Numerical Simulation for Moving Contact Line with Continuous Finite Element Schemes

In this paper, we compute a phase field (diffuse interface) model of Cahn-Hilliard type for moving contact line problems governing the motion of isothermal multiphase incompressible fluids. The generalized Navier boundary condition proposed by Qian et al. [1] is adopted here. We discretize model equations using a continuous finite element method in space and a modified midpoint scheme in time. We apply a penalty formulation to the continuity equation which may increase the stability in the pressure variable. Two kinds of immiscible fluids in a pipe and droplet displacement with a moving contact line under the effect of pressure driven shear flow are studied using a relatively coarse grid. We also derive the discrete energy law for the droplet displacement case, which is slightly different due to the boundary conditions. The accuracy and stability of the scheme are validated by examples, results and estimate order.     

An Adaptive Finite Element PML Method for the Open Cavity Scattering Problems

Consider the electromagnetic scattering of a time-harmonic plane wave by an open cavity which is embedded in a perfectly electrically conducting infinite ground plane. This paper is concerned with the numerical solutions of the transverse electric and magnetic polarizations of the open cavity scattering problems. In each polarization, the scattering problem is reduced equivalently into a boundary value problem of the two-dimensional Helmholtz equation in a bounded domain by using the transparent boundary condition (TBC). An a posteriori estimate based adaptive finite element method with the perfectly matched layer (PML) technique is developed to solve the reduced problem. The estimate takes account ofboththe finite element approximation error and the PML truncation error, where the latter is shown to decay exponentially with respect to the PML medium parameter and the thickness of the PML layer. Numerical experiments are presented and compared with the adaptive finite element TBC method for both polarizations to illustrate the competitive behavior of the proposed method.     

Sensitivity analysis approach for reduced‐order approximations of optimal control problems governed by Burgers equation

The reduced‐order model of the optimal control problem governed by Burgers equation is derived using the proper orthogonal decomposition (POD) method. The reduced‐order solution depending on parameters, which are different from the nominal values, may not be accurate if the POD basis functions depending on the nominal values are used to derive the reduced‐order model. It is known that Burgers equation is sensitive to the perturbations in the diffusion term, so we use the sensitivity information to improve the robustness of the POD solution by generating two new bases: extrapolated and expanded POD basis. We compare these different bases in terms of accuracy, robustness, and computational time. Copyright © 2015 John Wiley & Sons, Ltd.     

Applications of the exterior penalty method in constrained optimal control problems

This paper gives a theoretical analysis of the applications of the exterior penalty method in continuous-time non-linear programs and constrained optimal control problems. As an example, the Cauchy Inequality in the continuous case is proved. Also, the exterior penalty method is used to treat constrained optimal control problems. It is proved, under suitable assumptions, that a constrained optimal control problem can be solved by performing a sequence of unconstrained optimal control problems, and the constrained solution to the constrained optimal control problem can be obtained as the limit of the solutions to the sequence of unconstrained optimal control problems. In using the exterior penalty method to solve constrained optimal control problems it is usually assumed that each of the modified unconstrained optimal control problems has at least one solution. An existence theorem for these problems is also given.     

A Finite-Difference Lattice Boltzmann Approach for Gas Microflows

Finite-difference Lattice Boltzmann (LB) models are proposed for simulating gas flows in devices with microscale geometries. The models employ the roots of half-range Gauss-Hermite polynomials as discrete velocities. Unlike the standard LB velocity-space discretizations based on the roots of full-range Hermite polynomials, using the nodes of a quadrature defined in the half-space permits a consistent treatment of kinetic boundary conditions. The possibilities of the proposed LB models are illustrated by studying the one-dimensional Couette flow and the two-dimensional square driven cavity flow. Numerical and analytical results show an improved accuracy in finite Knudsen flows as compared with standard LB models.     

A Weighted Multi-Domain Spectral Penalty Method with Inhomogeneous Grid for Supersonic Injective Cavity Flows

In [J. Comput. Phys. 192(1), pp.325-354 (2003)], we have developed a multi-domain spectral method with stable and conservative penalty interface conditions for the numerical simulation of supersonic reactive recessed cavity flows with homogeneous grid. In this work, the previously developed methodology is generalized to inhomogeneous grid to simulate the two dimensional supersonic injector-cavity system. Non-physical modes in the solution generated at the domain interfaces due to the spatial grid inhomogeneity are minimized using the new weighted multi-domain spectral penalty method. The proposed method yields accurate and stable solutions of the injector-cavity system which agree well with experiments qualitatively. Through the direct numerical simulation of the injector-cavity system using the weighted method, the geometric effect of the cavity wall on pressure fluctuations is investigated. It is shown that the recessed slanted cavity attenuates pressure fluctuations inside cavity enabling the cavity to act potentially as a stable flameholder for scramjet engine.     

