A modified scaled boundary finite-element method for problems with parallel side-faces. Part II. Application and evaluation

The modified scaled boundary finite-element method (SBFEM), keeping the advantages of the original SBFEM, eliminates the restriction of the scaling center location so that this approach can solve two-dimensional problems with parallel side-faces. In this paper, the modified SBFEM is applied to solutions of two types of problems—wave diffraction by a single and twin surface rectangular obstacles and wave radiation induced by an oscillating mono-hull and twin-hull structures in a finite depth of water. For wave diffraction problems, numerical results agree extremely well with the analytic solution for the single obstacle case and other numerical results of a different approach for the twin obstacle case. For wave radiation problems, the particular solutions to the scaled boundary finite-element equation are presented for cases of heave, sway and roll motions. The added mass and damping coefficients for heave, sway and roll motions of a two-dimensional rectangular container are computed and the numerical results are compared with those from independent analytical solution and numerical solution using the boundary element method (BEM). It is found that the SBFEM method achieves equivalent accuracy to the conventional BEM with only a few degrees of freedom. In the last example, wave radiation by a two-dimensional twin-hull structure is analyzed. Comparisons of the results with those obtained using conventional Green's function method (GFM) demonstrate that the method presented in this paper is free from the irregular frequency problems.     

Scaled Boundary Finite Element Analysis of Wave Passing A Submerged Breakwater

The scaled boundary finite element method （SBFEM） is a novel semi-analytical technique combining the advantage of the finite element method （FEM） and the boundary element method （BEM） with its unique properties. In this paper, the SBFEM is used for computing wave passing submerged breakwaters, and the reflection coeffcient and transmission coefficient are given for the case of wave passing by a rectangular submerged breakwater, a rigid submerged barrier breakwater and a trapezium submerged breakwater in a constant water depth. The results are compared with the analytical solution and experimental results. Good agreement is obtained. Through comparison with the results using the dual boundary element method （DBEM）, it is found that the SBFEM can obtain higher accuracy with fewer elements. Many submerged breakwaters with different dimensions are computed by the SBFEM, and the changing character of the reflection coeffcient and the transmission coefficient are given in the current study.     

应用比例边界有限元法求解多种二维浮体水动力特性

比例边界有限元法(SBFEM)是一种半解析的数值方法,比完全数值方法具有更高的精度,该方法结合了有限元和边界元的优点,采用相对少的剖分单元就可以得到较高精度的模拟结果。通过改变有限子域内部比例中心的位置,使这种方法可以应用到多种形式浮体在波浪作用下的水动力特性的计算中。同时还给出了各种形式浮体的波浪力及反射、透射系数的数值结果,并与边界元方法(BEM)计算结果和特征函数展开方法得到解析解进行了比较,均吻合良好。研究表明比例边界有限元不仅可以计算矩形的浮体结构,而且对于多种结构形式的浮体都可以计算,这为多种结构形式浮体的水动力分析提供了一个可行的方法。     

Short-crested waves interaction with a concentric porous cylinder system with partially porous outer cylinder

In this paper, based on the linear wave theory, the interaction of short-crested waves with a concentric dual cylindrical system with a partially porous outer cylinder is studied by using the scaled boundary finite element method (SBFEM), which is a novel semi-analytical method with the advantages of combining the finite element method (FEM) with the boundary element method (BEM). The whole solution domain is divided into one unbounded sub-domain and one bounded sub-domain by the exterior cylinder. By weakening the governing differential equation in the circumferential direction, the SBFEM equations for both domains can be solved analytically in the radial direction. Only the boundary on the circumference of the exterior porous cylinder is discretized with curved surface finite elements. Meanwhile, by introducing a variable porous-effect parameter G, non-homogeneous materials caused by the complex configuration of the exterior cylinder are modeled without additional efforts. Comparisons clearly demonstrate the excellent accuracy and computational efficiency associated with the present SBFEM. The effects of the wide range wave parameters and the structure configuration are examined. This parametric study will help determine the various hydrodynamic effects of the concentric porous cylindrical structure.     

