Multivariate stress scenarios and solvency

We show how the probabilistic concepts of half-space trimming and depth may be used to define convex scenario sets Q_α for stress testing the risk factors that affect the solvency of an insurance company over a prescribed time period. By choosing the scenario in Q_α which minimizes net asset value at the end of the time period, we propose the idea of the least solvent likely event (LSLE) as a solution to the forward stress testing problem. By considering the support function of the convex scenario set Q_α, we establish theoretical properties of the LSLE when financial risk factors can be assumed to have a linear effect on the net assets of an insurer. In particular, we show that the LSLE may be interpreted as a scenario causing a loss equivalent to the Value-at-Risk (VaR) at confidence level α, provided the α-quantile is a subadditive risk measure on linear combinations of the risk factors. In this case, we also show that the LSLE has an interpretation as a per-unit allocation of capital to the underlying risk factors when the overall capital is determined according to the VaR. These insights allow us to define alternative scenario sets that relate in similar ways to coherent measures, such as expected shortfall. We also introduce the most likely ruin event (MLRE) as a solution to the problem of reverse stress testing.     

Study of wave–wind interaction at a seawall using a numerical wave channel

This paper presents the study on wind and waves interactions at a seawall using a numerical wave channel. The numerical experiments were conducted for wave overtopping of a 1/4 sloping seawall using several conditions of incident waves and wind speeds. The numerical results were verified against laboratory data in a case for wave overtopping without wind effects. The interaction of waves and wind was analyzed in term of mean wave quantities, overtopping rate and variation of wind velocity at some selected locations. The results showed that the overtopping rate was strongly affected by wind and the wind field was also significantly modified by waves. There exists an effective range of wind speed in comparison with the local shallow wave speed at the breaking location, which gives significant effects to the wave overtopping rates. The maximum of wind adjustment coefficient f_w for wave overtopping rate was strongly related to the mean overtopping rate in the case for no wind. This study also showed that when the mean overtopping rate was greater than 5 × 10⁻⁴ m³/s/m, the maximum of wind adjustment coefficient f_w approached to a specific value of about 1.25.     

A geometrically nonlinear (3,2) reduced degree-of-freedom unified zigzag laminated beam element for large deformation analysis

A geometrically nonlinear (3,2) unified zigzag beam element is developed with a reduced number of degree-of-freedom for the large deformation analysis. The main merit of the beam element model is the Kirchhoff and Cauchy shear stress solution for large deformation and large strain analysis is more accurate. The geometrically nonlinearity is considered in the calculation of the zigzag coefficients. Thus, the results of shear Cauchy stress are matching well with solid element analysis in case of the beam with aspect ratio greater than 20 under large deformation. The zigzag coefficients are derived explicitly. The Green strain and the second Piola Kirchhoff stress are used. The second Piola Kirchhoff shear stress is continuous at the interface between adjacent layers priori. The bottom surface second Piola Kirchhoff shear stress condition is used to determine the zigzag coefficient and the top surface second Piola Kirchhoff shear stress condition is used to reduce one degree-of-freedom. The nonlinear finite element equations are derived. In the numerical tests, several benchmark problems with large deformation are solved to verify the accuracy. It is observed that the proposed beam has accurate solution for beam with aspect ratio greater than 20. The second Piola Kirchhoff and Cauchy shear stress accuracy is also good. A convergence study is also presented.     

A non-Newtonian fluid flow model for blood flow through a catheterized artery—Steady flow

In this paper the effects catheterization and non-Newtonian nature of blood in small arteries of diameter less than 100 μm, on velocity, flow resistance and wall shear stress are analyzed mathematically by modeling blood as a Herschel–Bulkley fluid with parameters n and θ and the artery and catheter by coaxial rigid circular cylinders. The influence of the catheter radius and the yield stress of the fluid on the yield plane locations, velocity distributions, flow rate, wall shear stress and frictional resistance are investigated assuming the flow to be steady. It is shown that the velocity decreases as the yield stress increases for given values of other parameters. The frictional resistance as well as the wall shear stress increases with increasing yield stress, whereas the frictional resistance increases and the wall shear stress decreases with increasing catheter radius ratio k (catheter radius to vessel radius). For the range of catheter radius ratio 0.3–0.6, in smaller arteries where blood is modeled by Herschel–Bulkley fluid with yield stress θ = 0.1, the resistance increases by a factor 3.98–21.12 for n = 0.95 and by a factor 4.35–25.09 for n = 1.05. When θ = 0.3, these factors are 7.47–124.6 when n = 0.95 and 8.97–247.76 when n = 1.05.     

