Abstract

An adaptive Finite Point Method (FPM) for solving shallow water problems is presented. The numerical methodology we propose, which is based on weighted‐least squares approximations on clouds of points, adopts an upwind‐biased discretization for dealing with the convective terms in the governing equations. The viscous and source terms are discretized in a pointwise manner and the semi‐discrete equations are integrated explicitly in time by means of a multi‐stage scheme. Moreover, with the aim of exploiting meshless capabilities, an adaptive h‐refinement technique is coupled to the described flow solver. The success of this approach in solving typical shallow water flows is illustrated by means of several numerical examples and special emphasis is placed on the adaptive technique performance. This has been assessed by carrying out a numerical simulation of the 26th December 2004 Indian Ocean tsunami with highly encouraging results. Overall, the adaptive FPM is presented as an accurate enough, cost‐effective tool for solving practical shallow water problems. Copyright © 2011 John Wiley & Sons, Ltd.

Abstract

The finite point method (FPM) is a meshless technique, which is based on both, a weighted least‐squares numerical approximation on local clouds of points and a collocation technique which allows obtaining the discrete system of equations. The research work we present is part of a broader investigation into the capabilities of the FPM to deal with 3D applications concerning real compressible fluid flow problems. In the first part of this work, the upwind‐biased scheme employed for solving the flow equations is described. Secondly, with the aim of exploiting the meshless capabilities, an h‐adaptive methodology for 2D and 3D compressible flow calculations is developed. This adaptive technique applies a solution‐based indicator in order to identify local clouds where new points should be inserted in or existing points could be safely removed from the computational domain. The flow solver and the adaptive procedure have been evaluated and the results are encouraging. Several numerical examples are provided in order to illustrate the good performance of the numerical methods presented. 

Abstract

A finite point method for solving compressible flow problems involving moving boundaries and adaptivity is presented. The numerical methodology is based on an upwind‐biased discretization of the Euler equations, written in arbitrary Lagrangian–Eulerian form and integrated in time by means of a dual‐time steeping technique. In order to exploit the meshless potential of the method, a domain deformation approach based on the spring network analogy is implemented, and h‐adaptivity is also employed in the computations. Typical movable boundary problems in transonic flow regime are solved to assess the performance of the proposed technique. In addition, an application to a fluid–structure interaction problem involving static aeroelasticity illustrates the capability of the method to deal with practical engineering analyses. The computational cost and multi‐core performance of the proposed technique is also discussed through the examples provided.

Abstract

This paper presents a methodology for solving shape optimization problems in the context of fluid flow problems including adaptive remeshing. The method is based on the computation of the sensitivities of the geometrical design parameters, the mesh, the flow variables and the error estimator to project the refinement parameters from one design to the next. The efficiency of the proposed method is checked out in two 2D optimization problems using a full potential model coupled with a boundary layer model. The first one corresponds to an internal flow problem and the second one corresponds to an external flow problem. The work presented here can be considered as a continuation of previous work (Bugeda and Oñate, Int. J. Numer. Methods Fluids, 20, 915–924 (1995)). In that paper, the sensitivity analysis corresponding to both an incompressible potential flow and a Euler flow were derived together with a strategy for the adaptive remeshing. In the present paper, the sensitivity analysis is derived for a full potential flow coupled with a boundary layer model, and a new error estimator is employed.

Abstract

A mixed hierarchical approximation based on finite elements and meshless methods is presented. Two cases are considered. The first one couples regions where finite elements or meshless methods are used to interpolate: continuity and consistency is preserved. The second one enriches a finite element mesh with processes. In both cases the same formulation is used, convergence is studied and examples are shown.

Abstract

Two main ingredients are needed for adaptive finite element computations. First, the error of a given solution must be assessed, by means of either error estimators or errors indicators. After that, a new spatial discretization must be defined via h, p or r-adaptivity. In principle, any of the approaches for error assessment may be combined with any of the procedures for adapting the discretization. However, some combinations are elearly preferable. The advantages and limitations of the various alternatives are discussed. The most adequate strategies are illustrated by means of several applications in solid mechanics.  

Abstract

The present paper proposes a new technique for the definition of the shape design variables in 2D and 3D optimisation problems. It can be applied to the discrete model of the analysed structure or to the original geometry without any previous knowledge of the analytical expression of the CAD defining surfaces. The proposed technique allows the surface continuity to be preserved during the geometry modification process to be defined a priori. This capability allows for the definition of shape variables suitable for every kind of discipline involved in the optimisation process (structural analysis, fluid-dynamic analysis, crash analysis, aerodynamic analysis, etc.).

Abstract

In this work we present a general error estimator for the finite element solution of solid mechanics problems based on the Variational Multiscale method. The main idea is to consider a rich model for the subgrid scales as an error estimator. The subscales are considered to belong to a space orthogonal to the finite element space (Orthogonal Subgrid Scales) and we take into account their contribution both in the element interiors and on the element boundaries (Subscales on the Element Boundaries). A simple analysis shows that the upper bound for the obtained error estimator is sharper than in other error estimators based on the Variational Multiscale Method. Numerical examples show that the proposed error estimator is an accurate approximation for the energy norm error and can be used both in simple linear constitutive models and in more complex non-linear cases.

Abstract

Two implicit residual type estimators yielding upper bounds of the error are presented which do not require flux equilibration. One of them is based on the ideas introduced in [MAC 00, CAR 99, MOR 03, PRU 02]. The new approach introduced
here is based on using the estimated error function rather than the estimated error norms. Once the upper bounds are computed, also lower bounds for the error are obtained with little supplementary effort.

Abstract

This paper presents a formulation for the obtainment of the sensitivity analysis of a point wise error estimator with respect to the nodal coordinates using the adjoint state method. The proposed point wise error estimator is based on the SPR method.

The numerical accuracy of the presented sensitivity analysis has been tested by perturbing a mesh. The capability of the presented sensitivity analysis for the detection of pollution error has also been tested.

A new adaptive remeshing strategy based on the sensitivity analysis of the point wise error estimation has been developed and tested. This strategy produces very cheap meshes for the accurate evaluation of stresses at specific points.