Journal articles: 'Two-dimensional and three-dimensional parametric copulas'

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Kamaruzaman, Izzat Fakhruddin, Wan Zawiah Wan Zin, and Noratiqah Mohd Ariff. "A generalized bivariate copula for flood analysis in Peninsular Malaysia." Malaysian Journal of Fundamental and Applied Sciences 15, no. 1 (2019): 38–49. http://dx.doi.org/10.11113/mjfas.v15n2019.1275.

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This study generalized the best copula to characterize the joint probability distribution between rainfall severity and duration in Peninsular Malaysia using two dimensional copulas. Specifically, to construct copulas, Inference Function for Margins (IFM) and Canonical Maximum Likelihood (CML) methods were specially exploited. For the purpose of achieving copula fitting, the derived rainfall variables by making use of the Standardized Precipitation Index (SPI) were fitted into several distributions. Five copulas, namely Gaussian, Clayton, Frank, Joe and Gumbel were put to the tests to establish the best data fitted copula. The tests produced acknowledged and satisfactory results of copula fitting for rainfall severity and duration. Surveying the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC), only three copulas produced a better fit for parametric and semi parametric approaches. Finally, two consistency tests were conducted and the results shown that Frank Copula produced consistent results.

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Latif, Shahid, and Firuza Mustafa. "Parametric Vine Copula Construction for Flood Analysis for Kelantan River Basin in Malaysia." Civil Engineering Journal 6, no. 8 (2020): 1470–91. http://dx.doi.org/10.28991/cej-2020-03091561.

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The multivariate approach of flood characteristics such as flood peak flow (P), volume (V), and duration (D) is much beneficial in recognizing the critical behaviour of flood episodes at a river basin scale. The incorporation of 2-dimensional copulas for establishing bivariate flood dependency frequently appears, but it could be more comprehensive if we focus all the three flood characteristic simultaneously. In such circumstances, incorporation of vine or Pair-Copula Construction (PCC) could produce a better approximation of joint probability density and much practical approach in the uncertainty analysis, in comparison with conventional trivariate copula distribution. This study demonstrated the efficacy of parametric vine copula in the modelling of trivariate flood characteristics for the Kelantan River basin in Malaysia. The D-vine tree structure is selected where the Gaussian and Frank copula is recognized for bivariate flood pairs (P-V) and (P-D) pairs in the first stage, using the maximum-pseudo-likelihood (MPL) estimation procedure. Similarly, the Gumbel copula is selected in the modelling of conditioned data obtained through the conditional distribution function of bivariate copulas selected in the previous stage based on the partial differentiation, also called h-function. Finally, the full density function of the 3-dimension structure is derived and compared with the observed flood characteristics. Furthermore, tail dependence properties and behaviour of D-vine copula are also investigated, which reveals for well capturing the general behaviour of Gaussian and Frank copula fitted to flood pair (P-V) and (V-D) and reproduces the overall flood correlation structure fairely well. Both the primary ‘OR’ and ‘AND’ joint return periods for trivariate flood events are estimated which pointing that ‘AND’ joint case produces higher return value than ‘OR’ case.

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Samanthi, Ranadeera G. M., Wei Wei, and Vytaras Brazauskas. "Comparing the riskiness of dependent portfolios via nested L-statistics." Annals of Actuarial Science 11, no. 2 (2016): 237–52. http://dx.doi.org/10.1017/s1748499516000233.

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AbstractA non-parametric test based on nested L-statistics and designed to compare the riskiness of portfolios was introduced by Brazauskas et al. (2007). Its asymptotic and small-sample properties were primarily explored for independent portfolios, though independence is not a required condition for the test to work. In this paper, we investigate how performance of the test changes when insurance portfolios are dependent. To achieve that goal, we perform a simulation study where we consider three different risk measures: conditional tail expectation, proportional hazards transform, and mean. Further, three portfolios are generated from exponential, Pareto, and lognormal distributions, and their interdependence is modelled with the three-dimensional t and Gaussian copulas. It is found that the presence of strong positive dependence (comonotonicity) makes the test very liberal for all the risk measures under consideration. For types of dependence that are more common in an insurance environment, the effect of dependence is less dramatic but the results are mixed, i.e., they depend on the chosen risk measure, sample size, and even on the test’s significance level. Finally, we illustrate how to incorporate such findings into sensitivity analysis of the decisions. The risks we analyse represent tornado damages in different regions of the United States from 1890 to 1999.

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Ramnath, Vishal. "Analysis of approximations of GUM supplement 2 based non-Gaussian PDFs of measurement models with Rosenblatt Gaussian transformation mappings." International Journal of Metrology and Quality Engineering 11 (2020): 2. http://dx.doi.org/10.1051/ijmqe/2019018.

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In scientific metrology practise the application of Monte Carlo simulations with the aid of the GUM Supplement 2 (GS2) technique for performing multivariate uncertainty analyses is now more prevalent, however a key remaining challenge for metrologists in many laboratories is the implicit assumption of Gaussian characteristics for summarizing and analysing measurement model results. Whilst non-Gaussian probability density functions (PDFs) may result from Monte Carlo simulations when the GS2 is applied for more complex non-linear measurement models, in practice results are typically only reported in terms of multivariate expected and covariance values. Due to this limitation the measurement model PDF summary is implicitly restricted to a multivariate Gaussian PDF in the absence of additional higher order statistics (HOS) information. In this paper an earlier classical theoretical result by Rosenblatt that allows for an arbitrary multivariate joint distribution function to be transformed into an equivalent system of Gaussian distributions with mapped variables is revisited. Numerical simulations are performed in order to analyse and compare the accuracy of the equivalent Gaussian system of mapped random variables for approximating a measurement model’s PDF with that of an exact non-Gaussian PDF that is obtained with a GS2 Monte Carlo statistical simulation. Results obtained from the investigation indicate that a Rosenblatt transformation offers a convenient mechanism to utilize just the joint PDF obtained from the GS2 data in order to both sample points from a non-Gaussian distribution, and also in addition which allows for a simple two-dimensional approach to estimate coupled uncertainties of random variables residing in higher dimensions using conditional densities without the need for determining parametric based copulas.

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Nagler, Thomas, Christian Schellhase, and Claudia Czado. "Nonparametric estimation of simplified vine copula models: comparison of methods." Dependence Modeling 5, no. 1 (2017): 99–120. http://dx.doi.org/10.1515/demo-2017-0007.

