Perazzo Maggi, Franco OrlandoCOOPER VILLAGRAN, CHRISTOPHERMarchant Jara, Felipe Andrés2024-10-022024-10-022017https://repositorio.usm.cl/handle/123456789/19526Catalogado desde la version PDF de la tesis.In the past years, in the context of the meshfree method (MF), it has beenestablished a clear predominance of the methods based on a Weak or GalerkinFormulation (WF) over the methods based on a strong formulation (SF) (Liu,2003; Li and Liu, 2004; Liu and Gu, 2005). This has been mainly motivatedfor the fact that the WF has resulted in a more stable methodology, andthe only problem present in these methods are the scheme and domain ofintegration (Duan et al., 2012; Chen et al., 2013). In the case of SF the situationis completely different, because they have had to deal with the problem ofinstability and not robustness. One of the major factors responsible for this, isthe differential operator which is characterized as an error amplifier. Anotherdrawback of such methods is the difficulty of imposed the Neumann BoundaryCondition or Derivative condition, which involve another equations groupdifferent that those obtained for the domain . This problem has been reportedfor many authors (Patankar, 1980; Belytschko et al., 1994; Liu et al., 1997;Li and Liu, 2004; Liu and Gu, 2005). Among the solutions proposed to dealwith these issues of instabilities on SF, the main line of research is how the localsubdomains for the approximation are built, which has led to positive results insome particular case (O˜nate et al., 2001; Perazzo et al., 2008). We present inthis work a different type of solution consistent in the use of a different shapefunction instead of the traditional weight least squares (WLS) used on SF. Toobtain this shape function the Maximum Entropy Principle (MAXENT) is used(Jaynes, 1957a,b). The MAXENT is used in multiple fields, including graphicscomputer, geometric modelling, image processing and supervised learning. Thisshape function were obtained for Sukumar that established the relation betweenthe convex problem of the maximum-entropy (maxent) statistical inferenceand the meshfree basis functions with first order consistency (Sukumar, 2004;Sukumar and Wright, 2007). In the context of the Galerkin-based meshfreemethods these basis functions were used for compressible and near-incompressibleelasticity problems (Ortiz et al., 2010). The main interesting features of themaxent shape functions for SF, is the fact that their values are always positiveand smooth, in addition to have a reducing property on the boundary ofthe domain(Arroyo and Ortiz, 2006). Others secondary features, described in(Cyron et al., 2009) and not less relevant, are the variation diminishing property (this mean that the approximation do not create an extreme value) and themonotonicity property. These things are important because at least the firstpositivity criterion exposed in (Xiaozhong et al., 2004) for shape functions usedon SF is fulfilled, situation that not happen in the case of traditionalWeight LeastSquare (WLS). To prove the advantage of this alternative, in this work we usedthe maxent shape function in a SF meshless method, using the formulation forfirst and second order consistency proposed in the works of (Sukumar and Wright,2007) and (Cyron et al., 2009) respectively. I illustrate the performance of themethodology by mean four examples. There are two one-dimensional PDE valueproblem with Dirichlet and Neumman boundary condition, and in a 2D-domainother two problems: the first is a Poisson Equation using multiples source, andthe second is a cantilever beam that is working below Timoshenko solution.CD ROMFORMA MAXENTFORMULACION FUERTEFUNCIONES DE FORMALIBRE DE MALLAMESHFREETesis PregradoB - Solamente disponible para consulta en sala (opción por defecto)3560900257383