基于L^2-投影及H^1-投影进行不可压缩粘性流体数值模拟的比较:案例分析(英文)

本文的主要目的是讨论不可压缩粘性流体的Navier-Stokes方程的数值模拟。本文所用的方法是对时间用一阶精度算了分裂离散化,对空间度是用Uzawa方法对L2-投影及H1-投影求解Stokes问题,以及利用类波动方程方法求解平流问题。这两种投影格式都很容易实现。我们利用它们求解经典顶盖驱动方腔流问题直至雷诺数7500都取得了一致结果。当雷诺数处于区间[8575,8590](对应[8600,8625])时,运用L2-投影(对应H1-投影)得到的结果具有时间周期性,这表明Hopf分支的产生。当雷诺数为10000时,存在两个主导频率相互作用。

A DLM/FD/IB Method for Simulating Cell/Cell and Cell/Particle Interaction in Microchannels

A spring model is used to simulate the skeleton structure of the red blood cell (RBC) membrane and to study the red blood cell (RBC) rheology in Poiseuille flow with an immersed boundary method. The lateral migration properties of many cells in Poiseuille flow have been investigated. The authors also combine the above methodology with a distributed Lagrange multiplier/fictitious domain method to simulate the interaction of cells and neutrally buoyant particles in a microchannel for studying the margination of particles.

A Penalty-Regularization-Operator Splitting Method for the Numerical Solution of a Scalar Eikonal Equation

In this article, we discuss a numerical method for the computation of the minimal and maximal solutions of a steady scalar Eikonal equation. This method relies on a penalty treatment of the nonlinearity, a biharmonic regularization of the resulting variational problem, and the time discretization by operator-splitting of an initial value problem associated with the Euler-Lagrange equations of the regularized variational problem. A low-order finite element discretization is advocated since it is well-suited to the low regularity of the solutions. Numerical experiments show that the method sketched above can capture efficiently the extremal solutions of various two-dimensional test problems and that it has also the ability of handling easily domains with curved boundaries.

On the Numerical Solution of Some Eikonal Equations:An Elliptic Solver Approach

The steady Eikonal equation is a prototypical first-order fully nonlinear equation. A numerical method based on elliptic solvers is presented here to solve two different kinds of steady Eikonal equations and compute solutions, which are maximal and minimal in the variational sense. The approach in this paper relies on a variational argument involving penalty, a biharmonic regularization, and an operator-splitting-based time-discretization scheme for the solution of an associated initial-value problem. This approach allows the decoupling of the nonlinearities and differential operators.Numerical experiments are performed to validate this approach and investigate its convergence properties from a numerical viewpoint.

On the Numerical Solution to a Nonlinear Wave Equation Associated with the First Painlev Equation:an Operator-Splitting Approach

The main goal of this article is to discuss the numerical solution to a nonlinear wave equation associated with the first of the celebrated Painlev'e transcendent ordinary differential equations.In order to solve numerically the above equation,whose solutions blow up in finite time,the authors advocate a numerical methodology based on the Strang's symmetrized operator-splitting scheme.With this approach,one can decouple nonlinearity and differential operators,leading to the alternate solution at every time step of the equation as follows:(i) The first Painlevé ordinary differential equation,(ii) a linear wave equation with a constant coefficient.Assuming that the space dimension is two,the authors consider a fully discrete variant of the above scheme,where the space-time discretization of the linear wave equation sub-steps is achieved via a Galerkin/finite element space approximation combined with a second order accurate centered time discretization scheme.To handle the nonlinear sub-steps,a second order accurate centered explicit time discretization scheme with adaptively variable time step is used,in order to follow accurately the fast dynamic of the solution before it blows up.The results of numerical experiments are presented for different coefficients and boundary conditions.They show that the above methodology is robust and describes fairly accurately the evolution of a rather "violent" phenomenon.