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AbstractIn the last decade, simplified vine copula models have been an active area of research. They build a high dimensional probability density from the product of marginals densities and bivariate copula densities. Besides parametric models, several approaches to nonparametric estimation of vine copulas have been proposed. In this article, we extend these approaches and compare them in an extensive simulation study and a real data application. We identify several factors driving the relative performance of the estimators. The most important one is the strength of dependence. No method was found to be uniformly better than all others. Overall, the kernel estimators performed best, but do worse than penalized B-spline estimators when there is weak dependence and no tail dependence.

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Miyazawa, F. K., and Y. Wakabayashi. "Two- and three-dimensional parametric packing." Electronic Notes in Discrete Mathematics 19 (June 2005): 313–19. http://dx.doi.org/10.1016/j.endm.2005.05.042.

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Miyazawa, F. K., and Y. Wakabayashi. "Two- and three-dimensional parametric packing." Computers & Operations Research 34, no. 9 (2007): 2589–603. http://dx.doi.org/10.1016/j.cor.2005.10.001.

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Phu, Nguyen Dinh, Nguyen Nhut Hung, Ali Ahmadian, Soheil Salahshour, and Norazak Senu. "Geometric picture fuzzy numbers and three-dimensional copulas with the non-linear programming approach." Journal of Intelligent & Fuzzy Systems 40, no. 1 (2021): 1–12. http://dx.doi.org/10.3233/jifs-182519.

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This study presents a possible relationship between two main objects, which are three-dimensional copulas (3D-Cs) and geometric picture fuzzy numbers (GPFNs). This opens up a potential field for future studies for these two objects that three-dimensional copulas can become useful tools for handling uncertainty information in the form of a picture fuzzy set (PFS). Specifically, we define a GPFN as a base element of the PFS and a defined domain of three-dimensional copulas that contains a set of GPFNs, then we show some examples of three-dimensional copulas identified on this domain. In this framework, we present the theorems related to these two objects. At the same time, we provide some examples for three-dimensional semi-copulas, three-dimensional quasi-copulas, and three-dimensional empirical copulas defined on D, which is a defined domain of a three-dimensional copula and contains a set of GPFNs D g * . In addition, we also introduce a new approach to non-linear programming problems.

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Gladky, E. G. "Use of two-dimensional normal copula in parametric reliability estimation models." Technical mechanics 2018, no. 4 (2018): 105–18. http://dx.doi.org/10.15407/itm2018.04.105.

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Ballerini, Rocco. "Archimedean copulas, exchangeability, and max-stability." Journal of Applied Probability 31, no. 2 (1994): 383–90. http://dx.doi.org/10.2307/3215031.

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An exchangeable sequence of random variables is constructed with all finite-dimensional distribution functions having an Archimedean copula (as defined by Schweizer and Sklar (1983)). Through a monotone transformation of this exchangeable sequence, we obtain and characterize the class of exchangeable sequences possessing the max-stable property as defined by De Haan and Rachev (1989). Several parametric examples are given.

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Ballerini, Rocco. "Archimedean copulas, exchangeability, and max-stability." Journal of Applied Probability 31, no. 02 (1994): 383–90. http://dx.doi.org/10.1017/s0021900200044909.

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An exchangeable sequence of random variables is constructed with all finite-dimensional distribution functions having an Archimedean copula (as defined by Schweizer and Sklar (1983)). Through a monotone transformation of this exchangeable sequence, we obtain and characterize the class of exchangeable sequences possessing the max-stable property as defined by De Haan and Rachev (1989). Several parametric examples are given.

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Núnez Bootello, Juan Pablo, Monica Mier-Torrecilla, Manuel Doblaré, and Manuel Silva Pérez. "Aerodynamics of new solar parametric troughs: Two dimensional and three dimensional single module numerical analysis." Solar Energy 135 (October 2016): 742–49. http://dx.doi.org/10.1016/j.solener.2016.06.040.

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Kirsh, D. V., A. S. Shirokanev, and A. V. Kupriyanov. "Algorithm of reconstruction of a three-dimensional crystal structure from two-dimensional projections." Computer Optics 43, no. 2 (2019): 324–31. http://dx.doi.org/10.18287/2412-6179-2019-43-2-324-331.

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The article deals with a problem of three-dimensional crystal lattice reconstruction, which is an important stage in the X-ray structural analysis. The accuracy of parametric and structural identification of crystals directly depends on the quality of crystal lattice reconstruction. The proposed algorithm of reconstruction of a three-dimensional crystal lattice is based on minimizing the distances from each node to a line projected onto a specified plane. Three sets of two-dimensional node coordinates, obtained from three two-dimensional projections, are used as input data. We performed an analytical calculation of the reconstruction error, allowing the total reconstruction accuracy to be estimated. The results of computational experiments confirmed the high quality of the proposed reconstruction algorithms and its stability against the distortion of node coordinates. In addition, we revealed a problem of lattice system separability, with the identification accuracy for monoclinic, rhombic and tetragonal systems found to be 34%, 53% and 10%, respectively.

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Song, Quanrui, Jianxu Liu, and Songsak Sriboonchitta. "Risk Measurement of Stock Markets in BRICS, G7, and G20: Vine Copulas versus Factor Copulas." Mathematics 7, no. 3 (2019): 274. http://dx.doi.org/10.3390/math7030274.

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Multivariate copulas have been widely used to handle risk in the financial market. This paper aimed to adopt two novel multivariate copulas, Vine copulas and Factor copulas, to measure and compare the financial risks of the emerging economy, developed economy, and global economy. In this paper, we used data from three groups (BRICS, which stands for emerging markets, specifically, those of Brazil, Russia, India, China, and South Africa; G7, which refers to developed countries; and G20, which represents the global market), separated into three periods (pre-crisis, crisis, and post-crisis) and weighed Value at Risk (VaR) and Expected Shortfall (ES) (based on their market capitalization) to compare among three copulas, C-Vine, D-Vine, and Factor copulas. Also, real financial data demonstrated that Factor copulas have stronger stability and perform better than the other two copulas in high-dimensional data. Moreover, we showed that BRICS has the highest risk and G20 has the lowest risk of the three groups.

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Bozis, George, and Thomas A. Kotoulas. "Homogeneous two-parametric families of orbits in three-dimensional homogeneous potentials." Inverse Problems 21, no. 1 (2005): 343–56. http://dx.doi.org/10.1088/0266-5611/21/1/021.

