Search results for: conjugate gradient

Authors: Ahmad Alhawarat, Mustafa Mamat, Mohd Rivaie, Ismail Mohd

Abstract:

Conjugate gradient method has been enormously used to solve large scale unconstrained optimization problems due to the number of iteration, memory, CPU time, and convergence property, in this paper we find a new class of nonlinear conjugate gradient coefficient with global convergence properties proved by exact line search. The numerical results for our new βK give a good result when it compared with well-known formulas.

Keywords: conjugate gradient method, conjugate gradient coefficient, global convergence

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Authors: Belloufi Mohammed, Sellami Badreddine

Abstract:

This paper deals with a new nonlinear modified three-term conjugate gradient algorithm for solving large-scale unstrained optimization problems. The search direction of the algorithms from this class has three terms and is computed as modifications of the classical conjugate gradient algorithms to satisfy both the descent and the conjugacy conditions. An example of three-term conjugate gradient algorithm from this class, as modifications of the classical and well known Hestenes and Stiefel or of the CG_DESCENT by Hager and Zhang conjugate gradient algorithms, satisfying both the descent and the conjugacy conditions is presented. Under mild conditions, we prove that the modified three-term conjugate gradient algorithm with Wolfe type line search is globally convergent. Preliminary numerical results show the proposed method is very promising.

Keywords: unconstrained optimization, three-term conjugate gradient, sufficient descent property, line search

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Authors: B. Sellami, Y. Laskri, M. Belloufi

Abstract:

Conjugate gradient methods are an important class of methods for unconstrained optimization, especially for large-scale problems. Recently, they have been much studied. In this paper, a new family of conjugate gradient method is proposed for unconstrained optimization. This method includes the already existing two practical nonlinear conjugate gradient methods, which produces a descent search direction at every iteration and converges globally provided that the line search satisfies the Wolfe conditions. The numerical experiments are done to test the efficiency of the new method, which implies the new method is promising. In addition the methods related to this family are uniformly discussed.

Keywords: conjugate gradient method, global convergence, line search, unconstrained optimization

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Authors: B. Sellami, M. Belloufi

Abstract:

Conjugate gradient methods are an important class of methods for unconstrained optimization, especially for large-scale problems. Recently, they have been much studied. In this paper, we propose a new two-parameter family of conjugate gradient methods for unconstrained optimization. The two-parameter family of methods not only includes the already existing three practical nonlinear conjugate gradient methods, but also has other family of conjugate gradient methods as subfamily. The two-parameter family of methods with the Wolfe line search is shown to ensure the descent property of each search direction. Some general convergence results are also established for the two-parameter family of methods. The numerical results show that this method is efficient for the given test problems. In addition, the methods related to this family are uniformly discussed.

Keywords: unconstrained optimization, conjugate gradient method, line search, global convergence

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Authors: Belloufi Mohammed, Sellami Badreddine

Abstract:

Conjugate gradient methods have played a special role for solving large scale optimization problems due to the simplicity of their iteration, convergence properties and their low memory requirements. In this work, we propose a new class of conjugate gradient methods which ensures sufficient descent. Moreover, we propose a new search direction with the Wolfe line search technique for solving unconstrained optimization problems, a global convergence result for general functions is established provided that the line search satisfies the Wolfe conditions. Our numerical experiments indicate that our proposed methods are preferable and in general superior to the classical conjugate gradient methods in terms of efficiency and robustness.

Keywords: unconstrained optimization, conjugate gradient method, sufficient descent property, numerical comparisons

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Authors: Hussaini Doko Ibrahim, Hamilton Cyprian Chinwenyi, Henrietta Nkem Ude

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In this paper, efforts were made to examine and compare the algorithmic iterative solutions of the conjugate gradient method as against other methods such as Gauss-Seidel and Jacobi approaches for solving systems of linear equations of the form Ax=b, where A is a real n×n symmetric and positive definite matrix. We performed algorithmic iterative steps and obtained analytical solutions of a typical 3×3 symmetric and positive definite matrix using the three methods described in this paper (Gauss-Seidel, Jacobi, and conjugate gradient methods), respectively. From the results obtained, we discovered that the conjugate gradient method converges faster to exact solutions in fewer iterative steps than the two other methods, which took many iterations, much time, and kept tending to the exact solutions.

