Load Vector Research Articles

An [formula omitted][formula omitted]-version FEM in two dimensions: Preconditioning and post-processing

The Bernstein polynomials have been known for over a century and are widely used in the spline literature, computer aided geometric design, and computer graphics. However, the realization that the Bernstein basis has favorable properties allowing the efficient implementation of high order methods for the approximation of partial differential equations is a relatively recent development. For instance, it is known (Ainsworth et al., 2011) that the Bernstein basis can be exploited to compute all of the entries in the load vector in O(p3) operations even in the case of non-linear problems on curvilinear elements for a degree p approximation. Moreover, the element matrices can be assembled in O(1) operations per entry. We show that properties of the Bernstein polynomials can also be exploited to obtain O(p3) complexity procedures for all of the main components needed to implement a high order finite element code including: computation of the residuals needed for an iterative solution method; evaluating the action of a preconditioner for the global mass matrices; and, visualization and post-processing of the resulting finite element approximations.The construction of a preconditioner for the mass matrix whose condition number does not degenerate with the order p, at a cost of O(p3) operations, is one of the main contributions of the present work. The preconditioner is based on an abstract Additive Schwarz Method recently developed by the authors. The preconditioner can be implemented at a cost of O(p3) operations by exploiting properties of the Bernstein polynomials. In particular, we present an algorithm which allows one to invert the interior block of the element mass matrix in O(p3) operations. Numerical examples are provided to illustrate the applicability of the Bernstein basis to challenging non-linear reaction–diffusion problems, non-linear wave propagation of solitons and to robust approximation of problems exhibiting boundary layers.

Heterogeneous Network Selection Algorithm Based on Principal Component Analysis

Multiple attribute decision making (MADM) methods have shown effectiveness for network selection of heterogeneous network system, and analytic hierarchy process (AHP) is often applied to determine subjective attribute weights. However, APH method needs constructing appropriate judgment matrix and cannot address different business scenes. In this paper, we propose a novel selection algorithm for heterogeneous network system by introducing principal component analysis (PCA). By locating key attributes of different subjective demands, the whole loading vectors of PCA model can be divided into several blocks to fit different scenes. Then, the loading vectors can be used to construct subjective weights. Furthermore, we apply Criteria importance though intercrieria correlation (CITIC) method based objective weight to construct a combined weight. Therefore, the subjective demand and network objective feature can be considered jointly to select the optimal network. Simulation results show that the proposed approach can select the optimal network for different business scenes.

Advanced 3-D beam element including warping and distortional effects for the analysis of spatial framed structures

In this paper an advanced 32 × 32 stiffness matrix and the corresponding nodal load vector of a 3-D beam element of arbitrary cross section taking into account shear deformation, generalized warping (shear lag effects) and distortional effects due to both flexure and torsion is presented. Nonuniform distortional/warping distributions are taken into account by employing 10 additional degrees of freedom per node. Local stiffness matrix and local equivalent nodal load vector of the element are computed numerically applying the Analog Equation Method (AEM), a Boundary Element Method (BEM) based technique. Warping and distortional functions as well as geometric constants of the cross section are evaluated employing a 2-D BEM approach. The developed element is incorporated into a standard Direct Stiffness Method (DSM) algorithm employed for the solution of spatial frames. The problem of handling distortional/warping transmission conditions in non-aligned members is alleviated by using an approximate technique for a proper transformation of higher-order degrees of freedom at frame joints.

Stiffness matrix and loading vector of a two-layer Timoshenko composite beam

The stiffness matrix, transfer functions and the corresponding loading vector of a two-layer prismatic Timoshenko beam that includes the transversal and longitudinal relative displacements along the interface, the interaction between the normal and shear stresses and the effects of the coupling between the bending and shear deformations are derived in a classical manner. The proposed method can be used in the structural analysis and design of coupled elements and frame systems of composite elements such as simply and continuous steel-concrete composite beams. The proposed expressions for the load vector are general for any type or combination of transverse loads that fit a second-order polynomial curve, including uniformly, trapezoidal and parabolic distributed transverse loads. The transfer functions necessary to determine the adhesive stresses, axial and shear forces, bending moments, deflections and rotations along the member are also derived. Two comprehensive examples are presented to show the effectiveness and validity of the proposed method and corresponding equations.

