Order Taylor Expansion Research Articles

Efficient size and shape optimization of truss structures subject to stress and local buckling constraints using sequential linear programming

The advance in digital fabrication technologies and additive manufacturing allows for the fabrication of complex truss structure designs but at the same time posing challenging structural optimization problems to capitalize on this new design freedom. In response to this, an iterative approach in which Sequential Linear Programming (SLP) is used to simultaneously solve a size and shape optimization sub-problem subject to local stress and Euler buckling constraints is proposed in this work. To accomplish this, a first order Taylor expansion for the nodal movement and the buckling constraint is derived to conform to the SLP problem formulation. At each iteration a post-processing step is initiated to map a design vector to the exact buckling constraint boundary in order to facilitate the overall efficiency. The method is verified against an exact non-linear optimization problem formulation on a range of benchmark examples obtained from the literature. The results show that the proposed method produces optimized designs that are either close or identical to the solutions obtained by the non-linear problem formulation while significantly decreasing the computational time. This enables more efficient size and shape optimization of truss structures considering practical engineering constraints.

How to estimate the 3D power spectrum of the Lyman-α forest

We derive and numerically implement an algorithm for estimating the 3D power spectrum of the Lyman-α (Lyα) forest flux fluctuations. The algorithm exploits the unique geometry of Lyα forest data to efficiently measure the cross-spectrum between lines of sight as a function of parallel wavenumber, transverse separation and redshift. We start by approximating the global covariance matrix as block-diagonal, where only pixels from the same spectrum are correlated. We then compute the eigenvectors of the derivative of the signal covariance with respect to cross-spectrum parameters, and project the inverse-covariance-weighted spectra onto them. This acts much like a radial Fourier transform over redshift windows. The resulting cross-spectrum inference is then converted into our final product, an approximation of the likelihood for the 3D power spectrum expressed as second order Taylor expansion around a fiducial model. We demonstrate the accuracy and scalability of the algorithm and comment on possible extensions. Our algorithm will allow efficient analysis of the upcoming Dark Energy Spectroscopic Instrument dataset.

Description of nonideal Lorentz transformation for electromagnetic nulls

To identify characteristic topological features of the electromagnetic field in an arbitrary reference frame, Lorentz transformation properties of an electromagnetic field near a null point are explored under certain constraints, in different nonideal magnetohydrodynamics (MHD) situations for linear nulls, showing violations of topology accordingly. It is shown that Newcomb's condition for conservation of covariant magnetic surfaces does not necessarily mean conservation of field line topology under Lorentz transformation. Characterizations of the violation of magnetic topology under Lorentz transformation are given. A method describing local magnetic null webs by combination of the first and second order Taylor expansions is also proposed, whose transformation properties with possible nonideal influences are discussed in the frame of resistive MHD. These results are important for establishing a reasonable range of the fieldline picture and thus the dynamical analysis based on magnetic fieldlines.

A Design Method for an Optimal Pre-filter in FRIT Using Closed-loop Step Response Data

This paper gives a design method for an optimal pre-filter in FRIT using closed-loop step response data by extending the method for VRFT. The paper shows that both the data-driven cost function and the original one have the same second order Taylor expansion with extended parameter descriptions. Finally, a numerical example shows the effectiveness of the proposed method.

Analytic Continuation of Local (Un)Stable Manifolds with Rigorous Computer Assisted Error Bounds

We develop a validated numerical procedure for continuation of local stable/unstable manifold patches attached to equilibrium solutions of ordinary differential equations. The procedure has two steps. First we compute an accurate high order Taylor expansion of the local invariant manifold. This expansion is valid in some neighborhood of the equilibrium. An important component of our method is that we obtain mathematically rigorous lower bounds on the size of this neighborhood, as well as validated a posteriori error bounds for the polynomial approximation. In the second step we use a rigorous numerical integrating scheme to propagate the boundary of the local stable/unstable manifold as long as possible, i.e., as long as the integrator yields validated error bounds below some desired tolerance. The procedure exploits adaptive remeshing strategies which track the growth/decay of the Taylor coefficients of the advected curve. In order to highlight the utility of the procedure, we study the embedding of some...

