Interpolation Scheme Research Articles

Direct integration of measured viscoelastic relaxation data in time-domain finite element simulations

The current approach to modeling viscoelastic materials in most commercial finite element packages is based on the General Maxwell Model, which views these materials as combinations of spring and dashpot elements. However, the data can be incorporated more directly into a transient finite element study by direct interpolation of the relaxation function. This work explores a linear interpolation scheme to the inclusion of viscoelastic relaxation functions on an example problem. The results show several benefits over the General Maxwell Model for transient studies. Included in the analysis are displacement solutions utilizing both approaches, relaxation function error calculations for both approaches, and parametric runtime studies comparing speed of calculation. The variation in computational flop counts is considered and an argument is made for the preference of the proposed approach.

High order compact augmented methods for Stokes equations with different boundary conditions

This paper is devoted to fourth order compact schemes and fast algorithms for solving stationary Stokes equations with different boundary conditions numerically. One of the main ideas is to decouple the Stokes equations into three Poisson equations for the pressure and the velocity via the pressure Poisson equation (PPE). The augmented strategy is utilized to provide numerical boundary conditions for the pressure. Different velocity boundary conditions require different interpolation strategies for the augmented methods. The augmented variable is solved by the GMRES method. A new simple and efficient preconditioning strategy has also been developed to accelerate the convergence of the GMRES iteration. Numerical examples presented in this paper confirmed the designed convergence order and the efficiency of the new methods.

Planar quartic G2 Hermite interpolation for curve modeling

We study planar quartic G2 Hermite interpolation, that is, a quartic polynomial curve interpolating two planar data points along with the associated tangent directions and curvatures. When the two specified tangent directions are non-parallel, a quartic Bézier curve interpolating such G2 data is constructed using two geometrically meaningful shape parameters which denote the magnitudes of end tangent vectors. We then determine the two parameters by minimizing a quadratic energy functional or curvature variation energy. When the two specified tangent directions are parallel, a quartic G2 interpolating curve exists only when an additional condition on G2 data is satisfied, and we propose a modified optimization approach. Finally, we demonstrate the achievable quality with a range of examples and the application to curve modeling, and it allows to locally create G2 smooth complex shapes. Compared with the existing quartic interpolation scheme, our method can generate more satisfactory results in terms of approximation accuracy and curvature profiles.

Efficient Modeling of Water Adsorption in MOFs Using Interpolated Transition Matrix Monte Carlo.

With the specter of accelerating climate change, securing access to potable water has become a critical global challenge. Atmospheric water harvesting (AWH) through metal-organic frameworks (MOFs) emerges as one of the promising solutions. The standard numerical methods applied for rapid and efficient screening for optimal sorbents face significant limitations in the case of water adsorption (slow convergence and inability to overcome high energy barriers). To address these challenges, we employed grand canonical transition matrix Monte Carlo (GC-TMMC) methodology and proposed an efficient interpolation scheme that significantly reduces the number of required simulations while maintaining accuracy of the results. Through the example of water adsorption in three MOFs: MOF-303, MOF-LA2-1, and NU-1000, we show that the extrapolation of the free energy landscape allows for prediction of the adsorption properties over a continuous range of pressure and temperature. This innovative and versatile method provides rich thermodynamic information, enabling rapid, large-scale computational screening of sorbents for adsorption, applicable for a variety of sorbents and gases. As the presented methodology holds strong applicative potential, we provide alongside this paper a modified version of the RASPA2 code with a ghost swap move implementation and a Python library designed to minimize the user's input for analyzing data derived from the TMMC simulations.

Contractive Multivariate Zipper Fractal Interpolation Functions

In this paper we introduce a new general multivariate fractal interpolation scheme using elements of the zipper methodology. Under the assumption that the corresponding Read-Bajraktarevic operator is well-defined, we enlarge the previous frameworks occurring in the literature, considering the constitutive functions of the iterated function system whose attractor is the graph of the interpolant to be just contractive in the last variable (so, in particular, they can be Banach contractions, Matkowski contractions, or Meir-Keeler contractions in the last variable). The main difficulty that should be overcome in this multivariate framework is the well definedness of the above mentioned operator. We provide three instances when it is guaranteed. We also display some examples that emphasize the generality of our scheme.

