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REGULARIZATION METHODS FOR THE NUMERICAL SOLUTION OF THE DIVERGENCE EQUATION △. u = f＊

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Journal of Computational Mathematics 2012年第4期30卷 354-380页

作者： Alexandre Caboussat roland glowinski University of Houston Department of Mathematics Houston Texas USA

The problem of finding a L^∞-bounded two-dimensional vector field whose divergence is given in L2 is discussed from the numerical viewpoint. A systematic way to find such a vector field is to introduce a non-smooth variational problem involving a L^∞-norm. To solve this problem from calculus of variations, we use a method relying on a well- chosen augmented Lagrangian functional and on a mixed finite element approximation. An Uzawa algorithm allows to decouple the differential operators from the nonlinearities introduced by the L^∞-norm, and leads to the solution of a sequence of Stokes-like systems and of an infinite family of local nonlinear problems. A simpler method, based on a L2- regularization is also considered. Numerical experiments are performed, making use of appropriate numerical integration techniques when non-smooth data are considered; they allow to compare the merits of the two approaches discussed in this article and to show the ability of the related methods at capturing L^∞bounded solutions.

关键词： Divergence equation Bounded solutions Regularization methods AugmentedLagrangian Uzawa algorithm Nonlinear variational problems.

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

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Chinese Annals of Mathematics,Series B 2013年第2期34卷 237-254页

作者： roland glowinski Annalisa QUAINI Department of Mathematics University of Houston4800 Calhoun RdHouston(TX) 77204USA Laboratoire Jacques-Louis Lions Universit Pierre et Marie Curie4Place Jussieu75252 Paris CEDEX 05France

The main goal of this article is to discuss the numerical solution to a nonlinear wave equation associated with the first of the celebrated Painlevd 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 Painlevd 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.

关键词： Painlevd equation Nonlinear wave equation Blow-up solution Operator-SDlitting

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

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Chinese Annals of Mathematics,Series B 2010年第6期31卷 975-990页

作者： Tsorng-Whay PAN roland glowinski Department of Mathematics University of Houston

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.

关键词： Red blood cells Elastic spring model Margination Fictitious domain method Immersed boundary method Microchannel

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

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Communications in Computational Physics 2011年第3期9卷 587-606页

作者： roland glowinski Qiaolin He Department of Mathematics University of HoustonHoustonTX 77204USA Institute of Advanced Study The Hong Kong University of Science and TechnologyClear Water BayKowloonHong Kong Department of Mathematics The Hong Kong University of Science and TechnologyClear Water BayKowloonHong Kong Department of Mathematics Sichuan UniversityChengdu 610064China

In this article,we discuss a least-squares/fictitious domain method for the solution of linear elliptic boundary value problems with Robin boundary ***Ωandωbe 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 *** results obtained when applying our method to the solution of two-dimensional elliptic and parabolic problems are given;they suggest optimal order of convergence.

关键词： Least-Square methods fictitious domain methods finite element methods Robin boundary conditions

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

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Chinese Annals of Mathematics,Series B 2015年第5期36卷 689-702页

作者： Alexandre CABOUSSAT roland glowinski Tsorng-Whay PAN Haute Ecole de Gestion de Genève University of Applied Sciences Department of Mathematics University of Houston

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 *** experiments are performed to validate this approach and investigate its convergence properties from a numerical viewpoint.

关键词： Eikonal equations Maximal solutions Regularization methods Operator slalitting Finite element methods

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

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Chinese Annals of Mathematics,Series B 2018年第3期39卷 535-552页

作者： Shihai ZHAO Yao YU Tsorng-Whay PAN roland glowinski Department of Mathematics Dongbei University of Finance and EconomicsDalian 116023LiaoningChina CGG HoustonTX 77072USA Department of Mathematics University of HoustonHoustonTX 77204USA Department of Mathematics Hong Kong Baptist UniversityHong KongChina

In this article, a computational model and related methodologies have been tested for simulating the motion of a malaria infected red blood cell （iRBC 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 iRBC motion. Typically an iRBC oscillates in a Poiseuille flow due to the competition between these two factors. The interaction of an iRBC and several RBCs in a narrow channel shows that, at lower flow speed, the iRBC can be easily pushed toward the wall and stay there to block the channel. But, at higher flow speed, RBCs and iRBC stay in the central region of the channel since their migrations axe dominated by the motion of the RBC membrane.

关键词： Compound cell Red blood cells Elastic spring model Fictitious domainmethod Immersed boundary method Microchannel

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

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Journal of Computational Mathematics 2010年第2期28卷 149-169页

作者： Harbir Antil roland glowinski Ronald H.W.Hoppe Christopher Linsenmann Tsorng-Whay Pan Achim Wixforth Department of Mathematics University of Houston Institute of Mathematics University of Augsburg Institute of Physics University of Augsburg

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.

关键词： Microfluidic biochips Mathematical modeling Numerical simulation Shape optimization Multiphysics Multiscale problems.

Simulating the dynamics of fluid-cylinder interactions

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Journal of Zhejiang University-Science A(Applied Physics & Engineering) 2005年第2期6卷 97-109页

作者： PAN Tsorng-Whay glowinski roland JOSEPH Daniel D. Department of Mathematics University of Houston Houston Texas 77204-3008 USADepartment of Mathematics University of Houston Houston Texas 77204-3008 USADepartment of Aerospace Engineering and Mechanics University of Minnesota Minnesota MN 55455 USA

We present the simulation of the dynamics of fluid-cylinder interactions in a narrow three-dimensional channel filled with a Newtonian fluid, using a Lagrange multiplier based fictitious domain methodology combined with a finite element method and an operator splitting technique. As expected, a settling truncated cylinder turns its broadside perpendicular to the main stream direction and the center of mass moves to the central axis of the channel. In the case of two truncated cylinders, they first move around each other for a while and then stay together in a "T" shape. After the "T" shape has been formed for a long enough time, we found no vortex shedding behind the cylinders. When simulating the fluidization of 60 truncated cylinders, we captured the features of interactions among fluidized cylinders as observed in experiments.

关键词： Particulate flow Finite element methods Operator-splitting methods Fictitious domain methods

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)

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工程数学学报 2008年第5期25卷 761-778页

作者：王彤潘从辉 roland glowinski 休斯顿大学数学系

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... 详细信息

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。

关键词： Navier-Stokes方程有限元方法 L2-投影及H1-投影方法类波动方程方法顶盖驱动方腔流