Search results for: Density Dependent Nonlinear Reaction-Diffusion Equation

Authors: Pandian Elavarasan, Kishore Kondamudi, Sreedevi Upadhyayula

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Ionic liquids are well known as green solvents, reaction media and catalysis. Here, three different sulfonic acid functional ionic liquids prepared in the laboratory are used as catalysts in alkylation of p-cresol with tert-butyl alcohol. The kinetics on each of the catalysts was compared and a kinetic model was developed based on the product distribution over these catalysts. The kinetic parameters were estimated using Marquadt's algorithm to minimize the error function. The Arrhenius plots show a curvature which is best interpreted by the extended Arrhenius equation.

Keywords: Alkylation, p-cresol, tert-butyl alcohol, kinetics, activation parameter, extended Arrhenius equation.

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Authors: V. Kahya, A. Birinci, R. Erdol

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A frictionless contact problem for a two-layer orthotropic elastic medium loaded through a rigid flat stamp is considered. It is assumed that tensile tractions are not allowed and only compressive tractions can be transmitted across the interface. In the solution, effect of gravity is taken into consideration. If the external load on the rigid stamp is less than or equal to a critical value, continuous contact between the layers is maintained. The problem is expressed in terms of a singular integral equation by using the theory of elasticity and the Fourier transforms. Numerical results for initial separation point, critical separation load and contact stress distribution are presented.

Keywords: Frictionless contact, Initial separation, Orthotropicmaterial, Singular integral equation.

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Authors: M. Khaing, A. V. Tkacheva

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The work is devoted to solving the problem of temperature stresses, caused by the heating point of the round plate. The plate is made of elastoplastic material, so the Prandtl-Reis model is used. A piecewise-linear condition of the Ishlinsky-Ivlev flow is taken as the loading surface, in which the yield stress depends on the temperature. Piecewise-linear conditions (Treska or Ishlinsky-Ivlev), in contrast to the Mises condition, make it possible to obtain solutions of the equilibrium equation in an analytical form. In the problem under consideration, using the conditions of Tresca, it is impossible to obtain a solution. This is due to the fact that the equation of equilibrium ceases to be satisfied when the two Tresca conditions are fulfilled at once. Using the conditions of plastic flow Ishlinsky-Ivlev allows one to solve the problem. At the same time, there are also no solutions on the edge of the Ishlinsky-Ivlev hexagon in the plane-stressed state. Therefore, the authors of the article propose to jump from the edge to the edge of the mine edge, which gives an opportunity to obtain an analytical solution. At the same time, there is also no solution on the edge of the Ishlinsky-Ivlev hexagon in a plane stressed state; therefore, in this paper, the authors of the article propose to jump from the side to the side of the mine edge, which gives an opportunity to receive an analytical solution. The paper compares solutions of the problem of plate thermal deformation. One of the solutions was obtained under the condition that the elastic moduli (Young's modulus, Poisson's ratio) which depend on temperature. The yield point is assumed to be parabolically temperature dependent. The main results of the comparisons are that the region of irreversible deformation is larger in the calculations obtained for solving the problem with constant elastic moduli. There is no repeated plastic flow in the solution of the problem with elastic moduli depending on temperature. The absolute value of the irreversible deformations is higher for the solution of the problem in which the elastic moduli are constant; there are also insignificant differences in the distribution of the residual stresses.

Keywords: Temperature stresses, elasticity, plasticity, Ishlinsky-Ivlev condition, plate, annular heating, elastic moduli.

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Authors: S. Grishin

Abstract:

This article is dedicated to development of mathematical models for determining the dynamics of concentration of hazardous substances in urban turbulent atmosphere. Development of the mathematical models implied taking into account the time-space variability of the fields of meteorological items and such turbulent atmosphere data as vortex nature, nonlinear nature, dissipativity and diffusivity. Knowing the turbulent airflow velocity is not assumed when developing the model. However, a simplified model implies that the turbulent and molecular diffusion ratio is a piecewise constant function that changes depending on vertical distance from the earth surface. Thereby an important assumption of vertical stratification of urban air due to atmospheric accumulation of hazardous substances emitted by motor vehicles is introduced into the mathematical model. The suggested simplified non-linear mathematical model of determining the sought exhaust concentration at a priori unknown turbulent flow velocity through non-degenerate transformation is reduced to the model which is subsequently solved analytically.

