Phase Field Model of Thermo-Induced Marangoni Effects in the Mixtures and Its Numerical Simulations with Mixed Finite Element Method

In this paper, we study the Marangoni effects in the mixture of two Newtonian fluids due to the thermo-induced surface tension heterogeneity on the interface. We employ an energetic variational phase field model to describe its physical phenomena, and obtain the corresponding governing equations defined by a modified Navier-Stokes equations coupled with phase field and energy transport. A mixed Taylor-Hood finite element discretization together with full Newton's method are applied to this strongly nonlinear phase field model on a specific domain. Under different boundary conditions of temperature, the resulting numerical solutions illustrate that the thermal energy plays a fundamental role in the interfacial dynamics of two-phase flows. In particular, it gives rise to a dynamic interfacial tension that depends on the direction of temperature gradient, determining the movement of the interface along a sine/cosine-like curve.     

Numerical Simulations of Fiber Sedimentation in Navier-Stokes Flows

We perform numerical simulations of the sedimentation of rigid fibers suspended in a viscous incompressible fluid at nonzero Reynolds numbers. The fiber sedimentation system is modeled as a two-dimensional immersed boundary problem, which naturally accommodates the fluid-particle interactions and which allows the simulation of a large number of suspending fibers. We study the dynamics of sedimenting fibers under a variety of conditions, including differing fiber densities, Reynolds numbers, domain boundary conditions, etc. Simulation results are compared to experimental measurements and numerical results obtained in previous studies.     

Numerical Simulations of Particle Sedimentation Using the Immersed Boundary Method

We study the settling of solid particles in a viscous incompressible fluid contained within a two-dimensional channel, where the mass density of the particles is greater than that of the fluid. The fluid-structure interaction problem is simulated numerically using the immersed boundary method, where the added mass is incorporated using a Boussinesq approximation. Simulations are performed with a single circular particle, and also with two particles in various initial configurations. The terminal particle settling velocity and drag coefficient correspond closely with other theoretical, experimental and numerical results, and the particle trajectories reproduce the expected behavior qualitatively. In particular, simulations of a pair of interacting particles similar drafting-kissing-tumbling dynamics to that observed in other experimental and numerical studies.     

An Innovative Ski-Boot: Design,Numerical Simulations and Testing

The present work is concerned with the design of an innovative ski-boot. In order to optimize ergonomics and biomechanical behavior of the ski-boot it is important to take into account the orientation of the leg with respect to the ground. The SGS system (Stance Geometry System) developed in this work allows the skier to adjust for posture in the frontal plane by rotating the sole of the boot about the antero-posterior axis (ski-boot is then locked in the desired position before skiing). A simplified model of the effect of ski-boot deformation on skiing behavior is used to evaluate the minimal stiffness the system must have. An experimental analysis on the ski slopes was carried out to provide ski-boot deformations and loading data in different skiing conditions, to be used in numerical simulations. Finite Elements Method (FEM) simulations were performed for optimal design of the joint between ski-boot and sole. The active loads and local ski-boot deformations during small- and large-radius turns were experimentally determined and used to validate a FEM model of the ski-boot. The model was used to optimize the design for maximum stiffness and to demonstrate the efficacy of virtual design supported by proper experimental data. Mean loads up to 164% body weight were measured on the outer ski during turning. The new SGS design system allows the adjustment of lateral stance before using the ski-boot, optimizing the ski-boot stiffness through FEM analysis. Innovative aspects of this work included not only the stance geometry system ski-boot but also the setup of a virtual design environment that was validated by experimental evidence. An entire dataset describing loads during skiing has been obtained. The optimized SGS ski-boot increases intrinsic knee stability due to proper adjustment of lateral stance, guaranteeing appropriate stiffness of the ski-boot system.

Key Points

• Load acting during different phases of active skiing have been investigated in both qualitative and quantitative ways.
• The effects of ski-boot - ski-boot sole stiffness during skiing has been investigated.
• A ski-boot stance geometry system and an innovative design environment have been developed to make skiing easier and safer.

