High-Order Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for Euler Equations with Gravitation on Unstructured Meshes

In this paper, we propose a high-order accurate discontinuous Galerkin (DG) method for the compressible Euler equations under gravitational fields on unstructured meshes. The scheme preserves a general hydrostatic equilibrium state and provably guarantees the positivity of density and pressure at the same time. Comparing with the work on the well-balanced scheme for Euler equations with gravitation on rectangular meshes, the extension to triangular meshes is conceptually plausible but highly nontrivial. We first introduce a special way to recover the equilibrium state and then design a group of novel variables at the interface of two adjacent cells, which plays an important role in the well-balanced and positivity-preserving properties. One main challenge is that the well-balanced schemes may not have the weak positivity property. In order to achieve the well-balanced and positivity-preserving properties simultaneously while maintaining high-order accuracy, we carefully design DG spatial discretization with well-balanced numerical fluxes and suitable source term approximation. For the ideal gas, we prove that the resulting well-balanced scheme, coupled with strong stability preserving time discretizations, satisfies a weak positivity property. A simple existing limiter can be applied to enforce the positivity-preserving property, without losing high-order accuracy and conservation. Extensive one- and two-dimensional numerical examples demonstrate the desired properties of the proposed scheme, as well as its high resolution and robustness.     

A New Approach of High Order Well-Balanced Finite Volume WENO Schemes and Discontinuous Galerkin Methods for a Class of Hyperbolic Systems with Source Terms

Hyperbolic balance laws have steady state solutions in which the flux gradients are nonzero but are exactly balanced by the source terms. In our earlier work [31–33], we designed high order well-balanced schemes to a class of hyperbolic systems with separable source terms. In this paper, we present a different approach to the same purpose: designing high order well-balanced finite volume weighted essentially non-oscillatory (WENO) schemes and RungeKutta discontinuous Galerkin (RKDG) finite element methods. We make the observation that the traditional RKDG methods are capable of maintaining certain steady states exactly, if a small modification on either the initial condition or the flux is provided. The computational cost to obtain such a well balanced RKDG method is basically the same as the traditional RKDG method. The same idea can be applied to the finite volume WENO schemes. We will first describe the algorithms and prove the well balanced property for the shallow water equations, and then show that the result can be generalized to a class of other balance laws. We perform extensive one and two dimensional simulations to verify the properties of these schemes such as the exact preservation of the balance laws for certain steady state solutions, the non-oscillatory property for general solutions with discontinuities, and the genuine high order accuracy in smooth regions.     

A Well-Balanced Gas Kinetic Scheme for Navier-Stokes Equations with Gravitational Potential

The hydrostatic equilibrium state is the consequence of the exact balance between hydrostatic pressure and external force. Standard finite volume cannot keep this balance exactly due to their unbalanced truncation errors. In this study, we introduce an auxiliary variable which becomes constant at isothermal hydrostatic equilibria and propose a well-balanced gas kinetic scheme for the Navier-Stokes equations. Through reformulating the convection term and the force term via the auxiliary variable, zero numerical flux and zero numerical source term are enforced at the hydrostatic equilibrium state instead of the balance between hydrostatic pressure and external force. Several problems are tested to demonstrate the accuracy and the stability of the new scheme. The results confirm that, the new scheme can preserve the exact hydrostatic solution. The small perturbation riding on hydrostatic equilibria can be calculated accurately. More importantly, the new scheme is capable of simulating the process of converging towards hydrostatic equilibria from a highly unbalanced initial condition. The ultimate state of zero velocity and constant temperature is achieved up to machine accuracy. As demonstrated by the numerical experiments, the current scheme is very suitable for small amplitude perturbation and long time running under gravitational potential.     

A Well-Balanced Positivity-Preserving Quasi-Lagrange Moving Mesh DG Method for the Shallow Water Equations

A high-order, well-balanced, positivity-preserving quasi-Lagrange moving mesh DG method is presented for the shallow water equations with non-flat bottom topography. The well-balance property is crucial to the ability of a scheme to simulate perturbation waves over the lake-at-rest steady state such as waves on a lake or tsunami waves in the deep ocean. The method combines a quasi-Lagrange moving mesh DG method, a hydrostatic reconstruction technique, and a change of unknown variables. The strategies in the use of slope limiting, positivity-preservation limiting, and change of variables to ensure the well-balance and positivity-preserving properties are discussed. Compared to rezoning-type methods, the current method treats mesh movement continuously in time and has the advantages that it does not need to interpolate flow variables from the old mesh to the new one and places no constraint for the choice of a update scheme for the bottom topography on the new mesh. A selection of one- and two-dimensional examples are presented to demonstrate the well-balance property, positivity preservation, and high-order accuracy of the method and its ability to adapt the mesh according to features in the flow and bottom topography.     