Keywords: Urban ecology, time-dependent mathematical model, exhaust concentration, turbulent and molecular diffusion, airflow velocity.

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Authors: Ping He, Heng-You Lan, Gong-Quan Tan

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The multidelays linear control systems described by difference differential equations are often studied in modern control theory. In this paper, the delay-independent stabilization algebraic criteria and the theorem of delay-independent stabilization for linear systems with multiple time-delays are established by using the Lyapunov functional and the Riccati algebra matrix equation in the matrix theory. An illustrative example and the simulation result, show that the approach to linear systems with multiple time-delays is effective.

Keywords: Linear system, Delay-independent stabilization, Lyapunovfunctional, Riccati algebra matrix equation.

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Authors: V Krishnaveni, S Jayaraman, K Ramadoss

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The electrical potentials generated during eye movements and blinks are one of the main sources of artifacts in Electroencephalogram (EEG) recording and can propagate much across the scalp, masking and distorting brain signals. In recent times, signal separation algorithms are used widely for removing artifacts from the observed EEG data. In this paper, a recently introduced signal separation algorithm Mutual Information based Least dependent Component Analysis (MILCA) is employed to separate ocular artifacts from EEG. The aim of MILCA is to minimize the Mutual Information (MI) between the independent components (estimated sources) under a pure rotation. Performance of this algorithm is compared with eleven popular algorithms (Infomax, Extended Infomax, Fast ICA, SOBI, TDSEP, JADE, OGWE, MS-ICA, SHIBBS, Kernel-ICA, and RADICAL) for the actual independence and uniqueness of the estimated source components obtained for different sets of EEG data with ocular artifacts by using a reliable MI Estimator. Results show that MILCA is best in separating the ocular artifacts and EEG and is recommended for further analysis.

Keywords: Electroencephalogram, Ocular Artifacts (OA), Independent Component Analysis (ICA), Mutual Information (MI), Mutual Information based Least dependent Component Analysis(MILCA)

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Authors: M. Palizvan, M. T. Abadi, M. H. Sadr

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Due to variations in damage mechanisms in the microscale, the behavior of fiber-reinforced composites is nonlinear and difficult to model. To make use of computational advantages, homogenization method is applied to the micro-scale model in order to minimize the cost at the expense of detail of local microscale phenomena. In this paper, the effective stiffness is calculated using the homogenization of nonlinear behavior of a composite representative volume element (RVE) containing fiber-matrix debonding. The damage modes for the RVE are considered by using cohesive elements and contacts for the cohesive behavior of the interface between fiber and matrix. To predict more realistic responses of composite materials, different random distributions of fibers are proposed besides square and hexagonal arrays. It was shown that in some cases, there is quite different damage behavior in different fiber distributions. A comprehensive comparison has been made between different graphs.

Keywords: Homogenization, cohesive zone model, fiber-matrix debonding, RVE.

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Authors: Ya-Fen Lee, Yun-Yao Chi, Cing-Hong Hung

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Because of the global warming and the rising sea level, residents living in southwestern coastland, Taiwan are faced with the submerged land and may move to higher elevation area. It is desirable to discuss the key consideration factor for selecting the migration location under five dimensions of ಯ security”, “health”, “convenience”, “comfort” and “socio-economic” based on the document reviews. This paper uses the Structural Equation Modeling (SEM) and the questionnaire survey. The analysis results show that the convenience is the most key factor for residents in Taiwan. 

Keywords: Global warming, migration, structural equation modelling, questionnaire survey.

