Search results for: method of least square

Authors: Sattar Al-Jabair, Laith J. Habeeb

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In this work, the natural convection in a concentric annulus between a cold outer inclined square enclosure and heated inner circular cylinder is simulated for two-dimensional steady state. The Boussinesq approximation was applied to model the buoyancy-driven effect and the governing equations were solved using the time marching approach staggered by body fitted coordinates. The coordinate transformation from the physical domain to the computational domain is set up by an analytical expression. Numerical results for Rayleigh numbers 103 , 104 , 105 and 106, aspect ratios 1.5 , 3.0 and 4.5 for seven different inclination angles for the outer square enclosure 0o , -30o , -45o , -60o , -90o , -135o , -180o are presented as well. The computed flow and temperature fields were demonstrated in the form of streamlines, isotherms and Nusselt numbers variation. It is found that both the aspect ratio and the Rayleigh number are critical to the patterns of flow and thermal fields. At all Rayleigh numbers angle of inclination has nominal effect on heat transfer.

Keywords: natural convection, concentric annulus, square inclined enclosure

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Authors: Mahmoud Hassanlourad, Mehran Naghizadehrokni, Vahid Molaei

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In this study, the seismic behavior of a shallow tunnel with square cross section is investigated in a two layered and elastic heterogeneous environment using numerical method. To do so, FLAC finite difference software was used. Behavioral model of the ground and tunnel structure was assumed linear elastic. Dynamic load was applied to the model for 0.2 seconds from the bottom in form of a square pulse with maximum acceleration of 1 m/s². The interface between the two layers was considered at three different levels of crest, middle, and bottom of the tunnel. The stiffness of the two upper and lower layers was considered to be varied from 10 MPa to 1000 MPa. Deformation of cross section of the tunnel due to dynamic load propagation, as well as the values of axial force and bending moment created in the tunnel structure, were examined in the three states mentioned above. The results of analyses show that heterogeneity of the environment, its stratification, and positioning of the interface of the two layers with respect to tunnel height and the stiffness ratio of the two layers have significant effects on the value of bending moment, axial force, and distortion of tunnel cross-section.

Keywords: Dynamic analysis, shallow-buried tunnel, two-layered ground.

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Authors: S. Jani, M. Mahmoodi, M. Amini

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Magnetohydrodynamic free convection fluid flow and heat transfer in a square cavity filled with an electric conductive fluid with Prandtl number of 0.7 has been investigated numerically. The horizontal bottom wall of the cavity was kept at Th while the side and the top walls of the cavity were maintained at a constant temperature Tc with Th>Tc. The governing equations written in terms of the primitive variables were solved numerically using the finite volume method while the SIMPLER algorithm was used to couple the velocity and pressure fields. Using the developed code, a parametric study was performed, and the effects of the Rayleigh number and the Hartman number on the fluid flow and heat transfer inside the cavity were investigated. The obtained results showed that temperature distribution and flow pattern inside the cavity depended on both strength of the magnetic field and Rayleigh number. For all cases two counter rotating eddies were formed inside the cavity. The magnetic field decreased the intensity of free convection and flow velocity. Also it was found that for higher Rayleigh numbers a relatively stronger magnetic field was needed to decrease the heat transfer through free convection.

Keywords: Free Convection, Magnetic Field, Square Cavity, Numerical Simulation.

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Authors: Guangbin Wang, Fuping Tan, Deyu Sun

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In this paper, we present new preconditioned generalized mixed-type splitting (GMTS) methods for solving weighted linear least square problems. We compare the spectral radii of the iteration matrices of the preconditioned and the original methods. The comparison results show that the preconditioned GMTS methods converge faster than the GMTS method whenever the GMTS method is convergent. Finally, we give a numerical example to confirm our theoretical results.

Keywords: Preconditioned, GMTS method, linear system, convergence, comparison.

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Authors: C. B. Vishwakarma

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The author presented a method for model order reduction of large-scale time-invariant systems in time domain. In this approach, two modified Hankel matrices are suggested for getting reduced order models. The proposed method is simple, efficient and retains stability feature of the original high order system. The viability of the method is illustrated through the examples taken from literature.

Keywords: Model Order Reduction, Stability, Hankel Matrix, Time-Domain, Integral Square Error.