Mechanical factors influence the expression of endostatin--an inhibitor of angiogenesis--in tendons.

Avascular zones of tendons are predisposed for degenerative changes and spontaneous rupture. Therefore, we analyzed the expression of the endogenous angiogenesis inhibiting factor endostatin in human fetal and adult tendons by immunohistochemical and biochemical methods. Moreover, to elucidate factors involved in the regulation of vascularity, we exposed primary cultures of rat tendon cells to intermittent hydrostatic pressure (0.2 MPa, 0.1 Hz for 24 h), and measured the endostatin content by ELISA and the effect of the conditioned medium to the proliferation of human umbilical vein endothelial cells (HUVEC).In fetal tendons high endostatin levels could be quantified by ELISA whereas low levels were found in adult tissue. In fetal tendons endostatin could also be immunostained in endothelial cells but mainly in fibroblasts. In adult Achilles tendons endostatin immunostaining was restricted to endothelial cells. In the tibialis posterior tendon--as an example for "wrap around"--endostatin immunostaining remained positive in the fibrocartilage adjacent to the medial malleolus. Fibrochondrocytes of the type II collagen positive fibrocartilage were intensively stained with the endostatin antibody. Factor VIII immunostaining showed that this region was largely avascular. Monolayer cultures of tendon cells released measurable amounts of endostatin into their culture supernatants. Application of intermittent hydrostatic pressure increased endostatin expression significantly. The conditioned media of tendon fibroblasts cultivated under intermittent hydrostatic pressure inhibited the proliferation of HUVEC in a dose dependent way.The spatial expression of endostatin in adult gliding tendons suggests that mechanical factors are involved in the regulation of this anti-angiogenic factor. In accordance, tendon cells exposed to intermittent hydrostatic pressure inhibit endothelial cell proliferation via humoral factors and produce endostatin. These findings support the view that the development and maintenance of avascular zones in tendons might be caused by a mechanically induced upregulation of anti-angiogenic factors.     

An Approximate Riemann Solver for Fluid-Solid Interaction Problems with Mie-Grüneisen Equations of State

We propose an approximate solver for compressible fluid-elastoplastic solid Riemann problems. The fluid and hydrostatic components of the solid are described by a family of general Mie-Grüneisen equations of state, and the hypo-elastoplastic constitutive law we studied includes the perfect plasticity and linearly hardened plasticity. The approximate solver provides the interface stress and normal velocity by an iterative method. The well-posedness and convergence of our solver are verified with mild assumptions on the equations of state. The proposed solver is applied in computing the numerical flux at the phase interface for our compressible multi-medium flow simulation on Eulerian girds. Several numerical examples, including Riemann problems, underground explosion and high speed impact applications, are presented to validate the approximate solver.     

One-Dimensional Blood Flow with Discontinuous Properties and Transport: Mathematical Analysis and Numerical Schemes

In this paper we consider the one-dimensional blood flow model with discontinuous mechanical and geometrical properties, as well as passive scalar transport, proposed in [E.F. Toro and A. Siviglia. Flow in collapsible tubes with discontinuous mechanical properties: mathematical model and exact solutions. Communications in Computational Physics. 13(2), 361-385, 2013], completing the mathematical analysis by providing new propositions and new proofs of relations valid across different waves. Next we consider a first order DOT Riemann solver, proposing an integration path that incorporates the passive scalar and proving the well-balanced properties of the resulting numerical scheme for stationary solutions. Finally we describe a novel and simple well-balanced, second order, non-linear numerical scheme to solve the equations under study; by using suitable test problems for which exact solutions are available, we assess the well-balanced properties of the scheme, its capacity to provide accurate solutions in challenging flow conditions and its accuracy.     

