Analytical Derivatives Research Articles

Effects of the Geometric Uncertainties of Bladed Disks Through the Analytical Derivatives of the Finite Element Matrices

Effects of the Geometric Uncertainties of Bladed Disks Through the Analytical Derivatives of the Finite Element Matrices

Powder-diffraction-based structural comparison for crystal structure prediction without prior indexing

The objective of crystal structure prediction (CSP) is to predict computationally the thermodynamically stable crystal structure of a compound from its stoichiometry or its molecular diagram. Crystal similarity indices measure the degree of similarity between two crystal structures and are essential in CSP because they are used to identify duplicates. Powder-based indices, which are based on comparing X-ray diffraction patterns, allow the use of experimental X-ray powder diffraction data to inform the CSP search. Powder-assisted CSP presents two unique difficulties: (i) the experimental and computational structures are not entirely comparable because the former is subject to thermal expansion from lattice vibrations, and (ii) experimental patterns present features (noise, background contribution, varying peak shapes etc.) that are not easily predictable computationally. This work presents a powder-based similarity index (GPWDF) based on a modification of the index introduced by de Gelder, Wehrens & Hageman [J. Comput. Chem. (2001), 22, 273–289] using cross-correlation functions that can be calculated analytically. Based on GPWDF, a variable-cell similarity index (VC-GPWDF) is also proposed that assigns a high similarity score to structures that differ only by a lattice deformation and which takes advantage of the analytical derivatives of GPWDF with respect to the lattice parameters. VC-GPWDF can be used to identify similarity between two computational structures generated using different methods, between a computational and an experimental structure, and between two experimental structures measured under different conditions (e.g. different temperature and pressure). VC-GPWDF can also be used to compare crystal structures with experimental patterns in combination with an automatic pre-processing step. The proposed similarity indices are simple, efficient and fully automatic. They do not require indexing of the experimental pattern or a guess of the space group, they account for deformations caused by varying experimental conditions, they give meaningful results even when the experimental pattern is of very poor quality, and their computational cost does not increase with the flexibility of the molecular motif.

Analytical gradients of first-order diffraction and radiation forces for design optimization of floating structures

Gradient-based design optimization of floating structures with many design variables requires efficient and accurate computation of hydrodynamic coefficients including wave excitation forces and their derivatives with respect to design variables. For large-volume structures or structures with many component members, the only practical method to determine diffraction and radiation forces is to apply a boundary element method solver. This work presents the first known boundary element method solver with implicit analytic derivatives, allowing for total derivative computation alongside force evaluation. Two case studies are presented: a single floating circular column and a structure with multiple square columns attached to a horizontal pontoon. Force results from both case studies agree well with reference data, while derivative results agree with the best comparisons available. Computational requirements limit the applicability of this method and suggest further work is required to refine this approach.

Dxtb-An efficient and fully differentiable framework for extended tight-binding.

Automatic differentiation (AD) emerged as an integral part of machine learning, accelerating model development by enabling gradient-based optimization without explicit analytical derivatives. Recently, the benefits of AD and computing arbitrary-order derivatives with respect to any variable were also recognized in the field of quantum chemistry. In this work, we present dxtb-an open-source, fully differentiable framework for semiempirical extended tight-binding (xTB) methods. Developed entirely in Python and leveraging PyTorch for array operations, dxtb facilitates extensibility and rapid prototyping while maintaining computational efficiency. Through comprehensive code vectorization and optimization, we essentially reach the speed of compiled xTB programs for high-throughput calculations of small molecules. The excellent performance also scales to large systems, and batch operability yields additional benefits for execution on parallel hardware. In particular, energy evaluations are on par with existing programs, whereas the speed of automatically differentiated nuclear derivatives is only 2 to 5 times slower compared to their analytical counterparts. We showcase the utility of AD in dxtb by calculating various molecular and spectroscopic properties, highlighting its capacity to enhance and simplify such evaluations. Furthermore, the framework streamlines optimization tasks and offers seamless integration of semiempirical quantum chemistry in machine learning, paving the way for physics-inspired end-to-end differentiable models. Ultimately, dxtb aims to further advance the capabilities of semiempirical methods, providing an extensible foundation for future developments and hybrid machine learning applications. The framework is accessible at https://github.com/grimme-lab/dxtb.

