Journal articles: 'Fully nonlinear PDE'

1

Sirakov, Boyan. "Solvability of Uniformly Elliptic Fully Nonlinear PDE." Archive for Rational Mechanics and Analysis 195, no. 2 (May 6, 2009): 579–607. http://dx.doi.org/10.1007/s00205-009-0218-9.

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Lions, Pierre-Louis, and Panagiotis E. Souganidis. "Fully nonlinear stochastic pde with semilinear stochastic dependence." Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 331, no. 8 (October 2000): 617–24. http://dx.doi.org/10.1016/s0764-4442(00)00583-8.

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Wei-an, Liu, and Lu Gang. "Viscosity solutions of fully nonlinear functional parabolic PDE." International Journal of Mathematics and Mathematical Sciences 2005, no. 22 (2005): 3539–50. http://dx.doi.org/10.1155/ijmms.2005.3539.

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By the technique of coupled solutions, the notion of viscosity solutions is extended to fully nonlinear retarded parabolic equations. Such equations involve many models arising from optimal control theory, economy and finance, biology, and so forth. The comparison principle is shown. Then the existence and uniqueness are established by the fixed point theory.

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4

Wang, Falei. "Comparison Theorem for Nonlinear Path-Dependent Partial Differential Equations." Abstract and Applied Analysis 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/968093.

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We introduce a type of fully nonlinear path-dependent (parabolic) partial differential equation (PDE) in which the pathωton an interval [0,t] becomes the basic variable in the place of classical variablest,x∈[0,T]×ℝd. Then we study the comparison theorem of fully nonlinear PPDE and give some of its applications.

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Ikoma, Norihisa, and Hitoshi Ishii. "Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 29, no. 5 (September 2012): 783–812. http://dx.doi.org/10.1016/j.anihpc.2012.04.004.

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Katzourakis, Nikos. "Generalised solutions for fully nonlinear PDE systems and existence–uniqueness theorems." Journal of Differential Equations 263, no. 1 (July 2017): 641–86. http://dx.doi.org/10.1016/j.jde.2017.02.048.

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Wang, Zhi Yu. "Finite Strain Analysis of Crack Tip Fields in Yeoh-Model-Based Rubber-Like Materials which Are Loaded in Plane Stress." Applied Mechanics and Materials 127 (October 2011): 477–83. http://dx.doi.org/10.4028/www.scientific.net/amm.127.477.

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The dominant asymptotic stress feild near the tip of a Mode-I crack of Yeoh-model-based rubber-like materials is determined. The analysis bases on the fully nonlinear equilibrium theory of incompressible hyperelastic solids. First, The nonlinear PDE (partial differential equation) governig the leading behavior of y2 is transformed to a linear PDE. Then the linear PDE is solved and the solution of y2 is obtained. With the solution of y2 and boundery conditions the numerical solution of y1 is obtained,too.Finally,the analysis solution in polar coordinate of the first Kirchhoff Stress in plane stress state is obtained .

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8

Zhang, Jianfeng, and Jia Zhuo. "Monotone schemes for fully nonlinear parabolic path dependent PDEs." Journal of Financial Engineering 01, no. 01 (March 2014): 1450005. http://dx.doi.org/10.1142/s2345768614500056.

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In this paper, we extend the results of the seminal work Barles and Souganidis (1991) to path dependent case. Based on the viscosity theory of path dependent PDEs, developed by Ekren et al. (2012a, 2012b, 2014a and 2014b), we show that a monotone scheme converges to the unique viscosity solution of the (fully nonlinear) parabolic path dependent PDE. An example of such monotone scheme is proposed. Moreover, in the case that the solution is smooth enough, we obtain the rate of convergence of our scheme.

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Panayotounakos, D. E., and K. P. Zafeiropoulos. "General solutions of the nonlinear PDEs governing the erosion kinetics." Mathematical Problems in Engineering 8, no. 1 (2002): 69–85. http://dx.doi.org/10.1080/10241230211379.

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We present the construction of the general solutions concerning the one-dimensional (1D) fully dynamic nonlinear partial differential equations (PDEs), for the erosion kinetics. After an uncoupling procedure of the above mentioned equations a second–order nonlinear PDE of the Monge type governing the porosity is derived, the general solution of which is constructed in the sense that a full complement of arbitrary functions (as many as the order) is introduced. Afterwards, we specify the above solution according to convenient initial conditions.

