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.
According to the discretization manner and manipulation dimensionality, numerical methods can be classified into two big groups, the full dimensionality method and mesh reduction methods.The former includes the finite element method, finite block method, element differential method and so on, and the latter can be further divided into three types: surface-based, line-based and point-based methods.The surface-based method consists of the boundary element method, finite volume method, boundary face method and so on; the line-based method includes the finite difference method, finite element line method, finite line method and so on; and the point-based method mainly consists of the mesh free method, fundamental solution method, free element method and so on.The paper gives a detailed description on the progress achieved in recent years in the mesh reduction methods, among which a brief introduction will go into the surface-and point-based methods and a detailed one will be given to the line-based methods, especially the finite line method (FLM).FLM is a new type collocation method, which has a distinct feature that the solution scheme to solve a partial differential equation is established by using two or three lines crossing a collocation point for 2D or 3D problems.The Lagrange polynomial formulation is used to approximate the variation of the coordinates and physical variables over each line.And the directional derivative technique is used to derive various orders spatial partial derivatives of any physical variables.The derived partial derivatives can be directly substituted into the governing partial differential equations and related boundary conditions to set up the discretized system of equations.An example will be given for the mechanical problem to demonstrate the accuracy and efficiency of the descripted mesh reduction methods.
The paper deals with high frequency dosimetry analysis of a planar skin model exposed to a dipole antenna radiating in the GHz frequency range. The thin equivalent strip of a perfect electrically conducting (PEC) center-fed antenna is based on the electric field integral equation (EFIE) formulation in the frequency domain solved by means of method of moments (MoM), while the simple planar model for high frequency assessment is based on EFIE formulation for lossy homogeneous biological body solved by the same numerical approach. The electric field obtained using the EFIE-PEC part is subsequently utilized as the incident field in the EFIE-dielectric part of the proposed model. The numerical results for the surface antenna current, the equivalent surface current, and the induced electric field on the surface of the planar model are given, as well as a comparison of several field measures obtained at the averaging surface of planar models of varying thickness and width, respectively. The results could be found useful in the development of computational dosimetry models in the assessment of exposure of humans to electromagnetic fields in the GHz frequency range.
Periodic microstructures are often found in structures in the field of vibration acoustics and topology optimization is an effective method for the design of microstructures.Microstructural topology optimization of bi-material for minimizing the responses of the exterior acoustic-structure interaction system is investigated.The structural finite element method is combined with the acoustic boundary element method to analyse the response of the acoustic-structure interaction system, and the bidirectional coupling of the acoustic-structure system is considered.The equivalent macroscopic elastic matrix of microstructures is calculated by homogenization method.Topology optimization model is schemed based on the piecewise constant level set method.Considering the high efficiency of adjoint variable method in multi-variable and high complexity optimization problems, this research adopts the adjoint variable method to analyse the sensitivity of the objective function of the coupled system.Numerical results show that the response of the coupled systems can be reduced significantly, indicating the effectiveness of the optimization algorithm.
recently presented a quadrature rule to accurately evaluate singular and weakly singular integrals in the sense of the Cauchy Principal Value by an exclusively numerical procedure.The procedure was verified by solving engineering problems using the boundary element method with fundamental solutions that have singularities of type log(r) and 1/r.However, that quadrature does not handle the evaluation of the Hadamard Finite Part of hypersingular integrals.These types of singularity appear in several fundamental solutions and, also, when the hypersingular boundary element formulation is applied to the Green functions previously analysed by the authors.In this paper, the quadrature rule presented in is extended to accurately compute integrals with singularities of the type 1/r 2 .The quadrature weights are derived from a system of equations defined from the finite part of known integrals called generalised moments, which include the element shape functions.This novelty is included in the hypersingular formulation of the boundary element method to solve the Helmholtz equation, taking advantage of this methodology to consider null-thickness boundaries using the Dual BEM.
The paper focuses on deriving a local variant of the method of fundamental solutions (MFS) for the case of Stokes flow.Compared to the global and local basis variants, the local with global basis one leads to a sparse characteristic matrix as in fully localized variants but with a narrower system of equations and thus makes the solution of especially large-scale problems more efficient.It is also essential to keep the condition number of the characteristic matrix within reasonable bounds and remove the solution dependency on fictitious sources.A combination of MFS and finite collocation approach was used for the localization with a globally defined Stokeslet fundamental solution.The results of the particular local variant were compared on several examples, and the dependence of the solution on the density of the point network and the dimensions of the stencil used were also tested in the paper.
Exposure of humans to 5G mobile communication systems may result in a local surface temperature elevation, i.e. may cause heating skin, ears and eyes.For the frequencies less than transition frequency of 6 GHz the specific absorption rate (SAR) is used to quantify the volume heating.However, according to recently published ICNIRP 2020 safety guidelines, the surface heating above 6 GHz is quantified by absorbed power density (Sab).Furthermore, an alternative dosimetric quantity referred to as transmitted power density (TPD), is also used for internal dosimetry above 6 GHz.This paper aims to review some recently developed internal dosimetry methods based on the use of Galerkin-Bubnov indirect boundary element method for the assessment of human exposure to electromagnetic fields generated by 5G mobile systems.Different tissue models have been used in the paper.Some illustrative computational results have been presented.