Burnett Order Stress and Spatially-Dependent Boundary Conditions for the Lattice Boltzmann Method

Stress boundary conditions for the lattice Boltzmann equation that are consistent to Burnett order are proposed and imposed using a moment-based method. The accuracy of the method with complicated spatially-dependent boundary conditions for stress and velocity is investigated using the regularized lid-driven cavity flow. The complete set of boundary conditions, which involve gradients evaluated at the boundaries, are implemented locally. A recently-derived collision operator with modified equilibria and velocity-dependent collision rates to reduce the defect in Galilean invariance is also investigated. Numerical results are in excellent agreement with existing benchmark data and exhibit second-order convergence. The lattice Boltzmann stress field is studied and shown to depart significantly from the Newtonian viscous stress when the ratio of Mach to Reynolds numbers is not negligibly small.     

New Unconditionally Stable Schemes for the Navier-Stokes Equations

In this paper we propose some efficient schemes for the Navier-Stokes equations. The proposed schemes are constructed based on an auxiliary variable reformulation of the underlying equations, recently introduced by Li et al. [20]. Our objective is to construct and analyze improved schemes, which overcome some of the shortcomings of the existing schemes. In particular, our new schemes have the capability to capture steady solutions for large Reynolds numbers and time step sizes, while keeping the error analysis available. The novelty of our method is twofold: i) Use the Uzawa algorithm to decouple the pressure and the velocity. This is to replace the pressure-correction method considered in [20]. ii) Inspired by the paper [21], we modify the algorithm using an ingredient to capture stationary solutions. In all cases we analyze a first- and second-order schemes and prove the unconditionally energy stability. We also provide an error analysis for the first-order scheme. Finally we validate our schemes by performing simulations of the Kovasznay flow and double lid driven cavity flow. These flow simulations at high Reynolds numbers demonstrate the robustness and efficiency of the proposed schemes.     

Numerical solution of a class of two‐dimensional quadratic optimal control problems by using Ritz method

In this paper, we focus on a class of a two‐dimensional optimal control problem with quadratic performance index (cost function). We are going to solve the problem via the Ritz method. The method is based upon the Legendre polynomial basis. The key point of the Ritz method is that it provides greater flexibility in the initial and non‐local boundary conditions. By using this method, the given two‐dimensional continuous‐time quadratic optimal control problem is reduced to the problem of solving a system of algebraic equations. We extensively discuss the convergence of the method and finally present our numerical findings and demonstrate the efficiency and applicability of the numerical scheme by considering three examples. Copyright © 2015 John Wiley & Sons, Ltd.     

An Augmented Lagrangian Deep Learning Method for Variational Problems with Essential Boundary Conditions

This paper is concerned with a novel deep learning method for variational problems with essential boundary conditions. To this end, we first reformulate the original problem into a minimax problem corresponding to a feasible augmented Lagrangian, which can be solved by the augmented Lagrangian method in an infinite dimensional setting. Based on this, by expressing the primal and dual variables with two individual deep neural network functions, we present an augmented Lagrangian deep learning method for which the parameters are trained by the stochastic optimization method together with a projection technique. Compared to the traditional penalty method, the new method admits two main advantages: i) the choice of the penalty parameter is flexible and robust, and ii) the numerical solution is more accurate in the same magnitude of computational cost. As typical applications, we apply the new approach to solve elliptic problems and (nonlinear) eigenvalue problems with essential boundary conditions, and numerical experiments are presented to show the effectiveness of the new method.     

Reduced order model compensator control of species transport in a CVD reactor

We propose the use of proper orthogonal decomposition (POD) techniques as a reduced basis method for computation of feedback controls and compensators in a high‐pressure chemical vapour deposition (HPCVD) reactor. In this paper, we present a proof‐of‐concept computational implementation of this method with a simplified growth example for III–V layers in which we implement Dirichlet boundary control of a dilute Group III reactant transported by convection and diffusion to an absorbing substrate with no reactions. We implement the model‐based feedback control using a reduced order state estimator based on observations of the flux of reactant at the substrate centre. This is precisely the type of measurements available with current sensing technology. We demonstrate that the reduced order state estimator or compensator system is capable of substantial control authority when applied to a high‐order system. In principle, these ideas can be extended to more general HPCVD control situations by including multiple species with gas‐phase reactions and surface reactions. Copyright © 2000 John Wiley & Sons, Ltd.     