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Ayanbayev, Birzhan, and Nikos Katzourakis. "On the Inverse Source Identification Problem in $L^{\infty }$ for Fully Nonlinear Elliptic PDE." Vietnam Journal of Mathematics 49, no. 3 (July 22, 2021): 815–29. http://dx.doi.org/10.1007/s10013-021-00515-6.

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AbstractIn this paper we generalise the results proved in N. Katzourakis (SIAM J. Math. Anal. 51, 1349–1370, 2019) by studying the ill-posed problem of identifying the source of a fully nonlinear elliptic equation. We assume Dirichlet data and some partial noisy information for the solution on a compact set through a fully nonlinear observation operator. We deal with the highly nonlinear nonconvex nature of the problem and the lack of weak continuity by introducing a two-parameter Tykhonov regularisation with a higher order L2 “viscosity term” for the $L^{\infty }$ L ∞ minimisation problem which allows to approximate by weakly lower semicontinuous cost functionals.

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KOIKE, Shigeaki, and Andrzej ŚWIĘCH. "Weak Harnack inequality for fully nonlinear uniformly elliptic PDE with unbounded ingredients." Journal of the Mathematical Society of Japan 61, no. 3 (July 2009): 723–55. http://dx.doi.org/10.2969/jmsj/06130723.

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Ikoma, Norihisa, and Hitoshi Ishii. "Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls, II." Bulletin of Mathematical Sciences 5, no. 3 (July 25, 2015): 451–510. http://dx.doi.org/10.1007/s13373-015-0071-0.

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13

Barles, G., and Jérôme Busca. "EXISTENCE AND COMPARISON RESULTS FOR FULLY NONLINEAR DEGENERATE ELLIPTIC EQUATIONS WITHOUT ZEROTH-ORDER TERM1*." Communications in Partial Differential Equations 26, no. 11-12 (November 1, 2001): 2323–37. http://dx.doi.org/10.1081/pde-100107824.

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14

Bertens, M. W. M. C., M. J. H. Anthonissen, W. L. IJzerman, and J. H. M. ten Thije Boonkkamp. "Design of optical surfaces conform the hyperbolic Monge-Ampère equation." EPJ Web of Conferences 266 (2022): 02003. http://dx.doi.org/10.1051/epjconf/202226602003.

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We present a method for designing freeform optical surfaces for illumination optics. By the laws of reflection, refraction and conservation of energy, a fully nonlinear PDE, the Monge-Ampere equation, is derived for the optical surface. By the edge ray principle a transport boundary condition is obtained. We solve the hyperbolic variant of the PDE using a least-squares method, resulting in optical saddle surfaces for a parallel source and far-field target.

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15

Safdari, Mohammad. "Double obstacle problems and fully nonlinear PDE with non-strictly convex gradient constraints." Journal of Differential Equations 278 (March 2021): 358–92. http://dx.doi.org/10.1016/j.jde.2021.01.002.

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16

Jensen, Robert, and Andrzej Świech. "Uniqueness and existence of maximal and minimal solutions of fully nonlinear elliptic PDE." Communications on Pure & Applied Analysis 4, no. 1 (2005): 199–207. http://dx.doi.org/10.3934/cpaa.2005.4.187.

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17

Kong, Tao, Weidong Zhao, and Tao Zhou. "Probabilistic High Order Numerical Schemes for Fully Nonlinear Parabolic PDEs." Communications in Computational Physics 18, no. 5 (November 2015): 1482–503. http://dx.doi.org/10.4208/cicp.240515.280815a.

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AbstractIn this paper, we are concerned with probabilistic high order numerical schemes for Cauchy problems of fully nonlinear parabolic PDEs. For such parabolic PDEs, it is shown by Cheridito, Soner, Touzi and Victoir [4] that the associated exact solutions admit probabilistic interpretations, i.e., the solution of a fully nonlinear parabolic PDE solves a corresponding second order forward backward stochastic differential equation (2FBSDEs). Our numerical schemes rely on solving those 2FBSDEs, by extending our previous results [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751.]. Moreover, in our numerical schemes, one has the flexibility to choose the associated forward SDE, and a suitable choice can significantly reduce the computational complexity. Various numerical examples including the HJB equations are presented to show the effectiveness and accuracy of the proposed numerical schemes.