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Klostermeyer, J. "Two- and three-dimensional parametric instabilities in finite-amplitude internal gravity waves." Geophysical & Astrophysical Fluid Dynamics 61, no. 1-4 (1991): 1–25. http://dx.doi.org/10.1080/03091929108229035.

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Wee, Lai K., Hum Y. Chai, Sharul R. Bin Samsury, Naizaithull F. Binti Mujamil, and Eko Supriyanto. "Comparative studies of two dimensional and three dimensional ultrasonic nuchal translucency in trisomy assessments." Anais da Academia Brasileira de Ciências 84, no. 4 (2012): 1157–68. http://dx.doi.org/10.1590/s0001-37652012000400030.

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Current two-dimensional (2D) ultrasonic marker measurements are inherent with intra- and inter-observer variability limitations. The objective of this paper is to investigate the performance of conventional 2D ultrasonic marker measurements and proposed programmable interactive three-dimensional (3D) marker evaluation. This is essentially important to analyze that the measurement on 3D volumetric measurement possesses higher impact and reproducibility vis-à-vis 2D measurement. Twenty three cases of prenatal ultrasound examination were obtained from collaborating hospital after Ethical Committee's approval. The measured 2D ultrasonic marker is Nuchal Translucency or commonly abbreviated as NT. Descriptive analysis of both 2D and 3D ultrasound measurement were calculated. Three trial measurements were taken for each method. Both data were tested with One-Sample Kolmogorov-Smirnov Test and results indicate that markers measurements were distributed normally with significant parametric values at 0.621 and 0.596 respectively. Computed mean and standard deviation for both measurement methods are 1.4495 ± 0.46490 (2D) and 1.3561 ± 0.50994 (3D). ANOVA test shows that computerized 3D measurements were found to be insignificantly different from the mean of conventional 2D at the significance level of 0.05. With Pearson's correlation coefficient value or R = 0.861, the result proves strong positive linear correlation between 2D and 3D ultrasonic measurements. Reproducibility and accuracy of 3D ultrasound in NT measurement was significantly increased compared with 2D B-mode ultrasound prenatal assessment. 3D reconstructed imaging has higher clinical values compare to 2D ultrasound images with less diagnostics information.

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Yan, Yiming, Liqiang Zhang, and Xiaorong Luo. "Modeling Three-Dimensional Anisotropic Structures of Reservoir Lithofacies Using Two-Dimensional Digital Outcrops." Energies 13, no. 16 (2020): 4082. http://dx.doi.org/10.3390/en13164082.

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Reservoir heterogeneity is a key geological problem that restricts oil and gas exploration and development of clastic rocks from the early to late stages. Existing reservoir heterogeneity modeling methods such as multiple-point geostatistics (MPS) can accurately model the two-dimensional anisotropic structures of reservoir lithofacies. However, three-dimensional training images are required to construct three-dimensional reservoir lithofacies anisotropic structures models, and the method to use reservoir heterogeneity model of fewer-dimensional to obtain a three-dimensional model has become a much-focused research topic. In this study, the outcrops of the second member of Qingshuihe Formation (K1q2) in the northwestern margin of the Junggar Basin, which are lower cretaceous rocks, were the research target. The three-dimensional reservoir heterogeneity model of the K1q2 outcrop was established based on the unmanned aerial vehicle (UAV) digital outcrops model and MPS techniques, and the “sequential two-dimensional conditioning data” (s2Dcd) method was modified based on a sensitivity parameter analysis. Results of the parametric sensitivity analysis revealed that the isotropic multigrid simulations demonstrate poor performance because of the lack of three-dimensional training images, conditioning data that are horizontally discrete and vertically continuous, and distribution of lithofacies that are characterized by large horizontal continuities and small thicknesses. The reservoir lithofacies anisotropic structure reconstructions performed well with anisotropic multigrids. The simulation sequence of two-dimensional surfaces for generating the three-dimensional anisotropic structure of reservoir lithofacies models should be reasonably planned according to the actual geological data and limited hard data. In additional to this, the conditional probability density function of each two-dimensional training image should be fully utilized. The simulation results using only one two-dimensional section will have several types of noises, which is not consistent with the actual geological background. The anisotropic multigrid simulations and two-dimensional training image simulation sequence, proposed in this paper as “cross mesh, refinement step by step”, effectively reduced the noise generated, made full use of the information from the two-dimensional training image, and reconstructed the three-dimensional reservoir lithofacies anisotropic structures models, thus conforming to the actual geological conditions.

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Ansari, Mubashshir Ahmad, and Kwang-Yong kim. "Parametric study on mixing of two fluids in a three-dimensional serpentine microchannel." Chemical Engineering Journal 146, no. 3 (2009): 439–48. http://dx.doi.org/10.1016/j.cej.2008.10.006.

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Mackney, Michael D. A., and Carl T. F. Ross. "Preliminary Ship Design Using One and Two-Dimensional Models." Marine Technology and SNAME News 36, no. 02 (1999): 102–12. http://dx.doi.org/10.5957/mt1.1999.36.2.102.

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Computational studies of hull-superstructure interaction were carried out using one-, two-and three-dimensional finite element analyses. Simplification of the original three-dimensional cases to one- and two-dimensional ones was undertaken to reduce the data preparation and computer solution times in an extensive parametric study. Both the one- and two-dimensional models were evaluated from numerical and experimental studies of the three-dimensional arrangements of hull and superstructure. One-dimensional analysis used a simple beam finite element with appropriately changed sections properties at stations where superstructures existed. Two-dimensional analysis used a four node, first order quadrilateral, isoparametric plane elasticity finite element, with a corresponding increase in the grid domain where the superstructure existed. Changes in the thickness property reflected deck stiffness. This model was essentially a multi-flanged beam with the shear webs representing the hull and superstructure sides, and the flanges representing the decks One-dimensional models consistently and uniformly underestimated the three-dimensional behaviour, but were fast to create and run. Two-dimensional models were also consistent in their assessment, and considerably closer in predicting the actual behaviours. These models took longer to create than the one-dimensional, but ran in very much less time than the refined three-dimensional finite element models Parametric insights were accomplished quickly and effectively with the simplest model and processor, but two-dimensional analyses achieved closer absolute measure of the displacement behaviours. Although only static analysis with simple loading and support conditions were presented, it is believed that similar benefits would be found for other loadings and support conditions. Other engineering components and structures may benefit from similarly judged simplification using one- and two-dimensional models to reduce the time and cost of preliminary design.

