Versatile anisotropic mesh adaptation methodology applied to pure quantity of interest error estimator. Steady,laminar incompressible flow

We introduce a new flexible mesh adaptation approach to efficiently compute a quantity of interest by the finite element method. Efficiently, we mean that the method provides an evaluation of that quantity up to a predetermined accuracy at a lower computational cost than other classical methods. The central pillar of the method is our scalar error estimator based on sensitivities of the quantity of interest to the residuals. These sensitivities result from the computation of a continuous adjoint problem. The mesh adaptation strategy can drive anisotropic mesh adaptation from a general scalar error contribution of each element. The full potential of our error estimator is then reached. The proposed method is validated by evaluating the lift, the drag, and the hydraulic losses on a 2D benchmark case: the flow around a cylinder at a Reynolds number of 20.     

An ALE finite element method for a coupled Stefan problem and Navier–Stokes equations with free capillary surface

In this article, an ALE finite element method to simulate the partial melting of a workpiece of metal is presented. The model includes the heat transport in both the solid and liquid part, fluid flow in the liquid phase by the Navier–Stokes equations, tracking of the melt interface solid/liquid by the Stefan condition, treatment of the capillary boundary accounting for surface tension effects and a radiative boundary condition. We show that an accurate treatment of the moving boundaries is crucial to resolve their respective influences on the flow field and thus on the overall energy transport correctly. This is achieved by a mesh‐moving method, which explicitly tracks the phase boundary and makes it possible to use a sharp interface model without singularities in the boundary conditions at the triple junction. A numerical example describing the welding of a thin‐steel wire end by a laser, where all aforementioned effects have to be taken into account, proves the effectiveness of the approach.Copyright © 2012 John Wiley & Sons, Ltd.     

Evaluation of RBFs for volume data interpolation in CFD

A greedy method for choosing an optimum reduced set of control points is integrated with RBF interpolation and evaluated for the purpose of interpolating large‐volume data sets in CFD. Given a function defined at a set of points, the greedy method selects a small subset of these points that is sufficient to keep the interpolation error at all the remaining points below a chosen bound. This is equivalent to a type of data compression and would have useful storage, post‐processing, and computational applications in CFD. To test the method in terms of both the point selection scheme and the suitability of reduced control point volume interpolation, a trial application of the interpolation to velocity fields in CFD volume meshes is considered. To optimise the point selection process, and attempt to be able to capture multiple length scales, a variable support radius formulation has also been included. Structured and unstructured mesh cases are considered for aerofoils, a wing case and a wing‐body case. For smooth volume functions, the method is shown to work well, producing accurate velocity interpolations using a very small number of the cells in the mesh. For general complex fields including large gradients, the method is still shown to be effective, although large gradients require more interpolation points to be used.Copyright © 2013 John Wiley & Sons, Ltd.     

A meshless finite point method for three‐dimensional analysis of compressible flow problems involving moving boundaries and adaptivity

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. Copyright © 2013 John Wiley & Sons, Ltd.     

Strong Convergence Theorems for the Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in the Intermediate Sense in Hilbert Spaces

Abstract

The aim of this paper is to introduce the generalized viscosity implicit rules of one asymptotically nonexpansive mapping in the intermediate sense in Hilbert spaces. We obtain some strong convergence theorems under certain assumptions imposed on the parameters. We also give a numerical example to support our main results. The results obtained in this paper improve and extend many recent ones in this culture.     

Moving grid method for numerical simulation of stratified flows

We develop a second‐order accurate Navier–Stokes solver based on r‐adaptivity of the underlying numerical discretization. The motion of the mesh is based on the fluid velocity field; however, certain adjustments to the Lagrangian velocities are introduced to maintain quality of the mesh. The adjustments are based on the variational approach of energy minimization to redistribute grid points closer to the areas of rapid solution variation. To quantify the numerical diffusion inherent to each method, we monitor changes in the background potential energy, computation of which is based on the density field. We demonstrate on a standing interfacial gravity wave simulation how using our method of grid evolution decreases the rate of increase of the background potential energy compared with using the same advection scheme on the stationary grid. To further highlight the benefit of the proposed moving grid method, we apply it to the nonhydrostatic lock‐exchange flow where the evolution of the interface is more complex than in the standing wave test case. Naive grid evolution based on the fluid velocities in the lock‐exchange flow leads to grid tangling as Kelvin–Helmholtz billows develop at the interface. This is remedied by grid refinement using the variational approach. Copyright © 2012 John Wiley & Sons, Ltd.     

Three‐dimensional wave impact on a rigid structure using smoothed particle hydrodynamics

Navier–Stokes computations of a wave–structure interaction are performed with the aim of assessing the potential of smoothed particle hydrodynamics to accurately estimate impact loading time history. A three‐dimensional dam‐break flow with a rectangular column located downstream is considered. The net force and impulse exerted on the column is monitored throughout the simulation with the results correlating well with existing experimental data. Initial and boundary conditions and numerical parameters are varied and their effect on the column load investigated. The column load is found to be most sensitive to the choice of boundary treatment. Copyright © 2011 John Wiley & Sons, Ltd.     

Hybrid mesh adaptation applied to industrial numerical combustion

This paper presents an anisotropic mesh adaptation method applied to industrial combustion problems. The method is based on a measure of the distance between two Riemannian metrics called metric non‐conformity. This measure, which can be used to build a cost function to adapt meshes comprising several types of mesh elements, provides the basis for a generic mesh adaptation approach applicable to various types of physical problems governed by partial differential equations. The approach is shown to be applicable to industrial combustion problems, through the specification of a target metric computed as the intersection of several Hessian matrices reconstructed from the main variables of the governing equations. Numerical results show that the approach is cost effective in that it can drastically improve the prediction of temperature and species distributions in the flame region of a combustor while reducing computational cost. The results can be used as a basis for pollutant prediction models. Copyright © 2011 John Wiley & Sons, Ltd.     

Impact of tetrahedralization on parallel conforming octree mesh generation

This work discusses the performance impact from the tetrahedralization of non‐conforming meshes generated by a parallel octree method capable of handling arbitrary immersed geometries. First, we review conforming techniques for meshes created from octrees. Then we implement a tetrahedralization scheme based on templates in a parallel meshing algorithm that uses a linear octree with 2:1 balancing constraint. Besides, we also propose a change on the partitioning strategy for the same meshing algorithm to improve the octree refinement load balancing. Scalability analyses show that the chosen tetrahedralization technique preserves algorithm performance. Copyright © 2014 John Wiley & Sons, Ltd.     

Minimum-norm fixed point of nonexpansive mappings with applications

The purpose of this paper is to present two new iterative algorithms to find the minimum norm fixed point of nonexpansive nonself-mappings in the framework of Hilbert spaces. The results presented in this paper improve and extend the corresponding ones announced by Yao and Xu [Yao Y, Xu HK. Iterative methods for finding minimum norm fixed point of mappings with applications. Optimization. 2011;60:645–658] and many others. Some applications to convex minimization and split feast problems(SFP) are also included.