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Authors: Ahmad Seif, Karim Zare, Asadallah Boshra, Mehran Aghaie

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Density functional theory (DFT) calculations were performed to compute nitrogen-14 and boron-11 nuclear quadrupole resonance (NQR) spectroscopy parameters in the representative model of armchair boron nitride nanotube (BNNT) for the first time. The considered model consisting of 1 nm length of H-capped (5, 5) single-wall BNNT were first allowed to fully relax and then the NQR calculations were carried out on the geometrically optimized model. The evaluated nuclear quadrupole coupling constants and asymmetry parameters for the mentioned nuclei reveal that the model can be divided into seven layers of nuclei with an equivalent electrostatic environment where those nuclei at the ends of tubes have a very strong electrostatic environment compared to the other nuclei along the length of tubes. The calculations were performed via Gaussian 98 package of program.

Keywords: Armchair Nanotube, Density Functional Theory, Nuclear Quadrupole Resonance.

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Authors: D. K. Lee, S. M. Shin, J. H. Lee

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Topology optimization technique utilizes constant element densities as design parameters. Finally, optimal distribution contours of the material densities between voids (0) and solids (1) in design domain represent the determination of topology. It means that regions with element density values become occupied by solids in design domain, while there are only void phases in regions where no density values exist. Therefore the void regions of topology optimization results provide design information to decide appropriate depositions of web-opening in structure. Contrary to the basic objective of the topology optimization technique which is to obtain optimal topology of structures, this present study proposes a new idea that topology optimization results can be also utilized for decision of proper web-opening’s position. Numerical examples of linear elastostatic structures demonstrate efficiency of methodological design processes using topology optimization in order to determinate the proper deposition of web-openings.

Keywords: Topology optimization, web-opening, structure, element density, material.

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Authors: Rafael Jorge Menezes Santos, Ginalber Luiz de Oliveira Serra, Carlos César Teixeira Ferreira

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This paper proposes a methodology for analysis of the dynamic behavior of a robotic manipulator in continuous time. Initially this system (nonlinear system) will be decomposed into linear submodels and analyzed in the context of the Linear and Parameter Varying (LPV) Systems. The obtained linear submodels, which represent the local dynamic behavior of the robotic manipulator in some operating points were grouped in a Takagi-Sugeno fuzzy structure. The obtained fuzzy model was analyzed and validated through analog simulation, as universal approximator of the robotic manipulator.

Keywords: modeling of nonlinear dynamic systems, Takagi- Sugeno fuzzy model, Linear and Parameter Varying (LPV) System.

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Authors: O. A. Akinfenwa, N.M. Yao, S. N. Jator

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In this paper, a self starting two step continuous block hybrid formulae (CBHF) with four Off-step points is developed using collocation and interpolation procedures. The CBHF is then used to produce multiple numerical integrators which are of uniform order and are assembled into a single block matrix equation. These equations are simultaneously applied to provide the approximate solution for the stiff ordinary differential equations. The order of accuracy and stability of the block method is discussed and its accuracy is established numerically.

Keywords: Collocation and Interpolation, Continuous HybridBlock Formulae, Off-Step Points, Stability, Stiff ODEs.

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Authors: Udeme O. Ini, Edikan E. Akpanibah

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This paper studies the optimal investment strategies for a plan member (PM) in a defined contribution (DC) pension scheme with transaction cost, taxes on invested funds and couple risky assets (stocks) under the Ornstein-Uhlenbeck (O-U) process. The PM’s portfolio is assumed to consist of a risk-free asset and two risky assets where the two risky assets are driven by the O-U process. The Legendre transformation and dual theory is use to transform the resultant optimal control problem which is a nonlinear partial differential equation (PDE) into linear PDE and the resultant linear PDE is then solved for the explicit solutions of the optimal investment strategies for PM exhibiting constant absolute risk aversion (CARA) using change of variable technique. Furthermore, theoretical analysis is used to study the influences of some sensitive parameters on the optimal investment strategies with observations that the optimal investment strategies for the two risky assets increase with increase in the dividend and decreases with increase in tax on the invested funds, risk averse coefficient, initial fund size and the transaction cost.

Keywords: Ornstein-Uhlenbeck process, portfolio management, Legendre transforms, CARA utility.

