A fast multipole boundary element method for 2D multi-domain elastostatic problems based on a dual BIE formulation

A new fast multipole formulation for the hypersingular BIE (HBIE) for 2D elasticity is presented in this paper based on a complex-variable representation of the kernels, similar to the formulation developed earlier for the conventional BIE (CBIE). A dual BIE formulation using a linear combination of the developed CBIE and HBIE is applied to analyze multi-domain problems with thin inclusions or open cracks. Two pre-conditioners for the fast multipole boundary element method (BEM) are devised and their effectiveness and efficiencies in solving large-scale problems are discussed. Several numerical examples are presented to study the accuracy and efficiency of the developed fast multipole BEM using the dual BIE formulation. The numerical results clearly demonstrate the potentials of the fast multipole BEM for solving large-scale 2D multi-domain elasticity problems. The method can be applied to study composite materials, functionally-graded materials, and micro-electro-mechanical-systems with coupled fields, all of which often involve thin shapes or thin inclusions.     

A fast multipole boundary element method for 3D multi-domain acoustic scattering problems based on the Burton-Miller formulation

A fast multipole boundary element method (FMBEM) for 3D multi-domain acoustic scattering problems based on the Burton-Miller formulation is presented in this paper. A multi-tree structure is designed for the multi-domain FMBEM. It results in mismatch of leaves and well separate cells definition in different domains and complicates the implementation of the algorithm, especially for preconditioning. A preconditioner based on boundary blocks is devised for the multi-domain FMBEM and its efficiency in reducing the number of iterations in solving large-scale multi-domain scattering problems is demonstrated. In addition to the analytical moment, another method, based on the anti-symmetry of the moment kernel, is developed to reduce the moment computation further by a factor of two. Frequency sweep analysis of a penetrable sphere shows that the multi-domain FMBEM based on the Burton-Miller formulation can overcome the non-unique solution problem at the fictitious eigenfrequencies. Several other numerical examples are presented to demonstrate the accuracy and efficiency of the developed multi-domain FMBEM for acoustic problems. In spite of the high cost of memory and CPU time for the multi-tree structure in the multi-domain FMBEM, a large BEM model studied with a PC has 0.3 million elements corresponding to 0.6 million unknowns, which clearly shows the potential of the developed FMBEM in solving large-scale multi-domain acoustics problems.     

An adaptive fast multipole boundary element method for three-dimensional acoustic wave problems based on the Burton–Miller formulation

The high solution costs and non-uniqueness difficulties in the boundary element method (BEM) based on the conventional boundary integral equation (CBIE) formulation are two main weaknesses in the BEM for solving exterior acoustic wave problems. To tackle these two weaknesses, an adaptive fast multipole boundary element method (FMBEM) based on the Burton–Miller formulation for 3-D acoustics is presented in this paper. In this adaptive FMBEM, the Burton–Miller formulation using a linear combination of the CBIE and hypersingular BIE (HBIE) is applied to overcome the non-uniqueness difficulties. The iterative solver generalized minimal residual (GMRES) and fast multipole method (FMM) are adopted to improve the overall computational efficiency. This adaptive FMBEM for acoustics is an extension of the adaptive FMBEM for 3-D potential problems developed by the authors recently. Several examples on large-scale acoustic radiation and scattering problems are presented in this paper which show that the developed adaptive FMBEM can be several times faster than the non-adaptive FMBEM while maintaining the accuracies of the BEM.     

Acoustic problems analysis of 3D solid with small holes by fast multipole boundary face method

In this paper, an adaptive fast multipole boundary face method is introduced to implement acoustic problems analysis of 3D solids with open-end small tubular shaped holes. The fast multipole boundary face method is referred as FMBFM. These holes are modeled by proposed tube elements. The hole is open-end and intersected with the outer surface of the body. The discretization of the surface with circular inclusions is achieved by applying several special triangular elements or quadrilateral elements. In the FMBFM, the boundary integration and field variables approximation are both performed in the parametric space of each boundary face exactly the same as the B-rep data structure in standard solid modeling packages. Numerical examples for acoustic radiation in this paper demonstrated the accuracy, efficiency and validity of this method.     

A boundary element method applied to the elastostatic bending problem of beam-stiffened plates

The present paper proposes a basic formulation for the static bending problem of beam-stiffened elastic plates. This problem has been so far analyzed using the Timoshenko theory in which the equivalent shear force and bending moments are assumed to act on the beam stiffener. Since fourth-order derivatives of unknown displacements are included in the formulation, in its numerical implementation fourth-order polynomials must be used as the interpolation functions.

