A simplified two-dimensional boundary element method with arbitrary uniform mean flow

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.

Analytical solutions for the acoustic field in a thin annular duct with temperature gradient and mean flow

A new frequency-domain discretization technique that permits the numerical solution of the equation governing the propagation of small disturbances within nonuniform potential flows is described. The method can be applied successfully to the numerical solution of aeroacoustic problems because a good accuracy is preserved up to 3-4 points per period for three-dimensional unstructured meshes. The discretization scheme is based on a local interpolation formula that is strictly joined to the physics of wave propagation because It is constructed with the superimposition of elementary sources that are local solutions of the local convective wave equation. The method presents aspects in common with both finite difference and finite element methods, with the peculiarity that the local interpolation formula can be interpreted as a specific shape function introduced to take advantage of the physics of the problem. The method is applied to the potential equation linearized around an arbitrary aerodynamic mean flow, and several comparisons with theoretical results, as well as several convergence tests, are conducted to show that a good accuracy is preserved up to 3-4 points per period also for irregular meshes with both random and Systematic distortions. Numerical calculations are presented for three-dimensional problems with both uniform and nonuniform flow, and comparisons are made with theoretical and other available numerical results. With the appropriate generalizations, the method can be regarded as a new, efficient general approach for the discretization of generic partial differential equations.

This paper studies the dynamic foundation-soil-foundation interaction for two square rigid foundations embedded in a viscoelastic soil layer. The vibrations come from only one rigid foundation placed in the soil layer and subjected to harmonic loads of translation, rocking, and torsion. The required dynamic response of rigid surface foundations constitutes the solution of the wave equations obtained by taking account of the conditions of interaction. The solution is formulated using the frequency domain Boundary Element Method (BEM) in conjunction with the Kausel-Peek Green's s function for a layered stratum, with the aid of the Thin Layer Method (TLM), to study the dynamic interaction between adjacent foundations. This approach allows the establishment of a mathematical model that enables us to determine the dynamic displacements amplitude of adjacent foundations according to their different separations, the depth of the substratum, foundations masss, foundations embedded, and the frequencies of excitation. This paper attempts to introduce an approach based on a polynomial mathematical tool conducted from several results of numerical methods (BEM-TLM) so that practicing civil engineers can evaluation the dynamic foundations displacements more easy.

Numerical study on sub-harmonic generation due to interior and surface breaking cracks with contact boundary conditions using time-domain boundary element method

Acoustic nonlinearities in a quasi 1-D duct with arbitrary mean properties and mean flow

Boundary Element Analysis of Cracks in Shear Deformable Plates and Shells. Topics in Engineering, Vol 43

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

The spurious eigenvalues of an annular domain have been verified for the singular and hypersingular boundary‐element methods (BEMs) and circumvented by using the Burton‐Miller approach. Do they also occur in other formulations: continuous formulations such as the singular and hypersingular boundary integral equations (BIEs), the null‐field BIEs and the fictitious BIEs, or such discrete formulations as the null‐field BEMs and the fictitious BEMs? For the ten formulations of the multiply connected problem the study of otherwise the same issues is continued in the present paper. By using the degenerate kernels and the Fourier series, it is demonstrated analytically for the six continuous formulations of BIEs that spurious eigensolutions depend on the geometry of the inner boundary but not on that of the outer boundary. This conclusion can be extended to the six discrete formulations of BEMs. To filter out the spurious eigenvalues, the CHIEF (combined Helmholtz integral equation formulation) method is used here instead of the Burton‐Miller approach. The optimum number and appropriate positions of the CHIEF points are also addressed. It is then shown that, in the null‐field and fictitious BEMs, the spurious and true eigenvalues can be detected and distinguished by using the singular‐value‐decomposition‐updating techniques in conjunction with the Fredholm alternative theorem. Illustrative examples show the validity of the proposed methodologies.

