Order Interpolation Research Articles

A fundamental class of stress elements in lower bound limit analysis

There is a major restriction in the formulation of rigorous lower bound limit analysis by means of the finite-element method. Once the stress field has been discretized, the yield criterion and the equilibrium conditions must be applied at a finite number of points so that they are satisfied everywhere throughout the discretized structure. Until now, only the linear stress elements fulfil this requirement for several types of loads and structural conditions. However, there are also standard types of problems, like the one of plates under uniformly distributed loads, where the implementation of the lower bound theorem is still not possible. In this paper, it is proven for the first time that there is a class of stress interpolation elements which fulfils all the requirements of the lower bound theorem. Moreover, there is no upper restriction of the polynomial interpolation order. The efficiency is examined through plane strain, plane stress and Kirchhoff plate examples. The generalization to three-dimensional and other structural conditions is also straightforward. Thus, this interpolation scheme which is based on the Bernstein polynomials is expected to play a fundamental role in future developments and applications.

A combined finite element–finite volume framework for phase-field fracture

Numerical simulations of brittle fracture using phase-field approaches often employ a discrete approximation framework that applies the same order of interpolation for the displacement and phase-field variables. In particular, the use of linear finite elements to discretize both stress equilibrium and phase-field equations is widespread in the literature. However, the use of P1 Lagrange shape functions to model the phase-field is not optimal, as the latter contains cusps for fully developed cracks. These should in turn occur at locations corresponding to Gauss points of the associated FE model for the mechanics. Such a feature is challenging to reproduce accurately with low order elements, and element sizes must consequently be made very small relative to the phase-field regularization parameter in order to achieve convergence of results with respect to the mesh. In this paper, we combine a standard linear FE discretization of stress equilibrium with a cell-centered finite volume approximation of the phase-field evolution equation based on the two-point flux approximation constructed over the same simplex mesh. Compared to a pure FE formulation utilizing linear elements, the proposed framework results in looser restrictions on mesh refinement with respect to the phase-field length scale. This ability to employ coarser meshes relative to the traditional implementation allows for significant reductions on computational cost, as demonstrated in several numerical examples.

Numerical Investigation of Remote Ignition in Shock Tubes

Highly resolved two- and three-dimensional computational fluid dynamics (CFD) simulations are presented for shock-tube experiments containing hydrogen/oxygen (H2/O2) mixtures, to investigate mechanisms leading to remote ignition. The results of the reactive cases are compared against experimental results from Meyer and Oppenheim (Proc Combust Inst 13(1): 1153–1164, 1971. https://doi.org/10.1016/s0082-0784(71)80112-1) and Hanson et al. (Combust Flame 160(9): 1550–1558, 2013. https://doi.org/10.1016/j.combustflame.2013.03.026). The results of the non-reactive case are compared against shock tube experiments, recently carried out in Duisburg and Texas. The computational domain covers the end-wall region of the shock tube and applies high order numerics featuring an all-speed approximate Riemann scheme, combined with a 5th order interpolation scheme. Direct chemistry is employed using detailed reaction mechanisms with 11 species and up to 40 reactions, on a grid with up to 2.2 billion cells. Additional two-dimensional simulations are performed for non-reactive conditions to validate the treatment of boundary-layer effects at the inlet of the computational domain. The computational domain covers a region at the end part of the shock tube. The ignition process is analyzed by fields of localized, expected ignition times. Instantaneous fields of temperature, pressure, entropy, and dissipation rate are presented to explain the flow dynamics, specifically in the case of a bifurcated reflected shock. In all cases regions with locally increased temperatures were observed, reducing the local ignition-delay time in areas away from the end wall significantly, thus compensating for the late compression by the reflected shock and therefore leading for first ignition at a remote location, i.e., away from the end wall where the ignition would occur under ideal conditions. In cases without a bifurcated reflected shock, the temperature increase results from shock attenuation. In cases with a bifurcated reflected shock, the formation of a second normal shock and shear near the slip line is found to be crucial for the remote ignition to take place. Overall, the two- and three-dimensional simulations were found to qualitatively explain the occurrence of remote ignition and to be quantitatively correct, implying that they include the correct physics.

