Spectral Boundary Element Method Research Articles

Efficient spectral coupled boundary element method for fully nonlinear wave–structure interaction simulation

Accurately analyzing wave–structure interactions is crucial for the design and operational safety of ships and marine structures. This paper presents a fully nonlinear potential-flow approach for simulating wave–structure interactions using the newly proposed spectral coupled boundary element method (SCBEM). The SCBEM efficiently models an extensive water body that encompasses structures by establishing a boundary element method (BEM) computational domain solely around the object of interest while accurately simulating the far-field broad water by a spectral layer. To further improve efficiency, graphics processing unit acceleration is hired during iterative solving of the boundary value problem in the already small-sized interior BEM domain. Simulations are conducted to validate the accuracy of the method on cases with strong nonlinear phenomena, including wave run-up on a single cylinder, diffraction of a four-cylinder array, near-trapped modes for closely spaced columns, and gap resonance that occurred in side-by-side offloading. The wave run-up, diffraction wave pattern, near-trapped mode, and gap resonance frequency obtained by the proposed method are in good agreement with data from experiments and published literature. The quite good accuracy and the exceptional computational efficiency of the SCBEM demonstrate its promising potential for more application in practical marine problems.

A semi-analytical method for vibro-acoustic characteristics of orthogonal stiffened laminated cylindrical shells

This paper presents a semi-analytical model for vibro-acoustic characteristics of submerged orthogonal stiffened laminated cylindrical shells strengthened with orthogonal stiffeners. The first-order shear deformation theory (FSDT) combined with the Lekhnitsky smeared stiffener technique is adopted to describe the displacement field, and the spectral boundary element method is utilized to model the sound field. For modeling of a complex stiffened shell, the shell is divided into several main structures with the additional stiffness and mass of the smeared stiffeners. Then, the main structure is separated into sub-segments that match the boundary elements to ensure high accuracy of acoustic calculations. The continuous boundary conditions between sub-segments are simulated by using the artificial spring technique. The Fourier series and Legendre orthogonal polynomials are selected as admissible functions to describe the displacement field and the sound field. The vibration governing equations and Helmholtz integral equation are coupled with each other and solved simultaneously. High accuracy is demonstrated by comparing the results of the developed model with the literature. In numerical investigations, the influences of different external fluids, layup schemes, quantities and shapes of stiffeners (width and height) on vibro-acoustic characteristics of orthogonal stiffened laminated cylindrical shells (OSLCS) are investigated.

Vibration and critical pressure analyses of functionally graded combined shells submerged in water with external hydrostatic pressure

This paper describes a unified nondestructive semi-analytical formulation for the critical pressure of underwater functionally graded combined shells subjected to external hydrostatic pressure. The structural model of combined shells is constructed from the first-order shear deformation theory, Voigt mixing law, and virtual spring technology. The admissible displacement functions are uniformly expressed with Legendre polynomials along the generatrix direction and Fourier series along the circumferential direction. The Kirchhoff-Helmholtz boundary integral equation is discretized by using the Legendre-Gauss spectral boundary element method, and the external acoustic field model is derived by expanding the Green's function, sound pressure, and displacement of combined shells with one-dimensional Fourier transformation. The linear elastic material theory is applied to deduce the work done by the hydrostatic pressure. The key technology to predict the critical pressure is to create an accurate equivalent model of the hydrostatic pressure. Through solving the final coupled governing equation, the critical pressure is obtained from the linear relationship between the hydrostatic pressure and the square of natural frequency. The convergence and correctness are verified by comparison with the results from published papers and numerical methods. The effects of different hydrostatic pressures, power-law exponents, geometric parameters, and boundary conditions on the critical pressure are investigated in several examples. Computed results can provide security guidance for the nondestructive prediction of underwater functionally graded combined shells at the beginning of engineering application design.

Spectral wave modeling of tsunami waves in Pohang New Harbor (South Korea) and Paradip Port (India)

