Order Interpolation Research Articles

Experimental detection and shape reconstruction of inclusions in elastic bodies via a monotonicity method

We deal with the detection and shape reconstruction of inclusions in elastic bodies based on a monotonicity method and aim to reconstruct them with experimental measurements. Thus, we base our studies on the rigorously proven theory of the linearized monotonicity tests for noisy measurement data, where the so-called Neumann-to-Dirichlet operator, its Fréchet derivative and the corresponding monotonicity properties play an essential role. Further on, we give an insight into the lab experiment itself. More specifically, we take a look at Makrolon plates with one or two circular aluminium inclusions. Concerning the realization of the measurements, we have to deal with missing data which cannot be measured due to the set-up of the experiment. Hence, we take a look at a modified spline interpolation in order to determine this data. In doing so, we state the required steps for the implementation of the linearized monotonicity tests. Finally, we present our reconstructions based on experimental data and compare them with the simulations obtained from artificial data, where we want to highlight that all inclusions can be detected from the noisy experimental data, thus, we obtain accurate results. This paper combines the rigorously proven theory of the monotonicity methods developed for linear elasticity with the explicit application of the methods, i.e. the implementation and simulation of the reconstruction of inclusions in elastic bodies for both artificial and experimental data.

Transforming a Simple Structure Model to Represent a Complex Dynamic System with Unknown Boundary Restraints

Imposing a boundary condition on a structure can significantly alter its dynamic properties. However, sometimes the specifics of the new boundary conditions are not known. When the effects of a boundary condition are uncertain or there is not enough information, engineers need to excite the complex structure to obtain these modified properties. In order to experimentally obtain the new properties, engineers need multiple experiments and many outputs for interpolation in order to sufficiently represent the entire structure. The researchers attached a stinger to a cantilever beam, acting as a new transverse restraint of unknown properties. This paper presents a conversion expression that predicts the dynamic behavior of any point in the system with the new boundary condition. This expression relies only on one impact hammer experiment with one output and the model of the stinger-free cantilever beam, referred to as the simple structure. Researchers estimated the Transfer Function (TRF) of the beam and compared it with an experimentally measured TRF to validate the method. The mean absolute error of the estimated TRF compared to the experimental TRF is 1.99 dB. This demonstrates the use of the proposed method for estimating unmeasured TRFs in a system with an uncertain boundary condition using a single input, single output (SISO) test and a model of the simple structure.

Generalized finite difference schemes with higher order Whitney forms

Finite difference kind of schemes are popular in approximating wave propagation problems in finite dimensional spaces. While Yee’s original paper on the finite difference method is already from the sixties, mathematically there still remains questions which are not yet satisfactorily covered. In this paper, we address two issues of this kind. Firstly, in the literature Yee’s scheme is constructed separately for each particular type of wave problem. Here, we explicitly generalize the Yee scheme to a class of wave problems that covers at large physics field theories. For this we introduce Yee’s scheme for all problems of a class characterised on a Minkowski manifold by (i) a pair of first order partial differential equations and by (ii) a constitutive relation that couple the differential equations with a Hodge relation. In addition, we introduce a strategy to systematically exploit higher order Whitney elements in Yee-like approaches. This makes higher order interpolation possible both in time and space. For this, we show that Yee-like schemes preserve the local character of the Hodge relation, which is to say, the constitutive laws become imposed on a finite set of points instead of on all ordinary points of space. As a result, the usage of higher order Whitney forms does not compel to change the actual solution process at all. This is demonstrated with a simple example.