Exchange-correlation kernel for perturbation dependent auxiliary functions in auxiliary density perturbation theory

ContextAnalytic exchange-correlation kernel formulations are of the outermost importance for density functional theory (DFT) perturbation calculations. In this paper, the working equation for the exchange-correlation kernel of the generalized gradient approximation (GGA) for perturbation dependent auxiliary functions is derived and discussed in the framework of auxiliary density functional theory (ADFT). The presented new formulation is extended to the unrestricted approach, too. A comprehensive discussion of the implementation of the GGA ADFT kernel, using either the native exchange-correlation functional implementations in deMon2k or the ones from the LibXC library, is given. Calculations with analytic exchange-correlation kernels are compared to their finite difference counterparts. The obtained results are in quantitative agreement. Nevertheless, analytic GGA ADFT kernel implementations show substantial improvement in the computational performance. Similar results are reported for analytic second derivatives of effective core potential (ECP) and model core potential (MCP) matrix elements when compared to their finite difference counterparts in molecular frequency analyses.MethodAll calculations are performed in the framework of ADFT as implemented in deMon2k. In the ADFT analytic frequency calculations, auxiliary density perturbation theory was used. The underlying two-center exchange-correlation kernel matrix elements are calculated by numerical integration either with analytic or finite difference kernel expressions. Validation calculations are performed with the VWN and PBE functionals employing DFT-optimized DZVP basis sets in conjunction with automatically generated GEN-A2 auxiliary density function sets. In the (Pt3Cu)n cluster benchmark calculations, the RPBE functional was used. For Pt atoms, the quasi-relativistic LANL2DZ effective core potential with the corresponding valence basis set was employed, whereas for Cu atoms, the all-electron DFT-optimized TZVP basis was applied. The auxiliary density was expanded by the automatically generated GEN-A2* auxiliary function set. We run all benchmark calculations in parallel on 24 cores.

Analytic Gradients for Density Fitting MP2 Using Natural Auxiliary Functions.

The natural auxiliary function (NAF) approach is an approximation to decrease the size of the auxiliary basis set required for quantum chemical calculations utilizing the density fitting technique. It has been proven efficient to speed up various correlation models, such as second-order Møller-Plesset (MP2) theory and coupled-cluster methods. Here, for the first time, we discuss the theory of analytic derivatives for correlation methods employing the NAF approximation on the example of MP2. A detailed algorithm for the gradient calculation with the NAF approximation is proposed in the framework of the method of Lagrange multipliers. To assess the effect of the NAF approximation on gradients and optimized geometric parameters, a series of benchmark calculations on small and medium-sized systems was performed. Our results demonstrate that, for MP2, sufficiently accurate gradients and geometries can be achieved with a moderate time reduction of 15-20% for both small and medium-sized molecules.

Modeling Medium-Term Debris Cloud of Satellite Breakup via Probabilistic Method

Accurately modeling the evolution of debris clouds from satellite breakups is necessary for assessing the collision risk to space assets. Generally, the evolution of debris clouds can be divided into short-, medium-, and long-term phases. Compared to the short- and long-term phases, the study on the medium-term phase was insufficient. This paper proposes a probabilistic method to model the medium-term evolution of a debris cloud. First, a satellite breakup event can be characterized by a probabilistic density distribution of the initial velocity vectors of all fragments. Then, the distribution in the initial velocity can be transformed to that in three classic orbital elements, i.e., semimajor axis, eccentricity, and inclination. Given that the driving force of medium-term evolution is the J2 perturbation and that J2 has no secular contribution to these three classic orbital elements, the distribution of classic orbital elements remains stationary during the evolution. This feature can be exploited to express the probabilistic density distribution of any given target position in terms of the density distribution of the classic orbital elements by changing variables. Finally, 16 one-variable equations with analytical first derivatives were formulated to find the proper inverse mapping from the given position to the classic orbital elements, and the Jacobian matrix of the inverse mapping was derived. Medium-term debris clouds resulting from two typical breakup velocity distributions were tested using the proposed method. Results show that the proposed method can accurately simulate the evolution of the clouds’ geometrical extent and density flow.