Keywords: conjugate gradient, linear equations, symmetric and positive definite matrix, gauss-seidel, Jacobi, algorithm

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Authors: Mohammed Belloufi, Rachid Benzine, Badreddine Sellami

Abstract:

Conjugate gradient methods is useful for solving large scale optimization problems in scientific and engineering computation, characterized by the simplicity of their iteration and their low memory requirements. It is well known that the search direction plays a main role in the line search method. In this paper, we propose a search direction with the Wolfe line search technique for solving unconstrained optimization problems. Under the above line searches and some assumptions, the global convergence properties of the given methods are discussed. Numerical results and comparisons with other CG methods are given.

Keywords: unconstrained optimization, conjugate gradient method, strong Wolfe line search, global convergence

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Authors: Tsegay Giday Woldu, Haibin Zhang, Xin Zhang, Yemane Hailu Fissuh

Abstract:

It is well known that nonlinear conjugate gradient method is one of the widely used first order methods to solve large scale unconstrained smooth optimization problems. Because of the low memory requirement, attractive theoretical features, practical computational efficiency and nice convergence properties, nonlinear conjugate gradient methods have a special role for solving large scale unconstrained optimization problems. Large scale optimization problems are with important applications in practical and scientific world. However, nonlinear conjugate gradient methods have restricted information about the curvature of the objective function and they are likely less efficient and robust compared to some second order algorithms. To overcome these drawbacks, the new modified nonlinear conjugate gradient method is presented. The noticeable features of our work are that the new search direction possesses the sufficient descent property independent of any line search and it belongs to a trust region. Under mild assumptions and standard Wolfe line search technique, the global convergence property of the proposed algorithm is established. Furthermore, to test the practical computational performance of our new algorithm, numerical experiments are provided and implemented on the set of some large dimensional unconstrained problems. The numerical results show that the proposed algorithm is an efficient and robust compared with other similar algorithms.

Keywords: conjugate gradient method, global convergence, large scale optimization, sufficient descent property

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Authors: Sergey Kopysov, Nikita Nedozhogin, Leonid Tonkov

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The article presents a parallel iterative solver for large sparse linear systems which can be used on a heterogeneous platform. Traditionally, the problem of solving linear systems does not scale well on multi-CPU/multi-GPUs clusters. For example, most of the attempts to implement the classical conjugate gradient method were at best counted in the same amount of time as the problem was enlarged. The paper proposes the pipelined variant of the conjugate gradient method (PCG), a formulation that is potentially better suited for hybrid CPU/GPU computing since it requires only one synchronization point per one iteration instead of two for standard CG. The standard and pipelined CG methods need the vector entries generated by the current GPU and other GPUs for matrix-vector products. So the communication between GPUs becomes a major performance bottleneck on multi GPU cluster. The article presents an approach to minimize the communications between parallel parts of algorithms. Additionally, computation and communication can be overlapped to reduce the impact of data exchange. Using the pipelined version of the CG method with one synchronization point, the possibility of asynchronous calculations and communications, load balancing between the CPU and GPU for solving the large linear systems allows for scalability. The algorithm is implemented with the combined use of technologies: MPI, OpenMP, and CUDA. We show that almost optimum speed up on 8-CPU/2GPU may be reached (relatively to a one GPU execution). The parallelized solver achieves a speedup of up to 5.49 times on 16 NVIDIA Tesla GPUs, as compared to one GPU.

