A multi‐level adaptive mesh refinement method for level set simulations of multiphase flow on unstructured meshes

An adaptive mesh refinement (AMR) technique is proposed for level set simulations of incompressible multiphase flows. The present AMR technique is implemented for two‐dimensional/three‐dimensional unstructured meshes and extended to multi‐level refinement. Smooth variation of the element size is guaranteed near the interface region with the use of multi‐level refinement. A Courant–Friedrich–Lewy condition for zone adaption frequency is newly introduced to obtain a mass‐conservative solution of incompressible multiphase flows. Finite elements around the interface are dynamically refined using the classical element subdivision method. Accordingly, finite element method is employed to solve the problems governed by the incompressible Navier–Stokes equations, using the level set method for dynamically updated meshes. The accuracy of the adaptive solutions is found to be comparable with that of non‐adaptive solutions only if a similar mesh resolution near the interface is provided. Because of the substantial reduction in the total number of nodes, the adaptive simulations with two‐level refinement used to solve the incompressible Navier–Stokes equations with a free surface are about four times faster than the non‐adaptive ones. Further, the overhead of the present AMR procedure is found to be very small, as compared with the total CPU time for an adaptive simulation. Copyright © 2016 John Wiley & Sons, Ltd.     

A new preconditioning technique for implicitly coupled multidomain simulations with applications to hemodynamics

In cardiovascular blood flow simulations a large portion of computational resources is dedicated to solve the linear system of equations. Boundary conditions in these applications are critical for obtaining accurate and physiologically realistic solutions, and pose numerical challenges due to the coupling between flow and pressure. Using an implicit time integration setting can lead to an ill-conditioned tangent matrix that causes deterioration in performance of traditional iterative linear equation solvers (LS). In this paper we present a novel and efficient preconditioner (PC) for this class of problems that exploits the strong coupling between the flow and pressure. We implement this PC in a LS algorithm designed for solving systems of equations governing incompressible flows. Excellent efficiency and stability properties of the proposed method are illustrated on a set of clinically relevant hemodynamics simulations.     

Cartesian网格结合无网格法处理的二维柱体

本文发展了基于四叉树数据结构的网格生成和二维流动的N-S方程数值求解器及动边界问题的Euler方程求解方法。采用压力梯度或者密度梯度的绝对值作为网格自适应的控制参量,同时采用基于最小二乘法的无网格方法处理对于一般Cartesian网格难于处理的物面边界条件。文中采取了绕方柱流动和绕圆柱流动的经典算例对所发展的方法进行了验证。计算的结果验证了所发展的方法在处理绕流流动时的合理性和有效性。采用Naca0012翼型的几种工况验证了所发展的动网格技术在处理无粘流动的合理性和可行性。从而为数值模拟具有复杂几何外形的流动提供了一种网格布局合理、高效,边界处理简单易行的新思路。     

An AMG Preconditioner for Solving the Navier-Stokes Equations with a Moving Mesh Finite Element Method

AMG preconditioners are typically designed for partial differential equation solvers and divergence-interpolation in a moving mesh strategy. Here we introduce an AMG preconditioner to solve the unsteady Navier-Stokes equations by a moving mesh finite element method. A $4P1$ − $P1$ element pair is selected based on the data structure of the hierarchy geometry tree and two-layer nested meshes in the velocity and pressure. Numerical experiments show the efficiency of our approach.     

Computations of high‐speed compressible flows with adaptive cell‐centered finite element method

Abstract

An upwind cell‐centered finite element formulation is combined with an adaptive meshing technique to solve Navier‐Stokes equations for high‐speed inviscid and viscous compressible flows. The finite element formulation and the computational procedure are described. An adaptive meshing technique is applied to increase the analysis solution accuracy, as well as to minimize the computational time and the computer memory requirement. The efficiency of the combined method is evaluated by the examples of Mach 2.6 inviscid flow in a channel with compression and expansion ramps, Mach 6.47 inviscid and viscous flows past a cylinder, and Mach 4 viscous flow over a flat plate.     

