Boundary Element Method Algorithm Research Articles

An improved tolerance analysis method for gear mechanisms considering form deviations and elastic-plastic contact behaviors

The contact quality between geometric features significantly affects the assembly accuracy and dynamic performance of mechanical products. However, accurate and efficient contact problem solving is currently a common challenge in the fields of tolerance analysis and tooth contact analysis (TCA). In this paper, a tolerance analysis method for gear mechanisms considering form deviations and elastic-plastic contact behaviors is proposed. Specifically, tolerance modeling of systematic and random deviations on tooth surfaces is represented by skin model shapes (SMS). Elastic-plastic contact behaviors including contact deformation, status, and stress distribution are modeled and calculated by a modified boundary element method (BEM). The accuracy and efficiency of this BEM algorithm are verified by comparing with the finite element method (FEM), indicating that the BEM can greatly improve the computational efficiency while maintaining high solution accuracy. A case study for a pair of spur gears demonstrates the significant effects of contact behaviors on transmission performance. This research provides an effective method to introduce mechanical contact behaviors in tolerance analysis of gear mechanisms. In addition, the contact simulation results can also be adopted in the analysis of vibration, reliability, and other aspects.

Experimental Prediction Method of Free-Field Sound Emissions Using the Boundary Element Method and Laser Scanning Vibrometry

Acoustic emissions play a major role in the usability of many product categories. Therefore, mitigating the emitted sound directly at the source is paramount to improve usability and customer satisfaction. To reliably predict acoustic emissions, numerical methods such as the boundary element method (BEM) are employed, which allow for predicting, e.g., the acoustic emission into the free field. BEM algorithms need appropriate boundary conditions to couple the sound field with the structural motion of the vibrating body. In this contribution, firstly, an interpolation scheme is presented, which allows for appropriate interpolation of arbitrary velocity data to the computational grid of the BEM. Secondly, the free-field Helmholtz problem is solved with the open-source BEM software framework NiHu. The forward coupling between the device of interest and BEM is based on the surface normal velocities (i.e., a Neumann boundary condition). The BEM simulation results are validated using a previously established aeroacoustic benchmark problem. Furthermore, an application to a medical device (knee prosthesis frame) is presented. Furthermore, the radiated sound power is evaluated and contextualized with other low-cost approximations. Regarding the validation example, very good agreements are achieved between the measurements and BEM results, with a mean effective pressure level error of 0.63 dB averaged across three microphone positions. Applying the workflow to a knee prosthesis frame, the simulation is capable of predicting the acoustic radiation to four microphone positions with a mean effective pressure level error of 1.52 dB.

Efficient BEM Modeling of the Heat Transfer in the Turbine Blades of Aero-Parts

The modeling of the turbine blades in aero-parts presents difficulties in conventional domain solution techniques, especially when internal cooling air passages and a thermal barrier coating (TBC) are applied. This paper presents a very efficient 3D modeling of the anisotropic heat conduction in turbine blades with the boundary element method (BEM), where both the TBC and cooling air passages are considered. The BEM is very ideal for this modeling, since only boundary meshes are required for it; however, a serious problem of nearly singular integration will arise in modeling with coarse meshes. In this article, an efficient modeling and computational algorithm using the BEM is applied for the simulation of heat conduction in the turbine blades of aero-parts. The present work proposes a simplified BEM model to replace multiple thin coating layers on the top of the blade. In the end, the veracity of the implemented BEM code as well as its computational efficiency are illustrated with a few examples, showing that the settled temperature on the substrate can be reduced by 20% by employing a TBC. As compared to the analyses with ANSYS, the percentages of difference were within 2%, while the CPU time spent by the BEM algorithm was about 1/8 of that of ANSYS, not to mention the meshing efforts saved by adopting by a treatment of equivalent convection.

