Isogeometric Boundary Element Method Research Articles

Structural Shape Optimization Based on Multi-Patch Weakly Singular IGABEM and Particle Swarm Optimization Algorithm in Two-Dimensional Elastostatics

In this paper, a multi-patch weakly singular isogeometric boundary element method (WSIGABEM) for two-dimensional elastostatics is proposed. Since the method is based on the weakly singular boundary integral equation, quadrature techniques, dedicated to the weakly singular and regular integrals, are applied in the method. A new formula for the generation of collocation points is suggested to take full advantage of the multi-patch technique. The generated collocation points are essentially inside the patches without any correction. If the boundary conditions are assumed to be continuous in every patch, no collocation point lies on the discontinuous boundaries, thus simplifying the implementation. The multi-patch WSIGABEM is verified by simple examples with analytical solutions. The features of the present multi-patch WSIGABEM are investigated by comparison with the traditional IGABEM. Furthermore, the combination of the present multi-patch WSIGABEM and the particle swarm optimization algorithm results in a shape optimization method in two-dimensional elastostatics. By changing some specific control points and their weights, the shape optimizations of the fillet corner, the spanner, and the arch bridge are verified to be effective.

Uncertainty Quantification of Vibroacoustics with Deep Neural Networks and Catmull–Clark Subdivision Surfaces

This study proposes an uncertainty quantification method based on deep neural networks and Catmull–Clark subdivision surfaces for vibroacoustic problems. The deep neural networks are utilized as a surrogate model to efficiently generate samples for stochastic analysis. The training data are obtained from numerical simulation by coupling the isogeometric finite element method and the isogeometric boundary element method. In the simulation, the geometric models are constructed with Catmull–Clark subdivision surfaces, and meantime, the physical fields are discretized with the same spline functions as used in geometric modelling. Multiple deep neural networks are trained to predict the sound pressure response for various parameters with different numbers and dimensions in vibroacoustic problems. Numerical examples are provided to demonstrate the effectiveness of the proposed method.

A modified RI-IGABEM with only weakly singular integral for viscoelastic analysis

In this paper, an improved isogeometric boundary element method based on radial integration method (RI-IGABEM) is proposed to solve the viscoelastic problems. Prony series are utilized to discretize the stress with the two most recent time-step strains. The boundary integral equation (BIE) for the viscoelastic problems is derived. The domain integral related to stress is decomposed into the linear combination of two sub-integrals by volumetric–deviatoric split. The radial integration method (RIM) is applied to transform all the domain integrals into equivalent boundary integrals. The present method has several advantages. Firstly, the present method eliminates the need for stress calculation at each time step, thus avoiding the error accumulation associated with the traction recovery method (TRM). Secondly, the BIE for the viscoelastic problems contains only weakly singular integrals. In addition, we provide an analytical solution for the radial component of the equivalent boundary integrals, which is transformed using RIM. To sum up, the present method improves the accuracy and efficiency of RI-IGABEM by reducing the required computational steps, eliminating error accumulation, and providing analytical solutions for certain integrals. Numerical examples demonstrate that the proposed RI-IGABEM is an efficient and accurate method suitable for viscoelastic problems.

Isogeometric boundary element method for axisymmetric steady-state heat transfer

Isogeometric analysis (IGA) utilizes the Non-Uniform Rational B-Splines (NURBS) basis functions, which are commonly employed in Computer Aided Design (CAD), to both construct the problem’s geometry and approximate the field variables. In axisymmetric scenarios, the 3D problem can be simplified to 2D, requiring only the discretization of the 1D curve boundary for the boundary element method (BEM). This study proposes an isogeometric boundary element method (IGABEM) to simulate steady-state heat transfer in axisymmetric domains. Both the precise geometry of the boundary and the physical variables are approximated with the same NURBS basis functions, yielding higher-order continuity, superior accuracy per degree of freedom (DOF), and continuous nodal gradients with fewer DOFs than the traditional BEM. Additionally, a zoning method is introduced to handle heterogeneities. Five cases including a solid cylinder, a revolution hyperboloid with a coaxial cylindrical tube, an ellipsoid with a spherical cavity, a bimaterial ellipsoid and a complex geometry with a toroidal tube validate the accuracy, convergence, computational efficiency of IGABEM, and the applicability in addressing multiply-connected domains. The method outperforms Finite Element Method (FEM) and conventional BEM in accurately handling steady-state axisymmetric heat transfer issues under Dirichlet, Neumann, and Robin boundary conditions.