Continuous-energy perturbation methods in the MORET 5 code

The MORET code is a simulation tool that solves the transport equation for neutrons using the Monte Carlo method. A perturbation procedure using first order Taylor expansion series in the continuous-energy calculation for isotopic concentrations and density perturbations have been recently implemented. It relies on the calculation of an adjoint weighted sensitivity coefficient with respect to isotopic concentration. Thus, the MORET code now offers the users the possibility to use either the correlated sampling or taylor series as the perturbation method. The present paper provides a description of continuous-energy perturbation methods available in the MORET code focusing on the Taylor expansion series, which have been implemented in the last release of the code. Then, some considerations about the algorithm and implementation are presented. Finally the verification is presented using several ICSBEP and OECD/NEA benchmarks and some configurations of the French proprietary experimental program MIRTE.

Higher-order symmetry energy and neutron star core-crust transition with Gogny forces

We study the symmetry energy and the core-crust transition in neutron stars using the finite-range Gogny nuclear interaction and examine the deduced crustal thickness and crustal moment of inertia. We start by analyzing the second-, fourth- and sixth-order coefficients of the Taylor expansion of the energy per particle in powers of the isospin asymmetry for Gogny forces. These coefficients provide information about the departure of the symmetry energy from the widely used parabolic law. The neutron star core-crust transition is evaluated by looking at the onset of thermodynamical instability of the liquid core. The calculation is performed with the exact (i.e., without Taylor expansion) Gogny EoS for the core, and also with its Taylor expansion in order to assess the influence of isospin expansions on locating the inner edge of neutron star crusts. It is found that the properties of the core-crust transition derived from the exact EoS differ from the predictions of the Taylor expansion even when the expansion is carried through sixth order in the isospin asymmetry. Gogny forces, using the exact EoS, predict the ranges $0.094 \text{ fm}^{-3} \lesssim \rho_t \lesssim 0.118\text{ fm}^{-3}$ for the transition density and $0.339 \text{ MeV fm}^{-3} \lesssim P_t \lesssim 0.665 \text{ MeV fm}^{-3}$ for the transition pressure. The transition densities show an anticorrelation with the slope parameter $L$ of the symmetry energy. The transition pressures are not found to correlate with $L$. Neutron stars obtained with Gogny forces have maximum masses below $1.74M_\odot$ and relatively small moments of inertia. The crustal mass and moment of inertia are evaluated and comparisons are made with the constraints from observed glitches in pulsars.

Novel semi-implicit, locally conservative Galerkin (SILCG) methods: Application to blood flow in a systemic circulation

Three novel, locally conservative Galerkin (LCG) methods in their semi-implicit form are proposed for 1D blood flow modelling in arterial networks. These semi-implicit discretisations are: the second order Taylor expansion (SILCG-TE) method, the streamline upwind Petrov–Galerkin (SILCG-SUPG) procedure and the forward in time and central in space (SILCG–FTCS) method. In the LCG method, enforcement of the flux continuity condition at the element interfaces allows to solve the discretised system of equations at element level. For problems with a large number of degrees of freedoms, this offers a significant advantage over the standard continuous Galerkin (CG) procedure. The well established fully explicit LCG method is used for assessing the accuracy of the proposed new methods. Results presented in this work demonstrate that the proposed SILCG methods are stable and as accurate as the explicit LCG method. Among the three methods proposed, the SILCG–FTCS method requires considerably lower number of iterations per element, and thus requires lowest amount of CPU time. On the other hand, the SILCG-TE and SILCG-SUPG methods are stable and accurate for larger time step sizes. Although the standard Newton method requires evaluation of both the Jacobian matrix and the residual for every single iteration, which may be expensive for standard implicit solvers, the computed results show that the maximum number of iterations per element for SILCG-TE and SILCG-SUPG is less than unity (less than 0.3 and 0.7 respectively). Also, numerical experiments show that the Jacobian matrix can be calculated only once per time step, allowing to save a significant amount of computational time.