User friendly FE Formulation for anisotropic distortional hardening model based on non-associated flow plasticity and its application to springback prediction

Based on non-associated flow plasticity, a newly developed anisotropic distortional hardening model developed by Hu and Yoon [15] is implemented in finite element analysis in a user-friendly manner. The derivatives of complex hardening models are calculated using the Finite Difference Method (FDM), which is much more convenient than using the analytical derivatives. To further improve the accuracy of the proposed method, the step size analysis in FDM is performed by analyzing the derivative formation. To evaluate the accuracy and computational efficiency of a proposed step size for FDM, single element simulations are performed under different loading paths. It has been found that the maximum absolute error of the flow curves between the simulation and the theoretical result is less than 0.3%. The U-bending tests for DP600 and TRIP1180 are used to verify the ability of the distortional hardening model for springback prediction. The simulation result of the strain hardening model is in good agreement with the experiment. The computational efficiency is also increased by 24% due to the improved convergence rate.

Picking up Good Vibrations through Quartic Force Fields and Vibrational Perturbation Theory.

Quartic force fields (QFFs) define sparse potential energy surfaces (compared to semiglobal surfaces) that are the cheapest and easiest means of computing anharmonic vibrational frequencies, especially when utilized with second-order vibrational perturbation theory (VPT2). However, flat and shallow potential surfaces are exceedingly difficult for QFFs to treat through a combination of numerical noise in the often numerically computed derivatives and in competing energy factors in the composite energies often utilized to provide high-level spectroscopic predictions. While some of these issues can be alleviated with analytic derivatives, hybrid QFFs, and intelligent choices in coordinate systems, the best practice is for predicting good molecular vibrations via QFFs is to understand what they cannot do, and this manuscript documents such cases where QFFs may fail.

Current density functional framework for spin-orbit coupling: Extension to periodic systems.

Spin-orbit coupling induces a current density in the ground state, which consequently requires a generalization for meta-generalized gradient approximations. That is, the exchange-correlation energy has to be constructed as an explicit functional of the current density, and a generalized kinetic energy density has to be formed to satisfy theoretical constraints. Herein, we generalize our previously presented formalism of spin-orbit current density functional theory [Holzer et al., J. Chem. Phys. 157, 204102 (2022)] to non-magnetic and magnetic periodic systems of arbitrary dimension. In addition to the ground-state exchange-correlation potential, analytical derivatives such as geometry gradients and stress tensors are implemented. The importance of the current density is assessed for band gaps, lattice constants, magnetic transitions, and Rashba splittings. In the latter, the impact of the current density may be larger than the deviation between different density functional approximations.

Analytic Gradients for the Electrostatic Embedding QM/MM Model in Periodic Boundary Conditions Using Particle-Mesh Ewald Sums and Electrostatic Potential Fitted Charge Operators.

Long-range electrostatic effects are fundamental for describing chemical reactivity in the condensed phase. Here, we present the methodology of an efficient quantum mechanical/molecular mechanical (QM/MM) model in periodic boundary conditions (PBC) compatible with QM/MM boundaries at chemical bonds. The method combines electrostatic potential fitted charge operators and electrostatic potentials derived from the smooth particle-mesh Ewald (PME) sum approach. The total energy and its analytic first derivatives with respect to QM, MM, and lattice vectors allow QM/MM molecular dynamics (MD) in the most common thermodynamic ensembles. We demonstrate the robustness of the method by performing a QM/MM MD equilibration of methanol in water. We simulate the cis/trans isomerization free-energy profiles in water of proline amino acid and a proline-containing oligopeptide, showing a correct description of the reaction barrier. Our PBC-compatible QM/MM model can efficiently be used to study the chemical reactivity in the condensed phase and enzymatic catalysis.

