Piecewise Polynomial Interpolation Research Articles

A semi-analytical wavelet finite element method for wave propagation in rectangular rods

Prior knowledge of the dispersion curves and mode shapes of guided waves provides valuable information for wave mode selection and excitation in the field of non-destructive evaluation (NDE) and structural health monitoring (SHM). They are typically computed by the matrix methods, the finite element (FE) and semi-analytical finite element (SAFE) methods. However, the former is prone to numerical instability, and the latter two are limited by the refinement level of the FE mesh. In this paper, a semi-analytical wavelet finite element (SAWFE) method is presented to characterize wave propagation in rectangular rods. The piecewise polynomial interpolation functions of the SAFE method are replaced by two-dimensional scaling functions of the B-spline wavelet on the interval (BSWI). To demonstrate the accuracy of the proposed SAWFE technique, the propagation of guided waves in an aluminium plate is studied first. Then, the propagation of guided waves in rectangular rods of arbitrary aspect ratio is investigated. The results of this work clearly show that the SAWFE method presented here has higher accuracy and efficiency than the SAFE method.

Enhanced azimuth determination in drilling via piecewise polynomial fitting and interpolation

Enhanced azimuth determination in drilling via piecewise polynomial fitting and interpolation

Setochno-kharakteristicheskiy metod povyshennogo poryadka dlya sistem giperbolicheskikh uravneniy s kusochno-postoyannymi koeffitsientami

A new approach is considered for increasing the order of accuracy of the grid-characteristic method in the region of coefficient jumps. The approach is based on piecewise polynomial interpolation for schemes of the second and third orders of accuracy for the case where the interface between the media is consistent with a finite-difference grid. The method is intended for numerical simulation of the propagation of dynamic wave disturbances in heterogeneous media. Systems of hyperbolic equations with variable coefficients are used to describe the considered physical processes. The description of the numerical method and the results of its testing are given.

Local Cr interpolations on simplicial grids in any dimension

This paper presents a construction of local Cr interpolations on simplicial grids in Rd by using piecewise polynomials of degree k≥2dr+1. To this end, an interpolative decomposition of the set of corresponding multi-indices is introduced. It is shown that the piecewise polynomial interpolation is of Cr continuity provided that the corresponding function is of C2d−1r continuity.

Time-Optimal Trajectory Planning for Woodworking Manipulators Using an Improved PSO Algorithm

Woodworking manipulators are applied in wood processing to promote automatic levels in the wood industry. However, traditional trajectory planning results in low operational stability and inefficiency. Therefore, we propose a method combining 3-5-3 piecewise polynomial (composed of cubic and quintic polynomials) interpolation and an improved particle swarm optimization (PSO) algorithm to study trajectory planning and time optimization of woodworking manipulators. In trajectory planning, we conducted the kinematics analysis to determine the position information of joints at path points in joint space and used 3-5-3 piecewise polynomial interpolation to fit a point-to-point trajectory and ensure the stability. For trajectory time optimization, we propose an improved PSO that adapts multiple strategies and incorporates a golden sine optimization algorithm (Gold-SA). Therefore, the proposed improved PSO can be called GoldS-PSO. Using benchmark functions, we compared GoldS-PSO to four other types of PSO algorithms and Gold-SA to verify its effectiveness. Then, using GoldS-PSO to optimize the running time of each joint, our results showed that GoldS-PSO was superior to basic PSO and Gold-SA. The shortest running time obtained by using GoldS-PSO was 47.35% shorter than before optimization, 8.99% shorter than the basic PSO, and 6.23% shorter than the Gold-SA, which improved the running efficiency. Under optimal time for GoldS-PSO, our simulation results showed that the displacement and velocity of each joint were continuous and smooth, and the acceleration was stable without sudden changes, proving the method’s feasibility and superiority. This study can serve as the basis for the motion control system of woodworking manipulators and provide reference for agricultural and forestry engineering optimization problems.

