mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

the discrete least squares meshless (DLSM) method is a

meshless 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, origina ...

based on the

least squares The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The me ...

concept. The method is based on the minimization of a least squares

functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional s ...

, defined as the weighted summation of the squared residual of the governing differential equation and its boundary conditions at nodal points used to discretize the

domain A domain is a geographic area controlled by a single person or organization. Domain may also refer to: Law and human geography * Demesne, in English common law and other Medieval European contexts, lands directly managed by their holder rather ...

and its boundaries.

Description

While most of the existing meshless methods need background cells for

numerical integration In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral. The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integr ...

, DLSM did not require a numerical integration procedure due to the use of the

discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory * Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit * Discrete group, ...

least squares method to discretize the governing differential equation. A

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 r ...

(MLS) approximation method is used to construct the shape function, making the approach a fully least squares-based approach. Arzani and Afshar developed the DLSM method in 2006 for the solution of

Poisson's equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...

. Firoozjaee and Afshar proposed the collocated discrete least squares meshless (CDLSM) method to solve

elliptic In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in ...

partial differential equations, and studied the effect of the collocation points on the convergence and accuracy of the method. The method can be considered as an extension the earlier method of DLSM by the introduction of a set of collocation points for the calculation of the least squares functional. CDLSM was later used by Naisipour et al. to solve

elasticity Elasticity often refers to: *Elasticity (physics), continuum mechanics of bodies that deform reversibly under stress Elasticity may also refer to: Information technology * Elasticity (data store), the flexibility of the data model and the cl ...

problems regarding the irregular distribution of nodal points. Afshar and Lashckarbolok used the CDLSM method for the adaptive simulation of

hyperbolic Hyperbolic may refer to: * of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics ** Hyperbolic geometry, a non-Euclidean geometry ** Hyperbolic functions, analogues of ordinary trigonometric functions, defined u ...

problems. A simple a posteriori error indicator based on the value of the least squares functional and a node moving strategy was used and tested on 1-D hyperbolic problems. Shobeyri and Afshar simulated

free surface In physics, a free surface is the surface of a fluid that is subject to zero parallel shear stress, such as the interface between two homogeneous fluids. An example of two such homogeneous fluids would be a body of water (liquid) and the air in ...

problems using the DLSM method. The method was then extended for adaptive simulation of

two-dimensional A two-dimensional space is a mathematical space with two dimensions, meaning points have two degrees of freedom: their locations can be locally described with two coordinates or they can move in two independent directions. Common two-dimension ...

shocked hyperbolic problems by Afshar and Firoozjaee. Also, adaptive node-moving refinement and multi-stage node enrichment adaptive refinement are formulated in the DLSM for the solution of elasticity problems. Amani, Afshar and Naisipour.J. Amani, M.H.Afshar, M. Naisipour, Mixed Discrete Least Squares Meshless method for planar elasticity problems using regular and irregular nodal distributions, Engineering Analysis with Boundary Elements, 36, (2012) 894–902. proposed mixed discrete least squares meshless (MDLSM) formulation for solution of planar elasticity problems. In this approach, the differential equations governing the planar elasticity problems are written in terms of the stresses and displacements which are approximated independently using the same shape functions. Since the resulting governing

equation In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for ...

s are of the

first order In mathematics and other formal sciences, first-order or first order most often means either: * "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of high ...

, both the displacement and stress boundary conditions are of the

Dirichlet Johann Peter Gustav Lejeune Dirichlet (; ; 13 February 1805 – 5 May 1859) was a German mathematician. In number theory, he proved special cases of Fermat's last theorem and created analytic number theory. In Mathematical analysis, analysis, h ...

type, which is easily incorporated via a

penalty method In mathematical optimization, penalty methods are a certain class of algorithms for solving constrained optimization problems. A penalty method replaces a constrained optimization problem by a series of unconstrained problems whose solutions idea ...

. Because this is a least squares based

algorithm In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...

of the MDLSM method, the proposed method does not need to be satisfied by the Ladyzhenskaya– Babuška–Brezzi (LBB) condition.

Notes

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References

*H. Arzani, M.H. Afshar, Solving Poisson's equations by the discrete least square meshless method, WIT Transactions on Modelling and Simulation 42 (2006) 23–31. *M. H. Afshar, M. Lashckarbolok, Collocated discrete least square (CDLS) meshless method: error estimate and adaptive refinement, International Journal for Numerical Methods in Fluids 56 (2008) 1909–1928. *M. Naisipour, M. H. Afshar, B. Hassani, A.R. Firoozjaee, Collocation Discrete Least Square (CDLS) Method for Elasticity Problems. International Journal of Civil Engineering 7 (2009) 9–18.
A.R. Firoozjaee, M.H. Afshar, Discrete least squares meshless method with sampling points for the solution of elliptic partial differential equations. Engineering Analysis with Boundary Elements 33 (2009) 83–92.G. Shobeyri, M.H. Afshar, Simulating free surface problems using Discrete Least Squares Meshless method. Computers & Fluids 39 (2010) 461–470.M.H.Afshar, and A.R. Firoozjaee, Adaptive Simulation of Two Dimensional Hyperbolic Problems by Collocated Discrete Least Squares Meshless Method, Computer and Fluids, 39, (2010) 2030–2039.M.H.Afshar, M. Naisipour, J. Amani, Node moving adaptive refinement strategy for planar elasticity problems using discrete least squares meshless method, Finite Elements in Analysis and Design, 47, (2011) 1315–1325.M.H.Afshar, J. Amani, M. Naisipour, A node enrichment adaptive refinement by Discrete Least Squares Meshless method for solution of elasticity problems, Engineering Analysis with Boundary Elements, 36, (2012) 385–393.J. Amani, M.H.Afshar, M. Naisipour, Mixed Discrete Least Squares Meshless method for planar elasticity problems using regular and irregular nodal distributions, Engineering Analysis with Boundary Elements, 36, (2012) 894–902.
* ttp://scientiairanica.sharif.edu/article_4189.html Faraji, S. et al. (2018) Mixed discrete least squares meshless method for solving the linear and non-linear propagation problems Differential equations Least squares