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Vio, R., T. W. Nagler, and P. Andreani. "Modeling high-dimensional dependence in astronomical data." Astronomy & Astrophysics 642 (October 2020): A156. http://dx.doi.org/10.1051/0004-6361/202038585.

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Fixing the relationship of a set of experimental quantities is a fundamental issue in many scientific disciplines. In the 2D case, the classical approach is to compute the linear correlation coefficient ρ from a scatterplot. This method, however, implicitly assumes a linear relationship between the variables. Such an assumption is not always correct. With the use of the partial correlation coefficients, an extension to the multidimensional case is possible. However, the problem of the assumed mutual linear relationship of the variables remains. A relatively recent approach that makes it possible to avoid this problem is the modeling of the joint probability density function of the data with copulas. These are functions that contain all the information on the relationship between two random variables. Although in principle this approach also can work with multidimensional data, theoretical as well computational difficulties often limit its use to the 2D case. In this paper, we consider an approach based on so-called vine copulas, which overcomes this limitation and at the same time is amenable to a theoretical treatment and feasible from the computational point of view. We applied this method to published data on the near-IR and far-IR luminosities and atomic and molecular masses of the Herschel reference sample, a volume-limited sample in the nearby Universe. We determined the relationship of the luminosities and gas masses and show that the far-IR luminosity can be considered as the key parameter relating the other three quantities. Once removed from the 4D relation, the residual relation among the latter is negligible. This may be interpreted as the correlation between the gas masses and near-IR luminosity being driven by the far-IR luminosity, likely by the star formation activity of the galaxy.

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Abdallah, M., Fadi Hage Chehade, Walid Chehade, and A. Fawaz. "Interaction Deep Excavation – Adjacent Structure: Numerical Two and Three Dimensional Modeling." Advanced Materials Research 324 (August 2011): 344–47. http://dx.doi.org/10.4028/www.scientific.net/amr.324.344.

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Urban development often requires the construction of deep excavations near to buildings or other structures. We have to study complex material structure interactions where we should take into consideration several particularities. In this paper, we perform a numerical modeling with the finite element method, using PLAXIS software, of the interaction deep excavation-diaphragm wall-soil-structure in the case of non linear soil behavior. We focus our study on a comparison of the results given respectively by two and three dimensional modelings. This allows us to give some recommendations concerning the validity of twodimensional study. We perform a parametric study according to the initial loading on the structure and the struts number.

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Zhou, Yiyi, and Lianmen Chen. "Designing a new three-dimensional periodic cellular auxetic material." International Journal of Modern Physics B 31, no. 16-19 (2017): 1744088. http://dx.doi.org/10.1142/s021797921744088x.

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Auxetics are materials showing a negative Poisson’s ratio. Early research found several categories of auxetic materials in the chemical field. Later research identified the fundamental mechanism generating this behavior is rotation; a variety of two-dimensional auxetic material have been generated accordingly. Nevertheless, the successful example of three-dimensional auxetic material is still rare. This paper introduces a new design of three-dimensional periodic cellular auxetic material based on geometrical and mechanical methodology. The projections of the optimized periodic modules in two horizontal directions are geometrically same with auxetic hexahedral poem, so that the optimized periodic material can perform auxetic in both two horizontal directions under vertical compression. Parametric model is simulated to prove the design.

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LI, JIBIN, and XIAOHUA ZHAO. "KNOTTED PERIODIC SOLUTIONS OF A LINEAR NONAUTONOMOUS SYSTEM AND SOME RELATED THREE-DIMENSIONAL FLOWS." International Journal of Bifurcation and Chaos 22, no. 11 (2012): 1250280. http://dx.doi.org/10.1142/s021812741250280x.

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This paper considers a three-dimensional linear nonautonomous systems. It shows that for every integer frequency parameter value, this system has a distinct type of knotted periodic solutions, which lie in a bounded region of R3. Exact explicit parametric representations of the knotted periodic solutions are given. By using these parametric representations, two series of three-dimensional flows are constructed, such that these three-dimensional autonomous systems have knotted periodic orbits in the three-dimensional phase space.

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CARCASSES, JEAN-PIERRE, and ABDEL-KADDOUS TAHA. "STUDY OF A TWO-DIMENSIONAL ENDOMORPHISM BY USE OF THE PARAMETRIC SINGULARITIES." International Journal of Bifurcation and Chaos 10, no. 12 (2000): 2853–62. http://dx.doi.org/10.1142/s0218127400001900.

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A two-dimensional cubic endomorphism depending on three parameters is considered. The qualitative changes of the bifurcation curves in a parameter plane are studied when the third parameter varies. More particularly, the crossing through a "saddle-node" singularity of the parameter plane is analytically analyzed.

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STENFLO, LENNART, BENGT ELIASSON, and MATTIAS MARKLUND. "Three-dimensional instability of two nonlinearly coupled electromagnetic waves in a plasma." Journal of Plasma Physics 74, no. 3 (2008): 371–79. http://dx.doi.org/10.1017/s0022377807006666.

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AbstractThe three-dimensional instability of two coupled electromagnetic waves in an unmagnetized plasma is investigated theoretically and numerically. In the regime of two-plasmon decay, where one pump wave frequency is approximately twice the electron plasma frequency, we find that the coupled pump waves give rise to enhanced instability with wavevectors between those of the two beams. In the case of ion parametric decay instability, where the pump wave decays into one Langmuir wave and one ion acoustic wave, the instability regions are added with no distinct amplification. Our investigation can be useful in interpreting laser-plasma as well as ionospheric heating experiments.

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Nigro, N. J., F. J. Zhou, S. M. Heinrich, A. F. Elkouh, R. A. Fournelle, and P. S. Lee. "Parametric Finite Element Method for Predicting Shapes of Three-Dimensional Solder Joints." Journal of Electronic Packaging 118, no. 3 (1996): 142–47. http://dx.doi.org/10.1115/1.2792144.

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This paper discusses the application of the parametric finite element method for predicting shapes of three-dimensional solder joints. With this method, the surface of the joint is meshed (discretized) with finite elements. The spatial variables (x, y, z) are expanded over each element in terms of products of interpolation (blending) functions expressed in parametric form and element nodal coordinates. The element nodal coordinates which are not constrained by the boundary conditions are determined by minimizing the potential energy function which governs the joint formation problem. This method has been employed successfully in the past to predict the shapes of two dimensional fillet and axisymmetric joints. In this paper, the method is extended to three dimensional problems involving sessile drops formed on a rectangular pad and solder columns formed between two horizontal planes and subject to a vertical force.