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18

Caffarelli, Luis A., and Panagiotis E. Souganidis. "Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media." Inventiones mathematicae 180, no. 2 (January 8, 2010): 301–60. http://dx.doi.org/10.1007/s00222-009-0230-6.

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19

Philippin, G. A., and A. Safoui. "Some applications of the maximum principle to a variety of fully nonlinear elliptic PDE?s." Zeitschrift f�r Angewandte Mathematik und Physik (ZAMP) 54, no. 5 (September 1, 2003): 739–55. http://dx.doi.org/10.1007/s00033-003-3200-7.

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20

Han, Yuecai, and Chunyang Liu. "Asian Option Pricing under an Uncertain Volatility Model." Mathematical Problems in Engineering 2020 (April 21, 2020): 1–10. http://dx.doi.org/10.1155/2020/4758052.

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In this paper, we study the asymptotic behavior of Asian option prices in the worst-case scenario under an uncertain volatility model. We derive a procedure to approximate Asian option prices with a small volatility interval. By imposing additional conditions on the boundary condition and splitting the obtained Black–Scholes–Barenblatt equation into two Black–Scholes-like equations, we obtain an approximation method to solve a fully nonlinear PDE.

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21

Wang, Pei-Yong. "REGULARITY OF FREE BOUNDARIES OF TWO-PHASE PROBLEMS FOR FULLY NONLINEAR ELLIPTIC EQUATIONS OF SECOND ORDER. II. FLAT FREE BOUNDARIES ARE LIPSCHITZ." Communications in Partial Differential Equations 27, no. 7-8 (January 7, 2002): 1497–514. http://dx.doi.org/10.1081/pde-120005846.

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22

Lindstrom, Michael R., and Andrea L. Bertozzi. "Qualitative features of a nonlinear, nonlocal, agent-based PDE model with applications to homelessness." Mathematical Models and Methods in Applied Sciences 30, no. 10 (September 2020): 1863–91. http://dx.doi.org/10.1142/s0218202520400114.

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In this paper, we develop a continuum model for the movement of agents on a lattice, taking into account location desirability, local and far-range migration, and localized entry and exit rates. Specifically, our motivation is to qualitatively describe the homeless population in Los Angeles. The model takes the form of a fully nonlinear, nonlocal, non-degenerate parabolic partial differential equation. We derive the model and prove useful properties of smooth solutions, including uniqueness and [Formula: see text]-stability under certain hypotheses. We also illustrate numerical solutions to the model and find that a simple model can be qualitatively similar in behavior to observed homeless encampments.

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23

Katzourakis, Nikos. "On linear degenerate elliptic PDE systems with constant coefficients." Advances in Calculus of Variations 9, no. 3 (July 1, 2016): 283–91. http://dx.doi.org/10.1515/acv-2015-0004.

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AbstractLet ${\mathbf{A}}$ be a symmetric convex quadratic form on ${\mathbb{R}^{Nn}}$ and Ω $\subset$$\mathbb{R}^{n}$ a bounded convex domain. We consider the problem of existence of solutions u: Ω $\subset$$\mathbb{R}^{n}$$\to$$\mathbb{R}^{N}$ to the problem${}\left\{\begin{aligned} \displaystyle\sum_{\beta=1}^{N}\sum_{i,j=1}^{n}% \mathbf{A}_{\alpha i\beta j}D^{2}_{ij}u_{\beta}&\displaystyle=f_{\alpha}&&% \displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\end{% aligned}\right.\phantom{\}}$when ${f\in L^{2}(\Omega,\mathbb{R}^{N})}$. Problem (1) is degenerate elliptic and it has not been considered before without the assumption of strict rank-one convexity. In general, it may not have even distributional solutions. By introducing an extension of distributions adapted to (1), we prove existence, partial regularity and by imposing an extra condition uniqueness as well. The satisfaction of the boundary condition is also an issue due to the low regularity of the solution. The motivation to study (1) and the method of the proof arose from recent work of the author [10] on generalised solutions for fully nonlinear systems.

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Katzourakis, Nikos, and Tristan Pryer. "Second-order L∞ variational problems and the ∞-polylaplacian." Advances in Calculus of Variations 13, no. 2 (April 1, 2020): 115–40. http://dx.doi.org/10.1515/acv-2016-0052.