Nonlinear topology optimization of flexoelectric soft dielectrics at large deformation

We propose a novel nonlinear topology optimization framework tailored for flexoelectric soft dielectrics undergoing large deformation. A numerical method based on Isogeometric analysis (IGA) is introduced to nonlinear soft dielectrics at finite strain, ensuring the C1-continuity for flexoelectric problems. We outline the process of consistent linearizations and IGA discretizations. Additionally, we introduce an innovative and efficient Strain Density Function (SDF) interpolation scheme for optimizing electromechanical coupling factors (ECFs). In this scheme, the interpolation of electromechanical and hyper-elastic energy terms is grounded in the SIMP model, while a linear material interpolation model is employed for the dielectric energy. Our numerical analysis highlights the remarkable performance of the proposed SDF interpolation scheme in nonlinear electromechanical optimization scenarios. An energy remedy scheme is applied to void regions to eliminate the instability of nonlinear optimization, effectively preventing distortion deformations in low-stiffness elements. Furthermore, we investigate the influence of size effect and larger deformation on the optimization of flexoelectric soft materials. The proposed topology optimization framework adeptly leverages the interplay between size effect and larger deformation, resulting in a notable improvement in ECFs within the optimized structures. Finally, the electromechanical coupling factors (ECFs) of the optimized structures in all the examined cases exhibit enhancements up to 9 times as compared to the reference guess designs.

A Linear Interpolation and Curvature-Controlled Gradient Optimization Strategy Based on Adam

The Adam algorithm is a widely used optimizer for neural network training due to efficient convergence speed. The algorithm is prone to unstable learning rate and performance degradation on some models. To solve these problems, in this paper, an improved algorithm named Linear Curvature Momentum Adam (LCMAdam) is proposed, which introduces curvature-controlled gradient and linear interpolation strategies. The curvature-controlled gradient can make the gradient update smoother, and the linear interpolation technique can adaptively adjust the size of the learning rate according to the characteristics of the curve during the training process so that it can find the exact value faster, which improves the efficiency and robustness of training. The experimental results show that the LCMAdam algorithm achieves 98.49% accuracy on the MNIST dataset, 75.20% on the CIFAR10 dataset, and 76.80% on the Stomach dataset, which is more difficult to recognize medical images. The LCMAdam optimizer achieves significant performance gains on a variety of neural network structures and tasks, proving its effectiveness and utility in the field of deep learning.

Hermite kernel surrogates for the value function of high-dimensional nonlinear optimal control problems

Numerical methods for the optimal feedback control of high-dimensional dynamical systems typically suffer from the curse of dimensionality. In the current presentation, we devise a mesh-free data-based approximation method for the value function of optimal control problems, which partially mitigates the dimensionality problem. The method is based on a greedy Hermite kernel interpolation scheme and incorporates context knowledge by its structure. Especially, the value function surrogate is elegantly enforced to be 0 in the target state, non-negative and constructed as a correction of a linearized model. The algorithm allows formulation in a matrix-free way which ensures efficient offline and online evaluation of the surrogate, circumventing the large-matrix problem for multivariate Hermite interpolation. Additionally, an incremental Cholesky factorization is utilized in the offline generation of the surrogate. For finite time horizons, both convergence of the surrogate to the value function and for the surrogate vs. the optimal controlled dynamical system are proven. Experiments support the effectiveness of the scheme, using among others a new academic model with an explicitly given value function. It may also be useful for the community to validate other optimal control approaches.