Space-angle discontinuous Galerkin method for radiative transfer between concentric cylinders

The integro-differential radiative transfer equation (RTE) for concentric cylinders problem involving scattering, absorption and emission is solved using the discontinuous Galerkin (DG) finite element method (FEM). The space-angle DG method directly solves the cylindrically-symmetric RTE as a three-dimensional problem, where a 1D spatially domain in radial distance r is twice extruded in the cosine of polar angle (μ) and the difference in azimuthal angle (φ˜) directions. Thus, the method has a higher accuracy than hybrid FEM-Discrete Ordinate (SN) and FEM-Spherical Harmonic (PN) methods. This is reflected by numerically verified convergence rate of p+1 for smooth problems and space-angle polynomial interpolation order of p. The axisymmetric RTE formulation is more complicated than the plane-parallel formulation, for having two independent angle directions (μ and φ˜) and an extra derivative term with respect to φ˜ in the differential equation. This results in a complex characteristic structure in r−φ˜ plane with strong discontinuity lines in radiation intensity I. A method of characteristics is formulated and implemented to verify the DG formulation and demonstrate its accuracy when such strong discontinuities persist in the solution, specifically when there are no scattering and absorption terms. The relaxation of inter-element continuity constraint of continuous FEMs by this DG method implies its superiority in numerically capturing such discontinuities. Finally, a benchmark problem pertained to heat radiation in a gray gas and another one with nonzero phase function demonstrate the effectiveness of the method in modeling black-body and scattering angular integration terms.

Relative orientation constraints in the nonlinear large displacement analysis: application to soft materials

A new simple formulation of the orientation constraints for very flexible bodies is introduced in this investigation. This constraint formulation avoids using the tangent or cross-section frames previously used in the literature by directly using two position-gradient vectors to define the orientation constraint equations that eliminate the relative rotations between two bodies. The orientation constraints are commonly used to define clamped (rigid) joint between very flexible bodies, and rigid and less-flexible bodies. The very flexible bodies are modeled using the absolute nodal coordinate formulation (ANCF) which employs position gradients as nodal coordinates, while other rigid and less-flexible bodies have kinematics described in terms of orientation parameters. The less-flexible bodies can be modeled using the floating frame of reference formulation. Conventional clamped-end conditions eliminate all rigid body and deformation degrees of freedom because of the low-order of the finite element interpolation. When using a higher interpolation order, distinction is made between fully- and partially-clamped joints; the former eliminates all the degrees of freedom, while the latter eliminates only the relative translations and rotations, allowing for local deformations at the joint definition point. The singularity problem that arises, in the formulation of the orientation constraints, as the result of using ANCF orientation vectors derived using one position-gradient vector is demonstrated using simple, but common, example of a cantilever beam. The applicability of Saint–Venant principle, which states that the effect of the load diminishes away from the point of application of the load, to constraint forces that eliminate degrees of freedom is examined. Numerical results obtained in this study demonstrate that elimination of degrees of freedom, which leads to different kinematic structures and deformation basis-vectors, can be more significant in the case of soft materials.

High order conservative Lagrangian schemes for one-dimensional radiation hydrodynamics equations in the equilibrium-diffusion limit

Radiation hydrodynamics (RH) describes the interaction between matter and radiation which affects the thermodynamic states and the dynamic flow characteristics of the matter-radiation system. Its application areas are mainly in high-temperature hydrodynamics, including gaseous stars in astrophysics, combustion phenomena, reentry vehicles fusion physics and inertial confinement fusion (ICF). Solving the radiation hydrodynamics equations (RHE), even in the equilibrium-diffusion limit, is a difficult task. In this paper, we will discuss the methodology to construct fully explicit and implicit-explicit (IMEX) high order Lagrangian schemes solving one dimensional RHE in the equilibrium-diffusion limit respectively, which can be used to simulate multi-material problems with the coupling of radiation and hydrodynamics. The schemes are based on the HLLC numerical flux, the essentially non-oscillatory (ENO) reconstruction for the advection term, ENO reconstruction or high order central reconstruction and interpolation for the radiation diffusion term, the Newton iteration method (for the IMEX scheme), and the strong stability preserving (SSP) high order time discretizations. The schemes can maintain conservation and uniformly high order accuracy both in space and time. The issue of positivity-preserving for the high order explicit Lagrangian scheme is also discussed. Various numerical tests for the high order Lagrangian schemes are provided to demonstrate the desired properties of the schemes such as high order accuracy, non-oscillation, and positivity-preserving.