This paper presents a treatment of the topology optimization problem for two-dimensional fields governed by Laplace's equation.The study considers various boundary conditions, including Dirichlet, Neumann, Robin, and nonlinear radiation boundary conditions.Additionally, the topological derivative for a general objective functional comprising solely of boundary quantities is derived, with a special focus on the case of a radiation boundary condition in a black body.The accuracy of the derived adjoint problem and topological derivative is validated through several boundary element method calculations.
New modification of the fast algorithm based on the Barnes–Hut (BH) and multipole (FMM) methods is developed for the problem of velocities calculation in vortex particle method. It provides a quasilinear computational complexity and allows for the accuracy flexible adjustment, similarly to the classical Barnes–Hut method. Four schemes are developed with different number of terms being hold in multipole and local expansions. All the necessary formulae are presented, expressed in terms of operations with complex numbers. If extremely high accuracy is not required, the proposed algorithm is more efficient in comparison to the traditional FMM methods.
The paper presents new computational techniques based on coupled boundary and finite element methods to study fluid-structure interaction problems.Thin shells and plates are considered as structure elements interacting with an ideal and incompressible liquid.To describe the motion of both structural elements and the fluid, the basic relations of the continuous mechanics are incorporated.The liquid pressure is determined by applying the Laplace equation.Two kinds of boundary value problems are considered corresponding to one-sided and two-sided contact of structural elements with the liquid.Integral equations for numerical simulation of pressure are obtained.For a two-sided contact of the structural element with the liquid, hypersingular integral equations are received, whereas singular integral equations with logarithmic singularities describe the problems of one-sided contact.Considering the structure axial symmetry, the integral equations are reduced to one-dimensional ones.The finite element method for determining modes and frequencies of the elastic structure coupled with boundary element method for the hypersingular integral equation is implemented to find the fluid pressure on the structure element with two-sided contact with the liquid.The liquid pressure evaluation in axisymmetric problems is reduced to one-dimensional integral equations with kernels in the form of elliptic integrals.The effective technique is developed for numerical simulation of obtained singular integrals.The same technique is extended to hypersingular integral equations.The frequencies and modes of structure vibrations taking into account the added masses of the liquid are obtained.Thin circular plates and shells of revolution are considered as structure elements in numerical simulations.The accuracy and reliability of the proposed method are ascertained.
The Kansa method is one of the most popular meshless methods today.Its ease of implementation, high order of interpolation and ease of application to problems with complex geometry constitute its advantage over many other methods for solving partial differential equation-based problems.However, the Kansa method has a significant disadvantage -the need to find the shape parameter value despite these undeniable advantages.There are dozens of algorithms for finding a good shape parameter value, but none of them is proven to be optimal.Therefore, there is still a great scientific need to research new algorithms and improve those already known.In this work, an algorithm based on the study of the oscillation of certain shape parameter functions concerning the problems of two-dimensional heat flow in a material with spatially variable thermophysical parameters was investigated.It has been shown that algorithms of this type allow this class of problems to achieve solutions with high accuracy.At the same time, it was indicated that this direction of development of algorithms for searching for a good value of the shape parameter is auspicious.It is because this algorithm can be extended to a wide range of functions whose oscillation is studied and, consequently, its application to a broader range of problems.
The goal of the presented study is to provide a systematic approach for the efficient characterization of vector fields inside a defined region of interest.That means the vector field is described there with a set of coefficients that can be easily derived from the field values and that contains enough information to characterize the vector field accurately.A possible field of application of this approach is the design of defined distributions of vector fields for specific use cases based on optimization algorithms or machine learning approaches.For instance, the homogeneity of the magnetic B-field is an important measure in the context of nuclear magnetic resonance spectroscopy since it directly limits the achievable spectral resolution and applicability of this method.Here, we present a new combination of established techniques of modern boundary element methods, which are typically used for the solution of the field problem, with automatic analysis of the so-called local expansion of the fast multipole method to characterize a vector field based on a robust approach.The local expansion represents the field inside a defined domain, and the effect of all field sources outside this domain is replaced by a small set of local coefficients.Hence, we first discuss the meaning of these local coefficients and then show how they can be computed directly by a smart use of Green's theorem.Finally, we show the spectrum of local coefficients, which, in the next step, is the basis for a cost function of an optimization problem of the studied vector field.
The buckling of perforated plates considering the effect of shear deformation is analyzed with the boundary element method.The geometrical non-linearity (GNL) effect included rotation derivatives (curvatures) as well as the deflection derivatives.The importance of the shear deformation in the buckling of perforated plates appeared with the increase of the plate thickness and the effect of curvatures becomes greater in large hole diameter cases.
A convergence analysis technique in our previous work is extended to various theoretically proven convergent kernel-based least-squares collocation methods for surface elliptic equation, projection methods for surface elliptic equation, and recently for surface parabolic equations.These partial differential equations (PDEs) on surfaces closely resemble their Euclidean counterparts, except that the problem domains change from bulk regions with a flat geometry to some manifolds, on which curvatures plays an important role in the physical processes.We do not focus on proofs in this paper, but on implementation details instead.First, we present an embedding formulation to solve a surface PDE in a narrow-band domain containing the surface.Next, we present another extrinsic projection formulation that works solely on data points on the surface.Lastly, we solve surface diffusion problem using kernel and the method of lines.