A coupled numerical model is developed based on the spectral element method (SEM) and boundary element method (BEM) to predict the characteristics of tsunami wave response on Pohang New Harbor (PNH), South Korea, and Paradip Port, India. The current numerical model is developed to analyze the impacts on the coastal region with slowly varying bathymetry under the resonance conditions. In this method, the boundary integral equation of each boundary segment is discretized into the spectral elements using the Chebyshev’s polynomial, and the corresponding boundary integrals are transformed using the Jacobian of transformation. This leads to a highly accurate depiction of the boundary edges or corners of the complex domain. Convergence and error analyses are conducted on the rectangular port using the BEM and the present method, which shows that the later approach enhances the overall numerical accuracy of the model. The simulation results for normal day wave (ordinary wave propagating in the ocean under normal conditions) and tsunami waves are validated with previous studies and measurement data at Pohang New Harbor (PNH), South Korea. In addition, the calculated spectral density for tsunami waves at PNH is also compared with the Hokkaido tsunami (reported on July 12, 1993) data at tide gauge record station in Pohang. The amplification factor is determined for normal day wave and tsunami waves at key locations inside the PNH and Paradip Port for different directional incident waves. Further, the spectral density is also estimated at the same locations with respect to the wave period to understand the consequences of tsunami waves on the coastal ports. Therefore, the coupled numerical model is an efficient tool to predict tsunami wave impact on a realistic port. The causes and countermeasure are suggested to reduce the risk of normal day waves and tsunami waves.

A scalable parallel algorithm for direct-forcing immersed boundary method for multiphase flow simulation on spectral elements

In this work, we propose a highly scalable parallel double binned ghost particle (DBGP) algorithm for direct-forcing immersed boundary spectral element method for multiphase flow simulations. In particular, the DBGP algorithm is designed to obtain fully distributed data storage and scalable data transfer across hundreds of thousands of processors. The proposed algorithm uses a queen and worker data structure for fully resolved particles to demarcate particle-level and marker-level quantities and communication. In the DBGP algorithm, each particle’s centroid is represented by a queen marker and the particle surface is covered with a uniform distribution of surface worker markers. The queen marker contains information on the translational and rotational motion of a particle and integrates the force and torque computed at all the worker markers, while the worker marker implements the fluid–particle interaction. Ghost queen and ghost worker markers are generated for each real queen and real worker marker during computation for particle-level and marker-level communications, respectively. A double Cartesian binning process is introduced that divides the physical domain into a coarse queen-bin and a fine worker-bin structure in three dimensions. The queen-bin and worker-bin sizes are determined by their zone of influence at the particle-level and marker-level communication, respectively. Bin-to-rank maps that relate each queen-bin and worker-bin to all the MPI ranks that they interact with are created. By using the queen/worker marker representation and two-layer bin-to-rank maps, data communication across very large number of MPI ranks is efficiently carried out. A scaling analysis has been conducted, showing excellent performance of the DBGP algorithm for up to 16,384 MPI ranks in both weak and strong scaling studies. The proposed method has been demonstrated to accurately predict sedimentation of particle clouds. The simulated correlation between the mean settling velocity and volume fraction is in good agreement with empirical correlations from previous studies.

Dynamics of a Viscous Droplet in Return Bends of Microfluidic Channels.

Return bends are frequently encountered in microfluidic systems. In this study, a three-dimensional spectral boundary element method for interfacial dynamics in Stokes flow has been adopted to investigate the dynamics of viscous droplets in rectangular return bends. The droplet trajectory, deformation, and migration velocity are investigated under the influence of various fluid properties and operational conditions, which are depicted by the Capillary number, viscosity ratio, and droplet size, as well as the dimensions of the return bend. While the computational results provide information for the design of return bends in microfluidic systems in general, the computational framework shows potential to guide the design and operation of a droplet-based microfluidic delivery system for cell seeding.

Spectral boundary element modeling of water waves in Pohang New Harbor and Paradip Port

A novel mathematical model formulation based on spectral boundary element method (SBEM) is presented to examine the wave response in the Pohang New Harbor (PNH), South Korea and Paradip port, Odisha, India. In SBEM, the boundary element method (BEM) is coupled with the spectral element method (SEM) to enhance the numerical accuracy of the present numerical scheme. The numerical solution on each boundary element is obtained by using boundary integral associated with Chebyshev point discretization. The boundary Integrals are transformed using Jacobians and evaluated with Clenshaw Curtis Quadrature rule. Convergence analysis is performed on rectangular domain for present scheme, BEM and hybrid finite element method (HFEM), which shows that the present numerical scheme is better as compared to other traditional numerical schemes. Further, the simulation results are validated with BEM, analytical method and experimental data from Lee (1971) and Ippen and Goda (1963). The wave amplification is obtained at six record stations inside the PNH, South Korea and Paradip port, Odisha, India. In addition, the spectral density is also determined for multidirectional random waves propagating towards the PNH and Paradip port at the same record station. The resonant frequencies are estimated in PNH and Paradip port for the safe navigation of moored ship.

