Search results for: Boussinesq equations

Authors: Zahra Majd, Mohsen Kalantar

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Power management and voltage regulation is one of the most important issues in microgrid (MG) control and scheduling. This paper proposes a multiobjective scheduling formulation that consists of active power costs, voltage fluctuations summation, and technical constraints of MG. Furthermore, load flow and reserve constraints are considered to achieve proper voltage regulation. A modified Jacobian matrix is presented for calculating voltage variations and Mont Carlo simulation is used for generating and reducing scenarios. To convert the problem to a mixed integer linear program, a linearization procedure for nonlinear equations is presented. The proposed model is applied to a typical low-voltage MG and two different cases are investigated. The results show the effectiveness of the proposed model.

Keywords: microgrid, energy management system, voltage fluctuations, modified Jacobian matrix

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Authors: Waqas Ali

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Concrete is broadly used as a structural material in the construction of buildings. When the concrete is exposed to elevated temperature, its strength evaluation is very necessary in the existing structure. In this study, the effect of temperature and the reliability of the core test has been evaluated. For this purpose, the cylindrical cores were extracted from High strength concrete (HSC) specimens that were exposed to the temperature ranging from 300 ℃ to 900 ℃ with a constant duration of 4 hr. This study compares the difference between the standard heated cylinders and the cores taken from them after curing of 90 days. The difference of cylindrical control and binary mix samples and extracted cores revealed that there is 12.19 and 12.38% difference at 300℃, while this difference was found to increase up to 12.89%, 13.03% at 500 ℃. Furthermore, this value is recorded as 12.99%, 13.57% and 14.40%, 14.38% at 700 ℃ and 900 ℃, respectively. A total of four equations were developed through a regression model for the prediction of the strength of concrete for both standard cylinders and extracted cores whose R square values were 0.9733, 0.9627 and 0.9473, 0.9452, respectively.

Keywords: high strength, temperature, core, reliability

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Authors: Behzad Shahizare, N. Nik-Ghazali, Kannan M. Munisamy, Seyedsaeed Tabatabaeikia

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The main objective of this study is to clarify the effect of the adjustable outlet guide vane (OGV) on the axial fan. Three-dimensional Numerical study was performed to analyze the effect of adjustable guide vane for improving the airflow rate in axial fans. Grid independence test was done between five different meshes in order to choose the reliable mesh. In flow analyses, Reynolds averaged Navier-Stokes (RANS) equations was solved using three types of turbulence models named k-ɛ, k-ω and k-ω SST. The aerodynamic performances of the fan and guide vane were evaluated. Numerical method was validated by comparing with experimental test according to AMECA 210 standard. Results showed that, by using the adjustable guide vane the airflow rate is increased around 3% to 6 %. The maximum enhancement of the airflow rate was achieved when pressure was 374pa.

Keywords: axial fan, adjustable guide vane, CFD, turbo machinery

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Authors: N. M. Arifin, S. P. M. Isa, R. Nazar, N. Bachok, F. M. Ali, I. Pop

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In this paper, the steady forced convection boundary layer flow of a Casson fluid past a moving permeable semi-infinite flat plate beneath a uniform free stream is investigated. The mathematical problem reduces to a pair of noncoupled ordinary differential equations by similarity transformation, which is then solved numerically using the shooting method. Both the cases when the plate moves into or out of the origin are considered. Effects of the non-Newtonian (Casson) parameter, moving parameter, suction or injection parameter and Eckert number on the flow and heat transfer characteristics are thoroughly examined. Dual solutions are found to exist for each value of the governing parameters.

Keywords: forced convection, Casson fluids, moving flat plate, boundary layer

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Authors: Olukunle C. Olawole, Dilip Kumar De

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We have modified Richardson-Dushman equation considering thermal expansion of lattice and change of chemical potential with temperature in material. The corresponding modified Richardson-Dushman (MRDE) equation fits quite well the experimental data of thermoelectronic current density (J) vs T from carbon nanotubes. It provides a unique technique for accurate determination of W0 Fermi energy, EF0 at 0 K and linear thermal expansion coefficient of carbon nano-tube in good agreement with experiment. From the value of EF0 we obtain the charge carrier density in excellent agreement with experiment. We describe application of the equations for the evaluation of performance of concentrated solar thermionic energy converter (STEC) with emitter made of carbon nanotube for future applications.