High Resolution 2.5D PIV Measurements of the Flow Fields Generated by Small Fans

Abstract The free jets of an axial and a centrifugal fan have been scanned by a specialized particle image velocimetry (PIV) set-up, which allows for volumetric scans of the time-averaged velocity field. Both of these fans have similar dimensions of approximately 70 mm x 70 mm x 25 mm. A classic PIV set-up was combined with a precise linear stage to move the fans through the laser fan beam in small steps, creating a dense array of measurement planes. Two components of the time-averaged velocity field are captured by the first 2.5D scan. Another scan, with the fan rotated by 90° about its outlet surface normal, captures the missing third velocity component. This article describes the details of the measurement set-up, and mentions measures concerning seeding, reflections, and calibration. In the signal processing stage, two independent sets of gathered image data have to be processed, producing two sets of velocity image frames. These are subsequently combined using gridded interpolation in order to obtain a 3D velocity field. Specifically devised software tools allow for a CFD-like analysis and visualization of the flow field. Typical parameters of the generated jets, like the spreading and rotation rates, are calculated from the measurement data and details of the outlet flow fields are investigated. The interpolated data are also used to analyze the influence of an assumed coarser measurement grid resolution on the results for the obtained outlet flow fields.

Cavitation modelling using real gas state equation: A conceptual study

The prediction of the onset of cavitation to allow investigations of the flow in off-design conditions during design is unavoidable for a variety of applications. Numerical modeling of the cavitation phenomena is however computationally challenging due to the nature of the phenomena. High density ratios, steep pressure gradients, the presence of multiple phase including phase change and hence to occurrence of real gas thermophysical behavior would cause a computational effort unmanageable during design. Today’s models are therefore greatly simplified representations of the reality. Conservation equations are usually solved using a mixture approach with flow properties being a volume fraction weighted average of two incompressible fluids thus neglecting real gas effects. The source term from the phase exchange however usually poses significant stability problems. In this article we will therefore present an approach for cavitation modeling using high-order polynomial real-gas state equations to accurately predict the properties of water at any state. Computational performance is improved by the use of structured tabulation using high order interpolation methods. The numerical results are then compared against measurement data and conventional cavitation model results.

An edge-based smoothed three-node composite plate element with refined zigzag kinematics

The so called Refined Zigzag Theory (RZT) has proven to be one of the most promising theories for analyzing shear-elastic, laminated structures. Since its first appearance considerable efforts have been made in the development of numerical tools and the assessment of the theory. The present contribution focuses on the numerical assessment of a variant of a triangular plate element, which was presented for the first time in 2013 by Versino and co-workers. This element uses low order interpolation functions for the seven kinematic variables. While the deformations show a good convergence a very dense mesh is required to get stresses of satisfying accuracy especially near zones of warping constraints. An edge-based smoothed element technique (ES-FEM) is applied to improve the performance of this element. Several numerical tests are performed with regular and irregular meshes which have shown a significant improvement of the convergence rate of deflections for thick and thin plates respectively. In relation to the standard FEM the ES-version shows a lower sensitivity to mesh irregularities and more accurate stress results.

A fifth-order high-resolution shock-capturing scheme based on modified weighted essentially non-oscillatory method and boundary variation diminishing framework for compressible flows and compressible two-phase flows

First, a new reconstruction strategy is proposed to improve the accuracy of the fifth-order weighted essentially non-oscillatory (WENO) scheme. It has been noted that conventional WENO schemes still suffer from excessive numerical dissipation near-critical regions. One of the reasons is that they tend to under-use all adjacent smooth substencils thus fail to realize optimal interpolation. Hence in this work, a modified WENO (MWENO) strategy is designed to restore the highest possible order interpolation when three target substencils or two target adjacent substencils are smooth. Since the new detector is formulated under the original smoothness indicators, no obvious complexity and cost are added to the simulation. This idea has been successfully implemented into two classical fifth-order WENO schemes, which improve the accuracy near the critical region but without destroying essentially non-oscillatory properties. Second, the tangent of hyperbola for interface capturing (THINC) scheme is introduced as another reconstruction candidate to better represent the discontinuity. Finally, the MWENO and THINC schemes are implemented with the boundary variation diminishing algorithm to further minimize the numerical dissipation across discontinuities. Numerical verifications show that the proposed scheme accurately captures both smooth and discontinuous flow structures simultaneously with high-resolution quality. Meanwhile, the presented scheme effectively reduces numerical dissipation error and suppresses spurious numerical oscillation in the presence of strong shock or discontinuity for compressible flows and compressible two-phase flows.