Application of Adomian's approximation to blood flow through arteries in the presence of a magnetic field

The present investigation deals with the application of Adomian's decomposition method to blood flow through a constricted artery in the presence of an external transverse magnetic field which is applied uniformly. The blood flowing through the tube is assumed to be Newtonian in character. The expressions for the two-term approximation to the solution of stream function, axial velocity component and wall shear stress are obtained in this analysis. The numerical solutions of the wall shear stress for different values of Reynold number and Hartmann number are shown graphically. The solution of this theoretical result for a particular Hartmann number is compared with the integral method solution of Morgan and Young [17].     

梯形复式断面均匀流水力计算

对由主槽、斜边坡以及边滩三部分组成的梯形复式断面渠道均匀流的水力计算进行了研究,采用表观切应力表示各子断面间的横向动量交换,结合各部分力的平衡关系式,推导出了主槽、斜边坡和边滩平均流速的计算公式．采用UK-FCF的实验数据对公式进行了验证,滩槽相对水深为0.057~0.4的7组工况计算值与实测值的比较表明,不考虑表观切应力的计算值与实测值的一致性都较差．比较而言,斜边坡部分平均流速在水深较小时计算值大于实测值,在水深较大时计算值小于或接近实测值,表观切应力对边滩部分平均流速的影响不明显,而主槽部分若不考虑表观切应力的影响,则计算值与实测值差别较大．运用三段式模型与刘沛清等提出的两段式模型分别进行了有边坡梯形断面复式渠道的水力计算,结果表明对于有边坡段的梯形断面渠道（河道）,三段式模型更加有效．最后,将计算的表观切应力与实测表观切应力进行了比较,两者的一致性表明采用的动量输运系数是合适的．     

Assessment of an improved turbulence model in simulating the unsteady flows around a D-shaped cylinder and an open cavity

Most engineering flows are still predicted by the conventional Reynolds-averaged Navier-Stokes method because of the low requirements of the computational quantities. However, the resolution capability of Reynolds-averaged Navier-Stokes models is still open to deliberation, especially in the recirculation and wake regions, where the vortical flows dominate. In the present work, an improved turbulence model derived from the original shear stress transport k-ω model is proposed and its superiority is assessed by our modeling the unsteady flows around a D-shaped cylinder and an open cavity, corresponding to two different Reynolds numbers. The results are compared with results from experiments and other turbulence models in terms of the flow morphology and mean velocity profiles. This shows that the predictive accuracy of the modified turbulence model is increased significantly in the bluff body wake flows and in the shear layer and separation flows of the cavity. Some special vortex structures can be captured in the open cavity, in which the secondary vortex emerging from the shear layer and the separation vortex near the trailing edge can induce large flow instability, and this phenomenon should be eliminated in engineering applications. It is believed that this improved turbulence model can be used for the more complex turbomachinery flows with better prediction of the hydrodynamic/aerodynamic performance and the unsteady vortical flows, which can provide some guidelines to design or optimize rotating machines.     

Numerical study of wave overtopping of a seawall supported by porous structures

This paper presents a VOF-based two-phase flow model for the simulation of wave interactions with seawall supported by a porous terrace. Firstly, the model was verified against laboratory data in a simple case for wave overtopping of a vertical wall. Comparison of computed and measured wave properties showed reasonably good agreement. The model was then applied to study the interactions of waves and a seawall protected by porous structures with a permeable terrace. The application results showed that the overtopping rate was strongly related to the energy dissipation through the drag force; the porous reef and terrace were very effective to produce a low crest type seawall. It is concluded that there exist two optimum values of porosity of the submerged reef, about of 0.25 and 0.7, that give minimum overtopping rates. Whereas, there is an effective range of porosity of the permeable terrace varying from 0.4 to 0.65 for significantly reducing the overtopping rate. The verification results confirm that the VOF-based two-phase flow model is sufficient robust to simulate the wave overtopping of coastal structures with reasonable accuracy.     

