High-order Boundary Integral Equation Research Articles

OpenMP parallelism in computations of three-dimensional potential numerical wave tank for fully nonlinear simulation of wave-body interaction using NURBS

The open multi-processing parallelism was implemented into a serial code to compute a high-order boundary integral equation based on the second Green's identity. on symmetric multiprocessing computer platforms. The parallel code is augmented to a three-dimensional potential numerical wave tank for fully nonlinear wave-body interaction problems in the time domain. The non-uniform rational B-Spline surface was manipulated to define the geometry of the computational boundaries and the distribution of boundary values either as a high-order iso-geometric formulation. The material node approach and fourth-order Runge-Kutta time integration method were compounded for time marching the fully nonlinear free surface boundary. The propagation of a fifth-order Stokes wave was simulated to examine the scalability of the parallel constructs implementation. Afterwards, the second-order Stokes wave interaction with a cylindrical pile was modeled as a practical case study to show the benefits of the parallel code. The results were compared with former experiments and numerical studies.

Fully nonlinear and linear ship waves modelling using the potential numerical towing tank and NURBS

Waves induced by the ship advance in the calm water are simulated in a potential Numerical Towing Tank based on both the fully nonlinear approach and the linear theory. In the linear theory, a modified Green function, which was derived based on linearized free surface boundary conditions and the image method is used as a fundamental solution in the boundary integral equation. High-order boundary integral equation is used to solve the boundary value problem with Non-Uniform Rational B-Spline (NURBS) basis function. On the other hand, the time domain simulation of the fully nonlinear free surface flow due to the steady flow past over a ship model is conducted based on the Mixed Eulerian-Lagrangian approach. NURBS surface is used to describe the ship hull and the fully nonlinear free surface evolution. The numerical solutions are compared with a prior experimental study to verify both numerical models. In addition, the effect of nonlinear terms in fully nonlinear free surface boundary conditions is investigated. The wave making resistance as an essential criterion is compared with the experimental data and some numerical efforts to investigate the accuracy of both models. The capability of the modified Green function is also examined to simulate the ship waves in rectangular channels.

A boundary integral equation method for mode elimination and vibration confinement in thin plates with clamped points

We consider the bi-Laplacian eigenvalue problem for the modes of vibration of a thin elastic plate with a discrete set of clamped points. A high-order boundary integral equation method is developed for efficient numerical determination of these modes in the presence of multiple localized defects for a wide range of two-dimensional geometries. The defects result in eigenfunctions with a weak singularity that is resolved by decomposing the solution as a superposition of Green’s functions plus a smooth regular part. This method is applied to a variety of regular and irregular domains and two key phenomena are observed. First, careful placement of clamping points can entirely eliminate particular eigenvalues and suggests a strategy for manipulating the vibrational characteristics of rigid bodies so that undesirable frequencies are removed. Second, clamping of the plate can result in partitioning of the domain so that vibrational modes are largely confined to certain spatial regions. This numerical method gives a precision tool for tuning the vibrational characteristics of thin elastic plates.

Irregular wave transmission on bottom bumps using fully nonlinear NURBS numerical wave tank

In this study, a fully nonlinear three-dimensional numerical wave tank (NWT) is developed to simulate propagation of nonlinear random sea wave over the bottom ripple patches. Evolution of the free surface is performed by the non-uniform rational B-spline formulation (NURBS) and the mixed Eulerian–Lagrangian method (MEL). A high-order boundary integral equation is used to solve the Laplace equation in the Eulerian frame. The free surface is updated by the material node method and the fourth-order Runge–Kutta time integration scheme. Appropriate numerical solutions are obtained by deploying damping zones at the both tank end walls. Also, the NURBS approximation is applied to compute the kinematics of the free surface particles. Propagation of the irregular waves in an NWT is conducted and compared with the available experimental and numerical data. Bragg reflection of the random sea wave due to bottom ripple patches are compared with the prior numerical studies. The fully nonlinear free surface evolution of transmitted spectrum due to presence of the hemispherical bumps is modeled successfully. Also fifth-order Stokes wave is applied as a strong nonlinear incident wave to interact with the hemispherical bottom bumps.

