Numerical simulation of incompressible flows using adaptive unstructured meshes and the pseudo-compressibility hypothesis

The numerical simulation of incompressible viscous flows, using finite elements with automatic adaptive unstructured meshes and the pseudo-compressibility hypothesis, is presented in this work. Special emphasis is given to the automatic adaptive process of unstructured meshes with linear tetrahedral elements in order to get more accurate solutions at relatively low computational costs. The behaviour of the numerical solution is analyzed using error indicators to detect regions where some important physical phenomena occur. An adaptive scheme, consisting in a mesh refinement process followed by a nodal re-allocation technique, is applied to the regions in order to improve the quality of the numerical solution. The error indicators, the refinement and nodal re-allocation processes as well as the corresponding data structure (to manage the connectivity among the different entities of a mesh, such as elements, faces, edges and nodes) are described. Then, the formulation and application of a mesh adaptation strategy, which includes a refinement scheme, a mesh smoothing technique, very simple error indicators and an adaptation criterion based in statistical theory, integrated with an algorithm to simulate complex two and three dimensional incompressible viscous flows, are the main contributions of this work. Two numerical examples are presented and their results are compared with those obtained by other authors.     

On Resistive MHD Models with Adaptive Moving Meshes

In this paper we describe an adaptive moving mesh technique and its application to convection-diffusion models from magnetohydrodynamics (MHD). The method is based on a coordinate transformation between physical and computational coordinates. The transformation can be viewed as a solution of adaptive mesh partial differential equations (PDEs) which are derived from the minimization of a mesh-energy integral. For an efficient implementation we have used an approach in which the numerical solution of the physical PDE model and the adaptive PDEs are decoupled. Further, to avoid solving large nonlinear systems, an implicit-explicit method is applied for the time integration in combination with the iterative method Bi-CGSTAB. The adaptive mesh can be viewed as a 2D variant of the equidistribution principle, and it has the ability to track individual features of the physical solutions in the developing plasma flows. The results of a series of numerical experiments are presented which cover several aspects typifying resistive magnetofluid-dynamics.     

Application of a posteriori error estimation to finite element simulation of compressible Navier-Stokes flow

In this paper, we discuss how to improve the adaptive finite element simulation of compressible Navier-Stokes flow via a posteriori error estimate analysis. We use the moving space-time finite element method to globally discretize the time-dependent Navier-Stokes equations on a series of adapted meshes. The generalized compressible Stokes problem, which is the Stokes problem in its most generalized form, is presented and discussed. On the basis of the a posteriori error estimator for the generalized compressible Stokes problem, a numerical framework of a posteriori error estimation is established corresponding to the case of compressible Navier-Stokes equations. Guided by the a posteriori errors estimation, a combination of different mesh adaptive schemes involving simultaneous refinement/unrefinement and point-moving are applied to control the finite element mesh quality. Finally, a series of numerical experiments will be performed involving the compressible Stokes and Navier-Stokes flows around different aerodynamic shapes to prove the validity of our mesh adaptive algorithms.     

A priori anisotropic mesh adaptation driven by a higher dimensional embedding

We generalize the higher embedding approach proposed in Lévy and Bonneel (2013) to generate an adapted mesh matching the intrinsic directionalities of an assigned function. In more detail, the original embedding map between the physical (lower dimensional) and the embedded (higher dimensional) setting is modified to include information associated with the function and with its gradient. Then, we set an adaptive procedure, driven by the embedded metric but performed in the lower dimensional setting, which results into an anisotropic adapted mesh of the physical domain. The effectiveness of the proposed procedure is extensively investigated on several two-dimensional test cases, involving both analytical functions and finite element approximations of differential problems. The preliminary verification in three dimensions corroborates the robustness of the method.     