In this paper, the interactive forces and moments between the plate and the stiffener are treated as line distributed unknown loads. In the numerical implementation of the formulation, these forces can be approximated using a suitable family of interpolation functions. The formulation is presented in detail and a computer code is developed. The numerical results obtained by the computer code are discussed, whereby the usefulness of the proposed solution procedure is demonstrated.     

Analytical integration of the moments in the diagonal form fast multipole boundary element method for 3-D acoustic wave problems

A diagonal form fast multipole boundary element method (BEM) is presented in this paper for solving 3-D acoustic wave problems based on the Burton-Miller boundary integral equation (BIE) formulation. Analytical expressions of the moments in the diagonal fast multipole BEM are derived for constant elements, which are shown to be more accurate, stable and efficient than those using direct numerical integration. Numerical examples show that using the analytical moments can reduce the CPU time by a lot as compared with that using the direct numerical integration. The percentage of CPU time reduction largely depends on the proportion of the time used for moments calculation to the overall solution time. Several examples are studied to investigate the effectiveness and efficiency of the developed diagonal fast multipole BEM as compared with earlier p³ fast multipole method BEM, including a scattering problem of a dolphin modeled with 404,422 boundary elements and a radiation problem of a train wheel track modeled with 257,972 elements. These realistic, large-scale BEM models clearly demonstrate the effectiveness, efficiency and potential of the developed diagonal form fast multipole BEM for solving large-scale acoustic wave problems.     

A new fast multipole boundary element method for solving large‐scale two‐dimensional elastostatic problems

A new fast multipole boundary element method (BEM) is presented in this paper for large‐scale analysis of two‐dimensional (2‐D) elastostatic problems based on the direct boundary integral equation (BIE) formulation. In this new formulation, the fundamental solution for 2‐D elasticity is written in a complex form using the two complex potential functions in 2‐D elasticity. In this way, the multipole and local expansions for 2‐D elasticity BIE are directly linked to those for 2‐D potential problems. Furthermore, their translations (moment to moment, moment to local, and local to local) turn out to be exactly the same as those in the 2‐D potential case. This formulation is thus very compact and more efficient than other fast multipole approaches for 2‐D elastostatic problems using Taylor series expansions of the fundamental solution in its original form. Several numerical examples are presented to study the accuracy and efficiency of the developed fast multipole BEM formulation and code. BEM models with more than one million equations have been solved successfully on a laptop computer. These results clearly demonstrate the potential of the developed fast multipole BEM for solving large‐scale 2‐D elastostatic problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Fast multipole boundary element approaches for acoustic attenuation prediction of reactive silencers

The fast multipole boundary element method (FMBEM) is applied to predict the acoustic attenuation performance of reactive silencers. In order to overcome the difficulty of singular boundaries for the acoustic computation of reactive silencers with internal thin wall structure or/and perforated components, two approaches, the substructure FMBEM (Sub-FMBEM) and mixed-body FMBEM (MB-FMBEM) are proposed, and the theoretical foundations and numerical processes of the both approaches are introduced. The studies demonstrated that the ordering of column vectors and numbering of nodes in the Sub-FMBEM have great influence on the convergence of iteration, and the MB-FMBEM may reduce the number of elements and the computational complexity since it only needs to discretize one side boundary of the thin wall and perforated components and it is not necessary to create the interfaces. The Sub-FMBEM, MB-FMBEM and Sub-BEM are then employed to calculate the transmission loss of reactive silencers with thin wall components and perforated tubes, the computational accuracy and efficiency of the approaches are validated. The data of precomputing time and total iterative computational time demonstrated that, the computational efficiency of Sub-FMBEM will descend as the frequency arising, and the Sub-FMBEM may reveal higher computational efficiency than Sub-BEM only when the number of nodes is big enough.     

A wideband fast multipole boundary element method for three dimensional acoustic shape sensitivity analysis based on direct differentiation method

This paper presents a wideband fast multipole boundary element approach for three dimensional acoustic shape sensitivity analysis. The Burton-Miller method is adopted to tackle the fictitious eigenfrequency problem associated with the conventional boundary integral equation method in solving exterior acoustic wave problems. The sensitivity boundary integral equations are obtained by the direct differentiation method, and the concept of material derivative is used in the derivation. The iterative solver generalized minimal residual method (GMRES) and the wideband fast multipole method are employed to improve the overall computational efficiency. Several numerical examples are given to demonstrate the accuracy and efficiency of the present method.     

