Linear Interpolation Functions Research Articles

Derivative-Free Feasible Backtracking Search Methods for Nonlinear Multiobjective Optimization with Simple Boundary Constraint

In this paper, a derivative-free linear feasible direction models with backtracking search technique is considered for solving nonlinear multiobjective optimization problems subject to simple boundary constraint. The algorithm is designed to build linear interpolation models for each function of problem [Formula: see text]. We build the linear programming subproblem using linear interpolation function without the second-order derivative information. The new backtracking search step size function is given in our algorithm which guarantees both the monotone descent property of each function and the feasibility of the iterative point. Under reasonable assumptions, we prove that the algorithm converges to a weakly Pareto critical point of problem. The results of numerical experiments are reported to show the effectiveness of the proposed algorithm.

BOX DIMENSION OF WEYL–MARCHAUD FRACTIONAL DERIVATIVE OF LINEAR FRACTAL INTERPOLATION FUNCTIONS

This paper mainly explores Weyl–Marchaud fractional derivative of linear fractal interpolation functions (FIFs). We prove that Weyl–Marchaud fractional derivative of a linear FIF is still a linear FIFs. More generally, we get a conclusion that there exists some linear relationship between the order of Weyl–Marchaud fractional derivative and box dimension of linear FIFs.

Development of Positron Emission Tomography With Wobbling and Zooming for High Sensitivity and High-Resolution Molecular Imaging.

Demands for in-vivo human molecular imaging with high resolution and high sensitivity in positron emission tomography (PET) require several new design formulae. A classical problem of the PET design, however, was the trade-off between sensitivity and resolution. To satisfy both requirements, the brain-body convertible PET with wobbling and zooming is proposed. The features of this new proposed system are wobble sampling for high-resolution imaging and zooming mode for high sensitivity, especially for the brain dedicated imaging. For the high resolution, wobbling with a linear interpolation and line spread function (LSF) deconvolution reconstruction algorithm was introduced. The result of the proposed system provided resolution up to 1.56 mm full width at half maximum (FWHM) in the brain mode and resulting in the detector-to-resolution ratio (DRR) was 2.47. For both brain phantom and in-vivo rat brain imaging, the proposed system demonstrated superior image quality compared with the commercial PET systems. The newly designed PET with wobbling and zooming also demonstrated the possibility of developing practically usable high-resolution human brain PET-MRI fusion system, especially for the neuroscience research.

Dynamic instability of damped composite plates with embedded delaminations

A layerwise finite element formulation based on B-spline function is used to study dynamic instability of damped laminated composite plates with embedded delaminations in this article. The out-of-plane variation of in-plane displacement components are interpolated by Linear Lagrange interpolation function and the in-plane variation of in-plane and out-of-plane displacement fields are interpolated by B-spline functions. Delamination between layers of composite plate is modeled by jump discontinuities in displacement fields using Heaviside functions. In the delaminated region, virtual springs are added to prevent interpenetration of lower and upper sublaminates. The local buckling of thinner sublaminate is observed when delamination is placed near the top/bottom interface. The dynamic instability regions are computed by Bolotin's method. The effects of delamination lengths and its position on the natural frequencies, mode shapes, buckling loads and dynamic instability regions of the composite plates with various boundary conditions are examined. Both uniform and non-uniform periodic edge loading in the form of concentrated and partial edge loading are considered. Furthermore, the effect of damping on the dynamic instability regions of composite plate is studied. From the linear and nonlinear responses in the stable and unstable regions, various characteristics such as beats, its dependence on forcing frequency and damping are observed.

Efficient three-node triangular element based on a new mixed global-local higher-order theory for multilayered composite plates

An accurate and computationally attractive global-local higher-order theory (GLHT) is developed for the linearly elastic analysis of cross-ply multilayered composite plates. The theory is derived using the kinematic assumptions of GLHT in conjunction with the Reissner mixed variational principle. For a low-order linear element, it is difficult to accurately compute the transverse shear stresses even applying the three-dimensional equilibrium equation post-processing technique. The reason for this difficulty is that the higher-order derivatives of displacement variables are included in the transverse shear stress fields after using the post-processing technique. Thus, by employing the Reissner mixed variational principle, the higher-order derivatives of displacement variables have been removed from the transverse shear stress components before the finite element procedure is implemented. Based on the mixed GLHT, a computationally efficient C0-type three-node triangular plate element with linear interpolation function is proposed for the analysis of multilayered composite plates. The advantage of the present formulation is that no post-processing approach is needed to calculate the transverse shear stresses while maintaining the computational accuracy of a linear plate element. Performance of the proposed element is assessed by comparing with several benchmark solutions. Numerical results show that the present elements can robustly and accurately predict the displacements and stresses of multilayered composite plates.