Semi‐implicit material point method for simulating infiltration‐induced failure of unsaturated soil structures

AbstractThis study presents a semi‐implicit MPM to adequately characterize the mechanical behavior of unsaturated soil based on Biot's mixture theory. To represent the dependency of the degree of saturation on the suction, we employ the VG model along with a soil‐water characteristic curve, which determines a functional form of permeability called the Mualem model. Hencky's hyperelastic model and the Drucker‐Prager model assuming nonassociativity are adopted for elastic and plastic deformations, respectively. The novelty of this study is the incorporation of the fractional‐step method into the MPM framework so that the pore water pressure is obtained by implicitly solving the pressure Poisson's equation, which reduces numerical instability and improves computational efficiency. Also, because the drag force between solid and liquid phases is evaluated using the intermediate velocity of pore water relative to the intermediate velocity of solid skeleton, the time increment can be chosen without considering the magnitude of water permeability. In addition, to suppress “odd‐even” oscillation, we employ a sub‐grid method in which two grids with different spatial resolutions are used for the velocities and pore water pressure. Furthermore, considering the advantages and disadvantages of two different interpolation schemes for pore water pressure, we suggest switching the schemes depending on the model conditions. Several numerical examples are presented to demonstrate the performance of the proposed method. Specifically, unidirectional consolidation and leak flow analyses are performed for verification purposes, followed by validation analysis of a model experiment of infiltration‐induced landslides.

Acoustical Holography and Coherence-Based Noise Source Characterization of an Installed F404 Engine

Understanding the acoustic source characteristics of supersonic jets is vital to accurate noise field modeling and jet noise reduction strategies. This paper uses advanced, coherence-based partial field decomposition methods to characterize the acoustic sources in an installed, supersonic GE F404 engine. Partial field decomposition is accomplished using an equivalent source reconstruction via acoustical holography. Bandwidth is extended through the application of an array phase-unwrapping and interpolation scheme. The optimized-location virtual reference method is used. Apparent source distributions and source-related partial fields are shown as a function of frequency. Local maxima are observed in holography reconstructions at the nozzle lipline, distinct in frequency and space. The lowest-frequency local maximum may relate to noise generated by large-scale turbulence structures in the convectively subsonic region of the flow. Other local maxima are correlated primarily with Mach wave radiation originating from throughout the shear layer and into the fully mixed region downstream of the potential core tip. Source-elucidating decompositions show that the order and behavior of the decomposition lend to the local maxima being related to distinct subsources. Between the local maxima, however, there may be a combination of sources active, which is likely the cause of the spatiospectral lobes observed in other full-scale, supersonic jets.

Fast Detection and Classification of Microplastics below 10 μm Using CNN with Raman Spectroscopy.

In light of the growing awareness regarding the ubiquitous presence of microplastics (MPs) in our environment, recent efforts have been made to integrate Artificial Intelligence (AI) technology into MP detection. Among spectroscopic techniques, Raman spectroscopy is preferred for the detection of MP particles measuring less than 10 μm, as it overcomes the diffraction limitations encountered in Fourier transform infrared (FTIR). However, Raman spectroscopy's inherent limitation is its low scattering cross section, which often results in prolonged data collection times during practical sample measurements. In this study, we implemented a convolutional neural network (CNN) model alongside a tailored data interpolation strategy to expedite data collection for MP particles within the 1-10 μm range. Remarkably, we achieved the classification of plastic types for individual particles with a mere 0.4 s of exposure time, reaching an approximate confidence level of 85.47(±5.00)%. We postulate that the result significantly accelerates the aggregation of microplastic distribution data in diverse scenarios, contributing to the development of a comprehensive global microplastic map.