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AbstractIn this paper we initiate the study of second-order variational problems in {L^{\infty}}, seeking to minimise the {L^{\infty}} norm of a function of the hessian. We also derive and study the respective PDE arising as the analogue of the Euler–Lagrange equation. Given {\mathrm{H}\in C^{1}(\mathbb{R}^{n\times n}_{s})}, for the functional\mathrm{E}_{\infty}(u,\mathcal{O})=\|\mathrm{H}(\mathrm{D}^{2}u)\|_{L^{\infty}% (\mathcal{O})},\quad u\in W^{2,\infty}(\Omega),\mathcal{O}\subseteq\Omega,{}the associated equation is the fully nonlinear third-order PDE\mathrm{A}^{2}_{\infty}u:=(\mathrm{H}_{X}(\mathrm{D}^{2}u))^{\otimes 3}:(% \mathrm{D}^{3}u)^{\otimes 2}=0.{}Special cases arise when {\mathrm{H}} is the Euclidean length of either the full hessian or of the Laplacian, leading to the {\infty}-polylaplacian and the {\infty}-bilaplacian respectively. We establish several results for (1) and (2), including existence of minimisers, of absolute minimisers and of “critical point” generalised solutions, proving also variational characterisations and uniqueness. We also construct explicit generalised solutions and perform numerical experiments.

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25

Katzourakis, Nikos. "Weak vs. 𝒟-solutions to linear hyperbolic first-order systems with constant coefficients." Journal of Hyperbolic Differential Equations 15, no. 02 (June 2018): 329–47. http://dx.doi.org/10.1142/s0219891618500121.

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We establish a consistency result by comparing two independent notions of generalized solutions to a large class of linear hyperbolic first-order PDE systems with constant coefficients, showing that they eventually coincide. The first is the usual notion of weak solutions defined via duality. The second is the new notion of [Formula: see text]-solutions which we recently introduced and arose in connection to the vectorial calculus of variations in [Formula: see text] and fully nonlinear elliptic systems. This new approach is a duality-free alternative to distributions and is based on the probabilistic representation of limits of difference quotients.

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Grün, G., and P. Weiß. "On the field-induced transport of magnetic nanoparticles in incompressible flow: Modeling and numerics." Mathematical Models and Methods in Applied Sciences 29, no. 12 (November 2019): 2321–57. http://dx.doi.org/10.1142/s0218202519500477.

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By methods from non-equilibrium thermodynamics, we derive a class of nonlinear pde-models to describe the motion of magnetizable nanoparticles suspended in incompressible carrier fluids under the influence of external magnetic fields. Our system of partial differential equations couples Navier–Stokes and magnetostatic equations to evolution equations for the magnetization field and the particle number density. In the second part of the paper, a fully discrete mixed finite-element scheme is introduced which is rigorously shown to be energy-stable. Finally, we present numerical simulations in the 2D-case which provide first information about the interaction of particle density, magnetization and magnetic field.

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Zhang, Yanmei, Xia Cui, and Guangwei Yuan. "Nonlinear iteration acceleration solution for equilibrium radiation diffusion equation." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 5 (June 26, 2020): 1465–90. http://dx.doi.org/10.1051/m2an/2019095.

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This paper discusses accelerating iterative methods for solving the fully implicit (FI) scheme of equilibrium radiation diffusion problem. Together with the FI Picard factorization (PF) iteration method, three new nonlinear iterative methods, namely, the FI Picard-Newton factorization (PNF), FI Picard-Newton (PN) and derivative free Picard-Newton factorization (DFPNF) iteration methods are studied, in which the resulting linear equations can preserve the parabolic feature of the original PDE. By using the induction reasoning technique to deal with the strong nonlinearity of the problem, rigorous theoretical analysis is performed on the fundamental properties of the four iteration methods. It shows that they all have first-order time and second-order space convergence, and moreover, can preserve the positivity of solutions. It is also proved that the iterative sequences of the PF iteration method and the three Newton-type iteration methods converge to the solution of the FI scheme with a linear and a quadratic speed respectively. Numerical tests are presented to confirm the theoretical results and highlight the high performance of these Newton acceleration methods.

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Jandačka, Martin, and Daniel Ševčovič. "On the risk-adjusted pricing-methodology-based valuation of vanilla options and explanation of the volatility smile." Journal of Applied Mathematics 2005, no. 3 (2005): 235–58. http://dx.doi.org/10.1155/jam.2005.235.

