Second-order Taylor Expansion Research Articles

A deterministic approximation method in shape optimization under random uncertainties

This paper is concerned with the treatment of uncertainties in shape optimization. We consider uncertainties in the loadings, the material properties, the geometry and the vibration frequency, both in the parametric and geometric optimization setting. We minimize objective functions which are mean values, variances or failure probabilities of standard cost functions under random uncertainties. By assuming that the uncertainties are small and generated by a finite number N of random variables, and using first-or second-order Taylor expansions, we propose a deterministic approach to optimize approximate objective functions. The computational cost is similar to that of a multiple load problems where the number of loads is N. We demonstrate the effectiveness of our approach on various parametric and geometric optimization problems in two space dimensions.

Equivalence Assessment for Interchangeability Based on Two-Sided Tests

Interchangeability was originally developed in order to assess drug bioequivalence beyond average bioequivalence. In 2003, the Food and Drug Administration (FDA) published a Guidance documenting the procedures on using in vivo bioequivalence crossover trial to assess interchangeability between test and reference products. In general, this FDA Guidance describes interchangeability in terms of population and individual bioequivalence. The Guidance procedures were criticized for their lack of sampling distribution of the test statistics. As a result, the critical points were generated from simulation studies without adjusting for sample size. Further more, they lack consistency with average bioequivalence required in the 1992 FDA Guidance. Alternative interchangeability or interchangeability procedures were proposed to measure the probability of individual response difference under two treatments within prespecified lower and upper limits. Interchangeability is claimed if this probability is greater than a prespecified threshold. Tse et al. (2006) proposed an approximate distribution of the estimated probability based on the second-order Taylor expansion. For the same interchangeability probability hypothesis, Liu and Chow (1997) and Tsong and Shen (2007) also proposed a tolerance interval-based approach that can be extended to clinical trials with parallel arm design under the normality assumption. In this article, we first generalized the two-sided tolerance interval based interchangeability without equal sample size and variance assumption. We also derived a power function for the proposed method, and performed simulation studies to compare the type I error rate, power, and sample size between the Tse approximated test and the generalized tolerance interval approach for interchangeability assessment.

The breaking of transient inertio-gravity waves in a shear flow using the Gaussian beam approximation

AbstractThe propagation of transient inertio-gravity waves in a shear flow is examined using the Gaussian beam formulation. This formulation assumes Gaussian wavepackets in the spectral space and uses a second-order Taylor expansion of the phase of the wave field. In this sense, the Gaussian beam formulation is also an asymptotic approximation like spatial ray tracing; however, the first one is free of the singularities found in spatial ray tracing at caustics. Therefore, the Gaussian beam formulation permits the examination of the evolution of transient inertio-gravity wavepackets from the initial time up to the destabilization of the flow close to the critical levels. We show that the transience favours the development of the dynamical instability relative to the convective instability. In particular, there is a well-defined threshold for which small initial amplitude transient inertio-gravity waves never reach the convective instability criterion. This threshold does not exist for steady-state inertio-gravity waves for which the wave amplitude increases indefinitely towards the critical level. The Gaussian beam formulation is shown to be a powerful tool to treat analytically several aspects of inertio-gravity waves in simple shear flows. In more realistic shear flows, its numerical implementation is readily available and the required numerical calculations have a low computational cost.

Adaptable Center Detection of a Laser Line with a Normalization Approach using Hessian-matrix Eigenvalues