Raman signatures of ferroic domain walls captured by principal component analysis

Ferroic domain walls are currently investigated by several state-of-the art techniques in order to get a better understanding of their distinct, functional properties. Here, principal component analysis (PCA) of Raman maps is used to study ferroelectric domain walls (DWs) in LiNbO3 and ferroelastic DWs in NdGaO3. It is shown that PCA allows us to quickly and reliably identify small Raman peak variations at ferroelectric DWs and that the value of a peak shift can be deduced—accurately and without a priori—from a first order Taylor expansion of the spectra. The ability of PCA to separate the contribution of ferroelastic domains and DWs to Raman spectra is emphasized. More generally, our results provide a novel route for the statistical analysis of any property mapped across a DW.

Saddlepoint approximation reliability method for quadratic functions in normal variables

If the state of a component can be predicted by a limit-state function, the First and Second Order Reliability Methods are commonly used to calculate the reliability of the component. The latter method is more accurate because it approximates the limit-state function with a quadratic form in standard normal variables. To further improve the accuracy, this study develops a saddlepoint approximation reliability method that does not require additional transformations and approximations on the quadratic function. Analytical equations are derived for the cumulant generating function (CGF) of the limit-state function in standard normal variables, and then the saddlepoint is found by equating the derivative of the CGF to the limit state. Thereafter a closed form solution to the reliability is available. The method can also apply to general nonlinear limit-state functions after they are approximated by a second order Taylor expansion. Examples show the better accuracy than the traditional second order reliability methods.

Skewness and kurtosis of net baryon-number distributions at small values of the baryon chemical potential

We present results for the ratios of mean ($M_B$), variance ($\sigma_B^2$), skewness ($S_B)$ and kurtosis ($\kappa_B$) of net baryon-number fluctuations obtained in lattice QCD calculations with a physical light to strange quark mass ratio. Using next-to-leading order Taylor expansions in baryon chemical potential we find that qualitative features of these ratios closely resemble the corresponding experimentally measured cumulants ratios of net proton-number fluctuations for beam energies down to $\sqrt{s_{_{NN}}} \ge 19.6$ GeV. We show that the difference in cumulant ratios for the mean net baryon-number, $M_B/\sigma_B^2=\chi_1^B(T,\mu_B)/\chi_2^B(T,\mu_B)$ and the normalized skewness, $S_B\sigma_B=\chi_3^B(T,\mu_B)/\chi_2^B(T,\mu_B)$, naturally arises in QCD thermodynamics. Moreover, we establish a close relation between skewness and kurtosis ratios, $S_B\sigma_B^3/M_B=\chi_3^B(T,\mu_B)/\chi_1^B(T,\mu_B)$ and $\kappa_B\sigma_B^2=\chi_4^B(T,\mu_B)/\chi_2^B(T,\mu_B)$, valid at small values of the baryon chemical potential.

An enhanced subinterval analysis method for uncertain structural problems

Abstract This paper proposes an enhanced subinterval analysis method to predict the bounds of structural response with interval parameters, which could deal with problems with relatively large uncertainties of the parameters. The intervals are first divided into several subintervals, and two expansion routes are then constructed based on the sensitivity analysis. Two subinterval sets are selected according to the expansion points on the routes, and the first order Taylor expansion method is then adopted to complete the subinterval analysis. Based on the selected subinterval sets, the upper and lower bounds of the structural response are further obtained by employing the interval union operation. An adaptive convergence approach is presented to determine the appropriate number of subintervals. Four numerical examples are investigated to demonstrate the validity of the proposed method.

