Free‐surface tracking in 2D with the harmonic polynomial cell method: Two alternative strategies

Several cases of nonlinear wave propagation are studied numerically in two dimensions within the framework of potential flow. The Laplace equation is solved with the harmonic polynomial cell (HPC) method, which is a field method with high‐order accuracy. In the HPC method, the computational domain is divided into overlapping cells. Within each cell, the velocity potential is represented by a sum of harmonic polynomials. Two different methods denoted as immersed boundary (IB) and multigrid (MG) are used to track the free surface. The former treats the free surface as an IB in a fixed Cartesian background grid, while the latter uses a free‐surface fitted grid that overlaps with a Cartesian background grid. The simulated cases include several nonlinear wave mechanisms, such as high steepness and shallow‐water effects. For one of the cases, a numerical scheme to suppress local wave breaking is introduced. Such scheme can serve as a practical mean to ensure numerical stability in simulations where local breaking is not significant for the result. For all the considered cases, both the IB and MG method generally give satisfactory agreement with known reference results. Although the two free‐surface tracking methods mostly have similar performance, some differences between them are pointed out. These include aspects related to modeling of particular physical problems as well as their computational efficiency when combined with the HPC method.     

A high-order harmonic polynomial method for solving the Laplace equation with complex boundaries and its application to free-surface flows. Part I: Two-dimensional cases

A high-order harmonic polynomial method (HPM) is developed for solving the Laplace equation with complex boundaries. The “irregular cell” is proposed for the accurate discretization of the Laplace equation, where it is difficult to construct a high-quality stencil. An advanced discretization scheme is also developed for the accurate evaluation of the normal derivative of potential functions on complex boundaries. Thanks to the irregular cell and the discretization scheme for the normal derivative of the potential functions, the present method can avoid the drawback of distorted stencils, that is, the possible numerical inaccuracy/instability. Furthermore, it can involve stationary or moving bodies on the Cartesian grid in an accurate and simple way. With the proper free-surface tracking methods, the HPM has been successfully applied to the accurate and stable modeling of highly nonlinear free-surface potential flows with and without moving bodies, that is, sloshing, water entry, and plunging breaker.     

A multi‐level adaptive mesh refinement method for level set simulations of multiphase flow on unstructured meshes

An adaptive mesh refinement (AMR) technique is proposed for level set simulations of incompressible multiphase flows. The present AMR technique is implemented for two‐dimensional/three‐dimensional unstructured meshes and extended to multi‐level refinement. Smooth variation of the element size is guaranteed near the interface region with the use of multi‐level refinement. A Courant–Friedrich–Lewy condition for zone adaption frequency is newly introduced to obtain a mass‐conservative solution of incompressible multiphase flows. Finite elements around the interface are dynamically refined using the classical element subdivision method. Accordingly, finite element method is employed to solve the problems governed by the incompressible Navier–Stokes equations, using the level set method for dynamically updated meshes. The accuracy of the adaptive solutions is found to be comparable with that of non‐adaptive solutions only if a similar mesh resolution near the interface is provided. Because of the substantial reduction in the total number of nodes, the adaptive simulations with two‐level refinement used to solve the incompressible Navier–Stokes equations with a free surface are about four times faster than the non‐adaptive ones. Further, the overhead of the present AMR procedure is found to be very small, as compared with the total CPU time for an adaptive simulation. Copyright © 2016 John Wiley & Sons, Ltd.     

Vorticity field structure associated with the 3D Tollmien-Schlichting waves

Details of the vorticity field structure associated with the 3D Tollmien-Schlichting waves have been examined based upon the recent numerical studies of the subject. First, a single obliquet-s wave has been found to have the velocity component parallel to the wave front playing an overall dominant role, in particular, to create the longitudinal vorticity. The so-called Benney-Lin longitudinal vortices are then demonstrated to be, in fact, a minor consequence compared with the localized longitudinal vorticity field and its periodic pumping. Finally, the formation of the longitudinal vorticity field in the fundamental- and subharmonic-mode interactions is explained. The research reported in this paper has been supported in part by Bundesministerium für Forschung und Technologie and by Deutsche Forschungsgemeinschaft. The major part of the paper has been presented at the Third Asian Congress of Fluid Mechanics, 1–5 September 1986, in Tokyo, as a General Lecture by the senior author, FRH.     