Keywords: carbon nanotube, modified Richardson-Dushman equation, fermi energy at 0 K, charge carrier density

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Authors: Aziz Ahmad, Guillermo Ramirez-Prado

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Governments around the world encourage the use of solar water heating in residential houses due to the low maintenance requirements and efficiency of the solar collector water heating systems. The aim of this work is to study a domestic solar water heating system in a residential building to develop a model of the entire solar water heating system including flat-plate solar collector and storage tank. The proposed model is adaptable to any households and location. The model can be used to test different types of controllers and can provide efficiency as well as economic analysis. The proposed model is based on the heat and mass transfer equations along with assumptions applied in the model which can be modified for a variety of different solar water heating systems and sizes. Simulation results of the model were compared with the actual system which shows similar trends.

Keywords: solar thermal systems, solar water heating, solar collector model, hot water tank model, solar controllers

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Authors: Amor Jnifene, Cedric Cocaud

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One of the most constraining problems facing the development of autonomous vehicles is the limitations of current technologies. Guidance and navigation controllers need to be faster and more robust. Communication data links need to be more reliable and secure. For an Unmanned Aerial Vehicles (UAV) to be useful, and fully autonomous, one important feature that needs to be an integral part of the navigation system is autonomous trajectory planning. The work discussed in this paper presents a method for on-line trajectory planning for UAV’s. This method takes into account various constraints of different types including specific vectors of approach close to target points, multiple objectives, and other constraints related to speed, altitude, and obstacle avoidance. The trajectory produced by the proposed method ensures a smooth transition between different segments, satisfies the minimum curvature imposed by the dynamics of the UAV, and finds the optimum velocity based on available atmospheric conditions. Given a set of objective points and waypoints a skeleton of the trajectory is constructed first by linking all waypoints with straight segments based on the order in which they are encountered in the path. Secondly, vectors of approach (VoA) are assigned to objective waypoints and their preceding transitional waypoint if any. Thirdly, the straight segments are replaced by 3D curvilinear trajectories taking into account the aircraft dynamics. In summary, this work presents a method for on-line 3D trajectory generation (TG) of Unmanned Aerial Vehicles (UAVs). The method takes as inputs a series of waypoints and an optional vector of approach for each of the waypoints. Using a dynamic model based on the performance equations of fixed wing aircrafts, the TG computes a set of 3D parametric curves establishing a course between every pair of waypoints, and assembling these sets of curves to construct a complete trajectory. The algorithm ensures geometric continuity at each connection point between two sets of curves. The geometry of the trajectory is optimized according to the dynamic characteristics of the aircraft such that the result translates into a series of dynamically feasible maneuvers. In summary, this work presents a method for on-line 3D trajectory generation (TG) of Unmanned Aerial Vehicles (UAVs). The method takes as inputs a series of waypoints and an optional vector of approach for each of the waypoints. Using a dynamic model based on the performance equations of fixed wing aircraft, the TG computes a set of 3D parametric curves establishing a course between every pair of waypoints, and assembling these sets of curves to construct a complete trajectory. The algorithm ensures geometric continuity at each connection point between two sets of curves. The geometry of the trajectory is optimized according to the dynamic characteristics of the aircraft such that the result translates into a series of dynamically feasible maneuvers.

Keywords: trajectory planning, unmanned autonomous air vehicle, vector of approach, waypoints

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Authors: Mark Watson, J.-F. Bousquet, Adam Forget

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A magnetic induction based underwater communication link is evaluated using an analytical model and a custom Finite-Difference Time-Domain (FDTD) simulation tool. The analytical model is based on the Sommerfeld integral, and a full-wave simulation tool evaluates Maxwell’s equations using the FDTD method in cylindrical coordinates. The analytical model and FDTD simulation tool are then compared and used to predict the system performance for various transmitter depths and optimum frequencies of operation. To this end, the system bandwidth, signal to noise ratio, and the magnitude of the induced voltage are used to estimate the expected channel capacity. The models show that in seawater, a relatively low-power and small coils may be capable of obtaining a throughput of 40 to 300 kbps, for the case where a transmitter is at depths of 1 to 3 m and a receiver is at a height of 1 m.