High-Order Layerwise Formulation of Transverse Shear Stress Field for Laminated Composite Beams

This paper presents a high-order layerwise theoretical framework for laminated composite beams based on a piecewise description of the transverse shear-stress field. Linear and quadratic variations in the transverse shear-stress field are assumed in each discrete layer, and compatibility conditions of the shear stress are a priori fulfilled through the introduction of shear-stress variables defined on the layer surfaces. Based on the stress–strain relations, the transverse shear strain is obtained, and the in-plane displacement is determined by integrating the transverse shear strain along the thickness direction. By imposing continuity conditions of the displacements, a displacement field expression that contains only displacement variables is formulated. Moreover, a two-node beam element associated with the present model is developed, where Hermite cubic interpolation functions are used for the transverse displacement variable, and linear interpolation functions are employed for the remaining displacement variables. Comparisons with the exact solution, various displacement-based theories, and high-fidelity finite element models demonstrate the high accuracy of the present model in predicting the stress and displacement distributions. In addition, the introduction of sublayers to improve the modeling accuracy and the influence of the stiffness ratio between layers on the effectiveness of the proposed model are addressed.

Nonlinear finite element analysis of lattice core sandwich beams

A geometrically nonlinear finite element model is developed for the bending analysis of micropolar Timoshenko beams using the principle of virtual displacements and linear Lagrange interpolation functions. The nonlinearity enters the model via a nonlinear von Kármán strain term that allows the micropolar beam to undergo moderate rotations. The nonlinear micropolar Timoshenko beam is used as an equivalent single layer model to study four different lattice core sandwich beams. A two-scale energy method is used to derive the micropolar constitutive equations for web, hexagonal, Y-frame and corrugated core topologies. Various bending cases are studied numerically using the developed 1-D finite element model. Reduced integration techniques are used to overcome the shear and membrane locking. The present 1-D results are in good agreement with the corresponding 2-D finite element beam frame results for global bending.

Interpretation of Retrospective BG Measurements.

This study investigates blood glucose (BG) measurement interpolation techniques to represent intermediate BG dynamics, and the effect resampling of retrospective BG data has on key glycemic control (GC) performance results. GC protocols in the ICU have varying BG measurement intervals ranging from 0.5 to 4 hours. Sparse data pose problems, particularly in comparing GC performance or model fitting, and thus interpolation is required. Retrospective data from SPRINT in Christchurch Hospital Intensive Care Unit (ICU) (2005-2007) were used to analyze several interpolation techniques. Piecewise linear, spline, and cubic interpolation functions, which force interpolation through measured data, as well as 1st and 2nd Order B-spline basis functions, are used to identify the interpolated trace. Dense data were thinned to increase sparsity and obtain measurements (Hidden Measurements) for comparison after interpolation. Performance is assessed based on error in capturing hidden measurements. Finally, the effect of minutely versus hourly sampling of the interpolated trace on key GC performance statistics was investigated using retrospective data received from STAR GC in Christchurch Hospital ICU, New Zealand (2011-2015). All of the piecewise functions performed considerably better than the fitted interpolation functions. Linear piecewise interpolation performed the best having a mean RMSE 0.39 mmol/L, within 2 standard deviations of the BG sensor error. Minutely sampled BG resulted in significantly different key GC performance values when compared to raw sparse BG measurements. Linear piecewise interpolation provides the best estimate of intermediate BG dynamics and all analyses comparing GC protocol performance should use minutely linearly interpolated BG data.