Numerical simulation of a wave channel

The application of the Mixed Eulerian-Lagrangrian method to the simulation of transient free surface flows in the vicinity of a free surface piercing structure is considered. Particular attention is given to the validation of the numerical procedure.

Several applications are studied. Comparisons between the results of the numerical scheme and those of approximate theories or experiments are shown. They demonstrate the accuracy and versatility of the simulation that can be used as a “standard” to check the applicability of approximate theories.

The main limitation of the method is that it cannot account for viscous effects, in particular in the vicinity of the free surface. Approximate ways to simulate dissipative phenomena associated to breaking would be most useful.     

Evidence for two-dimensional solitary sound waves in a lipid controlled interface and its implications for biological signalling

Biological membranes by virtue of their elastic properties should be capable of propagating localized perturbations analogous to sound waves. However, the existence and the possible role of such waves in communication in biology remain unexplored. Here, we report the first observations of two-dimensional solitary elastic pulses in lipid interfaces, excited mechanically and detected by FRET. We demonstrate that the nonlinearity near a maximum in the susceptibility of the lipid monolayer results in solitary pulses that also have a threshold for excitation. These experiments clearly demonstrate that the state of the interface regulates the propagation of pulses both qualitatively and quantitatively. Finally, we elaborate on the striking similarity of the observed phenomenon to nerve pulse propagation and a thermodynamic basis of cell signalling in general.     

Automatic adaptive FE analysis of thin‐walled structures using 3D solid elements

In this study, a new automatic adaptive refinement procedure for thin‐walled structures using 3D solid elements is suggested. This procedure employs a specially designed superconvergent patch recovery (SPR) procedure for stress recovery, the Zienkiewicz and Zhu (Z–Z) error estimator for the a posteriori error estimation, a new refinement strategy for new element size prediction and a special mesh generator for adaptive mesh generation. The proposed procedure is different from other schemes in such a way that the problem domain is separated into two distinct parts: the shell part and the junction part. For stress recovery and error estimation in the shell part, special nodal coordinate systems are used and the stress field is separated into two components. For the refinement strategy, different procedures are employed for the estimation of new element sizes in the shell and the junction parts. Numerical examples are given to validate the effectiveness of the suggested procedure. It is found that by using the suggested refinement procedure, when comparing with uniform refinement, higher convergence rates were achieved and more accurate final solutions were obtained by using fewer degrees of freedoms and less amount of computational time. Copyright © 2008 John Wiley & Sons, Ltd.     

A direct coupling method for 3D hydroelastic analysis of floating structures

This paper presents a complete formulation for three‐dimensional hydrodynamic analysis of floating flexible structures subjected to surface regular waves, as well as other excitation forces, by employing a direct tight coupling method. The continuum mechanics‐based finite element method is employed to model floating structures with arbitrary geometries, which can account for the geometric nonlinearities and initial stress effects that result from the hydrostatic analysis, whereas the boundary element method is used for the fluid via total potential formulation. The simplicity and generality of the present formulation are revealed as compared with the conventional formulation. Numerical examples demonstrate the general capability of the formulation proposed. Copyright © 2013 John Wiley & Sons, Ltd.     

hp Fast multipole boundary element method for 3D acoustics

A fast multipole boundary element method (FMBEM) extended by an adaptive mesh refinement algorithm for solving acoustic problems in three‐dimensional space is presented in this paper. The Collocation method is used, and the Burton–Miller formulation is employed to overcome the fictitious eigenfrequencies arising for exterior domain problems. Because of the application of the combined integral equation, the developed FMBEM is feasible for all positive wave numbers even up to high frequencies. In order to evaluate the hypersingular integral resulting from the Burton–Miller formulation of the boundary integral equation, an integration technique for arbitrary element order is applied. The fast multipole method combined with an arbitrary order h‐p mesh refinement strategy enables accurate computation of large‐scale systems. Numerical examples substantiate the high accuracy attainable by the developed FMBEM, while requiring only moderate computational effort at the same time. Copyright © 2016 John Wiley & Sons, Ltd.     