In vision measurement systems based on structured light, the key point of detection precision is to determine accurately the central position of the projected laser line in the image. The purpose of this research is to extract laser line centers based on a decision function generated to distinguish the real centers from candidate points with a high recognition rate. First, preprocessing of an image adopting a difference image method is conducted to realize image segmentation of the laser line. Second, the feature points in an integral pixel level are selected as the initiating light line centers by the eigenvalues of the Hessian matrix. Third, according to the light intensity distribution of a laser line obeying a Gaussian distribution in transverse section and a constant distribution in longitudinal section, a normalized model of Hessian matrix eigenvalues for the candidate centers of the laser line is presented to balance reasonably the two eigenvalues that indicate the variation tendencies of the second-order partial derivatives of the Gaussian function and constant function, respectively. The proposed model integrates a Gaussian recognition function and a sinusoidal recognition function. The Gaussian recognition function estimates the characteristic that one eigenvalue approaches zero, and enhances the sensitivity of the decision function to that characteristic, which corresponds to the longitudinal direction of the laser line. The sinusoidal recognition function evaluates the feature that the other eigenvalue is negative with a large absolute value, making the decision function more sensitive to that feature, which is related to the transverse direction of the laser line. In the proposed model the decision function is weighted for higher values to the real centers synthetically, considering the properties in the longitudinal and transverse directions of the laser line. Moreover, this method provides a decision value from 0 to 1 for arbitrary candidate centers, which yields a normalized measure for different laser lines in different images. The normalized results of pixels close to 1 are determined to be the real centers by progressive scanning of the image columns. Finally, the zero point of a second-order Taylor expansion in the eigenvector's direction is employed to refine further the extraction results of the central points at the subpixel level. The experimental results show that the method based on this normalization model accurately extracts the coordinates of laser line centers and obtains a higher recognition rate in two group experiments.

Second-order homogenization of periodic materials based on asymptotic approximation of the strain energy: formulation and validity limits

In this paper a second-order homogenization approach for periodic material is derived from an appropriate representation of the down-scaling that correlates the micro-displacement field to the macro-displacement field and the macro-strain tensors involving unknown perturbation functions. These functions take into account of the effects of the heterogeneities and are obtained by the solution of properly defined recursive cell problems. Moreover, the perturbation functions and therefore the micro-displacement fields result to be sufficiently regular to guarantee the anti-periodicity of the traction on the periodic unit cell. A generalization of the macro-homogeneity condition is obtained through an asymptotic expansion of the mean strain energy at the micro-scale in terms of the microstructural characteristic size ɛ; the obtained overall elastic moduli result to be not affected by the choice of periodic cell. The coupling between the macro- and micro-stress tensor in the periodic cell is deduced from an application of the generalised macro-homogeneity condition applied to a representative portion of the heterogeneous material (cluster of periodic cell). The correlation between the proposed asymptotic homogenization approach and the computational second-order homogenization methods (which are based on the so called quadratic ansätze) is obtained through an approximation of the macro-displacement field based on a second-order Taylor expansion. The form of the overall elastic moduli obtained through the two homogenization approaches, here proposed, is analyzed and the differences are highlighted. An evaluation of the developed method in comparison with other recently proposed in literature is carried out in the example where a three-phase orthotropic material is considered. The characteristic lengths of the second-order equivalent continuum are obtained by both the asymptotic and the computational procedures here analyzed. The reliability of the proposed approach is evaluated for the case of shear and extensional deformation of the considered two-dimensional infinite elastic medium subjected to periodic body forces; the results from the second-order model are compared with those of the heterogeneous continuum.

On the Convergence of the Self-Consistent Field Iteration in Kohn--Sham Density Functional Theory

It is well known that the self-consistent field (SCF) iteration for solving the Kohn--Sham (KS) equation often fails to converge, yet there is no clear explanation. In this paper, we investigate the SCF iteration from the perspective of minimizing the corresponding KS total energy functional. By analyzing the second-order Taylor expansion of the KS total energy functional and estimating the relationship between the Hamiltonian and the part of the Hessian which is not used in the SCF iteration, we are able to prove global convergence from an arbitrary initial point and local linear convergence from an initial point sufficiently close to the solution of the KS equation under the assumptions that the gap between the occupied states and unoccupied states is sufficiently large and the second-order derivatives of the exchange correlation functional are uniformly bounded from above. Although these conditions are very stringent and are almost never satisfied in reality, our analysis is interesting in the sense th...

Analysis of Thermal Damage in a Laser-Irradiated Based on the Non-Fourier Model

A modified non-Fourier equation of bio-heat transfer based on the second-order Taylor expansion of dual-phase-lag model is proposed to estimating thermal damage in a laser-irradiated biological tissue. This non-Fourier bio-heat transfer equation involves the mixed-derivative terms and the high-order derivatives of temperature with respect to time. The thermal damage in the tissue is assessed with the rate process equation. The effects of blood perfusion and metabolic heat generation on thermal response and thermal damage are explored. There are mathematical difficulties in dealing with such a problem. A hybrid numerical scheme is extended to solve the present problem. The deviations of the results from the bio-heat transfer equation based on the linear form of the dual-phase-lag model are presented and discussed.