Variable High-Order Multiblock Overlapping Grid Methods for Mixed Steady and Unsteady Multiscale Viscous Flows,Part II: Hypersonic Nonequilibrium Flows

The variable high-order multiblock overlapping (overset) grids method of Sjögreen & Yee [CiCP, Vol. 5, 2009] for a perfect gas has been extended to nonequilibrium flows. This work makes use of the recently developed high-order well-balanced shock-capturing schemes and their filter counterparts [Wang et al., J. Comput. Phys., 2009, 2010] that exactly preserve certain non-trivial steady state solutions of the chemical nonequilibrium governing equations. Multiscale turbulence with strong shocks and flows containing both steady and unsteady components is best treated by mixing of numerical methods and switching on the appropriate scheme in the appropriate subdomains of the flow fields, even under the multiblock grid or adaptive grid refinement framework. While low dissipative sixth- or higher-order shock-capturing filter methods are appropriate for unsteady turbulence with shocklets, second- and third-order shock-capturing methods are more effective for strong steady or nearly steady shocks in terms of convergence. It is anticipated that our variable high-order overset grid framework capability with its highly modular design will allow for an optimum synthesis of these new algorithms in such a way that the most appropriate spatial discretizations can be tailored for each particular region of the flow. In this paper some of the latest developments in single block high-order filter schemes for chemical nonequilibrium flows are applied to overset grid geometries. The numerical approach is validated on a number of test cases characterized by hypersonic conditions with strong shocks, including the reentry flow surrounding a 3D Apollo-like NASA Crew Exploration Vehicle that might contain mixed steady and unsteady components, depending on the flow conditions.     

High‐order observers‐based LQ control scheme for wind speed and uncertainties estimation in WECSs

Tip speed ratio control is a popular method in wind energy conversion systems in order to capture the maximum power. This method, however, requires wind speed information, which is difficult in practice to accurately measure it. Therefore, estimation methods are usually applied, where a high‐precision estimate leads to a high‐efficient system. Based on the fact that the wind speed varies in a random way, this paper proposes a generalized high‐order observer to estimate the aerodynamic torque and the wind speed accordingly. This observer algorithm releases the assumption that the wind speed should be slowly varying, which is required in previous observer designs. Moreover, two other generalized high‐order observers are also applied to estimate the uncertainties, which depend on state variables and cannot be considered as slow‐varying disturbances. Using the outputs of these observers, a robust high‐performance optimal control system is developed for the rotor speed to keep the optimal tip speed ratio. The stability analysis of the designed control system is fully presented. The effectiveness of the proposed technique is validated via simulation studies.     

Numerical Validation for High Order Hyperbolic Moment System of Wigner Equation

A globally hyperbolic moment system up to arbitrary order for the Wigner equation was derived in [6]. For numerically solving the high order hyperbolic moment system therein, we in this paper develop a preliminary numerical method for this system following the NR$xx$ method recently proposed in [8], to validate the moment system of the Wigner equation. The developedmethod can keep both mass and momentum conserved, and the variation of the total energy under control though it is not strictly conservative. We systematically study the numerical convergence of the solution to the moment system both in the size of spatial mesh and in the order of the moment expansion, and the convergence of the numerical solution of the moment system to the numerical solution of the Wigner equation using the discrete velocity method. The numerical results indicate that the high order moment system in [6] is a valid model for the Wigner equation, and the proposed numerical method for the moment system is quite promising to carry out the simulation of the Wigner equation.     

Optimal proportional plus integral control for regulator and tracking problems

Two new design methods are proposed for optimal PI control which give fast recovery from load disturbances without producing large overshoots associated with set-point changes. Unlike existing methods, such as set-point filtering or any other scheme that essentially detunes the controller for set-point changes, the proposed methods do not increase the rise time significantly in order to reduce overshoot. The first method resets the integral term in the control law when the new set point is reached. The second method is a modified model-following approach that does not increase the overall dimension of the system. Two numerical examples demonstrate that the proposed methods are superior to alternatives such as set-point filtering and conventional model following.     

Space-Time Discontinuous Galerkin Method for Maxwell's Equations

A fully discrete discontinuous Galerkin method is introduced for solving time-dependent Maxwell's equations. Distinguished from the Runge-Kutta discontinuous Galerkin method (RKDG) and the finite element time domain method (FETD), in our scheme, discontinuous Galerkin methods are used to discretize not only the spatial domain but also the temporal domain. The proposed numerical scheme is proved to be unconditionally stable, and a convergent rate $\mathcal{O}((∆t)^{r+1}+h^{k+1/2})$ is established under the $L^2$ -norm when polynomials of degree at most $r$ and $k$ are used for temporal and spatial approximation, respectively. Numerical results in both 2-D and 3-D are provided to validate the theoretical prediction. An ultra-convergence of order $(∆t)^{2r+1}$ in time step is observed numerically for the numerical fluxes w.r.t. temporal variable at the grid points.     