The probabilistic mechanical-material modeling of complex non-homogeneous structures using the coupling of isogeometric boundary element method and reliability approach

Describing mechanical fields with precision in engineering analyses can be a challenging task. Analytical solutions fall short for real complex problems. The widespread use of numerical methods to overcome this obstacle has increased along with the advances in computational capacity. For instance, in geotechnical and composite materials problems, the use of a numerical approach becomes indispensable in several situations. For large volume-to-boundary ratio problems or for non-uniform fields evaluation, the boundary element method is a suitable tool. The method is based on the boundary discretization and on solutions to boundary integral equations. Thus, this article introduces the isogeometric boundary element method formulation and explores techniques for enhancing its physical problem descriptions, specially focusing in geotechnical issues and description of body forces. Two numerical examples are presented. Firstly, a multi-material dam, inspired by the real Itaipu dam section, is analyzed and the results are compared to the well-known ANSYS® FEM software. Furthermore, the article presents a tunnel case study in which a reliability assessment is conducted to account for the inherent uncertainties in the problem. This assessment is accomplished using the improved Hasofer, Lind, Rackwitz e Fiessler-first-order reliability method algorithm, which provides insight into the structural failure risk.

A generalized isogeometric boundary element method for the uncertain analysis of infinite domain two-dimensional acoustic problems

The key aim of this paper is to provide a new nth generalized order perturbed isogeometric fast multistage technique of boundary elements to compute the propagation of time harmonics in an infinite region. Structural geometry and boundary integral equations are constructed by using non-uniform rational B-splines. The source of system uncertainty is believed to be the incident plane wave number’s unpredictability. The actual field, depending on the input random variables, is simulated using the extended nth-order perturbation method. The field and kernel values for boundary integral formulas are generated via the nth-order generalized series of Taylor expansions using perturbation parameters. The fast multipole method (FMM) is utilized to speed up the process. The effectiveness and correctness of the proposed algorithm are verified by Monte Carlo simulations (MCs) with numerical examples.

Reduced order isogeometric boundary element methods for CAD-integrated shape optimization in electromagnetic scattering

This paper formulates a model order reduction method for electromagnetic boundary element analysis and extends it to computer-aided design integrated shape optimization of multi-frequency electromagnetic scattering problems. Firstly, a series expansion technique is adopted to decouple frequency-dependent terms from the integrands in boundary element formulation, and the second-order Arnoldi procedure is utilized to reduce the order of original systems. Secondly, to ensure geometric exactness and avoid re-meshing during shape optimization, the isogeometric boundary element method is employed, which uses the Non-Uniform Rational B-splines to represent the geometry and to discretize boundary integral equations. Lastly, we employ the Grey Wolf Optimization-Artificial Neural Network as a surrogate model for shape optimization in multi-frequency electromagnetic scattering problems, with the radar cross section as the objective function. Several examples are provided to demonstrate the accuracy and efficiency of the proposed algorithm.

Direct evaluation of stress intensity factors and T-stress for bimaterial interface cracks using the extended isogeometric boundary element method

This paper presents a new extended isogeometric boundary element method (XIGABEM) for the analysis of cracks in two-dimensional bimaterial interfaces. The classical NURBS approximations used in isogeometric formulations are augmented with functions based on the first two terms of the crack-tip stress and displacement series expansions. The first term is related to the complex stress intensity factor (SIF) and allows the numerical method to capture the singular and oscillatory near-tip behaviour, while the second accounts for the T-stress contribution to the solutions. The proposed enrichment strategy only introduces three additional degrees of freedom per crack tip, which are accommodated in a square linear system by a crack tip tying constraint and a novel condition on the stress parallel to the tip. These supplementary relations also cause the enrichment parameters to become proxies for the SIFs and T-stress. Therefore, the crack-tip factors can be obtained directly from the solution vector given by XIGABEM, eliminating the need for costly computational post-processing techniques. Several benchmark examples, including both straight and curved cracks, are presented to demonstrate the accuracy and convergence of the proposed method. The direct XIGABEM solutions for the SIFs and T-stress compare favourably against those from the literature.

Isogeometric Fast Multipole Boundary Element Method Based on Burton-Miller Formulation for 3D Acoustic Problems

An isogeometric boundary element method is applied to simulate wave scattering problems governed by the Helmholtz equation. The NURBS (non-uniform rational B-splines) widely used in the CAD (computer aided design) field is applied to represent the geometric model and approximate physical field variables. The Burton-Miller formulation is used to overcome the fictitious frequency problem when using a single Helmholtz boundary integral equation for exterior boundary-value problems. The singular integrals existing in Burton-Miller formulation are evaluated directly and accurately using Hadamard’s finite part integration. Fast multipole method is applied to accelerate the solution of the system of equations. It is demonstrated that the isogeometric boundary element method based on NURBS performs better than the conventional approach based on Lagrange basis functions in terms of accuracy, and the use of the fast multipole method both retains the accuracy for isogeometric boundary element method and reduces the computational cost.