Enhancement of the Lee–Kesler–Plöcker Equation of State for Calculating Thermodynamic Properties of Long-Chain Alkanes

Two optimization approaches to correct the physical limitations of the Lee–Kesler–Plöcker equation of state for the application to long-chain hydrocarbons are presented. The quality of the approaches is evaluated by comparisons with experimental data and reference equations of state. The calculation of thermodynamic properties for alkanes up to squalane is possible with both approaches, for some of which no highly accurate fundamental equation of state is available in the literature. For the first approach, the original parameter set was refitted with constraints guaranteeing correct behavior of the equation in the liquid state. The densities of long-chain hydrocarbons are reproduced with small deviations, while the results for some short-chain alkanes are worsened. For the second approach, existing Helmholtz energy equations of state are utilized, while keeping the linear interpolation scheme via the acentric factor. Significant improvements are achieved for all fluids considered.

Toward static and transient stress-constrained topology optimization for shell-infill structures

This paper contributes to a stress-constrained topology optimization approach for the design of shell-infill structures. Based on the density method, we utilize the two-step filtering and projection scheme and local volume constraint for generating the shell and non-uniform infills, respectively. The topology optimization formulation is defined as a Robust Minimum Compliance problem with Volume and Stress constraints (RMCVS), where the robust method is used to eliminate undesired topological characteristics. We globalize the static and transient stress constraints through the p-norm function, and tailor material interpolation and stress relaxation schemes for both cases, respectively, to avoid numerical difficulties. Sensitivity analysis is provided, and the derivatives of the objective function and constraints with respect to the design variable fields are derived. We then validate the suggested method through several 2D and 3D benchmarks. The results illustrate that the method is robust to generate various shell-infill structures while limiting the maximum static and transient stress.

Hexahedral Solid Element with Rotational Degrees of Freedom Based on a Novel Trail-Correction Displacement Interpolation Scheme

Hexahedral Solid Element with Rotational Degrees of Freedom Based on a Novel Trail-Correction Displacement Interpolation Scheme

Quadrotor attitude control by improved snake optimizer based adaptive switching disturbance rejection approach

In this paper, an adaptive switching anti-disturbance attitude control scheme based on improved snake optimizer (SO) is proposed for quadrotor attitude control when a quadrotor unmanned aerial vehicle is affected by measurement noise. The adaptive switching disturbance rejection controller (AWDRC) is composed of linear active disturbance rejection control and adaptive switching extended state observer which is used to achieve accurate signals reconstruction performance under measurement noise. Then, the improved SO (ISO) algorithm is developed with quadratic interpolation and comprehensive learning strategies to obtain the optimal parameters of the quadrotor attitude controller. The performance validity of ISO is demonstrated here by experiments on the CEC-2017 and the CEC-2020 benchmark functions with several state-of-the-art meta-heuristic algorithms. Secondly, the proposed ISO-based AWDRC algorithm is used in quadrotor attitude tracking control and compared with three other excellent active disturbance rejection controllers in a comparative experiment, and the experimental results show the effectiveness of the proposal. Finally, the robustness of the proposed method to parameters perturbation of the quadrotor attitude system is analyzed by Monte Carlo experiments.

Rapid simulation of two-dimensional spectra with correlated anisotropic dimensions.

A new algorithm has been developed to simulate two-dimensional (2D) spectra with correlated anisotropic frequencies faster and more accurately than previous methods. The technique uses finite-element numerical integration on the sphere and an interpolation scheme based on the Alderman-Solum-Grant algorithm. This method is particularly useful for numerical calculations of joint probability distribution functions involving quantities with a parametric orientation dependence. The technique's efficiency also allows for practical least-squares fitting of experimental 2D solid-state nuclear magnetic resonance (NMR) datasets. The simulation method is illustrated for select 2D NMR methods, and a least-squares analysis is demonstrated in the extraction of paramagnetic shift and quadrupolar coupling tensors and their relative orientation from the experimental shifting-d echo 2H NMR spectrum of a NiCl2 · 2D2O salt.