On modelling three-dimensional elastodynamic wave propagation with boundary spectral element method

ABSTRACTIn this paper, a boundary spectral element method (BSEM) for solving the problem of three-dimensional wave propagation is introduced. In the new formulation, elastodynamics of structures is computed by the Laplace transformed boundary element method (BEM), and boundaries of structures are discretised into high-order isoparametric spectral elements. Three types of spectral elements – Lobatto, Gauss–Legendre and Chebyshev elements – have been implemented. With a significantly higher computational efficiency than the conventional BEM, the BSEM provides a competitive alternative for modelling high-frequency wave propagation in engineering applications.

On modelling three-dimensional piezoelectric smart structures with boundary spectral element method

The computational efficiency of the boundary element method in elastodynamic analysis can be significantly improved by employing high-order spectral elements for boundary discretisation. In this work, for the first time, the so-called boundary spectral element method is utilised to formulate the piezoelectric smart structures that are widely used in structural health monitoring (SHM) applications. The resultant boundary spectral element formulation has been validated by the finite element method (FEM) and physical experiments. The new formulation has demonstrated a lower demand on computational resources and a higher numerical stability than commercial FEM packages. Comparing to the conventional boundary element formulation, a significant reduction in computational expenses has been achieved. In summary, the boundary spectral element formulation presented in this paper provides a highly efficient and stable mathematical tool for the development of SHM applications.

A coupled finite and boundary spectral element method for linear water-wave propagation problems

• A coupled finite and boundary spectral element method for linear water-wave propagation problems is proposed. • Boundary spectral element method (BSEM) is a new technique that combines the advantages of the spectral approach and the BEM. • BSEM has been applied to the mild-slope equation with variable bathymetry in one direction. • A convergence study has been made for the BSEM alone and coupled with finite spectral elements. • The proposed formulation has been validated by solving classical water-wave propagation problems. A coupled boundary spectral element method (BSEM) and spectral element method (SEM) formulation for the propagation of small-amplitude water waves over variable bathymetries is presented in this work. The wave model is based on the mild-slope equation (MSE), which provides a good approximation of the propagation of water waves over irregular bottom surfaces with slopes up to 1:3. In unbounded domains or infinite regions, space can be divided into two different areas: a central region of interest, where an irregular bathymetry is included, and an exterior infinite region with straight and parallel bathymetric lines. The SEM allows us to model the central region, where any variation of the bathymetry can be considered, while the exterior infinite region is modelled by the BSEM which, combined with the fundamental solution presented by Cerrato et al. [A. Cerrato, J. A. González, L. Rodríguez-Tembleque, Boundary element formulation of the mild-slope equation for harmonic water waves propagating over unidirectional variable bathymetries, Eng. Anal. Boundary Elem. 62 (2016) 22–34.] can include bathymetries with straight and parallel contour lines. This coupled model combines important advantages of both methods; it benefits from the flexibility of the SEM for the interior region and, at the same time, includes the fulfilment of the Sommerfeld’s radiation condition for the exterior problem, that is provided by the BSEM. The solution approximation inside the elements is constructed by high order Legendre polynomials associated with Legendre–Gauss–Lobatto quadrature points, providing a spectral convergence for both methods. The proposed formulation has been validated in three different benchmark cases with different shapes of the bottom surface. The solutions exhibit the typical p -convergence of spectral methods.

A Study on Fourier Spectral and Fourier-Hankel Spectral Field Variable Approximations in BEM for 2D Interior Helmholtz Problems

In this work the numerical modeling of the time harmonic problems in acoustics using boundary element method (BEM) is studied. The different computational techniques and methods for the solution of acoustic problems are reviewed. The formulation of some of the conventional BEM techniques is presented. The mathematical formulation and numerical implementation of Fourier Spectral BEM (FSBEM) and Fourier-Hankel Spectral BEM (FHSBEM) is presented for 2D problems. The numerical implementation of the conventional C-1 and linear C0 BEM formulations and FSBEM/FHSBEM is carried out by developing MATLAB codes. The verification of the numerical implementation of C-1 and linear C0 BEM formulations is carried out by comparing with analytical solutions and FEM. The comparison of FSBEM and FHSBEM is carried out with C-1 and linear C0 BEM formulations.