Keywords: conjugate gradient, GPU, parallel programming, pipelined algorithm

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Authors: Kuniyoshi Abe

Abstract:

Bi-conjugate gradient (Bi-CG) is a well-known method for solving linear equations Ax = b, for x, where A is a given n-by-n matrix, and b is a given n-vector. Typically, the dimension of the linear equation is high and the matrix is sparse. A number of hybrid Bi-CG methods such as conjugate gradient squared (CGS), Bi-CG stabilized (Bi-CGSTAB), BiCGStab2, and BiCGstab(l) have been developed to improve the convergence of Bi-CG. Bi-CGSTAB has been most often used for efficiently solving the linear equation, but we have seen the convergence behavior with a long stagnation phase. In such cases, it is important to have Bi-CG coefficients that are as accurate as possible, and the stabilization strategy, which stabilizes the computation of the Bi-CG coefficients, has been proposed. It may avoid stagnation and lead to faster computation. Motivated by a large number of processors in present petascale high-performance computing hardware, the scalability of Krylov subspace methods on parallel computers has recently become increasingly prominent. The main bottleneck for efficient parallelization is the inner products which require a global reduction. The resulting global synchronization phases cause communication overhead on parallel computers. The parallel variants of Krylov subspace methods reducing the number of global communication phases and hiding the communication latency have been proposed. However, the numerical stability, specifically, the convergence speed of the parallel variants of Bi-CGSTAB may become worse than that of the standard Bi-CGSTAB. In this paper, therefore, we compare the convergence speed between the standard Bi-CGSTAB and the parallel variants by numerical experiments and show that the convergence speed of the standard Bi-CGSTAB is faster than the parallel variants. Moreover, we propose the stabilization strategy for the parallel variants.

Keywords: bi-conjugate gradient stabilized method, convergence speed, Krylov subspace methods, linear equations, parallel variant

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Authors: Wikanda Phaphan

Abstract:

Mixture generalized gamma distribution is a combination of two distributions: generalized gamma distribution and length biased generalized gamma distribution. These two distributions were presented by Suksaengrakcharoen and Bodhisuwan in 2014. The findings showed that probability density function (pdf) had fairly complexities, so it made problems in estimating parameters. The problem occurred in parameter estimation was that we were unable to calculate estimators in the form of critical expression. Thus, we will use numerical estimation to find the estimators. In this study, we presented a new method of the parameter estimation by using the expectation – maximization algorithm (EM), the conjugate gradient method, and the quasi-Newton method. The data was generated by acceptance-rejection method which is used for estimating α, β, λ and p. λ is the scale parameter, p is the weight parameter, α and β are the shape parameters. We will use Monte Carlo technique to find the estimator's performance. Determining the size of sample equals 10, 30, 100; the simulations were repeated 20 times in each case. We evaluated the effectiveness of the estimators which was introduced by considering values of the mean squared errors and the bias. The findings revealed that the EM-algorithm had proximity to the actual values determined. Also, the maximum likelihood estimators via the conjugate gradient and the quasi-Newton method are less precision than the maximum likelihood estimators via the EM-algorithm.

Keywords: conjugate gradient method, quasi-Newton method, EM-algorithm, generalized gamma distribution, length biased generalized gamma distribution, maximum likelihood method

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Authors: Smita Sonker

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Various investigators have determined the degree of approximation of conjugate signals (functions) of functions belonging to different classes Lipα, Lip(α,p), Lip(ξ(t),p), W(Lr,ξ(t), (β ≥ 0)) by matrix summability means, lower triangular matrix operator, product means (i.e. (C,1)(E,1), (C,1)(E,q), (E,q)(C,1) (N,p,q)(E,1), and (E,q)(N,pn) of their conjugate trigonometric Fourier series. In this paper, we shall determine the degree of approximation of 2π-periodic function conjugate functions of f belonging to the function classes Lipα and W(Lr; ξ(t); (β ≥ 0)) by (C1.T) -means of their conjugate trigonometric Fourier series. On the other hand, we shall review above-mentioned work in the light of Lenski.

Keywords: signals, trigonometric fourier approximation, class W(L^r, \xi(t), conjugate fourier series

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Authors: Y. T. Tsai, Jin H. Huang

Abstract:

The sound pressure level (SPL) of the moving-coil loudspeaker (MCL) is often simulated and analyzed using the lumped parameter model. However, the SPL of a MCL cannot be simulated precisely in the high frequency region, because the value of cone effective area is changed due to the geometry variation in different mode shapes, it is also related to affect the acoustic radiation mass and resistance. Herein, the paper presents the inverse method which has a high ability to measure the value of cone effective area in various frequency points, also can estimate the MCL electroacoustic parameters simultaneously. The proposed inverse method comprises the direct problem, adjoint problem, and sensitivity problem in collaboration with nonlinear conjugate gradient method. Estimated values from the inverse method are validated experimentally which compared with the measured SPL curve result. Results presented in this paper not only improve the accuracy of lumped parameter model but also provide the valuable information on loudspeaker cone design.