Sensitivity to Gauss quadrature of isogeometric boundary element method for 2D potential problems

Sensitivity to Gauss quadrature of isogeometric boundary element method for 2D potential problems

Impedance Sensitivity Analysis Based on Discontinuous Isogeometric Boundary Element Method in Automotive Acoustics

Abstract Acoustic sensitivity analysis is an essential technique to determine the direction of structural-acoustic optimization by evaluating the gradient of the objective functions with respect to the design variables. However, acoustic sensitivity analysis with respect to acoustic impedance, which is an important parameter representing the interior absorbent material in automotive acoustics, is lacking in the study. Moreover, acoustic sensitivity analysis implemented with conventional numerical methods is time and effort-consuming in automotive acoustics, due to the large-scale mesh generation. In this work, the impedance sensitivity analysis for automotive acoustics based on the discontinuous isogeometric boundary element method is presented. The regularized boundary integral equation with impedance boundary conditions is established, then the sensitivity is derived by differentiating the boundary integral equation. The efficiency of the proposed method is improved by employing the parallel technique and generalized minimal residual solver. A long duct example with an analytical solution validates the accuracy of the proposed method, and an automotive passenger compartment subjecting to impedance boundary conditions illustrates that the computing time of the proposed method is one order of magnitude less than the conventional method. This work presents an easily implementable and efficient tool to investigate acoustic sensitivity with respect to impedance, showing great potential in the application of automotive acoustics.

A Spline-Based Partial Element Equivalent Circuit Method for Electrostatics

This contribution investigates the connection between isogeometric analysis (IgA) and the partial element equivalent circuit (PEEC) method for electrostatic problems. We demonstrate that using the spline-based geometry concepts from IgA allows for extracting circuit elements without an explicit meshing step. Moreover, the proposed IgA–PEEC method converges for complex geometries up to three times faster than the conventional PEEC approach and, in turn, it requires a significantly lower number of degrees of freedom to solve a problem with comparable accuracy. The resulting method is closely related to the isogeometric boundary element method. However, it uses lowest-order basis functions to allow for straightforward physical and circuit interpretations. The findings are validated by an analytical example with complex geometry, that is, significant curvature, and by a realistic model of a surge arrester.

Immersed isogeometric Boundary Elements: A user friendly method for the 3-D elasto-plastic simulation of underground excavations

The purpose of this paper is to present a simulation method for underground excavations that is more user friendly and efficient than existing methods. The immersed isogeometric Boundary Element Method is presented. Nonuniform rational B-splines (NURBS) are used for the accurate definition of complex geometries with few parameters. Immersed technology is applied to automatically generate cell meshes. This allows heterogeneous and anisotropic ground conditions as well as nonlinear material behaviour to be considered. On a practical example – that involves the elasto-plastic simulation of a borehole and where results are compared with a commercial Finite element program – the user friendliness and accuracy is demonstrated.

An isogeometric boundary element formulation for stress concentration problems in couple stress elasticity

An isogeometric boundary element method (IGABEM) is developed for the analysis of two-dimensional linear and isotropic elastic bodies governed by the couple stress theory. This theory is the simplest generalized continuum theory that can effectively model size effects in solids. The couple stress fundamental solutions are explicitly derived and used to construct the boundary integral equations. A new boundary integral equation arises to obtain the moments and rotations introduced by the couple stress formulation. A new analytical solution is also derived in the present work for an elliptical opening in an infinite sheet under uniaxial far-field stress. Several stress concentration problems are examined to illustrate and validate the application of the IGABEM in couple stress elasticity. It is shown that the IGABEM scheme exhibits advantageous convergence properties in comparison with the conventional BEM for boundary value problems within the framework of couple stress elasticity.

Isogeometric boundary element method for isotropic damage elastic mechanical problems

In this work, a complex isotropic damage mechanics structural model is developed using the isogeometric method to study the strain displacement and influence factors in isotropically damaged elastics. The proposed modeling method can be constructed dynamically by introducing control points and weight coefficients, which improves the accuracy and eliminates mesh errors, and the boundary element method is used for theoretical analysis and calculation of damage mechanics. For the description of irregular geometric models, the isogeometric method has unparalleled superiority. The isogeometric method can seamlessly connect the geometry generated by the advantages of CAD and CAE, and the combination of the two methods is very useful for the future extension of IGABEM to practical engineering applications. At the same time, we use the source point isolation method to establish the boundary-domain integral equations and use the radial integration method to solve the domain integrals. Compared with the traditional boundary element method, the isogeometric boundary element method makes the construction of complex models simple. The numerical results show that the numerical method has good stability and robustness, which provides strong support for further software integration.

