Piecewise Uniform Research Articles

An Initial-Boundary Value Problem With New Technique of Piecewise Uniform Mesh

An Initial-Boundary Value Problem With New Technique of Piecewise Uniform Mesh

Efficient Numerical Scheme for Generalized Black–Scholes Equations on Piecewise Uniform Shishkin-Type Mesh

Efficient Numerical Scheme for Generalized Black–Scholes Equations on Piecewise Uniform Shishkin-Type Mesh

Modelling of damage and plasticity phenomena in 3D printed materials via a multiscale approach

A multiscale technique able to describe the overall mechanical response of 3D printed materials taking into account the plastic behaviour and the occurrence of failure is presented. Reference is made to elements produced by extrusion-based additive manufacturing. At the microscale, the de-cohesion between filaments that constitute the printed material is described introducing a cohesive damage interface model that also takes into account the unilateral contact. The micromechanical problem is solved with a homogenization procedure based on the PieceWise Uniform Transformation Field Analysis properly extended to the case of interfaces. In detail, the plastic strains in the filaments and the inelastic relative displacements along the interfaces are approximated as piecewise constant functions and then are computed solving the evolutive problem. Some numerical applications are carried out, comparing the numerical results with the experimental ones and with results obtained from non-linear finite element analyses, to show the potentiality of the proposed multiscale technique.

Strategyproof and Proportional Chore Division for Piecewise Uniform Preferences

Strategyproof and Proportional Chore Division for Piecewise Uniform Preferences

Iterative Algorithm for Parameterization of Two-Region Piecewise Uniform Quantizer for the Laplacian Source

Motivated by the fact that uniform quantization is not suitable for signals having non-uniform probability density functions (pdfs), as the Laplacian pdf is, in this paper we have divided the support region of the quantizer into two disjunctive regions and utilized the simplest uniform quantization with equal bit-rates within both regions. In particular, we assumed a narrow central granular region (CGR) covering the peak of the Laplacian pdf and a wider peripheral granular region (PGR) where the pdf is predominantly tailed. We performed optimization of the widths of CGR and PGR via distortion optimization per border–clipping threshold scaling ratio which resulted in an iterative formula enabling the parametrization of our piecewise uniform quantizer (PWUQ). For medium and high bit-rates, we demonstrated the convenience of our PWUQ over the uniform quantizer, paying special attention to the case where 99.99% of the signal amplitudes belong to the support region or clipping region. We believe that the resulting formulas for PWUQ design and performance assessment are greatly beneficial in neural networks where weights and activations are typically modelled by the Laplacian distribution, and where uniform quantization is commonly used to decrease memory footprint.

Multiscale technique for the analysis of 3D-printed materials

The mechanical behaviour of structural elements made of 3D-printed materials is numerically investigated. With this aim a multiscale approach that allows to determine the macroscopic response, as depending on the material heterogeneity at the microscale level, is conceived. A non-linear laminate finite element that employs a reduced homogenization technique at each integration Gauss point is developed. In detail, the Piecewise Uniform Transformation Field Analysis is implemented. In order to validate the procedure, some numerical applications are developed. The obtained results are compared with evidence of experimental tensile and bending tests, available in literature. The application of a multiscale strategy, employing a reduced order method at the Gauss point level for the analysis of 3D-printed structural elements, could represent a good compromise in terms of accuracy of the results and computational efficiency.

Gaseous Slip Flow Forced Convection in a Micropipe and Parallel-Plate Microchannel with a Piecewise Uniform Wall Heat Flux and Axial Heat Conduction

Gaseous Slip Flow Forced Convection in a Micropipe and Parallel-Plate Microchannel with a Piecewise Uniform Wall Heat Flux and Axial Heat Conduction

Optimization clustering technique for PieceWise Uniform Transformation Field Analysis homogenization of viscoplastic composites

Aim of the present study is to propose an enhanced method for the domain decomposition (clustering) of the representative volume element (RVE) of composite materials to be used with homogenization techniques, based on the PieceWise Uniform Transformation Field Analysis (PWUTFA). With PWUTFA, both constitutive and evolutive equations for the constituents of the composite material are written in terms of averages in each cluster; moreover, it is not required to solve via FEM the nonlinear micro-mechanical problem, allowing to drastically reduce the number of internal variables. PWUTFA is founded on the idea that it is possible to divide the RVE into large subdomains (clusters) that should group together finite elements having, under any applied loading condition, the most similar values of strain. Accordingly, in this study a multi-objective optimization-based approach is proposed with the aim to simultaneously reduce both the error in the approximation of the strain fields and the number of clusters in which the domain is decomposed. Different clustering solutions, obtained through the proposed optimization approach are analyzed, and the corresponding mechanical responses are compared with the ones obtained by the finite element analysis and by uniform transformation field analysis.

