Second-order derivatives of a ray with respect to the variables of its source ray in optical systems containing spherical boundary surfaces

The second-order derivative matrix of a scalar function with respect to a variable vector is called a Hessian matrix, which is a square matrix. Our research group previously presented a method for determination of the first-order derivatives (i.e., the Jacobian matrix) of a skew ray with respect to the variable vector of an optical system. This paper extends our previous methodology to determine the second-order derivatives (i.e., the Hessian matrix) of a skew ray with respect to the variable vector of its source ray when this ray is reflected/refracted by spherical boundary surfaces. The traditional finite-difference methods using ray-tracing data to compute the Hessian matrix suffer from various cumulative rounding and truncation errors. The proposed method uses differential geometry, giving it an inherently greater accuracy. The proposed Hessian matrix methodology has potential use in optimization methods where the merit function is defined as ray aberrations. It also can be used to investigate the shape of the wavefront for a ray traveling through an optical system.     

A numerical green's function approach for boundary elements applied to fracture mechanics

The most accurate boundary element formulation to deal with fracture mechanics problems is obtained with the implementation of the associated Green's function acting as the fundamental solution. Consequently, the range of applications of this formulation is dependent on the availability of the appropriate Green's function for actual crack geometry. Analytical Green's functions have been presented for a few single crack configurations in 2-D applications and require complex variable theory. This work extends the applicability of the formulation through the introduction of efficient numerical means of computing the Green's function components for single or multiple crack problems, of general geometry, including the implementation to 3-D problems as a future development. Also, the approach uses real variables only and well-established boundary integral equations.     

面向精密制造的光学自由曲面在位偏折测量技术

复杂光学曲面的在位测量是当前精密工程领域面临的重要难题。偏折术对光学曲面的测量精度可与干涉仪相比,而且拥有更高的测量效率、稳定性及动态范围,因此具有广阔的应用前景。但是偏折测量本质上是一个标定问题,其测量精度直接取决于几何标定的可靠性。本文结合单点金刚石切削机床设计了原位偏折测量系统,采用机床中自带的气浮转台安装辅助反射镜,在两个姿态下进行光线追迹,通过数值优化计算各元件之间的相对位置,将标定精度提高一个数量级。根据反向投影偏差的统计规律,可有效分离工件的面形偏差与位姿误差。该方法有效利用了工件的名义面形信息,将传统的位置-面形单向映射转变为双向映射,显著提高了在位测量的灵活性与效率。对于复杂的自由曲面,采用子孔径拼接测量方法,对待测的局部区域发展了精准定位技术,有效保证了迭代重构过程的正确收敛。采用离轴抛物镜等光学曲面进行实验验证,所提出的偏折测量方法的精度优于150 nm RMS。     

Eigendecomposition-based convergence analysis of the Neumann series for laminated composites and discretization error estimation

In computational homogenization for periodic composites, the Lippmann-Schwinger integral equation constitutes a convenient formulation to devise numerical methods to compute local fields and their macroscopic responses. Among them, the iterative scheme based on the Neumann series is simple and efficient. For such schemes, a priori global error estimates on local fields and effective property are not available, and this is the concern of this article, which focuses on the simple, but illustrative, conductivity problem in laminated composites. The global error is split into an iteration error, associated with the Neumann series expansion, and a discretization error. The featured nonlocal Green's operator is expressed in terms of the averaging operator, which circumvents the use of the Fourier transform. The Neumann series is formulated in a discrete setting, and the eigendecomposition of the iterated matrix is performed. The ensuing analysis shows that the local fields are computed using a particular subset of eigenvectors, the iteration error being governed by the associated eigenvalues. Quadratic error bounds on the effective property are also discussed. The discretization error is shown to be related to the accuracy of the trapezoidal quadrature scheme. These results are illustrated numerically, and their extension to other configurations is discussed.     