Evaluation of stress intensity factors for multiple cracked circular disks under crack surface tractions with SBFEM

Stress intensity factors (SIFs) for the cracked circular disks under different distributing surface tractions are evaluated with the scaled boundary finite element method (SBFEM). In the SBFEM, the analytical advantage of the solution in the radial direction allows SIFs to be directly determined from its definition, therefore no special crack-tip treatment is necessary. Furthermore anisotropic material behavior can be treated easily. Different distributions of surface tractions are considered for the center and double-edge-cracked disks. The benchmark examples are modeled and an excellent agreement between the results in the present study and those in published literature is found. It shows that SBFEM is effective and possesses high accuracy. The SIFs of the cracked orthotropic material circular disks subjected to different surface tractions are also evaluated. The technique of substructure is applied to handle the multiple cracks problem.     

应用比例边界有限元法求解狭缝对双箱水动力的影响

比例边界有限元法(SBFEM)是一种半解析数值分析的新方法,既融合了有限元法和边界元法的优点,又有其特有的优点。用该方法可求解有限水深下狭缝对双箱水动力作用的影响,为波浪与多浮体超大型结构的相互作用探索一些规律。整个计算域划分成2个无限子域和4个有限子域,并利用加权余量法在各个子域上推导了SBFEM的积分方程;计算了4个数值算例并与边界元等其它数值方法进行了比较,验证了该方法是一种用很少单元便能得到精确结果的高效方法。应用SBFEM对不同箱体宽度、不同狭缝宽度、不同吃水深度条件的双箱作了计算,得出了狭缝对双箱水动力干涉影响的一些规律,对超大型浮体水动力分析和结构设计具有一定的参考价值。     

Vibration analysis of rectangular Mindlin plates on elastic foundations and vertically in contact with stationary fluid by the Ritz method

In this study Free vibration analysis of vertical rectangular Mindlin plates resting on Pasternak elastic foundation and fully or partially in contact with fluid on their one side is investigated for different combinations of boundary conditions. The plate is assumed to be one of vertical rectangular walls of a container in contact with fluid. In order to analyze the interaction of the Mindlin plate with the elastic foundation and fluid system, three displacement components of the plate are expressed in the Ritz method by adopting a set of static Timoshenko beam functions satisfying geometric boundary conditions in a Cartesian co-ordinate system. The method of separation of variables and the method of Fourier series expansion is used to model fluid and to obtain the exact expression of the motion of fluid in the form of integral equations. The fluid domain is finite in depth and width but infinite in the length direction. To demonstrate the accuracy of the present solution, convergence study is first carried out and then a few comparison studies are carried out with the available data in the literature. Finally, natural frequencies of rectangular plates are presented in tabular and graphical forms for different fluid levels, foundation parameters, aspect ratios, thickness to width ratios and boundary conditions.     

Parametric Analysis of A Submerged Cylindrical Shell Subjected to Shock Waves

In this study,an FEM-SBFEM(scaled boundary finite element method)coupling procedure proposed by Fan et al.(2005)is adopted to obtain the dynamic responses of a submerged cylindrical shell subjected to plane step or exponential acoustic shock waves.The coupling procedure can readily be applied to three-dimensional problem,however for clarity,the problems to be presented are limited to two-dimensional domain.In the analyses,the cylindrical shell is modeled by simple beam elements(using FEM),while the effects of the surrounding infinite fluid is modeled by the SBFEM.In it,no free surface and seabed are involved.Compared with Fan and his co-authors' works,the FEM-SBFEM coupling procedure is further verified to be feasible for shock waves by benchmark examples.Furthermore,parametric studies are performed and presented to gain insight into effects of the geometric and material properties of the cylindrical shell on its dynamic responses.     