Keywords: magnetic induction, FDTD, underwater communication, Sommerfeld

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Authors: Carlos Caro, Ernest Bladé, Pedro Acosta, Camilo Lesmes

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The drying-wet process is one of the topics to be more careful in distributed hydrological modeling using finite volume schemes as a means of solving the equations of Saint Venant. In a hydrologic and hydraulic computer model, surface flow phenomena depend mainly on the different flow accumulation and subsequent runoff generation. These accumulations are generated by routing, cell by cell, from the heights of water, which begin to appear due to the rain at each instant of time. Determine when it is considered a dry cell and when considered wet to include in the full calculation is an issue that directly affects the quantification of direct runoff or generation of flow at the end of a zone of contribution by accumulations flow generated from cells or finite volume.

Keywords: hydrology, transport processes, hydrological modelling, finite volume schemes

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Authors: Rogelio Luck, Yucheng Liu

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This paper proposes the continuous-time singular value decomposition (SVD) for the impulse response function, a special kind of Green’s functions e⁻⁽ᵗ⁻ ᵀ⁾, in order to find a set of singular functions and singular values so that the convolutions of such function with the set of singular functions on a specified domain are the solutions to the inhomogeneous differential equations for those singular functions. A numerical example was illustrated to verify the proposed method. Besides the continuous-time SVD, a discrete-time SVD is also presented for the impulse response function, which is modeled using a Toeplitz matrix in the discrete system. The proposed method has broad applications in signal processing, dynamic system analysis, acoustic analysis, thermal analysis, as well as macroeconomic modeling.

Keywords: singular value decomposition, impulse response function, Green’s function , Toeplitz matrix , Hankel matrix

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Authors: Daniel S. Brox

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Starting from earthquake fault dynamic equations, a correspondence between earthquake occurrence statistics in a seismic region before a major earthquake and eigenvalue statistics of a differential operator whose bound state eigenfunctions characterize the distribution of stress in the seismic region is derived. Modeling these eigenvalue statistics with a 2D Coulomb gas statistical physics model, previously reported deviation of seismic activation earthquake occurrence statistics from Gutenberg-Richter statistics in time intervals preceding the major earthquake is derived. It also explains how statistical physics modeling predicts a finite-dimensional nonlinear dynamic system that describes real-time velocity model evolution in the region undergoing seismic activation and how this prediction can be tested experimentally.

Keywords: seismic activation, statistical physics, geodynamics, signal processing

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Authors: Alexander Babin, Leonid Savin

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In the present paper, increasing of energy efficiency of a thrust hybrid bearing with a central feeding chamber is considered. The mathematical model was developed to determine the pressure distribution and the reaction forces, based on the Reynolds equation and static characteristics’ equations. The boundary problem of pressure distribution calculation was solved using the method of finite differences. For various types of lubricants, geometry and operational characteristics, axial gaps can be determined, where the minimal friction coefficient is provided. The next part of the study considers the application of servovalves in order to maintain the desired position of the rotor. The report features the calculation results and the analysis of the influence of the operational and geometric parameters on the energy efficiency of mechatronic fluid-film bearings.

Keywords: active bearings, energy efficiency, mathematical model, mechatronics, thrust multipad bearing

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Authors: P. Lauk, K. E. Seegel, T. Tähemaa

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The fuel consumption of modern, high wing loading, commercial aircraft in the first stage of flight is high because the usable flight level is lower and the weather conditions (jet stream) have great impact on aircraft performance. To reduce the fuel consumption, it is necessary to raise during first stage of flight the L/D ratio value within Cl 0.55-0.65. Different variable geometrical wing trailing edge modifications of SC(2)-410 airfoil were compared at M 0.78 using the CFD software STAR-CCM+ simulation based Reynolds-averaged Navier-Stokes (RANS) equations. The numerical results obtained show that by increasing the width of the airfoil by 4% and by modifying the trailing edge airfoil, it is possible to decrease airfoil drag at Cl 0.70 for up to 26.6% and at the same time to increase commercial aircraft L/D ratio for up to 5.0%. Fuel consumption can be reduced in proportion to the increase in L/D ratio.