High-order boundary integral equation solution of high frequency wave scattering from obstacles in an unbounded linearly stratified medium

We apply boundary integral equations for the first time to the two-dimensional scattering of time-harmonic waves from a smooth obstacle embedded in a continuously-graded unbounded medium. In the case we solve, the square of the wavenumber (refractive index) varies linearly in one coordinate, i.e. (Δ+E+x2)u(x1,x2)=0 where E is a constant; this models quantum particles of fixed energy in a uniform gravitational field, and has broader applications to stratified media in acoustics, optics and seismology. We evaluate the fundamental solution efficiently with exponential accuracy via numerical saddle-point integration, using the truncated trapezoid rule with typically 102 nodes, with an effort that is independent of the frequency parameter E. By combining with a high-order Nyström quadrature, we are able to solve the scattering from obstacles 50 wavelengths across to 11 digits of accuracy in under a minute on a desktop or laptop.

Simulation of irregular waves over submerged obstacle on a NURBS potential numerical wave tank

In this paper, a fully non-linear three-dimensional Numerical Wave Tank (NWT) is developed for studying propagation and scattering of non-linear random sea wave over bottom submerged bars. The simulation of fully non-linear free-surface is based on Non-Uniform Rational B-Spline formulation (NURBS) as a novel approach and Mixed Eulerian-Lagrangian method (MEL). High-order boundary integral equation is used to solve the Laplace equation in the Eulerian frame. To update the free-surface, time marching approach including material node method and fourth order Runge-Kutta time integration scheme is used. To obtain appropriate numerical solutions for wave propagation problem, damping zone is set at the downstream. Also, the NURBS approximation is employed to evaluate the velocity of the free-surface particles. Propagation of regular and irregular waves in a NWT is investigated and compared with the available experimental and numerical data. Transmission of the random sea wave over submerged bars is also compared with the experimental and prior numerical studies.

A fast and high-order method for the three-dimensional elastic wave scattering problem

In this paper we present a fast and high-order boundary integral equation method for the elastic scattering by three-dimensional large penetrable obstacles. The algorithm extends the method introduced in [5] for the acoustic surface scattering to the fully elastic case. In our algorithm, high-order accuracy is achieved through the use of the partition of unity and a semi-classical treatment of relevant singular integrals. The computational efficiency associated with the nonsingular integrals is attained by the method of equivalent source representations on a Cartesian grid and Fast Fourier Transform (FFT). The resulting algorithm computes one matrix–vector product associated with the discretization of the integral equation with O(N4/3logN) operations, and it shows algebraic convergence. Several numerical experiments are provided to demonstrate the efficiency of the method.

A high-order 3D boundary integral equation solver for elliptic PDEs in smooth domains

We present a high-order boundary integral equation solver for 3D elliptic boundary value problems on domains with smooth boundaries. We use Nyström’s method for discretization, and combine it with special quadrature rules for the singular kernels that appear in the boundary integrals. The overall asymptotic complexity of our method is O(N3/2), where N is the number of discretization points on the boundary of the domain, and corresponds to linear complexity in the number of uniformly sampled evaluation points. A kernel-independent fast summation algorithm is used to accelerate the evaluation of the discretized integral operators. We describe a high-order accurate method for evaluating the solution at arbitrary points inside the domain, including points close to the domain boundary. We demonstrate how our solver, combined with a regular-grid spectral solver, can be applied to problems with distributed sources. We present numerical results for the Stokes, Navier, and Poisson problems.

An effective high order interpolation scheme in BIEM for biharmonic boundary value problems

This paper presents an effective high order boundary integral equation method (BIEM) for the solution of biharmonic equations. All boundary values including geometries are approximated by high order radial basis function networks (RBFNs) rather than the conventional low order Lagrange interpolation schemes. For a better quality of approximation, the networks representing the boundary values and their derivatives are constructed by using integration processes. Prior conversions of network weights into nodal variable values are employed in order to form a square system of equations. Numerical results show that the proposed BIEM attains a great improvement in solution accuracy, convergence rate and computational efficiency over the linear- and quadratic-BIEMs.