轻质热防护系统多层材料组合结构的热应力分析

提出了轻质热防护系统外面板使用多层结构的概念,设计了2种热防护材料组合构成的3种铺层方案．通过模拟飞行器再入大气层时受到的机械和热载荷条件,数值计算得到了层间剪切力、底部温度和y方向位移．计算结果发现,层间剪切力发生在边缘部位且呈反对称分布;选用高热导率和高热容材料能够减少材料内的温度梯度,进而有效地降低结构的热应力和热变形;在均匀温度场情况下,两种材料的热膨胀系数之差越小,则层间剪切力越小．该研究表明不同的材料组合和铺层次序的多层结构,可以满足不同设计要求,具有优化设计潜力．     

Analytical and numerical results for the Swift-Hohenberg equation

The problem of the Swift-Hohenberg equation is considered in this paper. Using homotopy analysis method (HAM) the series solution is developed and its convergence is discussed and documented here for the first time. In particular, we focus on the roles of the eigenvalue parameter α and the length parameter l on the large time behaviour of the solution. For a given time t, we obtain analytical expressions for eigenvalue parameter α and length l which show how different values of these parameters may lead to qualitatively different large time profiles. Also, the results are presented graphically. The results obtained reveal many interesting behaviors that warrant further study of the equations related to non-Newtonian fluid phenomena, especially the shear-thinning phenomena. Shear thinning reduces the wall shear stress.     

Pulsatile flow of Herschel–Bulkley fluid through catheterized arteries – A mathematical model

The pulsatile flow of blood through catheterized artery has been studied in this paper by modeling blood as Herschel–Bulkley fluid and the catheter and artery as rigid coaxial circular cylinders. The Herschel–Bulkley fluid has two parameters, the yield stress θ and the power index n. Perturbation method is used to solve the resulting quasi-steady nonlinear coupled implicit system of differential equations. The effects of catheterization and non-Newtonian nature of blood on yield plane locations, velocity, flow rate, wall shear stress and longitudinal impedance of the artery are discussed. The existence of two yield plane locations is investigated and their dependence on yield stress θ, amplitude A, and time t are analyzed. The width of the plug core region increases with increasing value of yield stress at any time. The velocity and flow rate decrease, whereas wall shear stress and longitudinal impedance increase for increasing value of yield stress with other parameters held fixed. On the other hand, the velocity, flow rate and wall shear stress decrease but resistance to flow increases as the catheter radius ratio (ratio of catheter radius to vessel radius) increases with other parameters fixed. The results for power law fluid, Newtonian fluid and Bingham fluid are obtained as special cases from this model.     

Singular stress analysis near sharp corners in anisotropic notched plates subjected to bending loads

This paper presents an analytical approach to determine the singular stress fields near sharp corners in anisotropic notched plate subjected to bending. The first-order shear deformation plate theory (FSDPT) is used and the material anisotropy is modeled by the Stroh formalism. Based on the Mellin transform method, the stress singularity orders as well as the stress intensities are determined analytically. The results presented in this paper give us the singular behaviors for various plate angle α, material orientation angles ϕ and the anisotropic parameters ε. These new findings are helpful in reducing or even eliminating the singularities near the sharp corner.     

Inter-comparison and validation of computational fluid dynamics codes in two-stage meandering channel flows