Analysis of Non-Fourier Thermal Behavior in Layered Tissue with Pulse Train Heating

This paper investigates the thermal behavior in laser-irradiated layered tissue, which was stratified into skin, fat, and muscle. A modified nonFourier equation of bio-heat transfer was developed based on the second-order Taylor expansion of dual-phase lag model. This equation is a fourth order partial differential equation and can be simplified as the bio-heat transfer equations derived from Pennes model, thermal wave model, and the linearized form of dual-phase lag model. The boundary conditions at the interface between two adjacent layers become complicated. There are mathematical difficulties in dealing with such a problem. A hybrid numerical scheme is extended to solve the present problem. The deviations of the results from the bio-heat transfer equations based on Pennes model, thermal wave model and dual-phase lag model are presented and discussed.

Flexoelectricity from density-functional perturbation theory

We derive the complete flexoelectric tensor, including electronic and lattice-mediated effects, of an arbitrary insulator in terms of the microscopic linear response of the crystal to atomic displacements. The basic ingredient, which can be readily calculated from first principles in the framework of density-functional perturbation theory, is the quantum-mechanical probability current response to a long-wavelength acoustic phonon. Its second-order Taylor expansion in the wave vector $\mathbf{q}$ around the $\ensuremath{\Gamma}$ ($\mathbf{q}=0$) point in the Brillouin zone naturally yields the flexoelectric tensor. At order one in $\mathbf{q}$ we recover Martin's theory of piezoelectricity [Martin, Phys. Rev. B 5, 1607 (1972)], thus providing an alternative derivation thereof. To put our derivations on firm theoretical grounds, we perform a thorough analysis of the nonanalytic behavior of the dynamical matrix and other response functions in a vicinity of $\ensuremath{\Gamma}$. Based on this analysis, we find that there is an ambiguity in the specification of the ``zero macroscopic field'' condition in the flexoelectric case; such arbitrariness can be related to an analytic band-structure term, in close analogy to the theory of deformation potentials. As a by-product, we derive a rigorous generalization of the Cochran-Cowley formula [Cochran and Cowley, J. Phys. Chem. Solids $\mathbf{23}$, 447 (1962)] to higher orders in $\mathbf{q}$. This can be of great utility in building reliable atomistic models of electromechanical phenomena, as well as for improving the accuracy of the calculation of phonon dispersion curves. Finally, we discuss the physical interpretation of the various contributions to the flexoelectric response, either in the static or dynamic regime, and we relate our findings to earlier theoretical works on the subject.

Lattice Boltzmann method coupled with the Oldroyd-B constitutive model for a viscoelastic fluid

We developed a lattice Boltzmann method coupled with the Oldroyd-B constitutive equation to simulate a viscoelastic fluid. In this work, the flow field of the solvent is solved using an incompressible lattice Boltzmann Bhatnagar-Gross-Krook (BGK) model, while the advection operator of the polymer stress tensor is directly calculated with the help of the particle distribution functions. Specifically, we present a numerical scheme for the advection of the polymer stress tensor through the truncation of second-order Taylor expansion, which does not need to introduce the extra distribution functions and has better numerical accuracy. We consider two types of numerical tests to examine the performance of the presented method, including a two-dimensional (2D) channel flow and the 4:1 contraction problem. Our numerical results for the 2D channel flow agree well with the analytical results and some experimental results reported in the previous studies. Moreover, the numerical results also indicate that the current method can capture some complex rheological behaviors of the 4:1 contraction flow.