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Authors: Hailong Zhu, Zhaoxiang Li

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Semilinear elliptic equations are ubiquitous in natural sciences. They give rise to a variety of important phenomena in quantum mechanics, nonlinear optics, astrophysics, etc because they have rich multiple solutions. But the nontrivial solutions of semilinear equations are hard to be solved for the lack of stabilities, such as Lane-Emden equation, Henon equation and Chandrasekhar equation. In this paper, bifurcation method is applied to solving semilinear elliptic equations which are with homogeneous Dirichlet boundary conditions in 2D. Using this method, nontrivial numerical solutions will be computed and visualized in many different domains (such as square, disk, annulus, dumbbell, etc).

Keywords: Semilinear elliptic equations, positive solutions, bifurcation method, isotropy subgroups.

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Authors: S. Sukparungsee

Abstract:

Generally, the traditional Shewhart p chart has been developed by for charting the binomial data. This chart has been developed using the normal approximation with condition as low defect level and the small to moderate sample size. In real applications, however, are away from these assumptions due to skewness in the exact distribution. In this paper, a modified Exponentially Weighted Moving Average (EWMA) control chat for detecting a change in binomial data by improving square root transformations, namely ISRT p EWMA control chart. The numerical results show that ISRT p EWMA chart is superior to ISRT p chart for small to moderate shifts, otherwise, the latter is better for large shifts.

Keywords: Number of defects, Exponentially Weighted Moving Average, Average Run Length, Square root transformations.

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Authors: A. Danbaba

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Several experiments are conducted at different environments such as locations or periods (seasons) with identical treatments to each experiment purposely to study the interaction between the treatments and environments or between the treatments and periods (seasons). The commonly used designs of experiments for this purpose are randomized block design, Latin square design, balanced incomplete block design, Youden design, and one or more factor designs. The interest is to carry out a combined analysis of the data from these multi-environment experiments, instead of analyzing each experiment separately. This paper proposed combined analysis of experiments conducted via Sudoku square design of odd order with same experimental treatments.

Keywords: Sudoku designs, combined analysis, multi-environment experiments, common treatments.

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Authors: Zixin Liu, Huawei Yang, Fangwei Chen

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This paper studies the mean square exponential synchronization problem of a class of stochastic neutral type chaotic neural networks with mixed delay. On the Basis of Lyapunov stability theory, some sufficient conditions ensuring the mean square exponential synchronization of two identical chaotic neural networks are obtained by using stochastic analysis and inequality technique. These conditions are expressed in the form of linear matrix inequalities (LMIs), whose feasibility can be easily checked by using Matlab LMI Toolbox. The feedback controller used in this paper is more general than those used in previous literatures. One simulation example is presented to demonstrate the effectiveness of the derived results.

Keywords: Exponential synchronization, stochastic analysis, chaotic neural networks, neutral type system.

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Authors: Sunitha S. L., V. Udayashankara

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Hearing impairment is the number one chronic disability affecting many people in the world. Background noise is particularly damaging to speech intelligibility for people with hearing loss especially for sensorineural loss patients. Several investigations on speech intelligibility have demonstrated sensorineural loss patients need 5-15 dB higher SNR than the normal hearing subjects. This paper describes Discrete Hartley Transform Power Normalized Least Mean Square algorithm (DHT-LMS) to improve the SNR and to reduce the convergence rate of the Least Means Square (LMS) for sensorineural loss patients. The DHT transforms n real numbers to n real numbers, and has the convenient property of being its own inverse. It can be effectively used for noise cancellation with less convergence time. The simulated result shows the superior characteristics by improving the SNR at least 9 dB for input SNR with zero dB and faster convergence rate (eigenvalue ratio 12) compare to time domain method and DFT-LMS.

Keywords: Hearing Impairment, DHT-LMS, Convergence rate, SNR improvement.

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Authors: Noor Atinah Ahmad, Shazia Javed

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We present a discussion of three adaptive filtering algorithms well known for their one-step termination property, in terms of their relationship with the minimal residual method. These algorithms are the normalized least mean square (NLMS), Affine Projection algorithm (APA) and the recursive least squares algorithm (RLS). The NLMS is shown to be a result of the orthogonality condition imposed on the instantaneous approximation of the Wiener equation, while APA and RLS algorithm result from orthogonality condition in multi-dimensional minimal residual formulation. Further analysis of the minimal residual formulation for the RLS leads to a triangular system which also possesses the one-step termination property (in exact arithmetic)

Keywords: Adaptive filtering, minimal residual method, projection method.

