A three-dimensional shape optimization for transient acoustic scattering problems using the time-domain boundary element method

We develop a three-dimensional shape optimization (SO) framework for the wave equation with taking the unsteadiness into account. Resorting to the adjoint variable method, we derive the shape derivative (SD) with respect to a deformation (perturbation) of an arbitrary point on the target surface of acoustic scatterers. Successively, we represent the target surface with non-uniform rational B-spline patches and then discretize the SD in term of the associated control points (CPs), which are useful for manipulating a surface. To solve both the primary and adjoint problems, we apply the time-domain boundary element method (TDBEM) because it is the most appropriate when the analysis domain is the ambient air and thus infinitely large. The issues of the severe computational cost and instability of the TDBEM are resolved by exploiting the fast and stable TDBEM proposed by the present authors. Instead, since the TDBEM is mesh-based and employs the piecewise-constant element for space, we introduce some approximations in evaluating the discretized SD from the two solutions of TDBEM. By regarding the evaluation scheme as the computation of the gradient of the objective functional, given as the summation of the absolute value of the sound pressure over the predefined observation points, we can solve SO problems with a gradient-based non-linear optimization solver. To assess the developed SO system, we performed several numerical experiments from the perspective of verification and application with satisfactory results.     

Multiresolution subspace-based optimization method for inverse scattering problems

This paper investigates an approach to inverse scattering problems based on the integration of the subspace-based optimization method (SOM) within a multifocusing scheme in the framework of the contrast source formulation. The scattering equations are solved by a nested three-step procedure composed of (a) an outer multiresolution loop dealing with the identification of the regions of interest within the investigation domain through an iterative information-acquisition process, (b) a spectrum analysis step devoted to the reconstruction of the deterministic components of the contrast sources, and (c) an inner optimization loop aimed at retrieving the ambiguous components of the contrast sources through a conjugate gradient minimization of a suitable objective function. A set of representative reconstruction results is discussed to provide numerical evidence of the effectiveness of the proposed algorithmic approach as well as to assess the features and potentialities of the multifocusing integration in comparison with the state-of-the-art SOM implementation.     

Velocity field level-set method for topological shape optimization using freely distributed design variables

The velocity field level-set topological shape optimization method combines the implicit representation in the standard level-set method and the capabilities of general mathematical programming algorithms in handling multiple constraints and additional design variables. The key concept is to construct the normal velocity field using basis functions and the velocity design variables at specified points (referred to as velocity knots) in the entire design domain. In this study, the velocity design variables are decoupled from the level-set grid points. Making use of this property, we can adaptively change the arrangement of the velocity knots as the structural boundary evolves. This provides more design freedom in the optimization and allows for a significant reduction in the number of design variables. Several numerical examples in two- and three-dimensional design domains are presented to demonstrate the robustness and efficiency of the proposed method. We also show that changing the number of velocity knots may implicitly exert certain control on topological complexity and length scale.     

A wavelet method for reducing the computational cost of BE-based homogenization analysis

A wavelet BEM is applied to the evaluation of the effective elastic moduli of unidirectional composites, based on the homogenization theory. This attempt is devoted to the reduction of computational cost for the BE-based homogenization analysis. Truncation for matrix compression is carried out by the Beylkin-type algorithm. A thresholding value for the truncation is set such that the discretization error of BE solution is comparable to its truncation error. Besides, rearrangement of the BE equations is proposed to attain rapid convergence of iterative solutions. Through investigation of asymptotical convergence of the effective moduli, it is found that the BE-based homogenization analysis ensures the same rate of convergence for effective moduli as for characteristic functions. By applying the wavelet BEM to heterogeneous media which have microstructures with many voids, the effective moduli with agreement of 2–4 digits can be evaluated using 20–50% memory requirements of conventional BE approaches.     

Robust compliance topology optimization based on the topological derivative concept

In this paper, we present an approach for robust compliance topology optimization under volume constraint. The compliance is evaluated considering a point‐wise worst‐case scenario. Analogously to sequential optimization and reliability assessment, the resulting robust optimization problem can be decoupled into a deterministic topology optimization step and a reliability analysis step. This procedure allows us to use topology optimization algorithms already developed with only small modifications. Here, the deterministic topology optimization problem is addressed with an efficient algorithm based on the topological derivative concept and a level‐set domain representation method. The reliability analysis step is handled as in the performance measure approach. Several numerical examples are presented showing the effectiveness of the proposed approach. Copyright © 2015 John Wiley & Sons, Ltd.     