Topology Optimization of Sound-Absorbing Materials for Two-Dimensional Acoustic Problems Using Isogeometric Boundary Element Method

In this work, an acoustic topology optimization method for structural surface design covered by porous materials is proposed. The analysis of acoustic problems is performed using the isogeometric boundary element method. Taking the element density of porous materials as the design variable, the volume of porous materials as the constraint, and the minimum sound pressure or maximum scattered sound power as the design goal, the topology optimization is carried out by solid isotropic material with penalization (SIMP) method. To get a limpid 0–1 distribution, a smoothing Heaviside-like function is proposed. To obtain the gradient value of the objective function, a sensitivity analysis method based on the adjoint variable method (AVM) is proposed. To find the optimal solution, the optimization problems are solved by the method of moving asymptotes (MMA) based on gradient information. Numerical examples verify the effectiveness of the proposed topology optimization method in the optimization process of two-dimensional acoustic problems. Furthermore, the optimal distribution of sound-absorbing materials is highly frequency-dependent and usually needs to be performed within a frequency band.

Panel Acoustic Contribution Analysis in Automotive Acoustics Using Discontinuous Isogeometric Boundary Element Method

In automotive industries, panel acoustic contribution analysis (PACA) is used to investigate the contributions of the body panels to the acoustic pressure at a certain point of interest. Currently, PACA is implemented mostly by either experiment-based methods or traditional numerical methods. However, these schemes are effort-consuming and inefficient in solving engineering problems, thereby restraining the further development of PACA in automotive acoustics. In this work, we propose a PACA scheme using discontinuous isogeometric boundary element method (IGABEM) to build an easily implementable and efficient method to identify the relative acoustic contributions of each automotive body panel. Discontinuous IGABEM is more accurate and converges faster than continuous BEM and IGABEM in the interior sound pressure evaluation of automotive compartments. In this work, a contribution ratio is defined to estimate the relative acoustic contribution of the structure panels; it can be calculated by reusing the coefficient matrix that has already been generated in the sound pressure evaluation process. The utilization of the parallel technique enables the proposed method to be more efficient than conventional methods; it is validated in two numerical examples, including a car passenger compartment subjected to realistic boundary conditions. A sound pressure response experiment based on a steel box is conducted to verify the accuracy of the interior sound pressure calculation using discontinuous IGABEM. This work is expected to promote the practical process of IGABEM for application in automotive acoustic problems.

Broadband Electromagnetic Scattering Analysis with Isogeometric Boundary Element Method Accelerated by Frequency-Decoupling and Model Order Reduction Techniques

Broadband Electromagnetic Scattering Analysis with Isogeometric Boundary Element Method Accelerated by Frequency-Decoupling and Model Order Reduction Techniques

Sensitivity analysis of structural-acoustic fully-coupled system using isogeometric boundary element method

In many engineering challenges, the whole interaction between the structural domain and the acoustic domain must be taken into account, particularly for the acoustic analysis of thin structures submerged in water. The fast multipole boundary element approach is used in this work to simulate the external acoustic domain and the finite element method is used to describe the structural components. To improve coupling analysis accuracy, discontinuous higher-order boundary components are created for the acoustic domain. The isogeometric boundary element method (IGABEM) discretizes unknown physical fields by using CAD spline functions as basis functions. IGABEM is inherently compatible with CAD and can perform numerical analysis on CAD models without having to go through the time-consuming meshing process required by traditional FEM/BEM and volume parameterization in isogeometric finite element methods. IGABEM’s power in tackling infinite domain issues and combining CAD and numerical analysis is fully used when it is applied to structural form optimization of three-dimensional external acoustic problems. The structural-acoustic design and optimization procedures benefit from the use of structural-acoustic design sensitivity analysis because it may provide information on how design factors affect radiated acoustic performance. This paper provides adjoint operator-based equations for sound power sensitivity on structural surfaces and direct differentiation-based equations for sound power sensitivity on arbitrary closed surfaces surrounding the radiator. Numerical illustrations are provided to show the precision and viability of the suggested approach.

Acoustic shape optimization based on isogeometric boundary element method with subdivision surfaces

This study proposes a new acoustic shape optimization approach by combining isogeometric subdivision surfaces with the boundary element method. The geometry of the structural boundary is constructed by the Catmull–Clark subdivision surfaces, which are able to provide multiresolution subdivision hierarchy models to meet different precision requirements. The acoustic propagation in the unbounded domain is simulated using the boundary element method. The control points in subdivision meshes are set as design variables, and the minimization of sound pressure at the observation points is selected as the design objective. The acoustic shape sensitivities are calculated by the sensitivity boundary integral equation based on the direct differentiation method. In each iteration the geometry is updated from a direct optimized shape, then a secondary processing is considered to obtain smooth surfaces. Compared with the conventional optimization methods, we eliminate jagged geometry by multiresolution approach without needing the cumbersome meshing procedure and volume parameterization. Several numerical experiments demonstrate the effectiveness and validity of the developed shape optimization approach.