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Authors: H. Ghanbarpour Asl, S. H. Pourtakdoust

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Extended Kalman Filter (EKF) is probably the most widely used estimation algorithm for nonlinear systems. However, not only it has difficulties arising from linearization but also many times it becomes numerically unstable because of computer round off errors that occur in the process of its implementation. To overcome linearization limitations, the unscented transformation (UT) was developed as a method to propagate mean and covariance information through nonlinear transformations. Kalman filter that uses UT for calculation of the first two statistical moments is called Unscented Kalman Filter (UKF). Square-root form of UKF (SRUKF) developed by Rudolph van der Merwe and Eric Wan to achieve numerical stability and guarantee positive semi-definiteness of the Kalman filter covariances. This paper develops another implementation of SR-UKF for sequential update measurement equation, and also derives a new UD covariance factorization filter for the implementation of UKF. This filter is equivalent to UKF but is computationally more efficient.

Keywords: Unscented Kalman filter, Square-root unscentedKalman filter, UD covariance factorization, Target tracking.

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Authors: Merve Ermis, Murat Yılmaz, Nihal Eratlı, Mehmet H. Omurtag

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The static and dynamic analyses of hyperboloidal helix having the closed and the open square box sections are investigated via the mixed finite element formulation based on Timoshenko beam theory. Frenet triad is considered as local coordinate systems for helix geometry. Helix domain is discretized with a two-noded curved element and linear shape functions are used. Each node of the curved element has 12 degrees of freedom, namely, three translations, three rotations, two shear forces, one axial force, two bending moments and one torque. Finite element matrices are derived by using exact nodal values of curvatures and arc length and it is interpolated linearly throughout the element axial length. The torsional moments of inertia for close and open square box sections are obtained by finite element solution of St. Venant torsion formulation. With the proposed method, the torsional rigidity of simply and multiply connected cross-sections can be also calculated in same manner. The influence of the close and the open square box cross-sections on the static and dynamic analyses of hyperboloidal helix is investigated. The benchmark problems are represented for the literature.

Keywords: Hyperboloidal helix, squared cross section, thin walled cross section, torsional rigidity.

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Authors: Zeshan Ahmad, Matteo Zoppi, Rezia Molfino

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The objective of positioning the fixture elements in the fixture is to make the workpiece stiff, so that geometric errors in the manufacturing process can be reduced. Most of the work for optimal fixture layout used the minimization of the sum of the nodal deflection normal to the surface as objective function. All deflections in other direction have been neglected. We propose a new method for fixture layout optimization in this paper, which uses the element strain energy. The deformations in all the directions have been considered in this way. The objective function in this method is to minimize the sum of square of element strain energy. Strain energy and stiffness are inversely proportional to each other. The optimization problem is solved by the sequential quadratic programming method. Three different kinds of case studies are presented, and results are compared with the method using nodal deflections as objective function to verify the propose method.

Keywords: Fixture layout, optimization, strain energy, quadratic programming.

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Authors: Su Yi Ming, Hou Ying, Zou Guang Ping

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Metal-net rubber is a new dry friction damping material, compared with the traditional metal rubber, which has high mechanization degree, and the mechanical performance of metal-net rubber is more stable. Through the sine sweep experiment and random vibration experiment of metal-net rubber vibration isolator, the influence of several important factors such as the lines slope, relative density and wire diameter on the transfer rate, natural frequency and root-mean-square response acceleration of metal-net rubber vibration isolation system, were studied through the method of control variables. Also, several relevant change curves under different vibration levels were derived, and the effects of vibration level on the natural frequency and root-mean-square response acceleration were analyzed through the curves.

Keywords: Metal-net rubber vibration isolator, relative density, vibration level, wire diameter.

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Authors: Amir Rastegarnia, Mohammad Ali Tinati, Azam Khalili

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In this paper we consider the issue of distributed adaptive estimation over sensor networks. To deal with more realistic scenario, different variance for observation noise is assumed for sensors in the network. To solve the problem of different variance of observation noise, the proposed method is divided into two phases: I) Estimating each sensor-s observation noise variance and II) using the estimated variances to obtain the desired parameter. Our proposed algorithm is based on a diffusion least mean square (LMS) implementation with linear combiner model. In the proposed algorithm, the step-size parameter the coefficients of linear combiner are adjusted according to estimated observation noise variances. As the simulation results show, the proposed algorithm considerably improves the diffusion LMS algorithm given in literature.

Keywords: Adaptive filter, distributed estimation, sensor network, diffusion.