Autonomous supervision and optimization of product quality in a multi-stage manufacturing process based on self-adaptive prediction models

In modern manufacturing facilities, there are basically two essential phases for assuring high production quality with low (or even zero) defects and waste in order to save costs for companies. The first phase concerns the early recognition of potentially arising problems in product quality, the second phase concerns proper reactions upon the recognition of such problems. In this paper, we address a holistic approach for handling both issues consecutively within a predictive maintenance framework at an on-line production system. Thereby, we address multi-stage functionality based on (i) data-driven forecast models for (measure-able) product quality criteria (QCs) at a latter stage, which are established and executed through process values (and their time series trends) recorded at an early stage of production (describing its progress), and (ii) process optimization cycles whose outputs are suggestions for proper reactions at an earlier stage in the case of forecasted downtrends or exceeds of allowed boundaries in product quality. The data-driven forecast models are established through a high-dimensional batch time-series modeling problem. In this, we employ a non-linear version of PLSR (partial least squares regression) by coupling PLS with generalized Takagi–Sugeno fuzzy systems (termed as PLS-fuzzy). The models are able to self-adapt over time based on recursive parameters adaptation and rule evolution functionalities. Two concepts for increased flexibility during model updates are proposed, (i) a dynamic outweighing strategy of older samples with an adaptive update of the forgetting factor (steering forgetting intensity) and (ii) an incremental update of the latent variable space spanned by the directions (loading vectors) achieved through PLS; the whole model update approach is termed as SAFM-IF (self-adaptive forecast models with increased flexibility). Process optimization is achieved through multi-objective optimization using evolutionary techniques, where the (trained and updated) forecast models serve as surrogate models to guide the optimization process to Pareto fronts (containing solution candidates) with high quality. A new influence analysis between process values and QCs is suggested based on the PLS-fuzzy forecast models in order to reduce the dimensionality of the optimization space and thus to guarantee high(er) quality of solutions within a reasonable amount of time (→ better usage in on-line mode). The methodologies have been comprehensively evaluated on real on-line process data from a (micro-fluidic) chip production system, where the early stage comprises the injection molding process and the latter stage the bonding process. The results show remarkable performance in terms of low prediction errors of the PLS-fuzzy forecast models (showing mostly lower errors than achieved by other model architectures) as well as in terms of Pareto fronts with individuals (solutions) whose fitness was close to the optimal values of three most important target QCs (being used for supervision): flatness, void events and RMSEs of the chips. Suggestions could thus be provided to experts/operators how to best change process values and associated machining parameters at the injection molding process in order to achieve significantly higher product quality for the final chips at the end of the bonding process.

Prediction of Lignin Contents from Infrared Spectroscopy: Chemical Digestion and Lignin/Biomass Ratios of Cryptomeria japonica.

A method for the high-throughput analysis of the relative lignin contents of Cryptomeria japonica samples over a wide concentration range (3-73%), independent of the type of chemical pretreatment, was developed by using Fourier transform infrared spectroscopy. First, the assignments of the infrared absorbance related to lignin were reviewed. Then, various chemical treatments, including alkaline, acid, and hydrothermal processes, and a sodium chlorite oxidation treatment, were performed to prepare samples containing a wide range of different lignin contents. Principal component analysis indicated high variability among the chemical treatments in terms of the corresponding lignin contents as well as the resulting changes in the chemical structure of hemicellulose; this conclusion was supported by the loading vectors. The intensity of the key band of lignin at 1508cm-1 was calculated using the absorbance at 2900cm-1 as a reference; a reliable calibration curve with an R2 of 0.968 was obtained independent of the chemical treatment performed. This simple and rapid method for determining the lignin content is expected to be widely applicable for optimizing bioethanol production, as well as monitoring biomass degradation processes.

Sparse Principal Component Analysis With Preserved Sparsity Pattern.

Principal component analysis (PCA) is widely used for feature extraction and dimension reduction in pattern recognition and data analysis. Despite its popularity, the reduced dimension obtained from the PCA is difficult to interpret due to the dense structure of principal loading vectors. To address this issue, several methods have been proposed for sparse PCA, all of which estimate loading vectors with few non-zero elements. However, when more than one principal component is estimated, the associated loading vectors do not possess the same sparsity pattern. Therefore, it becomes difficult to determine a small subset of variables from the original feature space that have the highest contribution in the principal components. To address this issue, an adaptive block sparse PCA method is proposed. The proposed method is guaranteed to obtain the same sparsity pattern across all principal components. Experiments show that applying the proposed sparse PCA method can help improve the performance of feature selection for image processing applications. We further demonstrate that our proposed sparse PCA method can be used to improve the performance of blind source separation for functional magnetic resonance imaging data.