A three-dimensional hydrostatic model for coastal and ocean modelling using a generalised topography following co-ordinate system

A three-dimensional hydrostatic model is presented that combines a generalised vertical co-ordinate system with an efficient implicit solution technique for the free surface. The model is capable of maintaining high resolution in the surface and/or bottom boundary layers as well as dealing with steep topography. Horizontal diffusion is calculated using the Smagorinsky formulation and a k–ε turbulence model is used in the vertical. In addition the model uses higher-order advection routines. An important aspect in three-dimensional models is the choice of vertical discretisation. If one is mostly interested in problems which are governed by boundary layer flows, a terrain following or sigma co-ordinate system seems attractive. This paper focuses on the development of a generalised sigma-type grid in a three-dimensional hydrostatic model. The generalised grid offers a wide range of possibilities including grid refinement toward the bed or surface, a mixed layer transformation, and a constant layer transformation where the lowermost or uppermost grid cells can be specified to have a constant height above the bed or below the surface. A number of tests are presented which show that the model is capable of simulating both shallow nearshore, estuarine flows as well as large-scale geophysical flows. These include an extreme flooding event in the shallow North Sea and the Odden ice tongue formation in the Greenland Sea.     

Extension of the Rotated Elastic Parabolic Equation to Beach and Island Propagation

Improvements in the capability of handling sloping interfaces and boundaries with the parabolic equation method have been an active area of research. Recent progress in accurately treating range-dependent seismoacoustic problems has involved coordinate transformation techniques. The variable-rotated parabolic equation is among recent advances in this area. The solution rotates the coordinate axes to achieve greater accuracy in the presence of range-dependent bathymetry. At points of slope change the rotated solution interpolates and extrapolates the field into adjacent regions. This approach is extended to solve problems involving variable topography (above-ocean-surface sediments) by accounting for the transition and boundary conditions at the water/solid/air interfaces. It is applied to range-dependent problems of sound transmission up a beach and through an island. The method is benchmarked for accuracy against a finite-element solution.      

Wave-induced seabed response analysis by radial point interpolation meshless method

Wave-induced transient response of seabeds is numerically analyzed through a radial point interpolation meshless method (radial PIM). The Biot’s consolidation theory is employed and incorporated with virtual boundary conditions to describe this wave-induced transient response of the seabed. Displacement and pore water pressure are spatially discretized by the radial PIM with the same shape function. Compactly supported basis functions are proposed to obtain a banded system equation. Because the radial PIM passes through all nodal points within an influence domain, essential boundary conditions as well as virtual boundary conditions can be easily implemented at local level. Fully implicit integration scheme is used in time domain to avoid spurious ripple effect. The proposed algorithm is assessed through the comparison of numerical results with closed-form solution or finite element solutions.     

Parallel trajectory planning for shipborne Autonomous collision avoidance system

Collision at sea is always a significant issue affecting the safety of ship navigation. The shipborne autonomous collision avoidance system (SACAS) has the great advantage to minimize collision accidents in ship navigation. A parallel trajectory planning architecture is proposed in this paper for SACAS system. The fully-coupled deliberative planner based on the modified RRT algorithm is developed to search for optimal global trajectory in a low re-planning frequency. The fully-coupled reactive planner based on the modified DW algorithm is developed to generate the optimal local trajectory in a high re-planning frequency to counteract the unexpected behavior of dynamic obstacles in the vicinity of the vessel. The obstacle constraints, ship maneuvering constraints, COLREGs rules, trajectory optimality, and real-time requirements are satisfied simultaneously in both global and local planning to ensure the collision-free optimal navigation in compliance with COLREGs rules. The on-water tests of a trimaran model equipped with a model-scale SACAS system are presented to demonstrate the effectiveness and efficiency of the proposed algorithm. The good balance between the computational efficiency and trajectory optimality is achieved in parallel trajectory planning.     

Seasonal variability of surface and internal M<Subscript>2</Subscript> tides in the Arctic Ocean