Viscothermal Effects in a Two-Dimensional Acoustic Black Hole: A Boundary Element Approach

The acoustic analog of the quantum black hole for airborne sound in two dimensions was denominated as an omnidirectional acoustic absorber by Climente et al. [see Appl. Phys. Lett., 100, 144103 (2012)], who characterized its absorbing properties without providing any theoretical support. The viscothermal losses of the underlying structure, which consists of an absorbing core and a surrounding gradient-index (GRIN) lens both made of periodic distributions of cylindrical rods, are here comprehensively studied by using the boundary element method (BEM) in two dimensions. It is shown that the numerical simulations in two dimensions reproduce fairly well the increase in absorption of the core when the GRIN lens is added and reveal that the discrepancy between measured and calculated values of absorbance is an artifact of the experimental setup. The possibility of independent calculation of viscous and thermal losses contributions in the two-dimensional (2D) BEM algorithm is employed for the comparison with a homogenization theory in which the cluster of cylinders is represented by a single fluidlike viscous cylinder with effective parameters. We conclude that viscous losses represent about 90% of the total energy dissipated in the core. The homogenization approach results are only 2% below the results calculated with 2D BEM, indicating that the effective parameters obtained by the homogenization are very accurate.

A New BEM Modeling Algorithm for Size-Dependent Thermopiezoelectric Problems in Smart Nanostructures

The main objective of this paper is to introduce a new theory called size-dependent thermopiezoelectricity for smart nanostructures. The proposed theory includes the combination of thermoelastic and piezoelectric influences which enable us to describe the deformation and mechanical behaviors of smart nanostructures subjected to thermal, and piezoelectric loadings. Because of difficulty of experimental research problems associated with the proposed theory. Therefore, we propose a new boundary element method (BEM) formulation and algorithm for the solution of such problems, which involve temperatures, normal heat fluxes, displacements, couple-tractions, rotations, force-tractions, electric displacement, and normal electric displacement as primary variables within the BEM formulation. The computational performance of the proposed methodology has been demonstrated by using the generalized modified shift-splitting (GMSS) iteration method to solve the linear systems resulting from the BEM discretization. GMSS advantages are investigated and compared with other iterative methods. The numerical results are depicted graphically to show the size-dependent effects of thermopiezoelectricity, thermoelasticity, piezoelectricity, and elasticity theories of nanostructures. The numerical results also show the effects of the size-dependent and piezoelectric on the displacement components. The validity, efficiency and accuracy of the proposed BEM formulation and algorithm have been demonstrated. The findings of the current study contribute to the further development of technological and industrial applications of smart nanostructures.

FFT-Based Methods for Solving a Rough Adhesive Contact: Description and Convergence Study

The contact problem of a semi-infinite elastic body with a rigid rough surface with adhesive forces is modeled using a boundary element method (BEM). An original theoretical framework for this problem is presented. Four original BEM algorithms are studied, for both prescribed normal pressure and prescribed penetration, in primal and dual forms. They are all based on a conjugate gradient iterative solver. For each case, the solver is finely tuned to tackle the current problem. Using fast Fourier transforms (FFT) speeds up the computation of the matrix–vector multiplications. A new way of computing the coefficients of the conjugate gradient solver which reduces the number of FFTs at each step is presented. The stability and efficiency of the different methods are compared, showing a great sensitivity to the Tabor coefficient and to the contact area ratio. The most stable algorithm proved to be reliable for very large-scale computations through a participation to M. Mueser’s Contact Mechanics Challenge.

Boundary element method for 2D materials and thin films.

2D materials emerge as a viable platform for the control of light at the nanoscale. In this context the need has arisen for a fast and reliable tool capable of capturing their strictly 2D nature in 3D light scattering simulations. So far, 2D materials and their patterned structures (ribbons, discs, etc.) have been mostly treated as very thin films of subnanometer thickness with an effective dielectric function derived from their 2D optical conductivity. In this study an extension to the existing framework of the boundary element method (BEM) with 2D materials treated as a conductive interface between two media is presented. The testing of our enhanced method on problems with known analytical solutions reveals that for certain types of tasks the new modification is faster than the original BEM algorithm. Furthermore, the representation of 2D materials as an interface allows us to simulate problems in which their optical properties depend on spatial coordinates. Such spatial dependence can occur naturally or can be tailored artificially to attain new functional properties.

آنالیز عددی پروانه نیمه مغروق در حالت آب آزاد با استفاده از توسعه روش المان مرزی بر پایه جریان پتانسیل

Since the design algorithm of partially submerged propeller (PSP) is under the influence of the various geometrical and physical parameters; so a new convenient method and numerical tools is required to flow analysis on propeller and consider the effect of all parameters simultaneously. The aim of this study is the development of boundary element method (BEM) based on potential for PSP analysis under open water condition. Using the concept of material derivative and kinematic boundary condition, a BEM algorithm has been developed to analyze the growth, development and change in thickness of ventilation-cavitation regimes in both chord and radial directions on wide range of advance velocity coefficients. Based on the obtained results, in the high advance coefficients there are very good conformity between the values obtained from the numerical simulation compared with experimental data and observations. This adaptation is reduced by reducing the value of advance coefficient. For the low advance coefficients ((J 0.4), hydrodynamic coefficients obtained from numerical results and efficiency of propeller compared with experimental measurements are desirable.