A high-order combined finite element/interpolation approach for multidimensional nonlinear generalized Benjamin–Bona–Mahony–Burgers equation

A high-order combined finite element/interpolation approach is developed for solving a multidimensional nonlinear generalized Benjamin–Bona–Mahony–Burgers equation subjected to suitable initial and boundary conditions. In the proposed high-order scheme we approximate the time derivative with piecewise polynomial interpolation of second-order and use the finite element discretization of piecewise polynomials of degree q and q+1, where q≥2 is an integer, to approximating the space derivatives. The stability together with the error estimates of the constructed technique are established in W21(Ω)-norm. The analysis suggests that the developed numerical scheme is unconditionally stable, temporal second-order accurate and convergence in space with order q. Furthermore, the new procedure is faster and more efficient than a broad range of numerical methods discussed in the literature for the given initial–boundary value problem. A wide set of examples are performed to confirming the theoretical studies.

Monte Carlo integration of C^r functions with adaptive variance reduction: an asymptotic analysis

The theme of the present paper is numerical integration of C^r functions using randomized methods. We consider variance reduction methods that consist in two steps. First the initial interval is partitioned into subintervals and the integrand is approximated by a piecewise polynomial interpolant that is based on the obtained partition. Then a randomized approximation is applied on the difference of the integrand and its interpolant. The final approximation of the integral is the sum of both. The optimal convergence rate is already achieved by uniform (nonadaptive) partition plus the crude Monte Carlo; however, special adaptive techniques can substantially lower the asymptotic factor depending on the integrand. The improvement can be huge in comparison to the nonadaptive method, especially for functions with rapidly varying rth derivatives, which has serious implications for practical computations. In addition, the proposed adaptive methods are easily implementable and can be well used for automatic integration.

COMPUTATIONAL FEATURES OF CALCULAT-ING EXTENDED CONDUCTOR ANTENNAS BASED ON THE INTEGRAL EQUATIONS METHOD

The numerical solution of integral equations or a system of integral equations (for a system of conductors) consists in their discretization and reduction to a system of linear algebraic equations for the desired function. For discretization, it is possible to use projection methods, for example, the Galerkin method or the collocation method. A system of linear algebraic equations for the class of problems under consideration is characterized by a complete (filled) complex matrix. For conductor antenna structures that are quite arbitrary in geometry and size, systems of linear algebraic equations turn out to be of a high order, and special computational algorithms are required to solve them. The purpose of the work is to study adequate mathematical models for wire antennas with a subsequent description of uniform numerical algorithms based on methods of sampling and approximation of antenna current. With this method, the error in solving integral equations is determined by the error in calculating the elements of the matrix of a system of linear algebraic equations for a given piecewise polynomial approximation of the solution at step h. If the quadrature formula for numerical integration is chosen, then there are two ways to reduce the solution error. An increase in the number of collocation points leads to a rapid increase in the volume of calculations and, consequently, to a rapid increase in the amount of occupied computer memory. Therefore, the question arises of choosing the most profitable method of piecewise polynomial interpolation and discretization step h from the point of view of using computer resources, ensuring an acceptable number of solutions.

The MHM Method for Linear Elasticity on Polytopal Meshes

Abstract The multiscale hybrid-mixed (MHM) method consists of a multi-level strategy to approximate the solution of boundary value problems with heterogeneous coefficients. In this context, we propose a new family of finite elements for the linear elasticity equation defined on coarse polytopal partitions of the domain. The finite elements rely on face degrees of freedom associated with multiscale bases obtained from local Neumann problems with piecewise polynomial interpolations on faces. We establish sufficient conditions on the fine-scale interpolations such that the MHM method is well-posed and optimally convergent under local regularity conditions. Also, a multi-level error analysis demonstrates that the MHM method achieves convergence without refining the coarse partition. The upshot is that the Poincaré and Korn’s inequalities do not degenerate, and then the convergence arises on general meshes. Two- and three-dimensional numerical tests assess theoretical results and verify the robustness of the method on a multi-layer media case.