Key Words: Stance geometry system, stiffness, virtual design environment, FEM analysis, skiing performance     

Phase-Field Models for Biofilms II 2-D Numerical Simulations of Biofilm-Flow Interaction

We study the biofilm-flow interaction resulting in biofilm growth, deformation and detachment phenomena in a cavity and shear flow using the phase field model developed recently [28]. The growth of the biofilm and the biofilm-flow interaction in various flow and geometries are simulated using an extended Newtonian constitutive model for the biofilm mixture in 2-D. The model predicts growth patterns consistent with experimental findings with randomly distributed initial biofilm colonies. Shear induced deformation, and detachment in biofilms is simulated in a shear cell. Rippling, streaming, and ultimate detachment phenomena in biofilms are demonstrated in the simulations, respectively, in a shear cell. Possible merging of detached biofilm blobs in oscillatory shear is observed in simulations as well. Detachment due to the density variation is also investigated shedding light on the possible bacteria induced detachment.     

Full Wave Simulations of Lower Hybrid Waves in Toroidal Geometry with Non-Maxwellian Electrons

Analysis of the propagation of waves in the lower hybrid range of frequencies in the past has been done using ray tracing and the WKB approximation. Advances in algorithms and the availability of massively parallel computer architectures has permitted the solving of the Maxwell-Vlasov system for wave propagation directly [Wright et al., Phys. Plasmas (2004), 11, 2473-2479]. These simulations have shown that the bridging of the spectral gap (the difference between the high injected phase velocities and the slower phase velocity at which damping on electrons occurs) can be explained by the diffraction effects captured in the full wave algorithm - an effect missing in WKB based approaches. However, these full wave calculations were done with a Maxwellian electron distribution and the presence of RF power induces quasilinear velocity space diffusion that causes distortions away from an Maxwellian. With sufficient power, a flattened region or plateau is formed between the point of most efficient damping on electrons at about 2-3 ν_the and where collisional and quasilinear diffusion balance. To address this discrepancy and better model experiment, we have implemented [Valeo et al., "Full-wave Simulations of LH wave propagation in toroidal plasma with non-Maxwellian electron distributions", 18th Topical Conference on Radio Frequency Power in Plasmas, AIP Conference Proceedings (2007)] a non-Maxwellian dielectric in our full wave solver. We will show how these effects modify the electron absorption relative to what is found for a Maxwellian distribution.     

A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations

We develop a general framework for designing conservative numericalmethods based on summation by parts operators and split forms in space, combinedwith relaxation Runge-Kutta methods in time. We apply this framework to createnew classes of fully-discrete conservative methods for several nonlinear dispersivewave equations: Benjamin-Bona-Mahony (BBM), Fornberg-Whitham, Camassa-Holm,Degasperis-Procesi, Holm-Hone, and the BBM-BBM system. These full discretizationsconserve all linear invariants and one nonlinear invariant for each system. The spatialsemidiscretizations include finite difference, spectral collocation, and both discontinuous and continuous finite element methods. The time discretization is essentiallyexplicit, using relaxation Runge-Kutta methods. We implement some specific schemesfrom among the derived classes, and demonstrate their favorable properties throughnumerical tests.     

Numerical Simulations for Full History Recursive Multilevel Picard Approximations for Systems of High-Dimensional Partial Differential Equations

One of the most challenging issues in applied mathematics is to developand analyze algorithms which are able to approximately compute solutions of high-dimensional nonlinear partial differential equations (PDEs). In particular, it is veryhard to develop approximation algorithms which do not suffer under the curse of dimensionality in the sense that the number of computational operations needed by thealgorithm to compute an approximation of accuracy $ε$>0 grows at most polynomiallyin both the reciprocal 1/$ε$ of the required accuracy and the dimension $d∈mathbb{N}$of the PDE.Recently, a new approximation method, the so-called full history recursive multilevel Picard (MLP) approximation method, has been introduced and, until today, this approximation scheme is the only approximation method in the scientific literature which hasbeen proven to overcome the curse of dimensionality in the numerical approximationof semilinear PDEs with general time horizons. It is a key contribution of this articleto extend the MLP approximation method to systems of semilinear PDEs and to numerically test it on several example PDEs. More specifically, we apply the proposedMLP approximation method in the case of Allen-Cahn PDEs, Sine-Gordon-type PDEs,systems of coupled semilinear heat PDEs, and semilinear Black-Scholes PDEs in up to1000 dimensions. We also compare the performance of the proposed MLP approximation algorithm with a deep learning based approximation method from the scientificliterature.     