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We analyse a model for pricing derivative securities in the presence of both transaction costs as well as the risk from a volatile portfolio. The model is based on the Black-Scholes parabolic PDE in which transaction costs are described following the Hoggard, Whalley, and Wilmott approach. The risk from a volatile portfolio is described by the variance of the synthesized portfolio. Transaction costs as well as the volatile portfolio risk depend on the time lag between two consecutive transactions. Minimizing their sum yields the optimal length of the hedge interval. In this model, prices of vanilla options can be computed from a solution to a fully nonlinear parabolic equation in which a diffusion coefficient representing volatility nonlinearly depends on the solution itself giving rise to explaining the volatility smile analytically. We derive a robust numerical scheme for solving the governing equation and perform extensive numerical testing of the model and compare the results to real option market data. Implied risk and volatility are introduced and computed for large option datasets. We discuss how they can be used in qualitative and quantitative analysis of option market data.

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Di Francesco, Marco, and Simone Fagioli. "A nonlocal swarm model for predators–prey interactions." Mathematical Models and Methods in Applied Sciences 26, no. 02 (November 19, 2015): 319–55. http://dx.doi.org/10.1142/s0218202516400042.

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We consider a two-species system of nonlocal interaction PDEs modeling the swarming dynamics of predators and prey, in which all agents interact through attractive/repulsive forces of gradient type. In order to model the predator–prey interaction, we prescribed proportional potentials (with opposite signs) for the cross-interaction part. The model has a particle-based discrete (ODE) version and a continuum PDE version. We investigate the structure of particle stationary solution and their stability in the ODE system in a systematic form, and then consider simple examples. We then prove that the stable particle steady states are locally stable for the fully nonlinear continuum model, provided a slight reinforcement of the particle condition is required. The latter result holds in one space dimension. We complement all the particle examples with simple numerical simulations, and we provide some two-dimensional examples to highlight the complexity in the large time behavior of the system.

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Long, Kevin, Paul T. Boggs, and Bart G. van Bloemen Waanders. "Sundance: High-Level Software for PDE-Constrained Optimization." Scientific Programming 20, no. 3 (2012): 293–310. http://dx.doi.org/10.1155/2012/380908.

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Sundance is a package in the Trilinos suite designed to provide high-level components for the development of high-performance PDE simulators with built-in capabilities for PDE-constrained optimization. We review the implications of PDE-constrained optimization on simulator design requirements, then survey the architecture of the Sundance problem specification components. These components allow immediate extension of a forward simulator for use in an optimization context. We show examples of the use of these components to develop full-space and reduced-space codes for linear and nonlinear PDE-constrained inverse problems.

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Hamer, Russell D., and Christopher W. Tyler. "Phototransduction: Modeling the primate cone flash response." Visual Neuroscience 12, no. 6 (November 1995): 1063–82. http://dx.doi.org/10.1017/s0952523800006726.

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AbstractWe have developed a new model of phototransduction that accounts for the dynamics of primate and human cone flash responses in both their linear and saturating range. The model incorporates many of the known elements of the phototransduction cascade in vertebrate photoreceptors. The input stage is a new analytic expression for the activation and inactivation of cGMP-phosphodiesterase (PDE). Although the Lamb and Pugh (1992) model (of a delayed ramp for the rising phase of the PDE* response in amphibian rods) provided a good fit for the first 2 log units of stimulus intensity without parameter adjustments, the remaining 4 log units of the data required nonlinear modifications of both delay and gain (slope). We show that this nonlinear behavior is a consequence of the delay approximation and develop a completely linear model to account for the rising phase of amphibian rod photocurrent responses over the full intensity range (~6 log units). We use the same dynamic model to account for primate cone responses by decreasing the time constants of PDE activation and introducing an enhanced inactivation process. This PDE* response activates a nonlinear calcium feedback stage that modulates guanylate cyclase synthesis of cyclic GMP. By adjustment of the throughput and feedback parameters, the full model successfully captures most of the features of the primate and human cone flash responses throughout their dynamic range. Our analysis suggests that initial processes in the transduction cascade may be qualitatively different from comparable processes in rods.

