Case Studies of BEM Solver Accuracy With the Open-Source Code Capytaine

Rules of Classification Societies all around the world have made changes on design wave loads’ value and fatigue influence factor modification due to the influence of springing and whipping on ultra-large containerships. The paper firstly introduced 3-D linear hydroelastic theory in frequency domain and 3-D nonlinear hydroelastic theory in time domain, considering large amplitude motion nonlinearity and slamming force due to the severe relative motion between ship hull and wave. Then the spectrum analysis method and time domain statistical analysis method were introduced, which can make fatigue analysis under a series of standard steps in frequency and time domain, respectively. Finally, discussions on the influence factor of springing and whipping on fatigue damages of 8500TEU and 10000TEU containerships with different loading states were made. The fatigue assessment of different position on the midship section was done on the basis of nominal stress. The fatigue damage due to whipping can be the same as the fatigue damage due to springing and even sometimes can be larger than the springing damage. Besides, some suggestions on calculating load case selection were made to minimize the quantity of work in frequency and time domain. Thus, tools for fatigue influence factor modification were provided to meet the demand of IACS-UR.

Revisit of the equivalence of the displacement discontinuity method and boundary element method for solving crack problems

Identities for the fundamental solution of thin plate bending problems and the nonuniqueness of the hypersingular BIE solution for multi-connected domains

When a vibrating thin structure moves in a heavy flow field, the effect of sound waves on the structure cannot be ignored. This effect is especially pronounced when the relative velocity of the structure and the flow field increases. The hybrid numerical method of finite element method (FEM) and convective boundary element method (BEM) is proposed to solve acoustic-structural interaction problems in uniform relative motion between structure source and fluid. Besides, an acoustic-analogy Lorentz transform is proposed to handle the direct convective boundary integral equation (BIE), which makes the convective BIE has the same form as traditional BIE so that there are no more convective terms and extra singular integrals. The continuity condition at the fluid-structure interface is also proposed to generate the coupling equation. Compared with the traditional FEM, this hybrid method has the advantages of greatly reducing the number of finite elements outside the structure as well as automatically satisfying the radiation condition at far field. And compared with the direct convective BIE, this method reduces the computational complexity and time. It is potential to deliver a solver with higher accuracy and less memory consumption. Some numerical examples are designed to study the influence of the convective effect. Keywords: Acoustic-structural interaction, Finite element method, Boundary element method, FEM-BEM coupling

Sloshing, a violent behaviour of liquid contents in tanks submitted to the forced vessels’ motion on the sea represents one of the major considerations in LNG vessels design over several past decades. State of the art of sloshing analysis relies on small-scale sloshing model tests supported by extensive developments of CFD computation techniques, commonly studying one isolated tank submitted to the forced motion without their mutual interaction. In reality, wave-induced response of the vessel carrying liquid cargo is affected by internal liquid motion, and consequently, tank liquid flow is altered by the vessel motion in return. An efficient numerical model for dynamic coupling between motions exerted by tank liquid (sloshing) and rigid body motions of the vessel (seakeeping) was developed in Bureau Veritas, formulated under the assumptions of linear potential theory in frequency domain. As already experienced with anti-rolling tanks, strong coupling effect is perceived on the first order transverse motions. However, consequences of coupled motions on sloshing loads have not been explored yet. This paper presents comparative analysis of sloshing effects induced by coupled and non-coupled vessel motion, introduced as the excitation to 6 d.o.f. small-scale model test rig. Possible risk of coupled effects is demonstrated on the example of standard size of LNG carrier operating with partly filled cargo tanks.

