Shape optimization by conventional and extended isogeometric boundary element method with PSO for two-dimensional Helmholtz acoustic problems

NURBS-enhanced boundary element method based on independent geometry and field approximation for 2D potential problems

This paper explores an isogeometric boundary element method (IGA-BEM) for sloshing problems in cylindrical tanks with single and multiply connected domains. Instead of the Lagrange basis functions used in the standard BEM, the nonuniform rational B-splines (NURBS) basis functions are introduced to approximate the geometries of the problem boundaries and the unknown variables. Compared with the Lagrange basis functions, NURBS basis functions can accurately reconstruct the geometric boundary of analysis domain with almost no error, and all the data information for NURBS basis functions can be directly obtained from the computer-aided design (cad) or computer-aided engineering (cae) commercial software, which implies the modeling process of IGA-BEM is more simple than that of the standard BEM. NURBS makes it possible for the IGA-BEM to realize the seamless connection between cad and cae software with relative higher calculation accuracy than the standard BEM. Based on the weighted residual method as well as the divergence theorem, the IGA-BEM is developed for the single and multiply connected domains, whose boundaries are separately defined in the parameter space by different knot vectors. The natural sloshing frequencies of the liquid sloshing in a circular cylindrical tank with a coaxial or an off-center circular pipe, an elliptical cylindrical tank with an elliptical pipe, a circular cylindrical tank with multiple pipes are estimated with the introduced method by assuming an ideal (inviscid and incompressible) liquid, irrotational small-amplitude sloshing, and the linear free-surface condition. The comparison between the results obtained by the proposed method and those in the existing literatures shows very good agreements, which verifies the proposed model well. Meanwhile, the effects of radius ratio, liquid depth, number, and location of internal pipe (pipes) on the natural sloshing frequency and sloshing mode are analyzed carefully, and some conclusions are outlined finally.

A Shape Optimisation with the Isogeometric Boundary Element Method and Adjoint Variable Method for the Three-Dimensional Helmholtz Equation

The concept of isogeometric analysis, where functions that are used to describe geometry in CAD software are used to approximate the unknown fields in numerical simulations, has received great attention in recent years. The method has the potential to have profound impact on engineering design, since the task of meshing, which in some cases can add significant overhead, has been circumvented. Much of the research effort has been focused on finite element implementations of the isogeometric concept, but at present, little has been seen on the application to the Boundary Element Method. The current paper proposes an Isogeometric Boundary Element Method (BEM), which we term IGABEM, applied to two-dimensional elastostatic problems using Non-Uniform Rational B-Splines (NURBS). We find it is a natural fit with the isogeometric concept since both the NURBS approximation and BEM deal with quantities entirely on the boundary. The method is verified against analytical solutions where it is seen that superior accuracies are achieved over a conventional quadratic isoparametric BEM implementation.

We propose a gradient-based optimisation framework to design the shape of the interfaces in periodic layered structures, which can model photovoltaic devices, optical gratings, etc., according to a desired electromagnetic property. To this end, we first develop an isogeometric boundary element method (IGBEM) to analyse singly periodic boundary value problems for 2D Helmholtz equation in layered media. By nature, the IGBEM is not only numerically accurate but also suitable for the shape optimisation in comparison with the conventional boundary- and domain-type solvers. Next, we derive the shape derivative (or shape sensitivity) of the objective function in terms of the magnetic field strength or energy absorption rate on the basis of the adjoint variable method. The shape derivative can be computed by solving the primal and adjoint problems with the IGBEM. Subsequently, we map our shape optimisation problems to nonlinear programming problems in order to exploit a general-purpose software for the latter problems. After verifying our optimisation framework rigorously by comparing with the exact solutions, we demonstrate a shape optimisation of the silicon-metal interface in a model of thin-silicon photovoltaic devices: it is crucial to determine the interface so that the energy of the incident light can be confined in the silicon layer as much as possible when the thickness of the layer is unconventionally small; one micrometre in our model. The optimal shape achieved 8.6 times higher absorption rate than the reference (flat) shape. This simulation shows that the proposed framework is capable of making a fundamental contribution to analysing and designing photovoltaic devices as well as photonic and plasmonic devices that consist of singly periodic layers.

PetIGA: A framework for high-performance isogeometric analysis

Fast algorithm for multi-frequency numerical integration in the simulation of acoustic scattering from rigid object by the boundary element method is presented. Normal derivative of the free-space Greens function is partially approximated with the unknown variable by a set of shape functions. As a result, the numerical integral is independent of frequency and need be calculated only at the first frequency step. Singular integral can be computed using the same procedure as that applied in the conventional boundary element method. Computational efficiency and accuracy of the new technique are demonstrated by an example. Numerical results obtained using the new technique are compared with the corresponding analytical solutions and numerical results obtained using the conventional boundary element method. The new technique works well and saves a lot of computational time in the process of generation of coefficient matrices for multi-frequency analysis.