A general methodology for structural shape optimization problems using automatic adaptive remeshing

The proposed methodology is based on the use of the adaptive mesh refinement (AMR ) techniques in the context of 2D shape optimization problems analysed by the finite element method. A suitable and very general technique for the parametrization of the optimization problem, using B-splines to define the boundary, is first presented. Then mesh generation, using the advancing frontal method, the error estimator and the mesh refinement criterion are studied in the context of shape optimization problems In particular, the analytical sensitivity analysis of the different items ruling the problem (B-splines. finite element mesh, structural behaviour and error estimator) is studied in detail. The sensitivities of the finite element mesh and error estimator permit their projection from one design to the next one leading to an a priori knowledge of the finite element error distribution on the new design without the necessity of any additional structural analysis. With this information the mesh refinement criterion permits one to build up a finite element mesh on the new design with a specified and controlled level of error. The robustness and reliability of the proposed methodology is checked by means of several examples.     

Structural optimization using the boundary element method and topological derivative applied to a suspension trailing arm

This work investigates the optimization of elasticity problems using the boundary element method (BEM) as a numerical solver. A topological shape sensitivity approach is used to select the points showing the lowest sensitivities. As the iterative process evolves, the original domain has portions of material progressively removed in the less efficient areas until a given stop criterion is achieved. Two benchmark tests are investigated to demonstrate the influence of the boundary conditions on the final topology. Following this, a suspension trailing arm is optimized and a new design is proposed as an alternative to commercially available methods. A postprocedure of smoothing using Bézier curves was employed for the final topology of the trailing arm. This process allowed the external irregular shapes to be overcome. The BEM coupled with the topological derivative was shown to be an alternative to traditional optimization techniques using the finite element method. The present methodology was shown to be efficient for delivering optimal topologies with few iterations. All routines used were written in open code.     

Solutions to shape and topology eigenvalue optimization problems using a homogenization method

A solution strategy to find the shape and topology of structures that maximize a natural frequency is presented. The methodology is based on a homogenization method and the representation of the shape of the structure as a material property. The problem is formulated as a reinforcement problem in which a given structure is reinforced using a prescribed amount of material. Two dimensional, plane elasticity problems are considered. Examples are presented for illustration.     

A parallel two‐level domain decomposition based one‐shot method for shape optimization problems

A two‐level domain decomposition method is introduced for general shape optimization problems constrained by the incompressible Navier–Stokes equations. The optimization problem is first discretized with a finite element method on an unstructured moving mesh that is implicitly defined without assuming that the computational domain is known and then solved by some one‐shot Lagrange–Newton–Krylov–Schwarz algorithms. In this approach, the shape of the domain, its corresponding finite element mesh, the flow fields and their corresponding Lagrange multipliers are all obtained computationally in a single solve of a nonlinear system of equations. Highly scalable parallel algorithms are absolutely necessary to solve such an expensive system. The one‐level domain decomposition method works reasonably well when the number of processors is not large. Aiming for machines with a large number of processors and robust nonlinear convergence, we introduce a two‐level inexact Newton method with a hybrid two‐level overlapping Schwarz preconditioner. As applications, we consider the shape optimization of a cannula problem and an artery bypass problem in 2D. Numerical experiments show that our algorithm performs well on a supercomputer with over 1000 processors for problems with millions of unknowns. Copyright © 2014 John Wiley & Sons, Ltd.     

SIMP-ALL: A generalized SIMP method based on the topological derivative concept

Topology optimization has emerged in the last years as a promising research field with a wide range of applications. One of the most successful approaches, the SIMP method, is based on regularizing the problem and proposing a penalization interpolation function. In this work, we propose an alternative interpolation function, the SIMP-ALL method that is based on the topological derivative concept. First, we show the strong relation in plane linear elasticity between the Hashin-Shtrikman (H-S) bounds and the topological derivative, providing a new interpretation of the last one. Then, we show that the SIMP-ALL interpolation remains always in between the H-S bounds regardless the materials to be interpolated. This result allows us to interpret intermediate values as real microstructures. Finally, we verify numerically this result and we show the convenience of the proposed SIMP-ALL interpolation for obtaining auto-penalized optimal design in a wider range of cases. A MATLAB code of the SIMP-ALL interpolation function is also provided.     