Formulation of transition dipole gradients for non-adiabatic dynamics with polaritonic states.

A general formulation of the strong coupling between photons confined in a cavity and molecular electronic states is developed for the state-interaction state-average spin-restricted ensemble-referenced Kohn-Sham method. The light-matter interaction is included in the Jaynes-Cummings model, which requires the derivation and implementation of the analytical derivatives of the transition dipole moments between the molecular electronic states. The developed formalism is tested in the simulations of the nonadiabatic dynamics in the polaritonic states resulting from the strong coupling between the cavity photon mode and the ground and excited states of the penta-2,4-dieniminium cation, also known as PSB3. Comparison with the field-free simulations of the excited-state decay dynamics in PSB3 reveals that the light-matter coupling can considerably alter the decay dynamics by increasing the excited state lifetime and hindering photochemically induced torsion about the C=C double bonds of PSB3. The necessity of obtaining analytical transition dipole gradients for the accurate propagation of the dynamics is underlined.

Consistent Analytical Second Derivatives of the Kohn-Sham DFT Energy in the Framework of the Conductor-Like Screening Model through Gaussian Charge Distributions.

The use of implicit solvation models such as the conductor-like screening model (COSMO) in quantum chemical calculations is very common, as both a rough estimate of solvation effects as well as a general tool for stabilizing ionic molecular structures. In order to generate a smooth potential energy surface as well as consistent gradients, it is necessary to apply the Gaussian charge model (GCM) for the COSMO charges. This work introduces an efficient implementation for consistent analytical second derivatives of the electronic energy with COSMO-GCM in the framework of the Kohn-Sham density functional theory. This is used to investigate the infrared spectroscopy of amino acids in aqueous solution, where the impact of pH on the molecular structure and vibrational spectra is examined. Furthermore, the structure and stability of selected all-metal aromatic cluster ions are assessed.

Vee-Tail Preliminary Design Methodology for Class I mini- UAV

Unmanned Air Vehicles (UAVs) are becoming increasingly popular and widely used in a variety of industries. They can be used for tasks such as agriculture, construction, delivery, surveillance, rescue operations, mapping, wildlife tracking and many more. With the advancements in technology, UAVs are becoming more autonomous and able to perform tasks with minimal human intervention, so are widely used for military and law enforcement purposes. V-tail configurations are commonly used on UAVs due to their advantages in control and stability performance, as well as their ability to reduce drag and improve overall efficiency. However, research on V-tail design and sizing is limited, particularly for Class I mini-UAVs. The objective of this paper is to identify a methodology for a V-tail sizing of a Class I Mini UAV (NATO classification), which refers to the Conceptual and Preliminary Design of the UAV. The methodology will follow the design of a V-tail from the characteristics of the conventional tail of the UAV. Once the characteristics of the conventional tail were extracted, V-tail geometric characteristics (reference area, aspect ratio, mean aerodynamic chord, tail span, dihedral angle), were computed. Therefore, the aerodynamic characteristics of the V-tail have to be extracted, first as an isolated tail, and then as an installed tail. The stability derivatives of the V-tail are then calculated. The methodology for the analytical aerodynamic characteristics and stability derivatives, is a combination of NACA Report No.823 and Marcello R. Napolitano methodologies. Paul E. Purser and John P. Campbell provide design methods for V-tail on a NACA report, which include some of the desired stability derivatives. The rest of them will be calculated with Napolitano’s method. Marcello R. Napolitano gives a methodology for conventional tail sizing; thus, the equations of its methodology have to convert for a V-tail configuration. Furthermore, the aerodynamic characteristics and stability derivatives of the designed V-tail will be verified by Low Fidelity Aerodynamics simulation (XFLR5 software), and then by High Fidelity Aerodynamics by means of CFD. The results between low fidelity analytical values and High-Fidelity Aerodynamics values indicate a relative error lower than 20%.

