Narrow Results

A shape optimization approach based on isogeometric wideband fast multipole boundary element method (IGA WFMBEM) in 2D acoustics is developed in this study. The key treatment is shape sensitivity analysis by using the adjoint variable method under isogeometric analysis (IGA) conditions. A set of efficient parameters of the wideband fast multipole method has been identified for IGA boundary element method. Shape optimization is performed by applying the method of moving asymptotes. IGA WFMBEM is validated through an acoustic scattering example. The proposed optimization approach is tested on a sound barrier and two multiple structures to demonstrate its potential for engineering problems.

In this paper, a half-space fast multipole BEM is developed for the simulation of three-dimensional acoustic problems above an infinite impedance plane. The half-space impedance Green’s function involving a complex line source is used, so that both mass-like and spring-like impedance boundary conditions on the infinite plane can be explicitly satisfied and the infinite plane is not required to be discretized. The Burton–Miller method is employed to tackle the fictitious eigenfrequency problem involved in the conventional boundary integral equation method. Image relations of the multipole expansion coefficients are used and the half-space impedance Green’s function is modified to apply such relations to avoid calculating, translating and saving the multipole/local expansion coefficients in the image domain. An automatic integrator with adaptive interval subdivision is further adopted to calculate the line integral contained in the M2L translation formula accurately and efficiently. Numerical examples are given to show the validity and potential of the method.

The application of a boundary element technique in combination with a contour integral approach to the numerical analysis of acoustic resonances in exterior configurations is investigated in this paper. Similar to the boundary element analysis of exterior acoustic radiation or scattering problems, spurious eigenfrequencies also turn up in the boundary element solution to exterior acoustic resonance problems. To filter out the spurious eigenfrequencies, the Burton–Miller-type combined formulation is employed to shift them from the real axis to the complex domain. The shifting effect brought by the combined formulation with different types of coupling parameters is investigated. Unlike in acoustic radiation and scattering analyses for which $-\mathrm{\text{i}}/k$ is suggested as the coupling parameter, it will be shown that the coupling parameter specified as $\beta \cdot \mathrm{\text{i}}/k$ with $\beta >0$ (the time-dependent term herein is ${\mathrm{\text{e}}}^{-\mathrm{\text{i}}\omega t}$) is more desirable in distinguishing the spurious eigenfrequencies in the boundary element analysis of exterior acoustic resonances.

The interaction between the sound field outside an elastic structure and its boundary is of paramount importance in the field of acoustics. This paper describes the global acoustic impedance in the form of impedance integral operator and impedance matrix for the analysis of elastic structures. In general, the excitation applied locally to the structure can induce vibration on the entire structure surface and hence the global acoustic impedance is introduced to characterize the relationship between the resultant sound pressure and acoustic particle velocity. The main advantage of the global acoustic impedance over the local acoustic impedance is that it facilitates improved physical insights into the sound field due to the presence of excitation sources. Furthermore, the acoustic impedance of an elastic spherical shell in matrix form is solved and analyzed by solving the finite element equation of structural dynamics with programming techniques. The impedance matrix is subsequently taken as the boundary condition to investigate the acoustic scattering from the elastic shell structure using the acoustic boundary element method. Explanations for the difference between the acoustic scatterings calculated based on the global acoustic impedance and the local acoustic impedance are offered.

The implicit frequency dependence of linear systems arising from the acoustic boundary element method necessitates an efficient treatment for problems in a frequency range. Instead of solving the linear systems independently at each frequency point, this paper is concerned with solving them simultaneously at multiple frequency points within a single iteration scheme. The proposed concept is based on truncation of the frequency range solution and is incorporated into two well-known iterative solvers - BiCGstab and GMRes. The proposed method is applied to two acoustic interior problems as well as to an exterior problem in order to assess the underlying approximations and to study the convergence behavior. While this paper provides the proof of concept, its application to large-scale acoustic problems necessitates efficient preconditioning for multi-frequency systems, which are yet to be developed.