Direct forcing immersed boundary methods: Improvements to the ghost-cell method

The ghost-cell immersed boundary methods (IBMs) are widely used to implement boundary conditions on non-body fitted grids. It has been previously shown that they have two major drawbacks. Firstly, these methods tend to have a maximum stencil size larger than 1, yielding non-band matrices. Secondly, for a maximum stencil size of 2 the ghost-cell linear IBM [1] have a first-order convergence rate for Neumann immersed boundary conditions. To address these two shortcomings and in the pursuit of increased accuracy and increased order of convergence the current article proposes the linear/quadratic square shifting methods for the ghost-cell IBM for Cartesian grids. The linear square shifting method guarantees a maximum stencil size of 1 and also improves the accuracy and convergence while the quadratic square shifting method improves the accuracy and convergence while maintaining the same stencil size of 2 as the original linear method [1]. The quadratic ghost-cell method [2,3] further improves the accuracy and convergence whilst maintaining a maximum stencil size of 3 while the currently proposed quadratic square shifting method makes it possible to increase the Lagrange polynomial interpolation order whilst maintaining a maximum stencil of 2 resulting in an improved order of convergence for Neumann immersed boundary conditions. The proposed methods are evaluated by considering their accuracy and convergence thanks to a comprehensive verification and validation process. Firstly, the canonical verification 2D and 3D Poisson test problems for various analytical solutions and immersed boundaries are considered and are followed by various verification and validation test cases for the Navier-Stokes governing equations, with and without heat transfer, in 2D and 3D.

Numerical simulation of compressible fluid-particle flows in multimaterial Lagrangian hydrodynamics framework

In the numerical simulation of multimaterial flows, Lagrangian method was applied widely, and fluid-particle multiphase flows phenomenon often existed when elastic-plastic material was impacted intensively. Most of the previous studies about the fluid-particle flows in Lagrangian hydrodynamics framework only focused on the dilute flow. It is very necessary to establish a Lagrangian-Lagrangian coupling numerical simulation framework, which is suitable for both multimaterial and dense particle flows in engineering applications. In present study, a numerical model, in which both continuous fluid phase and dispersed particle phase are discretized in Lagrangian frameworks, is developed. In this model, the effect of particle volume fraction, collision force among particles, interaction and coupling between fluid and particle phases are all considered based on the physical meanings. Therefore it is well-suited to complex multiphase flows including transition between dense and dilute particles flows, and material interfaces and free boundaries in multimaterial flows can be solved spontaneously. The relevant numerical technology and approach are proposed and derived, including fixed background mesh, higher order interpolation function, discetezation of coupling source terms and so on. The above numerical model and approach are integrated to a multimaterial staggered-grid elastic-plastic Lagrangian hydrodynamics code. A series of tests, including single particle transport, shock-induced dense particles bed, and explosive dispersal of solid particles, are used to validate the model. The numerical results show excellent agreement with theoretical solutions or experimental measurements, providing confidence in the proposed numerical model and methods.

Matrix Factorization-Based RSS Interpolation for Radio Environment Prediction

This letter proposes matrix factorization (MF) based interpolation of received signal strength (RSS) from a transmitter mounted on moving robot in factory environment. For realizing the reliable wireless communication, machine learning based channel prediction methods have been intensively studied in the past decade. However, some traffic models will make the observation of RSS sequence be intermittent, and the missing values must be interpolated before input to the predictor. Classical interpolation such as linear interpolation cannot appropriately estimate the missing values because the result of the interpolation depends on the observation time. In this letter, we propose to apply an MF-based interpolation technique to RSS interpolation in order to restore the true RSS variation pattern. Moreover, the basic MF-based interpolation is improved by introducing a smoothing term in an objective function to represent the smooth variation of the RSS sequence. The simulation results show that the MF-based interpolation can improve the prediction accuracy of the machine learning based channel prediction method.