An element decomposition method for heat transfer analysis

In this work, an element decomposition method (EDM) is formulated to deal with heat transfer analysis. For this method, the quadrilateral elements are first divided into sub-triangular cells and the local temperature gradient in each sub-triangular cell is obtained using linear interpolation function. Then, the temperature gradient of whole quadrilateral can be calculated by averaging the local gradient in each sub-domain. As only one integration point is utilized for each element, the computational cost of presented method is much less and no mapping or coordinate transformation is involved. Several numerical examples are given to fully test the validity of this method and it is found that the EDM is very accurate, stable and can achieve high level of convergence when solving heat transfer problems.

Integer and Fractional Order-Based Viscoelastic Constitutive Modeling to Predict the Frequency and Magnetic Field-Induced Properties of Magnetorheological Elastomer

Magnetorheological elastomer (MRE)-based semi-active vibration mitigation device demands a mathematical representation of its smart characteristics. To model the material behavior over broadband frequency, the simplicity of the mathematical formulation is very important. Material modeling of MRE involves the theory of viscoelasticity, which describes the properties intermediate between the solid and the liquid. In the present study, viscoelastic property of MRE is modeled by an integer and fractional order derivative approaches. Integer order-based model comprises of six parameters, and the fraction order model is represented by five parameters. The parameters of the model are identified by minimizing the error between the response from the model and the dynamic compression test data. Performance of the model is evaluated with respect to the optimized parameters estimated at different sets of regularly spaced arbitrary input frequencies. A linear and quadratic interpolation function is chosen to generalize the variation of parameters with respect to the magnetic field and frequency. The predicted response from the model revealed that the fractional order model describes the properties of MRE in a simplest form with reduced number of parameters. This model has a greater control over the real and imaginary part of the complex stiffness, which facilitates in choosing a better interpolating function to improve the accuracy. Furthermore, it is confirmed that the realistic assessment on the performance of a model is based on its ability to reproduce the results obtained from optimized parameters.

STATISTICAL PROPERTIES OF LINEAR FRACTAL INTERPOLATION FUNCTIONS FOR RANDOM DATA SETS

Let [Formula: see text] be an integer greater than or equal to [Formula: see text] and let [Formula: see text] be numbers with [Formula: see text]. Denote that [Formula: see text] is the interval [Formula: see text] and [Formula: see text] is a set of points. Suppose that [Formula: see text] is a random perturbation of [Formula: see text] for [Formula: see text], and we set [Formula: see text]. Let [Formula: see text] and [Formula: see text] be linear fractal interpolation functions on [Formula: see text] corresponding to the set of points [Formula: see text] and [Formula: see text], respectively. The value [Formula: see text] is random for all [Formula: see text]. In this paper, we show that the expectation of [Formula: see text] is [Formula: see text]. We also establish estimations for the variance of [Formula: see text] and the expectation of [Formula: see text].

Optimization temperature in the room using air conditioner with finite element methods

Room temperature is one of the problems in everyday life that is often discussed. One of the tools in regulating the temperature is Air Conditioner (AC). As is known, that the AC work process in cooling the room is the process of heat transfer. This study aims to find out how the AC works in regulating the temperature in a room. In the process, this research is accomplished by implementing finite element method on the energy transfer equation. Where the energy transfer equation is the partial differential used for heat transfer. In the finite element method, the flow field is broken down into a set of small fluid elements (discrete domains). In the solution, use it in three-dimensional space, then select the linear interpolation function for the 3D element, and derive the matrix and vector elements by the Galerkin method to obtain global equations. Results from computer-assisted studies show the temperature distribution in the room of COMSOL Multiphysic 5.0. Simulation results with COMSOL show that there is a correlation between AC location and solar radiation indoors against indoor temperature.

Model for optimal management of the cooling system of a fuel cell-based combined heat and power system for developing optimization control strategies