Boundary element analyses on the adhesive contact between an elastic sphere and a rigid half-space

Boundary element method is used to analyze the adhesive contact between an elastic sphere and a rigid half-space. Lennard-Jones potential is used for the surface traction. In the past, the simulation for the adhesive contact between spheres usually used paraboloid approximation for sphere surface. In this paper, the adhesive contact is investigated using exact sphere. A new algorithm for boundary element method is proposed. The S-shaped load-approach curve and the whole solution can be obtained. The result is compared with the adhesive contact between a rigid paraboloid and an elastic half-space, and that between an elastic paraboloid and a rigid half-space. It is found that the Tabor parameter and the radius of sphere affect the adhesive contact. As the Tabor parameter is larger, the effect of the radius is larger. As the radius is smaller, the pull-off force and the pull-off distance for fixed-grips device are smaller. The adhesive contact for rigid paraboloid/elastic half-space can approximate that for elastic paraboloid/rigid half-space and that for elastic sphere/rigid half-space with small Tabor parameter and large radius. Finally, an approximate equation for the pull-off force vs. Tabor parameter is proposed.

Automated hybrid singularity superposition and anchored grid pattern BEM algorithm for the solution of inverse geometric problems

A method for solving an inverse geometric problem is presented by reconstructing the unknown subsurface cavity geometry with the boundary element method (BEM) and a genetic algorithm in combination with the Nelder-Mead non-linear simplex optimization method. The heat conduction problem is solved by the BEM which calculates the difference between the measured temperature at the exposed surface and the computed temperature under the current update of the unknown subsurface flaws and cavities. In a first step, clusters of singularities are utilized to solve the inverse problem and to identify the location of the centroid(s) of the subsurface cavity(ies)/flaw(s). In a second step, the reconstruction of the estimated cavity(ies)/flaw(s) geometry(ies) is accomplished by utilizing an anchored grid pattern upon which cubic spline knots are restricted to move in the search for the unknown geometry. The solution is achieved using a genetic algorithm accelerated with the Nelder-Mead non-linear simplex method. The automated algorithm successfully reconstructs single and multiple subsurface cavities within two dimensional mediums. The cavity detection was enhanced by applying multiple boundary condition sets (MBCS) to the same geometry. This extra information supplied on the boundary made the subsurface cavity easily detectable despite its low heat signature effect on the boundaries.

К РАСЧЕТУ КРУТИЛЬНЫХ КОЛЕБАНИЙ КОНСТРУКТИВНЫХ ЭЛЕМЕНТОВ ПОДЪЕМНО-ТРАНСПОРТНЫХ МАШИН МЕТОДОМ ГРАНИЧНЫХ ЭЛЕМЕНТОВ

The solutions of the Cauchy problem of torsional vibrations of thin rods, which are the constituent elements of the support system handling machines, including all members of V. Z. Vlasov's equations are given. The application of these solutions for the problems of the dynamics of torsional vibrations of thin-walled rod systems on the boundary element method algorithm is shown. Results of solutions in numerical and visual forms are shown

A fast boundary element method for the scattering analysis of high-intensity focused ultrasound.

High-intensity focused ultrasound (HIFU) techniques are promising modalities for the non-invasive treatment of cancer. For HIFU therapies of, e.g., liver cancer, one of the main challenges is the accurate focusing of the acoustic field inside a ribcage. Computational methods can play an important role in the patient-specific planning of these transcostal HIFU treatments. This requires the accurate modeling of acoustic scattering at ribcages. The use of a boundary element method (BEM) is an effective approach for this purpose because only the boundaries of the ribs have to be discretized instead of the standard approach to model the entire volume around the ribcage. This paper combines fast algorithms that improve the efficiency of BEM specifically for the high-frequency range necessary for transcostal HIFU applications. That is, a Galerkin discretized Burton-Miller formulation is used in combination with preconditioning and matrix compression techniques. In particular, quick convergence is achieved with the operator preconditioner that has been designed with on-surface radiation conditions for the high-frequency approximation of the Neumann-to-Dirichlet map. Realistic computations of acoustic scattering at 1 MHz on a human ribcage model demonstrate the effectiveness of this dedicated BEM algorithm for HIFU scattering analysis.