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Tholath, J., and V. Radhakrishnan. "Three-dimensional filtering of engineering surfaces using a least squares non-parametric B-spline surface." Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture 211, no. 7 (1997): 557–64. http://dx.doi.org/10.1243/0954405971516509.

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With the advent of cheap and powerful computers, three-dimensional evaluation of surfaces is possible. The three-dimensional surface requires a filtering procedure for waviness separation. This paper deals with the waviness plane construction through a least squares non-parametric B-spline surface. Roughness values obtained by two-, two-and-a-half- and three-dimensional evaluations of typical surfaces are also given.

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Tamborini, Gloria, Claudia Cefalù, Fabrizio Celeste, et al. "Multi-parametric “on board” evaluation of right ventricular function using three-dimensional echocardiography: feasibility and comparison to traditional two-and three dimensional echocardiographic measurements." International Journal of Cardiovascular Imaging 35, no. 2 (2018): 275–84. http://dx.doi.org/10.1007/s10554-018-1496-9.

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Shahbaba, Babak, Bo Zhou, Shiwei Lan, Hernando Ombao, David Moorman, and Sam Behseta. "A Semiparametric Bayesian Model for Detecting Synchrony Among Multiple Neurons." Neural Computation 26, no. 9 (2014): 2025–51. http://dx.doi.org/10.1162/neco_a_00631.

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We propose a scalable semiparametric Bayesian model to capture dependencies among multiple neurons by detecting their cofiring (possibly with some lag time) patterns over time. After discretizing time so there is at most one spike at each interval, the resulting sequence of 1s (spike) and 0s (silence) for each neuron is modeled using the logistic function of a continuous latent variable with a gaussian process prior. For multiple neurons, the corresponding marginal distributions are coupled to their joint probability distribution using a parametric copula model. The advantages of our approach are as follows. The nonparametric component (i.e., the gaussian process model) provides a flexible framework for modeling the underlying firing rates, and the parametric component (i.e., the copula model) allows us to make inferences regarding both contemporaneous and lagged relationships among neurons. Using the copula model, we construct multivariate probabilistic models by separating the modeling of univariate marginal distributions from the modeling of a dependence structure among variables. Our method is easy to implement using a computationally efficient sampling algorithm that can be easily extended to high-dimensional problems. Using simulated data, we show that our approach could correctly capture temporal dependencies in firing rates and identify synchronous neurons. We also apply our model to spike train data obtained from prefrontal cortical areas.

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Chen, Tiebing, and Yuwen Zhang. "Three-Dimensional Modeling of Selective Laser Sintering of Two-Component Metal Powder Layers." Journal of Manufacturing Science and Engineering 128, no. 1 (2005): 299–306. http://dx.doi.org/10.1115/1.2122947.

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Laser sintering of a metal powder mixture that contains two kinds of metal powders with significantly different melting points under a moving Gaussian laser beam is investigated numerically. The continuous-wave laser-induced melting accompanied by shrinkage and resolidification of the metal powder layer are modeled using a temperature-transforming model. The liquid flow of the melted low-melting-point metal driven by capillary and gravity forces is also included in the physical model. The numerical results are validated by experimental results, and a detailed parametric study is performed. The effects of the moving heat source intensity, the scanning velocity, and the thickness of the powder layer on the sintering depth, the configuration of the heat affected zone, and the temperature distribution are discussed.

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Jayakumar, B., G. A. Quadir, M. Z. Abdullah, and K. N. Seetharamu. "Three Dimensional CFD Conjugate Analysis of Two Inline PLCC Packages Horizontally Mounted." Journal of Microelectronics and Electronic Packaging 1, no. 4 (2004): 244–60. http://dx.doi.org/10.4071/1551-4897-1.4.244.

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A three dimensional conjugate analysis of heat and fluid flow of two 84-pin PLCC packages mounted horizontally on a printed circuit board in a wind tunnel is carried out using a commercial CFD code, FLUENTTM. Various inlet air velocities are used to emulate natural, mixed and forced convection conditions. Some parametric studies are carried out by varying the package chip power, gap between the packages and air inlet velocities. The results are presented in terms of the junction temperature, thermal resistance and top surface average heat transfer coefficient for each package under different operating conditions. The decrease in the junction temperature of the packages with the increase in air inlet velocity is clearly predicted. Further, the leading edge heat transfer coefficient of the packages is always higher than that at the trailing edge for all inlet air velocities considered. It is found that the variation in the package chip power does not influence the average heat transfer coefficient and the thermal resistance of the package at a particular inlet air velocity. Different correlations in terms of the junction temperature as well as the Nusselt number are presented for each package under different convection conditions.

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Chen, Wen-Hwa, Hsien-Chie Cheng, and Chih-Han Lin. "On the Thermal Performance Characteristics of Three-Dimensional Multichip Modules." Journal of Electronic Packaging 126, no. 3 (2004): 374–83. http://dx.doi.org/10.1115/1.1773198.

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The study explores the thermal performance of three-dimensional (3-D), vertically stacked multi-chip modules (the so-called MCM-V) in natural convection through finite element (FE) modeling and experimental validation. A modified Infrared (IR) thermography-based thermal characterization (IRTTC) technique that integrates a 3-D heat conduction FE modeling and a two-phased IR thermography measurement process is proposed. In contrast to the conventional IRTTC technique (Chen et al. [1]), the technique can improve the resolution of the captured thermal images so as to attain better characterization of the chip junction temperature. The effectiveness of the proposed modified IRTTC technique is confirmed by means of the thermal test die (TTD) measurement. Furthermore, for facilitating subsequent parametric thermal design, a direct FE approach (DFEA) is also introduced. The DFEA simply incorporates existing empirical models for heat transfer (HT) coefficients to describe the surface heat transfer to the ambient through convection and radiation in the proposed heat conduction FE model. Through the modified IRTTC technique and the TTD measurement, the validity of the proposed FE modeling, including the proposed heat conduction FE model and the applied empirical models for HT coefficients, is verified. With the validated FE modeling, four different chip stacking structures of MCM-V packages, including the thick-die-attach, pyramid, cross and dummy-die types, are investigated. In addition, some essential design factors, affecting the thermal performance of the MCM-V, are also extensively explored through parametric FE study. Eventually, an extensive thermal design guideline is accordingly provided.

