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Meshfree Method
In the field of numerical analysis, meshfree methods are those that do not require connection between nodes of the simulation domain, i.e. a mesh, but are rather based on interaction of each node with all its neighbors. As a consequence, original extensive properties such as mass or kinetic energy are no longer assigned to mesh elements but rather to the single nodes. Meshfree methods enable the simulation of some otherwise difficult types of problems, at the cost of extra computing time and programming effort. The absence of a mesh allows Lagrangian simulations, in which the nodes can move according to the velocity field. Motivation Numerical methods such as the finite difference method, finite-volume method, and finite element method were originally defined on meshes of data points. In such a mesh, each point has a fixed number of predefined neighbors, and this connectivity between neighbors can be used to define mathematical operators like the derivative. These operators a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Voronoi Diagram
Euclidean (or, less commonly, Euclidian) is an adjective derived from the name of Euclid, an ancient Greek mathematician. Geometry *Euclidean space, the two-dimensional plane and three-dimensional space of Euclidean geometry as well as their higher dimensional generalizations *Euclidean geometry, the study of the properties of Euclidean spaces *Non-Euclidean geometry, systems of points, lines, and planes analogous to Euclidean geometry but without uniquely determined parallel lines *Euclidean distance, the distance between pairs of points in Euclidean spaces * Euclidean ball, the set of points within some fixed distance from a center point Number theory *Euclidean division, the division which produces a quotient and a remainder *Euclidean algorithm, a method for finding greatest common divisors *Extended Euclidean algorithm, a method for solving the Diophantine equation ''ax'' + ''by'' = ''d'' where ''d'' is the greatest common divisor of ''a'' and ''b'' *Euclid's lemma: if a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dissipative Particle Dynamics
Dissipative particle dynamics (DPD) is an off-lattice mesoscopic simulation technique which involves a set of particles moving in continuous space and discrete time. Particles represent whole molecules or fluid regions, rather than single atoms, and atomistic details are not considered relevant to the processes addressed. The particles' internal degrees of freedom are integrated out and replaced by simplified pairwise dissipative and random forces, so as to conserve momentum locally and ensure correct hydrodynamic behaviour. The main advantage of this method is that it gives access to longer time and length scales than are possible using conventional MD simulations. Simulations of polymeric fluids in volumes up to 100 nm in linear dimension for tens of microseconds are now common. DPD was initially devised by Hoogerbrugge and Koelman to avoid the lattice artifacts of the so-called lattice gas automata and to tackle hydrodynamic time and space scales beyond those available with m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffuse Element Method
In numerical analysis the diffuse element method (DEM) or simply diffuse approximation is a meshfree method. The diffuse element method was developed by B. Nayroles, G. Touzot and Pierre Villon at the Universite de Technologie de Compiegne, in 1992. It is in concept rather similar to the much older smoothed particle hydrodynamics. In the paper they describe a "diffuse approximation method", a method for function approximation from a given set of points. In fact the method boils down to the well-known moving least squares for the particular case of a global approximation (using all available data points). Using this function approximation method, partial differential equations and thus fluid dynamic problems can be solved. For this, they coined the term diffuse element method (DEM). Advantages over finite element methods are that DEM doesn't rely on a grid, and is more precise in the evaluation of the derivatives of the reconstructed functions. See also * Computational fluid dyn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smoothed Particle Hydrodynamics
Smoothed-particle hydrodynamics (SPH) is a computational method used for simulating the mechanics of continuum media, such as solid mechanics and fluid flows. It was developed by Gingold and Monaghan and Lucy in 1977, initially for astrophysical problems. It has been used in many fields of research, including astrophysics, ballistics, volcanology, and oceanography. It is a meshfree Lagrangian method (where the co-ordinates move with the fluid), and the resolution of the method can easily be adjusted with respect to variables such as density. Method Advantages * By construction, SPH is a meshfree method, which makes it ideally suited to simulate problems dominated by complex boundary dynamics, like free surface flows, or large boundary displacement. * The lack of a mesh significantly simplifies the model implementation and its parallelization, even for many-core architectures. * SPH can be easily extended to a wide variety of fields, and hybridized with some other mod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frozen (2013 Film)