The boundary element method (BEM) is an efficient numerical method that approximates solutions of various boundary value problems. Despite its success little research has been performed on the conditioning of the linear systems that appear in the BEM. For a Laplace equation with Dirichlet boundary conditions a remarkable phenomenon is observed; the corresponding boundary integral equation (BIE) is singular for a certain critical size of the 2D domain. As a consequence the discrete counterpart of the BIE, the linear system, is singular too, or at least ill-conditioned. This is reflected by the condition number of the system matrix, which is infinitely large, or at least very large. When the condition number of the BEM-matrix is large, the linear system is difficult to solve and the solution of the system is very sensible to perturbations in the boundary data. For a Laplace equation with mixed boundary conditions a similar phenomenon is observed. The corresponding BEM-matrix consists of two blocks; one block originates from the BEM-matrix belonging to the Dirichlet problem, the other block originates from the BEM-matrix belonging to the Neumann problem. The composite matrix inherits the solvability problems from the Dirichlet block. In other words, for the Laplace equation with mixed boundary conditions there exists also a critical size of the 2D domain for which the BEM-matrix has an infinitely large condition number. Hence the size and shape of the domain affects the solvability of the BEM problem. The critical size of the domain for which the BIE becomes singular is related to the logarithmic capacity of the domain. The logarithmic capacity is a positive real number that is a function of the size and shape of the domain. If this logarithmic capacity is equal to one, the domain is a critical domain, and for this domain the BIE becomes singular. Thus by computing the logarithmic capacity we can a-priori determine whether the BIE will be singular or not. The logarithmic capacity depends linearly on the scale of the domain, and thus a domain with logarithmic capacity equal to one can always be found by rescaling the domain. Unfortunately the logarithmic capacity can only be computed analytically for a few simple domains; for more involved domains the logarithmic capacity can be estimated though. There are several possibilities to avoid large condition numbers, i.e. singular BIEs that appear at critical domains. The first option is to rescale the domain such that the logarithmic capacity is unequal to one. One can also add a supplementary condition to the BIE and the linear system. A drawback of this option is that the linear system has more equations than unknowns and different techniques are required to solve the system. A third option is to slightly modify the fundamental solution of the Laplace operator. This fundamental solution directly appears in the BIE and it can be shown that a suitable modification yields BIEs that do not become singular. The critical domains for which the BIEs become singular do not restrict to Laplace equations only. Also for BIEs applied to the biharmonic equation or the elastostatic equations and the Stokes equations such critical domains exist. As the last two equations are vectorial equations, also the corresponding BIE consists of two equations. As a consequence two critical domains can be found for which these BIEs become singular. To obtain nonsingular BIEs techniques similar to the Laplace case can be used. Unfortunately we cannot a-priori determine the sizes for which the BIEs becomes singular, and thus do not know to what size we should rescale the domain to obtain nonsingular BIEs. The existence of critical domains is in essence caused by the logarithmic term in the fundamental solutions for the elliptic boundary value problems in 2D. This logarithmic term does not depend linearly on the size of the domain. When a domain is scaled, i.e. multiplied by a scale factor, the argument of the logarithm is also multiplied by this scale factor, but the logarithm turns this into an additive term. Thus the logarithm transforms multiplication into addition. This affects the BIEs in such a way that critical domains can appear. The fundamental solutions of boundary value problems in 3D do not contain a logarithmic term. Hence scaling of the domain does not affect the fundamental solution, and consequently also the BIE is not affected. Hence we may safely rescale 3D domains without the risk to encounter a critical domain. An example in which a domain takes many different sizes and shapes is the blowing problem. In this problem a viscous fluid is blown to a desired shape. Typically the time is discretised into a set of discrete time steps, and at each step the shape of the fluid is computed by solving the Stokes equations. When attempting to simulate this problem in 2D, we meet a large number of 2D domains, and we risk that one of these domains is equal to or approaches a critical domain. In such a case the BEM will have difficulties with solving the Stokes equations for that particular domain. When simulating the blowing problem in 3D, no critical domains are encountered. It turns out that the BEM is a very efficient numerical method for this particular 3D problem with a moving boundary. As we are merely interested in the shape of the fluid, we only need to know the flow of its boundary. The BEM does exactly that; it does not compute the flow at the interior of the fluid. Furthermore it is rather easy to include other effects from the blowing problem in the model, such as gravity, surface tension and friction from the contact of the fluid with a wall. As only the boundary of the fluid is discretised, the system matrices that appear in the BEM are smaller than the system matrices that appear when solving the problem with a finite element method, for example. Though the BEM-matrices are dense, while the finite element matrices are sparse, the computational effort for the BEM is relatively low. In short, the BEMis a very appropriate numerical method when solving blowing problems.