Stiffened plate bending analysis by the boundary element method

 In this work, the plate bending formulation of the boundary element method (BEM) based on the Kirchhoff's hypothesis, is extended to the analysis of stiffened elements usually present in building floor structures. Particular integral representations are derived to take directly into account the interactions between the beams forming grid and surface elements. Equilibrium and compatibility conditions are automatically imposed by the integral equations, which treat this composite structure as a single body. Two possible procedures are shown for dealing with plate domain stiffened by beams. In the first, the beam element is considered as a stiffer region requiring therefore the discretization of two internal lines with two unknowns per node. In the second scheme, the number of degrees of freedom along the interface is reduced by two by assuming that the cross-section motion is defined by three independent components only. Received 6 November 2000     

A unified 3D‐based technique for plate bending analysis using scaled boundary finite element method

This paper presents a unified technique for solving the plate bending problems by extending the scaled boundary finite element method. The formulation is based on the three‐dimensional governing equation without enforcing the kinematics of plate theory. Only the in‐plane dimensions are discretised into finite elements. Any two‐dimensional displacement‐based elements can be employed. The solution along the thickness is expressed analytically by using a matrix function. The proposed technique is consistent with the three‐dimensional theory and applicable to both thick and thin plates without exhibiting the numerical locking phenomenon. Moreover, the use of higher order spectral elements allows the proposed technique to better represent curved boundaries and to achieve high accuracy and fast convergence. Numerical examples of various plate structures with different thickness‐to‐length ratios demonstrate the applicability and accuracy of the proposed technique. Copyright © 2012 John Wiley & Sons, Ltd.     

Degenerate scale for the analysis of circular thin plate using the boundary integral equation method and boundary element methods

In this paper, the degenerate scale for plate problem is studied. For the continuous model, we use the null-field integral equation, Fourier series and the series expansion in terms of degenerate kernel for fundamental solutions to examine the solvability of BIEM for circular thin plates. Any two of the four boundary integral equations in the plate formulation may be chosen. For the discrete model, the circulant is employed to determine the rank deficiency of the influence matrix. Both approaches, continuous and discrete models, lead to the same result of degenerate scale. We study the nonunique solution analytically for the circular plate and find degenerate scales. The similar properties of solvability condition between the membrane (Laplace) and plate (biharmonic) problems are also examined. The number of degenerate scales for the six boundary integral formulations is also determined. Tel.: 886-2-2462-2192-ext. 6140 or 6177     

基于快速多极子边界元法的齿轮箱声场分析

齿轮箱是广泛应用的工程机械零部件,准确地模拟其辐射声场对后续的降噪优化设计有着重要作用。边界元方法非常适合分析此类无限域下的声辐射问题。但传统边界元方法有着计算效率低、内存占用高的缺点。该研究发展了宽频的快速多极子边界元方法,并运用该方法计算了齿轮箱在特定频率下的场点声压以及辐射声场。通过对比商用软件的分析结果,验证了所提快速边界元方法的准确性。此外,运用多核并行计算方法,对计算量较大的扫频分析进行加速计算,最终快速、准确地获取了齿轮箱辐射声场的扫频结果。     

Statically admissible finite element solution of the thin plate bending problem with a posteriori error estimation

Two triangular elements of class C⁰ developed on the basis of the principle of complementary work are applied in the static analysis of a thin plate. Some techniques to widen the versatility of the equilibrium approach for the finite element method are presented. Plates of various shapes subjected to diverse types of loading are considered. The results are compared with outcomes obtained by use of the displacement-based finite element method. By use of these two dual types of solutions, the error of the approximate solution is calculated. The lower and upper bounds for the strain energy are found.     

A fast multipole boundary element method for solving two-dimensional thermoelasticity problems

A fast multipole boundary element method (BEM) for solving general uncoupled steady-state thermoelasticity problems in two dimensions is presented in this paper. The fast multipole BEM is developed to handle the thermal term in the thermoelasticity boundary integral equation involving temperature and heat flux distributions on the boundary of the problem domain. Fast multipole expansions, local expansions and related translations for the thermal term are derived using complex variables. Several numerical examples are presented to show the accuracy and effectiveness of the developed fast multipole BEM in calculating the displacement and stress fields for 2-D elastic bodies under various thermal loads, including thin structure domains that are difficult to mesh using the finite element method (FEM). The BEM results using constant elements are found to be accurate compared with the analytical solutions, and the accuracy of the BEM results is found to be comparable to that of the FEM with linear elements. In addition, the BEM offers the ease of use in generating the mesh for a thin structure domain or a domain with complicated geometry, such as a perforated plate with randomly distributed holes for which the FEM fails to provide an adequate mesh. These results clearly demonstrate the potential of the developed fast multipole BEM for solving 2-D thermoelasticity problems.     