An Adaptive Moving Mesh Finite Element Solution of the Regularized Long Wave Equation

An adaptive moving mesh finite element method is proposed for the numerical solution of the regularized long wave (RLW) equation. A moving mesh strategy based on the so-called moving mesh PDE is used to adaptively move the mesh to improve computational accuracy and efficiency. The RLW equation represents a class of partial differential equations containing spatial-time mixed derivatives. For the numerical solution of those equations, a \(C^0\) finite element method cannot apply directly on a moving mesh since the mixed derivatives of the finite element approximation may not be defined. To avoid this difficulty, a new variable is introduced and the RLW equation is rewritten into a system of two coupled equations. The system is then discretized using linear finite elements in space and the fifth-order Radau IIA scheme in time. A range of numerical examples in one and two dimensions, including the RLW equation with one or two solitary waves and special initial conditions that lead to the undular bore and solitary train solutions, are presented. Numerical results demonstrate that the method has a second order convergence and is able to move and adapt the mesh to the evolving features in the solution.     

Anisotropic mesh adaptation for finite element solution of Anisotropic Porous Medium Equation

Anisotropic Porous Medium Equation (APME) is developed as an extension of the Porous Medium Equation (PME) for anisotropic porous media. A special analytical solution is derived for APME for time-independent diffusion. Anisotropic mesh adaptation for linear finite element solution of APME is discussed and numerical results for two dimensional examples are presented. The solution errors using anisotropic adaptive meshes show second order convergence.     

Uniform convergence analysis of finite difference approximations for advection–reaction–diffusion problem on adaptive grids

A kind of singularly perturbed advection–reaction–diffusion problems with the exponential boundary layer are considered on an adaptive mesh. The existence, uniqueness and stability for the solution of the discrete problem are analysed with the maximum principle. The stability of the continuous problem is also considered. For the equidistribution problem composed by the difference scheme and equidistribution mesh equations, we establish a first-order ?-independent convergence rate for the numerical scheme defined on the equidistribution mesh and also an estimation for the accuracy of the solution computed on the final mesh generated by the adaptive algorithm. Numerical results are given to examine the validity of our theoretical analysis and the efficiency of the adaptive algorithm.     

Hierarchical Solutions for the Deformable Surface Problem in Visualization

In this paper we present a hierarchical approach for the deformable surface technique. This technique is a three dimensional extension of the snake segmentation method. We use it in the context of visualizing three dimensional scalar data sets. In contrast to classical indirect volume visualization methods, this reconstruction is not based on iso-values but on boundary information derived from discontinuities in the data. We propose a multilevel adaptive finite difference solver, which generates a target surface minimizing an energy functional based on an internal energy of the surface and an outer energy induced by the gradient of the volume. The method is attractive for preprocessing in numerical simulation or texture mapping. Red-green triangulation allows adaptive refinement of the mesh. Special considerations help to prevent self interpenetration of the surfaces. We will also show some techniques that introduce the hierarchical aspect into the inhomogeneity of the partial differential equation. The approach proves to be appropriate for data sets that contain a collection of objects separated by distinct boundaries. These kind of data sets often occur in medical and technical tomography, as we will demonstrate in a few examples.     

An ADI-based adaptive mesh Poisson solver for the MHD code NIRVANA

I describe a Poisson solver for the adaptive mesh magnetohydrodynamics (MHD) code NIRVANA using ADI techniques (ADI: Alternative Direction Implicit). The solver is fit to the mesh refinement framework of the code and utilizes its special block-structured design. The key part of the method is an algorithm for the intelligent clustering of subgrids which permits the application of numerical methods based on dimensional operator splitting like ADI. Test problems show the convergence of this ansatz.     

Adaptive moving mesh methods for two-dimensional resistive magneto-hydrodynamic PDE models

An adaptive moving mesh technique is applied to magneto-hydrodynamics (MHD) model problem. The moving mesh strategy is based on the approach proposed in Li et al. [Li R, Tang T, Zhang P. Moving mesh methods in multiple dimensions based on harmonic maps. J Comput Phys 2001;170:562-88] to separate the mesh-moving and PDE evolution at each time step. The Magneto-hydrodynamic equations are discretized by a finite-volume method in space, and the mesh-moving part is realized by solving the Euler-Lagrange equations to minimize a certain variation with the directional splitting monitor function. A conservative interpolation is used to redistribute the numerical solutions on the new meshes. Numerical results demonstrate the accuracy and effectiveness of the proposed algorithm.     