Keywords: inverse problem, cone effective area, loudspeaker, nonlinear conjugate gradient method

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Authors: Yuanhao Gao, Ping Lin, Kees Weijer

Abstract:

An optimal control system model is proposed for the cell flow in the process of chick embryo gastrulation in this paper. The target is to determine the cellular interaction forces which are hard to measure. This paper will take an approach to investigate the forces with the idea of the inverse problem. By choosing the forces as the control variable and regarding the cell flow as Stokes fluid, an objective functional will be established to match the numerical result of cell velocity with the experimental data. So that the forces could be determined by minimizing the objective functional. The Lagrange multiplier method is utilized to derive the state and adjoint equations consisting the optimal control system, which specifies the first-order necessary conditions. Finite element method is used to discretize and approximate equations. A conjugate gradient algorithm is given for solving the minimum solution of the system and determine the forces.

Keywords: optimal control model, Stokes equation, conjugate gradient method, finite element method, chick embryo gastrulation

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Authors: Yuanhao Gao, Pin Lin, Kees Weijer

Abstract:

In this paper, we will determine the external body force distribution with analysis of stokes fluid motion using mathematical modelling and numerical approaching. The body force distribution is regarded as the unknown variable and could be determined by the idea of optimal control theory. The Stokes flow motion and its velocity are generated by given forces in a unit square domain. A regularized objective functional is built to match the numerical result of flow velocity with the generated velocity data. So that the force distribution could be determined by minimizing the value of objective functional, which is also the difference between the numerical and experimental velocity. Then after utilizing the Lagrange multiplier method, some partial differential equations are formulated consisting the optimal control system to solve. Finite element method and conjugate gradient method are used to discretize equations and deduce the iterative expression of target body force to compute the velocity numerically and body force distribution. Programming environment FreeFEM++ supports the implementation of this model.

Keywords: optimal control model, Stokes equation, finite element method, conjugate gradient method

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Authors: Kejal Khatri, Vishnu Narayan Mishra

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We compute the degree of approximation of functions\tilde{f}\in H_w, a new Banach space using (T.E^1) summability means of conjugate Fourier series. In this paper, we extend the results of Singh and Mahajan which in turn generalizes the result of Lal and Yadav. Some corollaries have also been deduced from our main theorem and particular cases.

Keywords: conjugate Fourier series, degree of approximation, Hölder metric, matrix summability, product summability

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Authors: Tu-Chieh Hung, Wei-Mon Yan, Xiao-Dong Wang, Yu-Xian Huang

Abstract:

This work employs a combined optimization procedure including a simplified conjugate-gradient method and a three-dimensional fluid flow and heat transfer model to study the optimal geometric parameter design of double-layered microchannel heat sinks. The overall thermal resistance RT is the objective function to be minimized with number of channels, N, the channel width ratio, β, the bottom channel aspect ratio, αb, and upper channel aspect ratio, αu, as the search variables. It is shown that, for the given bottom area (10 mm×10 mm) and heat flux (100 W cm-2), the optimal (minimum) thermal resistance of double-layered microchannel heat sinks is about RT=0.12 ℃/m2W with the corresponding optimal geometric parameters N=73, β=0.50, αb=3.52, and, αu= 7.21 under a constant pumping power of 0.05 W. The optimization process produces a maximum reduction by 52.8% in the overall thermal resistance compared with an initial guess (N=112, β=0.37, αb=10.32 and, αu=10.93). The results also show that the optimal thermal resistance decreases rapidly with the pumping power and tends to be a saturated value afterward. The corresponding optimal values of parameters N, αb, and αu increase while that of β decrease as the pumping power increases. However, further increasing pumping power is not always cost-effective for the application of heat sink designs.