An isogeometric Burton-Miller method for the transmission loss optimization with application to mufflers with internal extended tubes

The conventional boundary element method (BEM) uses internal cells for the domain integrals when solving nonlinear problems or problems with domain effects. In the conventional BEM, however, the merit of the BEM, which is easy preparation of data, is lost. This paper presents numerical integration for a three-dimensional meshless BEM, which does not require the use of internal cells. This method uses arbitrary internal points instead of internal cells. In this paper, a three-dimensional interpolation method is proposed. Furthermore, three-dimensional numerical integration is proposed using the above-mentioned three-dimensional interpolation method. This method is based on an improved multiple-reciprocity BEM (triple-reciprocity BEM) for heat conduction analysis with heat generation. In order to investigate the efficiency of this method, several numerical examples are given.

A meshless interface integral BEM for solving heat conduction in multi-non-homogeneous media with multiple heat sources

The computation of the acoustic scattering from objects in the ocean is useful to analyze the characteristics of the targets, which can be numerically modelled by acoustic boundary element method (BEM). The BEM has a defect of fictitious frequency problem, Burton-Miller method is studied to overcome the singular problems. In this paper, the Hadamard finite-part integral method based on collocation method is studied to overcome the singularity. It is found that, the near singularity is also quite significant and deteriorates largely the accuracy of BEM. A method based on polar transformation is proposed in this paper to overcome such a defect. Once the problems of singularity and near-singularity are solved, the conventional BEM model is established to serve as a foundation for fast BEM.It is well known that conventional BEM yields a full matrix, which is of O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) memory complexity. Furthermore, the time complexity could be up to O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ), if Gaussian elimination method is adopted to solve the linear equation of BEM. Many fast methods are proposed to solve the problems, such as the GMRES iterative solver, fast multipole method and H-matrix (Hierarchical Matrices) method. H-matrix method is applied to overcome the faults of conventional BEM, by using the low-rank approximations to compress blocks of the matrix, which is a pure algebraic method. The ACA (Adaptive Cross Approximation) method is proposed to compute the outer-product form of low-rank matrices, instead of SVD (Singular Value Decomposition), which consumes high CPU time and is of no usability. ACA-BEM is used to predict the acoustic scattering from objects in the ocean, especially the large-scale problems. The directivities of a rigid sphere and a rigid ellipsoid are computed by ACA BEM, the time consumed is very short. For a problem of more than 20,000 DOFs, it can be solved within 2 minutes. Therefore, ACA BEM can be used to solve large-scale acoustic scattering problems very efficiently.

Numerical investigations with eXtended isogeometric boundary element analysis (XIBEM) for direct and inverse Helmholtz acoustic problems

The aim of this study is the numerical simulation of installation effects on acoustic radiation of aircraft power plant systems, taking into account all surfaces involved in airframe/engine interaction. More precisely, we are interested in determining the delta in acoustic pressure level and directivity for different deflected angles of the flaps. Compared to the numerical assessment of noise radiation of an isolated engine, the computational domain is becoming very large in terms of wavelengths. It is for this reason that a new methodology has been developed. The principle of the method lies in the breakdown of the initial problem, and so the initial study domain, into two non-connected parts: the first one contains the power plant system, the second one contains the rest of the aircraft. In the first domain, we use a conventional Boundary Element Method (BEM) code. In the second domain a modified BEM code is used: based on Fast Multipole Method (FMM), this code provides a drastic reduction of data storage and computational time. That is the reason why the code is appropriate to the large computational domain. The first step of the methodology presented here is a weak-coupling formulation as only flap position effects are studied.

In general, internal cells are required to solve deformation of thin plate with arbitrary distributed load using conventional boundary element methods (BEM). However, in this case, the merit of BEM, which is the easy preparation of data, is lost. In this paper, it is shown that deformation analysis of thin plate with arbitrary distributed load can be solved without the use of internal cells by using the triple-reciprocity boundary element method. Distribution of arbitrary load is interpolated using boundary integral equations. The problem of thin plate according to Kirchhoff's theory is formulated by means of two coupled Poisson equations to be expressed in integral form using the second theorem of Green in the classical way. A new computer program was developed and applied to several example problems.

In general, internal cells are required to solve elastoplastic problems using a conventional boundary element method (BEM). However, in this case, the merit of BEM, which is ease of data preparation, is lost. Triple-reciprocity BEM can be used to solve two-dimensional elastoplasticity problems with a small plastic deformation. It has been shown that three-dimensional elastoplastic problems can be solved, without the use of internal cells, by the triple-reciprocity BEM and initial strain method. In this study, an initial stress formulation is adopted and the initial stress distribution is interpolated using boundary integral equations. A new computer program was developed and applied to solving several problems.