Keywords: L/D ratio, miniflaps, mini-TED, supercritical airfoil

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Authors: Charline David, Alexandre Blondin Massé, Arnaud Zinflou

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In 2016, Clements, Hurn, and Li proposed a multiple equation time series approach for the short-term load forecasting, reporting an average mean absolute percentage error (MAPE) of 1.36% on an 11-years dataset for the Queensland region in Australia. We present an adaptation of their model to the electrical power load consumption for the whole Quebec province in Canada. More precisely, we take into account two additional meteorological variables — cloudiness and wind speed — on top of temperature, as well as the use of multiple meteorological measurements taken at different locations on the territory. We also consider other minor improvements. Our final model shows an average MAPE score of 1:79% over an 8-years dataset.

Keywords: short-term load forecasting, special days, time series, multiple equations, parallelization, clustering

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Authors: A. Ghafouri, N. Pourmahmoud, I. Mirzaee

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In this numerical study, effects of using Al2O3-water nanofluid on the rate of heat transfer have been investigated numerically. The physical model is a square enclosure with insulated top and bottom horizontal walls while the vertical walls are kept at different constant temperatures. Two appropriate models are used to evaluate the viscosity and thermal conductivity of nanofluid. The governing stream-vorticity equations are solved using a second order central finite difference scheme, coupled to the conservation of mass and energy. The study has been carried out for the nanoparticle diameter 30, 60, and 90 nm and the solid volume fraction 0 to 0.04. Results are presented by average Nusselt number and normalized Nusselt number in the different range of φ and D for mixed convection dominated regime. It is found that different heat transfer rate is predicted when the effect of nanoparticle diameter is taken into account.

Keywords: nanofluid, nanoparticle diameter, heat transfer enhancement, square enclosure, Nusselt number

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Authors: M. C. Lin, C. W. Su

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The paper investigates the optimization analysis to the heat exchanger design, mainly with response surface method and genetic algorithm to explore the relationship between optimal fluid flow velocity and temperature of the heat exchanger using field synergy principle. First, finite volume method is proposed to calculate the flow temperature and flow rate distribution for numerical analysis. We identify the most suitable simulation equations by response surface methodology. Furthermore, a genetic algorithm approach is applied to optimize the relationship between fluid flow velocity and flow temperature of the heat exchanger. The results show that the field synergy angle plays vital role in the performance of a true heat exchanger.

Keywords: optimization analysis, field synergy, heat exchanger, genetic algorithm

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Authors: Shohel Ahmed, Md. Abdul Alim, Sumaiya Rahman

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Optimal control can be helpful to test and compare different vaccination strategies of a certain disease. The mathematical model of HIV we consider here is a set of ordinary differential equations (ODEs) describing the interactions of CD4+T cells of the immune system with the human immunodeficiency virus (HIV). As an early treatment setting, we investigate an optimal chemotherapy strategy where control represents the percentage of effect the chemotherapy has on the system. The aim is to obtain a new optimal chemotherapeutic strategy where an isoperimetric constraint on the chemotherapy supply plays a crucial role. We outline the steps in formulating an optimal control problem, derive optimality conditions and demonstrate numerical results of an optimal control for the model. Numerical results illustrate how such a constraint alters the optimal vaccination schedule and its effect on cell-virus interactions.

Keywords: chemotherapy of HIV, optimal control involving ODEs, optimality conditions, Pontryagin’s maximum principle

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Authors: A. Suparmi, C. Cari, M. Yunianto, B. N. Pratiwi

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D-dimensional Dirac equations of q-deformed shape invariant potentials were solved using supersymmetric quantum mechanics (SUSY QM) in the case of exact spin symmetry. The D dimensional radial Dirac equation for shape invariant potential reduces to one-dimensional Schrodinger type equation by an appropriate variable and parameter change. The relativistic energy spectra were analyzed by using SUSY QM and shape invariant properties from radial D dimensional Dirac equation that have reduced to one dimensional Schrodinger type equation. The SUSY operator was used to generate the D dimensional relativistic radial wave functions, the relativistic energy equation reduced to the non-relativistic energy in the non-relativistic limit.