Boltzmann schemes for continuum gas dynamics

Many problems arising in the aerodynamic design of aerospace vehicles require the numerical solution of the Euler equations of gas dynamics. These are nonlinear partial differential equations admitting weak solutions such as shock waves and constructing robust numerical schemes for these equations is a challenging task. A new line of research called Boltzmann or kinetic schemes discussed in the present paper exploits the connection between the Boltzmann equation of the kinetic theory of gases and the Euler equations for inviscid compressible flows. Because of this connection, a suitable moment of a numerical scheme for the Boltzmann equation yields a numerical scheme for the Euler equations. This idea called the “moment method strategy” turns out to be an extremely rich methodology for developing robust numerical schemes for the Euler equations. The richness is demonstrated by developing a variety of kinetic schemes such as kinetic numerical method, kinetic flux vector splitting method, thermal velocity based splitting, multidirectional upwind method and least squares weak upwind scheme. A 3-D time-marching Euler code calledbheema based on the kinetic flux vector splitting method and its variants involving equilibrium chemistry have been developed for computing hypersonic reentry flows. The results obtained from the codebheema demonstrate the robustness and the utility of the kinetic flux vector splitting method as a design tool in aerodynamics. The work presented in this paper is based on the research work done by several graduate students at our laboratory and collaborators from research and development organizations within the country.     

Solvers for large-displacement fluid–structure interaction problems: segregated versus monolithic approaches

We compare the relative performance of monolithic and segregated (partitioned) solvers for large- displacement fluid–structure interaction (FSI) problems within the framework of oomph-lib, the object-oriented multi-physics finite-element library, available as open-source software at . Monolithic solvers are widely acknowledged to be more robust than their segregated counterparts, but are believed to be too expensive for use in large-scale problems. We demonstrate that monolithic solvers are competitive even for problems in which the fluid–solid coupling is weak and, hence, the segregated solvers converge within a moderate number of iterations. The efficient monolithic solution of large-scale FSI problems requires the development of preconditioners for the iterative solution of the linear systems that arise during the solution of the monolithically coupled fluid and solid equations by Newton’s method. We demonstrate that recent improvements to oomph-lib’s FSI preconditioner result in mesh-independent convergence rates under uniform and non-uniform (adaptive) mesh refinement, and explore its performance in a number of two- and three-dimensional test problems involving the interaction of finite-Reynolds-number flows with shell and beam structures, as well as finite-thickness solids.     

High-performance parallel computing for incompressible flow simulations

High-performance parallel computer systems have been employed to compute a variety of three-dimensional incompressible fluid flows. Three different numerical methods have been used for the discretization of the Navier-Stokes equations, with domain decomposition techniques employed for the parallel resolution of the discretized equations. These parallel flow solvers have been applied to the numerical simulation of different flows, ranging from basic flow studies to industrial applications. The results of these studies show that high-performance parallel computing has evolved from its initial investigative phase into a mature technology that can be employed for large-scale numerical flow simulation.     

Iterative techniques for 3-D boundary element method systems of equations

A key issue in the boundary element method (BEM) is the solution of the associated system of algebraic equations whose matrices are dense, nonsymmetric and sometimes ill conditioned. For large scale tridimensional problems, direct methods like Gauss elimination become too expensive and iterative methods may be preferable. This paper presents a comparison of the performances of some iterative techniques based on conjugate gradient solvers as conjugate gradient squared (CGS) and bi-conjugate gradient (Bi-CG) that seem to have the potential to be efficient and competitive for BEM algebraic systems of equations, specially when used with an appropriate preconditioner. A comparison with the direct application of the conjugate gradient method to the normalized systems of equations (CGNE and CGNR) is also presented.     

Numerical solution of the kinetic model equations for hypersonic flows

A numerical method for solving the model kinetic equations for hypersonic flows has been developed. The model equations for the distribution function are discretized in phase space using a second order upwind finite difference scheme for the spatial derivatives. The resulting system of ordinary differential equations in time is integrated by using a rational Runge-Kutta scheme. Calculations were carried out for hypersonic flow around a double ellipse under various free stream conditions. Calculated results are compared with the Navier-Stokes solutions and the Direct Simulation Monte Carlo (DSMC) method for the corresponding case. The agreement is quite excellent in general.     