On the calculation of boundary stresses in boundary elements

The discontinuity of boundary stresses evaluated using discretized boundary elements is discussed, and stress error bound is derived. A new procedure for calculating interelement stress is proposed to overcome stress discontinuity at the interelement boundary. By employing two middle nodes of the two adjacent elements and the interelement node, a new quadratic element is formed, which leads to a more accurate interelement stress. Two examples are used to study error distributions of displacements and stresses. The comparison study indicates that the new procedure provides a better solution for interelement stresses than the conventional method.     

Application of h‐adaptive FEM and Zarka's approach to analysis of shakedown problems

The Zarka shakedown approach and the h‐adaptive finite element method are applied to evaluate residual stresses resulting from arbitrary cyclic loading. Two error indicators are used to refine the mesh: the explicit residual one which controls accuracy of the momentum balance and the interpolation error indicator which controls approximation of the modified back stresses. Validation tests performed for the Zarka method of simplified shakedown analysis suggest that such an approach may be used to obtain a quick estimate of residual states with the error acceptable for engineering purposes. Thus, it has been applied to compute residual stresses arising from service load in railroad rails. Copyright © 2004 John Wiley & Sons, Ltd.     

Pure equilibrium tetrahedral finite elements for global error estimation by dual analysis

This study presents a general procedure of creating pure equilibrium tetrahedral finite elements for use under the elastostatic hypothesis. These pure equilibrium elements are of the Fraeijs de Veubeke type and the degree of the polynomial approximation functions of their internal stress field is the parameter generating this new elements family. The spurious kinematic modes (SKM), inherent in the equilibrium approach, are eliminated at the element level by converting each tetrahedron into a super‐element defined as an assembly of four tetrahedral primitive elements. A mathematical discussion on the number of SKM of the primitive elements as well as their elimination by the super‐element technique has been carried out. The development of first and second degree elements is presented here in detail and their efficiency in the frame of global error estimation by dual analysis is emphasized by two numerical applications. The main attribute of the error estimation by dual analysis is that it provides an upper bound on the global discretization error. Copyright © 2009 John Wiley & Sons, Ltd.     

A field‐based finite element method for magnetostatics derived from an error minimization approach

An enhanced version of a mixed field‐based formulation for magnetostatics developed in previous papers is presented under more general hypotheses and a deep discussion of its features is carried out. The approach relies upon the minimization of the residual of the constitutive equation under constraints represented by Maxwell's equations, which are exactly imposed with Lagrange multipliers. In order to obtain computed fields with correct continuity properties across interfaces between materials with different permeability, the magnetic and the magnetic induction fields are used in a complementary way, and discretized by edge and face elements. Moreover, it is discussed how the formulation can be decomposed into two separate sets of equations, highlighting the relationship with classical formulations. A preconditioned iterative scheme to solve the final algebraic linear system is also presented. Furthermore, a very natural refinement indicator is defined to guide an adaptive mesh refinement procedure. Copyright © 2000 John Wiley & Sons, Ltd.     

A DEM-based method for predicting the wear evolution of structural boundary composed of spherical boundary elements

In the discrete element method (DEM) simulation for wear prediction, structural boundary is now represented extensively by triangular meshes with high resolution, which brings a huge computational cost. A DEM-based method for predicting the wear evolution of structural boundary has been developed for computational efficiency. The structural boundary subjected to wear is represented by the spherical boundary elements in the DEM simulation in combination with the inside triangles and fitting curved surface in wear prediction. Wear prediction is performed through a series of evolution steps. In each evolution step, the collision energies by particles at structural boundary are collected via the DEM simulation and assigned to the boundary elements. Then, the volume losses of structural boundary are predicted across each relevant boundary element. Finally, the new geometry of structural boundary in response to wear is described by moving the boundary elements along the depths of wear individually. Through converting the contact detection between structural boundary and particles into between spherical boundary elements and particles, our method greatly reduces the computational cost in the DEM simulation. Through two numerical tests, our method has been verified to be an efficient and accurate method for the wear prediction of structural boundaries with different resolutions.     