Multilevel Leapfrogging Initialization Strategy for Quantum Approximate Optimization Algorithm

AbstractRecently, Zhou et al. have proposed an Interpolation‐based (INTERP) strategy to generate the initial parameters for Quantum Approximate Optimization Algorithm (QAOA). INTERP guesses the initial parameters at level by applying interpolation to the optimized parameters at level , achieving better performance than random initialization (RI). Nevertheless, INTERP consumes extensive costs for deep QAOA because it necessitates optimization at each level depth. To address it, a Multilevel Leapfrogging Interpolation (MLI) strategy is proposed. MLI produces initial parameters from level to () at level , omitting the optimization rounds from level to . MLI executes optimization at few levels rather than each level, and this operation is called Multilevel Leapfrogging optimization (M‐Leap). The performance of MLI is investigated on the Maxcut problem. The simulation results demonstrate MLI achieves the same quasi‐optima as INTERP while consuming 1/2 of costs required by INTERP. Besides, for MLI, where there is no RI except for level 1, the greedy‐MLI strategy is presented. The simulation results suggest greedy‐MLI has better stability than INTERP and MLI beyond obtaining the quasi‐optima. According to the efficiency of finding the quasi‐optima, the idea of M‐Leap might be extended to other training tasks, especially those requiring numerous optimizations.

A new Hermite finite element for nonlinear Kirchhoff rods: The plane case

AbstractThis article presents the formulation of a finite element method for nonlinear Kirchhoff rods, based on a interpolation of the rod's geometry in terms of Hermite shape functions. The critical use of the same interpolation scheme for both the geometry and the kinematics of the rod is shown to lead to the correct invariant properties of the final numerical formulation, thus leading to the correct resolution of the fundamental equilibrium relations along it (balance of forces and moments). The so‐called “self‐straining” is completely avoided. Several numerical examples are presented illustrating the adequacy of the proposed formulation for the analysis of thin rods undergoing large finite deformations.

An ANS/ATFs‐based unsymmetric solid‐shell finite element algorithm for high‐quality finite deformation analysis of hyper‐elastic shell

AbstractAn effective updated Lagrangian (UL) algorithm is designed for extending the recent distortion‐tolerant unsymmetric 8‐node, 24‐DOF hexahedral solid‐shell element, US‐ATFHS8, to finite deformation analysis of hyper‐elastic shell structures. The distinguishing feature of this unsymmetric element is that two different interpolation schemes are employed for virtual displacement and real stress calculations, respectively. The assumed natural strain (ANS) method with shell assumptions, referring to the current configuration, is introduced to modify the strain tensors derived from the assumed virtual displacement fields in terms of isoparametric coordinates, thereby mitigating shear locking and trapezoidal locking. On the other hand, the analytical trial functions (ATFs) derived from the general solutions of homogenous governing equations for linear elasticity are updated in each increment step to obtain the incremental deformation gradient, which is then utilized for calculating the real stresses for curing the numerical difficulties in large deformation problems. Numerical examples show that the proposed algorithm enables the hyper‐elastic solid‐shell element US‐ATFHS8 to exhibit excellent performance in both regular and distorted meshes and yield considerable results even when other models cannot work.

Multi‐Spacecraft Magnetic Field Reconstructions: A Cross‐Scale Comparison of Methods

AbstractSpace plasma studies frequently use in situ magnetic field measurements taken from many spacecraft simultaneously. A useful data product of these measurements is the reconstructed magnetic field in a volume near the spacecraft observatory. We compare a standard Linear method of computing the magnetic field at arbitrary spatial points to two novel approaches: a Radial Basis Function interpolation and a time‐dependent 2D inverse distance weighted interpolation scheme called Timesync. These three methods, which only require in situ measurements of the magnetic fields and bulk plasma velocities at a sparse set of spatial points, are implemented on synthetic data drawn from a time‐evolving numerical simulation of plasma turbulence. We compare both the topology of the reconstructed field to the ground truth of the simulation and the statistics of the fluctuations found in the reconstructed field to those from the simulated turbulence. We conclude that the Radial Basis Function and Timesync methods outperform the Linear method in both topological and statistical comparisons.