A Multipole Accelerated Desingularized Method for Computing Nonlinear Wave Forces on Bodies

1998 ◽  
Vol 120 (2) ◽  
pp. 71-76 ◽  
Author(s):  
S. M. Scorpio ◽  
R. F. Beck
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Wave ◽  
Mixed Boundary ◽  
Offshore Structures ◽  
Computational Effort ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Wave Forces ◽  
Surface Boundary ◽  
Time Step

Nonlinear wave forces on offshore structures are investigated. The fluid motion is computed using a Euler-Lagrange time-domain approach. Nonlinear free surface boundary conditions are stepped forward in time using an accurate and stable integration technique. The field equation with mixed boundary conditions that result at each time step are solved at N nodes using a desingularized boundary integral method with multipole acceleration. Multipole accelerated solutions require O(N) computational effort and computer storage, while conventional solvers require O(N2) effort and storage for an iterative solution and O(N3) effort for direct inversion of the influence matrix. These methods are applied to the three-dimensional problem of wave diffraction by a vertical cylinder.

Geometric Modeling for Fully Nonlinear Ship-Wave Interactions

1997 ◽  
Vol 41 (01) ◽  
pp. 17-25
Author(s):  
M.S. Celebi ◽  
R.F. Beck
Keyword(s):  
Body Surface ◽  
Geometric Modeling ◽  
Mixed Boundary ◽  
Boundary Integral ◽  
The Body ◽  
Boundary Integral Method ◽  
Lagrange Equations ◽  
Time Step ◽  
Fully Nonlinear ◽  
Node Placement

Using the desingularized boundary integral method to solve transient nonlinear water-wave problems requires the solution of a mixed boundary value problem at each time step. The problem is solved at nodes (or collocation points) distributed on an ever-changing body surface. In this paper, a dynamic node allocation technique is developed to distribute efficiently nodes on the body surface. A B-spline surface representation is employed to generate an arbitrary ship hull form in parametric space. A variational adaptive curve grid generation method is then applied on the hull station curves to generate effective node placement. The numerical algorithm uses a conservative form of the parametric variational Euler-Lagrange equations to perform adaptive gridding on each station. Numerical examples of node placement on typical hull cross sections and for fully nonlinear wave resistance computations are presented.

scholarly journals An integral equation–based numerical method for the forced heat equation on complex domains

2020 ◽  
Vol 46 (5) ◽  
Author(s):  
Fredrik Fryklund ◽  
Mary Catherine A. Kropinski ◽  
Anna-Karin Tornberg
Keyword(s):  
Integral Equation ◽  
Heat Equation ◽  
Helmholtz Equation ◽  
Singular Integrals ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Elliptic Pdes ◽  
Time Step ◽  
Modified Helmholtz Equation ◽  
High Regularity

Abstract Integral equation–based numerical methods are directly applicable to homogeneous elliptic PDEs and offer the ability to solve these with high accuracy and speed on complex domains. In this paper, such a method is extended to the heat equation with inhomogeneous source terms. First, the heat equation is discretised in time, then in each time step we solve a sequence of so-called modified Helmholtz equations with a parameter depending on the time step size. The modified Helmholtz equation is then split into two: a homogeneous part solved with a boundary integral method and a particular part, where the solution is obtained by evaluating a volume potential over the inhomogeneous source term over a simple domain. In this work, we introduce two components which are critical for the success of this approach: a method to efficiently compute a high-regularity extension of a function outside the domain where it is defined, and a special quadrature method to accurately evaluate singular and nearly singular integrals in the integral formulation of the modified Helmholtz equation for all time step sizes.

Numerical Simulation of Nonlinear Water Wave Propagation Over Rippled Bed

2007 ◽  
Author(s):  
Gerasimos A. Kolokythas ◽  
Athanassios A. Dimas
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Water Waves ◽  
Stokes Equations ◽  
Surface Flow ◽  
Surface Boundary ◽  
Computational Domain ◽  
Time Step ◽  
Outflow Boundary ◽  
Rippled Bed

In the present study, numerical simulations of the free-surface flow, developing by the propagation of nonlinear water waves over a rippled bottom, are performed assuming that the corresponding flow is two-dimensional, incompressible and viscous. The simulations are based on the numerical solution of the Navier-Stokes equations subject to the fully-nonlinear free-surface boundary conditions and the suitable bottom, inflow and outflow boundary conditions. The equations are properly transformed so that the computational domain becomes time-independent. For the spatial discretization, a hybrid scheme with finite-differences and Chebyshev polynomials is applied, while a fractional time-step scheme is used for the temporal discretization. A wave absorption zone is placed at the outflow region in order to efficiently minimize reflection of waves by the outflow boundary. The numerical model is validated by comparison to the analytical solution for the laminar, oscillatory, current flow which develops a uniform boundary layer over a horizontal bottom. For the propagation of finite-amplitude waves over a rigid rippled bed, the case with wavelength to water depth ratio λ/d0 = 6 and wave height to wavelength ratio H0/λ = 0.05 is considered. The ripples have parabolic shape, while their dimensions — length and height — are chosen accordingly to fit laboratory and field data. Results indicate that the wall shear stress over the ripples and the form drag forces on the ripples increase with increasing ripple height, while the corresponding friction force is insensitive to this increase. Therefore, the percentage of friction in the total drag force decreases with increasing ripple height.

