLPG) Methods of Atluri and the energy conservation laws of Noether and Eshelby [53]. MLPG is a truly meshless method that involves both non-element interpolation and non-element integration [54]. ...
... MLPG may alleviate problems such as element distortion, locking, remeshing during large deformation process. With the help of the "energy-momentum tensor" given by Eshelby, Han and Atluri solved the singular integrals appearing in the local boundary integral equation and proposed a numerical method called MLPGEM [53]. This method is simply described as follows: a body is assembled by spheres (in 3-D or circles in 2-D) centered at each meshless node. ...
... One may see further details in the references [53][54][55]. ...
In this study, a numerical approach—the discontinuous Meshless Local Petrov-Galerkin-Eshelby Method (MLPGEM)—was adopted to simulate and measure material plasticity in an Al 7075-T651 plate. The plate was modeled in two dimensions by assemblies of small particles that interact with each other through bonding stiffness. The material plasticity of the model loaded to produce different levels of strain is evaluated with the Lamb waves of S0 mode. A tone burst at the center frequency of 200 kHz was used as excitation. Second-order nonlinear wave was extracted from the spectrogram of a signal receiving point. Tensile-driven plastic deformation and cumulative second harmonic generation of S0 mode were observed in the simulation. Simulated measurement of the acoustic nonlinearity increased monotonically with the level of tensile-driven plastic strain captured by MLPGEM, whereas achieving this state by other numerical methods is comparatively more difficult. This result indicates that the second harmonics of S0 mode can be employed to monitor and evaluate the material or structural early-stage damage induced by plasticity.
... To overcome difficulties due to large mesh distortion, ALE formulation has been the only alternative to solve fluid structure interaction for engineering problems. For the last decade, SPH and DEM (Discrete Element Method), have been used usefully for engineering problems to simulate high velocity impact problems, high explosive detonation in soil, underwater explosion phenomena, and bird strike in aerospace industry, see Han et al.[2]for details description of DEM method. SPH is a mesh free Lagrangian description of motion that can provide many advantages in fluid mechanics and also for modelling large deformation in solid mechanics. ...
Design of fuel tanks requires the knowledge of hydrodynamic pressure distribution on the structure. These can be very useful for engineers and designers to define appropriate material properties and shell thickness of the structure to be resistant under sloshing or hydrodynamic loading. Data presented in current tank design codes as Eurocode, are based on simplified assumptions for the geometry and material tank properties. For complex material data and complex tank geometry, numerical simulations need to performed in order to reduce experimental tests that are costly and take longer time to setup. Different formulations have been used for sloshing tank analysis, including ALE (Arbitrary Lagrangian Eulerian) and SPH (Smooth Particle Hydrodynamic). The ALE formulation uses a moving mesh with a mesh velocity defined trough the structure motion. In this paper the mathematical and numerical implementation of the FEM and SPH formulations for sloshing problem are described. From different simulations, it has been observed that for the SPH method to provide similar results as ALE formulation, the SPH meshing, or SPH particle spacing needs to be finer than ALE mesh. To validate the statement, we perform a simulation of a sloshing analysis inside a partially filled tank. For this simple, the particle spacing of SPH method needs to be at least two times finer than ALE mesh. A contact algorithm is performed at the fluid structure interface and SPH particles. In the paper the efficiency and usefulness of two methods, often used in numerical simulations, are compared.
... Unlike ALE method, and because of the absence of the mesh, SPH method suffers from a lack of consistency than can lead to poor accuracy. In the literature there are different formulations of meshless methods, the Meshless Local Petrov-Galerkin Method for solving the bending problem of a thin, see Han and Atluri (2014) and Shuyao and Atluri (2002) and also Meshless Finite Volume Method developed for solving elasto-static, see Alturi, Han, and Rajendran (2004). In Meshless Local Petrov-Galerkin Method mixed approach, both the strains as well as displacements are interpolated, at randomly distributed points in the domain, through local meshless interpolation schemes such as the moving least squares or radial basis functions. ...
Simulation of airbag and membrane deployment under pressurized gas problems becomes more and more the focus of computational engineering, where FEM (Finite element Methods) for structural mechanics and Finite Volume for CFD are dominant. New formulations have been developed for FSI applications using mesh free methods as SPH method, (Smooth Particle Hydrodynamic). Up to these days very little has been done to compare different methods and assess which one would be more suitable. For small deformation, FEM Lagrangian formulation can solve structure interface and material boundary accurately, the main limitation of the formulation is high mesh distortion for large deformation and moving structure. One of the commonly used approach to solve these problems is the ALE formulation which has been used with success in the simulation of FSI (Fluid Structure Interaction) with large structure motion such as sloshing fuel tank in automotive industry and bird impact in aeronautic industry. For some applications, including bird impact and high velocity impact problems, engineers have switched from ALE to SPH method to reduce CPU time and save memory allocation. In this paper the mathematical and numerical implementation of the ALE and SPH formulations are described. From different simulation, it has been observed that for the SPH method to provide similar results as ALE or Lagrangian formulations, the SPH meshing, or SPH spacing particles needs to be finer than the ALE mesh. To validate the statement, we perform a simulation of membrane deployment generated by high pressurized gas. For this simple problem, the particle spacing of SPH method needs to be at least two times finer than ALE mesh. A contact algorithm is performed at the FSI for both SPH and ALE formulations.