Design of Switched Quantizers and Speech Coding Based on Quasi-Logarithmic Compandor

This article investigates switched quantizers for speech signal depicted with Gaussian probability density function (PDF). Gaussian PDF is better for smaller frame lengths that are represented here. Companding technique results in constant Signal to Quantization Noise Ratio (SQNR). In this paper two approaches are present: quasi-logarithmic (QL) and piecewise uniform (PU) compandor. Simpler compandor directly affects the complexity of hardware realization and expense of given solution, but, on the other hand, also brings to weaker performances. Therefore, a smart choice has to be made. Usage of switched technique leads to better performances. This way, the quality of quantization is improved by dividing the dynamic range of variances into multiple subranges. For each subrange a separate quantizer is designed, with its support region’s amplitude. The optimal amplitude is numerically determined, whereby a single criterion is obtaining the maximal SQNR. Bit rates of these quantizers don’t depend on signal variance, as the fixed length codes are used. The performances of proposed quantizers are demonstrated on real speech signals from the reliable database. Comparison of obtained results with other recent solutions is done in order to show the efficiency of this model.DOI: http://dx.doi.org/10.5755/j01.eie.24.6.22295

Optimal Quantization for Piecewise Uniform Distributions

Abstract Quantization for a probability distribution refers to the idea of estimating a given probability by a discrete probability supported by a finite number of points. In this paper, firstly a general approach to this process is outlined using independent random variables and ergodic maps; these give asymptotically the optimal sets of n-means and the nth quantization errors for all positive integers n. Secondly two piecewise uniform distributions are considered on R: one with infinite number of pieces and one with finite number of pieces. For these two probability measures, we describe the optimal sets of n-means and the nth quantization errors for all n ∈ N. It is seen that for a uniform distribution with infinite number of pieces to determine the optimal sets of n-means for n ≥ 2 one needs to know an optimal set of (n − 1)-means, but for a uniform distribution with finite number of pieces one can directly determine the optimal sets of n-means and the nth quantization errors for all n ∈ N.

Numerical Derivative Using the Piecewise Uniform Mesh

Numerical Derivative Using the Piecewise Uniform Mesh

Multiscale technique for nonlinear analysis of elastoplastic and viscoplastic composites

Abstract Aim of the present paper is to develop an efficient multiscale procedure for studying the mechanical response of structural elements made of elastoplastic or viscoplastic composite materials. The micro and the macro scales are considered separated. At the microscale a PieceWise Uniform Transformation Field Analysis (PWUTFA) homogenization technique is adopted to derive the overall response of a periodic composite. Thus, a Unit Cell (UC) containing all the properties of the heterogeneous material is analyzed and divided in subsets; in each one the inelastic strain is considered uniform, i.e. constant, and represents the history variable of the analysis. Elastoplastic and elasto-viscoplastic models with isotropic hardening are adopted in order to describe the nonlinear response of the constituents. A new numerical procedure is developed in order to solve the evolutive problem in all the subsets simultaneously adopting a predictor-corrector technique. The corrector phase is solved by means of a modified Newton-Raphson iterative procedure. Furthermore, the tangent consistent with the algorithm is computed and adopted in the multiscale computations. Numerical applications are carried out in order to assess the efficiency of the proposed multiscale approach.

Coding Algorithm for Grayscale Images – Design of Piecewise Uniform Quantizer with Golomb–Rice Code and Novel Analytical Model for Performance Analysis

Scalar quantizer selection for processing a signal with a unit variance is a difficult problem, while both selection and quantizer design for the range of variances is even tougher and to the authors’ best knowledge, it is not theoretically solved. Furthermore, performance estimation of various image processing algorithms is unjustifiably neglected and there are only a few analytical models that follow experimental analysis. In this paper, we analyse application of piecewise uniform quantizer with Golomb-Rice coding in modified block truncation coding algorithm for grayscale image compression, propose design improvements and provide a novel analytical model for performance analysis. Besides the nature of input signal, required compression rate and processing delay of the observed system have a strong influence on quantizer design. Consequently, the impact of quantizer range choice is analysed using a discrete designing variance and it was exploited to improve overall quantizer performance, whereas variable-length coding is applied in order to reduce quantizer’s fixed bit-rate. The analytical model for performance analysis is proposed by introducing Inverse Gaussian distribution and it is obtained by discussing a number of images, providing general closed-form solutions for peak-signal-to-noise ratio and the total average bit-rate estimation. The proposed quantizer design ensures better performance in comparison to the other similar methods for grayscale image compression, including linear prediction of pixel intensity and edge-based adaptation, whereas analytical model for performance analysis provides matching with the experimental results within the range of 1 dB for PSQNR and 0.2 bpp for the total average bit-rate.