Keywords: D-dimensional dirac equation, non-central potential, SUSY QM, radial wave function

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Authors: Xing Feng, Yuanbin Li

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Oil spills at sea can cause severe marine environmental damage, including bringing huge hazards to living resources and human beings. In situ burning or chemical dispersant methods can be used to handle the oil spills sometimes, but these approaches will bring secondary pollution and fail in some situations. Oil recovery techniques have also been developed to recover oil using oil skimmer equipment installed on ships, while the hydrodynamic process of the oil flowing through the oil skimmer is very complicated and important for evaluating the recovery efficiency. Based on this, a two-dimensional numerical simulation platform for simulating the hydrodynamic process of the oil flowing through the oil skimmer is established based on the Navier-Stokes equations for viscous, incompressible fluid. Finally, the influence of the design of the flow channel in the curved plane oil skimmer on the hydrodynamic process of the oil flowing through the oil skimmer is investigated based on the established simulation platform.

Keywords: curved plane oil skimmer, flow channel, CFD, VOF

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Authors: Nasser Mohamed Ramli, Mohamad Syafiq Mohamad

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Many types of controllers were applied on the continuous stirred tank reactor (CSTR) unit to control the temperature. In this research paper, Proportional-Integral-Derivative (PID) controller are compared with Fuzzy Logic controller for temperature control of CSTR. The control system for temperature non-isothermal of a CSTR will produce a stable response curve to its set point temperature. A mathematical model of a CSTR using the most general operating condition was developed through a set of differential equations into S-function using MATLAB. The reactor model and S-function are developed using m.file. After developing the S-function of CSTR model, User-Defined functions are used to link to SIMULINK file. Results that are obtained from simulation and temperature control were better when using Fuzzy logic control compared to PID control.

Keywords: CSTR, temperature, PID, fuzzy logic

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Authors: Ehsan Mehryaar

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The rate of penetration is (RoP) one of the vital factors in the cost and time of tunnel boring projects; therefore, predicting it can lead to a substantial increase in the efficiency of the project. RoP is heavily dependent geological properties of the project site and TBM properties. In this study, 151-point data from Queen’s water tunnel is collected, which includes unconfined compression strength, peak slope index, angle with weak planes, and distance between planes of weaknesses. Since the size of the data is small, it was observed that it is imbalanced. To solve that problem synthetic minority oversampling technique is utilized. The model based on the model tree is proposed, where each leaf consists of a support vector machine model. Proposed model performance is then compared to existing empirical equations in the literature.

Keywords: Model tree, SMOTE, rate of penetration, TBM(tunnel boring machine), SVM

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Authors: M. A. Khanday, Aasma Rafiq, Khalid Nazir

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This paper is an attempt to establish the mathematical models to understand the distribution of drug administration in the human body through oral and intravenous routes. Three models were formulated based on diffusion process using Fick’s principle and the law of mass action. The rate constants governing the law of mass action were used on the basis of the drug efficacy at different interfaces. The Laplace transform and eigenvalue methods were used to obtain the solution of the ordinary differential equations concerning the rate of change of concentration in different compartments viz. blood and tissue medium. The drug concentration in the different compartments has been computed using numerical parameters. The results illustrate the variation of drug concentration with respect to time using MATLAB software. It has been observed from the results that the drug concentration decreases in the first compartment and gradually increases in other subsequent compartments.