This paper is focused on the development of a model for achieving optimal control of the cooling system of a polymer electrolyte membrane fuel cell (PEMFC)-based cogeneration system. This model is developed to help facilitate the development and application of control strategies to maximize the energy efficiencies of PEMFCs, so that the costs associated with electric and thermal generation can be reduced. The results of experimental analysis conducted using an actual PEMFC-based combined heat and power system that can produce 600 W of electrical power are presented. Then, the development and validation of a simulation model of the experimental system are discussed. This model is based on a combination of an artificial neural network (ANN) with a non-linear autoregressive exogenous configuration and a 3D lookup table (LUT) that updates the data input into the ANN as a function of the electrical power demand and the flow rate and input temperature of the coolant fluid. Due to the nonlinearity of the data contained in the 3D LUT, an algorithm based on linear interpolation and shape-preserving piecewise cubic Hermite dynamic functions is implemented to interpolate the data in 3D. As a result, the model can predict the outlet temperature of the coolant fluid and hydrogen consumption rate of the PEMFC as functions of the inlet temperature and flow rate of the coolant fluid and the electrical power demand. The proposed model exhibits high accuracy and can be used as a black box for the development of new optimization strategies.

The density-of-states contributions to the negative field charge drift mobility effect in poly(3-hexylthiophene) organic semiconductor

The origin of the empirical linear dependence of Eint the electric field at the charge-injecting metal/poly(3-hexylthiophene), P3HT, interfaces on the externally applied bias, Ea, that governs its room temperature negative field charge drift mobility is investigated. The published electrostatic model is modified in the sense that the energetic shift φ of the Gaussian disordered hole and electron energy states is determined to be a linear function of the externally applied electric field, Ea. On this basis the distorted Gaussian shaped interfacial electric field Eint is obtained and the empirical Eint is identified as the linear interpolation function of positive slope through the inflection point of the calculated curve. The disordered energy states extend throughout the P3HT charge transport gap and the populations of interface charges follow unequal Ea dependences. The coupling between the deduced interfacial electric field Eint and the P3HT effective mobility enables the predictions to be compared to the published time-of-flight room temperature negative field drift mobility data of holes and electrons in various metal/P3HT sandwich-type organic structures.

A High-Order Accurate Numerical Scheme for the Caputo Derivative with Applications to Fractional Diffusion Problems

ABSTRACTIn this paper, using the piecewise linear and quadratic Lagrange interpolation functions, we propose a novel numerical approximate method for the Caputo fractional derivative. For the obtained explicit recursion formula, the truncation error is investigated, which shows the involved convergence order is O(τ3−β) with β∈(0,1). As an application, we use this proposed numerical approximation to solve the time fractional diffusion equations by the barycentric rational interpolations in space. The resultant systems of algebraic equations, truncation error, convergence, and stability are analyzed. Theoretical analysis and numerical examples show this constructed method enjoys accuracy of , where d is the degree of the rational polynomial.

A C0 three-node triangular element based on preprocessing approach for thick sandwich plates

This paper will try to overcome two difficulties encountered by the C0 three-node triangular element based on the displacement-based higher-order models. They are (i) transverse shear stresses computed from constitutive equations vanish at the clamped edges, and (ii) it is difficult to accurately produce the transverse shear stresses even using the integration of the three-dimensional equilibrium equation. Invalidation of the equilibrium equation approach ought to attribute to the higher-order derivations of displacement parameters involved in transverse shear stress components after integrating three-dimensional equilibrium equation. Thus, the higher-order derivatives of displacement parameters will be taken out from transverse shear stress field by using the three-field Hu–Washizu variational principle before the finite element procedure is implemented. Therefore, such method is named as the preprocessing method for transverse shear stresses in present work. Because the higher-order derivatives of displacement parameters have been eliminated, a C0 three-node triangular element based on the higher-order zig-zag theory can be presented by using the linear interpolation function. Performance of the proposed element is numerically evaluated by analyzing multilayered sandwich plates with different loading conditions, lamination sequences, material constants and boundary conditions, and it can be found that the present model works well in the finite element framework.

How should we interpret retrospective blood glucose measurements? Sampling and Interpolation