Keywords: optimization, double-layered microchannel heat sink, simplified conjugate-gradient method, thermal resistance

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Authors: Nadia Ben Youssef, Aicha Bouzid

Abstract:

Gradient estimation is one of the most fundamental tasks in the field of image processing in general, and more particularly for color images since that the research in color image gradient remains limited. The widely used gradient method is Di Zenzo’s gradient operator, which is based on the measure of squared local contrast of color images. The proposed gradient mechanism, presented in this paper, is based on the principle of the Di Zenzo’s approach using quaternion representation. This edge detector is compared to a marginal approach based on multiscale product of wavelet transform and another vector approach based on quaternion convolution and vector gradient approach. The experimental results indicate that the proposed color gradient operator outperforms marginal approach, however, it is less efficient then the second vector approach.

Keywords: gradient, edge detection, color image, quaternion

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Authors: Danni Cong, Meiping Wu, Hua Mu, Xiaofeng He, Junxiang Lian, Juliang Cao, Shaokun Cai, Hao Qin

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Gravity field is of great significance in geoscience, national economy and national security, and gravitational gradient measurement has been extensively studied due to its higher accuracy than gravity measurement. Gravity gradient sensor, being one of core devices of the gravity gradient instrument, plays a key role in measuring accuracy. Therefore, this paper starts from analyzing the working principle of the gravity gradient sensor by Newton’s law, and then considers the relative motion between inertial and non-inertial systems to build a relatively adequate mathematical model, laying a foundation for the measurement error calibration, measurement accuracy improvement.

Keywords: gravity gradient, gravity gradient sensor, accelerometer, single-axis rotation modulation

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Authors: A. K. Borah, A. K. Singh

Abstract:

In this paper we deal building blocks of the computer simulation of the multiphase flows. Whole simulation procedure can be viewed as two super procedures; The implementation of VOF method and the solution of Navier Stoke’s Equation. Moreover, a sequential code for a Navier Stoke’s solver has been studied.

Keywords: Bi-conjugate gradient stabilized (Bi-CGSTAB), ILUT function, krylov subspace, multifluid flows preconditioner, simple algorithm

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Authors: Abdul Rehman, Bo Liu

Abstract:

Axial flow turbines are commonly designed with high loads that generate strong secondary flows and result in high secondary losses. These losses contribute to almost 30% to 50% of the total losses. Non-axisymmetric endwall profiling is one of the passive control technique to reduce the secondary flow loss. In this paper, the non-axisymmetric endwall profile construction and optimization for the stator endwalls are presented to improve the efficiency of a high pressure turbine. The commercial code NUMECA Fine/ Design3D coupled with Fine/Turbo was used for the numerical investigation, design of experiments and the optimization. All the flow simulations were conducted by using steady RANS and Spalart-Allmaras as a turbulence model. The non-axisymmetric endwalls of stator hub and shroud were created by using the perturbation law based on Bezier Curves. Each cut having multiple control points was supposed to be created along the virtual streamlines in the blade channel. For the design of experiments, each sample was arbitrarily generated based on values automatically chosen for the control points defined during parameterization. The Optimization was achieved by using two algorithms i.e. the stochastic algorithm and gradient-based algorithm. For the stochastic algorithm, a genetic algorithm based on the artificial neural network was used as an optimization method in order to achieve the global optimum. The evaluation of the successive design iterations was performed using artificial neural network prior to the flow solver. For the second case, the conjugate gradient algorithm with a three dimensional CFD flow solver was used to systematically vary a free-form parameterization of the endwall. This method is efficient and less time to consume as it requires derivative information of the objective function. The objective function was to maximize the isentropic efficiency of the turbine by keeping the mass flow rate as constant. The performance was quantified by using a multi-objective function. Other than these two classifications of the optimization methods, there were four optimizations cases i.e. the hub only, the shroud only, and the combination of hub and shroud. For the fourth case, the shroud endwall was optimized by using the optimized hub endwall geometry. The hub optimization resulted in an increase in the efficiency due to more homogenous inlet conditions for the rotor. The adverse pressure gradient was reduced but the total pressure loss in the vicinity of the hub was increased. The shroud optimization resulted in an increase in efficiency, total pressure loss and entropy were reduced. The combination of hub and shroud did not show overwhelming results which were achieved for the individual cases of the hub and the shroud. This may be caused by fact that there were too many control variables. The fourth case of optimization showed the best result because optimized hub was used as an initial geometry to optimize the shroud. The efficiency was increased more than the individual cases of optimization with a mass flow rate equal to the baseline design of the turbine. The results of artificial neural network and conjugate gradient method were compared.