Second-Order Accurate Godunov Scheme for Multicomponent Flows on Moving Triangular Meshes

This paper presents a second-order accurate adaptive Godunov method for two-dimensional (2D) compressible multicomponent flows, which is an extension of the previous adaptive moving mesh method of Tang et al. (SIAM J. Numer. Anal. 41:487–515, 2003) to unstructured triangular meshes in place of the structured quadrangular meshes. The current algorithm solves the governing equations of 2D multicomponent flows and the finite-volume approximations of the mesh equations by a fully conservative, second-order accurate Godunov scheme and a relaxed Jacobi-type iteration, respectively. The geometry-based conservative interpolation is employed to remap the solutions from the old mesh to the newly resulting mesh, and a simple slope limiter and a new monitor function are chosen to obtain oscillation-free solutions, and track and resolve both small, local, and large solution gradients automatically. Several numerical experiments are conducted to demonstrate robustness and efficiency of the proposed method. They are a quasi-2D Riemann problem, the double-Mach reflection problem, the forward facing step problem, and two shock wave and bubble interaction problems.     

Adaptive mesh refinement for chevron nozzle jet flows

In this paper an adaptive mesh generation procedure is presented for improving the resolution of the numerical simulation of a turbulent jet exhausting from a chevron nozzle. This procedure is based on the minimization of a variational integral whose integrand depends on the metric (also called the monitor function) induced by a curvilinear grid generated in the physical domain. Specifically, it leads to solving parabolic equations involving the monitor function, which is carefully designed to resolve the flow gradients, and which, in the present instance, is determined by the time-averaged axial velocity profile within the jet. This mesh redistribution strategy is incorporated into a flow computation code (that solves the compressible three-dimensional Navier-Stokes equations using a prefactored optimized fourth-order compact difference scheme for spatial derivatives and the Beam-Warming method for the time derivative on a multi-block overset grid) and is demonstrated to be efficient and effective.     

求解任意波数的三维Helmholtz方程

提出通过Adomian分解法求解任意波数的三维Helmholtz方程。通过Adomian分解法可以把三维Helmholtz微分方程转换成递归代数公式,并进一步把其边界条件转换成适用符号计算的简单代数公式。利用边界条件可以很容易得到方程的解析解表达式。Adomian分解法的主要特点在于计算简单快速,并且不需要进行线性化或离散化。最后通过数值计算以验证Adomian分解法求解任意波数下三维Helmholtz方程的有效性。数值计算结果表明：Adomian分解法的计算结果非常接近精确解,并且该方法在大波数情况下还具有良好的收敛性。     

A robust conforming NURBS tessellation for industrial applications based on a mesh generation approach

A NURBS tessellation technique is presented with the goal to robustly approximate CAD surfaces that define the boundary of complicated three dimensional geometric shapes with a minimum number of triangles. The minimization is achieved by generating anisotropic triangles in the three dimensional space. New procedures are presented to handle numerical stability issues due to the anisotropy. The tessellation is generated using a mesh generation viewpoint, as opposed to the more classical viewpoint of graphical visualization of surfaces in CAD. This ensures topological conformity of the resulting mesh. A tiered approximation approach is used for speed and robustness. Degeneracies associated with NURBS curves and surfaces are given special attention as they occur frequently in naval and aerospace conceptual-to-early design process. Analogies with a classical mesh generation process are discussed and several numerical examples illustrate the method.     

