An L∞–Lp mesh‐adaptive method for computing unsteady bi‐fluid flows

This paper discusses the contribution of mesh adaptation to high‐order convergence of unsteady multi‐fluid flow simulations on complex geometries. The mesh adaptation relies on a metric‐based method controlling the L ^p‐norm of the interpolation error and on a mesh generation algorithm based on an anisotropic Delaunay kernel. The mesh‐adaptive time advancing is achieved, thanks to a transient fixed‐point algorithm to predict the solution evolution coupled with a metric intersection in the time procedure. In the time direction, we enforce the equidistribution of the error, i.e. the error minimization in L ^∞ norm. This adaptive approach is applied to an incompressible Navier–Stokes model combined with a level set formulation discretized on triangular and tetrahedral meshes. Applications to interface flows under gravity are performed to evaluate the performance of this method for this class of discontinuous flows. Copyright © 2010 John Wiley & Sons, Ltd.     

On the reinitialization procedure in a narrow‐band locally refined level set method for interfacial flows

A local level set algorithm for simulating interfacial flows described by the two‐dimensional incompressible Navier–Stokes equations is presented. The governing equations are solved using a finite‐difference discretization on a Cartesian grid and a second‐order approximate projection method. The level set transport and reinitialization equations are solved in a narrow band around the interface using an adaptive refined grid, which is reconstructed every time step and refined using a simple uniform cell‐splitting operation within the band. Instabilities at the border of the narrow band are avoided by smoothing the level set function in the outer part of the band. The influence of different PDE‐based reinitialization strategies on the accuracy of the results is investigated. The ability of the proposed method to accurately compute interfacial flows is discussed using different tests, namely the advection of a circle of fluid in two different time‐reversed vortex flows, the advection of Zalesak's rotating disk, the propagation of small‐amplitude gravity and capillary waves at the interface between two superposed viscous fluids in deep water, and a classical test of Rayleigh–Taylor instability with and without surface tension effects. The interface location error and area loss for some of the results obtained are compared with those of a recent particle level set method. Copyright © 2005 John Wiley & Sons, Ltd.     

An adaptive level set method based on two‐level uniform meshes and its application to dislocation dynamics

In this paper, we present an adaptive level set method for motion of high codimensional objects (e.g., curves in three dimensions). This method uses only two (or a few fixed) levels of meshes. A uniform coarse mesh is defined over the whole computational domain. Any coarse mesh cell that contains the moving object is further divided into a uniform fine mesh. The coarse‐to‐fine ratios in the mesh refinement can be adjusted to achieve optimal efficiency. Refinement and coarsening (removing the fine mesh within a coarse grid cell) are performed dynamically during the evolution. In this adaptive method, the computation is localized mostly near the moving objects; thus, the computational cost is significantly reduced compared with the uniform mesh over the whole domain with the same resolution. In this method, the level set equations can be solved on these uniform meshes of different levels directly using standard high‐order numerical methods. This method is examined by numerical examples of moving curves and applications to dislocation dynamics simulations. This two‐level adaptive method also provides a basis for using locally varying time stepping to further reduce the computational cost. Copyright © 2013 John Wiley & Sons, Ltd.     

On a two‐level finite element method for the incompressible Navier–Stokes equations

We consider the Galerkin finite element method for the incompressible Navier–Stokes equations in two dimensions, where the finite‐dimensional space(s) employed consist of piecewise polynomials enriched with residual‐free bubble functions. To find the bubble part of the solution, a two‐level finite element method (TLFEM) is described and its application to the Navier–Stokes equation is displayed. Numerical solutions employing the TLFEM are presented for three benchmark problems. We compare the numerical solutions using the TLFEM with the numerical solutions using a stabilized method. Copyright © 2001 John Wiley & Sons, Ltd.     

A three-step Oseen correction method for the steady Navier–Stokes equations

We present and analyze a two-grid scheme based on mixed finite element approximations for the steady incompressible Navier–Stokes equations. This numerical scheme aims at the simulations of high Reynolds number flows and consists of three steps: in the first step, we solve a finite element variational multiscale-stabilized nonlinear Navier–Stokes system on a coarse mesh, and then, in the second and third steps, we solve Oseen-linearized and -stabilized problems which have the same stiffness matrices with only different right-hand sides on a fine mesh. We provide error bounds for the approximate solutions, derive algorithmic parameter scalings from the analysis, and present some numerical results to verify the theoretical predictions and demonstrate the effectiveness of the proposed method.     