Keywords: artificial neural network, axial turbine, conjugate gradient method, non-axisymmetric endwall, optimization

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Authors: Mohamed El-Sayed Mosaad

Abstract:

An analytic approach is obtained for the steady heat transfer problem of two fluid systems, in thermal communication via heat conduction across a solid wall separating them. The two free convection layers created on wall sides are assumed to be in parallel flow. Fluid-solid interface temperature on wall sides is not prescribed in analysis in advance; rather, determined from conjugate solution among other unknown parameters. The analysis highlights the main conjugation parameters controlling thermal interaction process of involved heat transfer modes. Heat transfer results of engineering importance are obtained.

Keywords: conjugate heat transfer, boundary layer, convection, thermal systems

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Authors: M. Yaqub Khan, Usman Shabbir

Abstract:

History of plasma reveals that continuous struggle of experimentalists and theorists are not fruitful for confinement up to now. It needs a change to bring the research through entropy. Approximately, all the quantities like number density, temperature, electrostatic potential, etc. are connected to entropy. Therefore, it is better to change the way of research. In ion temperature gradient mode with the help of Braginskii model, Boltzmannian electrons, effect of velocity shear is studied inculcating entropy in the magnetoplasma. New dispersion relation is derived for ion temperature gradient mode, and dependence on entropy gradient drift is seen. It is also seen velocity shear enhances the instability but in anomalous transport, its role is not seen significantly but entropy. This work will be helpful to the next step of tokamak and space plasmas.

Keywords: entropy, velocity shear, ion temperature gradient mode, drift

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Authors: Ishak Hashim, Ammar Alsabery

Abstract:

The problem of conjugate free convection in a square cavity filled with nanofluid and heated from below by spatial wall temperature is studied numerically using the finite difference method. Water-based nanofluid with copper nanoparticles are chosen for the investigation. Governing equations are solved over a wide range of nanoparticle volume fraction (0 ≤ φ ≤ 0.2), wave number ((0 ≤ λ ≤ 4) and thermal conductivity ratio (0.44 ≤ Kr ≤ 6). The results presented for values of the governing parameters in terms of streamlines, isotherms and average Nusselt number. It is found that the flow behavior and the heat distribution are clearly enhanced with the increment of the non-uniform heating.

Keywords: conjugate free convection, square cavity, nanofluid, spatial temperature

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Authors: Mustafa Arda, Metin Aydogdu

Abstract:

Carbon nanotubes (CNTs) have many possible application areas because of their superior physical properties. Nonlocal Theory, which unlike the classical theories, includes the size dependency. Nonlocal Stress and Strain Gradient approaches can be used in nanoscale static and dynamic analysis. In the present study, torsional vibration of CNTs was investigated according to nonlocal stress and strain gradient theories. Effects of the small scale parameters to the non-dimensional frequency were obtained. Results were compared with the Molecular Dynamics Simulation and Lattice Dynamics. Strain Gradient Theory has shown more weakening effect on CNT according to the Stress Gradient Theory. Combination of both theories gives more acceptable results rather than the classical and stress or strain gradient theory according to Lattice Dynamics.

Keywords: torsional vibration, carbon nanotubes, nonlocal gradient theory, stress, strain

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Authors: Sachin Aggarwal, Sarah Kassinger, Nicholas Hoffman

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Geometry simplification is a key step in performing conjugate heat transfer analysis using CFD. This paper proposes a standard methodology for the geometry simplification of rotating machines, such as electrical generators and electrical motors (both air and liquid-cooled). These machines are extensively deployed throughout the aerospace and automotive industries, where optimization of weight, volume, and performance is paramount -especially given the current global transition to renewable energy sources and vehicle hybridization and electrification. Conjugate heat transfer analysis is an essential step in optimizing their complex design. This methodology will help in reducing convergence issues due to poor mesh quality, thus decreasing computational cost and overall analysis time.