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Authors: D. Toghraie, A. R. Azimian

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In this paper, the Lennard -Jones potential is applied to molecules of liquid argon as well as its vapor and platinum as solid surface in order to perform a non-equilibrium molecular dynamics simulation to study the microscopic aspects of liquid-vapor-solid interactions. The channel is periodic in x and y directions and along z direction it is bounded by atomic walls. It was found that density of the liquids near the solid walls fluctuated greatly and that the structure was more like a solid than a liquid. This indicates that the interactions of solid and liquid molecules are very strong. The resultant surface tension, liquid density and vapor density are found to be well predicted when compared with the experimental data for argon. Liquid and vapor densities were found to depend on the cutoff radius which induces the use of P3M (particle-particle particle-mesh) method which was implemented for evaluation of force and surface tension.

Keywords: Lennard-Jones Potential, Molecular DynamicsSimulation, Periodic Boundary Conditions (PBC), Non-EquilibriumMolecular Dynamics (NEMD).

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Authors: Jose William Porras Ferreira

Abstract:

This paper presents two solutions to the Fermat’s Last Theorem (FLT). The first one using some algebraic basis related to the Pythagorean theorem, expression of equations, an analysis of their behavior, when compared with power  and power  and using " the “Well Ordering Principle” of natural numbers it is demonstrated that in Fermat equation . The second one solution is using the connection between  and power  through the Pascal’s triangle or  Newton’s binomial coefficients, where de Fermat equation do not fulfill the first coefficient, then it is impossible that:

zⁿ=xⁿ+yⁿ for n>2 and (x, y, z) E Z⁺ - {0}

 

Keywords: Fermat’s Last Theorem, Pythagorean Theorem, Newton Binomial Coefficients, Pascal’s Triangle, Well Ordering Principle.

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Authors: P. Ramana , K. Alice Mary, M. Surya Kalavathi, M. Phani Kumar

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This paper discusses two observers, which are used for the estimation of parameters of PMSM. Former one, reduced order observer, which is used to estimate the inaccessible parameters of PMSM. Later one, full order observer, which is used to estimate all the parameters of PMSM even though some of the parameters are directly available for measurement, so as to meet with the insensitivity to the parameter variation. However, the state space model contains some nonlinear terms i.e. the product of different state variables. The asymptotic state observer, which approximately reconstructs the state vector for linear systems without uncertainties, was presented by Luenberger. In this work, a modified form of such an observer is used by including a non-linear term involving the speed. So, both the observers are designed in the framework of nonlinear control; their stability and rate of convergence is discussed.

Keywords: Permanent magnet synchronous motor, Mathematicalmodelling, Rotor reference frame, parameter estimation, Luenbergerobserver, reduced order observer, full order observer

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Authors: Richard Thai-Singama, Jean-Pierre Belin, Frédéric Du Burck, Marc Piette

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This paper presents a simple and original method for the generation of short monocycle pulses based on the transient response of a passive band-pass filter. The recorded sub-nanosecond pulses show a good symmetry and a small ringing (13 % of the peak amplitude). Their spectral density covers the range 3.1 GHz to 10.6 GHz. The possibility to adapt the pulse spectral density to the indoor FCC frequency mask is demonstrated with a prototype working at a reduced frequency (FCC/1000). A detection technique is proposed.

Keywords: Impulse, Monocycle, Transient, UWB.

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Authors: S.C. Chin, N. Shafiq, M.F. Nuruddin

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To date, theoretical studies concerning the Carbon Fiber Reinforced Polymer (CFRP) strengthening of RC beams with openings have been rather limited. In addition, various numerical analyses presented so far have effectively simulated the behaviour of solid beam strengthened by FRP material. In this paper, a two dimensional nonlinear finite element analysis is presented to validate against the laboratory test results of six RC beams. All beams had the same rectangular cross-section geometry and were loaded under four point bending. The crack pattern results of the finite element model show good agreement with the crack pattern of the experimental beams. The load midspan deflection curves of the finite element models exhibited a stiffer result compared to the experimental beams. The possible reason may be due to the perfect bond assumption used between the concrete and steel reinforcement.