Keywords: Laplace transform, diffusion, eigenvalue method, mathematical model

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Authors: Sung-Sik Park, Han Chang, Kyung-Hun Chae, Sae-Byeok Lee

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In this study, a three-dimensional (3D) Particle method without using grid was developed for analyzing large deformation of soils instead of using ordinary finite element method (FEM) or finite difference method (FDM). In the 3D Particle method, the governing equations were discretized by various particle interaction models corresponding to differential operators such as gradient, divergence, and Laplacian. The Mohr-Coulomb failure criterion was incorporated into the 3D Particle method to determine soil failure. The yielding and hardening behavior of soil before failure was also considered by varying viscosity of soil. First of all, an unconfined compression test was carried out and the large deformation following soil yielding or failure was simulated by the developed 3D Particle method. The results were also compared with those of a commercial FEM software PLAXIS 3D. The developed 3D Particle method was able to simulate the 3D large deformation of soils due to soil yielding and calculate the variation of normal and shear stresses following clay deformation.

Keywords: particle method, large deformation, soil column, confined compressive stress

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Authors: K. B. Ghazaryan, R. A. Ghazaryan

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Based on the constitutive model and anti-plane equations of anisotropic elastic body of monoclinic symmetry we consider the problem of shear wave propagation in multi-layered disordered composite structure with point defect. Using transfer matrix method the analytic expression is obtained providing solutions of shear Floquet wave propagation in periodic disordered anisotropic structure. The usefulness of the obtained analytical expression was discussed also in reflection and refraction problems from multi-layered reflector as well as in vibration problem of multi-layered waveguides. Numerical results are presented highlighting the effects arising in disordered periodic structure due to defects of multi-layered structure.

Keywords: shear elastic waves, monoclinic anisotropic media, periodic structure, disordered multilayer laminae, multi-layered waveguide

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Authors: Harpreet Kaur, Er. Arjun, Kirandeep Kaur, P. K. Mittal

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Simulation of a human body from a 20% gelatin & 80% water mixture is examined from a wound ballistics point of view. Parameters such as incapacitation energy & temporary to permanent cavity size & tools of hydrodynamics have been employed to arrive at a model of the human body similar to the one adopted by NATO. Calculations using equations of motion yield a value of 339 µs in which a temporary cavity with maximum size settles down to a permanent cavity. This occurs for 10mm size bullets & settles down to a permanent cavity in the case of 4 different bullets, i.e., 5.45, 5.56, 7.62,10 mm sizes. The obtained results are in excellent agreement with the body as a right circular cylinder of 15 cm height & 10 cm diameter. An effort is made here in this work to present a sound theoretical base to parameters commonly used in wound ballistics from field experience discussed by Col Coats & Major Beyer.

Keywords: gelatine, gunshot, hydrodynamic model, oscillation time, temporary and permanent cavity, wound ballistics

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Authors: Thomas Paris, Vincent Bruyere, Patrick Namy

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A numerical model is developed to simulate gas blowdowns through a thin tube and a filter (porous media), separating a high pressure gas filled reservoir to low pressure ones. Based on a previous work, a one-dimensional approach is developed by using the finite element method to solve the transient compressible flow and to predict the pressure and temperature evolution in space and time. Mass, momentum, and energy conservation equations are solved in a fully coupled way in the reservoirs, the pipes and the porous media. Numerical results, such as pressure and temperature evolutions, are firstly compared with experimental data to validate the model for different configurations. Couplings between porous media and pipe flow are then validated by checking mass balance. The influence of the porous media and the nature of the gas is then studied for different initial high pressure values.

Keywords: compressible flow, fluid mechanics, heat transfer, porous media

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Authors: Haoui Rabah

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The aim of this work is to analyze a viscous flow around the axisymmetric blunt body taken into account the mesh size both in the free stream and into the boundary layer. The resolution of the Navier-Stokes equations is realized by using the finite volume method to determine the flow parameters and detached shock position. The numerical technique uses the Flux Vector Splitting method of Van Leer. Here, adequate time stepping parameter, CFL coefficient and mesh size level are selected to ensure numerical convergence. The effect of the mesh size is significant on the shear stress and velocity profile. The best solution is obtained with using a very fine grid. This study enabled us to confirm that the determination of boundary layer thickness can be obtained only if the size of the mesh is lower than a certain value limits given by our calculations.