Evaluation of non-singular BEM algorithms for potential problems

Two non-singular boundary element method (BEM) algorithms for two-dimensional potential problems have been implemented using isoparametric quadratic, cubic and quartic elements. The first one is based on the self-regular potential boundary integral equation (BIE) and the second on the self-regular flux-BIE. The flux-BIE requires the C1,α continuity of the density functions, which is not satisfied by the standard isoparametric elements. This requirement is remedied by adopting the relaxed continuity strategy. The self-regular flux-BIE has presented some poor and oscillatory results, mainly with continuous quadratic elements. This odd behavior has completely disappeared when discontinuous elements, which satisfy the continuity requirement, were applied, and this suggests that the 'relaxed continuity hypothesis' seems to be the main cause of numerical errors in the implementation of the self-regular flux-BIE. On the other side, the potential algorithm has shown very reliable solutions.

A boundary element method/Brownian dynamics approach for simulating DNA electrophoresis in electrically insulating microfabricated devices

We present an approach for merging boundary element method (BEM) solutions of the electric field in electrically insulating complex geometries with Brownian dynamics (BD) simulations of DNA electrophoresis therein. Although a rote application of the standard BEM algorithm proves inaccurate and prohibitively expensive, we show that regularization of the near-wall electric field and an updating scheme commensurate with the characteristic length scale of the BD simulation furnishes a robust, efficient simulation protocol. The accuracy of the BEM-BD method is verified by simulating lambda-DNA collisions with an isolated, insulating cylindrical obstacle and comparing the results with equivalent BD simulations that employ the exact solution for the electric field. The computational overhead of our implementation of BEM-BD is comparable to an existing finite element method/BD approach. The BEM-BD algorithm is readily parallelized and well-suited to time-dependent and responsive electric fields, making it broadly applicable to simulating DNA electrophoresis in microfluidic devices.

A Preconditioned Hierarchical Algorithm for Impedance Extraction of Three-Dimensional Structures With Multiple Dielectrics

This paper presents the first boundary element method (BEM) impedance extraction algorithm for interconnects with multiple dielectrics. Multiple dielectrics are common in integrated circuits and packages. However, previous BEM algorithms, including FastImp and FastPep, assume uniform dielectric due to their limitation, thus causing considerable errors. Our algorithm introduces a circuit formulation which makes it possible to utilize either multilayer Green's function or equivalent charge method to extract impedance in multiple dielectrics. The novelty of the formulation is the reduction of the unknowns and the application of hierarchical data structure. The hierarchical data structure permits efficient sparsification transformation and preconditioners to accelerate the linear equation solver. Experimental results demonstrate that the new algorithm is accurate and efficient. For uniform dielectric problems, our algorithm is more accurate than FastImp while its number of unknowns is ten times less than that of FastImp. For multiple dielectric problems, its relative error with respect to HFSS is below 3%.

Fast single domain–subdomain BEM algorithm for 3D incompressible fluid flow and heat transfer

AbstractIn this paper acceleration and computer memory reduction of an algorithm for the simulation of laminar viscous flows and heat transfer is presented. The algorithm solves the velocity–vorticity formulation of the incompressible Navier–Stokes equations in 3D. It is based on a combination of a subdomain boundary element method (BEM) and single domain BEM. The CPU time and storage requirements of the single domain BEM are reduced by implementing a fast multipole expansion method. The Laplace fundamental solution, which is used as a special weighting function in BEM, is expanded in terms of spherical harmonics. The computational domain and its boundary are recursively cut up forming a tree of clusters of boundary elements and domain cells. Data sparse representation is used in parts of the matrix, which correspond to boundary‐domain clusters pairs that are admissible for expansion. Significant reduction of the complexity is achieved.The paper presents results of testing of the multipole expansion algorithm by exploring its effect on the accuracy of the solution and its influence on the non‐linear convergence properties of the solver. Two 3D benchmark numerical examples are used: the lid‐driven cavity and the onset of natural convection in a differentially heated enclosure. Copyright © 2008 John Wiley & Sons, Ltd.