Digit Recognition Based on Distance Metric by Gabor Transformation

Distance metric is important for image transform, it reflects the similarity of manifold between pre and post transformation. Tangent distance method is usually used to approximate nonlinear manifolds by its tangent hyperplane in digit recognition; however, it sometimes neglects the high-order statistic information of the image. A new image distance metric is proposed in this paper which is based on Gabor transformation to obtain the Gabor feature vector, then the feature manifold can be approximated by curve surfaces based on second-order Taylor expansion with respect to intrinsic variables, and the distance metric by Gabor transformation is defined as the minimum distance between the approximated curved surfaces in feature observation space. Experiments show that the distance metric based on Gabor transformation and manifold works well on digit recognition, it excelled tangent distance method.

Second-order Taylor expansion for backward doubly stochastic control system

We consider the second-order Taylor expansion for backward doubly stochastic control system. The results are obtained under no restriction on the convexity of control domain. Moreover, the control variable is allowed in the drift coefficient and the diffusion coefficient.

Study on NURBS Curve Interpolation Based on before Acceleration/Deceleration with Linear Jerk

In high speed machining, excessive vibration and shock growing out of acceleration and jerk should be suppressed. Therefore, based on before acceleration/deceleration with linear jerk, a new NURBS curve interpolation was proposed. Interpolation feed and acceleration were planned employing before acceleration/deceleration with linear jerk evolution. During real-time interpolation, the interpolation feed and acceleration were calculated with polynomial fitting method, and with the second-order Taylor expansion, the interpolation parameter was computed. A NURBS curve was interpolated with the proposed method and the one with S curve acceleration/deceleration as comparison. The results indicated the proposed method not only maintain high feed but also reduce the interpolation output acceleration and jerk significantly, and its output jerk evolves along with time linearly. The proposed method can be used to reduce the vibration and shock from the high speed moving parts and improve their smoothness.

Spin-splitting calculation for zincblende semiconductors using an atomic bond-orbital model

We develop a 16-band atomic bond-orbital model (16ABOM) to compute the spin splitting induced by bulk inversion asymmetry in zincblende materials. This model is derived from the linear combination of atomic-orbital (LCAO) scheme such that the characteristics of the real atomic orbitals can be preserved to calculate the spin splitting. The Hamiltonian of 16ABOM is based on a similarity transformation performed on the nearest-neighbor LCAO Hamiltonian with a second-order Taylor expansion over at the Γ point. The spin-splitting energies in bulk zincblende semiconductors, GaAs and InSb, are calculated, and the results agree with the LCAO and first-principles calculations. However, we find that the spin–orbit coupling between bonding and antibonding p-like states, evaluated by the 16ABOM, dominates the spin splitting of the lowest conduction bands in the zincblende materials.

Description of hard-sphere crystals and crystal-fluid interfaces: A comparison between density functional approaches and a phase-field crystal model

In materials science the phase-field crystal approach has become popular to model crystallization processes. Phase-field crystal models are in essence Landau-Ginzburg-type models, which should be derivable from the underlying microscopic description of the system in question. We present a study on classical density functional theory in three stages of approximation leading to a specific phase-field crystal model, and we discuss the limits of applicability of the models that result from these approximations. As a test system we have chosen the three-dimensional suspension of monodisperse hard spheres. The levels of density functional theory that we discuss are fundamental measure theory, a second-order Taylor expansion thereof, and a minimal phase-field crystal model. We have computed coexistence densities, vacancy concentrations in the crystalline phase, interfacial tensions, and interfacial order parameter profiles, and we compare these quantities to simulation results. We also suggest a procedure to fit the free parameters of the phase-field crystal model. Thereby it turns out that the order parameter of the phase-field crystal model is more consistent with a smeared density field (shifted and rescaled) than with the shifted and rescaled density itself. In brief, we conclude that fundamental measure theory is very accurate and can serve as a benchmark for the other theories. Taylor expansion strongly affects free energies, surface tensions, and vacancy concentrations. Furthermore it is phenomenologically misleading to interpret the phase-field crystal model as stemming directly from Taylor-expanded density functional theory.