''Frozen'' is a 2013 American animated musical fantasy film produced by Walt Disney Animation Studios and released by Walt Disney Pictures. Inspired by Hans Christian Andersen's 1844 fairy tale "The Snow Queen", it was directed by Chris Buck and Jennifer Lee and produced by Peter Del Vecho, from a screenplay by Lee, who also conceived the film's story with Buck and Shane Morris. The film stars the voices of Kristen Bell, Idina Menzel, Jonathan Groff, Josh Gad, and Santino Fontana. It follows Anna, the princess of Arendelle, who sets off on a journey with the iceman Kristoff, his reindeer Sven, and the snowman Olaf, to find her estranged sister Elsa after she accidentally traps their kingdom in eternal winter with her icy powers. ''Frozen'' underwent several story treatments before it was commissioned in 2011. Christophe Beck was hired to compose the film's orchestral score, and Robert Lopez and Kristen Anderson-Lopez wrote the songs. After its world premiere at the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Material Point Method
The material point method (MPM) is a numerical technique used to simulate the behavior of solids, liquids, gases, and any other continuum material. Especially, it is a robust spatial discretization method for simulating multi-phase (solid-fluid-gas) interactions. In the MPM, a continuum body is described by a number of small Lagrangian elements referred to as 'material points'. These material points are surrounded by a background mesh/grid that is used to calculate terms such as the deformation gradient. Unlike other mesh-based methods like the finite element method, finite volume method or finite difference method, the MPM is not a mesh based method and is instead categorized as a meshless/meshfree or continuum-based particle method, examples of which are smoothed particle hydrodynamics and peridynamics. Despite the presence of a background mesh, the MPM does not encounter the drawbacks of mesh-based methods (high deformation tangling, advection errors etc.) which makes it a promis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reproducing Kernel Particle Method
Reproduction (or procreation or breeding) is the biological process by which new individual organisms – "offspring" – are produced from their "parent" or parents. There are two forms of reproduction: asexual and sexual. In asexual reproduction, an organism can reproduce without the involvement of another organism. Asexual reproduction is not limited to single-celled organisms. The cloning of an organism is a form of asexual reproduction. By asexual reproduction, an organism creates a genetically similar or identical copy of itself. The evolution of sexual reproduction is a major puzzle for biologists. The two-fold cost of sexual reproduction is that only 50% of organisms reproduce and organisms only pass on 50% of their genes. John Maynard Smith ''The Evolution of Sex'' 1978. Sexual reproduction typically requires the sexual interaction of two specialized reproductive cells, called gametes, which contain half the number of chromosomes of normal cells and are created by m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ted Belytschko
Ted Bohdan Belytschko (January 13, 1943 – September 15, 2014) was an American mechanical engineer. He was Walter P. Murphy Professor and McCormick Professor of Computational Mechanics at Northwestern University. He worked in the field of computational solid mechanics and was known for development of methods like element-free Galerkin method and the Extended finite element method. Belytschko received his B.S. in Engineering Sciences (1965) and his Ph.D. in Mechanics (1968) from the Illinois Institute of Technology. He was named in ISI Database as the fourth most cited engineering researcher in January 2004. He was also the editor of the ''International Journal for Numerical Methods in Engineering''. He died at the age of 71 on September 15, 2014. Awards and honors * William Prager Medal, 2011. *Member of the National Academy of Sciences (2011) *Member of the National Academy of Engineering (1992) * John von Neumann Medal of the United States Association for Computational Mech ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moving Least Squares
Moving least squares is a method of reconstructing continuous functions from a set of unorganized point samples via the calculation of a weighted least squares measure biased towards the region around the point at which the reconstructed value is requested. In computer graphics, the moving least squares method is useful for reconstructing a surface from a set of points. Often it is used to create a 3D surface from a point cloud through either downsampling or upsampling. In numerical analysis to handle contributions of geometry where it is difficult to obtain discretizations, the moving least squares methods have also been used and generalized to solve PDEs on curved surfaces and other geometries. This includes numerical methods developed for curved surfaces for solving scalar parabolic PDEs and vector-valued hydrodynamic PDEs. In machine learning, moving least squares methods have also been used to develop model classes and learning methods. This includes function regression m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galerkin Method
In mathematics, in the area of numerical analysis, Galerkin methods are a family of methods for converting a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete problem by applying linear constraints determined by finite sets of basis functions. They are named after the Soviet mathematician Boris Galerkin. Often when referring to a Galerkin method, one also gives the name along with typical assumptions and approximation methods used: * Ritz–Galerkin method (after Walther Ritz) typically assumes symmetric and positive-definite bilinear form in the weak formulation, where the differential equation for a physical system can be formulated via minimization of a quadratic function representing the system energy and the approximate solution is a linear combination of the given set of the basis functions.A. Ern, J.L. Guermond, ''Theory and practice of finite elements'', Springer, 2004, * Bubnov–Galerkin method (after Ivan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
