Analytical Derivatives Research Articles

Optimization of planar mechanisms by using a minimum distance function

The paper examines the application of a general minimum distance error function to the dimensional kinematic synthesis of bidimensional mechanisms. The minimum distance approach makes it possible to solve the problem maintaining the same generality as that of the minimum deformation energy method while solving the problems that occasionally appear in the former method involving low stiffness mechanisms. It is a general method that can deal both with unprescribed and prescribed timing problems, and is applicable for path generation problems, function generation, solid guidance, and any combination of the aforementioned requirements as introduced in the usual precision point scheme. The method exhibits good convergence and computational efficiency. The minimum distance error function is solved with a sequential quadratic programming (SQP) approach. In the study, the synthesis problem is also optimized by using SQP, and the function can be easily adapted to other methods such as genetic algorithms.In the study, the minimum distance approach is initially presented. Subsequently, an efficient SQP method is developed by using analytic derivatives for solving. The next point addresses the application of the concept for the synthesis of mechanisms by using an SQP approach with approximate derivatives. This delivers a situation where the optimization is performed on an error function that itself consists of an inner optimization function. A few examples are presented and are also compared with the minimum deformation energy method. Finally, a few conclusions and future studies are discussed.

PageRank as a collective variable to study complex chemical transformations and their energy landscapes.

A reduced set of reaction coordinates is often employed in chemistry to describe the collective change between reactants and products within the context of rare event theories and the exploration of energy landscapes. Yet selecting the proper collective variable becomes increasingly challenging as the systems under study become more complex. Recent advancement of new descriptions of collective molecular coordinates has included graph-theoretical metrics, including social permutation invariant and PageRank (PR) coordinates, based upon the network of interactions about molecules and atoms within a system. Herein we continue the development of PR by (1) presenting a new formulation that is continuous along a reaction path, (2) illustrating that the fluctuations in PR are demonstrative of the fundamental motions of the atoms/molecules, and (3) providing the analytical derivatives with respect to atomic coordinates. The latter is subsequently combined with a harmonic bias to create the potential of mean force (PMF). As an example, we first consider the transformation of tetrahedral [Al(OH)4](aq) - to octahedral [Al(OH)4(H2O)2](aq) - using the PR PMF. Second, we explore the interchange of contact ion pair and solvent separated ion pairs of aqueous Na⋯OH, where the distance-biased PMF is projected onto PR space. In turn, this reveals where solvent rearrangement has the most impact upon the reaction pathway.

Field-Aware Interfaces in Continuum Solvation.

Continuum models of solvation are widespread tools for the prediction of solvation free energies of small molecular compounds from first principles. However, the continuum approximation at the core of these approaches limits their accuracy for the modeling of the aqueous solvation of compounds with highly polarized residue or of charged species. This is due to the fact that straightforward definitions of the continuum interface do not account for the reorganization effects induced by these solutes on the positions of surrounding solvent molecules. This kind of problem is usually overcome by stretching the definition of the continuum model, i.e., by using chemical intuition to adjust (usually shrinking) the size of the solvation interface close to the polarized/charged atoms. Nonetheless, this strategy introduces a significant number of additional parameters that need to be tuned and, at the same time, deters the model's transferability. A transferable solution is instead represented by an improved definition of the continuum interface, able to automatically account for the polarization/charge state of the embedded system. Following recent approaches in the literature, the component of the solute's electric field normal to the interface can be used as an effective proxy for the net charge of the embedded system. Here we show a simple definition of this field-aware approach as applied to the recently proposed soft-sphere continuum solvation (SSCS) method. In this model, each soft sphere composing the interface is allowed to readjust as a function of the value of the field flux through its surface. This effect introduces a complex dependence of the interface function on both the electronic and the ionic degrees of freedom of the solute. To account for this dependence during optimization procedures (e.g., the SCF loop and geometry optimization algorithms), the analytic derivatives of the new interface are reported and validated with their numerical counterparts. Application of the field-aware procedure to molecular compounds showing pathological behaviors with the standard SSCS approach show that significant improvements can be achieved by specifically tuning the newly introduced parameters.