New Coding Algorithm Based on Variable-Length Codewords for Piecewise Uniform Quantizers

New Coding Algorithm Based on Variable-Length Codewords for Piecewise Uniform Quantizers

A nonuniform TFA homogenization technique based on piecewise interpolation functions of the inelastic field

The present paper deals with a homogenization technique based on the Transformation Field Analysis (TFA) for the study of heterogeneous composite media characterized by nonlinear response. According to the TFA, the behavior of the representative volume element (RVE) is studied accounting for the nonlinear effects by means of the presence of a uniform inelastic strain distribution in the nonlinear constituent of the heterogeneous material. In order to improve the TFA, the assumption of uniformity of the inelastic strain distribution is removed, so that a nonuniform inelastic strain field, better representing the inelasticity distribution in the composite, is considered. In particular, the inelastic strain is represented as a piecewise linear combination of analytical functions of the spatial variable. The theory, presented in a general framework, can be successfully adopted for deriving the overall constitutive response for a wide range of nonlinear composite materials. Furthermore, the procedure is tailored to investigate the response of composites whose constituents are Shape Memory Alloys (SMA) and materials characterized by plastic behavior. Finally, numerical applications are developed in order to assess the effectiveness of the proposed nonuniform TFA procedure, comparing the results with the ones carried out performing Uniform and Piecewise Uniform TFA homogenizations and nonlinear finite element micromechanical analyses.

Design of Forward Adaptive Piecewise Uniform Scalar Quantizer with Optimized Reproduction Level Distribution per Segments

The problem we address in this paper is the design of nearly optimal scalar quantizer in a wide variance range of the Laplacian input signals, using the piecewise uniform quantizers while restricting the class of quantizers to be forward adaptive. Particularly, the design procedure of the piecewise uniform quantizer with an equidistant support region partion and the optimized reproduction level distribution per segments is presented along with the design procedure of its forward adaptive version. Reproduction level optimization is performed by optimizing the granular distortion of the proposed quantizer using the method of the Lagrange multipliers. For the proposed model we study the influence of the segment number on the SQNR, as well as the SQNR robustness in a wide variance range. Since the results obtained for the assumed Laplacian distribution indicate the SQNR improvement over the G.711 standard, one can expect that the proposed quantizer will be effective in the quantization of signals having the same distribution and the time varying characteristics. Ill. 4, bibl. 10 (in English; abstracts in English and Lithuanian).DOI: http://dx.doi.org/10.5755/j01.eee.119.3.1356

Switched Nonuniform and Piecewise Uniform Scalar Quantization of Laplacian Source

In this paper switched nonuniform and piecewise uniform scalar quantization of Laplacian source are analyzed. This scalar quantization techniques are used in order to obtain higher signal quality by increasing signal-to-quantization noise ratio (SNRQ) with respect to it’s necessary robustness over a broad range of input variances in a wide range of signal volumes. We observe μ-law compandor implementation to achieve compromise between high-rate digitalization and variance adaptation. The main contribution of this model is kipping almost the same quality as the nonuniform compandor model, with simpler realization structure and the possibility of it’s applying for digitalization of wide range continuous signals.

Coding Algorithm for Grayscale Images Based on Piecewise Uniform Quantizers

In this paper, a piecewise uniform quantizer for input samples with discrete amplitudes for Laplacian source is designed and analyzed, and its forward adaptation is done. This type of quantizers is very often used in practice for the purpose of compression and coding of already quantized signals. It is shown that the design and the adaptation of quantizers for discrete input samples are different from the design and the adaptation of quantizers for continual input samples. A weighting function for PSQNR (peak signal-to-quantization noise ratio), which is obtained based on probability density function of variance of standard test images is introduced. Experiments are done, applying these quantizers for compression of grayscale images. Experimental results are very well matched to the theoretical results, proving the theory. Adaptive piecewise uniform quantizer designed for discrete input samples gives for 9 to 20 dB higher PSQNR compared to the fixed piecewise uniform quantizer designed for discrete input samples. Also it is shown that the adaptive piecewise uniform quantizer designed for discrete input samples gives higher PSQNR for 1.46 to 3.45 dB compared the adaptive piecewise uniform quantizer designed for continual input samples, which proves that the discrete model is more appropriate for image quantization than continual model.

Convergence Analysis of Weighted Difference Approximations on Piecewise Uniform Grids to a Class of Singularly Perturbed Functional Differential Equations

We consider the numerical approximation of a singularly perturbed time delayed convection diffusion problem on a rectangular domain. Assuming that the coefficients of the differential equation be smooth, we construct and analyze a higher order accurate finite difference method that converges uniformly with respect to the singular perturbation parameter. The method presented is a combination of the central difference spatial discretization on a Shishkin mesh and a weighted difference time discretization on a uniform mesh. A priori explicit bounds on the solution of the problem are established. These bounds on the solution and its derivatives are obtained using a suitable decomposition of the solution into regular and layer components. It is shown that the proposed method is \(L_{2}^{h}\)-stable. The analysis done permits its extension to the case of adaptive meshes which may be used to improve the solution. Numerical examples are presented to demonstrate the effectiveness of the method. The convergence obtained in practical satisfies the theoretical predictions.

A Class of One-dimensional Nonlinear Functions with Multi Return Patterns and with Piecewise Uniform Invariant Density

A Class of One-dimensional Nonlinear Functions with Multi Return Patterns and with Piecewise Uniform Invariant Density