Element-boundary Integral Method Research Articles

Large-scale thermal analysis of fiber composites using a line-inclusion model by the fast boundary element method

Abstract The fast boundary element method is applied for the three-dimensional large-scale thermal analysis of fiber-reinforced composites based on a line inclusion model. In this approach, fibers are treated as inclusions with temperature assumed constant over the circular cross-section and varying along the length direction. Therefore, fibers can be meshed with line elements, making both the modeling complexity and the number of unknowns significantly reduced. An interface integral boundary element method introduced by Gao in 2009 (Engineering Analysis with Boundary Elements 2009; 33: 539–546) is extended to generate a single-domain boundary integral equation for governing this line-inclusion problem. Thus in principle, fibers with arbitrary length can be modeled. The fast multipole method is employed for the fast analysis of such problems with large-scales. The largest composite model in a personal desktop computer has the number of fibers reaching 20,000. Numerical results clearly demonstrate validity of the proposed model and its potential for large-scale analysis of fiber-reinforced composites.

The trapezoidal rule for computing supersingular integral on interval

The modified trapezoidal rule for the computation of supersingular integrals in boundary element methods is discussed. For the case of the mesh-point coinciding with singular point, a new quadrature rule is presented and the asymptotic expansion of error function is obtained. Some numerical results are also reported to confirm the theoretical results and show the efficiency of the algorithms.

Stability of slender beams and frames resting on 2D elastic half-space

Making use of a mixed variational formulation based on the Green function of the substrate, which assumes as independent fields the structure displacements and the contact pressure, a simple and efficient finite element-boundary integral equation coupling method is derived and applied to the stability analysis of beams and frames resting on an elastic half-plane. Slender Euler–Bernoulli beams with different combinations of end constraints are considered. The examples illustrate the convergence to the existing exact solutions and provide new estimates of the buckling loads for different boundary conditions. Finally, nonlinear incremental analyses of rectangular pipes with compressed columns and free or pinned foundation ends are performed, showing that pipes stiffer than the soil may exhibit snap-through instability.

Coupling interactions in electromagnetic scattering from finite array of cavities with stratified dielectric coating

The coupling interactions in electromagnetic scattering from a finite array of two-dimensional identical cavities engraved in a perfectly electric conducting screen covered with stratified dielectric coating are presented using an algorithm based on the hybrid finite element-boundary integral method. The solution to the scattering from a finite array of cavities is approximated using the array factor method, which does not take into account the coupling between the cavities, and is compared to the recently developed finite element-boundary element method to demonstrate the importance of the inclusion of the coupling effect. Dependence of the coupling interactions between the cavities on various parameters such as separation periods, incident angle of the plane-wave excitation, and permittivity and thickness of the dielectric coating is demonstrated quantitatively through several numerical examples.

Multi-hybrid method for investigation of EM scattering from inhomogeneous object above a dielectric rough surface

An iterative strategy combining Kirchhoff approximation^(KA) with the hybrid finite element-boundary integral (FE-BI) method is presented in this paper to study the interactions between the inhomogeneous object and the underlying rough surface. KA is applied to study scattering from underlying rough surfaces, whereas FE-BI deals with scattering from the above target. Both two methods use updated excitation sources. Huygens equivalence principle and an iterative strategy are employed to consider the multi-scattering effects. This hybrid FE-BI-KA scheme is an improved and generalized version of previous hybrid Kirchhoff approximation-method of moments (KA-MoM). This newly presented hybrid method has the following advantages: (1) the feasibility of modeling multi-scale scattering problems (large scale underlying surface and small scale target); (2) low memory requirement as in hybrid KA-MoM; (3) the ability to deal with scattering from inhomogeneous (including coated or layered) scatterers above rough surfaces. The numerical results are given to evaluate the accuracy of the multi-hybrid technique; the computing time and memory requirements consumed in specific numerical simulation of FE-BI-KA are compared with those of MoM. The convergence performance is analyzed by studying the iteration number variation caused by related parameters. Then bistatic scattering from inhomogeneous object of different configurations above dielectric Gaussian rough surface is calculated and the influences of dielectric compositions and surface roughness on the scattering pattern are discussed.