Performance study for a set of BLUE based Filters applied to amplitude estimation using as a reference the single photoelectron signal of the ν-Angra Experiment

This document presents a detailed study of the performance of a set of digital filters whose implementations are based on the best linear unbiased estimator theory interpreted as a constrained optimization problem that could be relaxed depending on the input signal characteristics. This approach has been employed by a number of recent particle physics experiments for measuring the energy of particle events interacting with their detectors. The considered filters have been designed to measure the peak amplitude of signals produced by their detectors based on the digitized version of such signals. This study provides a clear understanding of the characteristics of those filters in the context of particle physics and, additionally, it proposes a phase related constraint based on the second derivative of the Taylor expansion in order to make the estimator less sensitive to phase variation (phase between the analog signal shaping and its sampled version), which is stronger in asynchronous experiments. The asynchronous detector developed by the ν-Angra Collaboration is used as the basis for this work. Nevertheless, the proposed analysis goes beyond, considering a wide range of conditions related to signal parameters such as pedestal, phase, sampling rate, amplitude resolution, noise and pile-up; therefore crossing the bounds of the ν-Angra Experiment to make it interesting and useful for different asynchronous and even synchronous experiments.

In vitro evaluation of cell signalling processes associated with the potential genotoxicity of metal oxide nanoparticles

The assessment of the mean annual runoff and its interannual variability in a basin is the first and fundamental task for several activities related to water resources management and water quality analysis. The scarcity of observed runoff data is a common problem worldwide so that the runoff estimation in ungauged basins is still an open question. In this context, the main aim of this work is to propose and test a simple tool able to estimate the probability distribution of the annual surface runoff in ungauged river basins in arid and semi-arid areas using a simplified Fu's parameterization of the Budyko's curve at regional scale. Starting from a method recently developed to derive the distribution of annual runoff, under the assumption of negligible inter-annual change in basin water storage, we here generalize the application to any catchment where the parameter of the Fu's curve is known. Specifically, we provide a closed-form expression of the annual runoff distribution as a function of the mean and standard deviation of annual rainfall and potential evapotranspiration, and the Fu's parameter. The proposed method is based on a first order Taylor expansion of the Fu's equation and allows calculating the probability density function of annual runoff in seasonally dry arid and semi-arid geographic context around the world by taking advantage of simple easy-to-find climatic data and the many studies with estimates of the Fu's parameter worldwide. The computational simplicity of the proposed tool makes it a valuable supporting tool in the field of water resources assessment for practitioners, regional agencies and authorities.

Analytical estimation of annual runoff distribution in ungauged seasonally dry basins based on a first order Taylor expansion of the Fu's equation

Abstract The assessment of the mean annual runoff and its interannual variability in a basin is the first and fundamental task for several activities related to water resources management and water quality analysis. The scarcity of observed runoff data is a common problem worldwide so that the runoff estimation in ungauged basins is still an open question. In this context, the main aim of this work is to propose and test a simple tool able to estimate the probability distribution of the annual surface runoff in ungauged river basins in arid and semi-arid areas using a simplified Fu's parameterization of the Budyko's curve at regional scale. Starting from a method recently developed to derive the distribution of annual runoff, under the assumption of negligible inter-annual change in basin water storage, we here generalize the application to any catchment where the parameter of the Fu's curve is known. Specifically, we provide a closed-form expression of the annual runoff distribution as a function of the mean and standard deviation of annual rainfall and potential evapotranspiration, and the Fu's parameter. The proposed method is based on a first order Taylor expansion of the Fu's equation and allows calculating the probability density function of annual runoff in seasonally dry arid and semi-arid geographic context around the world by taking advantage of simple easy-to-find climatic data and the many studies with estimates of the Fu's parameter worldwide. The computational simplicity of the proposed tool makes it a valuable supporting tool in the field of water resources assessment for practitioners, regional agencies and authorities.