Efficient implementation of the analytical second derivatives of hartree-fock and hybrid DFT energies within the framework of the conductor-like polarizable continuum model.

Calculation of vibrational frequencies for solvated systems is essential to study reactions in complex environments. In this paper, we report the implementation of the analytical self-consistent field Hessian at the Hartree-Fock and density functional theory levels in the framework of the conductor-like polarizable continuum model (C-PCM) into the ORCA quantum chemistry suite. The calculated vibrational frequencies agree very well with those computed through numerical differentiation of the analytical gradients. The deviation between both sets of data is smaller than 3 cm-1 for frequencies larger than 200 cm-1 and smaller than 5 cm-1 for the low-frequency regime (100 cm-1 < ω < 200 cm-1 ). The accuracy of the frequencies is not significantly affected by the size of the density functional theory (DFT) integration grid, with a deviation lower than 0.5 cm-1 between data computed with the smallest and that with the largest DFT grid size. The calculation of the analytical Hessian is between 3 and 12 times faster than its numerical counterpart. The C-PCM terms only add an overhead of 10-30% relative to the gas phase calculations. Finally, for acetone, the (B3LYP) values for the frequency shifts obtained in going from the gas phase to liquid acetone are in agreement with experiment. © 2019 Wiley Periodicals, Inc.

USING THE COMBINATION OF THE 3D GRAVITY INVERSION METHOD WITH THE DIRECTIONAL ANALYTIC SIGNAL DERIVATIVES AND THE CURVATURE GRAVITY GRADIENT TENSOR METHOD TO DETERMINE STRUCTURE OF THE PRE-CENOZOIC BASEMENT ON SOUTHEAST CONTINENTAL SHELF OF VIETNAM

In this paper, we present some new results of the Pre-Cenozoic basement structure on the Southeast continental shelf of Vietnam based on the combination of some modern methods to analyse and interpret gravity data. They are the 3D gravity inversion method, the directional analytic signal derivatives and the curvature gravity gradient tensor. The results obtained include the density distribution, the fault system and the main structural blocks inside of the Pre-Cenozoic basement on Southeast continental shelf of Vietnam. The initial results about density distribution show that it relatively clearly reflects the shape of basins in the area: The contours that have value σ =2.7 g/cm3 and value σ = 2.76 g/cm3 are near the edges of the Cuu Long basin and the Nam Con Son basin, respectively. The density value reaches the maximum at the center of basins. For the Cuu Long basin, the maximum value is σmax= 2.76 g/cm3 and the Nam Con Son has maximum value σmax= 3.0 g/cm3. Many faults that appear in the study area have the existence depth in the wide range from 6 km to 30 km, even above 30 km and the faults that have the existence depth from 8 to 10 km are in the majority. In particular, the boundary of anomalous sources existing in the Pre-Cenozoic basement is shown by the better resolution, at which more edge points are identified than the maximum horizontal gradient amplitude method that is widely used. The results also show that the combination of individual results complements each other and creats the sufficient and clearer picture inside of the Pre-Cenozoic basement.

A predicted-Newton's method for solving the interface positioning equation in the MoF method on general polyhedrons

Solving the interface positioning equation is the key procedure of the PLIC-VoF methods. Most of previous research only focused on the planar constant calculation and paid less attention to the relationship between the planar constant and the approximate interface orientation. The latter issue is important especially for the second order iteration based PLIC-VoF method, such as the MoF and LVIRA method. In these methods, the most accurate interface orientation is calculated through an iterative procedure, so the interface positioning equation has to be solved multiple times for the given volume fraction with different interface orientations. In this situation, if the incremental relation between the planar constant and the interface orientation is known, a predicted planar constant can be estimated. In this paper, we deduce the analytical partial derivatives of the planar constant with respect to the interface orientation and use them to predict the planar constant. A predicted-Newton's method is proposed to solve the interface positioning equation which takes the predicted planar constant as the initial guess. A great deal of numerical tests are also presented in this paper to verify the robustness of the new scheme. The efficiency of the proposed predicted-Newton's method is compared with the commonly used secant/bisection method by Ahn and Shashkov, and the numerical results indicate that the new method can reduce the iteration steps by 60%∼66% in solving the interface positioning equation and reduce the CPU time by 32%∼39% when implemented in the MoF method.