Extended isogeometric analysis for cohesive fracture

SummaryThe objective of this study is to present an extended isogeometric formulation for cohesive fracture. The approach exploits the higher order interelement continuity property of nonuniform rational B‐splines (NURBS), in particular the higher accuracy that results for the stress prediction, which yields an improved estimate for the direction of crack propagation compared to customary Lagrangian interpolations. Shifting is used to ensure compatibility with the surrounding discretization, where, different from extended finite element methods, the affected elements stretch over several rows perpendicular to the crack path. To avoid fine meshes around the crack tip in case of cohesive fracture, a blending function is used in the extension direction of the crack path. To comply with standard finite element data structures, Bézier extraction is used. The absence of the Kronecker‐delta property in the higher order interpolations of isogeometric analysis impedes the enrichment scheme and compatibility enforcement. These issues are studied comprehensively at the hand of several examples, while crack propagation analyses show the viability of the approach.

A distributed framework for extracting maritime traffic patterns

ABSTRACT All the modern surveillance systems take advantage of the Automatic Identification System (AIS), a compulsory tracking system for many types of vessels. Ships that carry AIS transponders on board transmit their position and status in order to alert nearby vessels and ground stations, but this information can well be used to identify events of interest and support decision making. The detection of anomalies (i.e. unexpected sailing behavior) in vessels’ trajectories is such an event, which is of utmost importance. Approaches for detecting such anomalies vary from extracting normality models to searching for individual cases, such as AIS switch-off or collision avoidance maneuvers. The current research work follows the former method; it employs sparse historic AIS data and polynomial interpolation in order to extract shipping lanes. It modifies the DB-Scan clustering algorithm in order to achieve more coherent trajectory clusters, which are then composed to create the shipping lanes. The proposed approach implements distributed processing on Apache Spark in order to improve processing speed and scalability and is evaluated using real-world AIS data collected from terrestrial AIS receivers. The evaluation shows that the biggest part (i.e. more than 90%) of any future vessel trajectory falls within the extracted shipping lanes.

An edge-based smoothed finite element method for the assessment of human exposure to extremely low frequency electric fields

This paper presents an edge-based smoothed finite element method (ES-FEM) for the assessment of human exposure to extremely low frequency (ELF) electric fields. The problem domain is first discretized into a set of four-node tetrahedral elements, and linear shape functions are employed to interpolate the field variables. Then, the smoothing domains are further formed with respect to the edges of elements. In order to improve the accuracy of lower order interpolation, a gradient smoothing technique (GST) is introduced to perform the electric potential gradient smoothing operation over each smoothing domain. Based on the generalized smoothed Galerkin weakform, the discretized system equations are finally obtained. Numerical examples, including both benchmark and practical complex cases, demonstrate that the present method possesses the following interesting properties in assessing human exposure to ELF electric fields: (1) super accuracy and faster convergence rate; (2) higher computational efficiency; (3) insensitive to mesh distortion.