A variation-gradient method for optimization of the shape of blade cascades

A variation-gradient method for solving the problem on optimization of the shape of blade cascades has been developed and tested. This method is based on calculation of the gradient of the functional of the optimization problem with the use of the parameters of a fluid flow varied relative to the design parameters determined from the system of variation gasdynamic equations. The efficiency of the indicated method was tested, as compared to the efficiency of the analogous gradient method in which the gradient of the functional of the optimization problem is calculated using the finite-difference method, in solving concrete problems on optimization of blade cascades. It is shown that the method proposed allows one to substantially decrease the time of determining the extremum of the indicated functional. __________ Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 80, No. 2, pp. 79–83, March–April, 2007.     

A velocity field level set method for shape and topology optimization

In this paper, we propose a new implementation of the level set shape and topology optimization, the velocity field level set method. Therein, the normal velocity field is constructed with specified basis functions and velocity design variables defined on a given set of points that are independent of the finite element mesh. A general mathematical programming algorithm can be employed to find the optimal normal velocities on the basis of the sensitivity analysis. As compared with conventional level set methods, mapping the variational boundary shape optimization problem into a finite‐dimensional design space and the use of a general optimizer makes it more efficient and straightforward to handle multiple constraints and additional design variables. Moreover, the level set function is updated by the Hamilton‐Jacobi equation using the normal velocity field; thus, the inherent merits of the implicit representation is retained. Therefore, this method combines the merits of both the general mathematical programming and conventional level set methods. Integrated topology optimization of structures with embedded components of designable geometries is considered to show the capability of this method to deal with general design variables. Several numerical examples in 2D or 3D design domains illustrate the robustness and efficiency of the method using different basis functions.     

Fast spectral-domain method for acoustic scattering problems

This paper presents the application of the conjugate-gradient (CG) fast Fourier transform (FFT) (CG-FFT) method and the CG nonuniform FFT (CG-NUFFT) method for the integral equation arising from acoustic scattering problems. In the conventional method of moments (MoM) for integral equations, the CPU and memory requirements are O(N³) and O(N²), respectively, where N is the number of unknowns in the problem. The CG-FFT method, which combines the iterative conjugate-gradient method with FFT, reduces these requirements to O(KN log₂N) and O(N), respectively, where K is the number of CG iterations. The CG-NUFFT method differs from the CG-FFT method in that it makes use of nonuniform FFT algorithms instead of FFT to allow a nonuniform discretization. Therefore, the CG-NUFFT method can solve the integral equation with both uniform and nonuniform grid while retaining the efficiency of the CG-FFT method. These two methods are applied to solve for two-dimensional constant density acoustic scattering problems. Numerical. results demonstrate that they can solve much larger problems than the MoM     

A modified indirect boundary element method for solving two-dimensional sound radiation problems

This paper proposes an engineering remedy to circumvent numerical shortcomings inherent in the indirect boundary element method (IBEM) to two-dimensional sound radiation problems. It is shown that when the acoustic sources are arranged on a shrunk internal boundary, which is produced by a uniform scaling factor applied to the actual vibrating boundary, then fictitious eigenfrequencies `move' to larger values being inversely proportional to this factor. In this way, IBEM analysis becomes capable of treating the problem of the nonuniqueness in a simple and efficient practical manner, which makes the method applicable to praxis. A conservative `a-priori' known scaling factor is established. The proposed method is applied to a circular, a square and rectangular vibrators. Dieser Beitrag beschreibt ein technisches Hilfsmittel, durch dessen Anwendung numerische Schwierigkeiten, die mit dem indirekten Randelementverfahren (IBEM) in zweidimensionalen Schallabstrahlungsproblemen verbunden sind, vermieden werden Können. Es wird hiermit folgendes aufgezeigt. Wenn Schallquellen an einen geschrumpften internen Rand gebracht sind, der entsteht, wenn der real schwingende Rand um einen Stufenfaktor gleichmäßig reduziert wird, dann gehen künstliche Eigenfrequenzen zu höheren Werten, die in umgekehrtem Verhältnis zu diesem Faktor stehen. Dadurch kann die IBEM-Analyse das Problem der nicht Einzigartigkeit auf eine einfache und effiziente Weise behandeln, was die Methode praktisch anwendbar macht. Ein konservativer, von vornherein bekannter Stufenfaktor ist bestimmt. Die vorgeschlagene Methode wird auf kreisförmige, viereckige und rechtwinklige Oszillatoren angewendet.     