Keywords: CFRP, large opening, RC beam, strengthening

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Authors: Osea Zebua, Norikazu Ikoma, Hiroshi Maeda

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Estimation of voltage stability based on optimal filtering method is presented. PV curve is used as a tool for voltage stability analysis. Dynamic voltage stability estimation is done by using particle filter method. Optimum value (nose point) of PV curve can be estimated by estimating parameter of PV curve equation optimal value represents critical voltage and condition at specified point of measurement. Voltage stability is then estimated by analyzing loading margin condition c stimating equation. This maximum loading ecified dynamically.

Keywords: normalized PV curve, optimal filtering method particle filter, voltage stability.

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Authors: Muhammad Shafi, Paul W. H. Chung

Abstract:

This paper proposes a hybrid method for eyes localization in facial images. The novelty is in combining techniques that utilise colour, edge and illumination cues to improve accuracy. The method is based on the observation that eye regions have dark colour, high density of edges and low illumination as compared to other parts of face. The first step in the method is to extract connected regions from facial images using colour, edge density and illumination cues separately. Some of the regions are then removed by applying rules that are based on the general geometry and shape of eyes. The remaining connected regions obtained through these three cues are then combined in a systematic way to enhance the identification of the candidate regions for the eyes. The geometry and shape based rules are then applied again to further remove the false eye regions. The proposed method was tested using images from the PICS facial images database. The proposed method has 93.7% and 87% accuracies for initial blobs extraction and final eye detection respectively.

Keywords: Erosion, dilation, Edge-density

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Authors: M. A. Torkamani, S. Mahmoodzadeh, S. Pourroostaei, C. Lucas

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Daily production of information and importance of the sequence of produced data in forecasting future performance of market causes analysis of data behavior to become a problem of analyzing time series. But time series that are very complicated, usually are random and as a result their changes considered being unpredictable. While these series might be products of a deterministic dynamical and nonlinear process (chaotic) and as a result be predictable. Point of Chaotic theory view, complicated systems have only chaotically face and as a result they seem to be unregulated and random, but it is possible that they abide by a specified math formula. In this article, with regard to test of strange attractor and biggest Lyapunov exponent probability of chaos on several foreign exchange rates vs. IRR (Iranian Rial) has been investigated. Results show that data in this market have complex chaotic behavior with big degree of freedom.

Keywords: Chaos, Exchange Rate, Nonlinear Models.

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Authors: Khalid S. Hashim, Andy Shaw, Rafid Alkhaddar, Montserrat Ortoneda Pedrola

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A flow column has been innovatively used in the design of a new electrocoagulation reactor (ECR1) that will reduce the temperature of water being treated; where the flow columns work as a radiator for the water being treated. In order to investigate the performance of ECR1 and compare it to that of traditional reactors; 600 mL water samples with an initial temperature of 350C were pumped continuously through these reactors for 30 min at current density of 1 mA/cm2. The temperature of water being treated was measured at 5 minutes intervals over a 30 minutes period using a thermometer. Additional experiments were commenced to investigate the effects of initial temperature (15-350C), water conductivity (0.15 – 1.2 S) and current density (0.5 -3 mA/cm2) on the performance of ECR1. The results obtained demonstrated that the ECR1, at a current density of 1 mA/cm2 and continuous flow model, reduced water temperature from 350C to the vicinity of 280C during the first 15 minutes and kept the same level till the end of the treatment time. While, the temperature increased from 28.1 to 29.80C and from 29.8 to 31.90C in the batch and the traditional continuous flow models respectively. In term of initial temperature, ECR1 maintained the temperature of water being treated within the range of 22 to 280C without the need for external cooling system even when the initial temperatures varied over a wide range (15 to 350C). The influent water conductivity was found to be a significant variable that affect the temperature. The desirable value of water conductivity is 0.6 S. However, it was found that the water temperature increased rapidly with a higher current density.

Keywords: Water temperature, flow column, electrocoagulation.