Numerical Solution of Blow-Up Problems for Nonlinear Wave Equations on Unbounded Domains

The numerical solution of blow-up problems for nonlinear wave equations on unbounded spatial domains is considered. Applying the unified approach, which is based on the operator splitting method, we construct the efficient nonlinear local absorbing boundary conditions for the nonlinear wave equation, and reduce the nonlinear problem on the unbounded spatial domain to an initial-boundary-value problem on a bounded domain. Then the finite difference method is used to solve the reduced problem on the bounded computational domain. Finally, a broad range of numerical examples are given to demonstrate the effectiveness and accuracy of our method, and some interesting propagation and behaviors of the blow-up problems for nonlinear wave equations are observed.     

A Modified Crank-Nicolson Numerical Scheme for the Flory-Huggins Cahn-Hilliard Model

In this paper we propose and analyze a second order accurate numerical scheme for the Cahn-Hilliard equation with logarithmic Flory Huggins energy potential. A modified Crank-Nicolson approximation is applied to the logarithmic nonlinear term, while the expansive term is updated by an explicit second order Adams-Bashforth extrapolation, and an alternate temporal stencil is used for the surface diffusion term. A nonlinear artificial regularization term is added in the numerical scheme, which ensures the positivity-preserving property, i.e., the numerical value of the phase variable is always between -1 and 1 at a point-wise level. Furthermore, an unconditional energy stability of the numerical scheme is derived, leveraging the special form of the logarithmic approximation term. In addition, an optimal rate convergence estimate is provided for the proposed numerical scheme, with the help of linearized stability analysis. A few numerical results, including both the constant-mobility and solution-dependent mobility flows, are presented to validate the robustness of the proposed numerical scheme.     

A Direct Numerical Simulation (DNS) Study on the Effect of Particle Configuration on Drag

Drag correlations are very important in particle-laden two-phase flow simulations. Some statistical studies have investigated extracting particle configurationfactors from simulation data to improve drag correlations. However, little attentionhas been paid to studying particle configuration effects on drag from the perspectiveof the flow mechanism. In this paper, a direct numerical simulation (DNS) methodbased on the second-order accurate immersion interface method is developed to provide highly reliable data. Then, the 'shielding' effect of the two-particle configurationon drag is comprehensively analysed under different angles, distances, and Reynoldsnumber $(Re)$ values, revealing that the complex configuration dependence of the draginfluence is attributed to the dominant flow mechanism, such as the 'pressure regionunit', 'nozzle', and 'wake' effects. Moreover, we study the 'superposition' effect of thethree-particle configuration on drag in a finite $Re$ range. The results show that whenthe surrounding particles do not directly shield each other, the drag influence calculated by pairwise linear superposition is close to the drag influence revealed by DNS.Otherwise, when the shielding phenomenon of the surrounding particles is obviousand the $Re$ is high, the drag influence of the nearest particle can represent the DNSresult.     

Numerical Modeling of Elastic Wave Propagation in a Fluid-Filled Borehole

We review the methods of simulating elastic wave propagation in a borehole. We considered two different approaches: a quasi-analytic approach using the Discrete Wavenumber Summation Method, and the purely numerical Finite Difference Method. We consider the special geometry of the borehole and discuss the problem in cylindrical coordinates. We point out some numerical difficulties that are particularly unique to this problem in cylindrical coordinates.     

A Biological Model of Acupuncture and Its Derived Mathematical Modeling and Simulations