A comparative analysis of contact algorithms in contact shape optimization problems

This paper presents a comparative analysis of contact algorithms used for solving contact shape optimization problems. Specifically, a nonlinear, feasible direction interior point method (FDIPM) for the frictional contact analysis of hyperelastic materials has been implemented in which the friction is introduced using the return mapping approach. This comparative investigation aims to find the cause of instability in sensitivity of the contact pressure nonuniformity (CPN). The results obtained by the FDIPM are found to be comparable with those by the penalty methods (PM) and the augmented Lagrange multiplier methods (ALMM); however, the FDIPM possesses advantages, including good convergence and convenience in modeling. Furthermore, the basic cause of the unstable sensitivity is revealed to be the discretization of the finite element method, which causes the discontinuous increase of contact area with respect to the continuous increase of contact load. To improve the stability of the CPN, an adaptive post-processing technique is proposed.     

Comparison among the variants of subspace-based optimization method for addressing inverse scattering problems: transverse electric case

The subspace-based optimization method (SOM) is an efficient approach to addressing the inverse scattering problem. In this paper, a comparative study, on the basis of numerical experiments, is conducted to evaluate the performances of variants of SOM, so as to find the optimal method for the determination of the ambiguous portion, which has a dominant influence on the computational cost and the reconstruction capability of the algorithm.     

A BEM sensitivity and shape identification analysis for acoustic scattering in fluid–solid problems

In this paper a boundary element formulation for the sensitivity analysis of structures immersed in an inviscide fluid and illuminated by harmonic incident plane waves is presented. Also presented is the sensitivity analysis coupled with an optimization procedure for analyses of flaw identification problems. The formulation developed utilizes the boundary integral equation of the Helmholtz equation for the external problem and the Cauchy–Navier equation for the internal elastic problem. The sensitivities are obtained by the implicit differentiation technique. Examples are presented to demonstrate the accuracy of the proposed formulations. © 1998 John Wiley & Sons, Ltd.     

A novel method for fabricating the surface-enhanced Raman scattering substrates and its enhanced properties

We have got large area surface-enhanced Raman scattering (SERS) substrates with uniform high enhancement factors by the so-called moulage method for the first time. A silver film (99.99%) with several millimeters thickness was thermally evaporated on the porous anodic alumina templates and the SERS substrate was got after moving off the templates. Surface-enhanced Raman scattering spectra of pyridine (0.01 Mol/L) were measured under 632.8 nm excitation. The experimental enhancement factors were more than 10(5) and S/N(p-p) around 100 was obtained. We have compared the SERS spectra of pyridine collected from different locations on the same SERS substrate and different substrates, which illustrate the well uniform enhance properties and the reproducibility of this method, respectively. The comparison of the SERS spectra, obtained from the SERS substrates and Ag film evaporated directly on glass slide, have proved that the electromagnetic coupling between two adjacent nanoparticles was important to the SERS effect. We also used rhodamine 6G as the probe molecules and found that the different molecules were very sensitive to the morphology of the SERS substrates.     

求解水下非圆弹性环声散射问题的一种半解析方法

基于传递矩阵法、齐次扩容精细积分法和复数矢径虚拟边界谱方法 ,提出了一种求解水下非圆弹性环声散射问题的半解析方法。该方法具有以下几个优点 :(1)采用复数矢径虚拟边界谱方法 ,不仅能保证在全波数域内Helmholtz外问题解的唯一性 ,而且由于虚拟源强密度函数采用 Fourier级数展开 ,克服了用单元离散解法不能用于较高频率范围的缺点 ;(2 )采用齐次扩容精细积分法求解非圆弹性环的状态微分方程 ,其计算结果具有很高的精度 ;(3)耦合方程不需要交错迭代求解 ,提高了计算效率。文中给出了两个典型非圆弹性环在平面声波激励下的声散射算例 ,计算结果表明本文方法是一种求解二维非圆弹性环声散射问题非常有效的半解析法。