Validated integration of differential equations with state-dependent delay

We present an implicit method of steps for differential equations with state-dependent delays and validated numerics to rigorously enclose solutions of initial-value problems. Our approach uses a combination of contraction mapping arguments based on a Newton–Kantorovich type theorem and piecewise polynomial interpolation. Completing multiple steps of integration is challenging, and we resolve it by smooth interpolation of the previous solution, resulting in an interval-valued polynomial initial condition for the subsequent step. A set of examples is provided.

On [formula omitted] matrix-based approximate inverse preconditioning technique for diagonal-plus-Toeplitz linear systems from spatial fractional diffusion equations

In this paper, we firstly explore the special structure of the discretized linear systems from the spatial fractional diffusion equations. The coefficient matrices of the resulting discretized systems have a diagonal-plus-Toeplitz structure. Because the resulting Toeplitz matrix is symmetric positive definite (SPD), then we can employ the τ matrix to approximate it. By making use of the piecewise interpolation polynomials, we propose a new approximate inverse preconditioner to handle the diagonal-plus-Toeplitz coefficient matrices. The τ matrix-based approximate inverse (TAI) preconditioning technique can be implemented very efficiently by using discrete sine transforms(DST). Theoretically, we have proved that the spectrum of the resulting preconditioned matrices are clustered around one. Thus, Krylov subspace methods with the proposed preconditioners converge very fast. To demonstrate the efficiency of the new preconditioners, numerical experiments are implemented. The numerical results show that with the proper interpolation node numbers, the performance of the τ-matrix based preconditioning technique is better than the other tested preconditioners.

Exploring Optimality of Piecewise Polynomial Interpolation Functions for Lung Field Modeling in 2D Chest X-Ray Images

In this paper, a landmark based approach, using five different interpolating polynomials (linear, cubic convolution, cubic spline, PCHIP, and Makima) for modeling of lung field region in 2D chest X-ray images have been presented. Japanese Society of Radiological Technology (JSRT) database which is publicly available has been used for evaluation of the proposed method. Selected radiographs are anatomically landmarked using 17 and 16 anatomical landmark points to represent left and right lung field regions, respectively. Local, piecewise polynomial interpolation is then employed to create additional semilandmark points to form the lung contour. Jaccard similarity coefficients and Dice coefficients have been used to find accuracy of the modeled shape through comparison with the prepared ground truth. With the optimality condition of three intermediate semilandmark points, PCHIP interpolation method with an execution time of 5.04873 s is found to be the most promising candidate for lung field modeling with an average Dice coefficient (DC) of 98.20 and 98.54% (for the left and right lung field, respectively) and with the average Jaccard similarity coefficient (JSC) of 96.47 and 97.13% for these two lung field regions. While performance of Makima and cubic convolution is close to the PCHIP with the same optimality condition, i.e., three intermediate semilandmark points, the optimality condition for the cubic spline method is of at least seven intermediate semilandmark points which, however, does not result in better performance in terms of accuracy or execution time.

FEM-PGD BASED TECHNIQUE FOR COLUMN SHAPE OPTIMIZATION AGAINST BUCKLING

This paper presents a simple numerical procedure based upon the projected gradient descent (PGD) and finite element method (FEM) for the shape optimization of laterally restrained columns to attain the maximum elastic buckling load under the specified volumetric constraint. The analysis of the buckling load is achieved via the formulation based on Euler-Bernoulli beam theory, the discretization by the standard finite element technique, and the determination of the least eigenvalue and the corresponding eigenvector via the power method with Rayleigh quotient. In the optimization, the profile of the cross-sectional area of the column is represented by piecewise polynomial interpolation functions. The gradient information and the projection operator required in PGD iterations are obtained explicitly in a closed form. A selected set of results is reported to demonstrate not only the good convergence behavior and accuracy of numerical solutions, but also the capability of the proposed technique to attain the optimal shape of columns for various scenarios.