Theory, analysis and applications of 2D global illumination

We investigate global illumination in 2D and show how this simplified problem domain leads to practical insights for 3D rendering. We first derive a full theory of 2D light transport by introducing 2D analogs to radiometric quantities such as flux and radiance, and deriving a 2D rendering equation. We use our theory to show how to implement algorithms such as Monte Carlo raytracing, path tracing, irradiance caching, and photon mapping in 2D, and demonstrate that these algorithms can be analyzed more easily in this domain while still providing insights for 3D rendering. We apply our theory to develop several practical improvements to the irradiance caching algorithm. We perform a full second-order analysis of diffuse indirect illumination, first in 2D, and then in 3D by deriving the irradiance Hessian, and show how this leads to increased accuracy and performance for irradiance caching. We propose second-order Taylor expansion from cache points, which results in more accurate irradiance reconstruction. We also introduce a novel error metric to guide cache point placement by analyzing the error produced by irradiance caching. Our error metric naturally supports anisotropic reconstruction and, in our preliminary study, resulted in an order of magnitude less error than the “split-sphere” heuristic when using the same number of cache points.

Some Relations Between Extended and Unscented Kalman Filters

The unscented Kalman filter (UKF) has become a popular alternative to the extended Kalman filter (EKF) during the last decade. UKF propagates the so called sigma points by function evaluations using the unscented transformation (UT), and this is at first glance very different from the standard EKF algorithm which is based on a linearized model. The claimed advantages with UKF are that it propagates the first two moments of the posterior distribution and that it does not require gradients of the system model. We point out several less known links between EKF and UKF in terms of two conceptually different implementations of the Kalman filter: the standard one based on the discrete Riccati equation, and one based on a formula on conditional expectations that does not involve an explicit Riccati equation. First, it is shown that the sigma point function evaluations can be used in the classical EKF rather than an explicitly linearized model. Second, a less cited version of the EKF based on a second-order Taylor expansion is shown to be quite closely related to UKF. The different algorithms and results are illustrated with examples inspired by core observation models in target tracking and sensor network applications.

A novel curvature-based method for analyzing the second-order immobility of frictionless grasp

SUMMARYThis paper presents a new method to analyze frictionless grasp immobility based on defined surface-to-surface signed distance function. Distance function's differential properties are analyzed and its second-order Taylor expansion with respect to differential motion is deduced. Based on the non-negative condition of the signed distance function, the first- and second-order free motions are defined and the corresponding conditions for immobility of frictionless grasp are derived. As one benefit of the proposed method, the second-order immobility check can be formulated as a nonlinear programming problem. Numerical examples are used to verify the proposed method.

The transmission performance degradation of the optical millimeter-wave signals by fiber chromatic dispersion

The influence of the fiber chromatic dispersion on double sideband (DSB), optical carrier suppression (OCS), and single sideband (SSB) optical mm-wave signals is investigated based on the Taylor expansion of the propagation constant and is verified by simulation. According to our theoretical results, the fading effect suppresses the signal power of the DSB optical mm-wave periodically in a cosine-like pattern, and it can be described by the zero-order Taylor expansion of the propagation constant. For the optical mm-wave with the signal modulated on two or more tones, the bit pulses of the mm-wave signal are distorted by the dispersion-inducing bit walk-off effect between tones, which is expressed by the first-order Taylor expansion of the propagation constant. Moreover, as the signal rate and the transmission distance are increased further, higher-order Taylor expansion of the propagation constant still degrades the optical mm-wave signal even if both the fading effect and the bit walk-off effect are eliminated completely. The distortion of the signal pulses of SSB optical mm-wave is derived based on the second-order Taylor expansion of the propagation constant. This degradation is verified by the simulation with the eye diagram evolution of the SSB optical mm-wave signal.

Fuzzy rules emulated networks with adaptive controller for nonaffine discrete-time systems

An adaptive controller for a class of nonaffine discrete-time systems is developed as the main contribution of this article. With the system’s properties obtained by the second-order Taylor expansion, muti-input fuzzy rules emulated networks or MIFRENs are implemented to approximate the unknown plant under control. The closed-loop performance is guaranteed by an on-line learning algorithm developed to tune the parameters inside MIFRENs. According to the computation management, only linear parameters are adjusted with the constraints issued by the main theorem. Furthermore, the suitable learning rate can be determined with the information provided by the MIFREN approximation. The computer simulation system demonstrates the validation of the proposed controller. Moreover, the system robustness is described both nominal system and uncertain system.