A Probabilistic Automatic Steady State Detection Method for the Direct Simulation Monte Carlo

The statistical error associated with sampling in the DSMC method can be categorized as type I and II, which are caused by the incorrect rejection and acceptance of the null hypothesis, respectively. In this study, robust global and local automatic steady state detection methods were developed based on an ingenious method based purely on the statistics and kinetics of particles. The key concept is built upon probabilistic automatic reset sampling (PARS) to minimize the type II error caused by incorrect acceptance of the samples that do not belong to the steady state. The global steady state method is based on a relative standard variation of collisional invariants, while the local steady state method is based on local variations in the distribution function of particles at each cell. In order to verify the capability of the new methods, two benchmark cases–the one-dimensional shear-driven Couette flow and the two-dimensional high speed flow past a vertical wall–were extensively investigated. Owing to the combined effects of the automatic detection and local reset sampling, the local steady state detection method yielded a substantial gain of 30-36% in computational cost for the problem studied. Moreover, the local reset feature outperformed the automatic detection feature in overall computational savings.     

A High Order Central DG Method of the Two-Layer Shallow Water Equations

In this paper, we focus on the numerical simulation of the two-layer shallow water equations over variable bottom topography. Although the existing numerical schemes for the single-layer shallow water equations can be extended to two-layer shallow water equations, it is not a trivial work due to the complexity of the equations. To achieve the well-balanced property of the numerical scheme easily, the two-layer shallow water equations are reformulated into a new form by introducing two auxiliary variables. Since the new equations are only conditionally hyperbolic and their eigenstructure cannot be easily obtained, we consider the utilization of the central discontinuous Galerkin method which is free of Riemann solvers. By choosing the values of the auxiliary variables suitably, we can prove that the scheme can exactly preserve the still-water solution, and thus it is a truly well-balanced scheme. To ensure the non-negativity of the water depth, a positivity-preserving limiter and a special approximation to the bottom topography are employed. The accuracy and validity of the numerical method will be illustrated through some numerical tests.     

人工关节凝胶软骨材料的动态破坏研究

目的研究人工关节凝胶软骨材料在高应变率下的特性，探讨该材料应用于膝关节修补的可行性。方法通过霍普金森压力杆（SHPB）技术的应用，来研究凝胶材料在高应变率下的破坏情形。结果凝胶在冲击应变率达到1200s^-1时，凝胶并无任何破坏发生；当应变率为2000s。时，凝胶中心开始出现裂缝；达到3000s。时，凝胶中心部分已完全破裂了。结论未膨胀的凝胶可承受20MPa的瞬间压应力而不会发生破坏，所以凝胶用来作为软骨的修补材料是十分有潜力的。     

Responsiveness of two methods for measuring foot and ankle volume

BACKGROUND: Measurements of volume are taken in the clinical environment to determine the extent of swelling and to evaluate the effects of treatment interventions. In the research setting, volume measurements are taken to determine experimental outcomes. Water displacement and figure-of-eight methods are highly reliable for measuring foot and ankle volumes, but the responsiveness of the two methods has not been compared. This study was designed to investigate effects of manipulating hydrostatic and blood pressures on foot and ankle volume and responsiveness of the two methods to induced changes in foot and ankle volume. METHODS: Intervention effects on volume were compared using water displacement and tape (figure-of-eight) methods. Foot and ankle volume was measured in each of the 30 participants while they were supine, sitting, and sitting with a sphygmomanometer cuff inflated around the lower thigh to occlude blood flow into the leg. These variations allowed manipulation of the hydrostatic and blood pressures acting on the foot and ankle. RESULTS: Data from the water displacement method showed that a significant increase in volume of 31 mL (p < 0.002) occurred with the cuff in place, but this was not detected using the figure-of-eight method. No significant difference (p > 0.136) between the sitting and supine positions was detected using either method. CONCLUSIONS: Limb dependency while sitting or lying had no effect on measures from either method, but volumetry had a higher responsiveness to changes induced by application of the cuff. This effect was interpreted as arterial leakage past the cuff. CLINICAL RELEVANCE: The figure-of-eight tape method and the water displacement technique for measuring ankle and foot volume may not be interchangeable. Changes in volume of the ankle and foot are better measured by the water displacement technique, but for measurements of ankle volume alone, the tape method is appropriate. The responsiveness of the tape method to changes in volume is yet to be determined.     