Beamforming for measurements under disturbed propagation conditions using numerically calculated Green’s functions

Boundary element method in nonlinear transient heat transfer of reactor solids with convection and radiation on surfaces

A u-p format 2.5D finite element method (FEM) is proposed to investigate the ground vibration and distribution of pore water pressure in transversely isotropic saturated medium subjected to high-speed train load. The governing equations are derived based on Biot’s theory in frequency domain by applying Fourier transform with respect to time. The Galerkin method is then employed to develop the 2.5D FE model. Verification of the model is carried out by comparing the predictions with published data. Effects of soil parameters on the dynamic responses of ground are investigated in detail. Results indicate that the increase in vertical elastic modulus contributes much more to the decrease in displacement amplitude of ground and maximum amplitude of pore water pressure in soil, compared to that of horizontal elastic modulus. Both horizontal and vertical Poisson ratios have insignificant effects on ground vibration, while the change of Poisson ratio lead to a great change in the maximum amplitude of pore water pressure. Effects of shear modulus on both ground vibration and maximum amplitude of pore water pressure are negligible.

A direct (potential based) boundary element method (BEM) developed in our company (Romanian Research and Design Institute for Shipbuilding) is briefly presented. It may be applied for the calculation of the potential flow paste the ship hull, also around the lifting bodies like: hydrofoil, nozzle, rudder, propeller. The lifting bodies hydrodynamic calculation has some modelling particularities, e.g.: the wake shape determination, and numerical difficulties, e.g.: the BEM integrals calculation. Therefore the BEM application to such kind of bodies substantially differs from standard BEM application.

The Laplace problem subject to the Dirichlet or Neumann boundary condition in the direct and indirect boundary element methods (BEM) sometimes both may result in a singular or ill-conditioned system (some special situations) for the interior problem. In this paper, the direct and indirect BEMs are revisited to examine the uniqueness of the solution by introducing the Fichera’s idea and the self-regularized technique. In order to construct the complete range of the integral operator in the BEM lacking a constant term in the case of a degenerate scale, the Fichera’s method is provided by adding the constraint and a slack variable to circumvent the problem of degenerate scale. We also revisit the Fredholm alternative theorem by using the singular value decomposition (SVD) in the discrete system and explain why the direct BEM and the indirect BEM are not indeed equivalent in the solution space. According to the relation between the SVD structure and Fichera’s technique, a self-regularized method is proposed in the matrix level to deal with non-unique solutions of the Neumann and Dirichlet problems which contain rigid body mode and degenerate scale, respectively, at the same time. The singularity and proportional influence matrices of 3 by 3 are studied by using the property of the symmetric circulant matrix. Finally, several examples are demonstrated to illustrate the validity and the effectiveness of the self-regularized method.

This paper presents the development of a machine learning (ML) algorithm to predict the hydrodynamic coefficients to calculate the motion behavior of a vessel in frequency domain. The aim is to develop a less laborious and faster method compared to the conventional procedure for determining the hydrodynamic coefficients for added mass, damping, and excitation forces by radiation-diffraction models with linear potential flow solvers based on Boundary Element Methods (BEM). Physics-informed machine learning was utilized to incorporate physical laws directly into the ML models, ensuring accurate and physically consistent predictions. The ML-algorithm is trained and validated with a database created by the software ANSYS AQWA. The approach is performed on the Jack-Up Vessel(JUV) Innovation, owned and operated by DEME.