A Well-Conditioned Hypersingular Boundary Element Method for Electrostatic Potentials in the Presence of Inhomogeneities within Layered Media

In this paper, we will present a high-order, well-conditioned boundary element method (BEM) based on Müller's hypersingular second kind integral equation formulation to accurately compute electrostatic potentials in the presence of inhomogeneity embedded within layered media. We consider two types of inhomogeneities: the first one is a simple model of an ion channel which consists of a finite height cylindrical cavity embedded in a layered electrolytes/membrane environment, and the second one is a Janus particle made of two different semi-spherical dielectric materials. Both types of inhomogeneities have relevant applications in biology and colloidal material, respectively. The proposed BEM gives$\mathcal{O}$(1) condition numbers, allowing fast convergence of iterative solvers compared to previous work using first kind of integral equations. We also show that the second order basis converges faster and is more accurate than the first order basis for the BEM.     

An extended penalty function approach to the numerical solution of constrained optimal control problems

This paper presents the extended penalty function method for solving constrained optimal control problems. Here, equality and inequality constraints on the state and control variables are considered. Using the extended penalty function method, the original constrained optimal control problem is transformed into a sequence of optimal control problems without inequality constraints. This is accomplished by adding to the cost functional a penalty term that takes on large values when the inequality constraints are violated and small values when the constraints are satisfied. Also presented is a continuation method for solving the sequence of differential-algebraic boundary value problems arising from the transformed optimal control problems. The effectiveness of the approach is demonstrated via examples.     

Near-Field Imaging of Interior Cavities

A novel method is developed for solving the inverse problem of reconstructing the shape of an interior cavity. The boundary of the cavity is assumed to be a small and smooth perturbation of a circle. The incident field is generated by a point source inside the cavity. The scattering data is taken on a circle centered at the source. The method requires only a single incident wave at one frequency. Using a transformed field expansion, the original boundary value problem is reduced to a successive sequence of two-point boundary value problems and is solved in a closed form. By dropping higher order terms in the power series expansion, the inverse problem is linearized and an explicit relation is established between the Fourier coefficients of the cavity surface function and the total field. A nonlinear correction algorithm is devised to improve the accuracy of the reconstruction. Numerical results are presented to show the effectiveness of the method and its ability to obtain subwavelength resolution.     

Development of normal-suction boundary control method based on inflow cannula pressure waveform for the undulation pump ventricular assist device

It is desirable to obtain the maximum assist without suction in ventricular assist devices (VADs). However, high driving power of a VAD may cause severe ventricle suction that can induce arrhythmia, hemolysis, and pump damage. In this report, an appropriate VAD driving level that maximizes the assist effect without severe systolic suction was explored. The target driving level was set at the boundary between low driving power without suction and high driving power with frequent suction. In the boundary range, intermittent mild suction may occur. Driving power was regulated by the suction occurrence. The normal-suction boundary control method was evaluated in a female goat implanted with an undulation pump ventricular assist device (UPVAD). The UPVAD was driven in a semipulsatile mode with heartbeat synchronization control. Systolic driving power was adjusted using a normal-suction boundary control method developed for this study. We confirmed that driving power could be maintained in the boundary range. Occurrences of suction were evaluated using the suction ratio. We defined this ratio as the number of suction occurrences divided by the number of heartbeats. The suction ratio decreased by 70% when the normal-suction boundary control method was used.     

Fluoroscopy-Based Surgical Navigation versus Fluoroscopic Guidance to Control Guide Wire Insertion for Osteosynthesis of Femoral Neck Fractures

AbstractBackground and Purpose: Long fluoroscopic times and related radiation exposure are a universal concern when C-arm fluoroscopy is used to guide percutaneous procedures. Fluoroscopy-based surgical navigation has been proposed as an alternative guidance method requiring limited fluoroscopic times to achieve precision. The purpose of this experimental study was to compare fluoroscopy-based surgical navigation with C-arm fluoroscopy for guidance with respect to the precision achieved, the fluoroscopic time, and the resources needed.Material and Methods: 114 guide wires were placed in 38 synthetic bone models using either C-arm fluoroscopy (group A) or fluoroscopy-based surgical navigation (group B) for guidance. Precision of guide wire placement was rated on the basis of an individual CT scan on all fracture models of both groups. The fluoroscopic time, the procedure time, and the number of attempts required to place the guide wires were documented as well.Results: An average fluoroscopic time of 26 s was needed with C-arm fluoroscopy to place three guide wires compared with an average fluoroscopic time of 2 s that was needed when fluoroscopy-based surgical navigation was used for guidance (p < 0.0001). Precision of guide wire placement and procedure times required to place the guide wires did not differ significantly between both groups. The number of attempts required for correct placement was found significantly reduced with fluoroscopy-based surgical navigation when compared with fluoroscopic guidance (p = 0.04).Conclusion: Fluoroscopic times to achieve precision are reduced with fluoroscopy-based surgical navigation compared with C-arm fluoroscopy. The impact of this new technique on minimally invasive, percutaneous procedures has to be evaluated in controlled prospective clinical studies.     