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Authors: Hassan Manouzi

Abstract:

We present in this paper a useful strategy to solve stochastic partial differential equations (SPDEs) involving stochastic coefficients. Using the Wick-product of higher order and the Wiener-Itˆo chaos expansion, the SPDEs is reformulated as a large system of deterministic partial differential equations. To reduce the computational complexity of this system, we shall use a decomposition-coordination method. To obtain the chaos coefficients in the corresponding deterministic equations, we use a least square formulation. Once this approximation is performed, the statistics of the numerical solution can be easily evaluated.

Keywords: Least squares, Wick product, SPDEs, finite element, Wiener chaos expansion, gradient method.

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Authors: A. Ghafouri, A. Falavand Jozaei, M. Salari

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In this paper, effects of using Alumina-water nanofluid on the rate of heat transfer have been investigated numerically. 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 Richardson number 0.1 to 10 and the solid volume fraction 0 to 0.04. Results are presented by isotherms lines, average Nusselt number and normalized Nusselt number in different range of φ and Ri for forced, combined and natural convection dominated regime. It is found that higher heat transfer rate is predicted when the effects of nanoparticle is taken into account.

Keywords: Nanofluid, Heat Transfer Enhancement, Square Enclosure, Nusselt number.

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Authors: Navneet Singh Deora, Aastha Deswal, H. N. Mishra

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Pasta is one of the most widely consumed food products around the world. Rapid determination of the moisture content in pasta will assist food processors to provide online quality control of pasta during large scale production. Rapid Fourier transform near-infrared method (FT-NIR) was developed for determining moisture content in pasta. A calibration set of 150 samples, a validation set of 30 samples and a prediction set of 25 samples of pasta were used. The diffuse reflection spectra of different types of pastas were measured by FT-NIR analyzer in the 4,000-12,000cm^-1 spectral range. Calibration and validation sets were designed for the conception and evaluation of the method adequacy in the range of moisture content 10 to 15 percent (w.b) of the pasta. The prediction models based on partial least squares (PLS) regression, were developed in the near-infrared. Conventional criteria such as the R², the root mean square errors of cross validation (RMSECV), root mean square errors of estimation (RMSEE) as well as the number of PLS factors were considered for the selection of three pre-processing (vector normalization, minimum-maximum normalization and multiplicative scatter correction) methods. Spectra of pasta sample were treated with different mathematic pre-treatments before being used to build models between the spectral information and moisture content. The moisture content in pasta predicted by FT-NIR methods had very good correlation with their values determined via traditional methods (R² = 0.983), which clearly indicated that FT-NIR methods could be used as an effective tool for rapid determination of moisture content in pasta. The best calibration model was developed with min-max normalization (MMN) spectral pre-processing (R² = 0.9775). The MMN pre-processing method was found most suitable and the maximum coefficient of determination (R²) value of 0.9875 was obtained for the calibration model developed.

Keywords: FT-NIR, Pasta, moisture determination.

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Authors: Wisam F. Al-Azzo, Borhanuddin Mohd. Ali

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This paper addresses the problem of peak-to-average power ratio (PAPR) in orthogonal frequency division multiplexing (OFDM) systems. It also introduces a new PAPR reduction technique based on adaptive square-rooting (SQRT) companding process. The SQRT process of the proposed technique changes the statistical characteristics of the OFDM output signals from Rayleigh distribution to Gaussian-like distribution. This change in statistical distribution results changes of both the peak and average power values of OFDM signals, and consequently reduces significantly the PAPR. For the 64QAM OFDM system using 512 subcarriers, up to 6 dB reduction in PAPR was achieved by square-rooting technique with fixed degradation in bit error rate (BER) equal to 3 dB. However, the PAPR is reduced at the expense of only -15 dB out-ofband spectral shoulder re-growth below the in-band signal level. The proposed adaptive SQRT technique is superior in terms of BER performance than the original, non-adaptive, square-rooting technique when the required reduction in PAPR is no more than 5 dB. Also, it provides fixed amount of PAPR reduction in which it is not available in the original SQRT technique.

Keywords: complementary cumulative distribution function(CCDF), OFDM, peak-to-average power ratio (PAPR), adaptivesquare-rooting PAPR reduction technique.

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Authors: Kariyappa B. S., Hariprasad S. A., R. Nagaraj

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In this paper, a new method of controlling position of AC Servomotor using Field Programmable Gate Array (FPGA). FPGA controller is used to generate direction and the number of pulses required to rotate for a given angle. Pulses are sent as a square wave, the number of pulses determines the angle of rotation and frequency of square wave determines the speed of rotation. The proposed control scheme has been realized using XILINX FPGA SPARTAN XC3S400 and tested using MUMA012PIS model Alternating Current (AC) servomotor. Experimental results show that the position of the AC Servo motor can be controlled effectively. KeywordsAlternating Current (AC), Field Programmable Gate Array (FPGA), Liquid Crystal Display (LCD).