Regularized p-version collocation BEM algorithms for two-dimensional heat conduction

Based on conditioning, expectations should be that the boundary element method (BEM) is at least as sensitive to local modeling error as the finite element method (FEM), thus requiring comparable surface mesh densities when using the same order approximations. While the FEM appears most efficient as a sparse, low order, large scale approximation, the dense BEM formulation is more attractive as a small scale, high order approximation. Results show that p-version refinement in the BEM using interpolations as high as sixteenth order can reduce errors by five orders of magnitude for a given system size and can reduce system sizes by an order of magnitude for a given level of accuracy, thus reducing computational expense by three orders of magnitude. The algorithm presented includes a regularized form for the gradient boundary integral equation (BIE) and a super-regularized form for the standard BIE along with a comparison of the convergence behavior for the two forms. Examples are taken from two-dimensional heat conduction.

2.5D scattering of waves by rigid inclusions buried under a fluid channel via BEM

Abstract A 2.5D Boundary Element Method (BEM) formulation, applied in the frequency domain, is developed to compute the scattering of waves by rigid inclusions buried in a semi-infinite solid under a fluid layer, when this system is excited by a spatially-sinusoidal harmonic load. The BEM algorithm includes Green's functions for a horizontal fluid layer over a semi-infinite solid, which avoids the discretrization of the horizontal surfaces, and thus only the rigid inclusion needs to be discretized by boundary elements. The model uses complex frequencies with a small imaginary part to avoid aliasing phenomena. Time domain responses are obtained by applying an inverse Fourier Transform to the frequency results. The source is modeled as a Ricker pulse. The simulations are performed for three different properties of the solid medium: a fast formation, a slow formation and a sediment formation.

A Divide-and-Conquer Algorithm for 3-D Capacitance Extraction

We present a divide-and-conquer algorithm to improve the three-dimensional (3-D) boundary element method (BEM) for capacitance extraction. We divide large interconnect structures into small sections, set new boundary conditions using the border for each section, solve each section, and then combine the results to derive the capacitance. The target application is critical nets, clock trees, or packages where 3-D accuracy is required. Our algorithm is a significant improvement over the traditional BEMs and their enhancements, such as the "window" method, where conductors far away are dropped, and the "shield" method where conductors hidden behind other conductors are dropped. Experimental results show that our algorithm is a magnitude faster than the traditional BEM and the window+shield method, for medium to large structures. The error of the capacitance computed by the new algorithm is within 2% for self capacitance and 7% for coupling capacitance, compared with the results obtained by solving the entire system using BEM. Furthermore, our algorithms gives accurate distributed RC, where none of the previous 3-D BEM algorithms and their enhancements can.

An implicit and explicit BEM sensitivity approach for thermo-structural optimization

A computer-automated shape optimization methodology has been developed for the purpose of providing internal cooling systems designers the ability to optimize the internal cooling configuration, geometry and heat transfer enhancements for greater cooling efficiency and more durable turbine airfoils. The methodology presents the theory and practical programming requirements for coupling existing computer design and analysis tools together into a new and powerful design system. The goal of this paper is to demonstrate the computational advantages of using implicit sensitivity with the boundary element method (BEM) within this system over other more brute force methods. For this research, BEM algorithms for nonlinear heat conduction and thermo-elasticity were developed and coupled to an unstructured finite volume CFD code for the hot gas flow and a quasi-one-dimensional thermo-fluid system for the analysis of the internal coolant network. These computational tools were controlled by a constrained hybrid optimization algorithm to provide aerodynamic, thermal and internal fluid flow analyses on modified designs. The coolant supply total pressure, turbine inlet temperature, coolant wall thickness, thickness of ribs, rib positions, rib orientations, pin fin diameters and trip strip heights were incorporated into the set of optimization design variables. In order to improve performance, sensitivity gradients of the objective and constraint functions with respect to the geometric and heat transfer enhancement design variables were obtained using implicit differentiation of the boundary element system of equations. A three-to-one improvement in the optimization convergence rate and greater gradient accuracy were obtained for the two-dimensional thermal optimization problems. An order of magnitude larger computing time reduction was realized for three-dimensional thermal optimizations at the expense of additional memory, and another order of magnitude is expected for thermo-elastic optimization problems. Examples include studies of the accuracy of the design sensitivities with respect to forward and central finite differences, and validation of the optimization process using a symmetric cooled configuration.