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Zhao, Xiaodong, Jianzhong Cao, Zuofeng Zhou, and Jijiang Huang. "A Novel PDE-Based Single Image Super-Resolution Reconstruction Method." International Journal of Pattern Recognition and Artificial Intelligence 31, no. 06 (March 30, 2017): 1754010. http://dx.doi.org/10.1142/s0218001417540106.

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For applications such as remote sensing imaging and medical imaging, high-resolution (HR) images are urgently required. Image Super-Resolution (SR) reconstruction has great application prospects in optical imaging. In this paper, we propose a novel unified Partial Differential Equation (PDE)-based method to single image SR reconstruction. Firstly, two directional diffusion terms calculated by Anisotropic Nonlinear Structure Tensor (ANLST) are constructed, combing information of all channels to prevent singular results, making full use of its directional diffusion feature. Secondly, by introducing multiple orientations estimation using high order matrix-valued tensor instead of gradient, orientations can be estimated more precisely for junctions or corners. As a unique descriptor of orientations, mixed orientation parameter (MOP) is separated into two orientations by finding roots of a second-order polynomial in the nonlinear part. Then, we synthesize a Gradient Vector Flow (GVF) shock filter to balance edge enhancement and de-noising process. Experimental results confirm the validity of the method and show that the method enhances image edges, restores corners or junctions, and suppresses noise robustness, which is competitive with the existing methods.

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Chen, Lidao, and Yong Liu. "Differential Quadrature Method for Fully Intrinsic Equations of Geometrically Exact Beams." Aerospace 9, no. 10 (October 12, 2022): 596. http://dx.doi.org/10.3390/aerospace9100596.

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In this paper, a differential quadrature method of high-order precision (DQ−Pade), which is equivalent to the generalized Pade approximation for approximating the end of a time or spatial interval, is used to solve nonlinear fully intrinsic equations of beams. The equations are a set of first-order differential equations with respect to time and space, and the explicit unknowns of the equations involve only forces, moments, velocity and angular velocity, without displacements and rotations. Based on the DQ−Pade method, the spatial and temporal discrete forms of fully intrinsic equations were derived. To verify the effectiveness and applicability of the proposed method for discretizing the fully intrinsic equations, different examples available in the literatures were considered. It was found that when using the DQ−Pade method, the solutions of the intrinsic beam equations are obviously superior to those found by some other usual algorithms in efficiency and computational accuracy.

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Yang, Haijian, Feng-Nan Hwang, and Xiao-Chuan Cai. "Nonlinear Preconditioning Techniques for Full-Space Lagrange--Newton Solution of PDE-Constrained Optimization Problems." SIAM Journal on Scientific Computing 38, no. 5 (January 2016): A2756—A2778. http://dx.doi.org/10.1137/15m104075x.

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El hajaji, Abdelmajid, Khalid Hilal, Abdelhafid Serghini, and El bekkey Mermri. "Pricing American bond options using a cubic spline collocation method." Boletim da Sociedade Paranaense de Matemática 32, no. 2 (September 11, 2014): 189. http://dx.doi.org/10.5269/bspm.v32i2.21354.

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In this paper, American options on a discount bond are priced under the Cox-Ingrosll-Ross (CIR) model. The linear complementarity problem of the option value is solved numerically by a penalty method. The problem is transformed into a nonlinear partial differential equation (PDE) by adding a power penalty term. The solution of the penalized problem converges to the one of the original problem. To numerically solve this nonlinear PDE, we use the horizontal method of lines to discretize the temporal variable and the spatial variable by means of trapezoidal method and a cubic spline collocation method, respectively. We show that this full discretization scheme is second order convergent, and hence the convergence of the numerical solution to the viscosity solution of the continuous problem is guaranteed. Numerical results are presented and compared with other collocation methods given in the literature.

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Garde, Smita, Michael G. Regalado, Vicki L. Schechtman, and Michael C. K. Khoo. "Nonlinear dynamics of heart rate variability in cocaine-exposed neonates during sleep." American Journal of Physiology-Heart and Circulatory Physiology 280, no. 6 (June 1, 2001): H2920—H2928. http://dx.doi.org/10.1152/ajpheart.2001.280.6.h2920.