This paper presents a study in the inter-comparison and validation of three-dimensional computational fluid dynamics codes which are currently used in river engineering. Finite volume codes PHOENICS, FLUENT and SSIIM; and finite element code TELEMAC3D are considered in this study. The work has been carried out by competent hydraulic modellers who are users of the codes and not involved in their development. This paper is therefore written from the perspective of independent practitioners of the techniques. In all codes, the flow calculations are performed by solving the three-dimensional continuity and Reynolds-averaged Navier–Stokes equations with the k–ε turbulence model. The application of each code was carried out independently and this led to slightly different, but nonetheless valid, models. This is particularly seen in the different boundary conditions which have been applied and which arise in part from differences in the modelling approaches and methodology adopted by the different research groups and in part from the different assumptions and formulations implemented in the different codes. Similar finite volume meshes are used in the simulations with PHOENICS, FLUENT and SSIIM while in TELEMAC3D, a triangular finite element mesh is used. The ASME Journal of Fluids Engineering editorial policy is taken as a minimum framework for the control of numerical accuracy. In all cases, grid convergence is demonstrated and conventional criteria, such as Y⁺, are satisfied. A rigorous inter-comparison of the codes is performed using large-scale experimental data from the UK Flood Channel Facility for a two-stage meandering channel. This example data set shows complex hydraulic behaviour without the additional complications found in natural rivers. Standardised methods are used to compare each model with the available experimental data. Results are shown for the streamwise and transverse velocities, secondary flow, turbulent kinetic energy, bed shear stress and free surface elevation. They demonstrate that the models produce similar results overall, although there are some differences in the predicted flow field and greater differences in turbulent kinetic energy and bed shear stress. This study is seen as an essential first step in the inter-comparison of some of the computational fluid dynamics codes used in the field of river engineering.     

一种非矩形平行平板流动腔的定常流分析

高度远小于横向和纵向几何尺寸的平行平板流动腔是当前用以体外研究细胞力学行为的主要工具之一·人们常用的是上下两平行平板是矩形边界的流动腔·这种流动腔内（除入口端和出口端的邻近区域外）的流场是均匀的［１］，因而每次实验只能观测底部培养细胞受一种切应力值的影响情况·本文提出一种上下两平行平板不是矩形的平行平板流动腔·通过对腔内流体作定常流动的流场进行详细分析，给出了流动腔内流速和腔底切应力值的分布·结果发现，利用这种非矩形平行平板流动腔可以同时研究底部培养细胞在多种切应力值下的力学行为·本文理论结果与超声多普勒技术检测流速的实验结果吻合得相当好·     

Exact analytic solutions for the unsteady flow of a non-Newtonian fluid between two cylinders with fractional derivative model

The velocity field and the associated shear stress corresponding to the torsional oscillatory flow of a generalized Maxwell fluid, between two infinite coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. Initially, the fluid and cylinders are at rest and after some time both cylinders suddenly begin to oscillate around their common axis with different angular frequencies of their velocities. The solutions that have been obtained are presented under integral and series forms in terms of generalized G and R functions. Moreover, these solutions satisfy the governing differential equation and all imposed initial and boundary conditions. The respective solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for the similar flow of ordinary Maxwell fluid are also obtained as limiting cases of our general solutions. At the end, flows corresponding to the ordinary Maxwell and generalized Maxwell fluids are shown and compared graphically by plotting velocity profiles at different values of time and some important results are remarked.     

A dynamical model of storm surges along island coasts

A time-independent dynamical model of storm surge along island coasts using orthogonal curvilinear coordinates is presented. The curved annulus between an island coast and an arbitrary deep-water boundary is mapped conformally onto a rectangular image. Two configurations of island coasts are investigated; circular and elliptic coasts. The corresponding coordinates are circular polar and elliptic respectively. The linearized vertically-integrated equations of motion are used to model storm surges with two assumptions: (i) bottom stress is proportional to horizontal transport, and (ii) storm forces are shear stresses on water surface. Analytical solutions are presented for three dynamical cases: (i) a constant-depth basin acted upon by a uniform storm stress, (ii) variable-depth basin acted upon by a uniform-direction variable-magnitude stress, and. (iii) a basin with closed depth contours acted upon by vortex-shaped storm stress. The obtained solutions clarify the relative importance of the various parameters and variables that affect surge height distribution along island coasts. These solutions may be used to test a time-dependent, numerical dynamical storm model.     

