Featured Cover Zhang, Chao; Fu, Zhuojia; Zhang, Yaoming
International journal for numerical methods in engineering,
02/2023, Volume:
124, Issue:
3
Journal Article
Peer reviewed
The cover image is based on the Research Article A novel high‐order collocation indirect boundary element method based on the Leis formulation for three‐dimensional high frequency exterior acoustic ...problems by Chao Zhang et al., https://doi.org/10.1002/nme.7138.
Using the standard approach of the Finite Element Method, numerical values of the spatial derivatives of the primal variable present low accuracy. It is due to the order reduction in the ...interpolation functions used for the approximation of the derivatives. Aiming to overcome this problem, this work uses the integral equation of the Boundary Element Method to recalculate new values of internal variables, using the nodal boundary values obtained by the Finite Element solution. The mathematical foundation for this procedure is based on the idea that the boundary integral equation is equivalent to a weighted residual sentence, and its reuse implies a new minimization of numerical errors. This strategy was previously used within the Boundary Element context to successfully recalculate nodal variables in scalar problems governed by Laplace and Poisson Equations, also used successfully for solving linear elastic problems expressed by Navier's Equation. Here, to confirm the consistency of the proposed model, computational tests are performed, in which the standard FEM results are compared with those obtained by the proposed procedure. Internal potential derivatives and also internal potential values were recalculated. To assess the quality of the solution, the results are compared to a benchmark.
The Burton-Miller method is widely used to solve the problem of solution non-uniqueness in a conventional boundary element method (CBEM). Although this method is very robust, the hypersingular ...integral kernel contained in its normal derivative equation increases the computational complexity and reduces the computational accuracy. In this paper, a coupled double boundary Burton-Miller method with unique solutions at full wave numbers is proposed. The aforementioned process is done by replacing the normal integral equation of the Burton-Miller method with the virtual indirect boundary element method (VIBEM) integral equation of combined layer potential and using the equivalent relationship between their coefficient matrices. The proposed method inherits the high-precision advantages of VIBEM and avoids the hypersingular integrals of traditional Burton-Miller methods. In particular, no singular integral is present when the plane element is used for discretization. The numerical results of acoustic radiation and scattering show that the calculation accuracy of the coupled double boundary Burton-Miller method is higher than that of CBEM and the conventional Burton-Miller method. Moreover, the condition number of the coefficient matrix is much lower than that of VIBEM. Lastly, the computation time is less than that of the conventional Burton-Miller method.
•Avoiding the calculation of hypersingular integrals in the Burton-Miller method.•Significantly improved the computational accuracy of the Burton-Miller method.•The non-uniqueness of the solution of the boundary integral equation is overcome.•The matrix condition number of the virtual boundary element method is reduced.
A new hybrid finite element method-boundary element method (FEM-BEM) scheme is proposed for the solution of the nonlinear 2-D Laplace problem. The novelty is an original approach of the BEM where the ...domain integrals are eliminated at the discrete level by using the FEM approximation of the fundamental solution at every node of the related mesh in the linear regions. The implementation of this FEM-Green approach requires less computational burden than the standard BEM. The coupling with FEM is straightforward and appears to be more natural. The validity of the method is examined through numerical examples.
A novel method of combination of wavelet‐based boundary element method (WBEM) with frequency‐independent fundamental solutions is proposed to determine the band structures of fluid–solid phononic ...crystals (PCs) with square and triangular lattices. Integral equations established are based on the frequency‐independent fundamental solutions, which can avoid nonlinear eigenvalue problems and reduce computing time. Domain integral terms arising from the use of frequency‐independent fundamental solutions are handled with the radial integration method (RIM) and dual reciprocity method (DRM), respectively. The results show the lower precision in high frequency domain of using RIM to handle domain integral terms than that of using DRM, which can be solved by increasing Gauss point. The B‐spline wavelet on the interval and wavelet coefficients are applied to approximate the physical boundary conditions. It is proved that coupling conditions between matrix and scatterers and Bloch theorem are also applicable to wavelet coefficients. Some small matrix entries generated by wavelet vanishing moment characteristics are truncated by the provided matrix compression technique, and the influence of compressed matrices on the results is studied. Furthermore, the final Eigen equations constructed are modified to avoid numerical instability. Some examples are provided to demonstrate the accuracy and efficiency of the proposed method.
