Browsing by Author "Li, Zhilin"

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A Coupled Immersed Interface and Level Set Method for Three-Dimensional Interfacial Flows with Insoluble Surfactant
(2014-02-01)
Xu, Jian-Jun
;
Huang, Yunqing
;
Lai, Ming-Chih
;
Li, Zhilin
;
應用數學系
In this paper, a numerical method is presented for simulating the 3D interfacial flows with insoluble surfactant. The numerical scheme consists of a 3D immersed interface method (IIM) for solving Stokes equations with jumps across the interface and a 3D level-set method for solving the surfactant convection-diffusion equation along a moving and deforming interface. The 3D IIM Poisson solver modifies the one in the literature by assuming that the jump conditions of the solution and the flux are implicitly given at the grid points in a small neighborhood of the interface. This assumption is convenient in conjunction with the level-set techniques. It allows standard Lagrangian interpolation for quantities at the projection points on the interface. The interface jump relations are re-derived accordingly. A novel rotational procedure is given to generate smooth local coordinate systems and make effective interpolation. Numerical examples demonstrate that the IIM Poisson solver and the Stokes solver achieve second-order accuracy. A 3D drop with insoluble surfactant under shear flow is investigated numerically by studying the influences of different physical parameters on the drop deformation.
A least squares augmented immersed interface method for solving Navier-Stokes and Darcy coupling equations
(2018-05-15)
Li, Zhilin
;
Lai, Ming-Chih
;
Peng, Xiaofei
;
Zhang, Zhiyue
;
應用數學系
A new finite difference method based on Cartesian meshes and fast Poisson/Helmholtz solvers is proposed to solve the coupling of a fluid flow modeled by the incompressible Navier-Stokes equations and a porous media modeled by the Darcy's law. The finite difference discretization in time is based on the pressure Poisson equation formulation. At each time step, several augmented variables along the interface between the fluid flow and the porous media are introduced so that the coupled equations can be decoupled into several Poisson/Helmholtz equations with those augmented variables acting as jumps of the unknown solution and some directional derivatives. The augmented variables should be chosen so that the Beavers-Joseph-Saffman (BJS) or Beavers-Joseph (BJ)and other interface conditions are satisfied. It has been tested that a direct extension of the augmented idea in [27] does not work well when the fluid flow is modeled by the Navier-Stokes equations. One of the new ideas of this paper is to enforce the divergence free condition at the interface from the fluid side. In this way, the Schur complement matrix for the augmented variables is over-determined and the least squares solution is used for the coupling problem. The new augmented approach enables us to solve the Navier-Stokes and Darcy coupling efficiently with second order accurate velocity and pressure in the L-infinity norm for tested problems. The proposed new idea in enforcing the divergence free condition at the interface from the fluid side has also been utilized to solve the Stokes and Darcy coupling equations and shown to outperform the original method in [27]. In additional to the detailed accuracy check for the present method, some interesting numerical simulations for Navier-Stokes and Darcy coupling have been conducted in this paper as well. (C) 2018 Elsevier Ltd. All rights reserved.
A well-conditioned augmented system for solving Navier-Stokes equations in irregular domains
(2009-04-20)
Ito, Kazufumi
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Lai, Ming-Chih
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Li, Zhilin
;
應用數學系
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Department of Applied Mathematics
An augmented method based on a Cartesian grid is proposed for the incompressible Navier-Stokes equations in irregular domains. The irregular domain is embedded into a rectangular one so that a fast Poisson solver can be utilized in the projection method. Unlike several methods suggested in the literature that set the force strengths as unknowns, which often results in an ill-conditioned linear system, we set the jump in the normal derivative of the velocity as the augmented variable. The new approach improves the condition number of the system for the augmented variable significantly. Using the immersed interface method, we are able to achieve second order accuracy for the velocity. Numerical results and comparisons to benchmark tests are given to validate the new method. A lid-driven cavity flow with multiple obstacles and different geometries are also presented. (C) 2008 Elsevier Inc. All rights reserved.
ADAPTIVE MESH REFINEMENT FOR ELLIPTIC INTERFACE PROBLEMS USING THE NON-CONFORMING IMMERSED FINITE ELEMENT METHOD
(2011)
Wu, Chin-Tien
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Li, Zhilin
;
Lai, Ming-Chih
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應用數學系
;
數學建模與科學計算所(含中心)
In this paper, an adaptive mesh refinement technique is developed and analyzed for the non-conforming immersed finite element (IFE) method proposed in [27]. The IFE method was developed for solving the second order elliptic boundary value problem with interfaces across which the coefficient may be discontinuous. The IFE method was based on a triangulation that does not need to fit the interface. One of the key ideas of IFE method is to modify the basis functions so that the natural jump conditions are satisfied across the interface. The IFE method has shown to be order of O(h(2)) and O(h) in L(2) norm and H(1) norm, respectively. In order to develop the adaptive mesh refinement technique, additional priori and posterior error estimations are derived in this paper. Our new a-priori error estimation shows that the generic constant is only linearly proportional to ratio of the diffusion coefficient beta(-) and beta(+), which improves the corresponding result in [27]. We also show that a-posteriori error estimate similar to the one obtained by Bernardi and Verfurth [4] holds for the IFE solutions. Numerical examples support our theoretical results and show that the adaptive mesh refinement strategy is effective for the IFE approximation.