The modeling results of surface and internal M₂ tides for summer and winter periods in the Arctic Ocean (AO) are presented. We employed a modified version of the three-dimensional finite-element hydrothermodynamic model QUODDY-4 differing from the original model by using a rotated (instead of spherical) coordinate system and by considering the equilibrium-tide effects. It has been shown that the modeling results for the surface tide differs little from the results obtained earlier by other authors. According to these results, the amplitudes of internal tidal waves (ITWs) in the AO are significantly lower than in other oceans and the ITWs proper have the character of trapped waves. Their source of generation is located at the continental slope northwest of the New Siberian Islands. Our results are consistent with the fields of average (over a tidal cycle) and integral (by depth) densities of baroclinic tidal energy, the maximum baroclinic tidal velocity, and the coefficient of diapycnic mixing. The local rate of baroclinic tidal energy dissipation at the AO ridges increases as it approaches the bottom, as was observed on Mid-Atlantic and Hawaii ridges (but merely within the bottom boundary layer) and is two to three orders of magnitude lower than in other oceans. The ITW degeneration scale in the AO is several hundreds of kilometers in summer and winter, remaining within the range of its values between 100 and 1000 km in mid- and low-latitude oceans. In both seasons, the integral (over the AO area) rate of baroclinic tidal energy dissipation is two orders of magnitude lower than the global estimate (2.5 × 10¹² W).     

Local scour prediction around piers with complex geometry

A laboratory study of local scour at complex piers under steady clear-water conditions is presented. The term complex piers is used to define a bridge pier comprising of a column, pile cap, and pile group. Comprehensive data over the full range of possible pile cap elevations for complex piers with different geometries were obtained using five complex pier models, which were scaled down from existing bridges in Malaysia. The data are used to evaluate existing methodologies for characterizing the effective width of complex piers with varying pile cap location relative to the undisturbed streambed. The effect of pile cap location on scour depth is also addressed. To improve the predictions of local scour at complex piers, the new data and some previous data are used to propose a new method to predict local scour depth at complex piers.     

A Novel Control Algorithm for Interaction Between Surface Waves and A Permeable Floating Structure

An analytical solution is undertaken to describe the wave-induced flow field and the surge motion of a permeable platform structure with fuzzy controllers in an oceanic environment. In the design procedure of the controller, a parallel distributed compensation (PDC) scheme is utilized to construct a global fuzzy logic controller by blending all local state feedback controllers. A stability analysis is carried out for a real structure system by using Lyapunov method. The corresponding boundary value problems are then incorporated into scattering and radiation problems. They are analytically solved, based on separation of variables, to obtain series solutions in terms of the harmonic incident wave motion and surge motion. The dependence of the wave-induced flow field and its resonant frequency on wave characteristics and structure properties including platform width, thickness and mass has been thus drawn with a parametric approach. From which mathematical models are applied for the wave-induced displacement of the surge motion. A nonlinearly inverted pendulum system is employed to demonstrate that the controller tuned by swarm intelligence method can not only stabilize the nonlinear system, but has the robustness against external disturbance.     

An integrated sensory-intelligent system for underwater acoustic signal-processing applications

A generic integrated sensory-intelligent system (ISIS) is developed for underwater acoustic signal-processing applications. ISIS constantly monitors the current acoustic channel conditions and smoothly integrates the outputs of the most appropriate signal-processing procedures or algorithms available to it for those conditions. The system is based on a generalization of a tuneable approximate piecewise linear (TAPL) model derived from the modified probabilistic neural network (MPNN). This model was designed to seamlessly integrate a set of local linear signal-processing algorithms within a given multidimensional data space. Depending on the input signal distortions, which are determined by environmental effects, ISIS automatically weighs and adds the outputs from a set of processing algorithms working in parallel. The weighting is related to the "closeness" of each algorithm to the sensed input signal characteristics or some other measured environmental state. A single tuning parameter is used to smoothly and seamlessly select appropriately among the parallel processing algorithm outputs. A very small tuning-parameter value selects the closest most appropriate algorithm output. At the other extreme, a fixed weighted average of all the algorithm outputs is produced with a very large value. Otherwise, a dynamic weighed average of all algorithm outputs is achieved with values in between. Some features and benefits of ISIS are demonstrated with an illustrative linear sweep chirp signal-detector estimation problem characterized by extremely variable Doppler conditions.     