A Novel Improved Maximum Entropy Regularization Technique and Application to Identification of Dynamic Loads on the Coal Rock

A new signal processing algorithm was proposed to identify the dynamic load acting on the coal-rock structure. First, the identification model for dynamic load is established through the relationship between the uncertain load vector, and the assembly matrix of the responses was measured by the machinery dynamic system. Then, the entropy item of maximum entropy regularization (MER) is redesigned using the robust estimation method, and the elongated penalty function according to the ill-posedness characteristics of load identification, which was named as a novel improved maximum entropy regularization (IMER) technique, was proposed to process the dynamic load signals. Finally, the load identification problem is transformed into an unconstrained optimization problem and an improved Newton iteration algorithm was proposed to solve the objective function. The result of IMER technique is compared with MER technique, and it is found that IMER technique is available for analyzing the dynamic load signals due to higher signal-noise ratio, lower restoration time, and fewer iterative steps. Experiments were performed to investigate the effect on the performance of dynamic load signals identification by different regularization parameters and calculation parameters, pi, respectively. Experimental results show that the identified dynamic load signals are closed to the actual load signals using IMER technique combined with the proposed PSO-L regularization parameter selection method. Selecting optimal calculated parameters pi is helpful to overcome the ill-condition of dynamic load signals identification and to obtain the stable and approximate solutions of inverse problems in practical engineering. Meanwhile, the proposed IMER technique can also play a guiding role for the coal-rock interface identification.

Cross Wind Effects on Vehicle–Track Interactions: A Methodology for Dynamic Model Construction

An efficient and accurate model for the dynamic assessment of vehicle-track behavior subjected to cross wind actions is developed in this paper, where the wind–vehicle–track interaction is regarded as a coupled vibration system. First, a vehicle–track interaction model is proposed by taking the hypotheses of wheel/rail rigid contact and displacement complementarity. Unlike explicit force-based methods, the vehicle-track systems are wholly coupled by interaction matrices and load vectors, which are computationally more efficient than most of the existing methods and fairly accurate in low frequency vibrations. Then, the fluctuating cross winds are simulated by the fast Fourier transform technique from spectral representations with the consideration of spatial correlation of multipoint wind time histories and vehicle movement. The unsteady cross wind forces are obtained by introducing weighting function. Finally, a modeling framework, with the coupled interactions between cross winds, vehicle, and the tracks included, is built effectively. Through the validated dynamic model, the cross wind effects on vehicle-track dynamic performance can be fully revealed. Besides, it is concluded that the dynamic performance of vehicle-track systems differs significantly in various excitation modes, i.e., average cross wind, fluctuating cross wind, and track irregularities.

Learning an Unknown Network State in Routing Games

We study learning dynamics induced by myopic travelers who repeatedly play a routing game on a transportation network with an unknown state. The state impacts cost functions of one or more edges of the network. In each stage, travelers choose their routes according to Wardrop equilibrium based on public belief of the state. This belief is broadcasted by an information system that observes the edge loads and realized costs on the used edges, and performs a Bayesian update to the prior stage’s belief. We show that the sequence of public beliefs and edge load vectors generated by the repeated play converge almost surely. In any rest point, travelers have no incentive to deviate from the chosen routes and accurately learn the true costs on the used edges. However, the costs on edges that are not used may not be accurately learned. Thus, learning can be incomplete in that the edge load vector at rest point and complete information equilibrium can be different. We present some conditions for complete learning and illustrate situations when such an outcome is not guaranteed.

Free and forced vibration analysis of multiple cracked FGM multi span continuous beams using dynamic stiffness method

Abstract In the present paper, the free and forced vibration of multiple cracked multi span continuous beams made of functionally graded material (FGM) is investigated by the dynamic stiffness method. First, there are constructed dynamic stiffness matrix and nodal load vector of multiple cracked FGM beam element using Timoshenko beam theory and massless double spring model of crack. Then, effect of cracks parameters on free and forced vibration of the FGM continuous beams is examined. The theoretical developments are validated by numerical examples. The obtained results provide an efficient method to analyze free and forced vibration of multiple cracked FGM framed structures and assessment of the behavior of damaged structures.

基于特征值分析的多尺度结构优化设计方法

A multi-scale structure optimization method was proposed based on eigenvlue analysis, to find the macrostructure and microstructure of maximum macro stiffness under the worst load. The constraint that the Euclidian norm of the uncertain load is 1 was introduced, the structural compliance was calculated according to the Rayleigh-Ritz theorem, and the compliance was transformed to a symmetric matrix with the same dimensions as the local load vector. In this way, the compliance minimization problem under the worst load was transformed to the minimum problem of the maximum eigenvalue of the symmetric matrix. Moreover, the worst load case was determined with the eigenvector corresponding to the maximum eigenvalue of the matrix. Several numerical experiments demonstrated the validity of the proposed method, and illustrated the reasonability of the macro topological structure and the micro material distribution. The proposed multi-scale optimization method has virtues of iterative stability and rapid convergence. The update of the density function in the topological optimization was performed based on sensitivity analysis and the method of moving asymptotes (MMA).