Optimised electromagnetic 3d field solver for frequencies below the first resonance

An optimised low-frequency solution algorithm is proposed that reduces the electromagnetic wave equation to a small number of scalar potential problems. The latter can be solved using black-box algebraic multigrid solvers. Within the framework of the finite integration technique, the reduction of the wave equation is accomplished by tree/cotree decompositions of primary and dual grids, which allow for highly efficient solutions of Ampere's and Faraday's equations. The example of a ferrite-loaded accelerator component demonstrates that the scheme is much more efficient than the direct approach using Krylov subspace solvers     

Parallel implementation of the boundary element method for linear elastic problems on a MIMD parallel computer

The parallel implementation of the boundary element method (BEM) for linear elastic 2D-problems on a MIMD parallel computer is treated. The parallelization is performed by data decomposition. The use of collocation method leads to a non-symmetric system of linear equations which is solved by direct or iterative methods. Focusing on the parallel implementation, these solvers are compared with respect to their efficiency using a Parsytec MultiCluster2 with 32 T805 transputers.     

Conjugate gradient methods for three-dimensional BEM systems of equations

A fundamental advantage of the boundary element method (BEM) is that the dimensionality of the problems is reduced by one. However, this advantage has to be weighted against the difficulty in solving the resulting systems of algebraic linear equations whose matrices are dense, non-symmetric and sometimes ill conditioned. For large three-dimensional problems the application of the classical direct methods becomes too expensive.This paper studies the comparative performance of iterative techniques based on conjugate gradient solvers as bi-conjugate gradient (Bi-CG), generalized minimal residual (GMRES), conjugate gradient squared (CGS), quasi-minimal residuals (QMR) and bi-conjugate gradient stabilized (Bi-CGStab) for potential and exterior problems. Preconditioning is also considered and assessed.Two examples, one from electrostatics and other from fluid mechanics, were employed to test these methods, which proved to be effective and competitive as solvers for BEM linear algebraic systems of equations.     

Probabilistic crack growth analyses using a boundary element model: Applications in linear elastic fracture and fatigue problems

This paper addresses the numerical solution of random crack propagation problems using the coupling boundary element method (BEM) and reliability algorithms. Crack propagation phenomenon is efficiently modelled using BEM, due to its mesh reduction features. The BEM model is based on the dual BEM formulation, in which singular and hyper-singular integral equations are adopted to construct the system of algebraic equations. Two reliability algorithms are coupled with BEM model. The first is the well known response surface method, in which local, adaptive polynomial approximations of the mechanical response are constructed in search of the design point. Different experiment designs and adaptive schemes are considered. The alternative approach direct coupling, in which the limit state function remains implicit and its gradients are calculated directly from the numerical mechanical response, is also considered. The performance of both coupling methods is compared in application to some crack propagation problems. The investigation shows that direct coupling scheme converged for all problems studied, irrespective of the problem nonlinearity. The computational cost of direct coupling has shown to be a fraction of the cost of response surface solutions, regardless of experiment design or adaptive scheme considered.     

Viscous-inviscid interaction schemes for external aerodynamics

For aerofoils a calculation, which involves the coupling of the external inviscid flow with the viscous flow in the boundary layer and the wake, still provides a worthwhile alternative to the solution of the ‘time-averaged’ Navier-Stokes equations. Classical viscous-inviscid interaction methods which can be extended to include flows with separations and significant pressure gradients across the boundary layer are described. Basic theoretical principles of interactive methods in two dimensions are discussed. The extension of the classical methods leads to generalisations of the concept of displacement thickness and the momentum integral equation. The boundary conditions for the equivalent inviscid flow (EIF) are also described and these also include the effect of normal pressure gradients. An integral method based on the lag-entrainment method for the calculation of the turbulent boundary layer is described. The correlations associated with the method are extended to include separated flow. Two methods of solving the boundary-layer equations through a separation region are described: the inverse method and the quasi-simultaneous method. Principles of techniques for coupling the flows are described and the properties of the direct, fully inverse, semi-inverse and quasi-simultaneous methods are discussed. Results from a method for incompressible flow about a stalled aerofoil, a method for compressible flow about a high-lift aerofoil and a method for compressible flow about a transonic aerofoil are compared with experimental results. The current situation regarding the development of viscous-inviscid interaction methods is briefly summarized and future possibilities are considered.     