Common‐refinement‐based data transfer between non‐matching meshes in multiphysics simulations

In multiphysics simulations using a partitioned approach, each physics component solves on its own mesh, and the interfaces between these meshes are in general non‐matching. Simulation data (e.g. jump conditions) must be exchanged across the interface meshes between physics components. It is highly desirable for such data transfers to be both numerically accurate and physically conservative. This paper presents accurate, conservative, and efficient data transfer algorithms utilizing a common refinement of two non‐matching surface meshes. Our methods minimize errors in a certain norm while achieving strict conservation. Some traditional methods for data transfer and related problems are also reviewed and compared with our methods. Numerical results demonstrate significant advantages of common‐refinement based methods, especially for repeated transfers. While the comparisons are performed with matching geometries, this paper also addresses additional complexities associated with non‐matching surface meshes and presents some experimental results from 3‐D simulations using our methods. Copyright © 2004 John Wiley & Sons, Ltd.     

Coupling of non‐conforming meshes in a component mode synthesis method

A common mesh refinement‐based coupling technique is embedded into a component mode synthesis method, Craig–Bampton. More specifically, a common mesh is generated between the non‐conforming interfaces of the coupled structures, and the compatibility constraints are enforced on that mesh via L2‐minimization. This new integrated method is suitable for structural dynamic analysis problems where the substructures may have non‐conforming curvilinear and/or surface interface meshes. That is, coupled substructures may have different element types such as shell, solid, and/or beam elements. The proposed method is implemented into a commercial finite element software, B2000++, and its demonstration is carried out using an academic and industry oriented test problems. Copyright © 2013 John Wiley & Sons, Ltd.     

An essentially non‐oscillatory semi‐Lagrangian method for tidal flow simulations

We develop an essentially non‐oscillatory semi‐Lagrangian method for solving two‐dimensional tidal flows. The governing equations are derived from the incompressible Navier–Stokes equations with assumptions of shallow water flows including bed frictions, eddy viscosity, wind shear stresses and Coriolis forces. The method employs the modified method of characteristics to discretize the convective term in a finite element framework. Limiters are incorporated in the method to reconstruct an essentially non‐oscillatory algorithm at minor additional cost. The central idea consists in combining linear and quadratic interpolation procedures using nodes of the finite element where departure points are localized. The resulting semi‐discretized system is then solved by an explicit Runge–Kutta Chebyshev scheme with extended stages. This scheme adds in a natural way a stabilizing stage to the conventional Runge–Kutta method using the Chebyshev polynomials. The proposed method is verified for the recirculation tidal flow in a channel with forward‐facing step. We also apply the method for simulation of tidal flows in the Strait of Gibraltar. In both test problems, the proposed method demonstrates its ability to handle the interaction between water free‐surface and bed frictions. Copyright © 2009 John Wiley & Sons, Ltd.     

The enriched space–time finite element method (EST) for simultaneous solution of fluid–structure interaction

The paper introduces a weighted residual‐based approach for the numerical investigation of the interaction of fluid flow and thin flexible structures. The presented method enables one to treat strongly coupled systems involving large structural motion and deformation of multiple‐flow‐immersed solid objects. The fluid flow is described by the incompressible Navier–Stokes equations. The current configuration of the thin structure of linear elastic material with non‐linear kinematics is mapped to the flow using the zero iso‐contour of an updated level set function. The formulation of fluid, structure and coupling conditions uniformly uses velocities as unknowns. The integration of the weak form is performed on a space–time finite element discretization of the domain. Interfacial constraints of the multi‐field problem are ensured by distributed Lagrange multipliers. The proposed formulation and discretization techniques lead to a monolithic algebraic system, well suited for strongly coupled fluid–structure systems. Embedding a thin structure into a flow results in non‐smooth fields for the fluid. Based on the concept of the extended finite element method, the space–time approximations of fluid pressure and velocity are properly enriched to capture weakly and strongly discontinuous solutions. This leads to the present enriched space–time (EST) method. Numerical examples of fluid–structure interaction show the eligibility of the developed numerical approach in order to describe the behavior of such coupled systems. The test cases demonstrate the application of the proposed technique to problems where mesh moving strategies often fail. Copyright © 2007 John Wiley & Sons, Ltd.     

Adaptive mesh refinement applied to the scalar transported PDF equation in a turbulent jet

This paper describes a novel solution method for the transported probability density function (PDF) equation for scalars (compositions). In contrast to conventional solution methods based on the Monte Carlo approach, we use a finite‐volume method combined with adaptive mesh refinement (AMR) applied in both physical and compositional space. The obvious advantage of this over a uniform grid is that fine meshes are only used where the solution requires high resolution. The efficiency of the method is demonstrated by a number of tests involving a turbulent jet flow with up to two scalars (both reacting and non‐reacting). We find that the AMR calculation can be at a fraction of the computer cost of a uniform grid calculation with the same accuracy. Copyright © 2010 John Wiley & Sons, Ltd.     