Solving Nonsmooth and Discontinuous Optimal Power Flow problems via interior-point [formula omitted]-penalty approach

The Nonsmooth and Discontinuous Optimal Power Flow (ND-OPF) is a large-scale, nonsmooth, discontinuous, nonconvex and multi-modal problem. In this problem, discontinuity is related to the representation of Prohibited Operating Zones (POZ), whereas nonsmoothness is related to the representation of Valve-Point Loading Effect (VPLE) in the fuel costs. Due to such features, all approaches that have been proposed for solving this problem are based on heuristics or on mixed integer nonlinear programming reformulations. In this paper, we propose the Piecewise Polynomial Interpolation (PPI) function approach for handling discontinuities related to the POZ constraints, which consists in replacing such constraints by an equivalent set of smooth and continuous equality and inequality constraints. The PPI function is of class C1 and assumes null values at all allowed operating zones and non-null values at all forbidden zones. The model resulting from the PPI approach is the Equivalent Intermediate Model (EIM). For handling nonsmoothness on the EIM, we recast it as an equivalent model, where the VPLE term in the objective function becomes linear, and nonlinear box inequality constraints are introduced. The optimization model that results from such recasts is the Equivalent Smooth and Continuous OPF(ESC-OPF) model proposed, which can be solved by strictly gradient-based methods. Finally, we propose a primal–dual interior point ℓp-penalty approach (with 0<p≤1) for solving the ESC-OPF model, where inequality constraints are penalized by using the modified log-barrier function of Jittortrum–Osborne–Meggido and the equality constraints associated with the PPI function approach are penalized via ℓp lower-order exact penalty functions. Numerical results involving systems with up to 2007 buses, with 282 generating units and 846 POZ have shown that the proposed approach has been able to solve large-scale ND-OPF problems, with acceptable computation times.

Automatic approximation using asymptotically optimal adaptive interpolation

We present an asymptotic analysis of adaptive methods for Lp approximation of functions f ∈ Cr([a, b]), where 1le ple +infty . The methods rely on piecewise polynomial interpolation of degree r − 1 with adaptive strategy of selecting m subintervals. The optimal speed of convergence is in this case of order m−r and it is already achieved by the uniform (nonadaptive) subdivision of the initial interval; however, the asymptotic constant crucially depends on the chosen strategy. We derive asymptotically best adaptive strategies and show their applicability to automatic Lp approximation with a given accuracy ε.

Industrial robot trajectory planning based on improved pso algorithm

A particle swarm optimization (PSO) algorithm which can dynamically adjust learning factors is proposed to solve the problems of low efficiency and unstable operation of traditional industrial robots. The method uses piecewise polynomial interpolation to fit the trajectory, and uses an improved particle swarm algorithm to optimize the trajectory of industrial robots with time as a fitness function. This method effectively combines the piecewise polynomial interpolation function with PSO, avoids the complex process of particle swarm algorithm to construct the adaptation function, and improves the problem that the traditional PSO is more likely to fall into local extreme value in the early stage and convergence speed is slow in the later stage. Through experiments to obtain the motion pose, velocity and acceleration trajectory of each joint of the manipulator, we can know that this method can effectively realize the trajectory optimization of the industrial robot, and ensure the stability of the overall operation while improving the operating efficiency.

ECG Beat Representation and Delineation by Means of Variable Projection.

The electrocardiogram (ECG) follows a characteristic shape, which has led to the development of several mathematical models for extracting clinically important information. Our main objective is to resolve limitations of previous approaches, that means to simultaneously cope with various noise sources, perform exact beat segmentation, and to retain diagnostically important morphological information. We therefore propose a model that is based on Hermite and sigmoid functions combined with piecewise polynomial interpolation for exact segmentation and low-dimensional representation of individual ECG beat segments. Hermite and sigmoidal functions enable reliable extraction of important ECG waveform information while the piecewise polynomial interpolation captures noisy signal features like the baseline wander (BLW). For that we use variable projection, which allows the separation of linear and nonlinear morphological variations of the according ECG waveforms. The resulting ECG model simultaneously performs BLW cancellation, beat segmentation, and low-dimensional waveform representation. We demonstrate its BLW denoising and segmentation performance in two experiments, using synthetic and real data. Compared to state-of-the-art algorithms, the experiments showed less diagnostic distortion in case of denoising and a more robust delineation for the P and T wave. This work suggests a novel concept for ECG beat representation, easily adaptable to other biomedical signals with similar shape characteristics, such as blood pressure and evoked potentials. Our method is able to capture linear and nonlinear wave shape changes. Therefore, it provides a novel methodology to understand the origin of morphological variations caused, for instance, by respiration, medication, and abnormalities.