The Impact of Second-Order Analytical Derivatives on Regularized Direct Multiple Shooting Method for Impulsive Spacecraft Trajectory Optimization

While the first-order derivatives of an objective function and constraints have often been analytically provided in the gradient-based optimization algorithms, the joint use of the second-order derivatives can further improve the computational efficiency and robustness. This paper implements the first- and second-order analytical derivatives in the direct multiple shooting-based, regularized method of minimizing the fuel expenditure for impulsive space trajectories. The high-order dynamical information expresses the Hessian matrix of the Lagrange function in the nonlinear programming problem. The result is an efficient tool for robustly computing fuel-optimal, multi-impulse trajectories in the regularized framework of removing singularities associated with null thrust impulses from the derivatives of the objective function. The computational performance is compared for the cases of optimizing impulsive transfers between cislunar periodic orbits within the regularized framework implementing the analytical derivatives in different ways over various initial guess trajectories and computational conditions. The results indicate the superiority of adopting both the first- and second-order analytical derivatives in terms of the efficiency and robustness.

The Studying of Magnetic Anomalies Due to Shallow Underground Targets and the Environmental Applications.

The magnetic method of investigation is a well-known and dependable technique for detecting underground ferro-materials. However, the application of the magnetic method to identify the exact position of underground targets such as buried explosive devices (BED), illegally buried waste (IBW), metallic storage tanks, utility pipes, and communication cables is quite challenging in both civil and environmental engineering. This article presents the results of the investigation performed at a 55m by 55m area within the Ahmadu Bello University Geophysical Test Site (ABUGTS) to study the signals generated by similar buried targets to simulate situations encountered in environmental engineering. The total magnetic intensity (TMI) of the site without buried targets was obtained to identify the possible existing underground materials or faults that could generate false geophysical data. The magnetic susceptibility of the buried targets was determined in the laboratory before the burial to validate the field results. The influence of the buried materials over the magnetic signature of the site was examined, showing the TMI values vary between 34966 nT and 44510 nT before burial and 351607 nT to 49795 nT after burial. The Euler deconvolution (ED) and residual maps alongside the signal enhancement maps like analytical signal and first vertical derivative (FVD) were employed to study the magnetic anomalies due to buried targets. The results of the data interpretation confirmed the existence of cluster anomaly source bodies due to buried targets. The ED delineates the depth of the buried targets varying between 0.7m – 3.3m corresponding with the actual depth of burial. The analytical signal map produced a perfect behaviour of the magnetic anomalies to buried targets, capturing all the buried targets buried at the site. However, the shape of the anomaly signal is almost independent of the buried target orientation with a spherical pattern of anomalies. However, the exact size and shape of the buried targets were not achieved. Nevertheless, the FVD and analytical signal maps perform better than the residual and TMI maps due to the substantially smaller magnetic anomalies of the buried targets similar to expected. In addition, stronger magnetic anomaly signals occur maximally around buried larger ferro-materials with precise prediction of the buried target position. The magnetic method is a very powerful, convenient, cheap, and time-saving geophysical tool for investigating underground targets, which is highly applicable in engineering and environmental contexts.

Multivariate Density Estimation by Neural Networks.

We propose nonparametric methods to obtain the Probability Density Function (PDF) to assess the properties of the underlying data generating process (DGP) without imposing any assumptions on the DGP, using neural networks (NNs). The proposed NN has advantages compared to well-known parametric and nonparametric density estimators. Our approach builds on literature on cumulative distribution function (CDF) estimation using NN. We extend this literature by providing analytical derivatives of this obtained CDF. Our approach hence removes the numerical approximation error in differentiating the CDF output, leading to more accurate PDF estimates. The proposed solution applies to any NN model, i.e., for any number of hidden layers or hidden neurons in the multilayer perceptron (MLP) structure. The proposed solution applies the PDF estimation by NN to continuous distributions as well as discrete distributions. We also show that the proposed solution to obtain the PDF leads to good approximations when applied to correlated variables in a multivariate setting. We test the performance of our method in a large Monte Carlo simulation using various complex distributions. Subsequently, we apply our method to estimate the density of the number of vehicle counts per minute measured with road sensors for a time window of 24 h.