Smart embedded system based on demosaicking for enhancement of surveillance systems

Demosaicking and denoising are essential elements in digital photography pipelines. The use of convolutional neural networks (CNN)-based image demosaicking and denoising methods has been very successful. However, still there is a room for improvement in the network performance in terms of efficiency and accuracy. The main challenge that remains to be addressed is to guarantee the visual quality of reconstructed images, particularly in the presence of noise. To address these challenges, this paper introduces a novel demosaicking and denoising conjunct strategy using deep adaptive residual learning. The proposed framework has three stages. Initially, zero padding is performed to increase processing speed and preserve the edges of the image. In the second phase, we perform demosaicking using interpolation in order to find missing values using information about neighboring pixels. Finally, the reconstructed image is created using the original image. To evaluate the feasibility of the proposed scheme, we used Pytorch and Google Colab with 400 images for training and 100 images for validation The outcomes show that the proposed scheme beats cutting edge joint demosaicking and denoising schemes regarding both structural similarity index metrics (SSIM) and peak signal-to-noise ratio (PSNR) and basic similitude record measurements (SSIM).

Stability Analysis of Multi-Discrete Delay Milling with Helix Effects Using a General Order Full-Discretization Method Updated with a Generalized Integral Quadrature

A tensor-based general order full-discretization method is enhanced with the capacity to handle multiple discrete delays and helix effects leading to a unique automated algorithm in the stability analysis of milling process chatter. The automated algorithm is then exploited in investigating the effects of interpolation order of chatter states and helix-induced terms on the convergence of milling stability lobes. The enhanced capacity to handle the distributed helix effects is based on a general order formulation of the Newton-Coates integral quadrature method. Application to benchmark milling models showed that high order methods are necessary for convergence of the low speed domain of stability lobes while all the numerically stable orders converge in the high speed domain where the ultra-high order methods are prone to numerical instability. Also, composite numerical integration of the helix-induced integrand beyond the usual zero-th order method leads to higher accuracy of stability lobes especially in the low speed domain.

Compressible multiphase particle-in-cell method (CMP-PIC) for full pattern flows of gas-particle system

The compressible multiphase flows in a gas-particle system often arise in many engineering applications and nature phenomena. This study focuses on the development of a novel method, i.e. the compressible multiphase particle-in-cell (CMP-PIC), which can be capable of simulating all pattern flows of gas-particle system from dilute to dense and granular flows. The equivalent relation of the momentum equations of the particle phase between the Baer and Nunziato (B-N) model and the MP-PIC model is the key support for the modeling of gas phase, particle phase and coupling effects. For the gas phase, the governing equations are constructed in the Eulerian frame and the B-N model is taken as reference to derive the equations of gas phase and two-way coupling terms. A transport five-equation model for multi-material compressible flows is adopted in gas equation. Additionally, the disperse particle is tracked in the Lagrangian coordinate based on the particle-in-cell (PIC) method. The collisions among particles are simulated with coarse-grained discrete element method (DEM) model which provides more physical manner rather than empirical collision stress model. Moreover, a two-way coupling model is derived based on a comparative study with B-N model. Thereafter a set of unified HLL/HLLC solvers is developed for the discretization of the convective and nozzling terms of gas phase equations. A high order interpolation operator with certain enlarged smooth length is applied for the robust computation of coupling effects. Overall, the advantages of traditional compressible two-fluid model, DEM and PIC model are integrated into the current CMP-PIC model, and it can simulate the transition flows from dilute to dense or vice versa, which is difficult for traditional methods. This Eulerian-Lagrangian approach is implemented in an in-house parallel code and validated against experimental observations involving different shock induced multiple phase flows. The qualitative and quantitative comparison shows that numerical results perform good consistence with experimental and theoretical results.

Considerations on a phase‐field model for adhesive fracture

SummaryA recently proposed phase‐field model for cohesive fracture is examined. Previous investigations have shown stress oscillations to occur when using unstructured meshes. It is now shown that the use of nonuniform rational B‐splines (NURBS) as basis functions rather than traditional Lagrange polynomials significantly reduces this oscillatory behavior. Moreover, there is no effect on the global structural behavior, as evidenced through load‐displacement curves. The phase‐field model imposes restrictions on the interpolation order of the NURBS used for the three different fields: displacement, phase field, and crack opening. This holds within the Bézier element, but also at the boundaries, where a reduction to ‐continuity yields optimal results. Application to a range of cases, including debonding of a hard fiber embedded in a soft matrix, illustrates the potential of the cohesive phase‐field model.