(Aims) Acupuncture was employed since 2 millenaries, but the underlying mechanisms are not globally handled. The present study is aimed at proposing an explanation by pointing out involved processes and a convincing modeling to demonstrate its efficiency when carried out by trained practitioners.(Method) In the absence of global knowledge of any mechanism explaining the acupuncture process, a biological model is first developed, based on stimulation in a given domain around the needle tip of a proper mastocyte population by a mechanical stress, electrical, electromagnetic, or heat field. Whatever the type of mechanical or physical stimuli, mastocytes degranulate. Released messengers either facilitate the transfer of main mediators, or target their cognate receptors of local nerve terminals or after being conveyed by blood their receptors on cerebral cells. Signaling to the brain is fast by nervous impulses and delayed by circulating messengers that nevertheless distribute preferentially in the brain region of interest due to hyperemia. The process is self-sustained due to mastocyte chemotaxis from the nearby dense microcirculatory circuit and surrounding mastocyte pools, which are inadequate for acupuncture, but serve as a signal relay. A simple mathematical model is solved analytically. Numerical simulations are also carried out using the finite element method with mesh adaptivity.(Results) The analytical solution of the simple mathematical model demonstrates the conditions filled by a mastocyte population to operate efficiently. A theorem gives the blow-up condition. This analytical solution serves for validation of numerical experiments. Numerical simulations show that when the needle is positioned in the periphery of the acupoint or outside it, the response is too weak. This explains why along training is necessary as the needle implantation requires a precision with a magnitude of the order of 1 mm.(Conclusion) The acupoint must contain a highly concentrated population of mastocytes (e.g., very-high–amplitude, small-width Gaussian distribution) to get an initial proper response. Permanent signaling is provided by chemotaxis and continuous recruitment of mastocytes. Therefore, the density and distribution of mastocytes are crucial factors for efficient acupuncture as well as availability of circulating and neighboring pools of mastocytes.     

A Parallel Computational Model for Three-Dimensional,Thermo-Mechanical Stokes Flow Simulations of Glaciers and Ice Sheets

This paper focuses on the development of an efficient, three-dimensional, thermo-mechanical, nonlinear-Stokes flow computational model for ice sheet simulation. The model is based on the parallel finite element model developed in [14] which features high-order accurate finite element discretizations on variable resolution grids. Here, we add an improved iterative solution method for treating the nonlinearity of the Stokes problem, a new high-order accurate finite element solver for the temperature equation, and a new conservative finite volume solver for handling mass conservation. The result is an accurate and efficient numerical model for thermo-mechanical glacier and ice-sheet simulations. We demonstrate the improved efficiency of the Stokes solver using the ISMIP-HOM Benchmark experiments and a realistic test case for the Greenland ice-sheet. We also apply our model to the EISMINT-II benchmark experiments and demonstrate stable thermo-mechanical ice sheet evolution on both structured and unstructured meshes. Notably, we find no evidence for the "cold spoke" instabilities observed for these same experiments when using finite difference, shallow-ice approximation models on structured grids.     

On Energy Stable Runge-Kutta Methods for the Water Wave Equation and Its Simplified Non-Local Hyperbolic Model

Although interest in numerical approximations of the water wave equation grows in recent years, the lack of rigorous analysis of its time discretization inhibits the design of more efficient algorithms. In practice of water wave simulations, the trade-off between efficiency and stability has been a challenging problem. Thus to shed light on the stability condition for simulations of water waves, we focus on a model simplified from the water wave equation of infinite depth. This model preserves two main properties of the water wave equation: non-locality and hyperbolicity. For the constant coefficient case, we conduct systematic stability studies of the fully discrete approximation of such systems with the Fourier spectral approximation in space and general Runge-Kutta methods in time. As a result, an optimal time discretization strategy is provided in the form of a modified CFL condition, i.e. $∆t = \mathcal{O}(\sqrt{∆x}).$ Meanwhile, the energy stable property is established for certain explicit Runge-Kutta methods. This CFL condition solves the problem of efficiency and stability: it allows numerical schemes to stay stable while resolves oscillations at the lowest requirement, which only produces acceptable computational load. In the variable coefficient case, the convergence of the semi-discrete approximation of it is presented, which naturally connects to the water wave equation. Analogue of these results for the water wave equation of finite depth is also discussed. To validate these theoretic observation, extensive numerical tests have been performed to verify the stability conditions. Simulations of the simplified hyperbolic model in the high frequency regime and the water wave equation are also provided.     

