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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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01 Jan 2016
TL;DR: The present work outlines a framework that can be used to define and study meshfree geometry representations, and a particular mesh free geometry representation called the Meshfree Correction Implicit Geometry is introduced and studied under the guidelines of the framework.
Abstract: Patient-specific biomechanical analysis is an important tool used to understand the complex processes that occur in the body due to physical stimulation. Patient-specific models are generated by processing medical images; once an object from the image is identified via segmentation, a point cloud representation of the object is extracted. Generating an analysis suitable representation from the point cloud has traditionally required generating a finite element mesh, which often requires a well defined surface to accomplish. Point clouds lack a well defined geometry, meaning that the surface definition is incomplete at best. Point clouds that have been generated from images have a fuzzy boundary, meaning that no direct sampling of the boundary is guaranteed to exist and any method for completing the geometry definition requires assumptions on the part of the modeler. The process of generating a finite element mesh of the point cloud is difficult and tedious often requiring manual editing to alleviate poorly constructed elements. An alternative to generating a finite element mesh is to use meshfree analysis to solve the boundary value problem. The geometric representation of meshfree analysis is a point cloud, thus making it a natural choice for patient-specific analysis. When using meshfree analysis, it is common to skip the geometric reconstruction and use the point cloud directly as an analysis suitable geometry. The lack of a well defined surface can be problematic for a variety of reasons, namely the visualization of results and solving contact driven problems. vi The goal of this dissertation is to exploit some characteristics of the meshfree analysis to generate a well defined geometry for point clouds. Meshfree methods are commonly used for the solution of boundary value problems; their lack of a well defined geometry representation is a hindrance that is often remedied by accompanying the meshfree particle distribution with a secondary geometry representation, such as a mesh. The present work outlines a framework that can be used to define and study meshfree geometry representations. A particular meshfree geometry representation called the Meshfree Correction Implicit Geometry is introduced and studied under the guidelines of the framework.
2 citations
01 May 2010
TL;DR: The practical implication of the mesh size with reference to waves coming from earthquake action and hence natural hazards are explained and outlined and the effects of mesh size used in the numerical modeling of engineering problems on the accuracy of the results are discussed.
Abstract: Numerical methods like the Finite Element Method or Finite Difference Method are popular in the analysis of engineering structures which are submitted to dynamic impact that induce wave propagation. Dynamic impact for structures often emanate from earthquake and blasting. The waves which propagate as a result of these actions stay only for short durations. Because of the extremely short duration of the resulting waves and the energy transmission is accomplished through the different sized grids in the numerical modeling, the numerical results are sensitive to the finite element mesh size. Practical experiences in numerical simulations show that a mesh size which fits to one dynamic impact might not be appropriate for another case dictating that the convergence criterion of a numerical model with a given numerical mesh size might not be enough to guarantee accurate numerical simulation results. Therefore, both coarse mesh and fine mesh are used in different dynamic loading cases to investigate the mesh size effect on numerical results of impact wave propagation and interaction with structures. Based on the numerical results, the effects of mesh size used in the numerical modeling of engineering problems on the accuracy of the results are discussed. Moreover, the practical implication of the mesh size with reference to waves coming from earthquake action and hence natural hazards are explained and outlined.
2 citations
01 Dec 1976
TL;DR: Finite element and finite difference methods are examined in this article, and it is shown that both methods use two types of discrete representations of continuous functions, i.e., functional approximations and finite volume difference methods.
Abstract: Finite element and finite difference methods are examined in order to bring out their relationship. It is shown that both methods use two types of discrete representations of continuous functions. They differ in that finite difference methods emphasize the discretization of independent variable, while finite element methods emphasize the discretization of dependent variable (referred to as functional approximations). An important point is that finite element methods use global piecewise functional approximations, while finite difference methods normally use local functional approximations. A general conclusion is that finite element methods are best designed to handle complex boundaries, while finite difference methods are superior for complex equations. It is also shown that finite volume difference methods possess many of the advantages attributed to finite element methods.
2 citations
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01 Jan 2019
TL;DR: A novel numerical method based on strong form point collocation methods for solving problems with geometric non-linearity including membrane problems is developed, where the basis functions possess some advantages such as the weak Kronecker-Delta property on boundaries.
Abstract: Conventional mesh-based methods for solid mechanics problems suffer 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 effciently. This geometrically non-linear method is finally 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 findings of this thesis.
2 citations
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01 Jan 2019
Abstract: The main aim of the dissertation is to develop a meshless model that describes the solidification and macrosegregation phenomena during the direct chill casting (DCC) of aluminium alloys under the influence of a low-frequency electromagnetic field. Macrosegregation is an undesired consequence of alloy solidification. It represents one of the major casting defects and substantially reduces the quality of the finished product. On the other hand, low-frequency electromagnetic casting (LFEC) is a process that promises to increase greatly the product quality, including the reduction of macrosegregation. The modelling of both processes is of tremendous importance to the metallurgical industry, due to the high costs of experiments during production.
The volume-averaging formulation is used for the modelling of the solid-liquid interaction. The conservation equations for mass, energy, momentum, and species are used to model the solidification of aluminium-alloy billets in axysimmetry. The electromagnetic-induction equation is coupled with the melt flow. It is used to calculate the magnetic vector potential and the Lorentz force. The Lorentz force is time-averaged and included in the momentum-conservation equation, which intensifies the melt flow. The effect of Joule heating is neglected in the energy conservation due to its insignificant contribution. The semi-continuous casting process is modelled with the Eulerian approach. This implies that the global computational domain is fixed in space. The inflow of the liquid melt is assumed at the top boundary and the outflow of the solid metal is assumed at the bottom. It is assumed that the whole mushy area is a rigid porous media, which is modelled with the Darcy law. The Kozeny-Carman relation is used for the permeability definition. The incompressible mass conservation is ensured by the pressure correction, which is performed with the fractional step method. The conservation equations and the induction equation are posed in the cylindrical coordinate system. A linearised eutectic binary phase diagram is used to predict the solute redistribution in the solid and liquid phases. The micro model uses the lever rule to determine the temperature and the liquid fraction field from the transport equations.
The partial differential equations are solved with the meshless-diffuse-approximate method (DAM). The DAM uses weighted least squares to determine a locally smooth approximation from a discrete set of data. The second-order polynomials are used as the trial functions, while the Gaussian function is used as the weight function. The method is localised by defining a smooth approximation for each computational node separately. This is performed by associating each node with a unique local neighbourhood, which is used for the minimisation. There are 14 nodes included in the local subdomains for the DCC and LFEC simulations. The stability of the advective term is achieved with a shift of the Gaussian weight in the upwind direction. This approach is called the adaptive upwind weight function and is used in the DAM for the first time. The Explicit-Euler scheme is used for temporal discretisation.
The use of a meshless method and the automatic node-arrangement generation makes it possible to investigate the complicated flow structures, which are formed in geometrically complex inflow conditions in a straightforward way. A realistic inflow geometry and mould can therefore be included in the model. The number of computational nodes is increased in the mushy zone and decreased in the solid phase, due to the optimisation of the computational time and memory. The computational node arrangement is automatically adapted with time, as the position of the mushy zone is changed in shape and position.
2 citations