Systematic assessment of the Method of Moments with Interpolative Closure and guidelines for its application to soot particle dynamics in laminar and turbulent flames

The Method of Moments with Interpolative Closure (MOMIC) is a widely used approach for numerical modeling of aerosol dynamics, where soot formation is an important area of application. Two basic variants of the algorithm were proposed in the past: In the original method, transport equations are solved for several non-negative order integer moments of the soot Number Density Function and one for the moment of order minus infinity. Based on these moments, two interpolation functions are constructed and used for the evaluation of positive and negative fractional order moments required for closure of the moment equation source terms. In the other variant, which is a simplification of the first, equations are solved only for non-negative order integer moments. Negative order moments then need to be evaluated by extrapolation. In addition to the choice between these two algorithms, it is not clear a-priori what is the optimal choice regarding the number of transported moments and hence the order of the interpolation polynomial. In this work, the effects of different choices in the setup of the MOMIC algorithm on the evaluation of soot source terms and the prediction of the soot number density and volume fraction are assessed systematically. To this end, MOMIC is combined with a state-of-the-art physico-chemical soot model and applied to the simulation of soot formation in a laminar premixed flame and along Lagrangian trajectories extracted from a Direct Numerical Simulation of a turbulent sooting flame. Monte Carlo simulations serve as reference solutions for a combined a-priori and a-posteriori study. Including the moment of order minus infinity is shown to be beneficial for an accurate calculation of the coagulation source term and hence the soot number density, because moments of negative order can be evaluated more accurately. Furthermore, a high interpolation order yields a more accurate prediction of the surface growth source term. This source term is systematically overpredicted, but errors are substantially reduced to a few percent with increasing interpolation order.

Thermodynamic potential for quark-qluon plasma with finite quark mass and chemical potential

We summarize the derivation of the finite temperature, finite chemical potential thermodynamic potential in the bag-model approximation to quantum chromodynamics (QCD) that includes a finite s-quark mass in the Feynman diagram contributions for both zero-order and two-loop corrections to the quark interaction. The thermodynamic potential for quarks in QCD is a desired ingredient for computations of the equation of state in the early universe, supernovae, neutron stars, and heavy-ion collisions. The 2-loop contributions are normally divergent and become even more difficult in the limit of finite quark masses and finite chemical potential. We describe various means to interpolate between the low and high chemical potential limits. Although physically well motivated, we show that the infinite series Padé rational polynomial interpolation scheme introduces spurious poles. Nevertheless, we show that lower order interpolation schemes such as polynomial interpolation reproduce the Padé result without the presence of spurious poles. We propose that in this way one can determine the equation of state for the two-loop corrections for arbitrary chemical potential, temperature and quark mass. This provides a new realistic bag-model treatment of the QCD equation of state. We compute the QCD phase diagram with up to the two-loop corrections. We show that the two-loop corrections decrease the pressure of the quark-gluon plasma and therefore increase the critical temperature and chemical potential of the phase transition. We also show, however, that the correction for finite s-quark mass in the two-loop correction serves to decrease the critical temperature for the quark-hadron phase transition in the early universe.

Effective and fast prediction of milling stability using a precise integration-based third-order full-discretization method

Chatter stability prediction is of great practical importance for stable machining because regenerative chatter in the milling process will result in poor surface quality and low machining efficiency. Full-discretization method and its variants have been demonstrated to be effective for the prediction of milling stability. However, the main shortcoming of such methods is that they can predict milling stability but involve inverse matrix calculation, which would lead to increases in computational complexity and reductions in numerical stability. In addition, there may not necessarily exist good inverse matrix for these methods. This study proposes a precise integration-based third-order full-discretization method that can be both accurate and efficient in milling stability prediction without the need of any inverse matrix calculation. The performance evaluation performed by the simulation demonstrates that the proposed method outperforms the conventional methods with respect to stability prediction accuracy and speed. Extensive simulation is also carried out to investigate the effects of interpolation order for the simplified state term on the performance of the proposed method. Three demonstrative examples are employed to demonstrate how the proposed method can function effectively in the prediction of milling chatter stability. Although a chatter stability prediction tool for the milling process is the particular application presented here, the proposed method can be applied to other machining processes, such as turning, boring, and drilling.