Numerical Solution of the Upper-Convected Maxwell Model for Three-Dimensional Free Surface Flows

This work presents a finite difference technique for simulating three-dimensional free surface flows governed by the Upper-Convected Maxwell (UCM) constitutive equation. A Marker-and-Cell approach is employed to represent the fluid free surface and formulations for calculating the non-Newtonian stress tensor on solid boundaries are developed. The complete free surface stress conditions are employed. The momentum equation is solved by an implicit technique while the UCM constitutive equation is integrated by the explicit Euler method. The resulting equations are solved by the finite difference method on a 3D-staggered grid. By using an exact solution for fully developed flow inside a pipe, validation and convergence results are provided. Numerical results include the simulation of the transient extrudate swell and the comparison between jet buckling of UCM and Newtonian fluids.     

住院患者心理应激反应及影响因素关系的模型构建

目的建立住院患者心理应激反应及其影响因素关系的结构方程模型,对住院患者的应激过程进行全面整体的研究。方法采取分层、随机、整群抽样方法,使用相应问卷调查782例住院患者的应激源、应对方式、自我效能、社会支持、人格和心理应激反应。结果直接关系的检验进入回归方程的变量为：医院应激（β=0．357）、社会支持（β=-0．109）、病程（β=0．106）、经济状况（β=-0．120）、自我效能（r=-0．315）、神经质（β=0．388）、内外向（β=-0．153）、回避（β=0．095）、屈服（β=0．370）,均P〈0．01。医院应激与回避、屈服、神经质3个中介因素存在显著性相关（r=0．125、0．140、0．223,均P〈0．01）,自我效能与回避、屈服、社会支持、神经质、内外向5个中介因素存在显著性相关（r=0．113、-0．102、0．189、-0．192、0．196,均P〈0．01）。自我效能对各原因变量（神经质、内外向、屈服、社会支持）的口值下降,但仍然显著。由此建立的应激结构方程模型拟合较好,进入结构方程模型的中介变量为：医院应激、社会支持、病程、经济、自我效能、神经质、内外向、回避。自我效能是一个重要的中介变量;医院应激可直接和间接作用于心理反应。结论可以通过加强正面影响因素如自我效能,削弱负面影响因素如神经质、消极应对方式等措施,降低或缓冲患者心理应激强度,提高患者心理健康水平。     

住院患者心理应激反应及影响因素关系的模型构建

目的 建立住院患者心理应激反应及其影响因素关系的结构方程模型,对住院患者的应激过程进行全面整体的研究.方法 采取分层、随机、整群抽样方法,使用相应问卷调查782例住院患者的应激源、应对方式、自我效能、社会支持、人格和心理应激反应.结果 直接关系的检验进入回归方程的变量为:医院应激(β=0.357)、社会支持(β=-0.109)、病程(β=0.106)、经济状况(β=-0.120)、自我效能(r=-0.315)、神经质(β=0.388)、内外向(β=-0.153)、回避(β=0.095)、屈服(β=0.370),均P<0.01.医院应激与回避、屈服、神经质3个中介因素存在显著性相关(r=0.125、0.140、0.223,均P<0.01),自我效能与回避、屈服、社会支持、神经质、内外向5个中介因素存在显著性相关(r=0.113、-0.102、0.189、-0.192、0.196,均P<0.01).自我效能对各原因变量(神经质、内外向、屈服、社会支持)的口值下降,但仍然显著.由此建立的应激结构方程模型拟合较好,进入结构方程模型的中介变量为:医院应激、社会支持、病程、经济、自我效能、神经质、内外向、回避.自我效能是一个重要的中介变量;医院应激可直接和间接作用于心理反应.结论 可以通过加强正面影响因素如自我效能,削弱负面影响因素如神经质、消极应对方式等措施,降低或缓冲患者心理应激强度,提高患者心理健康水平.     

Numerical Simulations of Rarefied Gases in Curved Channels: Thermal Creep,Circulating Flow,and Pumping Effect

We present numerical simulations of a new system of micro-pump based on the thermal creep effect described by the kinetic theory of gases. This device is made of a simple smooth and curved channel with a periodic temperature distribution. Using the Boltzmann-BGK model as the governing equation for the gas flow, we develop a numerical method based on a deterministic finite volume scheme, implicit in time, with an implicit treatment of the boundary conditions. This method is comparatively less sensitive to the slow flow velocity than the usual Direct Simulation Monte Carlo method in case of long devices, and turns out to be accurate enough to compute macroscopic quantities like the pressure field in the channel. Our simulations show the ability of the device to produce a one-way flow that has a pumping effect.