Sound radiation from vibrating structures is a crucial concern in the vehicle design process. One effective tool to recover vibration patterns on surfaces is the surface contribution analysis. Recent implementations, however, focus on surface contributions with respect to single evaluation points. For a contribution analysis regarding an entire volume, the tedious volume integration is required. This study aims to develop an efficient contribution analysis technique for the acoustic evaluation of an entire cavity. In order to circumvent the cumbersome volume integral, the acoustic quantities are evaluated at regularly distributed field points. For this purpose, the three-dimensional Helmholtz equation is solved by using the boundary element method. Moreover, the eigendecomposition of the accompanying coupling matrices is involved in the proposed method. In contrast to traditional techniques, the sound energy is deployed as the objective function, since the sound energy is not only sensitive to the sound pressure but also to the particle velocity. Another beneficial aspect is that the energy-based contributions are nonnegative. In this way, acoustic short circuits are avoided. The proposed method is validated for two numerical examples: the inward radiating sphere and the vehicle interior noise problem. Initial findings already reveal that entire volumes can be analyzed with the energy-based contribution analysis. By this means, our method designates an efficient method to evaluate contributing surfaces with regard to entire cavities. This research emphasizes the relevance of an energy-based contribution analysis, since they provide deep insights into the acoustic behavior of cavities.

The primary disadvantage of the aerodynamic sound field analysis based on the Lighthill’s equation using the boundary element method (BEM) is the computational time; this is mainly because contributions from numerous aerodynamic sound sources are computed at all boundary element nodes and sound-receiving points. This study proposes a fast method for computing source contributions based on the fast multipole method (FMM). Along with the fast multipole BEM, which is already in practical use as a fast BEM, the analysis is substantially accelerated. The use of a common hierarchical cell structure for grouping boundary element nodes, sound-receiving points and aerodynamic sound sources, enables coefficients of the FMM to be reused, thereby further accelerating the analysis. To deal with the increasing hierarchical level, a wideband FMM is applied. In the sound radiation analysis of a quadrupole source located in a free field, the accuracy is validated. Sound radiation from a cylinder located in a flow is analyzed as a practical problem; the accuracy and numerical settings are discussed. Finally, the proposed method is applied to a problem with more than 0.4 million degrees-of-freedom and more than 3 million aerodynamic sound sources to demonstrate its applicability to large-scale problems.

When modeling sound waves in fluids it can be important to include the viscous and thermal losses originating from the fluids’ interaction with boundaries. In the audible frequency range, the thickness of the boundary layers is between a micrometer and a millimeter. As such the viscous and thermal losses are important when simulating the properties of small acoustical devices such as e.g. hearing aids or transducers. However, the inclusion of viscous and thermal losses is a computationally demanding task as it requires a fine discretization of the boundary layer in order to fully capture the complicated physical phenomena happening on the microscale. Recently, there has been developments to ease the computational demands using both the Finite Element Method and the Boundary Element Method, by approximating the losses using the Boundary Layer Impedance (BLI) boundary condition. In this paper, we extend previous developments for multi-frequency analysis using the Reduced Order Series Expansion Boundary Element Method to handle the BLI condition. This model follows a two-step procedure: Using a series expansion to decrease the assembly time of the BEM matrices and a projection to reduce the overall memory consumption of the model. Results from two acoustic interior problems show that the model decreases the total computational time by around 96% while using less than 15% of the memory. For both test setups the limiting factor of the accuracy was the reduction and not the series expansion.

To calculate the acoustic problems of relative uniform motion between the acoustic source and the fluid, we propose a boundary element method (BEM) strategy that can calculate various forms of relative uniform motion in subsonic conditions in a unified framework and is simple to implement. The acceleration algorithm for the BEM, like the fast multipole method (FMM), in the relative motionless state between the source and the fluid can be directly used without major modifications to the program. We propose a two-step transformation method to unify the wave equations of different relative motion forms into the classical form. In the first step, we transform the wave equations for various forms of relative motion into the equation where the convective terms are present only in the source part. Then, in the second step, we propose an acoustic-analogy Lorentz (a-a Lorentz) transformation to apply Lorentz covariance further to eliminate the convection term and establish the wave equation with classical form in a-a Lorentz space. We implement the boundary integration in the transformed a-a Lorentz space and derive a transformation method to transform discretized geometry and boundary conditions in the original space to the a-a Lorentz space. The problem that the boundary conditions are difficult to apply when solving the boundary integral equation (BIE) after the time-space coordinate transformation is solved. Numerical validations for the proposed method are performed by comparing with analytical results over a wide range of relative velocities. The results show that the proposed method can efficiently compute such problems with high accuracy and concise formulation.