Stabilised Variational Multi-scale Finite Element Formulations for Viscoelastic Fluids

The objective of this article is to summarise the work that we have been doing as a group in the context of stabilised finite element formulations for viscoelastic fluid flows. Viscoelastic fluids are complex non-Newtonian fluids, characterised by having an irreducible constitutive equation that needs to be solved coupled with the momentum and continuity equations. The finite element approximation of this kind of fluids presents several numerical difficulties. It inherits obviously the problems associated with the approximation of the incompressible Navier–Stokes equations. But, on top of that, now the constitutive equation is highly non-linear, with an advective term that may lead to both global and local oscillations in the numerical approximation. Moreover, even in the case of smooth solutions, it is necessary to meet some additional compatibility conditions between the velocity and the stress interpolation in order to ensure control over velocity gradients. The stabilised methods detailed in this work allow one to use equal order or even arbitrary interpolation for the problem unknowns ( $${\varvec{\sigma }}$$ - $${\varvec{u}}$$ -p) (elastic deviatoric stress-velocity-pressure) and to stabilise dominant convective terms, and all of them can be framed in the context of variational multi-scale methods. Some additional numerical ingredients that are introduced in this article are the treatment of the non-linearities associated with the problem and the possibility to introduce a discontinuity-capturing technique to prevent local oscillations. Concerning the constitutive equation, both the standard as the logarithmic conformation reformulation are discussed for stationary and time-dependent problems, and different versions of stabilised finite element formulations are presented in both cases.

Modeling of spillage and debris floods as Newtonian and viscoplastic Bingham flows with free surface with mixed stabilized finite elements

• Numerical modelling of Newtonian and viscoplastic Bingham flows with free surface. • Study of flow due to dam-break resulting in spillage and debris floods. • Mixed velocity/pressure element with Orthogonal Subgrid Scale stabilization. • Accurate determination of the location of the free surface of the flow. • Analytical solution to a Poiseuille flow. This work deals with the numerical modeling of spillage and debris floods as Newtonian and viscoplastic Bingham flows with free surface using mixed stabilized finite elements. The main objective is the determination of the location of the free surface of the flow. A mixed stabilized velocity/pressure finite element formulation is used and the Orthogonal Subgrid Scale stabilization (OSS) and the Split Orthogonal Subgrid Scale stabilization (Split-OSS) techniques enable equal order interpolation for velocity and pressure. A double viscosity regularized Bingham model is introduced as an alternative to the regularized Papanastasiou model. The analytical solution to a Poiseuille flow has been obtained and a convergence test has been performed to compare both models. A simplified Eulerian method solved on a fixed mesh is presented to track the movement of the free surface. This is done by transporting the free surface with a stabilized level set method. The numerical solutions are compared with analytical, experimental results and numerical benchmarks from the literature. They are also compared with field data in a real case study, the flow due to the failure of a mining reservoir, which is solved in 3D and as a 2D simplified model.

3D Airborne EM Forward Modeling Based on Time-Domain Spectral Element Method

Airborne electromagnetic (AEM) method uses aircraft as a carrier to tow EM instruments for geophysical survey. Because of its huge amount of data, the traditional AEM data inversions take one-dimensional (1D) models. However, the underground earth is very complicated, the inversions based on 1D models can frequently deliver wrong results, so that the modeling and inversion for three-dimensional (3D) models are more practical. In order to obtain precise underground electrical structures, it is important to have a highly effective and efficient 3D forward modeling algorithm as it is the basis for EM inversions. In this paper, we use time-domain spectral element (SETD) method based on Gauss-Lobatto-Chebyshev (GLC) polynomials to develop a 3D forward algorithm for modeling the time-domain AEM responses. The spectral element method combines the flexibility of finite-element method in model discretization and the high accuracy of spectral method. Starting from the Maxwell's equations in time-domain, we derive the vector Helmholtz equation for the secondary electric field. We use the high-order GLC vector interpolation functions to perform spectral expansion of the EM field and use the Galerkin weighted residual method and the backward Euler scheme to do the space- and time-discretization to the controlling equations. After integrating the equations for all elements into a large linear equations system, we solve it by the multifrontal massively parallel solver (MUMPS) direct solver and calculate the magnetic field responses by the Faraday's law. By comparing with 1D semi-analytical solutions for a layered earth model, we validate our SETD method and analyze the influence of the mesh size and the order of interpolation functions on the accuracy of our 3D forward modeling. The numerical experiments for typical models show that applying SETD method to 3D time-domain AEM forward modeling we can achieve high accuracy by either refining the mesh or increasing the order of interpolation functions.