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The aim of this study was to determine the effects of prenatal cocaine exposure (PCE) on the dynamics of heart rate variability in full-term neonates during sleep. R-R interval (RRI) time series from 9 infants with PCE and 12 controls during periods of stable quiet sleep and active sleep were analyzed using autoregressive modeling and nonlinear dynamics. There were no differences between the two groups in spectral power distribution, approximate entropy, correlation dimension, and nonlinear predictability. However, application of surrogate data analysis to these measures revealed a significant degree of nonlinear RRI dynamics in all subjects. A parametric model, consisting of a nonlinear delayed-feedback system with stochastic noise as the perturbing input, was employed to estimate the relative contributions of linear and nonlinear deterministic dynamics in the data. Both infant groups showed similar proportional contributions in linear, nonlinear, and stochastic dynamics. However, approximate entropy, correlation dimension, and nonlinear prediction error were all decreased in active versus quiet sleep; in addition, the parametric model revealed a doubling of the linear component and a halving of the nonlinear contribution to overall heart rate variability. Spectral analysis indicated a shift in relative power toward lower frequencies. We conclude that 1) RRI dynamics in infants with PCE and normal controls are similar; and 2) in both groups, sympathetic dominance during active sleep produces primarily periodic low-frequency oscillations in RRI, whereas in quiet sleep vagal modulation leads to RRI fluctuations that are broadband and dynamically more complex.

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Hall, E., S. Kessler, and S. Hanagud. "Use of Fractal Dimension in the Characterization of Chaotic Structural Dynamic Sytems." Applied Mechanics Reviews 44, no. 11S (November 1, 1991): S107—S113. http://dx.doi.org/10.1115/1.3121342.

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The purpose of this paper is to investigate the use of fractal dimensions in the characterization of chaotic systems in structural dynamics. The investigation focuses on the example of a simply-supported, Euler-Bernoulli beam which when subjected to a transverse forcing function of a particular amplitude responds chaotically. Three different nonlinear models of the system are studied: a complex partial differential equation (PDE) model, a simplified PDE model, and a Galerkin approximation to the simpler PDE model. The responses of each model are examined through zero velocity Poincare´ sections. To characterize and compare the chaotic trajectories, the box counting fractal dimension of the Poincare´ sections are computed. The results demonstrate that the fractal dimension is a spatial invariant along the length of the beam for the specific class of forcing function studied, and thus it can be used to characterize chaotic motions. In addition, the three models yield different fractal dimensions for the same forcing which indicates that fractal dimensions can also be used to quantify whether a simplification of a chaotic model accurately predicts the chaotic behavior of the full-blown model. Thus the conclusion of the paper is that fractal dimensions may play an important role in the characterization of chaotic structural dynamic systems.

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Hwang, Feng-Nan. "Three-dimensional trajectory optimization for multi-stage launch vehicle mission using a full-space quasi-Lagrange–Newton method." ANZIAM Journal 60 (August 30, 2019): C172—C186. http://dx.doi.org/10.21914/anziamj.v60i0.14067.

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Many aerospace industrial applications require robust and efficient numerical solutions of large sparse nonlinear constrained parameter optimization problems arising from optimal trajectory problems. A three-dimensional multistage launcher problem is taken as a numerical example for studying the performance and applicability of the full-space Lagrange–Newton–Krylov method. The typical optimal trajectory, control history and other important physical qualities are presented, and the efficiency of the algorithm is also investigated. References J. T. Betts. Practical methods for optimal control and estimation using nonlinear programming. Advances in Design and Control. SIAM, 2nd edition, 2010. doi:10.1137/1.9780898718577. R. T. Marler and J. S. Arora. Survey of multi-objective optimization methods for engineering. Struct. Multidiscip. Opt., 26(6):369395, 2004. doi:10.1007/s00158-003-0368-6. W. Roh and Y. Kim. Trajectory optimization for a multi-stage launch vehicle using time finite element and direct collocation methods. Eng. Opt., 34:1532, 2002. doi:10.1080/03052150210912. G. D. Silveira and V. Carrara. A six degrees-of-freedom flight dynamics simulation tool of launch vehicles. J. Aero. Tech. Manag., 7:231239, 2015. doi:10.5028/jatm.v7i2.433. H.-H. Wang, Y.-S. Lo, F.-T. Hwang, and F.-N. Hwang. A full-space quasi-LagrangeNewtonKrylov algorithm for trajectory optimization problems. Electron. T. Numer. Anal., 49:103125, 2018. doi:10.1553/etna_vol49s103. H. Yang, F.-N. Hwang, and X.-C. Cai. Nonlinear preconditioning techniques for full-space Lagrange-Newton solution of PDE-constrained optimization problems. SIAM J. Sci. Comput., 38:A2756A2778, 2016. doi:10.1137/15M104075X.