Keywords: CFD, electrical machines, Geometry simplification, heat transfer

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Authors: M. Mosaad, J. H. Almutairi, A. S. Almutairi

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In this paper, saturated-vapour film condensation on a vertical wall with the backside cooled by forced convection is analyzed as a conjugate problem. In the analysis, the temperature and heat flux at the wall sides are assumed unknown and determined from the solution. The model is presented in a dimensionless form to take a broad view of the solution. The dimensionless variables controlling this coupled heat transfer process are discovered from the analysis. These variables explain the relative impact of the interactive heat transfer mechanisms of forced convection and film condensation. The study shows that the conjugate treatment of film condensation process yields results different from that predicted by a non-conjugate Nusselt-type solution, wherein the effect of the cooling fluid is neglected.

Keywords: film condensation, forced convection, coupled heat transfer, analytical modelling

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Authors: A. Selmi, A. Bisharat

Abstract:

Using Fourier transform and based on the Mindlin's 2^nd gradient model that involves two length scale parameters, the Green's function, the Eshelby tensor, and the Eshelby-like tensor for a spherical inclusion are derived. It is proved that the Eshelby tensor consists of two parts; the classical Eshelby tensor and a gradient part including the length scale parameters which enable the interpretation of the size effect. When the strain gradient is not taken into account, the obtained Green's function and Eshelby tensor reduce to its analogue based on the classical elasticity. The Eshelby tensor in and outside the inclusion, the volume average of the gradient part and the Eshelby-like tensor are explicitly obtained. Unlike the classical Eshelby tensor, the results show that the components of the new Eshelby tensor vary with the position and the inclusion dimensions. It is demonstrated that the contribution of the gradient part should not be neglected.

Keywords: Eshelby tensor, Eshelby-like tensor, Green’s function, Mindlin’s 2nd gradient model, spherical inclusion

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Authors: Changqiao Wu, Guoqing Ding, Xin Chen

Abstract:

In this paper, ways of modeling dynamic measurement systems are discussed. Specially, for linear system with single-input single-output, it could be modeled with shallow neural network. Then, gradient based optimization algorithms are used for searching the proper coefficients. Besides, method with normal equation and second order gradient descent are proposed to accelerate the modeling process, and ways of better gradient estimation are discussed. It shows that the mathematical essence of the learning objective is maximum likelihood with noises under Gaussian distribution. For conventional gradient descent, the mini-batch learning and gradient with momentum contribute to faster convergence and enhance model ability. Lastly, experimental results proved the effectiveness of second order gradient descent algorithm, and indicated that optimization with normal equation was the most suitable for linear dynamic models.

Keywords: dynamic system modeling, neural network, normal equation, second order gradient descent

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Authors: Mantai Chen, Johnny Ching Ming Ho

Abstract:

The stress-strain relationship of concrete under flexure is one of the essential parameters in assessing ultimate flexural strength capacity of RC beams. Currently, the concrete stress-strain curve in flexure is obtained by incorporating a constant scale-down factor of 0.85 in the uniaxial stress-strain curve. However, it was revealed that strain gradient would improve the maximum concrete stress under flexure and concrete stress-strain curve is strain gradient dependent. Based on the strain-gradient-dependent concrete stress-strain curve, the investigation of the combined effects of strain gradient and concrete strength on flexural strength of RC beams was extended to high strength concrete up to 100 MPa by theoretical analysis. As an extension and application of the authors’ previous study, a new flexural strength design method incorporating the combined effects of strain gradient and concrete strength is developed. A set of equivalent rectangular concrete stress block parameters is proposed and applied to produce a series of design charts showing that the flexural strength of RC beams are improved with strain gradient effect considered.

Keywords: beams, equivalent concrete stress block, flexural strength, strain gradient

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