A fast convergent iterative boundary element method on PVM cluster

This paper reports a fast convergent boundary element method on a Parallel Virtual Machine (PVM) (Geist et al., PVM: Parallel Virtual Machine, A Users' Guide and Tutorial for Networked Parallel Computing. MIT Press, Cambridge, 1994) cluster using the SIMD computing model (Single Instructions Multiple Data). The method uses the strategy of subdividing the domain into a number of smaller subdomains in order to reduce the size of the system matrix and to achieve overall speedup. Unlike traditional subregioning methods, where equations from all subregions are assembled into a single linear algebraic system, the present scheme is iterative and each subdomain is handled by a separate PVM node in parallel. The iterative nature of the overall solution procedure arises due to the introduction of the artificial boundaries. However, the system equations for each subdomain is now smaller and solved by direct Gaussian elimination within each iteration. Furthermore, the boundary conditions at the artificial interfaces are estimated from the result of the previous iteration by a reapplication of the boundary integral equation for internal points. This method provides a consistent mechanism for the specification of boundary conditions on artificial interfaces, both initially and during the iterative process. The method is fast convergent in comparison with other methods in the literature. The achievements of this method are therefore: (a) simplicity and consistency of methodology and implementation; (b) more flexible choice of type of boundary conditions at the artificial interfaces; (c) fast convergence; and (d) the potential to solve large problems on very affordable PVM clusters. The present parallel method is suitable where (a) one has a distributed computing environment; (b) the problem is big enough to benefit from the speedup achieved by coarse-grained parallelisation; and (c) the subregioning is such that communication overhead is only a small percentage of total computation time.     

A dual BIE approach for large‐scale modelling of 3‐D electrostatic problems with the fast multipole boundary element method

A dual boundary integral equation (BIE) formulation is presented for the analysis of general 3‐D electrostatic problems, especially those involving thin structures. This dual BIE formulation uses a linear combination of the conventional BIE and hypersingular BIE on the entire boundary of a problem domain. Similar to crack problems in elasticity, the conventional BIE degenerates when the field outside a thin body is investigated, such as the electrostatic field around a thin conducting plate. The dual BIE formulation, however, does not degenerate in such cases. Most importantly, the dual BIE is found to have better conditioning for the equations using the boundary element method (BEM) compared with the conventional BIE, even for domains with regular shapes. Thus the dual BIE is well suited for implementation with the fast multipole BEM. The fast multipole BEM for the dual BIE formulation is developed based on an adaptive fast multiple approach for the conventional BIE. Several examples are studied with the fast multipole BEM code, including finite and infinite domain problems, bulky and thin plate structures, and simplified comb‐drive models having more than 440 thin beams with the total number of equations above 1.45 million and solved on a PC. The numerical results clearly demonstrate that the dual BIE is very effective in solving general 3‐D electrostatic problems, as well as special cases involving thin perfect conducting structures, and that the adaptive fast multipole BEM with the dual BIE formulation is very efficient and promising in solving large‐scale electrostatic problems. Copyright © 2007 John Wiley & Sons, Ltd.     

hp Fast multipole boundary element method for 3D acoustics

A fast multipole boundary element method (FMBEM) extended by an adaptive mesh refinement algorithm for solving acoustic problems in three‐dimensional space is presented in this paper. The Collocation method is used, and the Burton–Miller formulation is employed to overcome the fictitious eigenfrequencies arising for exterior domain problems. Because of the application of the combined integral equation, the developed FMBEM is feasible for all positive wave numbers even up to high frequencies. In order to evaluate the hypersingular integral resulting from the Burton–Miller formulation of the boundary integral equation, an integration technique for arbitrary element order is applied. The fast multipole method combined with an arbitrary order h‐p mesh refinement strategy enables accurate computation of large‐scale systems. Numerical examples substantiate the high accuracy attainable by the developed FMBEM, while requiring only moderate computational effort at the same time. Copyright © 2016 John Wiley & Sons, Ltd.