This paper presents an acoustic topology optimization approach using isogeometric boundary element methods based on subdivision surfaces to optimize the distribution of sound adsorption materials ...adhering to structural surfaces. The geometries are constructed from triangular control meshes through Loop subdivision scheme, and the associated Box-spline functions that generate limit smooth subdivision surfaces are employed to discretize the acoustic boundary integral equations. The effect of sound-absorbing materials on the acoustic response is characterized by acoustic impedance boundary conditions. The optimization problem is formulated in the framework of Solid Isotropic Material with Penalization methods and the sound absorption coefficients on elements are selected as design variables. The potential of the proposed topology optimization approach for engineering prototyping is illustrated by numerical examples.
•Isogeometric boundary element method is applied to exterior acoustic analysis.•Subdivision surfaces are used in both CAD and numerical analysis.•The sound absorbing material distribution is optimized.•Burton-Miller method is adopted to prevent instabilities.•Adjoint variable method is employed for sensitivity analysis.
The paper presents a novel approach for multi-frequency acoustic topology optimization of sound-absorption materials. In this work, the isogeometric boundary element method based on subdivision ...surfaces is used to solve Helmholtz equations. To avoid time-consuming frequency sweep, we adopt a series expansion method to decouple the frequency-dependent terms from the integrand in the boundary element method, including the terms associated with the impedance boundary conditions that were introduced to model the absorption materials. Moreover, the second-order Arnoldi (SOAR) approach is employed to reduce the order of the systems. Three dimensional numerical examples were given to demonstrate the effectiveness of the proposed algorithm.
•Series expansions are utilized for frequency decoupling of acoustic BEM.•The model order is reduced by the second-order Arnoldi (SOAR) algorithm.•The isogeometric BEM is employed to integrate CAD and numerical analysis.•Multi-frequency topology optimization is performed with the isogeometric BEM.
The effect of in-plane loads on plate behavior can be studied by buckling analyses. The domain integral containing the effect of the in-plane loads is the essential feature in the boundary element ...method (BEM) formulations. The dual reciprocity method (DRM) is a technique used in the literature to convert domain integrals to equivalent boundary integrals in the BEM. In this study, the DRM is modified with a meshless solution using similar equations to the method. The plate buckling parameters obtained with the present DRBEM formulation, which considers the effect of shear deformation in the bending model, are compared to results available in the literature.
A nonconformal twofold domain decomposition method (TDDM) based on the hybrid finite element method-boundary element method (FEM-BEM) is proposed for analyzing 3-D multiscale composite structures. ...The proposed TDDM starts by partitioning the composite object into a closed exterior boundary domain and an interior volume domain. The interior and exterior boundary value problems are coupled to each other through the Robin transmission conditions (RTCs). Both domains are then independently decomposed into subregions to facilitate computation. Specifically, FEM-DDM with the second order transmission conditions (SOTCs) is employed for the interior domain, and BEM-discontinuous Galerkin (BEM-DG) based on the combined field integral equation (CFIE) is applied for the exterior boundary domain. The proposed TDDM allows for nonconformal discretization between any touching subdomains. Without the introduction of a stabilization term that relies on a line integral over intersection of nonmatching meshes and relevant terms involving surface-line integrals, the proposed TDDM provides an effective domain decomposition (DD) preconditioner for the global system. Numerical examples are presented, and the comparisons of the simulation results with FEM-BEM confirm the validity and accuracy of TDDM. Moreover, its ability to model practical large-scale and multiscale targets is also demonstrated.
•Simultaneous equations can overcome the non-uniqueness of boundary element method.•The non-uniqueness of eigenfrequency can be overcome by adding complex damping.•The complex damping method is more ...accurate than Burton-Miller method.•Singular integrals can be calculated indirectly by matrix substitution.•The accuracy of boundary element method can be improved by using indirect matrix.
In this paper, a high-precision combined coupled double boundary element method (CCD-BEM) with a full wavenumber unique solution is proposed by combining the boundary element method (BEM) equation with the virtual indirect BEM (VIBEM) equation and using the equivalent relationship between their coefficient matrices. This method only requires overcoming the non-uniqueness of the VIBEM equation to obtain the unique solution of full-wave number, thus avoiding the failure of interior-point configuration in the combined Helmholtz integral equation formulation (CHIEF) method and the calculation of hyper-singular integral in Burton-Miller method. In addition, the singular matrix in BEM can be calculated indirectly using the equivalent relationship between the BEM and VIBEM coefficient matrix, which completely avoids the direct calculation of weak singular or singular integral. The numerical results of acoustic radiation and scattering show that CCD-BEM has a lower condition number of the coefficient matrix and higher calculation accuracy than BEM.
