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Generalized-strain Mesh-free Formulation
The generalized-strain mesh-free (GSMF) formulation is a local meshfree method in the field of numerical analysis, completely integration free, working as a weighted-residual weak-form collocation. This method was first presented by Oliveira and Portela (2016), in order to further improve the computational efficiency of meshfree methods in numerical analysis. Local meshfree methods are derived through a weighted-residual formulation which leads to a local weak form that is the well known work theorem of the theory of structures. In an arbitrary local region, the work theorem establishes an energy relationship between a statically-admissible stress field and an independent kinematically-admissible strain field. Based on the independence of these two fields, this formulation results in a local form of the work theorem that is reduced to regular boundary terms only, integration-free and free of volumetric locking. Advantages over finite element methods are that GSMF doesn't rely on a gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heaviside Step Function
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Different conventions concerning the value are in use. It is an example of the general class of step functions, all of which can be represented as linear combinations of translations of this one. The function was originally developed in operational calculus for the solution of differential equations, where it represents a signal that switches on at a specified time and stays switched on indefinitely. Heaviside developed the operational calculus as a tool in the analysis of telegraphic communications and represented the function as . Formulation Taking the convention that , the Heaviside function may be defined as: * a piecewise function: H(x) := \begin 1, & x \geq 0 \\ 0, & x * an indicator function: H(x) := \mathbf_=\mathbf 1_(x) For the al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meshfree Methods
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boundary Element Method
The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i.e. in ''boundary integral'' form), including fluid mechanics, acoustics, electromagnetics (where the technique is known as method of moments or abbreviated as MoM), fracture mechanics, and contact mechanics. Mathematical basis The integral equation may be regarded as an exact solution of the governing partial differential equation. The boundary element method attempts to use the given boundary conditions to fit boundary values into the integral equation, rather than values throughout the space defined by a partial differential equation. Once this is done, in the post-processing stage, the integral equation can then be used again to calculate numerically the solution directly at any desired point in the interior of the solution domain. BEM is applicable to problems for which Green's functions can be calculated. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Element Method
Finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. Computers are usually used to perform the calculations required. With high-speed supercomputers, better solutions can be achieved and are often required to solve the largest and most complex problems. FEM is a general numerical method for solving partial differential equations in two- or three-space variables (i.e., some boundary value problems). There are also studies about using FEM to solve high-dimensional problems. To solve a problem, FEM subdivides a large system into smaller, simpler parts called finite elements. This is achieved by a particular space discretization in the space dimensions, which is implemented by the construction of a mesh of the object: the numer ... [...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] |
Euler-Cauchy Stress Principle
In continuum mechanics, the Cauchy stress tensor (symbol \boldsymbol\sigma, named after Augustin-Louis Cauchy), also called true stress tensor or simply stress tensor, completely defines the state of stress (mechanics), stress at a point inside a material in the Deformation (engineering), deformed state, placement, or configuration. The second order tensor consists of nine components \sigma_ and relates a unit-length direction vector e to the ''traction vector'' T(e) across an imaginary surface perpendicular to e: :\mathbf^ = \mathbf e \cdot\boldsymbol\quad \text \quad T_^= \sum_\sigma_e_i. The SI base units of both stress tensor and traction vector are newton per square metre (N/m2) or pascal (unit), pascal (Pa), corresponding to the stress scalar. The unit vector is Dimensionless quantity, dimensionless. The Cauchy stress tensor obeys the Covariant transformation, tensor transformation law under a change in the system of coordinates. A graphical representation of this transform ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distributional Derivative
Distributions, also known as Schwartz distributions are a kind of generalized function in mathematical analysis. Distributions make it possible to derivative, differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions (weak solutions) than Solution of a differential equation, classical solutions, or where appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are singular, such as the Dirac delta function, Dirac delta function. A Function (mathematics), function f is normally thought of as on the in the function Domain (function), domain by "sending" a point x in the domain t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirac Delta Function
In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. Thus it can be Heuristic, represented heuristically as \delta (x) = \begin 0, & x \neq 0 \\ , & x = 0 \end such that \int_^ \delta(x) dx=1. Since there is no function having this property, modelling the delta "function" rigorously involves the use of limit (mathematics), limits or, as is common in mathematics, measure theory and the theory of distribution (mathematics), distributions. The delta function was introduced by physicist Paul Dirac, and has since been applied routinely in physics and engineering to model point masses and instantaneous impulses. It is called the delta function because it is a continuous analogue of the Kronecker delta function, which is usually defined on a discrete domain and takes values ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meshfree Methods
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 ... [...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] |