A new stochastic residual error based homotopy approach for stability analysis of structures with large fluctuation of random parameters

The large fluctuation of uncertain parameters introduces a great challenge in the stability analysis of structures. To address this problem, a novel stochastic residual error based homotopy method is proposed in this article. This new method used the concept of homotopy to reconstruct a new governing equation for stochastic elastic buckling analysis, and the closed-form solutions of the isolated buckling eigenvalues and eigenvectors are obtained by the stochastic homotopy analysis method. On this basis, a pth order origin moment of the stochastic residual error with respect to the elastic buckling equation is defined. Then, the optimal form of the homotopy series can be determined automatically by minimizing the pth order origin moment, which overcomes the disadvantage of highly relying on sample values of the existing homotopy stochastic finite element method. Moreover, the proposed method is developed to deal with the stochastic closely spaced buckling eigenvalue problem. Three mathematical examples and three buckling eigenvalue examples, including a variable cross-section column, a 7-story frame, and a Kiewitt single-layer latticed spherical shell, are performed to illustrate the accuracy and effectiveness of the proposed method by comparing with the existing methods when dealing with large fluctuation of random parameters.     

A posteriori analysis and adaptive error control for operator decomposition solution of coupled semilinear elliptic systems

In this paper, we develop an a posteriori error analysis for operator decomposition iteration methods applied to systems of coupled semilinear elliptic problems. The goal is to compute accurate error estimates that account for the combined effects arising from numerical approximation (discretization) and operator decomposition iteration. In an earlier paper, we considered ‘triangular’ systems that can be solved without iteration. In contrast, operator decomposition iterative methods for fully coupled systems involve an iterative solution technique. We construct an error estimate for the numerical approximation error that specifically addresses the propagation of error between iterates and provide a computable estimate for the iteration error arising because of the decomposition of the operator. Finally, we develop an adaptive discretization strategy to systematically reduce the discretization error.Copyright © 2013 John Wiley & Sons, Ltd.     

边界点法计算振动声辐射的误差研究

文章导出了边界点法分析结构振动声辐射问题的误差估计理论公式，揭示了边界点法分析结构振动声辐射问题的物理意义，在此基础上，给出了几个在边界点法的误差分析中很实用的结论，并通过算例进行了验证。     

A posteriori error estimates and adaptive procedures for the meshless Galerkin boundary node method for 3D potential problems

The Galerkin boundary node method (GBNM) is a boundary only meshless method that combines variational formulations of boundary integral equations with the moving least-squares approximations. This paper presents the mathematical derivation of a posteriori error estimates and adaptive refinement procedures for the GBNM for 3D potential problems. Two types of error estimators are developed in detail. One is a perturbation error estimator that is formulated based on the difference between numerical solutions obtained using two successive nodal arrangements. The other is a projection error estimator that is formulated based on the difference between the GBNM solution itself and its L²-orthogonal projection. The reliability and efficiency of both types of error estimators is established. That is, these error estimators are proven to have an upper and a lower bound by the constant multiples of the exact error in the energy norm. A localization technique is introduced to accommodate the non-local property of integral operators for the needed local and computable a posteriori error indicators. Convergence analysis results of corresponding adaptive meshless procedures are also given. Numerical examples with high singularities illustrate the theoretical results and show that the proposed adaptive procedures are simple, effective and efficient.     

Lumping analysis in the presence of measurement error

A review of the analysis of our earlier paper is presented, where systems mathematics is utilized to define in a direct way the conditions for exact lumping of monomolecular first-order kinetic systems. A new extension is to show that a system that is exactly lumpable in a batch reactor will follow the same lumping in any reactor type. In addition, our new approach shows how cluster analysis provides a useful way to analyse approximate lumping. This is especially important when measurement error is present, since the implementation of a successful lumping scheme must be viewed in the context of the errors. The connection of this work with Dr Doraiswamy is that the senior author was invited to present a plenary lecture at ICREC-1 in 1984, where a very preliminary form of the work (unpublished) was presented.     