Computing Large Amplitude Motions of a Floating Offshore Structure in the Time Domain

2008 ◽  
Author(s):  
Wei Qiu ◽  
Hongxuan Peng
Keyword(s):  
Time Domain ◽  
Large Amplitude ◽  
Offshore Structures ◽  
The Body ◽  
Surface Boundary ◽  
Offshore Structure ◽  
Floating Structure ◽  
Time Step ◽  
Large Amplitude Motions ◽  
The Time Domain

Based on the panel-free method, large-amplitude motions of floating offshore structures have been computed by solving the body-exact problem in the time domain using the exact geometry. The body boundary condition is imposed on the instantaneous wetted surface exactly at each time step. The free surface boundary is assumed linear so that the time-domain Green function can be applied. The instantaneous wetted surface is obtained by trimming the entire NURBS surfaces of a floating structure. At each time step, Gaussian points are automatically distributed on the instantaneous wetted surface. The velocity potentials and velocities are computed accurately on the body surface by solving the desingularized integral equations. Nonlinear Froude-Krylov forces are computed on the instantaneous wetted surface under the incident wave profile. Validation studies have been carried out for a Floating Production Storage and Offloading (FPSO) vessel. Computed results were compared with experimental results and solutions by the panel method.

Time-Domain Second-Order Wave Radiation in Two Dimensions

1993 ◽  
Vol 37 (01) ◽  
pp. 25-33 ◽  
Author(s):  
Michael Isaacson ◽  
Joseph Y. T. Ng
Keyword(s):  
Circular Cylinder ◽  
Time Domain ◽  
Body Position ◽  
Two Dimensions ◽  
Boundary Integral ◽  
Second Order ◽  
The Body ◽  
Wave Radiation ◽  
Surface Boundary ◽  
Time Step

This paper presents a time-domain second-order method to study the nonlinear wave radiation problem in two dimensions. A time-stepping scheme is adopted to obtain the resulting flow development which satisfies the nonlinear free-surface boundary conditions and the radiation condition to second order, and the numerical procedure utilizes a boundary integral equation method based on Green's theorem to calculate the field solution at each time step. The body surface boundary condition is expanded about the mean body position to second order by a Taylor series. The method is applied to the cases of a semi-submerged circular cylinder and a rectangular cylinder undergoing sinusoidal sway, heave and roll motions. For the case of the circular cylinder, comparisons of the computed hydrodynamic forces at first and second order are made with previous theoretical and experimental results and a favorable agreement is indicated. The importance of second-order effects in the calculation of the hydrodynamic force is discussed.

3D finite-difference dynamic-rupture modeling along nonplanar faults

Geophysics ◽  
2007 ◽  
Vol 72 (5) ◽  
pp. SM123-SM137 ◽  
Author(s):  
Víctor M. Cruz-Atienza ◽  
Jean Virieux ◽  
Hideo Aochi
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Heterogeneous Media ◽  
Slip Rate ◽  
Quantitative Agreement ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Weight Functions ◽  
Dynamic Rupture ◽  
Fault Surface

Proper understanding of seismic emissions associated with the growth of complexly shaped microearthquake networks and larger-scale nonplanar fault ruptures, both in arbitrarily heterogeneous media, requires accurate modeling of the underlying dynamic processes. We present a new 3D dynamic-rupture, finite-difference model called the finite-difference, fault-element (FDFE) method; it simulates the dynamic rupture of nonplanar faults subjected to regional loads in complex media. FDFE is based on a 3D methodology for applying dynamic-rupture boundary conditions along the fault surface. The fault is discretized by a set of parallelepiped fault elements in which specific boundary conditions are applied. These conditions are applied to the stress tensor, once transformed into a local fault referenceframe. Numerically determined weight functions multiplying particle velocities around each element allow accurate estimates of fault kinematic parameters (i.e., slip and slip rate) independent of faulting mechanism. Assuming a Coulomb-like slip-weakening friction law, a parametric study suggests that the FDFE method converges toward a unique solution, provided that the cohesive zone behind the rupture front is well resolved (i.e., four or more elements inside this zone). Solutions are free of relevant numerical artifacts for grid sizes smaller than approximately [Formula: see text]. Results yielded by the FDFE approach are in good quantitative agreement with those obtained by a semianalytical boundary integral method along planar and nonplanar parabola-shaped faults. The FDFE method thus provides quantitative, accurate results for spontaneous-rupture simulations on intricate fault geometries.

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