Starry: Analytic Occultation Light Curves

Abstract We derive analytic, closed form, numerically stable solutions for the total flux received from a spherical planet, moon, or star during an occultation if the specific intensity map of the body is expressed as a sum of spherical harmonics. Our expressions are valid to arbitrary degree and may be computed recursively for speed. The formalism we develop here applies to the computation of stellar transit light curves, planetary secondary eclipse light curves, and planet–planet/planet–moon occultation light curves, as well as thermal (rotational) phase curves. In this paper, we also introduce starry, an open-source package written in C++ and wrapped in Python that computes these light curves. The algorithm in starry is six orders of magnitude faster than direct numerical integration and several orders of magnitude more precise. starry also computes analytic derivatives of the light curves with respect to all input parameters for use in gradient-based optimization and inference, such as Hamiltonian Monte Carlo (HMC), allowing users to quickly and efficiently fit observed light curves to infer properties of a celestial body’s surface map. (Please see https://github.com/rodluger/starry, https://rodluger.github.io/starry/, and https://doi.org/10.5281/zenodo.1312286).

Constrained Burn Optimization for the International Space Station

Long-term trajectory planning for the International Space Station (ISS) uses simple rules of thumb embedded in a manual framework to target translation burns. This approach is limited by coarse knowledge of the problem sensitivities, it lacks automation, and it is unable to optimize these burns to minimize total . These issues can be resolved by casting the ISS trajectory planning problem as a constrained burn optimization problem. Unfortunately, the six-month to two-year time frames for long-term ISS trajectory planning preclude the use of numeric partial derivatives for accurate constraint sensitivities. A framework is presented to solve the constrained burn optimization problem for the ISS in a robust, automated fashion. The cost function for this problem is total , with constraints imposed on longitude of ascending node and semimajor axis altitude. The framework uses analytic derivatives for the cost function and variational equations using state transition matrices for the constraints. A local minimum solution is found using this framework, assuming fixed dates for the burns. Burn dates are then randomly varied to find other local solutions, which are searched for a candidate global minimum solution. This process is shown to work on real-world ISS burn-planning problems.

NMR Shielding Tensors and Chemical Shifts in Scalar-Relativistic Local Exact Two-Component Theory.

An efficient formulation of scalar-relativistic NMR shielding tensors based on (one-electron) spin-free exact two-component theory (X2C) is presented. It utilizes the diagonal local approximation to the unitary decoupling matrix (DLU), which we recently applied to analytical derivatives [ J. Chem. Phys. 2018 , 148 , 104110 ]. This allows for routine calculations of large molecules with heavy atoms. Here, the computation times of the NMR shielding tensors of all nuclei formally scale cubically with the size of the system, while memory demands increase quadratically. Efficiency is demonstrated for heavy-element clusters and organometallic complexes with more than 120 atoms and 3200 contracted basis functions. The accuracy of the DLU scheme is evaluated based on 13C, 17O, 29Si, 73Ge, 119Sn, 129Xe, 183W, and 207Pb NMR shielding constants and chemical shifts using different basis sets. The finite nucleus model is available throughout.

Anharmonic excited state frequencies of para-difluorobenzene, toluene and catechol using analytic RI-CC2 second derivatives.