Novel Imaging Analysis System to Measure the Spatial Dimension of Engineered Tissue Construct

The measurement of the spatial dimensions of tissue‐engineered constructs is very important for their clinical applications. In this study, a novel method to measure the volume of tissue‐engineered constructs was developed using iterative mathematical computations. The method measures and analyzes three‐dimensional (3D) parameters of a construct to estimate its actual volume using a sequence of software‐based mathematical algorithms. The mathematical algorithm is composed of two stages: the shape extraction and the determination of volume. The shape extraction utilized 3D images of a construct: length, width, and thickness, captured by a high‐quality camera with charge coupled device. The surface of the 3D images was then divided into fine sections. The area of each section was measured and combined to obtain the total surface area. The 3D volume of the target construct was then mathematically obtained using its total surface area and thickness. The accuracy of the measurement method was verified by comparing the results with those obtained from the hydrostatic weighing method (Korea Research Institute of Standards and Science [KRISS], Korea). The mean difference in volume between two methods was 0.0313 ± 0.0003% (n = 5, P = 0.523) with no significant statistical difference. In conclusion, our image‐based spatial measurement system is a reliable and easy method to obtain an accurate 3D volume of a tissue‐engineered construct.     

Intravesical distension in the corrugated tuberculotic bladder

The treatment problem of urinary tract post-tuberculous changes is not completely solved. Current methods do not always give good results and are accompanied by some complications. The authors describe their own modified method of hydrostatic intravesical distension of the tuberculotic-cirrhotic bladder, using epidural anesthesia, and worked out indications for it. The results show an increase in the bladder volume from 100-120 to 250-300 ml with a decreased urination frequency in 8 patients. The method occurred to be very convenient-to-use due to the absence of any complications.     

A Geometry-Preserving Finite Volume Method for Compressible Fluids on Schwarzschild Spacetime

We consider the relativistic Euler equations governing spherically symmetric, perfect fluid flows on the outer domain of communication of Schwarzschild spacetime, and we introduce a version of the finite volume method which is formulated from the geometric formulation (and thus takes the geometry into account at the discretization level) and is well-balanced, in the sense that it preserves steady solutions to the Euler equations on the curved geometry under consideration. In order to formulate our method, we first derive a closed formula describing all steady and spherically symmetric solutions to the Euler equations posed on Schwarzschild spacetime. Second, we describe a geometry-preserving, finite volume method which is based on the family of steady solutions to the Euler system. Our scheme is second-order accurate and, as required, preserves the family of steady solutions at the discrete level. Numerical experiments are presented which demonstrate the efficiency and robustness of the proposed method even for solutions containing shock waves and nonlinear interacting wave patterns. As an application, we investigate the late-time asymptotics of perturbed steady solutions and demonstrate its convergence for late time toward another steady solution, taking the overall effect of the perturbation into account.     

Intussusception: clinical prediction of outcome of barium reduction

Patients with intussusception who have necrotic bowel requiring bowel resection or who are at major risk of perforation from attempted hydrostatic reduction may be better managed without barium reduction. Ideally, the clinician would like to identify such patients at presentation. Two groups of patients, representing the extreme ends of the intussusception treatment spectrum were investigated in order to highlight the clinical features of patients in whom an attempted barium reduction is not justified. A group of 200 patients who had successful and safe hydrostatic reduction was compared with 104 patients who ultimately required bowel resection. Four features were found to be indicators of an increased likelihood of resection: age 3 months or less, or greater than 2 years, duration of symptoms greater than 24 h, presence of small bowel obstruction on plain radiology, and clinical assessment of dehydration greater than 5%. The validity of these features as prognostic indicators was assessed by applying them to all patients who had attempted barium reduction to see how they predicted patient outcome. In isolation, each feature was found to be compatible with a safe and successful hydrostatic reduction. The rate of resection was increased in patients with multiple adverse features and in these patients the enema technique may require modification. Patients with three or four adverse features had an unacceptably high incidence of gangrenous bowel requiring resection and a low likelihood of successful hydrostatic reduction. It is believed that attempts at reduction are not appropriate in these patients.