Dynamics of fluid bridges between a rising capillary tube and a substrate

Micro/nanolithography is an emerging technique to create micro-/nano-features on substrate. New capillary-based lithography method has been developed to overcome the limitations (e.g., the direct contact with the substrate) of existing lithography techniques including dip-pen nanolithography, nano-imprint lithography, and electron-beam lithography. The understanding of the behavior of the liquid bridge formed between a capillary tube and a substrate is essential for the recently developed capillary-based lithography method that is non-invasive to the substrate. A three-dimensional spectral boundary element method has been employed and modified to describe the dynamics of the liquid bridge. Starting with a steady-state liquid bridge shape, the transient bridge deformation is computed as the capillary tube is being lifted away from the substrate. The motion of the three-phase contact line on the substrate is taken into consideration. The computational results are validated with the experimental findings. The influences of the lifting speed of the capillary, liquid properties, and contact line slip conditions on the bridge dynamics are investigated. We conclude by computations that with a higher lifting speed of the capillary tube, the contact line radius on the substrate and the bridge neck radius reduce in a faster manner; a smaller residual droplet tends to form on the substrate if the viscosity ratio (the bridge liquid versus the surrounding medium) is larger or the substrate is more slippery to the fluids.

A modified scaled boundary approach in frequency domain with diagonal coefficient matrices

In order to solve the scaled boundary differential equation in dynamic stiffness, an initial value is needed. This initial value can be obtained using high frequency asymptotic expansion of dynamic stiffness matrix. Expanded dynamic stiffness matrix of unbounded mediums at high frequency was presented by previous researchers based on the fully populated coefficient matrices. In this paper, lumped coefficient matrices are used to modify the scaled boundary procedure. Some extra computational efforts of the original scaled boundary method can be eliminated using the proposed approach. The scaled boundary spectral element method (SBSEM) is used to achieve lumped coefficient matrices. It is shown that the proposed method leads to correct dynamic stiffness matrix. Therefore, it can be applied to solve scaled boundary differential equation of unbounded mediums, efficiently. A comparison between the results of the modified and the original methods is presented and accuracy of the modified method is investigated.

Sparse tensor product spectral Galerkin BEM for elliptic problems with random input data on a spheroid

We introduce and analyze a sparse tensor product spectral Galerkin Boundary Element Method based on spherical harmonics for elliptic problems with random input data on a spheroid. Problems of this type appear in geophysical applications, in particular in data acquisition by satellites. Aiming at a deterministic computation of the k-th order statistical moments of the random solution, we establish convergence theorems showing that the sparse tensor product spectral Galerkin discretization is superior to the full tensor product spectral Galerkin discretization in the case of mixed regularity of the data's k-th order moments, naturally implying mixed regularity of the k-th order moments of the random solution. We prove that analytic regularity of the data's k-th order moments implies analytic regularity of the solution's k-th order moments. We illustrate performance of the sparse and full tensor product discretization schemes on several numerical examples.

Deformation and migration of a leaky-dielectric droplet in a steady non-uniform electric field

This work numerically investigates the dynamics of an initially uncharged droplet freely suspended in another immiscible fluid and influenced by a steady non-uniform electric field. Both the droplet and the suspending fluid are assumed leaky dielectric. A three-dimensional spectral boundary element method is employed. It is validated by comparing with other numerical results and experimental findings for droplet deformation in a uniform electric field. This work reveals the dominant influence of the relative conductivity of fluids on the droplet migration direction in a non-uniform electric field. We have also explored the complicated behavior of the droplet deformation and speed affected by the relative permittivity and conductivity. In addition, the impact of the electric capillary number and viscosity ratio on the droplet dynamics has also been investigated. Results found and numerical algorithms developed in this study pave ways for investigations on droplet motion in an electric field created by co-planar electrodes, which has direct applications in digital microfluidics.

Dynamics of concentric and eccentric compound droplets suspended in extensional flows

The motion, deformation, and stability of compound droplets in extensional flows are investigated numerically via a three-dimensional spectral boundary element method. We examine the droplet stability under the influences of the capillary number, the inner droplet size and the relative magnitude of the surface tension of the two interfaces composing the compound droplet. The influence of viscosity on the droplet deformation is also discussed. We conclude that a compound droplet with a larger inner droplet and/or smaller inner surface tension is less stable and cannot withstand strong flow. For moderate viscosity ratios, a compound droplet with a more viscous “shell” exhibits larger deformation at steady state. In addition, for an eccentric compound droplet, both the inner and outer droplets tend to migrate away from its original location due to the asymmetry of the problem. The initial location of the inner droplet also influences the droplet stability as well as the migration velocity of the compound droplet.