A reduced-order model based on the coupled 1D-3D finite element simulations for an efficient analysis of hemodynamics problems

A reduced-order model for an efficient analysis of cardiovascular hemodynamics problems using multiscale approach is presented in this work. Starting from a patient-specific computational mesh obtained by medical imaging techniques, an analysis methodology based on a two-step automatic procedure is proposed. First a coupled 1D-3D Finite Element Simulation is performed and the results are used to adjust a reduced-order model of the 3D patient-specific area of interest. Then, this reduced-order model is coupled with the 1D model. In this way, three-dimensional effects are accounted for in the 1D model in a cost effective manner, allowing fast computation under different scenarios. The methodology proposed is validated using a patient-specific aortic coarctation model under rest and non-rest conditions.     

A class of model equations for bi-directional propagation of capillary-gravity waves

A class of model equations that describe the bi-directional propagation of small amplitude long waves on the surface of shallow water is derived from two-dimensional potential flow equations at various orders of approximation in two small parameters, namely the amplitude parameter α=a/h₀ and wavelength parameter β=(h₀/l)², where a and l are the actual amplitude and wavelength of the surface wave, and h₀ is the height of the undisturbed water surface from the flat bottom topography. These equations are also characterized by the surface tension parameter, namely the Bond number τ=Γ/ρgh₀², where Γ is the surface tension coefficient, ρ is the density of water, and g is the acceleration due to gravity.The traveling solitary wave solutions are explicitly constructed for a class of lower order Boussinesq system. From the Boussinesq equation of higher order, the appropriate equations to model solitary waves are derived under appropriate scaling in two specific cases: (i) β?(1/3−τ)?1/3 and (ii) (1/3−τ)=O(β). The case (i) leads to the classical Boussinesq equation whose fourth-order dispersive term vanishes for τ=1/3. This emphasizes the significance of the case (ii) that leads to a sixth-order Boussinesq equation, which was originally introduced on a heuristic ground by Daripa and Hua [Appl. Math. Comput. 101 (1999) 159] as a dispersive regularization of the ill-posed fourth-order Boussinesq equation.     

A Method of Solution for the 3D Problem of the Elasticity Theory in Displacements

A new method of solution for the 3D problem of the elasticity theory in displacements is proposed. The method is based on the Tedone equilibrium equation. In contrast to the Betti and Cerrutti–Boussinesq methods, the described approach does not require volume expansion to be predetermined. The first and second boundary-value problems for an elastic isotropic half-space are considered to illustrate this method.     

一类吸引玻色-爱因斯坦凝聚的坍塌性质

本文讨论一类描述吸引玻色-爱因斯坦凝聚的阻尼非线性Schr(?)dinger方程。运用能量方法,我们证明了在某些条件下解将发生坍塌。     

Numerical evaluation of harmonic Green's functions for triclinic half‐space with embedded sources—Part II: a 3D model

The displacement and stress Green's functions for a 3D triclinic half‐space with embedded harmonic point load is considered. The resulting displacement and stress fields are expressed in terms of triple Fourier integrals. The first integral was evaluated using contour integration and the 3D Green's functions were obtained as a superposition of 2D results over the azimuthal angle. The resulting algorithm developed for evaluation of the Green's functions avoids repeated calculations of the same quantities and it utilizes the vectorized manipulation within MATLAB environment. The algorithm places no restriction on material properties, frequency and location of source and observation points. Extensive testing of the numerical results was performed for both displacement and stress. The tests confirm the accuracy of the numerical results. Copyright © 2006 John Wiley & Sons, Ltd.     

A new 3D finite element for adaptive h-refinement in 1-irregular meshes

A new finite element, viable for use in the three-dimensional simulation of transient physical processes with sharply varying solutions, is presented. The element is intended to function in adaptive h-refinement schemes as a versatile transition between regions of different refinement levels, ensuring interelement continuity by constructing a piecewise linear solution at the element boundaries, and retaining all degrees of freedom in the solution phase. Construction of the element shape functions is described, and a numerical example is presented which illustrates the advantages of using such an element in an adaptive refinement problem. The new element can be used in moving-front problems, such as those found in reservoir engineering and groundwater flow applications.     