Keywords: Alternating Current (AC), Field Programmable Gate Array (FPGA), Liquid Crystal Display (LCD).

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Authors: A. H. Fatah, A. H. Ahmed

Abstract:

Levenberg-Marquardt method (LM) was proposed to be applied as a non-linear least-square fitting in the analysis of a natural gamma-ray spectrum that was taken by the Hp (Ge) detector. The Gaussian function that composed of three components, main Gaussian, a step background function and tailing function in the lowenergy side, has been suggested to describe each of the y-ray lines mathematically in the spectrum. The whole spectrum has been analyzed by determining the energy and relative intensity for the strong y-ray lines.

Keywords: Gamma-Ray, Spectrum analysis, Non-linear leastsquare fitting.

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Authors: Manoochehr Gholipoor

Abstract:

The response of growth and yield of rainfed-chickpea to population density should be evaluated based on long-term experiments to include the climate variability. This is achievable just by simulation. In this simulation study, this evaluation was done by running the CYRUS model for long-term daily weather data of five locations in Iran. The tested population densities were 7 to 59 (with interval of 2) stands per square meter. Various functions, including quadratic, segmented, beta, broken linear, and dent-like functions, were tested. Considering root mean square of deviations and linear regression statistics [intercept (a), slope (b), and correlation coefficient (r)] for predicted versus observed variables, the quadratic and broken linear functions appeared to be appropriate for describing the changes in biomass and grain yield, and in harvest index, respectively. Results indicated that in all locations, grain yield tends to show increasing trend with crowding the population, but subsequently decreases. This was also true for biomass in five locations. The harvest index appeared to have plateau state across low population densities, but decreasing trend with more increasing density. The turning point (optimum population density) for grain yield was 30.68 stands per square meter in Isfahan, 30.54 in Shiraz, 31.47 in Kermanshah, 34.85 in Tabriz, and 32.00 in Mashhad. The optimum population density for biomass ranged from 24.6 (in Tabriz) to 35.3 stands per square meter (Mashhad). For harvest index it varied between 35.87 and 40.12 stands per square meter.

Keywords: Rainfed-chickpea, biomass, harvest index, grain yield, simulation.

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Authors: Tsai-Sheng Kao

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The ideal sinc filter, ignoring the noise statistics, is often applied for generating an arbitrary sample of a bandlimited signal by using the uniformly sampled data. In this article, an optimal interpolator is proposed; it reaches a minimum mean square error (MMSE) at its output in the presence of noise. The resulting interpolator is thus a Wiener filter, and both the optimal infinite impulse response (IIR) and finite impulse response (FIR) filters are presented. The mean square errors (MSE-s) for the interpolator of different length impulse responses are obtained by computer simulations; it shows that the MSE-s of the proposed interpolators with a reasonable length are improved about 0.4 dB under flat power spectra in noisy environment with signal-to-noise power ratio (SNR) equal 10 dB. As expected, the results also demonstrate the improvements for the MSE-s with various fractional delays of the optimal interpolator against the ideal sinc filter under a fixed length impulse response.

Keywords: Interpolator, minimum mean square error, Wiener filter.

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Authors: Asma Mehan

Abstract:

Under the thrust of technological changes, population growth and vehicular traffic, Iranian historical squares have lost their significance and they are no longer the main social nodes of the society. This research focuses on how historical public squares can inspire designers to enhance social interactions among citizens in Iranian urban context. Moreover, the recent master plan of Tehran demonstrates the lack of public spaces designed for the purpose of people’s social gatherings. For filling this gap, first the current situation of 7 selected primary historical public squares in Tehran including Sabze Meydan, Arg, Topkhaneh, Baherstan, Mokhber-al-dole, Rah Ahan and Hassan Abad have been compared. Later, the influencing elements on social interactions of the public squares such as subjective factors (human relationships and memories) and objective factors (natural and built environment) have been investigated. As a conclusion, some strategies are proposed for improving social interactions in historical public squares like; holding cultural, national, athletic and religious events, defining different and new functions in public squares’ surrounding, increasing pedestrian routs, reviving the collective memory, demonstrating the historical importance of square, eliminating visual obstacles across the square, organization the natural elements of the square, appropriate pavement for social activities. Finally, it is argued that the combination of all influencing factors which are: human interactions, natural elements and built environment criteria will lead to enhance the historical public squares’ potential for social interaction.

