The boundary element method (BEM) and the method of fundamental solutions (MFS) are well-known fundamental solution-based methods for solving a variety of problems.Both methods are boundarytype techniques and can provide accurate results.In comparison to the finite element method (FEM), which is a domain-type method, the BEM and the MFS need less manual effort to solve a problem.The aim of this study is to compare the accuracy and reliability of the BEM and the MFS.This comparison is made for 2D potential and elasticity problems with different boundary and loading conditions.In the comparisons, both convex and concave domains are considered.Both linear and quadratic elements are employed for boundary element analysis of the examples.The discretization of the problem domain in the BEM, i.e., converting the boundary of the problem into boundary elements is relatively simple; however, in the MFS, obtaining appropriate locations of collocation and source points need more attention to obtain reliable solutions.The results obtained from the presented examples show that both methods lead to accurate solutions for convex domains, whereas the BEM is more suitable than the MFS for concave domains.
The present article introduces a novel boundary integral method (BIM), adapted from an earlier method of Hansen and Kelmanson (1992, 1994) and suitable for the solution of creeping flow boundary value problems where the boundary presents singularities in the stresses.We use the new BIM to solve the problem of the planar extrusion of a Newtonian fluid at zero Reynolds number and, in particular, to determine the shape of the free surface in the immediate neighbourhood of the separation point for a range of capillary numbers.The proposed method incorporates the singular solution near the separation point, thus overcoming one limitation of a classical BIM to the problem (see, for example, Kelmanson (1983)).In a recent article, Owens (2022) also incorporated the singular solution into his BIM formulation.However, since the integration path used in the present BIM passes directly through the separation point this leads to an important improvement on the method of Owens (2022), who was obligated to skirt the singularity due to the non-integrability there of the normal derivative of the vorticity.Results presented for the extrudate swell ratio, the angle of separation and the leading exponent in the asymptotic expression for the stream function are shown to be in convincing agreement with others in the theoretical, numerical and experimental literature.
The collocation boundary element method has recently been entirely revisited by the author.Arbitrary rigid-body displacements, as for elasticity, are naturally taken into account, and traction force parameters are always in balance independently of problem scale and mesh discretization.For generally curved boundaries, the correct definition of traction force interpolation functions enables the enunciation of a general convergence theorem, the introduction of patch and cut-out tests, and, not least, a considerable simplification of the numerical implementations.Simple code schemes for the 2D formulation are proposed exclusively in terms of Gauss-Legendre quadrature for arbitrarily high -actually only machine-precision dependent -computational accuracy of results independently of a problem's geometry and topology.On the other hand, the complex-variable formulation of the problem leads to more simplicity of implementation and numerical results that seem less liable to round-off errors.We propose in this contribution the comparative assessment of the real-and complexvariable formulations regarding coding efficiency, computational effort, error estimation, and numerical precision and accuracy of results for applications that are topologically highly challenging, with sourcefield distances in the subnanometer range.
Glass façades are widely used in modern high-rise buildings because of their aesthetic merits and environmentally friendly characteristics. However, collapsing glass in an accidental fire can lead to huge casualties and economic losses. Here, we develop a meshfree computational framework based on peridynamics to investigate the fire resistance and mechanical performance of glass façades. The proposed model is well verified and applied to explore the damage and failure mechanisms of glass façades. The effects of the temperature distribution on the thermomechanical fracture behaviour of glazing are also explored. Results show that cracks tend to initiate at fixed supporting points such as bolts and their propagation paths are greatly influenced by the temperature distributions. Both temperature gradient and local boundary conditions can play a significant role in the thermomechanical cracking of glass façades. This work provides important insights on cracking mechanisms of glass façades during a fire and provide the building and construction companies with new recommendations and design guidelines for improved fire safety.
The application of the boundary element method to the buckling problem requires a domain integration and the use of the gradient boundary integral equation to perform the analysis.One of the most used techniques to convert the domain integral to equivalent boundary integrals is the dual reciprocity method (DRM).The present study employs the DRM in conjunction with a meshless solution using radial functions to obtain the gradient of the deflection instead of employing the gradient boundary integral equation.The plate bending model considered the effect of shear deformation and the results obtained are compared to those available in the literature.
The problem of 2D incompressible flow simulation around airfoils with sharp edges and corner points is considered.The solution of the boundary integral equation with respect to vortex sheet intensity, arising in Lagrangian vortex method, has weak singularity that cannot be resolved correctly in the framework of the existing Galerkin-type numerical schemes.Known numerical schemes with piecewise-constant or piecewise-linear numerical solution representation provide solution reconstruction with high quality and the second order of accuracy only for piecewise-smooth bounded solutions.For singular solutions their order of accuracy goes down to the first.It is shown that wrong behaviour of numerical solution takes place only on the panels that adjust to the corner point.Modified numerical scheme is developed that is based on the Galerkin-Petrov approach and allows us to obtain integral characteristics of solution (the components of the added masses tensor) with the second order of accuracy.The scheme can be easily implemented in codes developed for flow simulations using the vortex particle method.
Meshless Lagrangian vortex methods that are characterized by considering vorticity as a primary computational variable are discussed, including their modern modifications for 2D and 3D flow simulation.Original mathematical models developed by the authors are described, that allow for significant improvement of the accuracy of the flow simulation around the airfoils/bodies.The hierarchy of numerical schemes based on the Galerkin approach is developed for numerical solution of the boundary integral equation.The quality of the surface mesh is not essential, rather high quality of the numerical solution can be achieved even for low-quality mesh consists of triangular cells with high aspect ratio.The open-source parallel codes (for CPU and GPU, using OpenMP, MPI and Nvidia CUDA technologies) are developed, that implement viscous vortex domains method and closed vortex loops method for 2D and 3D cases, respectively In 3D cases, the numerical scheme allows to satisfy the divergence-free condition for vorticity field (in 2D it is done trivially).The suggested methods can be applied for unsteady hydrodynamic load computation at rather low computational cost of the algorithm.The developed models and algorithms are suitable for numerical simulation in coupled problems, including for light movable bodies.Both weakly-coupled and strongly-coupled strategies are implemented, the last one requires several iterations; at each of them the boundary integral equation is solved.In addition to flow simulation and hydrodynamic load estimation, the suggested technique allows for added masses tensor calculation with high accuracy.Efficient fast method of quasilinear numerical complexity, both well-known and developed by the authors for the integral equation solution and vortex particles (that simulate the vorticity distribution in the flow domain) evolution simulation are discussed.A number of numerical examples are presented, being performed for validation of the developed mathematical models, numerical algorithms and parallel codes.