A DLM/FD/IB Method for Simulating Compound Cell Interacting with Red Blood Cells in a Microchannel

In this article,a computational model and related methodologies have been tested for simulating the motion of a malaria infected red blood cell(i RBC for short)in Poiseuille flow at low Reynolds numbers. Besides the deformability of the red blood cell membrane,the migration of a neutrally buoyant particle(used to model the malaria parasite inside the membrane) is another factor to determine the i RBC motion. Typically an i RBC oscillates in a Poiseuille flow due to the competition between these two factors.The interaction of an i RBC and several RBCs in a narrow channel shows that,at lower flow speed,the i RBC can be easily pushed toward the wall and stay there to block the channel.But,at higher flow speed,RBCs and i RBC stay in the central region of the channel since their migrations are dominated by the motion of the RBC membrane.

MODELING,SIMULATION,AND OPTIMIZATION OF SURFACE ACOUSTIC WAVE DRIVEN MICROFLUIDIC BIOCHIPS

We will be concerned with the mathematical modeling, numerical simulation, and shape optimization of micro fluidic biochips that are used for various biomedical applications. A particular feature is that the fluid flow in the fluidic network on top of the biochips is in- duced by surface acoustic waves generated by interdigital transducers. We are thus faced with a multiphysics problem that will be modeled by coupling the equations of piezoelectricity with the compressible Navier-Stokes equations. Moreover, the fluid flow exhibits a multiscale character that will be taken care of by a homogenization approach. We will discuss and analyze the mathematical models and deal with their numerical solution by space-time discretizations featuring appropriate finite element approximations with respect to hierarchies of simplicial triangulations of the underlying computational domains. Simulation results will be given for the propagation of the surface acoustic waves on top of the piezoelectric substrate and for the induced fluid flow in the microchannels of the fluidic network. The performance of the operational behavior of the biochips can be significantly improved by shape optimization. In particular, for such purposes we present a multilevel interior point method relying on a predictor-corrector strategy with an adaptive choice of the continuation steplength along the barrier path. As a specific example, we will consider the shape optimization of pressure driven capillary barriers between microchannels and reservoirs.

A Least-Squares/Fictitious Domain Method for Linear Elliptic Problems with Robin Boundary Conditions

In this article,we discuss a least-squares/fictitious domain method for the solution of linear elliptic boundary value problems with Robin boundary conditions.LetΩandωbe two bounded domains of R d such thatω⊂Ω.For a linear elliptic problem inΩ\ωwith Robin boundary condition on the boundaryγofω,our goal here is to develop a fictitious domain method where one solves a variant of the original problem on the fullΩ,followed by a well-chosen correction overω.This method is of the virtual control type and relies on a least-squares formulation making the problem solvable by a conjugate gradient algorithm operating in a well chosen control space.Numerical results obtained when applying our method to the solution of two-dimensional elliptic and parabolic problems are given;they suggest optimal order of convergence.

Application of the Alternating Direction Method of Multipliers to Control Constrained Parabolic Optimal Control Problems and Beyond

Control constrained parabolic optimal control problems are generally challenging,from either theoretical analysis or algorithmic design perspectives.Conceptually,the well-known alternating direction method of multipliers(ADMM)can be directly applied to such problems.An attractive advantage of this direct ADMM application is that the control constraints can be untied from the parabolic optimal control problem and thus can be treated individually in the iterations.At each iteration of the ADMM,the main computation is for solving an unconstrained parabolic optimal control subproblem.Because of its inevitably high dimensionality after space-time discretization,the parabolicoptimal control subproblem at each iteration can be solved only inexactly by implementing certain numerical scheme internally and thus a two-layer nested iterative algorithm is required.It then becomes important to find an easily implementable and efficient inexactness criterion to perform the internal iterations,and to prove the overall convergence rigorously for the resulting two-layer nested iterative algorithm.To implement the ADMM efficiently,we propose an inexactness criterion that is independent of the mesh size of the involved discretization,and that can be performed automatically with no need to set empirically perceived constant accuracy a priori.The inexactness criterion turns out to allow us to solve the resulting parabolic optimal control subproblems to medium or even low accuracy and thus save computation significantly,yet convergence of the overall two-layer nested iterative algorithm can be still guaranteed rigorously.Efficiency of this ADMM implementation is promisingly validated by some numerical results.Our methodology can also be extended to a range of optimal control problems modeled by other linear PDEs such as elliptic equations,hyperbolic equations,convection-diffusion equations,and fractional parabolic equations.