Keywords: Historical Square, Iranian Public Square, Social Interaction, Tehran.

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Authors: Seok Hyoung Lee, Vladimir Shin

Abstract:

An optimal mean-square fusion formulas with scalar and matrix weights are presented. The relationship between them is established. The fusion formulas are compared on the continuous-time filtering problem. The basic differential equation for cross-covariance of the local errors being the key quantity for distributed fusion is derived. It is shown that the fusion filters are effective for multi-sensor systems containing different types of sensors. An example demonstrating the reasonable good accuracy of the proposed filters is given.

Keywords: Kalman filtering, fusion formula, multi-sensor, mean-square error.

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Authors: Bharatendra Rai

Abstract:

Predictive data analysis and modeling involving machine learning techniques become challenging in presence of too many explanatory variables or features. Presence of too many features in machine learning is known to not only cause algorithms to slow down, but they can also lead to decrease in model prediction accuracy. This study involves housing dataset with 79 quantitative and qualitative features that describe various aspects people consider while buying a new house. Boruta algorithm that supports feature selection using a wrapper approach build around random forest is used in this study. This feature selection process leads to 49 confirmed features which are then used for developing predictive random forest models. The study also explores five different data partitioning ratios and their impact on model accuracy are captured using coefficient of determination (r-square) and root mean square error (rsme).

Keywords: Housing data, feature selection, random forest, Boruta algorithm, root mean square error.

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Authors: S. Channgam, A. Sae-Tang, T. Termsaithong

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In this research, the occurrences of large size events in various system sizes of the Bak-Tang-Wiesenfeld sandpile model are considered. The system sizes (square lattice) of model considered here are 25×25, 50×50, 75×75 and 100×100. The cross-correlation between the ratio of sites containing 3 grain time series and the large size event time series for these 4 system sizes are also analyzed. Moreover, a prediction method of the large-size event for the 50×50 system size is also introduced. Lastly, it can be shown that this prediction method provides a slightly higher efficiency than random predictions.

Keywords: Bak-Tang-Wiesenfeld sandpile model, avalanches, cross-correlation, prediction method.

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Authors: Carla E. O. de Moraes, Gladson O. Antunes, Mauro A. Rincon

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The aim of this paper is to study the internal stabilization of the Bernoulli-Euler equation numerically. For this, we consider a square plate subjected to a feedback/damping force distributed only in a subdomain. An algorithm for obtaining an approximate solution to this problem was proposed and implemented. The numerical method used was the Finite Difference Method. Numerical simulations were performed and showed the behavior of the solution, confirming the theoretical results that have already been proved in the literature. In addition, we studied the validation of the numerical scheme proposed, followed by an analysis of the numerical error; and we conducted a study on the decay of the energy associated.

Keywords: Bernoulli-Euler Plate Equation, Numerical Simulations, Stability, Energy Decay, Finite Difference Method.

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Authors: Sanaz Ahmadzadeh Siyahrood, Arghavan Ebrahimi, Mohammadjavad Mahdavinejad

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Islamic culture was accountable for a plethora of development in astronomy and science in the medieval term, and in geometry likewise. Geometric patterns are reputable in a considerable number of cultures, but in the Islamic culture the patterns have specific features that connect the Islamic faith to mathematics. In Islamic art, three fundamental shapes are generated from the circle shape: triangle, square and hexagon. Originating from their quiddity, each of these geometric shapes has its own specific structure. Even though the geometric patterns were generated from such simple forms as the circle and the square, they can be combined, duplicated, interlaced, and arranged in intricate combinations. So in order to explain geometrical interaction principles between square and equal triangle, in the first definition step, all types of their linear forces individually and in the second step, between them, would be illustrated. In this analysis, some angles will be created from intersection of their directions. All angles are categorized to some groups and the mathematical expressions among them are analyzed. Since the most geometric patterns in Islamic art and architecture are based on the repetition of a single motif, the evaluation results which are obtained from a small portion, is attributable to a large-scale domain while the development of infinitely repeating patterns can represent the unchanging laws. Geometric ornamentation in Islamic art offers the possibility of infinite growth and can accommodate the incorporation of other types of architectural layout as well, so the logic and mathematical relationships which have been obtained from this analysis are applicable in designing some architecture layers and developing the plan design.

Keywords: Angle, architecture, design, equal triangle, generating, grammar, square and structural hierarchy.

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