Scattering of Gaussian beam by arbitrarily shaped inhomogeneous particles

A hybrid finite element-boundary integral method is applied to characterize the scattering of an arbitrarily incident-focused Gaussian beam by arbitrarily shaped inhomogeneous particles. Specifically, the Davis–Barton fifth-order approximation in combination with rotation Euler angles is used to represent the arbitrarily incident Gaussian beams. The finite element method is employed to formulate the fields in the interior region of the inhomogeneous particle, while the boundary integral equation is applied to represent the fields in the exterior region. The interior and exterior fields are coupled by means of the field continuity conditions. To reduce the computational burden, the frontal method and the multilevel fast multipole algorithm are adopted to solve the resultant matrix equation. Numerical results for differential scattering cross sections of several selected inhomogeneous particles are presented and can be served as further study on this subject.

Solving Scattering from Conducting Body Coated by Thin-Layer Material by Hybrid Shell Vector Element with Boundary Integral Method

The finite element boundary integral (FEM-BI) method is widely used in the scattering and radiating problems. But for the conducting body coated by thin-layer material, plenty of fine meshes are required to discretize the geometry in the traditional FEM. It requires very expensive storage and CPU time. In this paper, the hybrid shell vector element with the boundary integral method is used to expedite the solution of thin coating problems. The shell vector elements are used to discretize thin-layer material instead of traditional tetrahedral elements. Consequently, the volume integral can be simplified into surface integral. This method reduces the number of unknowns greatly and is also extended into the complicated case of multi-thin-layer coating materials. Several numerical results are presented to prove the accuracy and efficiency of this present method.

A Domain Decomposition of the Finite Element-Boundary Integral Method for Scattering by Deep Cavities

An efficient domain decomposition method (DDM) is employed to improve upon the efficiency and capability of the finite element-boundary integral (FE-BI) method for calculation of electromagnetic (EM) scattering from deep cavities. This method first subdivides the original cavity into many sub-domains along its depth and classifies these sub-domains into a few building blocks. It then employs the substructuring method to deal with the different types of sub-domains. The resulting Schur complement system is solved by a special method which has low memory requirements because the formation of the global Schur complement matrix is not necessary. Numerical results indicate that the presented method is an effective approach for scattering by deep cavities.

Eddies and interface deformations induced by optical streaming

AbstractWe study flows and interface deformations produced by the scattering of a laser beam propagating through non-absorbing turbid fluids. Light scattering produces a force density resulting from the transfer of linear momentum from the laser to the scatterers. The flow induced in the direction of the beam propagation, called ‘optical streaming’, is also able to deform the interface separating the two liquid phases and to produce wide humps. The viscous flow taking place in these two liquid layers is solved analytically, in one of the two liquid layers with a stream function formulation, as well as numerically in both fluids using a boundary integral element method. Quantitative comparisons are shown between the numerical and analytical flow patterns. Moreover, we present predictive simulations regarding the effects of the geometry, of the scattering strength and of the viscosities, on both the flow pattern and the deformation of the interface. Finally, theoretical arguments are put forth to explain the robustness of the emergence of secondary flows in a two-layer fluid system.

A posteriori error estimates for the Johnson–Nédélec FEM–BEM coupling

Only very recently, Sayas [The validity of Johnson–Nédélec's BEM-FEM coupling on polygonal interfaces. SIAM J Numer Anal 2009;47:3451–63] proved that the Johnson–Nédélecone-equation approach from [On the coupling of boundary integral and finite element methods. Math Comput 1980;35:1063–79] provides a stable coupling of finite element method (FEM) and boundary element method (BEM). In our work, we now adapt the analytical results for different a posteriori error estimates developed for the symmetric FEM–BEM coupling to the Johnson–Nédélec coupling. More precisely, we analyze the weighted-residual error estimator, the two-level error estimator, and different versions of (h−h/2)-based error estimators. In numerical experiments, we use these estimators to steer h-adaptive algorithms, and compare the effectivity of the different approaches.

