mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

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, original ...

based on the

least squares The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the res ...

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 sy ...

, defined as the weighted summation of the squared residual of the governing

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

and its boundary conditions at nodal points used to discretize the

domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...

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, and by extension, the term is also sometimes used to describe the numerical solution of differential equations ...

, 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, a ...

least squares method to discretize the governing

. 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 ...

(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 th ...

. 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 is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...

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 In mathematics, a plane is a Euclidean (flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as s ...

shocked hyperbolic problems by Afshar and Firoozjaee. Also,

adaptive Adaptation, in biology, is the process or trait by which organisms or population better match their environment Adaptation may also refer to: Arts * Adaptation (arts), a transfer of a work of art from one medium to another ** Film adaptation, a ...

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 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 example, in ...

s are of the first order, 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 who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...

type, which is easily incorporated via a

penalty method 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 ideally converge to the solution o ...

. Because this is a least squares based

algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...

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