The Helmholtz equation is a reliable model for acoustics in inviscid fluids. Real fluids, however, experience viscous and thermal dissipation that impact the sound propagation dynamics. The viscothermal losses primarily arise in the boundary region between the fluid and solid, the acoustic boundary layers. To preserve model accuracy for structures housing acoustic cavities of comparable size to the boundary layer thickness, meticulous consideration of these losses is essential. Recent research efforts aim to integrate viscothermal effects into acoustic boundary element methods (BEM). While the reduced discretization of BEM is advantageous over finite element methods, it results in fully populated system matrices whose conditioning deteriorates when extended with additional degrees of freedom to account for viscothermal dissipation. Solving such a linear system of equations becomes prohibitively expensive for large-scale applications, as only direct solvers can be used. This work proposes a revised formulation for the viscothermal BEM employing the Schur complement and a change of basis for the boundary coupling. We demonstrate that static condensation significantly improves the conditioning of the coupled problem. When paired with an iterative solution scheme, the approach lowers the algorithmic complexity and thus reduces the computational costs in terms of runtime and storage requirements. The results demonstrate the favorable performance of the new method, indicating its usability for applications of practical relevance in thermoviscous acoustics.

Common multi-frequency solution strategies for acoustic systems are either associated with high computational costs or resort to model reduction strategies, which potentially induce considerable approximation errors. In that context, this work proposes a fast multi-frequency solver for acoustic Boundary Element Method (BEM) analyses that combines conventional preconditioners with an accelerated recycling strategy, achieving in that way a fast and accurate iterative solution. Specifically, by approximating both the system and the related preconditioner by individual affine expressions, an upfront Galerkin projection on a global deflation basis is facilitated, therefore enabling the deployment of the accelerated recycling scheme. The latter acts by accelerating the construction of the related deflation projectors, therefore leading to a significantly faster recycling procedure. Finally, the efficiency of the global deflation basis is guaranteed through an Automatic Krylov subspaces Recycling for Deflation (AKR-D) [D. Panagiotopoulos, W. Desmet and E. Deckers, An accelerated subspaces recycling strategy for the deflation of parametric linear systems based on model order reduction, Comput. Methods Appl. Mech. Eng. 403 (2023) 115765] algorithm and thus, the cumulative number of iterations required by an iterative solver within a frequency sweep is drastically decreased. The proposed multi-frequency solver is tested on two industrially relevant examples of an interior and an exterior scattering problem, benchmarked against the traditional solution strategies.

This paper presents the thermo-acoustic frequency response of an un-baffled rectangular panel subjected to an external excitation load. A boundary element method BEM has been employed taking into account the Kirchhoff-Helmholtz K-H integral equation for the acoustic pressure and with the fluid-plate interface boundary condition the acoustic pressure jump over the panel is calculated. The thermal effects are considered regarding in the form of a uniform increment of temperature of the panel and are analysed in order to prevent the buckling phenomena. The excitation force considered is in the form of a concentrated load at some point of the panel and the deformation modes correspond to the vacuum case. Applying a collocation method for the panel equation, a frequency transfer function is obtained that relates the deflexion of the panel with the applied load. The effect of several geometric parameters, different thermal loads and location of the load applied on the acoustic radiated power and the acoustic efficiency spectrum are evaluated. Furthermore, the influence of the excitation frequency on the sound directivity is evaluated. The verification of the method is proven with other works.

This paper presents an accelerated higher order boundary element method for wave response analysis of VLFS (Very Large Floating Structures). The Fast Multipole Method (FMM) has been implemented on the higher order boundary element code using 8-node quadrilateral element. The method utilizes an iterative solver, multipole expansion of Green's function, and hierarchical algorithm using quadrant-tree. The numerical benchmark calculations have shown the efficiency of the method both in storage requirement of O(N) and computation time of O(N log N), where N is the number of unknowns for the velocity potential.

Lamb wave scattering analysis by an internal crack in a plate is carried out using the mode exciting method combined with the partial analysis method, which is our original method for solving a scattering problem of Lamb waves. Reflection and transmission coefficients are calculated for frequencies below the cut-off frequency of S₁ mode and for crack lengths less than 10 times of the plate thickness. Peak values in the reflection coefficients show dispersive properties for various frequencies and crack lengths. Using these dispersive characteristics, the applicability of the Lamb wave method to nondestructive evaluation of an internal crack is discussed.

A hybrid boundary point interpolation method (HBPIM) is presented for solving boundary value problems of two-dimensional solid mechanics. In the HBPIM, the boundary of a problem domain is represented by properly scattered nodes. The point interpolation method (PIM) is used to construct shape functions with Kronecker delta function properties based on arbitrary distributed nodes. In HBPIM, the ‘stiffness’ matrix so obtained is symmetric. This property of symmetry can be an added advantage in coupling the HBPIM with other established meshfree methods. A novel coupled EFG/HBPIM method for 2-D solids is then presented.