Finite difference based stress integration algorithm for crystal plasticity finite element method

In this study, we present a Finite Difference Method (FDM)-based stress integration algorithm for Crystal Plasticity Finite Element Method (CPFEM). It addresses the complexity of computing the first derivative of resolved shear stress in the Euler backward stress integration algorithm with Newton-Raphson method. The proposed FDM-based model was verified by evaluating its accuracy, convergence and computational efficiency through single-element simulations. The developed FDM-based model can be easily applied to various constitutive models for CPFEM, overcoming the problem of deriving complex derivative regardless of constitutive models. Additionally, the proposed FDM-based model was validated with the reduced texture approach using AA 2090-T3. Specific parameters including crystallographic orientations were calibrated and the plastic anisotropy was successfully described. In addition, the earing profiles were compared using various stress integration methods. As a result, the proposed FDM-based model can be used as an alternative to the Euler backward method using analytic derivatives with the compatible accuracy, convergence, computational efficiency along with easy implementation within the CPFEM framework.

Weighted Hermite Variable Projection Networks for Classifying Visually Evoked Potentials.

The occipital cortex responds to visual stimuli regardless of a patient's level of consciousness or attention, offering a noninvasive diagnostic tool for both ophthalmologists and neurologists. This response signal manifests as a unique waveform referred to as the visually evoked potential (VEP), which can be extracted from the electroencephalogram (EEG) activity of a human being. We propose a trainable VEP representation to disentangle the underlying explanatory factors of the data. To enhance the learning process with domain knowledge, we present an innovative parameterization of classical Hermite functions that effectively captures VEP pattern variations arising from patient-specific factors, disorders, and measurement setup influences. Then, we introduce a differentiable variable projection (VP) layer to fuse Hermite basis function expansions (BFEs) of VEP signals with machine learning (ML) approaches. We prove the existence of an optimal set of parameters in the least-squares sense, assess the representation power of such layers, and calculate their analytical derivatives, which allows us to utilize backpropagation for training. Finally, we evaluate the effectiveness of the proposed learning framework in VEP-based color classification. To achieve this, we have designed a novel measurement system dedicated to intraoperative clinical use cases, which presents new ways for patient monitoring during neurosurgical procedures.

Z-Splat: Z-Axis Gaussian Splatting for Camera-Sonar Fusion.

Differentiable 3D-Gaussian splatting (GS) is emerging as a prominent technique in computer vision and graphics for reconstructing 3D scenes. GS represents a scene as a set of 3D Gaussians with varying opacities and employs a computationally efficient splatting operation along with analytical derivatives to compute the 3D Gaussian parameters given scene images captured from various viewpoints. Unfortunately, capturing surround view (360° viewpoint) images is impossible or impractical in many real-world imaging scenarios, including underwater imaging, rooms inside a building, and autonomous navigation. In these restricted baseline imaging scenarios, the GS algorithm suffers from a well-known 'missing cone' problem, which results in poor reconstruction along the depth axis. In this paper, we demonstrate that using transient data (from sonars) allows us to address the missing cone problem by sampling high-frequency data along the depth axis. We extend the Gaussian splatting algorithms for two commonly used sonars and propose fusion algorithms that simultaneously utilize RGB camera data and sonar data. Through simulations, emulations, and hardware experiments across various imaging scenarios, we show that the proposed fusion algorithms lead to significantly better novel view synthesis (5dB improvement in PSNR) and 3D geometry reconstruction (60% lower Chamfer distance).