The method of fundamental solutions for solving incompressible Navier–Stokes problems

A novel meshless numerical procedure based on the method of fundamental solutions (MFS) is proposed to solve the primitive variables formulation of the Navier–Stokes equations. The MFS is a meshless method since it is free from the mesh generation and numerical integration. We will transform the Navier–Stokes equations into simple advection–diffusion and Poisson differential operators via the operator-splitting scheme or the so-called projection method, instead of directly using the more complicated fundamental solutions (Stokeslets) of the unsteady Stokes equations. The resultant velocity advection–diffusion equations and the pressure Poisson equation are then calculated by using the MFS together with the Eulerian–Lagrangian method (ELM) and the method of particular solutions (MPS). The proposed meshless numerical scheme is a first attempt to apply the MFS for solving the Navier–Stokes equations in the moderate-Reynolds-number flow regimes. The lid-driven cavity flows at the Reynolds numbers up to 3200 for two-dimensional (2D) and 1000 for three-dimensional (3D) are chosen to validate the present algorithm. Through further simulating the flows in the 2D circular cavity with an eccentric rotating cylinder and in the 3D cube with a fixed sphere inside, we are able to demonstrate the advantages and flexibility of the proposed meshless method in the irregular geometry and multi-dimensional flows, even though very coarse node points are used in this study as compared with other mesh-dependent numerical schemes.     

Investigation of fluid–structure interactions using a velocity‐linked P2/P1 finite element method and the generalized‐ α method

A velocity‐linked algorithm for solving unsteady fluid–structure interaction (FSI) problems in a fully coupled manner is developed using the arbitrary Lagrangian–Eulerian method. The P2/P1 finite element is used to spatially discretize the incompressible Navier–Stokes equations and structural equations, and the generalized‐ α method is adopted for temporal discretization. Common velocity variables are employed at the fluid–structure interface for the strong coupling of both equations. Because of the velocity‐linked formulation, kinematic compatibility is automatically satisfied and forcing terms do not need to be calculated explicitly. Both the numerical stability and the convergence characteristics of an iterative solver for the coupled algorithm are investigated by solving the FSI problem of flexible tube flows. It is noteworthy that the generalized‐ α method with small damping is free from unstable velocity fields. However, the convergence characteristics of the coupled system deteriorate greatly for certain Poisson's ratios so that direct solvers are essential for these cases. Furthermore, the proposed method is shown to clearly display the advantage of considering FSI in the simulation of flexible tube flows, while enabling much larger time‐steps than those adopted in some previous studies. This is possible through the strong coupling of the fluid and structural equations by employing common primitive variables. Copyright © 2012 John Wiley & Sons, Ltd.     

Time evolution of three-dimensional nonlinear gravity–capillary free-surface flows

The evolution in time of three-dimensional gravity and gravity–capillary free-surface flows generated by a moving pressure distribution is considered. Solutions of the fully nonlinear equations in deep water are calculated by boundary-integral-equation methods and marching in time. Comparisons between unsteady and steady solutions are discussed.     

Hybrid finite element/volume method for shallow water equations

A hybrid numerical scheme based on finite element and finite volume methods is developed to solve shallow water equations. In the recent past, we introduced a series of hybrid methods to solve incompressible low and high Reynolds number flows for single and two‐fluid flow problems. The present work extends the application of hybrid method to shallow water equations. In our hybrid shallow water flow solver, we write the governing equations in non‐conservation form and solve the non‐linear wave equation using finite element method with linear interpolation functions in space. On the other hand, the momentum equation is solved with highly accurate cell‐center finite volume method. Our hybrid numerical scheme is truly a segregated method with primitive variables stored and solved at both node and element centers. To enhance the stability of the hybrid method around discontinuities, we introduce a new shock capturing which will act only around sharp interfaces without sacrificing the accuracy elsewhere. Matrix‐free GMRES iterative solvers are used to solve both the wave and momentum equations in finite element and finite volume schemes. Several test problems are presented to demonstrate the robustness and applicability of the numerical method. Copyright © 2010 John Wiley & Sons, Ltd.     

On the general solution by a direct method of a large-scale singular system of linear equations: application to the analysis of floating structures

Finding the general solution of a singular system of linear equations requires computing a particular solution and a basis of the null space of the corresponding singular matrix. In this paper, we consider the case where the singular matrix is large and sparse, and the application calls for a direct solution method. We highlight the dependence of straightforward factorization algorithms on an arbitrary constant that can influence the correctness of the computed solution, and describe a family of improved direct solution methods that alleviate this problem. For structural mechanics applications, we propose a hybrid geometric–algebraic method that is more robust than the purely algebraic direct methods that are currently used for solving singular sparse systems of equations. We illustrate the potential of our proposed solution algorithms with examples from structural mechanics and domain-decomposition-based iterative solvers. © 1998 John Wiley & Sons, Ltd.