A Domain Decomposition of the Finite Element-Boundary Integral Method for Scattering by Multiple Objects

A domain decomposition method based on the hybrid finite element–boundary integral method is presented for analyzing electromagnetic scattering problems involving multiple separable objects. In this method, the finite element formulation is applied inside each object to derive a linear system of equations associated with edge field values. The boundary integral equation is then applied on the surfaces of the objects to truncate the computational domain and to connect the matrix subsystem generated from each object. The coupling system of equations is solved by a hybrid domain decomposition algorithm. Numerical results are presented to demonstrate the accuracy, efficiency, and capability of the proposed method.

Characterization of the light scattering by ensembles of randomly distributed soot aggregates

Soot particles formed in combustion processes commonly exist in the form of ensembles of randomly distributed aggregates of small, nearly spherical monomers. In this paper, these randomly distributed aggregates are numerically generated using a combination of the cluster–cluster aggregation algorithm with the Monte Carlo method. Moreover, an efficient and accurate numerical method is presented for characterizing the light scattering by these complex soot particles illuminated by plane wave and Gaussian beam. This method exploits the unique features of the hybrid finite element-boundary integral method and, more importantly, the unique features of soot aggregates. It is designed in such a manner that it first decomposes the original problem into many sub-regions, where each primary particle is regarded as a sub-region, and then it employs the edge-based finite element method to deal with each sub-region. The sub-regions communicate through the near-field Green’s function. To reduce computational burdens, an iterative domain decomposition method in combination with parallel conjugate gradient method is adopted to solve the coupling system of equations. As an illustration, we present some of our preliminary numerical results. The results are expected to provide useful insights into the optical properties of soot particles formed in combustion processes.

Using analytical expressions in radial integration BEM for variable coefficient heat conduction problems

In this paper, a new approach using analytical expressions in the radial integration boundary element method (RIBEM) is presented for solving variable coefficient heat conduction problems. This approach can improve the computational efficiency considerably and can overcome the time-consuming deficiency of RIBEM in computing involved radial integrals. The fourth-order spline RBF is employed to approximate unknowns appearing in domain integrals arising from the varying heat conductivity. The radial integration method is utilized to convert domain integrals to the boundary resulting in a pure boundary discretization algorithm. Numerical examples are given to demonstrate the efficiency of the presented approach.

Hybrid FE-BI-KA method in analysing scattering from dielectric object above sea surface

A new and more powerful algorithm combining Kirchhoff approximation (KA) with the hybrid finite element-boundary integral (FE-BI) method is presented to study the interactions between an inhomogeneous object and the underlying sea surface. This hybrid FE-BI-KA scheme is an improved and generalised version of the previous KA-MoM. A reduced model composed of a PEC object above the conducting sea surface is simulated to evaluate the validity of the multi-hybrid technique by comparing with MoM. Then the bistatic scattering pattern of a more complex layered object above the dielectric sea surface is calculated at different wind speeds.

Coupled 2D electric and mechanical fields in piezoelectric solids containing cracks and thin inhomogeneities

This paper develops integral equations and boundary element method for determination of 2D electro-elastic state of solids containing cracks, thin voids and inclusions. It proves that stress and electric displacement field near the tip of thin inhomogeneity possesses square root singularity. Thus, for determination of electromechanical fields near thin defects new special base functions are introduced. The interpolation quadratures along with the polynomial transformations are adopted for efficient numerical evaluation of singular and hypersingular integrals. Presented numerical examples show high efficiency and accuracy of the proposed approach.