Simulation of two‐ and three‐dimensional viscoplastic flows using adaptive mesh refinement

This paper presents a finite element solver for the simulation of steady non‐Newtonian flow problems, using a regularized Bingham model, with adaptive mesh refinement capabilities. The solver is based on a stabilized formulation derived from the variational multiscale framework. This choice allows the introduction of an a posteriori error indicator based on the small scale part of the solution, which is used to drive a mesh refinement procedure based on element subdivision. This approach applied to the solution of a series of benchmark examples, which allow us to validate the formulation and assess its capabilities to model 2D and 3D non‐Newtonian flows.     

A Cartesian ghost‐cell multigrid Poisson solver for incompressible flows

In this paper, a simple Cartesian ghost‐cell multigrid Poisson solver is proposed for simulating incompressible fluid flows. The flow field is discretized efficiently on a rectangular mesh, in which solid bodies are immersed. A small number of ghost mesh cells and their symmetric image cells are distributed in the vicinity of the solid boundary. With the aid of the ghost and image cells, the Dirichlet and Neumann boundary conditions can be implemented effectively. Chorin's fractional‐step projection method is adopted for the coupling of velocity and pressure for the solution of the Navier–Stokes equations. Point‐wise Gauss–Seidel iteration is used to solve the pressure Poisson equation. To speed up the convergence of the solution to the corresponding linear system, sub‐level coarse meshes embedded with ghost and image cells are also introduced and operated in a sequential V‐cycle. Several test cases including the classical ideal incompressible flow around a cylinder, a lid‐driven cavity flow and viscous flow past a fixed/rotating cylinder are presented to demonstrate the accuracy and efficiency of the current approach. Copyright © 2010 John Wiley & Sons, Ltd.     

Non‐conservative and conservative formulations of characteristics‐based numerical reconstructions for incompressible flows

An investigation of characteristics‐based (CB) schemes for solving the incompressible Navier–Stokes equations in conjunction with the artificial‐compressibility approach, is presented. Both non‐conservative and conservative CB numerical reconstructions are derived and their accuracy and convergence properties are assessed analytically and numerically. We demonstrate by means of eigenvalue analysis that there are differences in the spectral characteristics of these formulations that result in different convergence properties. Numerical tests for two‐ and three‐dimensional flows reveal that the two formulations provide similar accuracy but the non‐conservative formulation converges faster. Copyright © 2005 John Wiley & Sons, Ltd.     

Local boundary value problems for the error in FE approximation of non‐linear diffusion systems

In this work we investigate the a posteriori error estimation for a class of non‐linear, multicomponent diffusion operators, which includes the Stefan–Maxwell equations. The local error indicators for the global error are based on local boundary value problems, which are chosen to approximate either the global residual of the finite element approximation or the global linearized error equation. Using representative numerical examples, it is shown that the error indicators based on the latter approach are more accurate for estimating the global error for this problem class as the problem becomes more non‐linear, and can even produce better adaptive mesh refinement (AMR). In addition, we propose a new local error indicator for the error in output functionals that is accurate, inexpensive to compute, and is suitable for AMR, as demonstrated by numerical examples. Copyright © 2007 John Wiley & Sons, Ltd.     

Three‐dimensional grayscale‐free topology optimization using a level‐set based r‐refinement method

In this paper, we propose a three‐dimensional (3D) grayscale‐free topology optimization method using a conforming mesh to the structural boundary, which is represented by the level‐set method. The conforming mesh is generated in an r‐refinement manner; that is, it is generated by moving the nodes of the Eulerian mesh that maintains the level‐set function. Although the r‐refinement approach for the conforming mesh generation has many benefits from an implementation aspect, it has been considered as a difficult task to stably generate 3D conforming meshes in the r‐refinement manner. To resolve this task, we propose a new level‐set based r‐refinement method. Its main novelty is a procedure for minimizing the number of the collapsed elements whose nodes are moved to the structural boundary in the conforming mesh; in addition, we propose a new procedure for improving the quality of the conforming mesh, which is inspired by Laplacian smoothing. Because of these novelties, the proposed r‐refinement method can generate 3D conforming meshes at a satisfactory level, and 3D grayscale‐free topology optimization is realized. The usefulness of the proposed 3D grayscale‐free topology optimization method is confirmed through several numerical examples. Copyright © 2017 John Wiley & Sons, Ltd.     