Ultrasonic guided wave mode conversion was studied for thin-plate defects by using lamb wave scattering. For lamb wave scattering analysis, the phase velocity dispersion curve and group velocity dispersion curve of the material were calculated. A mode conversion was investigated by the boundary element method (BEM) in terms of reflection and the transmission factor. Time domain wave analysis was performed with scattering field modeling and inverse fast Fourier transform (IFFT). The feasibility of the lamb wave technique was discussed for various practical applications, including the classification of defects.

In recent years, increased attention has been shifted to the design of cooling systems in the injection molding process, as it becomes clear that the cooling systems affect significantly both productivity and the part quality. In order to systematically improve the performance of a cooling system the mold designer may need a computer aided optimal design system for designing the injection mold cooling systems and determining the process conditions during the cooling stage. In this chapter, an efficient optimization procedure for this problem is proposed utilizing (i) the special boundary element analysis, (ii) the corresponding design sensitivity analysis using the direct differentiation approach, and (iii) the optimization algorithm. For this optimal design, an objective function is proposed to minimize a weighted combination of the cooling time and the temperature nonuniformity over the part surface. The former has to do with the warpage in the final part, while the latter is directly related to the overall productivity of the injection molding process. In this optimization program, various design variables are considered as follows: (i) (design variables related to processing conditions) the inlet coolant bulk temperature and inlet coolant volumetric flow rate of each cooling channel and (ii) (design variables related to mold cooling system design) the radius and location of each cooling channel. Each step of the proposed optimization procedure will be briefly explained below. First, in thermal analysis, mold heat transfer is considered as a cyclic-steady, three-dimensional conduction; heat transfer within the melt region is treated as a transient, one-dimensional conduction; heat exchange between the cooling channel surfaces and the coolant is considered to be steady; heat exchange between ambient air and mold exterior surfaces is also considered steady. Numerical implementation includes the application of a hybrid scheme consisting of a modified three-dimensional boundary element method for the mold region and a finite difference method with a variable mesh for the melt region. However, it was found that seemingly negligible inaccuracy in the thermal analysis result sometimes lead to a meaningless sensitivity analysis result. In this study, the thermal analysis system based on the above-mentioned modified boundary element method has been improved and rigorous treatments of boundary conditions appropriate for sensitivity analysis have been developed by considering the following issues: (i) numerical convergency, (ii) the series solution in part thermal analysis, (iii) the treatment of tip surface of line elements, (iv) the treatment of coolant, and (v) the treatment of mold exterior surface. Using an example, the importance of these issue is amply demonstrated. Next, the sensitivity analysis program developed in the present studies utilizes the implicit differentiation of the boundary integral equations and the boundary conditions presented in thermal analysis with respect to all design variables to obtain the sensitivity equations. A sample problem is solved to demonstrate the accuracy and the efficiency of the present sensitivity analysis formulation as well as to discuss the characteristics of each design variable. Finally, the CONMIN algorithm is applied for the optimization program with the help of the above thermal analysis and corresponding design sensitivity analysis. In this optimization program, the proper constraints imposed upon the design variables are considered to maintain design reality. The CONMIN algorithm employs the augmented Lagrangian multiplier method to deal with the equality constraints and the Davidon–Fletcher–Powell method for the unconstrained minimization during the successive unconstrained minimization procedure. Two sample problems were solved to demonstrate the efficiency and the usefulness of the objective function. The developed computer aided optimal design system would be very useful for injection mold designers in obtaining an optimal configuration of an injection mold cooling system in terms of radii and locations of cooling channels, as well as determining the optimal processing conditions of the cooling stage in terms of the inlet coolant bulk temperature and the inlet coolant volumetric flow rate of each cooling channel by minimizing certain objective functions related to the part quality and/or the productivity in the injection molding processes.

The deformation prediction is significant for plastic injection mold design. Whereas the existing deformation simulation base on finite element method (FEM) encounters many constraints on mesh discretizations and boundary condition definitions in actual injection mold production. A deformation simulation approach based on boundary element method (BEM) is proposed to overcome these constraints. In contrast to the FEM, only surface meshes of the mold are required. The cavity pressure and the contact constraints can be automatic defined on the mold boundary in the proposed approach. Therefore it saves a great deal of time for the mold engineers on laborious preprocessing task in deformation simulations.