Comparison of interpolation-based sampling frequency offset compensation schemes for practical OFDM-VLC systems.

Sampling frequency offset (SFO) is an important issue in the orthogonal frequency-division multiplexing (OFDM)-based visible light communication (VLC) systems with low-cost analog-to-digital or digital-to-analog converters (ADCs/DACs). A digital interpolation or resampling filter can be used to effectively compensate the SFO. In such case, oversampling at the receiver ADC is required to mitigate the aliasing effect due to imperfect DACs and nonlinearity of visible light sources that cause extra frequency components inside/outside the OFDM signal spectrum. The oversampling factor (rate) is mainly determined by the order of the digital interpolation filter and nonlinear VLC links. The design of the OFDM-VLC receiver incorporating the digital interpolation filter is vital as it affects not only the transmission performance but also the complexity of digital signal processing (DSP). To evaluate the feasibility of the digital interpolation-based SFO compensation schemes for cost-sensitive VLC applications, in this paper, a real-time OFDM-VLC receiver incorporating the 2nd/3rd/4th order interpolation filters is experimentally demonstrated. An OFDM frame structure is designed for the synchronization including SFO estimation and compensation, in which the precision and latency of DSP are considered. On the basis of the real-time OFDM-VLC receiver, the comparison in the VLC transmission performance and DSP complexity between different interpolation-based SFO compensation schemes is discussed.

Generalization of the strain-split method and evaluation of the nonlinear ANCF finite elements

This paper discusses the generalization of the strain-split method (SSM) for the locking alleviation of curved structures. The generalization is achieved by using proper definitions of the stress and strain tensors along the curved-coordinate lines using the matrix of position vector gradients in the reference configuration. This matrix, which accurately captures the element geometry at the integration points, allows using consistent gradient transformation in the calculation of the stress and strain tensors. The generalized SSM implementation is used to verify the results and evaluate the performance of the absolute nodal coordinate formulation (ANCF) finite elements (FE). The focus of this study is on the Poisson locking that characterizes fully parameterized ANCF elements that employ different orders of interpolation in different directions. ANCF beam and plate nonlinear problems are presented, and the obtained simulation results are compared with analytical solutions as well as results obtained using commercial FE computer programs. These results are also compared with the results obtained using ANCF beam and curved plate elements in the case of nonzero Poisson ratio in order to demonstrate the SSM effectiveness in alleviating the Poisson locking. It is shown that a much smaller number of ANCF plate elements is required to achieve approximately 0.9% difference from the results of commercial FE computer programs.

The Reliability of Numerical Modeling in Remote Sensing Data Analysis

The article deals with the problem of calculating reliable estimates of empirical distribution functions under conditions of small sample and data uncertainty. To study these issues, we develope computational probabilistic analysis as a new area in computational statistics. We propose a new approach based on random interpolation polynomials and order statistics. Arithmetic operations on probability density functions and procedures for constructing the probabilistic extensions are used.

FRACTIONAL HERMITE DEGENERATE KERNEL METHOD FOR LINEAR FREDHOLM INTEGRAL EQUATIONS INVOLVING ENDPOINT WEAK SINGULARITIES

In this article, the Fredholm integral equation of the second kind with endpoint weakly singular kernel is considered and suppose that the kernel possesses fractional Taylor's expansions about the endpoints of the interval. For this type kernel, the fractional order interpolation is adopted in a small interval involving the singularity and piecewise cubic Hermite interpolation is used in the remaining part of the interval, which leads to a kind of fractional degenerate kernel method. We discuss the condition that the method can converge and give the convergence order. Furthermore, we design an adaptive mesh adjusting algorithm to improve the computational accuracy of the degenerate kernel method. Numerical examples confirm that the fractional order hybrid interpolation method has good computational results for the kernels involving endpoint weak singularities.