Analytic second nuclear derivatives for excited electronic state energies have been implemented for the resolution-of-the-identity accelerated CC2, CIS(D∞) and ADC(2) models. Our efficient implementation with O(N2) memory demands enables the treatment of medium sized molecules with large basis sets and high numerical precision and thereby paves the way for semi-numerical evaluation of the higher-order derivatives required for anharmonic corrections to excited state vibrational frequencies. We compare CC2 harmonic and anharmonic excited state frequencies with experimental values for para-difluorobenzene, toluene and catechol. Basis set problems occur for out-of-plane bending vibrations due to intramolecular basis set superposition error. For non-planar molecules and in plane modes of planar molecules, the agreement between theory and experiment is better than 30 cm-1 on average and we reassign a number of experimental bands on the basis of the ab initio predictions.

A new method to analyze and understand molecular linear and nonlinear optical responses via field-induced functions: a straightforward alternative to sum-over-states (SOS) analysis.

The sum-over-states (SOS) method allows the computation of polarizabilities and hyperpolarizabilities additively from the contributions of different electronic excited states in a given molecule or cluster. Subsequent analysis of the main excited configurations contributing to the relevant excited states allows characterizing the orbitals involved in the linear and nonlinear optical response. Unfortunately, the chemically relevant information that can be obtained by SOS is hindered by a series of methodological and computational drawbacks. Among these drawbacks, we can highlight the high computational cost, problems arising from nonconvergent series and errors caused by the inaccurate description of excitation energies and/or higher excited state matrix elements. For this reason, coupled-perturbed schemes are currently widely used to determine the NLO potential of molecules and materials. However, such a choice limits the amount of intuitive chemical information that, on the other hand, can be retrieved by a successful SOS computation. In this work, we present and discuss a novel computational strategy that offers the means to extract the useful chemical insights from a coupled-perturbed calculation at almost negligible extra computational cost providing a transparent picture about orbital contributions to the properties of interest. The proposed method is based on the generation and further analysis of field-induced orbitals, FIOs, from the analytic or numerical derivatives of the dipole moment. Orbital symmetry rules are derived using group theory and the method is tested for a series of small and medium size systems.

Application of spline interpolation to speckle shearography measurements for damage identification

This paper aims to explore a new technique for structural damage identification using cubic spline interpolation. The method is based on the interpolation of modal rotations measured with speckle shearography. In order to locate the damaged areas, we make use of the analytical derivative of the cubic spline function to compute the modal curvature, which is known to be very sensitive to damage. A comprehensive parametric study of the spatial sampling interval is carried out to find its influence on noise filtering. Furthermore, the identification quality dependency on the mode shape and respective noise is also examined. The results obtained with the proposed method show the consistency of the localizations. Additionally, the challenging tasks of identifying small and multiple damage are tackled, yielding a good performance.

An investigation of initialisation strategies for dynamic temperature optimisation in beer fermentation

A wide range of optimisation methodologies exist for solving optimal control trajectory problems. With most approaches it is necessary to solve iteratively, starting from an initialising solution (often referred to as an initial guess). In this paper we investigate the performance of two different dynamic optimisation strategies for batch beer fermentation temperature control, using different initialisation profiles. A sequential method, Control Vector Parameterisation (CVP), is demonstrated as unable to produce solution profiles satisfying the industrially imposed undesirable by-product species concentration constraints. An alternative simultaneous method, Complete Parameterisation (CP), is employed in order to determine solutions satisfying these essential industrial production constraints. Blind initialisation guesses (isothermal profiles) have been shown to produce solution profiles not suitable for implementation on real fermentors; more promising candidate profile initialisations yield superior solutions. The use of a state-of-art NLP solver (IPOPT) with analytical first derivatives achieves remarkable solution robustness, eliminating initialisation and discretisation level dependence.