A generalized probabilistic edge-based smoothed finite element method for elastostatic analysis of Reissner–Mindlin plates

Probabilistic analysis is becoming more important in mechanical science and real-world engineering applications. In this work, a novel generalized stochastic edge-based smoothed finite element method is proposed for Reissner–Mindlin plate problems. The edge-based smoothing technique is applied in the standard FEM to soften the over-stiff behavior of Reissner–Mindlin plate system, aiming to improve the accuracy of predictions for deterministic response. Then, the generalized nth order stochastic perturbation technique is incorporated with the edge-based S-FEM to formulate a generalized probabilistic ES-FEM framework (GP_ES-FEM). Based upon a general order Taylor expansion with random variables of input, it is able to determine higher order probabilistic moments and characteristics of the response of Reissner–Mindlin plates. The significant feature of the proposed approach is that it not only improves the numerical accuracy of deterministic output quantities with respect to a given random variable, but also overcomes the inherent drawbacks of conventional second-order perturbation approach, which is satisfactory only for small coefficients of variation of the stochastic input field. Two numerical examples for static analysis of Reissner–Mindlin plates are presented and verified by Monte Carlo simulations to demonstrate the effectiveness of the present method.

Target localization based on structured total least squares with hybrid TDOA-AOA measurements

In this paper, we focus on the target localization problem which finds broad applications in radar, sonar and wireless sensor networks. A pseudolinear overdetermined system of equations is constructed from the nonlinear hybrid TDOA-AOA measurements about target location. Considering the matrix and vector in the constructed pseudolinear system are both contaminated by the measurement noise, a new weight least squares (WLS) method which is based on the first order Taylor expansions of the noise terms is developed in this paper and it can reduce the estimation bias that arise from the least squares (LS) method. In particular we focus on constructing a localization algorithm to reduce the bias that easily arise from the traditional methods. Thus in addition, a novel structured total least squares (STLS) method is also developed in this paper to further reduce the estimation bias specially when the target is outside the convex hull formed by sensors. Numerical examples show the superiority of the proposed STLS method in estimation accuracy compared with the LS method, total least squares (TLS) method and the proposed WLS method.

Homogenizing atomic dynamics by fractional differential equations

In this paper, we propose two ways to construct fractional differential equations (FDE) for approximating atomic chain dynamics. Taking harmonic chain as an example, we add a power function of fractional order to Taylor expansion of the dispersion relation, and determine the parameters by matching two selected wave numbers. This approximate function leads to an FDE after considering both directions for wave propagation. As an alternative, we consider the symbol of the force term, and approximate it by a similar function. It also induces an FDE. Both approaches produce excellent agreement with the harmonic chain dynamics. The accuracy may be improved by optimizing the selected wave numbers, or starting with higher order Taylor expansions. When resolved in the lattice constant, the resulting FDE's faithfully reproduce the lattice dynamics. When resolved in a coarse grid instead, they systematically generate homogenized algorithms. Numerical tests are performed to verify the proposed approaches. Moreover, FDE's are also constructed for diatomic chain and anharmonic lattice, to illustrate the generality of the proposed approaches.

Nonsmooth Morse–Sard theorems

We prove that every function f:Rn→R satisfies that the image of the set of critical points at which the function f has Taylor expansions of order n−1 and non-empty subdifferentials of order n is a Lebesgue-null set. As a by-product of our proof, for the proximal subdifferential ∂P, we see that for every lower semicontinuous function f:R2→R the set f({x∈R2:0∈∂Pf(x)}) is L1-null.

Analysis of spacecraft disposal solutions from LPO to the Moon with high order polynomial expansions

This paper presents the analysis of disposal trajectories from libration point orbits to the Moon under uncertainty. The paper proposes the use of polynomial chaos expansions to quantify the uncertainty in the final conditions given an uncertainty in initial conditions and disposal manoeuvre. The paper will compare the use of polynomial chaos expansions against high order Taylor expansions computed with point-wise integration of the partials of the dynamics, the use of the covariance matrix propagated using a unscented transformation and a standard Monte Carlo simulation. It will be shown that the use of the ellipsoid of uncertainty, that corresponds to the propagation of the covariance matrix with a first order Taylor expansions, is not adequate to correctly capture the dispersion of the trajectories that can intersect the Moon. Furthermore, it will be shown that polynomial chaos expansions better represent the distribution of the final states compared to Taylor expansions of equal order and are comparable to a full scale Monte Carlo simulations but at a fraction of the computational cost.