Effect of spatial inhomogeneity of the resistance coefficient on the dynamics of the <Emphasis Type="Italic">M</Emphasis><Subscript>2</Subscript> tidal wave in the White Sea

To find variations in the dynamics of the surface M ₂ tide in the White Sea induced by the spatially inhomogeneity of the resistance coefficient, we use a modified version of the QUODDY-4 three-dimensional finite-element hydrostatic model. This version differs from the original version in that it has a module introduced to calculate the resistance coefficient in the bottom boundary layer (BBL). The resistance coefficient is found from resistance laws for an oscillating rotating turbulent BBL over hydrodynamically rough and partially rough (smoothly rough) underlying surfaces describing the dependence of the resistance coefficient and other integral characteristics of resistance on dimensionless similarity parameters: the sea-bottom Rossby number Ro, the streaming Reynolds number Re, and the relative (normalized to tidal frequency) inertial frequency f/σ. The use of spatial inhomogeneity of the resistance coefficient was shown not to lead to considerable changes in tidal characteristics. The values of these characteristics are several times larger than the instrumental measurement errors for the level and velocity but less than the errors in their calculation.     

Open boundary conditions for regional tidal simulations

The performance of regional tide model simulations is examined in relation to the choice of open boundary conditions. Three barotropic open boundary conditions, clamped elevation, clamped normal velocity, and Flather, give similar results when the prescribed values are exact; however, Flather is much less sensitive to errors in the prescribed values. Of particular concern, it was found that with a phase error between the two boundaries, both the clamped conditions resulted in magnitude errors in the unclamped variable (although the simulation remained stable).A modified flow relaxation scheme for the depth-varying prognostic variables is presented. This implementation allows the transmission of a range of vertical modes while retaining realistic topography at the boundary. It was found to be an excellent internal tide boundary condition in tests comparing simulations of different domain length encompassing a ridge and sloping bottom, and in a comparison to an analytical solution. Mass is conserved without any artificial volume constraint.     

Analytic study to wave scattering by a general Homma island using the explicit modified mild-slope equation

In this paper, an exact analytic solution in terms of Taylor series to the explicit modified mild-slope equation (EMMSE) for wave scattering by a general Homma island is constructed and the convergence of the series solution is analyzed. To validate the new analytic solution, comparisons are made against the existing solutions including analytic solutions to both the long-wave equation and Helmholtz equation, approximate analytic solutions to the modified mild-slope equation, numerical solutions to the mild-slope equation and experimental solutions. Because of the use of the governing equation EMMSE together with mass-conserving matching conditions along the toe of the shoal, the present model is valid for not only waves in the whole spectrum from long waves to short waves but also bathymetries with the maximal seabed slope being as high as 4.27:1. Since the general Homma island is an extension of the original Homma island, the present solution can be very conveniently used to study the effects of bottom topography on combined refraction and diffraction. It is found that the larger the shoal size is, the more significant the wave amplification against the cylinder is.     

A time‐split, forward‐backward numerical model for solving a nonhydrostatic and compressible system of equations

In this paper, we present a numerical procedure for solving a 2‐dimensional, compressible, and nonhydrostatic system of equations. A forward‐backward integration scheme is applied to treat high‐frequency and internal gravity waves explicitly. The numerical procedure is shown to be neutral in time as long as a Courant–Friedrichs–Lewy criterion is met. Compared to the leap‐frog‐scheme most models use, this method involves only two time steps, which requires less memory and is also free from unstable computational modes. Hence, a time‐filter is not needed. Advection and diffusion terms are calculated with a time step longer than sound‐wave related terms, so that extensive computer time can be saved. In addition, a new numerical procedure for the free‐slip bottom boundary condition is developed to avoid using inaccurate one‐sided finite difference of pressure in the surface horizontal momentum equation when the terrain effect is considered. We have demonstrated the accuracy and stability of this new model in both linear and nonlinear situations. In linear mountain wave simulations, the model results match the corresponding analytical solution very closely for all three cases presented in this paper. The analytical streamlines for uniform flow over a narrow mountain range were obtained through numerical integration of Queney's mathematical solution. It was found Queney's original diagram is not very accurate. The diagram had to be redrawn before it was used to verify our model results. For nonlinear tests, we simulated the famous 1972 Boulder windstorm and a bubble convection in an isentropic enviroment. Although there are no analytical solutions for the two nonlinear tests, the model results are shown to be very robust in terms of spatial resolution, lateral boundary conditions, and the use of the time-split scheme.