Multigrid solver with automatic mesh refinement for transient elastoplastic dynamic problems

This paper presents an adaptive refinement strategy based on a hierarchical element subdivision dedicated to modelling elastoplastic materials in transient dynamics. At each time step, the refinement is automatic and starts with the calculation of the solution on a coarse mesh. Then, an error indicator is used to control the accuracy of the solution and a finer localized mesh is created where the user‐prescribed accuracy is not reached. A new calculation is performed on this new mesh using the non‐linear ‘Full Approximation Scheme’ multigrid strategy. Applying the error indicator and the refinement strategy recursively, the optimal mesh is obtained. This mesh verifies the error indicator on the whole structure. The multigrid strategy is used for two purposes. First, it optimizes the computational cost of the solution on the finest localized mesh. Second, it ensures information transfer among the different hierarchical meshes. A standard time integration scheme is used and the mesh is reassessed at each time step. Copyright © 2010 John Wiley & Sons, Ltd.     

Quality improvements in extruded meshes using topologically adaptive generalized elements

This paper describes a method to extrude near‐body volume meshes that exploits topologically adaptive generalized elements to improve local mesh quality. Specifically, an advancing layer algorithm for extruding volume meshes from surface meshes of arbitrary topology, appropriate for viscous fluid flows, is discussed. First, a two‐layer reference mesh is generated from the layer initial surface mesh by extruding along the local surface normals. The reference mesh is then smoothed using a Poisson equation. Local quality improvement operations such as edge collapse, face refinement, and local reconnection are performed in each layer to drive the mesh toward isotropy and improve the transition from the extruded mesh to a void‐filling tetrahedral mesh. A few example meshes along with quality plots are presented to demonstrate the efficacy of this approach. Copyright © 2004 John Wiley & Sons, Ltd.     

Automated refinement of conformal quadrilateral and hexahedral meshes

Conformal refinement using a shrink and connect strategy, known as pillowing or buffer insertion, contracts and reconnects contiguous elements of an all‐quadrilateral or an all‐hexahedral mesh in order to locally increase vertex density without introducing hanging nodes or non‐cubical elements. Using layers as shrink sets, the present method automates the anisotropic refinement of such meshes according to a prescribed size map expressed as a Riemannian metric field. An anisotropic smoother further enhances vertex clustering to capture the features of the metric. Both two‐ and three‐dimensional test cases with analytic control metrics confirm the feasibility of the present approach and explore strategies to minimize the trade‐off between element shape quality and size conformity. Additional examples using discrete metric maps illustrate possible practical applications. Although local vertex removal and reconnection capabilities have yet to be developed, the present refinement method is a step towards an automated tool for conformal adaptation of all‐quadrilateral and all‐hexahedral meshes. Copyright © 2004 John Wiley & Sons, Ltd.     

A parallel two‐level domain decomposition based one‐shot method for shape optimization problems

A two‐level domain decomposition method is introduced for general shape optimization problems constrained by the incompressible Navier–Stokes equations. The optimization problem is first discretized with a finite element method on an unstructured moving mesh that is implicitly defined without assuming that the computational domain is known and then solved by some one‐shot Lagrange–Newton–Krylov–Schwarz algorithms. In this approach, the shape of the domain, its corresponding finite element mesh, the flow fields and their corresponding Lagrange multipliers are all obtained computationally in a single solve of a nonlinear system of equations. Highly scalable parallel algorithms are absolutely necessary to solve such an expensive system. The one‐level domain decomposition method works reasonably well when the number of processors is not large. Aiming for machines with a large number of processors and robust nonlinear convergence, we introduce a two‐level inexact Newton method with a hybrid two‐level overlapping Schwarz preconditioner. As applications, we consider the shape optimization of a cannula problem and an artery bypass problem in 2D. Numerical experiments show that our algorithm performs well on a supercomputer with over 1000 processors for problems with millions of unknowns. Copyright © 2014 John Wiley & Sons, Ltd.     

Remeshing strategies for large deformation problems with frictional contact and nearly incompressible materials

We present a framework to efficiently solve large deformation contact problems with nearly incompressible materials by implementing adaptive remeshing. Specifically, nodally integrated elements are employed to avoid mesh locking when linear triangular or tetrahedral elements are used to facilitate mesh re‐generation. Solution variables in the bulk and on contact surfaces are transferred between meshes such that accuracy is maintained and re‐equilibration on the new mesh is possible. In particular, the displacement transfer in the bulk is accomplished through a constrained least squares problem based on nodal integration, while the transfer of contact tractions relies on parallel transport. Finally, a residual‐based error indicator is chosen to drive adaptive mesh refinement. The proposed strategies are applicable to both two‐dimensional or three‐dimensional analysis and are demonstrated to be robust by a number of numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.