An adaptive Rothe method for the wave equation

The adaptive Rothe method approaches a time-dependent PDE as an ODE in function space. This ODE is solved virtually using an adaptive state-of-the-art integrator. The actual realization of each time-step requires the numerical solution of an elliptic boundary value problem, thus perturbing the virtual function space method. The admissible size of that perturbation can be computed a priori and is prescribed as a tolerance to an adaptive multilevel finite element code, which provides each time-step with an individually adapted spatial mesh. In this way, the method avoids the well-known difficulties of the method of lines in higher space dimensions. During the last few years the adaptive Rothe method has been applied successfully to various problems with infinite speed of propagation of information. The present study concerns the adaptive Rothe method for hyperbolic equations in the model situation of the wave equation. All steps of the construction are given in detail and a numerical example (diffraction at a corner) is provided for the 2D wave equation. This example clearly indicates that the adaptive Rothe method is appropriate for problems which can generally benefit from mesh adaptation. This should be even more pronounced in the 3D case because of the strong Huygens' principle. Accepted: 12 August 1997     

An adaptive phase field method for the mixture of two incompressible fluids

This paper develops an adaptive moving mesh method to solve a phase field model for the mixture of two incompressible fluids. The projection method is implemented on a half-staggered, moving quadrilateral mesh to keep the velocity field divergence-free, and the conjugate gradient or multigrid method is employed to solve the discrete Poisson equations. The current algorithm is composed by two independent parts: evolution of the governing equations and mesh-redistribution. In the first part, the incompressible Navier-Stokes equations are solved on a fixed half-staggered mesh by the rotational incremental pressure-correction scheme, and the Allen-Cahn type of phase equation is approximated by a conservative, second-order accurate central difference scheme, where the Lagrangian multiplier is used to preserve the mass-conservation of the phase field. The second part is an iteration procedure. During the mesh redistribution, the phase field is remapped onto the newly resulting meshes by the high-resolution conservative interpolation, while the non-conservative interpolation algorithm is applied to the velocity field. The projection technique is used to obtain a divergence-free velocity field at the end of this part. The resultant numerical scheme is stable, mass conservative, highly efficient and fast, and capable of handling variable density and viscosity. Several numerical experiments are presented to demonstrate the efficiency and robustness of the proposed algorithm.     

Lossy Compression in Optimal Control of Cardiac Defibrillation

This paper presents efficient computational techniques for solving an optimization problem in cardiac defibrillation governed by the monodomain equations. Time-dependent electrical currents injected at different spatial positions act as the control. Inexact Newton-CG methods are used, with reduced gradient computation by adjoint solves. In order to reduce the computational complexity, adaptive mesh refinement for state and adjoint equations is performed. To reduce the high storage and bandwidth demand imposed by adjoint gradient and Hessian-vector evaluations, a lossy compression technique for storing trajectory data is applied. An adaptive choice of quantization tolerance based on error estimates is developed in order to ensure convergence. The efficiency of the proposed approach is demonstrated on numerical examples.     

Numerical investigation of effectivity indices of space-time error indicators for Navier–Stokes equations

In this paper we propose two error indicators aimed at estimating the space discretization error and the time discretization error for the unsteady Navier–Stokes equations. We define a space error indicator for evaluating the quality of the mesh and a time error indicator for evaluating the time discretization error. Moreover, we verify the reliability of the estimations through numerical experiments and we propose an effective space-time adaptive strategy for the unsteady Navier–Stokes equations. Such technique is based on two residual-based error indicators that suitably drive the mesh and the timestep-length modifications. Adaptive simulations show that the presented strategy allows to obtain accurate solutions in efficient way.     

数值求解中网格自适应加密和合并技术的研究

在计算流体力学(CFD)领域,几乎所有的方法都离不开网格,网格是各种数值方法的基础,网格质量的好坏直接影响数值结果的精度,甚至影响到数值计算的成败,随着计算流体力学解决的问题越来越复杂,对网格质量的要求也越来越高,传统的统一网格技术已不能适应这一不断发展的需求,为此CFD工作者发展了许多方法,如迭合网格、贴体网格和非