Numerical studies of geometry effects of a two-dimensional microfluidic four-roll mill on droplet elongation and rotation

Abstract A microfluidic analog of the four-roll mill has been designed to generate all flow types including extensional flow, shear flow and rotational flow [ Lee et al. Appl. Phys. Lett. 90, 074103 (2007) ]. Boundary element methods are powerful tools to study interfacial dynamics of droplets in microchannels due to its high facility in describing the complex geometries and the various boundary conditions. Through a two-dimensional (2D) spectral boundary element method, effects of the geometry and the volume flow rate on the elongation and rotation of droplets in this microfluidic four-roll mill (MFRM) are investigated in this paper. Behaviors of deforming or rotating droplets trapped in a MFRM are determined by the size of orifices connecting the channels (inlets and outlets) and the central cavity, and the volume flow rate of the continuous phases (CP) pumped into and out of the device, especially for the rotation of droplets. Through an approximate analysis of order of magnitude, a simple conclusion that the average angular velocity of the rotation is proportional to the volume flow rate of CP at inlets and the radius of the central cavity is presented, which is testified by the numerical calculation. Besides, the appropriate radius of the central cavity which directly determines the size of the orifices is specified through the analysis of the flow fields. These results are helpful in designing a MFRM with a much larger depth than the width of the micro-channel for the purpose of microfluidic rheometry.

Low-Reynolds-number droplet motion in a square microfluidic channel

In this study, we investigate computationally the low-Reynolds-number droplet motion in a square micro-channel, a problem frequently encountered in microfluidic devices, enhanced oil recovery and coating processes. The droplet deformation and motion are determined via a three-dimensional spectral boundary element method for wall-bounded flows. The effects of the flow rate, viscosity ratio and droplet size on the interfacial dynamics are identified for droplets smaller and larger than the channel size and for a wide range of viscosity ratio. Owing to the stronger hydrodynamic forces in the thin lubrication film between the droplet interface and the solid walls, large droplets exhibit larger deformation and smaller velocity. Under the same average velocity, a droplet in a channel shows a significantly smaller deformation and higher velocity than in a cylindrical tube with the same size, owing to the existence of the corners’ area in the channel which permits flow of the surrounding fluid. A suitable periodic boundary implementation for our spectral element method is developed to study the dynamics of an array of identical droplets moving in the channel. In this case, the droplet deformation and velocity are reduced as their separation decreases; the reduction is influenced by the flow rate, viscosity ratio and more significantly the droplet size.

Particle mesh Ewald Stokesian dynamics simulations for suspensions of non-spherical particles

A particle mesh Ewald (PME) Stokesian dynamics algorithm has been developed to model hydrodynamic interactions in suspensions of non-spherical dicolloidal particles. Dicolloids, which have recently been synthesized by a number of independent research groups (Johnson, van Kats & van Blaaderen (Langmuir, vol. 21, 2005, p. 11510), Mocket al. (Langmuir, vol. 22, 2006, p. 4037), Kim, Larsen & Weitz (J. Am. Chem. Soc., vol. 128, 2006, p. 14374)), consist of two intersecting spheres of varying radii and centre-to-centre separation. One-body resistance tensors and disturbance velocity fields are computed for general linear flows using a superposition of Stokes singularities along the symmetry axis of the dicolloid particles. The coefficients and the locations of the singularities are optimized to minimize the norm of the velocity error on the particle surface. The one-body solution provides all coefficients required for the far-field many-body interactions in the Stokesian dynamics algorithm. These generalize the analytical results for spheres employed in the classic algorithm. Modified lubrication interaction tensors are developed for dicolloids for the singular near-field lubrication interactions. Accuracy of the one-body solutions and two-body generalized Stokesian dynamics solutions are validated by comparison with high-precision numerical solutions computed with the spectral boundary element method of Muldowney & Higdon (J. Fluid Mech., vol. 298, 1995, p. 167). The newly developed PME Stokesian dynamics algorithm was used to study transport properties in dicolloidal suspensions over a range of volume fractions (φ ≤ 0.5). The effects of the degree of anisotropy on the properties of the suspension are discussed. For these mildly anisotropic particles, the transport properties remain close to those of spheres, however certain interesting trends emerge, with non-monotonic viscosity dependence as a function of increasing aspect ratio. The minimum viscosity in concentrated suspensions is lower than that for spheres with equal volume fraction over a range of volume fractions.

Droplet motion in a microconfined shear flow via a three-dimensional spectral boundary element method

A 3D spectral boundary element method is employed to compute the dynamics of a single droplet in a microconfined shear flow. Comparisons have been made for the motion of an initially spherical droplet near a single wall and that between two parallel plates. Investigations are conducted for the influences of the capillary number, viscosity ratio, and initial location of the droplet on the droplet deformation, orientation, velocities, as well as the transition between the initial rapid deformation and the subsequent relaxation stage. Computational results for the deformation and velocities are compared with analytical predictions. It is found that the analytical predictions are limited for small deformations, large droplet-wall distances, and near equiviscous droplets.