Treatment of domain integrals in boundary element methods

A systematic and rigorous technique to calculate domain integrals without a volume-fitted mesh has been developed and validated in the context of a boundary element approximation. In the proposed approach, a domain integral involving a continuous or weakly-singular integrand is first converted into a surface integral by means of straight-path integrals that intersect the underlying domain. Then, the resulting surface integral is carried out either via analytic integration over boundary elements or by use of standard quadrature rules. This domain-to-boundary integral transformation is derived from an extension of the fundamental theorem of calculus to higher dimension, and the divergence theorem. In establishing the method, it is shown that the higher-dimensional version of the first fundamental theorem of calculus corresponds to the well-known Poincaré lemma.The proposed technique can be employed to evaluate integrals defined over simply- or multiply-connected domains with Lipschitz boundaries which are embedded in an Euclidean space of arbitrary but finite dimension. Combined with the singular treatment of surface integrals that is widely available in the literature, this approach can also be utilized to effectively deal with boundary-value problems involving non-homogeneous source terms by way of a collocation or a Galerkin boundary integral equation method using only the prescribed surface discretization. Sample problems associated with the three-dimensional Poisson equation and featuring the Newton potential are successfully solved by a constant element collocation method to validate this study.

An integral-equation formulation of nonlinear deformation in a stack of buffered plates

An integral-equation formulation has been derived for nonlinear deformation in a stack of buffered Kirchhoff plates. The plates are assumed to follow a nonlinear bending moment-curvature law and the buffer material to follow the generalized Hooke’s law. By employing the recently derived special Green’s function for multilayers with interfacial membrane and flexural rigidities as the kernel, the integral-equation formulation only involves the surface loading area (for application to an indentation problem) and the portion of plates undergoing nonlinear deformation. Based on the integral equation, an efficient and accurate boundary element method has been derived to numerically solve the cylindrical indentation problem of the material with a bilinear flexural bending law for the plates. Numerical examples are presented to show a progressive damage process of yielding across a stack of plates as well as to demonstrate the validity and accuracy of the present integral-equation formulation.

A rectangle rule for the computation of hypersingular integral

The composite rectangle rule for the computation of Hadamard finite-part integrals in boundary element methods with the hypersingular kernel 1/(x − s)2 is discussed. For the case of singular point coinciding with the mesh point, a new quadrature rule which is based on the classical finite-part definition is presented. A kind of the hypersingular integral equation is solved by collocation methods and the maximal error estimation is given. Some numerical results are also illustrated to confirm the theoretical results and show the efficiency of the algorithms.

Parallelizing a hybrid finite element-boundary integral method for the analysis of scattering and radiation of electromagnetic waves

This paper presents the practical experience of parallelizing a simulator of general scattering and radiation electromagnetic problems. The simulator stems from an existing sequential simulator in the frequency domain, which is based on a finite element analysis. After the analysis of a test case, two steps were carried out: first, a “hand-crafted” code parallelization of a convolution-type operation was developed within the kernel of the simulator. Second, the sequential HSL library, used in the existing simulator, was replaced by the parallel MUMPS (MUltifrontal Massively Parallel sparse direct Solver) library in order to solve the associated linear algebra problem in parallel. Such a library allows for the distribution of the factorized matrix and some of the computational load among the available processors. A test problem and three realistic (in terms of the number of unknowns) cases have been run using the parallelized version of the code, and the results are presented and discussed focusing on the memory usage and achieved speed-up.

Dynamic rupture modeling on unstructured meshes using a discontinuous Galerkin method

We introduce the application of an arbitrary high‐order derivative (ADER) discontinuous Galerkin (DG) method to simulate earthquake rupture dynamics. The ADER‐DG method uses triangles as computational cells which simplifies the process of discretization of very complex surfaces and volumes by using external automated tools. Discontinuous Galerkin methods are well suited for solving dynamic rupture problems in the velocity‐stress formulation as the variables are naturally discontinuous at the interface between two elements. Therefore, the fault has to be honored by the computational mesh. The so‐called Riemann problem can be solved to obtain well defined values of the variables at the discontinuity itself. Fault geometries of high complexity can be modeled thanks to the flexibility of unstructured meshes, which solves a major bottleneck of other high‐order numerical methods. Additionally, element refinement and coarsening are easily controlled in the meshing process to better resolve the near‐fault area of the model. The fundamental properties of the method are shown, as well as a series of validating exercises with reference solutions and a comparison with the well‐established finite difference, boundary integral, and spectral element methods, in order to test the accuracy of our formulation. An example of dynamic rupture on a nonplanar fault based upon the Landers 1992 earthquake fault system is presented to illustrate the main potentials of the new method.