Real-Time Acceleration-Continuous Path-Constrained Trajectory Planning With Built-In Tradeoff Between Cruise and Time-Optimal Motions

In this article, a novel real-time acceleration-continuous path-constrained trajectory planning algorithm is proposed with an appealing built-in tradeoff mechanism between the cruise motion and time-optimal motion. Different from existing approaches, the proposed approach smoothens time-optimal trajectories with bang-bang input structures to generate acceleration-continuous trajectories while preserving the completeness property. More importantly, a novel built-in tradeoff mechanism is proposed and embedded into the trajectory planning framework so that the proportion of the cruise motion and time-optimal motion can be flexibly adjusted by changing a user-specified functional parameter. Thus, the user can easily apply the trajectory planning algorithm for various tasks with different requirements on motion efficiency and cruise proportion. Moreover, it is shown that feasible trajectories are computed more quickly than optimal trajectories. Rigorous mathematical analysis and proofs are presented for those aforementioned theoretical results. Comparative simulations and experimental results on an omnidirectional wheeled mobile robot demonstrate that flexible tunings between the cruise and time-optimal motions can be achieved in a higher computational efficiency manner by the proposed algorithm. Note to Practitioners —This article is motivated by the time-optimal and smooth motion planning problem for mobile robots along given paths. Existing approaches generally use the piecewise polynomial interpolations to smoothen and adjust feasible trajectories. This article proposes a novel path-constrained trajectory planning approach, which preserves properties of completeness and a high-efficient tradeoff mechanism between the optimal and cruise motions when achieving a globally optimal and acceleration-continuous trajectory. Comparative experimental results with other methods show the effectiveness of the proposed approach. In future research, we will attempt to integrate the proposed approach with typical path planning methods to achieve a complete and high-efficient motion planning framework.

Smooth reconstruction method for surfaces defined by a continuous set of triangles

The paper considers the problem of triangular grids distortion in three-dimensional space, approximating the boundaries and interfaces of interacting solid, liquid and gaseous media. Unacceptable deformations of the mesh cells size and shape can occur during the numerical solution of continuous media dynamics problems. A method and algorithm for rebuilding grids using smooth piecewise polynomial interpolation by algebraic polynomials of the third degree is proposed.

Two novel prediction methods for milling stability analysis based on piecewise polynomial interpolations

Chatter has been one of the major limiting factors leading to poor productivity and surface finish. The prediction of the stable cutting zones of cutting parameters is thus vital in machining. On the basis of piecewise polynomial interpolations, two efficient and accurate methods are developed in this paper for calculating the stability lobes. Firstly, the dynamic milling process with regenerative effect is modeled by delay differential equations (DDEs) with time-periodic coefficients. The principal period of system is easily separated based on the existence of any non-zero elements of coefficient matrix, and the forced vibration phase is equally discretized into a series of subintervals. Piecewise polynomial interpolations are adopted in two contiguous subintervals to estimate all parts of the DDEs. Secondly, the state transition matrix is formed to search for the borderline of stable cutting region based on Floquet theory. Thirdly, the convergence rates of the two proposed methods are analyzed through comparison with four existing algorithms. The results validate that the two proposed approaches achieve higher convergence rate, accuracy, and efficiency than the others within the same time intervals. Finally, the operability of the two proposed methods is verified by cutting tests, which indicate that the two proposed methods are of effectiveness and practicability.