A 3D implicit unstructured-grid finite volume method for structural dynamics

In this work, a new vertex-based finite volume method (FVM) using unstructured grids and cell-based data structure is proposed for computational analysis of two-and three-dimensional (2D/3D) general structural dynamic problems. The governing equations are spatially discretized by the FVM and an implicit dual time stepping scheme is employed to integrate the equations in time. The proposed method is applied to calculate deformations and dynamics of 2D and 3D cantilevers, as well as simply supported and clamped square plates. Computational results obtained are found to agree well with analytical solutions. It can be a viable alternative to the traditional finite element method (FEM) for structural dynamic calculations. And it can be seamlessly integrated into FVM-based Computational Fluid Dynamics (CFD) solver for simulating fluid-structure interaction (FSI).     

QALE‐FEM for numerical modelling of non‐linear interaction between 3D moored floating bodies and steep waves

This paper presents further development of the quasi arbitrary Lagrangian–Eulerian finite element method (QALE‐FEM) based on a fully non‐linear potential theory to numerically simulate non‐linear responses of 3D moored floating bodies to steep waves. In the QALE‐FEM (recently developed by the authors and applied to 2D floating bodies), the complex unstructured mesh is generated only once at the beginning of calculation and is moved to conform to the motion of boundaries at other time steps by using a robust spring analogy method specially suggested for these kind of problems, avoiding the necessity of high‐cost remeshing. In order to tackle challenges associated with 3D floating bodies, several new numerical techniques are developed in this paper. These include the technique for moving the mesh near body surfaces, the scheme for calculating velocity on 3D body surfaces and the modified semi‐implicit time integration method for floating bodies procedure (ISITIMFB‐M) that is more efficient for dealing with the full coupling between waves and bodies. Using the newly developed techniques and methods, various cases for 3D floating bodies with motions of up to six degrees of freedom (DoFs) are simulated. These include a SPAR platform, a barge‐type floating body and one or two Wigley Hulls in head seas or in oblique waves. For some selected cases, the numerical results are compared with experimental data available in the public domain and satisfactory agreements are achieved. Many results presented in this paper have not been found elsewhere to the best knowledge of the authors. Copyright © 2008 John Wiley & Sons, Ltd.     

Mesh refinement strategies without mapping of nonlinear solutions for the generalized and standard FEM analysis of 3‐D cohesive fractures

A robust and efficient strategy is proposed to simulate mechanical problems involving cohesive fractures. This class of problems is characterized by a global structural behavior that is strongly affected by localized nonlinearities at relatively small‐sized critical regions. The proposed approach is based on the division of a simulation into a suitable number of sub‐simulations where adaptive mesh refinement is performed only once based on refinement window(s) around crack front process zone(s). The initialization of Newton‐Raphson nonlinear iterations at the start of each sub‐simulation is accomplished by solving a linear problem based on a secant stiffness, rather than a volume mapping of nonlinear solutions between meshes. The secant stiffness is evaluated using material state information stored/read on crack surface facets which are employed to explicitly represent the geometry of the discontinuity surface independently of the volume mesh within the generalized finite element method framework. Moreover, a simplified version of the algorithm is proposed for its straightforward implementation into existing commercial software. Data transfer between sub‐simulations is not required in the simplified strategy. The computational efficiency, accuracy, and robustness of the proposed strategies are demonstrated by an application to cohesive fracture simulations in 3‐D. Copyright © 2016 John Wiley & Sons, Ltd.     

层状饱和半空间中沉积谷地对斜入射平面P1波的三维散射

在作者给出层状饱和场地三维精确动力刚度矩阵和层状饱和半空间中移动荷载动力格林函数基础上,采用间接边界元方法在频域内求解了层状流体饱和场地中沉积谷地对斜入射平面P1波的三维散射问题。该方法的特点在于虚拟移动均布荷载和斜线孔隙水压可以直接施加在沉积与层状饱和半空间交界面而不存在奇异性。该文通过与已有结果的比较验证了方法的正确性,并以均匀饱和半空间和弹性基岩上单一饱和土层中沉积谷地为例进行了数值计算分析。研究表明,沉积谷地对平面P1波的三维散射与二维散射之间存在本质差别,入射角度、孔隙率、饱和土层刚度和饱和土层厚度等参数对沉积谷地附近动力响应有着显著影响。