Fekete-Gauss Spectral Elements for Incompressible Navier-Stokes Flows: The Two-Dimensional Case

Spectral element methods on simplicial meshes, say TSEM, show both the advantages of spectral and finite element methods, i.e., spectral accuracy and geometrical flexibility. We present a TSEM solver of the two-dimensional (2D) incompressible Navier-Stokes equations, with possible extension to the 3D case. It uses a projection method in time and piecewise polynomial basis functions of arbitrary degree in space. The so-called Fekete-Gauss TSEM is employed, i.e., Fekete (resp. Gauss) points of the triangle are used as interpolation (resp. quadrature) points. For the sake of consistency, isoparametric elements are used to approximate curved geometries. The resolution algorithm is based on an efficient Schur complement method, so that one only solves for the element boundary nodes. Moreover, the algebraic system is never assembled, therefore the number of degrees of freedom is not limiting. An accuracy study is carried out and results are provided for classical benchmarks: the driven cavity flow, the flow between eccentric cylinders and the flow past a cylinder.     

植入型丹参缓释剂治疗股骨头坏死的实验研究

目的研究丹参缓释剂植入股骨头内治疗股骨头坏死的效果.方法参照Rich等方法,用马血清、醋酸强的松龙复制家兔股骨头坏死的动物模型,造模8周后实验组(A组)将丹参缓释剂40 mg植入到右侧股骨头内.对照组(B组)用同样方法植入赋形剂40 mg.正常对照(C组).术后不同时间做X线摄片、ECT扫描、组织学及扫描电镜检查.结果 X线片示A组右侧股骨头骨密度增高,透光区较小,B组骨密度减低,有大小不等透光区;ECT扫描示A组静态放射性分布中等浓聚,而B组静态图像放射性分布呈炸面包圈样改变,欠均匀;组织学检查显示A组骨空陷窝率、血管数、骨髓腔内脂肪细胞直径较B组明显改善(P＜0.01);扫描电镜检查显示A组松质骨结构较规则,骨陷窝结构完整,部分骨板断裂,骨小管可见.B组结构紊乱不清,骨板断裂,骨陷窝少,骨小管消失.结论丹参缓释剂植入到坏死的股骨头模型内,能改善股骨头血供,促进坏死骨的吸收、修复与重建.为股骨头坏死这种新的治疗方法提供实验依据.     

3D simulation of needle-tissue interaction with application to prostate brachytherapy.

This paper presents a needle-tissue interaction model that is a 3D extension of prior work based on needle and tissue models discretized using the Finite Element Method. The use of flexible needles necessitates remeshing the tissue during insertion, since simple mesh-node snapping to the tip can be detrimental to the simulation. In this paper, node repositioning and node addition are the two methods of mesh modification examined for coarse meshes. Our focus is on numerical approaches for fast implementation of these techniques. Although the two approaches compared, namely the Woodbury formula (matrix inversion lemma) and the boundary condition switches, have the same computational complexity, the Woodbury formula is shown to perform faster due to its cache-efficient order of operations. Furthermore, node addition is applied in constant time for both approaches, whereas node repositioning requires longer and variable computational times. A method for rendering the needle forces during simulated insertions into a 3D prostate model has been implemented. Combined with a detailed anatomical segmentation, this will be useful in teaching the practice of prostate brachytherapy. Issues related to discretization of such coupled (e.g., needle-tissue) models are also discussed.     

A Diffuse-Interface Lattice Boltzmann Method for the Dendritic Growth with Thermosolutal Convection

In this work, we proposed a diffuse-interface model for the dendritic growth with thermosolutal convection. In this model, the sharp boundary between the fluid and solid dendrite is firstly replaced by a thin but nonzero thickness diffuse interface, which is described by the order parameter, and the diffuse-interface based governing equations for the dendritic growth are presented. To solve the model for the dendritic growth with thermosolutal convection, we also developed a diffuse-interface multi-relaxation-time lattice Boltzmann (LB) method. In this method, the order parameter in the phase-field equation is combined into the force caused by the fluid-solid interaction, and the treatment on the complex fluid-solid interface can be avoided. In addition, four LB models are designed for the phase-field equation, concentration equation, temperature equation and the Navier-Stokes equations in a unified framework. Finally, we performed some simulations of the dendritic growth to test the present diffuse-interface LB method, and found that the numerical results are in good agreements with some previous works.