This study investigates blood glucose (BG) measurement interpolation techniques to represent intermediate BG dynamics, and the effect resampling of retrospective BG data has on key glycemic control (GC) performance results. Many GC protocols in the ICU have varying BG measurement intervals with gaps ranging from 0.5 to 4 hrs. Sparse data poses problems in model fitting techniques and GC performance comparisons, and thus interpolation is required to assume a continuous solution.Retrospective data from SPRINT in the Christchurch Hospital Intensive Care Unit (ICU) (2005-2007) was used to analyze various interpolation techniques. Piece-wise linear, spline and cubic interpolation functions, which force lines through data, as well as 1st and 2nd Order B-spline basis functions, used to identify the data, are investigated. Dense data was thinned to increase sparsity and obtain measurements (Hidden measurements) for comparison after interpolation. All of the piece-wise functions performed considerably better than the fitted interpolation functions. Linear piece-wise interpolation performed the best having a mean RMSE 0.39 mmol/L, within 2 standard deviations of the BG sensor error.The effect of minutely vs hourly sampling of the interpolated trace on key GC performance statistics was investigated using the retrospective data received from STAR GC in the Christchurch Hospital Intensive Care Unit (ICU), New Zealand (2011-2015). Minutely sampled BG resulted in significantly different key GC performance when compared to raw sparse BG measurements. Linear piece-wise interpolation provides the best estimate of intermediate BG dynamics and all analyses comparing GC protocol performance should use minutely linearly interpolated BG data.

An Assessment of Mean Areal Precipitation Methods on Simulated Stream Flow: A SWAT Model Performance Assessment

Accurate mean areal precipitation (MAP) estimates are essential input forcings for hydrologic models. However, the selection of the most accurate method to estimate MAP can be daunting because there are numerous methods to choose from (e.g., proximate gauge, direct weighted average, surface-fitting, and remotely sensed methods). Multiple methods (n = 19) were used to estimate MAP with precipitation data from 11 distributed monitoring sites, and 4 remotely sensed data sets. Each method was validated against the hydrologic model simulated stream flow using the Soil and Water Assessment Tool (SWAT). SWAT was validated using a split-site method and the observed stream flow data from five nested-scale gauging sites in a mixed-land-use watershed of the central USA. Cross-validation results showed the error associated with surface-fitting and remotely sensed methods ranging from −4.5 to −5.1%, and −9.8 to −14.7%, respectively. Split-site validation results showed the percent bias (PBIAS) values that ranged from −4.5 to −160%. Second order polynomial functions especially overestimated precipitation and subsequent stream flow simulations (PBIAS = −160) in the headwaters. The results indicated that using an inverse-distance weighted, linear polynomial interpolation or multiquadric function method to estimate MAP may improve SWAT model simulations. Collectively, the results highlight the importance of spatially distributed observed hydroclimate data for precipitation and subsequent steam flow estimations. The MAP methods demonstrated in the current work can be used to reduce hydrologic model uncertainty caused by watershed physiographic differences.

3-D Modeling for Airborne EM using the Spectral-element Method

The 3-D spectral-element method (SE) based on Gauss–Lobatto–Legendre (GLL) polynomials is a very efficient and accurate solver for high-frequency computational electromagnetic (EM) modeling. Although the SE method is based on a weighted residual technique similar to the finite-element method (FE), it utilizes polynomial interpolation functions rather than linear interpolation functions. The GLL polynomials have the characteristic of exponential convergence with the order of polynomial that helps improve the modeling accuracy. We apply the SE method in the frequency range (900 Hz to 56 kHz) to airborne EM modeling. Starting from the vector Maxwell's equations, we use the SE method to establish spatial-discrete forms of 3-D vector Helmholtz equations and adopt GLL integration to calculate the matrix elements. The magnetic field is calculated using Faraday's law. To check the accuracy of the SE algorithm, we compare our results with semi-analytical solutions for a homogeneous half-space and a layered earth model. We further analyze the influence of grids, expansion of outside boundary of model domain, and the order of interpolation polynomial on EM modeling accuracy and reveal that increasing the SE polynomial order can especially improve the accuracy for complex models. Finally, we demonstrate the feasibility of our SE algorithm for modeling an airborne EM system by numerical 3-D experiments.

Oscillation of Runge-Kutta methods for advanced impulsive differential equations with piecewise constant arguments

The purpose of this paper is to study oscillation of Runge-Kutta methods for linear advanced impulsive differential equations with piecewise constant arguments. We obtain conditions of oscillation and nonoscillation for Runge-Kutta methods. Moreover, we prove that the oscillation of the exact solution is preserved by the θ-methods. It turns out that the zeros of the piecewise linear interpolation functions of the numerical solution converge to the zeros of the exact solution. We give some numerical examples to confirm the theoretical results.