By G.R. Liu
This booklet goals to give meshfree equipment in a pleasant and simple demeanour, in order that newbies can with ease comprehend, understand, application, enforce, follow and expand those tools. It presents first the basics of numerical research which are really vital to meshfree equipment. average meshfree tools, similar 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 hooked up with priceless descriptions. the applying of the codes may be comfortably played utilizing the examples with enter and output records given in desk shape. those codes include lots of the easy meshfree ideas, and will be simply prolonged to different diversifications of extra complicated strategies of meshfree equipment. Readers can simply perform with the codes supplied to potent examine and understand the fundamentals of meshfree tools.
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Extra info for An Introduction to Meshfree Methods and Their Programming
1 Meshfree methods based on weak-forms These are called MFree weak-form methods in this book. In MFree weakform methods, the governing partial differential equations (PDEs) with derivative boundary conditions are first transformed to a set of so-called weakform integral equations using different techniques discussed in Chapter 1. The weak-forms are then used to derive a set of algebraic system equations through a numerical integration process using sets of background cells that may be constructed globally or locally in the problem domain.
The weighted residual methods will possess the convergence property, meaning that the approximate solution of the weighted residual methods will approach the exact solution when the number of the basis functions used increases, as long as 1) The weight functions Wi , Vi and the basis functions linearly independent. 2) The basis functions i( i( ) are ) have a certain order of continuity. 3) The weight functions and the basis functions have a certain degree of overlapping. The simple example solved using these five different methods (collocation, subdomain, moment, least squares and Galerkin) confirmed the convergence property.
The density of the nodes depends on the accuracy required and resources Chapter 2 42 available. The nodal distribution is usually not uniform. Since adaptive algorithms can be used in MFree methods, the density is eventually controlled automatically and adaptively in the code; the initial nodal distribution becomes not important. An MFree method should be able to work for an arbitrary nodal distribution. In the finite element method, this step is different: meshing needs to be performed to discretize the geometry and create the elements.
An Introduction to Meshfree Methods and Their Programming by G.R. Liu