Simulation of airbag and membrane deployment under pressurized gas problems becomes more and more the focus of computational engineering, where finite element methods (FEMs) for structural mechanics and finite volume for computational fluid dynamics are dominant. New formulations have been developed for fluid structure interaction (FSI) applications using mesh free methods as smooth particle hydrodynamic (SPH) method. Up to these days very little has been done to compare different methods and assess which one would be more suitable. For small deformation, FEM Lagrangian formulation can solve structure interface and material boundary accurately, the main limitation of the formulation is high mesh distortion for large deformation and moving structure. One of the commonly used approaches to solve these problems is the arbitrary Lagrangian Eulerian (ALE) formulation which has been used with success in the simulation of FSI with large structure motion such as sloshing fuel tank in automotive industry and bird impact in aeronautic industry. For some applications, including bird impact and high velocity impact problems, engineers have switched from ALE to SPH method to reduce central processing unit (CPU) time and save memory allocation. Both ALE and SPH methods are described and compared here using similar mesh size, each ALE element is replaced by an SPH particle at the element center. From different simulation, it has been observed that for the SPH method to provide similar results as ALE or Lagrangian formulations, the SPH meshing needs to be finer than the ALE mesh. A contact algorithm is performed at the FSI for both SPH and ALE formulations. A simulation of airbag membrane deployment generated by high pressurized gas is performed.
The local radial basis function collocation method (LRBFCM) is proposed for plate bending analysis in orthorhombic quasicrystals (QCs) under static and transient dynamic loads. Three common types of the plate bending problems are considered: (1) QC plates resting on Winkler foundation (2) QC plates with variable thickness and (3) QC plates under a transient dynamic load. According to the Reissner–Mindlin plate bending theory, there is allowed to simulate the behavior of the two excitations in QC plates, phonon and phason, by 2D strong formulations for the system of governing equations. The governing equations, which describe the phason displacements, are based on Agiasofitou and Lazar elastodynamic model. Numerical results demonstrate the effect of the elastic foundation, as well as plate thickness on the phonon and phason characteristics in this paper. For the transient dynamic analysis, the influence of the phason friction coefficients on the responses of QC plate to transient dynamic loads is also studied.
A local reproducing kernel method based on spatial trial space spanned by the Newton basis functions in the native Hilbert space of the reproducing kernel is proposed. It is a truly meshless approach which uses the local sub clusters of domain nodes for approximation of the arbitrary field. It leads to a system of ordinary differential equations (ODEs) for the time-dependent partial differential equations (PDEs). An adaptive algorithm, so-called adaptive residual subsampling, is used to adjust nodes in order to remove oscillations which are caused by a sharp gradient. The method is applied for solving the Allen-Cahn and Burgers' equations. The numerical results show that the proposed method is efficient, accurate and be able to remove oscillations caused by sharp gradient.
During hydraulic fracturing in coal seams, fractures expand to coal-rock interface and the direction of fracture growth may change because of the effects of many mechanical factors. Based on the 2D model of coal-rock interface in hydraulic fracturing, the theoretical analysis and numerical simulation were conducted to investigate the failure mechanism of coal-rock interface and the propagation characteristics of fractures induced by hydraulic fracturing. It is shown that: the main factors which affect the development direction of fractures are intersecting angle between coal-rock interface and horizontal profile, horizontal stress difference, minimum horizontal in-situ stress, cohesive strength of coal-rock interface and elastic modulus difference between coal and rock. With the decrease of intersecting angle or horizontal stress difference, the probability of fractures crossing coal-rock interface will raise; while the minimum horizontal in-situ stress increases, the probability of fractures crossing coal-rock interface will reduce; with the decrease of cohesive strength of coal-rock interface or increase of elastic modulus difference between coal and rock, the fractures will tend to extend along the coal-rock interface.
The present work is to investigate the failure mechanisms in the deformation of silicon carbide (SiC) particle reinforced aluminum Metal Matrix Composites (MMCs). To better deal with crack growth, a new numerical approach: the MLPG-Eshelby Method is used. This approach is based on the meshless local weak-forms of the Noether/Eshelby Energy Conservation Laws and it achieves a faster convergent rate and is of good accuracy. In addition, it is much easier for this method to allow material to separate in the material fracture processes, comparing to the conventional popular FEM based method. Based on a statistical method and physical observations, the hard SiC particles are distributed randomly over the cubic space of the matrix. Four failure mechanisms are found to be critical to the accurate prediction of the mechanical properties of MMCs: a) the failure inside the matrix; b) the failure between the interface of aluminum matrix and the SiC particles; c) the fracture of the SiC particles; and d) the separation of two neighboring SiC particles. Plastic work is used as a failure criterion. It is found that the current approach can accurately predict the mechanical behavior of MMCs, including Young's moduls, stress strain curve, tensile strength, and limit strain. When the SiC volume fraction is low, the interface failure is more important; while for the case of high SiC volume fraction, all the four failure mechanisms work together to affect the mechanical property for the composite structure.