A new efficient algorithm to determine three-phase equilibrium conditions in the presence of aqueous phase: Phase stability and computational cost

Three-phase flash calculation is a crucial part of the modeling and simulation of the transport phenomenon in various chemical and petroleum processes. To obtain the compositions of phases in equilibrium, utilization of Equations of State (EOSs) and thermodynamic phase equilibria rules/laws (e.g., Henry's law, Lewis/Randall rule, and modified Raoult's law) seem inevitable to model phase behaviors of liquid, vapor, and aqueous phases. This study plans to develop two thermodynamic modeling approaches (on the basis of Henry's law and EOS) to conduct three-phase flash calculations in the presence of water. It is found that both developed algorithms are capable of determining the equilibrium conditions in the three-phase region such that a desired precision is obtained. In this research study, a new stability algorithm based on a negative flash procedure is introduced to forecast the formation and/or condition of the stable phases. The modeling results reveal that the aqueous phase equilibrium calculations by using EOS lead to an inaccurate prediction of stable phases in a vicinity of phase change regions. Improving the accuracy and reliability of the phase equilibria calculations, a new analytical derivative of fugacity coefficient with respect to mole fraction is obtained so that a better convergence and less runtime (compared to numerical derivatives) are achieved. An appropriate method to guess the initial equilibrium ratio is presented in this research investigation. The best roots of the cubic EOS are then selected for all three phases by minimizing Gibbs free energy. The proposed procedure can be implemented in different multiphase systems (and simulation softwares) to increase the accuracy of multiphase equilibrium calculations with an acceptable computational cost.

A new approach to speed up the conjugate gradient method by applying the complex variable differentiation method

ABSTRACTThis study represents a novel approach to speed up the solution of nonlinear inverse heat conduction problems (IHCPs) by the implementation of the complex variable differentiation method (CVDM). A difficulty appeared in the solution of nonlinear problems is the great amount of computational time. To handle this problem, a new scheme is introduced to improve the conjugate gradient method (CGM). The main contribution in the improvement of conventional CGM is the simultaneous solution of direct and sensitivity problems by applying the complex variable method. The derivation of sensitivity problem by utilizing the analytic derivative is difficult or impossible, since the direct problem is complicated mathematically in many cases. By this approach, the analytic derivation of sensitivity problem is circumvented, while the Jacobian matrix components are obtained accompanied by the solution of the direct problem. Therefore, the developed scheme results in the reduction of mathematical manipulations. Due to the high nonlinearity of moving boundary inverse problems, the ablation problem is considered as the benchmark to examine the accuracy and effectiveness of the proposed scheme. The simulated results illustrate that the developed scheme has the potential to significantly reducing of the computational expenses while maintaining the quality of numerical solution.

Selective guided sampling with complete light transport paths

Finding good global importance sampling strategies for Monte Carlo light transport is challenging. While estimators using local methods (such as BSDF sampling or next event estimation) often work well in the majority of a scene, small regions in path space can be sampled insufficiently (e.g. a reflected caustic). We propose a novel data-driven guided sampling method which selectively adapts to such problematic regions and complements the unguided estimator. It is based on complete transport paths, i.e. is able to resolve the correlation due to BSDFs and free flight distances in participating media. It is conceptually simple and places anisotropic truncated Gaussian distributions around guide paths to reconstruct a continuous probability density function (guided PDF). Guide paths are iteratively sampled from the guided as well as the unguided PDF and only recorded if they cause high variance in the current estimator. While plain Monte Carlo samples paths independently and Markov chain-based methods perturb a single current sample, we determine the reconstruction kernels by a set of neighbouring paths. This enables local exploration of the integrand without detailed balance constraints or the need for analytic derivatives. We show that our method can decompose the path space into a region that is well sampled by the unguided estimator and one that is handled by the new guided sampler. In realistic scenarios, we show 4× speedups over the unguided sampler.

