By G.R. Liu
This booklet goals to offer meshfree equipment in a pleasant and simple demeanour, in order that novices can comfortably comprehend, understand, application, enforce, observe and expand those equipment. It offers first the basics of numerical research which are rather very important to meshfree tools. ordinary meshfree tools, equivalent to EFG, RPIM, MLPG, LRPIM, MWS and collocation tools are then brought systematically detailing the formula, numerical implementation and programming. Many well-tested desktop resource codes constructed via the authors are connected with invaluable descriptions. the applying of the codes will be without difficulty played utilizing the examples with enter and output documents given in desk shape. those codes encompass many of the simple meshfree strategies, and will be simply prolonged to different diversifications of extra advanced approaches of meshfree tools. Readers can simply perform with the codes supplied to powerful examine and understand the fundamentals of meshfree equipment.
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Extra resources for An Introduction to Meshfree Methods and Their Programming
Readers are also encouraged to obtain the solution using more than 3 terms. 98) 27 1. 68). 69). The same conclusion can be drawn when more than 3 terms are used. For quantitative analysis, the following norm is defined as the error indicator. 100) where u(( j ) num and u(( j )exact are, respectively, displacements at point xj ( =1,2, …, N (j N) obtained using the numerical methods and the analytical method, N is the number of uniform points used to study the error, and N=21 N is used here. 10 plots the convergence curves of different weighted residual methods using different terms in the approximate solution.
Note that to obtain the 21 1. 45) is often used, in which only the residual for the governing equation is considered. The boundary conditions (BCs) are treated separately for the essential BCs and the natural BCs. The former is handled after obtaining the discretized system equations, and the latter is implemented after performing integration by parts. This procedure will also be followed in forming the MFree weak-form methods (Chapters 4 and 5). 6 Examples In order to illustrate these approximation methods, consider a simple example problem of a truss member.
The integrand in the first integral term can be written as follows using the chain rule of variation. 110) 33 1. 112) *t The minimum total potential energy principle requires G Ȇ 0 . 113) can also be viewed as the principle of virtual work, which states that if a solid body is in its equilibrium states, the total virtual work performed by all the stresses in the body and all the external forces applied on the body vanishes, when the body is subjected to a virtual displacement. The virtual work can be viewed as an alternative statement of equilibrium equation.
An Introduction to Meshfree Methods and Their Programming by G.R. Liu