A Koiter‐Newton approach for nonlinear structural analysis

A new approach termed the Koiter‐Newton is presented for the numerical solution of a class of elastic nonlinear structural response problems. It is a combination of a reduction method inspired by Koiter's post‐buckling analysis and Newton arc‐length method so that it is accurate over the entire equilibrium path and also computationally efficient in the presence of buckling. Finite element implementation based on element independent co‐rotational formulation is used. Various numerical examples of buckling sensitive structures are presented to evaluate the performance of the method. The examples demonstrate that the method is robust and completely automatic and that it outperforms traditional path‐following techniques. This improved efficiency will open the door for the direct use of detailed nonlinear finite element models in the design optimization of next generation flight and launch vehicles. Copyright © 2013 John Wiley & Sons, Ltd.     

An efficient ray tracing algorithm for the simulation of light trapping effects in Si solar cells with textured surfaces

Optimizing the design of the surface texture is an essential aspect of Si solar cell technology as it can maximize the light trapping efficiency of the cells. The proper simulation tools can provide efficient means of designing and analyzing the effects of the texture patterns on light confinement in an active medium. In this work, a newly devised algorithm termed Slab-Outline, based on a ray tracing technique, is reported. The details of the intersection searching logic adopted in Slab-Outline algorithm are also discussed. The efficiency of the logic was tested by comparing the computing time between the current algorithm and the Constructive Solid Geometry algorithm, and its superiority in computing speed was proved. The validity of the new algorithm was verified by comparing the simulated reflectance spectra with the measured spectra from a textured Si surface.     

A novel framework based on a data-driven approach for modelling the behaviour of organisms in chemical plume tracing

We propose a data-driven approach for modelling an organism''s behaviour instead of conventional model-based strategies in chemical plume tracing (CPT). CPT models based on this approach show promise in faithfully reproducing organisms’ CPT behaviour. To construct the data-driven CPT model, a training dataset of the odour stimuli input toward the organism is needed, along with an output of the organism’s CPT behaviour. To this end, we constructed a measurement system comprising an array of alcohol sensors for the measurement of the input and a camera for tracking the output in a real scenario. Then, we determined a transfer function describing the input–output relationship as a stochastic process by applying Gaussian process regression, and established the data-driven CPT model based on measurements of the organism’s CPT behaviour. Through CPT experiments in simulations and a real environment, we evaluated the performance of the data-driven CPT model and compared its success rate with those obtained from conventional model-based strategies. As a result, the proposed data-driven CPT model demonstrated a better success rate than those obtained from conventional model-based strategies. Moreover, we considered that the data-driven CPT model could reflect the aspect of an organism’s adaptability that modulated its behaviour with respect to the surrounding environment. However, these useful results came from the CPT experiments conducted in simple settings of simulations and a real environment. If making the condition of the CPT experiments more complex, we confirmed that the data-driven CPT model would be less effective for locating an odour source. In this way, this paper not only poses major contributions toward the development of a novel framework based on a data-driven approach for modelling an organism’s CPT behaviour, but also displays a research limitation of a data-driven approach at this stage.     

Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation

When applying numerical methods for the computation of stationary waves from the Helmholtz equation, one obtains ‘numerical waves’ that are dispersive also in non-dispersive media. The numerical wave displays a phase velocity that depends on the parameter k of the Helmholtz equation. In dispersion analysis, the phase difference between the exact and the numerical solutions is investigated. In this paper, the authors' recent result on the phase difference for one-dimensional problems is numerically evaluated and discussed in the context of other work directed to this topic. It is then shown that previous error estimates in H¹-norm are of nondispersive character but hold for medium or high wavenumber on extremely refined mesh only. On the other hand, recently proven error estimates for constant resolution contain a pollution term. With certain assumptions on the exact solution, this term is of the order of the phase difference. Thus a link is established between the results of dispersion analysis and the results of numerical analysis. Throughout the paper, the presentation and discussion of theoretical results is accompanied by numerical evaluation of several model problems. Special attention is given to the performance of the Galerkin method with a higher order of polynomial approximation p(h-p-version).