An enhanced VEM formulation for plane elasticity

In this paper, an enhanced Virtual Element Method (VEM) formulation is proposed for plane elasticity. It is based on the improvement of the strain representation within the element, without altering the degree of the displacement interpolating functions on the element boundary. The idea is to fully exploit polygonal elements with a high number of sides, a peculiar VEM feature, characterized by many displacement degrees of freedom on the element boundary, even if a low interpolation order is assumed over each side. The proposed approach is framed within a generalization of the classic VEM formulation, obtained by introducing an energy norm in the projection operator definition. Although such generalization may mainly appear to have a formal value, it allows to effectively point out the mechanical meaning of the quantities involved in the projection operator definition and to drive the selection of the enhanced representations. Various enhancements are proposed and tested through several numerical examples. Numerical results successfully show the capability of the enhanced VEM formulation to (i) considerably increase accuracy (with respect to standard VEM) while keeping the optimal convergence rate, (ii) bypass the need of stabilization terms in many practical cases, (iii) obtain natural serendipity elements in many practical cases, and (vi) effectively treat also nearly incompressible materials.

Statistical Analysis and Evaluation of Drinking Water Consumption Needs in Dakar from 2016 to 2025

In Dakar, a considerable deficit is observed in the distribution of drinking water. The demand for drinking water is increasing day by day and the available resources are insufficient. The objective of this article is first to assess the drinking water consumption needs of the Dakar region by performing a statistical analysis of the data. Statistical analysis of the data made it possible to assess the state of coverage of drinking water consumption needs. In addition, this statistical analysis will make it possible to make a comparison between the evolution of drinking water consumption needs and the current available resources. The results obtained show that after interpolations of the data by different polynomials (interpolation of degree 1, interpolation of degree 5, and interpolation of degree 10), the best interpolation in order to have an optimal projection of the evolution of drinking water consumption as a function of the population is the interpolation polynomial of degree 10. This equation called the Dakar equation which gives the consumption as a function of the population is as follows: Cons = 8,959.102Pop2 – 5,2592.108Pop + 1,382.1014.

A Numerical Scheme for the Quantile Hedging Problem

We consider numerical approximations to the quantile hedging price of a European claim in a nonlinear market with Markovian dynamics. We study an equivalent stochastic target problem with the conditional probability of success as a new state variable, in addition to the asset value process. We propose numerical approximations based on piecewise constant policy time stepping coupled with novel finite difference schemes. We prove convergence in the monotone case combining backward stochastic differential equation arguments with the Barles and Jakobsen and Barles and Souganidis approaches for nonlinear PDEs. The difficulties compared to the classical setting consist in the construction of monotone schemes under degeneracy due to the perfectly correlated joint process, the unboundedness of the control variable, and the effect of the boundaries in the probability variable on the analysis. We extend the method to a class of nonmonotone schemes using higher order interpolation and prove convergence for linear drivers. In a numerical section, we illustrate the performance of our schemes by considering an example in a financial market with imperfections, and show that a standard nonmonotone scheme produces financially counterintuitive solutions.