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LI, JIBIN, and FENGJUAN CHEN. "KNOTTED PERIODIC ORBITS AND CHAOTIC BEHAVIOR OF A CLASS OF THREE-DIMENSIONAL FLOWS." International Journal of Bifurcation and Chaos 21, no. 09 (2011): 2505–23. http://dx.doi.org/10.1142/s0218127411029896.

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This paper considers a class of three-dimensional systems constructed by rotating some planar symmetric polynomial vector fields. It shows that this class of systems has infinitely many distinct types of knotted periodic orbits, which lie on a family of invariant torus. For two three-dimensional systems, exact explicit parametric representations of the knotted periodic orbits are given. For their perturbed systems, the chaotic behavior is discussed by using two different methods.

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Sun, Xiankun, He Liu, Wanqing Song, and Francesco Villecco. "Modeling of Eddy Current Welding of Rail: Three-Dimensional Simulation." Entropy 22, no. 9 (2020): 947. http://dx.doi.org/10.3390/e22090947.

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In this paper is given a three-dimensional numerical simulation of the eddy current welding of rails where the longitudinal two directions are not ignored. In fact, usually it is considered a model where, in the two-dimensional numerical simulation of rail heat treatment, the longitudinal directions are ignored for the magnetic induction strength and temperature, and only the axial calculation is performed. Therefore, we propose the electromagnetic-thermal coupled three-dimensional model of eddy current welding. The induced eddy current heat is obtained by adding the z-axis spatial angle to the two-dimensional electromagnetic-thermal, thus obtaining some new results by coupling the numerical simulation and computations of the electric field and magnetic induction intensity of the three-dimensional model. Moreover, we have considered the objective function into a weak formulation. The three-dimensional model is then meshed by the finite element method. The electromagnetic-thermal coupling has been numerically computed, and the parametric dependence to the eddy current heating process has been fully studied. Through the numerical simulation with different current densities, frequencies, and distances, the most suitable heat treatment process of U75V rail is obtained.

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Yang, Chun Hui, Peter D. Hodgson, Qian Chu Liu, and Lin Ye. "Three-Dimensional Finite Element Modelling of Laser Shock Peening Process." Materials Science Forum 561-565 (October 2007): 2261–64. http://dx.doi.org/10.4028/www.scientific.net/msf.561-565.2261.

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Laser shock peening (LSP) is an innovative surface treatment technique for metal alloys, with the great improvement of their fatigue, corrosion and wear resistance performance. Finite element method has been widely applied to simulate the LSP to provide the theoretically predictive assessment and optimally parametric design. In the current work, 3-D numerical modelling approaches, combining the explicit dynamic analysis, static equilibrium analysis algorithms and different plasticity models for the high strain rate exceeding 106s-1, are further developed. To verify the proposed methods, 3-D static and dynamic FEA of AA7075-T7351 rods subject to two-sided laser shock peening are performed using the FEA package–ABAQUS. The dynamic and residual stress fields, shock wave propagation and surface deformation of the treated metal from different material modelling approaches have a good agreement.

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Ghaemsaidi, Sasan John, and Manikandan Mathur. "Three-dimensional small-scale instabilities of plane internal gravity waves." Journal of Fluid Mechanics 863 (January 29, 2019): 702–29. http://dx.doi.org/10.1017/jfm.2018.921.

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We study the evolution of three-dimensional (3-D), small-scale, small-amplitude perturbations on a plane internal gravity wave using the local stability approach. The plane internal wave is characterised by its non-dimensional amplitude, $A$, and the angle the group velocity vector makes with gravity, $\unicode[STIX]{x1D6F7}$. For a given $(A,\unicode[STIX]{x1D6F7})$, the local stability equations are solved on the periodic fluid particle trajectories to obtain growth rates for all two-dimensional (2-D) and 3-D perturbation wave vectors. For small $A$, the local stability approach recovers previous results of 2-D parametric subharmonic instability (PSI) while offering new insights into 3-D PSI. Higher-order triadic resonances, and associated deviations from them, are also observed at small $A$. Moreover, for small $A$, purely transverse instabilities resulting from parametric resonance are shown to occur at select values of $\unicode[STIX]{x1D6F7}$. The possibility of a non-resonant instability mechanism for transverse perturbations at finite $A$ allows us to derive a heuristic, modified gravitational instability criterion. We then study the extension of small $A$ to finite $A$ internal wave instabilities, where we recover and build upon existing knowledge of small-scale, small-amplitude internal wave instabilities. Four distinct regions of the $(A,\unicode[STIX]{x1D6F7})$-plane based on the dominant instability modes are identified: 2-D PSI, 3-D oblique, quasi-2-D shear-aligned, and 3-D transverse. Our study demonstrates the local stability approach as a physically insightful and computationally efficient tool, with potentially broad utility for studies that are based on other theoretical approaches and numerical simulations of small-scale instabilities of internal waves in various settings.

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CARCASSES, JEAN-PIERRE. "SINGULARITIES OF THE PARAMETRIC PLANE OF AN n-DIMENSIONAL MAP." International Journal of Bifurcation and Chaos 05, no. 02 (1995): 419–47. http://dx.doi.org/10.1142/s0218127495000351.

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This paper uses the notion of “contour lines” in a parameter plane. A given contour line is related to a constant value of a “reduced multiplier” constructed from the elements of the Jacobian matrix associated with a given periodic point. The singularities type of such lines permit to determine a point of intersection of two bifurcation curves of same nature (flip or fold) and a point of tangency between a fold bifurcation curve and a flip bifurcation curve. When a third parameter varies, these singularities permit to determine the appearance (or disappearance) of a closed fold or flip bifurcation curve. Three different configurations of fold and flip bifurcation curves, centred round a cusp point of a fold curve, are considered. They are called saddle area, spring area, and crossroad area. The singularities type of the contour lines define the configuration types of these areas and, when a third parameter varies, the qualitative changes of such areas are directly identified.

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Liew, K. M., and X. B. Liu. "The Maximal Lyapunov Exponent for a Three-Dimensional Stochastic System." Journal of Applied Mechanics 71, no. 5 (2004): 677–90. http://dx.doi.org/10.1115/1.1782648.