An augmented approach for Stokes equations with a discontinuous viscosity and singular forces
(2007-03-01)
Li, Zhilin
;
Ito, Kazufumi
;
Lai, Ming-Chih
;
應用數學系
;
Department of Applied Mathematics
For Stokes equations with a discontinuous viscosity across an arbitrary interface or/and singular forces along the interface, it is known that the pressure is discontinuous and the velocity is non-smooth. It has been shown that these discontinuities are coupled together, which makes it difficult to obtain accurate numerical solutions. In this paper, a new numerical method that decouples the jump conditions of the fluid variables through two augmented variables has been developed. The GMRES iterative method is used to solve the Schur complement system for the augmented variables that are only defined on the interface. The augmented approach also rescales the Stokes equations in such a way that a fast Poisson solver can be used in each iteration. Numerical tests using examples that have analytic solutions show that the new method has average second order accuracy for the velocity in the infinity norm. An example of a moving interface problem is also presented. (c) 2006 Elsevier Ltd. All rights reserved.
An augmented method for free boundary problems with moving contact lines
(2010-06-01)
Li, Zhilin
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Lai, Ming-Chih
;
He, Guowei
;
Zhao, Hongkai
;
應用數學系
An augmented immersed interface method (IIM) is proposed for simulating one-phase moving contact line problems in which a liquid drop spreads or recoils on a solid substrate. While the present two-dimensional mathematical model is a free boundary problem, in our new numerical method, the fluid domain enclosed by the free boundary is embedded into a rectangular one so that the problem can be solved by a regular Cartesian grid method. We introduce an augmented variable along the free boundary so that the stress balancing boundary condition is satisfied. A hybrid time discretization is used in the projection method for better stability. The resultant Helmholtz/Poisson equations with interfaces then are solved by the IIM in an efficient way. Several numerical tests including an accuracy check, and the spreading and recoiling processes of a liquid drop are presented in detail. (C) 2010 Elsevier Ltd. All rights reserved.
New Conservative Finite Volume Element Schemes for the Modified Regularized Long Wave Equation
(2017-04)
Yan, Jinliang
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Lai, Ming-Chih
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Li, Zhilin
;
Zhang, Zhiyue
;
應用數學系
In this paper, we propose a new energy-preserving scheme and a new momentum-preserving scheme for the modified regularized long wave equation. The proposed schemes are designed by using the discrete variational derivative method and the finite volume element method. For comparison, we also propose a finite volume element scheme. The conservation properties of the proposed schemes are analyzed and we find that the energy-preserving scheme can precisely conserve the discrete total mass and total energy, the momentum-preserving scheme can precisely conserve the discrete total mass and total momentum, while the finite volume element scheme merely conserve the discrete total mass. We also analyze their linear stability property using the Von Neumann theory and find that the proposed schemes are unconditionally linear stable. Finally, we present some numerical examples to illustrate the effectiveness of the proposed schemes.
New Finite Difference Methods Based on IIM for Inextensible Interfaces in Incompressible Flows
(2011-05-01)
Li, Zhilin
;
Lai, Ming-Chih
;
應用數學系
;
數學建模與科學計算所(含中心)
;
Department of Applied Mathematics
In this paper, new finite difference methods based on the augmented immersed interface method (IIM) are proposed for simulating an inextensible moving interface in an incompressible two-dimensional flow. The mathematical models arise from studying the deformation of red blood cells in mathematical biology The governing equations are incompressible Stokes or Navier-Stokes equations with an unknown surface tension, which should be determined in such a way that the surface divergence of the velocity is zero along the interface. Thus, the area enclosed by the interface and the total length of the interface should be conserved during the evolution process. Because of the nonlinear and coupling nature of the problem, direct discretization by applying the immersed boundary or immersed interface method yields complex nonlinear systems to be solved. In our new methods, we treat the unknown surface tension as an augmented variable so that the augmented IIM can be applied. Since finding the unknown surface tension is essentially an inverse problem that is sensitive to perturbations, our regularization strategy is to introduce a controlled tangential force along the interface, which leads to a least squares problem. For Stokes equations, the forward solver at one time level involves solving three Poisson equations with an interface. For Navier-Stokes equations, we propose a modified projection method that can enforce the pressure jump condition corresponding directly to the unknown surface tension. Several numerical experiments show good agreement with other results in the literature and reveal some interesting phenomena.