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Authors: Kewei Xu, Jens Friedrich, Kevin Dwinger, Wei Fan, Xijin Zhang

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Parallel Compressor Model (PCM) is a simplified approach to predict compressor performance with inlet distortions. In PCM calculation, it is assumed that the sub-compressors’ outlet static pressure is uniform and therefore simplifies PCM calculation procedure. However, if the compressor’s outlet duct is not long and straight, such assumption frequently induces error ranging from 10% to 15%. This paper provides a revised calculation method of PCM that can correct the error. The revised method employs energy equation, momentum equation and continuity equation to acquire needed parameters and replace the equal static pressure assumption. Based on the revised method, PCM is applied on two compression system with different blades types. The predictions of their performance in non-uniform inlet conditions are yielded through the revised calculation method and are employed to evaluate the method’s efficiency. Validating the results by experimental data, it is found that although little deviation occurs, calculated result agrees well with experiment data whose error ranges from 0.1% to 3%. Therefore, this proves the revised calculation method of PCM possesses great advantages in predicting the performance of the distorted compressor with limited exhaust duct.

Keywords: Parallel Compressor Model (PCM), Revised Calculation Method, Inlet Distortion, Outlet Unequal Pressure Distribution.

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Authors: Ali Safdari-Vaighani

Abstract:

Pricing financial contracts on several underlying assets received more and more interest as a demand for complex derivatives. The option pricing under asset price involving jump diffusion processes leads to the partial integral differential equation (PIDEs), which is an extension of the Black-Scholes PDE with a new integral term. The aim of this paper is to show how basket option prices in the jump diffusion models, mainly on the Merton model, can be computed using RBF based approximation methods. For a test problem, the RBF-PU method is applied for numerical solution of partial integral differential equation arising from the two-asset European vanilla put options. The numerical result shows the accuracy and efficiency of the presented method.

Keywords: Radial basis function, basket option, jump diffusion, RBF-PUM.

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Authors: B. Chetti, W. A. Crosby

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The work described in this paper is an investigation of the static and dynamic characteristics of two-lobe journal bearings taking into consideration the thermal effects. A thermo-hydrodynamic solution of a finite two-lobe journal bearing is performed by solving the generalized form Reynolds equation with the energy equation, taking into consideration viscosity variation across the film thickness. The static and dynamic characteristics were numerically obtained. The results are evaluated for different values of viscosity-temperature coefficient and Peclet number. The results show that considering the thermal effects in the solution of the two-lobe journal bearing has a marked on the study of its stability.

Keywords: Two-lobe bearing, thermal effect, static and dynamic characteristics.

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Authors: K. Karuppasamy

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In this paper, analysis of an infinite beam resting on multilayer tensionless extensible geosynthetic reinforced granular fill-poor soil system overlying soft soil strata under moving load with constant velocity is presented. The beam is subjected to a concentrated load moving with constant velocity. The upper reinforced granular bed is modeled by a rough membrane embedded in Pasternak shear layer overlying a series of compressible nonlinear winkler springs representing the underlying the very poor soil. The multilayer tensionless extensible geosynthetic layer has been assumed to deform such that at interface the geosynthetic and the soil have some deformation. Nonlinear behaviour of granular fill and the very poor soil has been considered in the analysis by means of hyperbolic constitutive relationships. Governing differential equations of the soil foundation system have been obtained and solved with the help of appropriate boundary conditions. The solution has been obtained by employing finite difference method by means of Gauss-Siedal iterative scheme. Detailed parametric study has been conducted to study the influence of various parameters on the response of soil–foundation system under consideration by means of deflection and bending moment in the beam and tension mobilized in the geosynthetic layer. These parameters include magnitude of applied load, velocity of load, damping, ultimate resistance of poor soil and granular fill layer. Range of values of parameters has been considered as per Indian Railway conditions. This study clearly observed that the comparisons of multilayer tensionless extensible geosynthetic reinforcement with poor foundation soil and magnitude of applied load, relative compressibility of granular fill and ultimate resistance of poor soil has significant influence on the response of soil–foundation system.

Keywords: Infinite beams, multilayer tensionless extensible geosynthetic, granular layer, moving load, nonlinear behavior of poor soil.