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This paper examines the almost-sure asymptotic stability condition of a linear multiplicative stochastic system, which is a linear part of a co-dimension two-bifurcation system that is on a three-dimensional central manifold and subjected to parametric excitation by an ergodic real noise. The excitation is assumed to be an integrable function of an n-dimensional Ornstein-Uhlenbeck vector process which is the output of a linear filter system, while both the detailed balance condition and the strong mixing condition are removed. Through a perturbation method and the spectrum representations of the Fokker Planck operator and its adjoint operator of the linear filter system, the explicit asymptotic expressions of the maximal Lyapunov exponent for three case studies, in which different forms of the coefficient matrix included in the noise excitation term are assumed, are obtained.

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Vu, Hung Q., and Delwyn G. Fredlund. "The prediction of one-, two-, and three-dimensional heave in expansive soils." Canadian Geotechnical Journal 41, no. 4 (2004): 713–37. http://dx.doi.org/10.1139/t04-023.

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The prediction of heave in unsaturated, expansive soils has historically been studied primarily as a one-dimensional type analysis. This paper proposes a methodology that can be used for the prediction of one-, two-, or three-dimensional heave. It is suggested that negative pore-water pressures (i.e., soil suctions) can be estimated through a saturatedunsaturated seepage analysis. The results of the seepage analysis are then used as input for the prediction of displacements in a stressdeformation analysis. The formulation of the governing partial differential equations for both seepage and stressdeformation is based on the general theory of unsaturated soils using two independent stress state variables. The elasticity parameter functions required for a stressdeformation analysis can be calculated from various tests, including conventional one-dimensional oedometer tests. The proposed method is studied and tested against data collected on a case history involving a slab-on-ground floor on Regina clay. The predicted results from the two-dimensional analysis agree well with the measured data in terms of both total vertical displacements and final water contents in the soil. The results of a parametric study are also presented to show the effect of measured soil parameters (i.e., swelling index and initial void ratio) and assumed parameters (i.e., Poisson's ratio and coefficient of earth pressure at rest) on the predicted displacements.Key words: heave prediction, unsaturated soil, expansive soil, seepage analysis, stress analysis.

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Tsamasphyros, George J., George N. Kanderakis, Nikos K. Furnarakis, and Zaira P. Marioli-Riga. "Two - Dimensional Finite Element Analysis of Composite Patch Repairs Using Shell Laminate Elements." Advanced Composites Letters 12, no. 2 (2003): 096369350301200. http://dx.doi.org/10.1177/096369350301200202.

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A host of one and two-sided composite patch reinforcements of metallic structures with different patch thickness were considered, in order to compare two-dimensional finite element analysis using shell laminate elements with three-dimensional finite element analysis. In order to verify the accuracy of this approach a parametric study has been performed and the 2D results were compared to the outcome of the corresponding three-dimensional finite element analysis, which accuracy has been experimentally verified in previous works. It was found that for the case of two-sided reinforcement the results obtained by the two methods were in very good agreement. For the case of one-sided reinforcement some deviation of the results of two-dimensional analysis has been observed, which was due to the tendency of the structure for out-of-plane bending, resulting from the bonding of a reinforcing patch to only one face of the structure. According to the results of this parametric study and since most aircraft structures are constrained against local out-of-plane bending (e.g. aircraft skins through stringers) two-dimensional finite elements analysis using shell laminate elements is proposed as an accurate and easy to use analysis tool for the design of both one and two-sided composite patch repairs of relatively simple structures.

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Kim, Sungmin. "Development of a Parametric Design Method for Various Woven Fabric Structures." Journal of Engineered Fibers and Fabrics 6, no. 4 (2011): 155892501100600. http://dx.doi.org/10.1177/155892501100600405.

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An integrated woven fabric system has been developed. A series of parametric design rules have been defined by analyzing famous woven structures. An expandable script language and its compiler have been developed to generate scalable weave diagrams using those rules. A calculation algorithm has also been developed to generate draft and peg plans based on the weave diagrams. Finally, a two-dimensional realistic fabric image rendering function using the actual image of various yarns, a three-dimensional fabric modeling method based on B-Spline approximation, and a pseudo three-dimensional mapping method for simulated images of final products were all integrated into the system.

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Song, Seongjin, Wooyoung Jeon, and Sunho Park. "Parametric Study on Strength Characteristics of Two-Dimensional Ice Beam Using Discrete Element Method." Applied Sciences 11, no. 18 (2021): 8409. http://dx.doi.org/10.3390/app11188409.

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Strength characteristics of a two-dimensional ice beam were studied using a discrete element method (DEM). The DEM solver was implemented by the open-source discrete element method libraries. Three-point bending and uniaxial compressive tests of the ice beam were simulated. The ice beam consisted of an assembly of disk-shaped particles with a particular thickness. The connection of the ice particles was modelled using a cuboid element, which represents a bond. If the stress acting on the bond exceeded the bond strength criterion, the bond started to break, explaining the cracking of the ice beam. To find out the effect of the local parameters of the contact and bond models on the ice fracture, we performed numerical simulations for various bond Young‘s modulus of the particles, the bond strength, and the relative particle size ratio.

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Krarti, M., D. E. Claridge, and J. F. Kreider. "ITPE Technique Applications to Time-Varying Three-Dimensional Ground-Coupling Problems." Journal of Heat Transfer 112, no. 4 (1990): 849–56. http://dx.doi.org/10.1115/1.2910491.

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Approximate analytical solutions for the three-dimensional heat transfer between slab-on-grade floors and rectangular basements under steady-periodic conditions are developed using the Interzone Temperature Profile Estimation (ITPE) method. The slab-on-grade solution is the first analytical slab-on-grade solution that treats the presence of insulation on/under the floor, while the basement solution is the first analytical solution of the time-dependent three-dimensional problem for basements. Solutions are given for the temperature field and expressions are derived for the annual heat loss. Parametric analysis is used to emphasize the effect of geometric dimensions on the magnitude and phase of heat loss relative to ambient temperature. The results obtained are compared with those from the two-dimensional model, and the three-dimensional characteristics of heat flow from slabs and basements are examined.

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Kolibal, Joseph, and Daniel Howard. "Alternative Parametric Boundary Reconstruction Method for Biomedical Imaging." Journal of Biomedicine and Biotechnology 2008 (2008): 1–7. http://dx.doi.org/10.1155/2008/623475.