Practical constrained dynamic positioning control for uncertain ship through the minimal learning parameter technique

In this note, a novel adaptive constrained control algorithm is proposed for dynamic positioning vessels in the presence of input amplitude and rate saturations. In the scheme, both kinematics and kinetics loops are augmented by the auxiliary system to deal with the saturation impact. Meanwhile, for merits of the command filters and minimal leering parameter technique, the cumbersome analytic derivatives of the virtual control law need not be required in the control design, and only two adaptive parameters are updated online to stabilise the effect of the model uncertainty and the external disturbance for each subsystem. Using the Lyapunov theory, the semi-global uniformly ultimately bounded is proved for the closed-loop system. The effectiveness and robustness of the proposed algorithm are demonstrated by the numerical example in the presence of the simulated marine disturbance.

Stress integration-based on finite difference method and its application for anisotropic plasticity and distortional hardening under associated and non-associated flow rules

Stress integration algorithm based on finite difference method (FDM) was proposed to effectively deal with both first and second derivatives of yield and potential functions which are the lengthiest component in stress integration procedure. With the proposed numerical algorithm, both first and second derivatives of yield function are approximated by central difference method, so that finite element modeling using advanced constitutive model could be easily performed no matter how complicated its derivatives are. For the verification purpose, the algorithm was applied for advanced constitutive models: Plastic anisotropy model under associated (AFR) and non-associated flow rule (non-AFR), the homogeneous anisotropic hardening (HAH) model under associated (AFR) and non-associated flow rule (non-AFR). The proposed algorithm was verified with single element loading–unloading and cup-drawing simulations. The Euler backward method based on both the proposed numerical algorithm and analytical derivatives were employed for verification purpose. The accuracy and time efficiency of the proposed numerical algorithm were evaluated by comparing the simulation results from analytical derivatives. In addition, the applicability of the proposed numerical algorithm for the HAH models was estimated with single element loading–unloading, loading–reloading, and deep-drawing/springback simulations. Non-associated flow plasticity for the HAH model is newly proposed to improve numerical efficiency with finite difference method by keeping the same level of accuracy as associated flow rule plasticity. All the simulation results proved that the proposed numerical algorithm can be widely used for the implementation of advanced constitutive models.

Density functionals for nondynamical correlation constructed from an upper bound to the exact exchange energy density

ABSTRACTHyper-generalised-gradient approximations (hGGAs) for the exact exchange-correlation functional are increasingly popular in density functional theory. HGGAs model nondynamical correlation using a flexible local combination of exact (Hartree–Fock, HF) exchange and approximate exchange. We present a simplified ‘Rung 3.5’ upper bound to the HF exchange energy density, the essential ingredient of hGGAs. We also present a nonempirical generalised gradient approximation for this upper bound. Both upper bounds go to zero in the high-density and density tail limits, facilitating the construction of hGGAs that recover HF exchange in these limits. The ‘Rung 3.5’ construction enables facile evaluation of analytic derivatives and calculations in periodic boundary conditions. Extensive numerical tests show that the upper bounds capture the critical difference between HF and approximate exchange, showing these ingredients' promise for building simple hGGAs. The tests also indicate a need for more sophisticated semi-local upper bounds.

Acceleration of Gradient-Based Algorithms for Array Antenna Synthesis With Far-Field or Near-Field Constraints

This paper presents a technique for the acceleration of gradient-based algorithms that employ finite differences in the calculation of the gradient for the optimization of array antennas. It is based on differential contributions, which takes advantage of the fact that when an array is optimized, each element is analyzed independently of the rest. Thus, the computation of the gradient of the cost function, which is typically the most time-consuming operation of the algorithm, can be accelerated. A time cost study is presented and the technique is implemented, as an example, in the generalized intersection approach algorithm for array optimization in near and far fields. Several syntheses are performed to assess the improvement of this technique. In the far field, it is compared with periodic and aperiodic arrays using different approaches for the computation of the gradient, including the analytic derivative. A reflectarray is also optimized in the near field with the goal of improving its quiet zone. The technique of differential contributions shows the important reductions in the time per iteration in all three syntheses, especially in that of aperiodic arrays and near-field optimization, where the time saved in the evaluation of the gradient is greater than 99%.