Keywords: supersonic flow, viscous flow, finite volume, blunt body

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Authors: H. Bayat, M. Majidi, M. Bolhasani, A. Karbalaie Alilou, A. Mirabdolah Lavasani

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Unsteady flow and heat transfer from a circular cylinder in cross-flow is studied numerically. The governing equations are solved by using finite volume method. Reynolds number varies in range of 50 to 200, in this range flow is considered to be laminar and unsteady. Al2O3 nanoparticle with volume fraction in range of 5% to 20% is added to pure water. Effects of adding nanoparticle to pure water on lift and drag coefficient and Nusselt number is presented. Addition of Al2O3 has inconsiderable effect on the value of drags and lift coefficient. However, it has significant effect on heat transfer; results show that heat transfer of Al2O3 nanofluid is about 9% to 36% higher than pure water.

Keywords: nanofluid, heat transfer, unsteady flow, forced convection, cross-flow

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Authors: Nur Rokhman

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A mathematical function gives relationship between the variables composing the function. Interpolation can be viewed as a process of finding mathematical function which goes through some specified points. There are many interpolation methods, namely: Lagrange method, Newton method, Spline method etc. For some specific condition, such as, big amount of interpolation points, the interpolation function can not be written explicitly. This such function consist of computational steps. The solution of equations involving the interpolation function is a problem of solution of non linear equation. Newton method will not work on the interpolation function, for the derivative of the interpolation function cannot be written explicitly. This paper shows the use of Secton method to determine the numerical solution of the function involving the interpolation function. The experiment shows the fact that Secton method works better than Newton method in finding the root of Lagrange interpolation function.

Keywords: Secton method, interpolation, non linear function, numerical solution

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Authors: Stojan Kravanja, Andrej Ivanič, Tomaž Žula

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This contribution focuses on structural optimization in civil engineering using mixed integer non-linear programming (MINLP). MINLP is characterized as a versatile method that can handle both continuous and discrete optimization variables simultaneously. Continuous variables are used to optimize parameters such as dimensions, stresses, masses, or costs, while discrete variables represent binary decisions to determine the presence or absence of structural elements within a structure while also calculating discrete materials and standard sections. The optimization process is divided into three main steps. First, a mechanical superstructure with a variety of different topology-, material- and dimensional alternatives. Next, a MINLP model is formulated to encapsulate the optimization problem. Finally, an optimal solution is searched in the direction of the defined objective function while respecting the structural constraints. The economic or mass objective function of the material and labor costs of a structure is subjected to the constraints known from structural analysis. These constraints include equations for the calculation of internal forces and deflections, as well as equations for the dimensioning of structural components (in accordance with the Eurocode standards). Given the complex, non-convex and highly non-linear nature of optimization problems in civil engineering, the Modified Outer-Approximation/Equality-Relaxation (OA/ER) algorithm is applied. This algorithm alternately solves subproblems of non-linear programming (NLP) and main problems of mixed-integer linear programming (MILP), in this way gradually refines the solution space up to the optimal solution. The NLP corresponds to the continuous optimization of parameters (with fixed topology, discrete materials and standard dimensions, all determined in the previous MILP), while the MILP involves a global approximation to the superstructure of alternatives, where a new topology, materials, standard dimensions are determined. The optimization of a convex problem is stopped when the MILP solution becomes better than the best NLP solution. Otherwise, it is terminated when the NLP solution can no longer be improved. While the OA/ER algorithm, like all other algorithms, does not guarantee global optimality due to the presence of non-convex functions, various modifications, including convexity tests, are implemented in OA/ER to mitigate these difficulties. The effectiveness of the proposed MINLP approach is demonstrated by its application to various structural optimization tasks, such as mass optimization of steel buildings, cost optimization of timber halls, composite floor systems, etc. Special optimization models have been developed for the optimization of these structures. The MINLP optimizations, facilitated by the user-friendly software package MIPSYN, provide insights into a mass or cost-optimal solutions, optimal structural topologies, optimal material and standard cross-section choices, confirming MINLP as a valuable method for the optimization of structures in civil engineering.

Keywords: MINLP, mixed-integer non-linear programming, optimization, structures

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