Improved finite element of a transversely cracked straight beam with an additional degree of freedom

Abstract A new finite element with an additional degree of freedom at the crack location is derived for static bending analysis of a transversely cracked uniform slender beam. In the simplified computational model, which is based on Euler-Bernoulli's theory of small displacements, the crack is represented by a linear rotational spring connecting two elastic parts. The derivation of the transverse displacements, the coefficients of the stiffness matrix as well as the load vector for uniformly distributed load along the whole beam element was based on the utilization of polynomial interpolation functions of the fourth degree and all derived expressions were obtained in the closed form. The novelty of the new model, by comparison to the previously presented simplified finite element models, is that the transverse displacements functions obtained by utilization of the newly presented interpolation functions for the case of uniform continuous transverse load along whole beam element, as well as the functions of the bending moments and transverse forces, are accurate. The values obtained by the simplified model also exhibited good agreement in additional comparison with the results from more demanding and more detailed 2D models.

Active Vibration Control of Launch Vehicle on Satellite Using Piezoelectric Stack Actuator

Satellites are subject to various severe vibration during different phases of flight. The concept of satellite smart adapter is proposed in this study to achieve active vibration control of launch vehicle on satellite. The satellite smart adapter has 18 active struts in which the middle section of each strut is made of piezoelectric stack actuator. Comprehensive conceptual design of the satellite smart adapter is presented to indicate the design parameters, requirements and philosophy applied which are based on the reliability and durability criterions to ensure successful functionality of the proposed system, also fabrication process of the proposed piezoelectric stack actuator is discussed in detail. The coupled electromechanical virtual work equation for the piezoelectric stack actuator in each active strut is derived by applying D'Alembert's principle to calculate the consistent mass matrix, the stiffness matrix and the load vector using finite element approximation. Modal analysis is performed to characterize the inherent properties of the smart adapter and extraction of a mathematical model of the system. Active vibration control analysis was conducted using fuzzy logic control with triangular membership functions and acceleration feedback. The control results conclude that the proposed satellite smart adapter configuration which benefits from piezoelectric stack actuator as elements of its 18 active struts has high strength and shows excellent robustness and effectiveness in vibration suppression of launch vehicle on satellite.

Improvements in Shear Locking and Spurious Zero Energy Modes Using Chebyshev Finite Element Method

In this paper, the authors present Chebyshev finite element (CFE) method for the analysis of Reissner–Mindlin (RM) plates and shells. Chebyshev polynomials are a sequence of orthogonal polynomials that are defined recursively. The values of the polynomials belong to the interval [−1,1] and vanish at the Gauss points (GPs). Therefore, high-order shape functions, which satisfy the interpolation condition at the points, can be performed with Chebyshev polynomials. Full gauss quadrature rule was used for stiffness matrix, mass matrix and load vector calculations. Static and free vibration analyses of thick and thin plates and shells of different shapes subjected to different boundary conditions were conducted. Both regular and irregular meshes were considered. The results showed that by increasing the order of the shape functions, CFE automatically overcomes shear locking without the formation of spurious zero energy modes. Moreover, the results of CFE are in close agreement with the exact solutions even for coarse and irregular meshes.

Damage detection in frame structures using noisy accelerometers and Damage Load Vectors (DLV)

In the domain of damage detection, many notable methods have been introduced over the past years. Damage Load Vectors (DLV) is among the most powerful methods, which computes a set of load vectors from variations in exibility matrices of a frame in undamaged and damaged conditions. These exibility matrices are derived from acceleration responses of the frame that can be captured using accelerometers. The DLV method then scrutinizes this shift among exibility matrices, which ultimately enables locating the damaged member(s). This study holistically conducted seven experimental tests with seven damage scenarios of a test frame installed on a semi-harmonic shaking table. The DLV method was subsequently employed to locate the damaged members using recorded frame vibration data that were obtained from 'noisy' accelerometers positioned on the frame at eight predefined locations. The Eigen Realization Algorithm (ERA) alongside Pandy's recommendations was adopted herein to facilitate the generation of accurate exibility matrices derived from the noisy accelerometers. The outcome is very encouraging with the accurate identification of damaged members in all seven damage scenarios without any 'positive-false' and 'negative-false' findings. Additionally, there is a decrease (from 0.045 to 0.289) in the accuracy of Weighted Stress Indices (WSI) index when the number of damaged members is increased. © 2020 Sharif University of Technology.