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Determining the outline or boundary contour of a two-dimensional object, or the surface of a three-dimensional object poses difficulties particularly when there is substantial measurement noise or uncertainty. By adapting the mathematical approach of stochastic function recovery to this task, it is possible to obtain usable estimates for these boundaries, even in the presence of large amounts of noise. The technique is applied to parametric boundary data and has potential applications in biomedical imaging. It should be considered as one of several techniques to improve the visualization of images.

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Tian, Ying, Qing Jian Liu, Lei Zhang, Song Ling Tian, and Peng Lin Li. "Research on a Type of Program-Controlled Three-Dimensional Vibration Platform." Key Engineering Materials 693 (May 2016): 1604–10. http://dx.doi.org/10.4028/www.scientific.net/kem.693.1604.

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Conventional mechanical vibration platform vibrates not only in the direction of back and forth but also in the direction of up and down. Vibrations in both directions are achieved by two mechanisms, respectively. Three-dimensional vibration cannot be realized. In this paper, the servo motor, the gear drive mechanism, and the parallel four-bar linkage mechanism are combined with the NC system. And a type of program-controlled three-dimensional vibration platform is designed. The platform can vibrate in three directions, such as the directions of up and down, left and right, back and forth. The parametric adjustment of amplitude and frequency in three directions can be realized. The vibration platform is suitable for high-speed and light-load three-dimensional vibration. Compared with the existing mechanical vibration platform, the designed vibration platform overcomes some deficiencies to realize three-dimensional vibration with variable frequency and amplitude.

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Li, Lin, Zexuan Liu, Xifeng Ren, et al. "Metalens-array–based high-dimensional and multiphoton quantum source." Science 368, no. 6498 (2020): 1487–90. http://dx.doi.org/10.1126/science.aba9779.

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The development of two-dimensional metasurfaces has shown great potential in quantum-optical technologies because of the excellent flexibility in light-field manipulation. By integrating a metalens array with a nonlinear crystal, we demonstrate a 100-path spontaneous parametric down-conversion photon-pair source in a 10 × 10 array, which shows promise for high-dimensional entanglement and multiphoton-state generation. We demonstrate two-, three- and four-dimensional two-photon path entanglement with different phases encoded by metalenses with fidelities of 98.4, 96.6, and 95.0%, respectively. Furthermore, four-photon and six-photon generation is observed with high indistinguishability of photons generated from different metalenses. Our metalens-array–based quantum photon source is compact, stable, and controllable, indicating a new platform for integrated quantum devices.

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Eid, Hisham T. "Two- and three-dimensional analyses of translational slides in soils with nonlinear failure envelopes." Canadian Geotechnical Journal 47, no. 4 (2010): 388–99. http://dx.doi.org/10.1139/t09-110.

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This paper aims to assess the effect of nonlinearity of the soil shear strength envelope on the sliding mass configurations and the minimum two- and three-dimensional factors of safety for slopes susceptible to translational failure in both static and seismic conditions. An extensive parametric study was conducted using slope models and nonlinear soil strength envelopes that simulate translational failure case histories. Analysis of the results led to the development of stability charts that do not require an iterative procedure when determining the factors of safety. The introduction of a two-dimensional stability number and three-dimensional stability function, the values of which depend on the degrees of nonlinearity of the involved soils’ strength envelopes, has made it possible to develop such charts. These charts give the practicing geotechnical engineer a reliable and fast method to analyze translational failures. Using available software in such an analysis can be complicated if shear resistance along the sliding mass vertical sides and nonlinearity of soil shear strength envelopes are considered. Numerical examples and a case history are given to verify the reliability and illustrate the different applications of these charts.

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Lopes, António M., J. A. Tenreiro Machado, and Alexandra M. Galhano. "Multidimensional Scaling Visualization Using Parametric Entropy." International Journal of Bifurcation and Chaos 25, no. 14 (2015): 1540017. http://dx.doi.org/10.1142/s0218127415400179.

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This paper studies complex systems using a generalized multidimensional scaling (MDS) technique. Complex systems are characterized by time-series responses, interpreted as a manifestation of their dynamics. Two types of time-series are analyzed, namely 18 stock markets and the gross domestic product per capita of 18 countries. For constructing the MDS charts, indices based on parametric entropies are adopted. Multiparameter entropies allow the variation of the parameters leading to alternative sets of charts. The final MDS maps are then assembled by means of Procrustes’ method that maximizes the fit between the individual charts. Therefore, the proposed method can be interpreted as a generalization to higher dimensions of the standard technique that represents (and discretizes) items by means of single “points” (i.e. zero-dimensional “objects”). The MDS plots, involving one-, two- and three-dimensional “objects”, reveal a good performance in capturing the correlations between data.

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Korakianitis, T. "Hierarchical Development of Three Direct-Design Methods for Two-Dimensional Axial-Turbomachinery Cascades." Journal of Turbomachinery 115, no. 2 (1993): 314–24. http://dx.doi.org/10.1115/1.2929237.

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The direct and inverse blade-design iterations for the selection of isolated airfoils and gas turbine blade cascades are enormously reduced if the initial blade shape has performance characteristics near the desirable ones. This paper presents the hierarchical development of three direct blade-design methods of increasing utility for generating two-dimensional blade shapes. The methods can be used to generate inputs to the direct- or inverse-blade-design sequences for subsonic or supersonic airfoils for compressors and turbines, or isolated airfoils. The examples included for illustration are typical modern turbine cascades, and they have been designed by the direct method exclusively. The first method specifies the airfoil shapes with analytical polynomials. It shows that continuous curvature and continuous slope of curvature are necessary conditions to minimize the possibility of flow separation, and to lead to improved blade designs. The second method specifies the airfoil shapes with parametric fourth-order polynomials, which result in continuous-slope-of-curvature airfoils, with smooth Mach number and pressure distributions. This method is time consuming. The third method specifies the airfoil shapes by using a mixture of analytical polynomials and mapping the airfoil surfaces from a desirable curvature distribution. The third method provides blade surfaces with desirable performance in very few direct-design iterations. In all methods the geometry near the leading edge is specified by a thickness distribution added to a construction line, which eliminates the leading edge overspeed and laminar-separation regions. The blade-design methods presented in this paper can be used to improve the aerodynamic and heat transfer performance of turbomachinery cascades, and they can result in high-performance airfoils in very few iterations.

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Journal articles on the topic 'Two-dimensional and three-dimensional parametric copulas'

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