The time-resolved reflectance of photons from a homogeneous tissue was modeled using a Monte Carlo simulation. The data was then converted by fast Fourier transform (FFT) into the frequency domain. In the frequency domain, the phase, (Phi) , and modulation, M, of collected light from a frequency-modulated light source was determined. A comparison of Monte Carlo and diffusion theory was made for various separation distances between the source and collector on the tissue surface. The results showed that Monte Carlo and diffusion theory agreed in the time domain only for times larger than 500 ps after injection of an impulse of photons. In the frequency domain, Monte Carlo and diffusion theory agreed only if the probe separation, r, was at least 2 cm apart for (mu) s' equals (mu) s(1 - g) equals 5 cm-1, or in dimension less units r(mu) s' > 10. The effect of buried absorbed is also tested in the time and frequency domains. A semi-infinite volume of absorber is placed at 0, 3 mm, 6 mm, or (infinity) from the surface of a nonabsorbing tissue. The presence of a deep absorber on the time and frequency domain data show that attenuation of longer pathlength photons causes the phase of collected photons to reduce and the modulation of collected photons to increase. Both effects are indicative of the net shorter pathlength of the ensemble of collected photons.© (1993) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

This work reviews models and methods for determining the dynamic response of pavements to moving vehicle loads in the framework of continuum-based three dimensional models and linear theories. This review emphasizes the most representative models and methods of analysis in the existing literature and illustrates all of them by numerical examples. Thus, 13 such examples are presented here in some detail. Both flexible and rigid (concrete) pavement models involving simple and elaborate cases with respect to geometry and material behavior are considered. Thus, homogeneous or layered half-spaces with isotropic or cross-anisotropic and elastic, viscoelastic or poroelastic properties are considered. The vehicles are modeled as simple point or distributed loads or discrete spring-mass-dashpot system moving with constant or variable velocity. The dynamic response of the above pavement-vehicle systems is obtained by analytical/numerical or purely numerical methods of solution. Analytical/numerical methods have mainly to do with Fourier transforms or complex Fourier series with respect to both space and time. Purely numerical methods involve the finite element method (FEM) and the boundary element method (BEM) working in time or frequency domain. Critical discussions on the advantages and disadvantages of the various pavement-vehicle models and their methods of analysis are provided and the effects of the main parameters on the pavement response are determined through parametric studies and presented in the examples. Finally, conclusions are provided and suggestions for future research are made. • A review on models and methods for dynamic analysis of road pavements under moving vehicle loads is presented. • Flexible and rigid pavements are considered under 3D continuum-based models with linear theories and conditions. • Representative examples serve to illustrate models and methods of analysis.

Within the wave energy community, hydrodynamic coefficients obtained from boundary element methods (BEMs) are commonly used to predict the behaviour of wave energy converters (WECs) in response to incident waves. A number of commercially-available BEM solvers exist, with a number of open-source alternatives also available. While open-source solvers have an obvious cost advantage compared to their commercial counterparts, the results from such solvers are often susceptible to so-called ‘irregular frequencies’, which arise from ill-conditioning in boundary integral problems, and result in large under- or over-estimation of hydrodynamic parameters at certain excitation frequencies. Furthermore, while commercial solvers may employ techniques to suppress the effects of irregular frequencies, such solvers may, under certain circumstances, exhibit other problems in the hydrodynamic results produced. For example, the results obtained for the added mass at high frequencies, and the infinite frequency added mass for a water column, may be incorrect. The current work first focusses on an approach to remove the effects of irregular frequencies from the results obtained for the radiation damping of a particular WEC geometry. The use of radiation damping results to obtain values for the added mass, through the use of the Ogilvie relations, is then considered.The technique described herein has been implemented in BEMRosetta, an open-source tool which allows a user to view the results from various BEM solvers, as well as converting input files between solvers. The results presented in this paper have been obtained using the BEMRosetta implementation.