A new state-space method for exponentially damped linear systems

A new state-space method is presented in this paper for linear systems with multiple exponential damping models. By introducing displacement vector and its derivatives as state variables, the original second-order differential equations involving convolution integrals are transformed into a state-space formulism. The iterative relationships are given to calculate all the coefficient matrices, loading vectors and initial conditions. Based on the extended state-space formulism, the eigensolutions, frequency response functions and dynamic responses can be calculated easily using mature techniques for first-order linear systems. Besides, a direct integration method for time-domain analysis based on the proposed stated-space formulism is presented. Two numerical examples are used to illustrate the proposed approach.

Linear Analysis of Stability of Pitched Roof Frames

Background: In this paper, geometric nonlinear analysis of pitched roof frames was carried out by the stiffness matrix method using stability functions. Objective: This study contributes to a better knowledge of the stability of pitched roof frames, not braced, and therefore of the efficiency in their dimensioning. Method: At first, the argument of the stability functions was set as 0.01. The stiffness matrix of the frame has been assembled, as well as the nodal load vector of the frame. The boundary conditions (support restraint and wind bracing restraint) were introduced for the reduction of this matrix and the nodal load vector. At this stage, the determinant of the reduced stiffness matrix and the reduced nodal displacement vector are calculated. The argument of the stability functions is incremented by 0.01 and the operations are repeated until the determinant of the reduced stiffness matrix changes sign. The argument of the iteration preceding the sign change of the determinant and corresponding to its positive value is taken and refined by a process described in the paper. The buckling loads of the frame members are determined at this stage. Results and Conclusion: The analysis focused on four frames; the obtained results show that the increase in the inclination of the crossbar makes it possible to take full advantage of the “arch effect”. Arch effect is due to the presence of crossbars which have a linear arch shape. Furthermore, the angle as well as the length ratio, between the crossbar and post, influence critical load value.

Spectral model for diagnosis of acute leukemias in whole blood and plasma through Raman spectroscopy.

Acute leukemias are oncohematological diseases that compromise the bone marrow and have a complex diagnostic definition, leading to a high mortality when diagnosed late. This study proposed to determine the spectral differences between whole blood and plasma samples of healthy and leukemic subjects based on Raman spectroscopy (RS), correlating these differences with their resulting biochemical alterations and performing discriminant analysis of the samples (n = 38 whole blood and n = 40 plasma samples). Raman spectra were obtained using a dispersive Raman spectrometer (830-nm wavelength, 280-mW laser power, 30-s exposure time) with a Raman probe. The exploratory analysis based on principal component analysis (PCA) of the blood and plasma sample's spectra showed loading vectors with peaks related to amino acids, proteins, carbohydrates, lipids, and carotenoids, being the spectral differences related to amino acids and proteins for whole blood samples, and mainly carotenoids for plasma samples. Discriminant models based on partial least squares (PLS) and PCA were developed and classified the spectra as healthy or leukemic, with sensitivity of 91.9% (PLS) and 83.9% (PCA), specificity of 100% (both PLS and PCA), and overall accuracy of 96.5% (PLS) and 93.0% (PCA) for the whole blood spectra. In plasma, the sensitivity was 95.7% (PLS) and 11.6% (PCA), specificity of 98% (PLS) and 100% (PCA), and overall accuracy of 97.1% (PLS) and 64.1% (PCA). The study demonstrated that RS is a technique with potential to be applied in the diagnosis of acute leukemias in whole blood samples.

Linear Hypothesis Testing in Dense High-Dimensional Linear Models

ABSTRACTWe propose a methodology for testing linear hypothesis in high-dimensional linear models. The proposed test does not impose any restriction on the size of the model, that is, model sparsity or the loading vector representing the hypothesis. Providing asymptotically valid methods for testing general linear functions of the regression parameters in high-dimensions is extremely challenging—especially without making restrictive or unverifiable assumptions on the number of nonzero elements. We propose to test the moment conditions related to the newly designed restructured regression, where the inputs are transformed and augmented features. These new features incorporate the structure of the null hypothesis directly. The test statistics are constructed in such a way that lack of sparsity in the original model parameter does not present a problem for the theoretical justification of our procedures. We establish asymptotically exact control on Type I error without imposing any sparsity assumptions on model parameter or the vector representing the linear hypothesis. Our method is also shown to achieve certain optimality in detecting deviations from the null hypothesis. We demonstrate the favorable finite-sample performance of the proposed methods, via a number of numerical and a real data example. Supplementary materials for this article are available online.