A stabilized meshless method for the solution of the lagrangian equations of newtonian fluids

En este artículo, presentamos métodos numéricos para resolver problemas de fluidos incompresibles. Tres son los problemas que se abordan: el flujo de Stokes estacionario, el flujo de Stokes transitorio y el problema general del movimiento de los fluidos newtonianos. En los tres casos, se emplea una discretización que no requiere una malla del dominio, sino que se utilizan funciones de aproximación de la máxima entropía. Además, para garantizar la robustez de la solución, se emplea una técnica de estabilización. El problema más general, el del movimiento de fluidos newtonianos, se formula de forma lagrangiana y, con los resultados presentados, se comprueba que el uso de métodos sin malla estabilizados puede ser una alternativa competitiva a otras técnicas que se emplean en la actualidad.In this article we present numerical methods for the approximation of incompressible flows. We have addressed three problems: the stationary Stokes’ problem, the transient Stokes’ problem, and the general motion of newtonian fluids. In the three cases a discretization is employed that does not require a mesh of the domain but uses maximum entropy approximation functions. To guarantee the robustness of the solution a stabilization technique is employed. The most general problem, that of the motion of newtonian fluids, is formulated in lagrangian form. The results presented verify that stabilized meshless methods can be a competitive alternative to other approached currently in use.Peer Reviewe

A Pressure-Stabilized Lagrange-Galerkin Method in a Parallel Domain Decomposition System

A pressure-stabilized Lagrange-Galerkin method is implemented in a parallel domain decomposition system in this work, and the new stabilization strategy is proved to be effective for large Reynolds number and Rayleigh number simulations. The symmetry of the stiffness matrix enables the interface problems of the linear system to be solved by the preconditioned conjugate method, and an incomplete balanced domain preconditioner is applied to the flow-thermal coupled problems. The methodology shows good parallel efficiency and high numerical scalability, and the new solver is validated by comparing with exact solutions and available benchmark results. It occupies less memory than classical product-type solvers; furthermore, it is capable of solving problems of over 30 million degrees of freedom within one day on a PC cluster of 80 cores

A stabilized formulation with maximum entropy meshfree approximants for viscoplastic flow simulation in metal forming

The finite element method is the reference technique in the simulation of metal forming and provides excellent results with both Eulerian and Lagrangian implementations. The latter approach is more natural and direct but the large deformations involved in such processes require remeshing-rezoning algorithms that increase the computational times and reduce the quality of the results. Meshfree methods can better handle large deformations and have shown encouraging results. However, viscoplastic flows are nearly incompressible, which poses a challenge to meshfree methods. In this paper we propose a simple model of viscoplasticity, where both the pressure and velocity fields are discretized with maximum entropy approximants. The inf-sup condition is circumvented with a numerically consistent stabilized formulation that involves the gradient of the pressure. The performance of the method is studied in some benchmark problems including metal forming and orthogonal cutting

Penalty-free Nitsche method for interface problems in computational mechanics

Nitsche’s method is a penalty-based method to enforce weakly the boundary conditions in the finite element method. In this thesis, we consider a penalty-free version of Nitsche’s method, we prove its stability and convergence in various frameworks. The idea of the penalty-free method comes from the nonsymmetric version of the Nitsche’s method where the penalty parameter has been set to zero; it can be seen as a Lagrange multiplier method, where the Lagrange multiplier has been replaced by the boundary fluxes of the discrete elliptic operator. The main observation is that although coercivity fails, inf-sup stability can be proven. The study focuses on compressible and incompressible elasticity. An unfitted framework is considered when the computational mesh does not fit with the physical domain (fictitious domain method). The penalty-free Nitsche’s method is also used to enforce the coupling for interface problems when the mesh fits the interface (nonconforming domain decomposition) or not (unfitted domain decomposition). Fluid structure interaction is also investigated, a new fully discrete implicit scheme is introduced

Computer Science Research Institute 2004 annual report of activities.

This report summarizes the activities of the Computer Science Research Institute (CSRI) at Sandia National Laboratories during the period January 1, 2004 to December 31, 2004. During this period the CSRI hosted 166 visitors representing 81 universities, companies and laboratories. Of these 65 were summer students or faculty. The CSRI partially sponsored 2 workshops and also organized and was the primary host for 4 workshops. These 4 CSRI sponsored workshops had 140 participants--74 from universities, companies and laboratories, and 66 from Sandia. Finally, the CSRI sponsored 14 long-term collaborative research projects and 5 Sabbaticals

APPLICATION OF NODE BASED SMOOTHED POINT INTERPOLATION METHODS IN SMALL AND LARGE DEFORMATION PROBLEMS OF GEOMECHANICS

This thesis aims to study the application of node-based smoothed point interpolation methods (NSPIMs) in small and large deformation problems of geomechanics. NSPIMs are a sub-class of the recently developed family of smoothed point interpolation methods (SPIMs) in which the gradient smoothing technique is utilised in the framework of the generalised smoothed Galerkin method. NSPIMs require background mesh in their formulations, however, unlike traditional mesh-basedmethods like the finite element method (FEM), NSPIMs’ solutions are not heavily dependent on the quality of the background mesh and they maintain desirable features of mesh-based methods such as simplicity of imposing boundary conditions. In NSPIMs, all the calculations and numerical integrations are conducted over the nodes of the discretised domain, and all the parameters and variables are also calculated and stored at the nodes of the domain. Such a node-basedformulation renders NSPIMs suitable for a variety of small and large deformation problems of geomechanics which are often accompanied by complex constitutive modellings and considerable changes in the geometry during the analysis. The contributions made in this thesis include: i) providing theoretical and numerical discussions on the softness of the behaviour of NSPIMs compared to the FEM with linear triangular elements in coupled problems of geomechanics; ii) developing a novel and highly accurate hybrid NSPIM-FEM for geotechnical engineering applications, exploiting the contradicting features of the solutions of the both numerical techniques; iii) developing an improved formulation of NSPIMs for coupled flow-deformation problems in porous media to overcome the numerical instabilities associated with the original NSPIMs; and iv) developing a particle-based numerical method formulated based on NSPIMs for large deformation problems of geomechanics, extending the efficiency and suitability of this numerical method to a wide range of geotechnical engineering problems