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Authors: Dongdong Zhang, Deran Zhang

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In this paper, a Markovian risk model with two-type claims is considered. In such a risk model, the occurrences of the two type claims are described by two point processes {Ni(t), t ¸ 0}, i = 1, 2, where {Ni(t), t ¸ 0} is the number of jumps during the interval (0, t] for the Markov jump process {Xi(t), t ¸ 0} . The ruin probability ª(u) of a company facing such a risk model is mainly discussed. An integral equation satisfied by the ruin probability ª(u) is obtained and the bounds for the convergence rate of the ruin probability ª(u) are given by using key-renewal theorem.

Keywords: Risk model, ruin probability, Markov jump process, integral equation.

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Authors: Sami Mutair, Yasuyuki Ikegami

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This paper reports on the results of experimental investigations of flash evaporation from superheated jet issues vertically upward from a round straight nozzle of 81.3 mm diameter. For the investigated range of jet superheat degree and velocity, it was shown that flash evaporation enhances with initial temperature increase. Due to the increase of jet inertia and subsequently the delay of jet shattering, increase of jet velocity was found to result in increase of evaporation "delay period". An empirical equation predicts the jet evaporation completion height was developed, this equation is thought to be useful in designing the flash evaporation chamber. In attempts for enhancement of flash evaporation, use of steel wire mesh located at short distance downstream was found effective with no consequent pressure drop.

Keywords: Enhancement; Flash Evaporation; OTEC; superheated jet

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Authors: Shuenn-Yih Chang, Chiu-Li Huang

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It has been shown that a load discontinuity at the end of an impulse will result in an extra impulse and hence an extra amplitude distortion if a step-by-step integration method is employed to yield the shock response. In order to overcome this difficulty, three remedies are proposed to reduce the extra amplitude distortion. The first remedy is to solve the momentum equation of motion instead of the force equation of motion in the step-by-step solution of the shock response, where an external momentum is used in the solution of the momentum equation of motion. Since the external momentum is a resultant of the time integration of external force, the problem of load discontinuity will automatically disappear. The second remedy is to perform a single small time step immediately upon termination of the applied impulse while the other time steps can still be conducted by using the time step determined from general considerations. This is because that the extra impulse caused by a load discontinuity at the end of an impulse is almost linearly proportional to the step size. Finally, the third remedy is to use the average value of the two different values at the integration point of the load discontinuity to replace the use of one of them for loading input. The basic motivation of this remedy originates from the concept of no loading input error associated with the integration point of load discontinuity. The feasibility of the three remedies are analytically explained and numerically illustrated.

Keywords: Dynamic analysis, load discontinuity, shock response, step-by-step integration

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Authors: Cheng Zeng, George Michailidis, Hitoshi Iyatomi, Leo L Duan

Abstract:

The conditional density characterizes the distribution of a response variable y given other predictor x, and plays a key role in many statistical tasks, including classification and outlier detection. Although there has been abundant work on the problem of Conditional Density Estimation (CDE) for a low-dimensional response in the presence of a high-dimensional predictor, little work has been done for a high-dimensional response such as images. The promising performance of normalizing flow (NF) neural networks in unconditional density estimation acts a motivating starting point. In this work, we extend NF neural networks when external x is present. Specifically, they use the NF to parameterize a one-to-one transform between a high-dimensional y and a latent z that comprises two components [zP , zN]. The zP component is a low-dimensional subvector obtained from the posterior distribution of an elementary predictive model for x, such as logistic/linear regression. The zN component is a high-dimensional independent Gaussian vector, which explains the variations in y not or less related to x. Unlike existing CDE methods, the proposed approach, coined Augmented Posterior CDE (AP-CDE), only requires a simple modification on the common normalizing flow framework, while significantly improving the interpretation of the latent component, since zP represents a supervised dimension reduction. In image analytics applications, AP-CDE shows good separation of x-related variations due to factors such as lighting condition and subject id, from the other random variations. Further, the experiments show that an unconditional NF neural network, based on an unsupervised model of z, such as Gaussian mixture, fails to generate interpretable results.

Keywords: Conditional density estimation, image generation, normalizing flow, supervised dimension reduction.

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