Accurate prediction of loads on mechanical couplings is crucial in assessment of loads on coupled structures, and in reliable prediction of the motions of coupled bodies. This thesis presents a numerical articulation method, which is capable of modeling various mechanical couplings. The proposed numerical method relies on contact elements. The first objective of the work is to prove the eligibility of contact elements in representation of mechanical couplings, and to highlight the shortcomings and positive aspects of contact elements for this representation. The second objective of the thesis is to study the influence of different hydrodynamic, kinematic, and coupling models on the motion behavior of articulated bodies and coupling forces. To accomplish these objectives extensive and systematic studies on a twofold pushing convoy are conducted. The convoy bodies are interconnected with hinge joints. The hinges are modeled by means of kinematic constraints, and with contact elements. Frequency and time domain boundary element methods with linearized motions of equation, as well as a Reynolds-averaged Navier-Stokes (RANS) code coupled with a nonlinear kinematic solver are used to compute the body motion responses and coupling forces. The simulation results are compared to model basin measurements. It is shown that the contact element model can simulate mechanical couplings efficiently and it provides a suitable method to idealize free and suppressed modes at articulation locations. Furthermore, it is shown, that RANS in association with the nonlinear equation of motion and kinematically constrained joints, as well as boundary element methods in adherence to the contact element model provide more accurate prediction of articulation forces, and motion responses of coupled bodies. However, frequency and time domain boundary element methods with kinematically constrained joints still deliver agreeable and reliable results. Moreover, a case study is presented, in which a river-sea pushing convoy is applied in time domain numerical simulations. The convoy bodies are interconnected with prismatic joints and are free to heave and pitch relative to each other. Prismatic joints are modeled by means of contact elements. The case study demonstrates a successful application of contact elements in simulation of complexly articulated multibody systems. Furthermore, the findings obtained in this work contribute sustainably to the numerical approaches in assessment of hydrodynamic and articulation loads on mechanically coupled floating bodies.

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

In the boundary element method (BEM) for stress analysis, it is well known that thermal loads give rise to an additional volume integral in the primary form of the boundary integral equation (BIE). This volume integral needs to be further transformed to surface ones in order to retain the characteristic of the BEM as a boundary solution technique. In this study of the BEM for 3D thermoelasticity in general anisotropy, the fundamental solutions are expressed as Fourier series with coefficients calculated using an explicit-form Green’s function. In the exact volume-to-surface integral transformation associated with the term for the thermal effects in the BIE, a new kernel function is constructed. All formulations are implemented in an existing BEM code for 3D elastostatic analysis. Some numerical examples are presented to demonstrate the veracity of the formulations and the implementation, where the numerical results are compared with those obtained using the finite element method (FEM).

Transient elastodynamic crack analysis by time-domain boundary element method (BEM) is presented. Time-dependent traction boundary integral equations (BIEs) are used for this purpose. Both the hypersingular and the non-hypersingular formulations are dealt with. The time-domain traction BIEs are solved numerically by a time-stepping scheme. The efficiency and the accuracy of the time-domain BEM for transient elastodynamic crack analysis are demonstrated by a numerical example. Advantages and drawbacks of the hypersingular and the non-hypersingular BEM formulations are compared and discussed.

A coupled system of finite elements and boundary elements is presented for the modelling of sound propagation in a nonuniform mean flow. Finite elements are used to discretize the acoustic pressure in a bounded computational domain including the nonuniform flow field. The radiation of acoustic waves in the exterior uniform flow field is modelled by a boundary integral representation formula. The interface condition on the shared boundary yields an integral equation which is solved using boundary elements. The coupling of finite elements and boundary elements is described in detail for sound propagation in a stationary fluid where the acoustic pressure is governed by the Helmholtz equation. The coupled computational system is applied to estimate scattering and refractive effects on the acoustic pressure waves through a boundary layer surrounding an aircraft’s fuselage. The coupled mathematical formulation for three dimensional sound radiation around an engine in subsonic flow is discussed.

In linear poroelasticity so far only collocation boundary element methods have been available. However, in some applications, e.g., when coupling with finite elements is desired, a symmetric formulation is preferable. Choosing a Galerkin approach which involves the second boundary integral equation, such a formulation is possible. Here, a previously presented integration by part technique for the regularization of the first boundary integral equation is extended to the second boundary integral equation as well. While the weakly singular representation of the double layer operator has been presented before, the emphasis lies here on the so called hyper‐singular boundary integral operator. Due to the regularization, this operator can be evaluated numerically and, hence, be used within a numerical scheme for the first time. Different numerical studies will be presented to show the behavior of the established symmetric Galerkin boundary element method, also comparing it with collocation boundary element methods. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)