Milling stability prediction based on the hybrid interpolation scheme of the Newton and Lagrange polynomials

The stability lobe diagram (SLD) is commonly used to determine the suitable cutting parameters of the machining system in order to achieve a chatter-free machining process. An improved full-discretization method (FDM) is proposed to predict the SLD based on the hybrid interpolation scheme of the Newton and Lagrange polynomials. In order to solve the SLD, a third-order Newton polynomial is employed to interpolate the state term of the physical space equation of the system. Meanwhile, to investigate the influence of the interpolation order on predicting the SLD, the delayed term is estimated using the Lagrange polynomials of orders one to four successively. Subsequently, after constructing the transition matrix, a series of calculation for the stability prediction are carried out by applying Floquet theory. The calculated results from these constructed methods demonstrate that the FDM with a second-order Lagrange polynomial is optimal, and further has better computational performance compared with some existing discretization methods. Lastly, the influences of the dynamic parameters on chatter stability are analyzed based on the proposed FDM. When the stiffness and damping ratio increase, the limit cutting depth will be enhanced. The increasing natural frequency not only causes an obvious shift of the lobes to the right, but also raises the limit cutting depth to some extent. These theoretical analyses can guide the prediction and improvement of the chatter stability of a machining system.

Domain decomposition based on the spectral element method for frequency-domain computational elastodynamics

We propose a domain decomposition method based on the spectral element method (DDM-SEM) for elastic wave computation in frequency domain. It combines the high accuracy of the spectral element method and the high degree of parallelism of a domain decomposition technique, which makes this method suitable for accurate and efficient simulations of large scale problems in elastodynamics. In the DDM-SEM, the original large-scale problem is divided into a number of well designed subdomains. We use the spectral element method independently for each subdomain, and the neighboring subdomains are connected by a frequency-domain version of Riemann transmission condition (RTC) for elastic waves. For the proposed method, we can employ the non-conforming meshes and different interpolation orders in different subdomains to maximize the efficiency. By separating the internal and boundary unknowns of each subdomain, an efficient and naturally parallelizable block LDU direct solver is developed to solve the final system matrix. Numerical experiments verify its accuracy and efficiency, and show that the proposed DDM-SEM can be a promising numerical tool for accurately and effectively solving large and multi-scale problems of elastic waves. It is potentially valuable for the frequency domain seismic inversion where multiple source illuminations are required.

Quasi-geometric integration of guiding-center orbits in piecewise linear toroidal fields

A numerical integration method for guiding-center orbits of charged particles in toroidal fusion devices with three-dimensional field geometry is described. Here, high order interpolation of electromagnetic fields in space is replaced by a special linear interpolation, leading to locally linear Hamiltonian equations of motion with piecewise constant coefficients. This approach reduces computational effort and noise sensitivity, while the conservation of total energy, magnetic moment and phase space volume is retained. The underlying formulation treats motion in piecewise linear fields exactly and, thus, preserves the non-canonical symplectic form. The algorithm itself is only quasi-geometric due to a series expansion in the orbit parameter. For practical purposes, an expansion to the fourth order retains geometric properties down to computer accuracy in typical examples. When applied to collisionless guiding-center orbits in an axisymmetric tokamak and a realistic three-dimensional stellarator configuration, the method demonstrates stable long-term orbit dynamics conserving invariants. In Monte Carlo evaluation of transport coefficients, the computational efficiency of quasi-geometric integration is an order of magnitude higher than with a standard fourth order Runge–Kutta integrator.

On the competition for ultimately stiff and strong architected materials

Advances in manufacturing techniques may now realize virtually any imaginable microstructures, paving the way for architected materials with properties beyond those found in nature. This has lead to a quest for closing gaps in property-space by carefully designed metamaterials. Development of mechanical metamaterials has gone from open truss lattice structures to closed plate lattice structures with stiffness close to theoretical bounds. However, the quest for optimally stiff and strong materials is complex. Plate lattice structures have higher stiffness and (yield) strength but are prone to buckling at low volume fractions. Hence here, truss lattice structures may still be optimal. To make things more complicated, hollow trusses or structural hierarchy bring closed-walled microstructures back in the competition. Based on analytical and numerical studies of common microstructures from the literature, we provide higher order interpolation schemes for their effective stiffness and (buckling) strength. Furthermore, we provide a case study based on multi-property Ashby charts for weight-optimal porous beams under bending, that demonstrates the intricate interplay between structure and microarchitecture that plays the key role in the design of ultimate load carrying structures. The provided interpolation schemes may also be used to account for microstructural yield and buckling in multiscale design optimization schemes.