Analytical and numerical solutions of the expected number of occurrences for combinations of event sequences due to variability

This paper investigates variability in the occurrence of different event sequences on an annual basis during the operation of a proposed nuclear facility. During the operational period of a nuclear facility, the annual radiological dose received by workers or members of the public depends on the number of event sequence occurrences. Based on the facility design, some combinations of event sequences will be expected to occur at least once during the operational period, and some combinations will not. This paper provides analytical solutions for calculating the expected number of combinations of independent event sequences. These analytical solutions agree with numerical solutions for an example problem. Although uncertainties can be incorporated into the method, only point-estimate parameter values are used in the example problem presented. The main objectives of this paper are to present calculational approaches to (i) identify which combinations of event sequences within the same year are expected to occur at least once during the operational life of a proposed facility and (ii) determine the annual doses from those identified combinations. Facility performance based on some proposed design is evaluated against the operational dose limits. Because the operational dose limits tend to be annual quantities that may not be exceeded in any year of operation, calculation of the doses resulting from combinations of event sequences that are expected to occur at least once can provide insight on the maximum annual dose expected during the operation of a proposed facility.     

On the flow of an electrically conducting,incompressible fluid near an accelerated plate in the presence of a parallel plate,under transverse magnetic field

This paper deals with hydromagnetic flow of an electrically conducting, incompressible viscous fluid near an accelerated flat, non-conducting plate, in the presence of another parallel plate, when there is a transversely applied magnetic field. Induced magnetic field is neglected in comparison with the applied magnetic field. Laplace transform techniques are used. The equations are integrated by applying residue principle, and expressions for velocity profiles and skin-friction at both plates are derived for different values of Hartmann number M. It is observed that, with the increase of the value of the Hartmann number M, the velocity profiles are flattened, the shear stress at the stationary plate decreases, as the value of the time T and Hartmann number M increases, but the shear stress at the accelerated plate increases directly in proportion with the increase in time and Hartmann number.     

Instability of power-law fluid flows down an incline subjected to wind stress

In this paper we investigate the effect of a prescribed superficial shear stress on the generation and structure of roll waves developing from infinitesimal disturbances on the surface of a power-law fluid layer flowing down an incline. The unsteady equations of motion are depth integrated according to the von Kármán momentum integral method to obtain a non-homogeneous system of nonlinear hyperbolic conservation laws governing the average flow rate and the thickness of the fluid layer. By conducting a linear stability analysis we obtain an analytical formula for the critical conditions for the onset of instability of a uniform and steady flow in terms of the prescribed surface shear stress. A nonlinear analysis is performed by numerically calculating the nonlinear evolution of a perturbed flow. The calculation is carried out using a high-resolution finite volume scheme. The source term is handled by implementing the quasi-steady wave propagation algorithm. Conclusions are drawn regarding the effect of the applied surface shear stress parameter and flow conditions on the development and characteristics of the roll waves arising from the instability. For a Newtonian flow subjected to a prescribed superficial shear stress, using an analytical theory, we show that the nonlinear governing equations do not admit roll waves solutions under conditions when the uniform and steady flow is linearly stable. For the case of a general power-law fluid flow with zero shear stress applied at the surface, the analytical investigation leads to a procedure for calculating the characteristics of a roll waves flow. These results are compared with those yielded by the numerical procedure.     

Mathematical modeling of pulsatile flow of non-Newtonian fluid in stenosed arteries

The pulsatile flow of blood through mild stenosed artery is studied. The effects of pulsatility, stenosis and non-Newtonian behavior of blood, treating the blood as Herschel–Bulkley fluid, are simultaneously considered. A perturbation method is used to analyze the flow. The expressions for the shear stress, velocity, flow rate, wall shear stress, longitudinal impedance and the plug core radius have been obtained. The variations of these flow quantities with different parameters of the fluid have been analyzed. It is found that, the plug core radius, pressure drop and wall shear stress increase with the increase of yield stress or the stenosis height. The velocity and the wall shear stress increase considerably with the increase in the amplitude of the pressure drop. It is clear that for a given value of stenosis height and for the increasing values of the stenosis shape parameter from 3 to 6, there is a sharp increase in the impedance of the flow and also the plots are skewed to the right-hand side. It is observed that the estimates of the increase in the longitudinal impedance increase with the increase of the axial distance or with the increase of the stenosis height. The present study also brings out the effects of asymmetric of the stenosis on the flow quantities.