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Said, Hamid. "An analytical mechanics approach to the first law of thermodynamics and construction of a variational hierarchy." Theoretical and Applied Mechanics, no. 00 (2020): 11. http://dx.doi.org/10.2298/tam200315011s.

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A simple procedure is presented for the study of the conservation of energy equation with dissipation in continuum mechanics in 1D. This procedure is used to transform this nonlinear evolution-diffusion equation into a hyperbolic PDE; specifically, a second-order quasi-linear wave equation. An immediate implication of this procedure is the formation of a least action principle for the balance of energy with dissipation. The corresponding action functional enables us to establish a complete analytic mechanics for thermomechanical systems: a Lagrangian-Hamiltonian theory, integrals of motion, bracket formalism, and Noether?s theorem. Furthermore, we apply our procedure iteratively and produce an infinite sequence of interlocked variational principles, a variational hierarchy, where at each level or iteration the full implication of the least action principle can be shown again.

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Sieber, J., M. Radžiūnas, and K. R. Schneider. "DYNAMICS OF MULTISECTION SEMICONDUCTOR LASERS." Mathematical Modelling and Analysis 9, no. 1 (March 31, 2005): 51–66. http://dx.doi.org/10.3846/13926292.2004.9637241.

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We investigate the longitudinal dynamics of multisection semiconductor lasers based on a model, where a hyperbolic system of partial differential equations is nonlinearly coupled with a system of ordinary differential equations. We present analytic results for that system: global existence and uniqueness of the initial‐boundary value problem, and existence of attracting invariant manifolds of low dimension. The flow on these manifolds is approximately described by the so‐called mode approximations which are systems of ordinary differential equations. Finally, we present a detailed numerical bifurcation analysis of the two-mode approximation system and compare it with the simulated dynamics of the full PDE model.

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COMAN, Calin-Dumitru. "The Influence of Temperature on the Strength of Hybrid Metal-Composite Multi-Bolts Joints." INCAS BULLETIN 12, no. 3 (September 1, 2020): 49–64. http://dx.doi.org/10.13111/2066-8201.2020.12.3.4.

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This paper presents the temperature influence on the strength of the hybrid metal-composite multi-bolted joints. A detailed 3D finite element model, incorporating all possible nonlinearities as large deformations, in plane nonlinear shear deformations, elastic properties degradation of the composite material and friction-based full contact, is developed to anticipate the temperature changing effects on the progressive damage analysis (PDA) at lamina level and failure modes of metal-composite multi-bolted joints. The PDA material model accounts for lamina nonlinear shear deformation, Hashin-type failure criteria and strain-based continuum degradation rules being developed using the UMAT user subroutine in Nastran commercial software. In order to validate the temperature effects on the failure modes of the joint with protruding and countersunk bolts, experiments were conducted using the SHM (Structural Health Monitoring) technique in the temperature controlled chamber. The results showed that the temperature effects on damage initiation and failure modes have to be taken into account in the design process in order to fructify the high specific strength of the composites. Experimental results were quite accurately predicted by the PDA material model, which proved to be computational efficient and can predict failure propagation and damage mechanism in hybrid metal-composite multi-bolted joints.

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Djouadi, Seddik M., R. Chris Camphouse, and James H. Myatt. "Empirical Reduced-Order Modeling for Boundary Feedback Flow Control." Journal of Control Science and Engineering 2008 (2008): 1–11. http://dx.doi.org/10.1155/2008/154956.

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This paper deals with the practical and theoretical implications of model reduction for aerodynamic flow-based control problems. Various aspects of model reduction are discussed that apply to partial differential equation- (PDE-) based models in general. Specifically, the proper orthogonal decomposition (POD) of a high dimension system as well as frequency domain identification methods are discussed for initial model construction. Projections on the POD basis give a nonlinear Galerkin model. Then, a model reduction method based on empirical balanced truncation is developed and applied to the Galerkin model. The rationale for doing so is that linear subspace approximations to exact submanifolds associated with nonlinear controllability and observability require only standard matrix manipulations utilizing simulation/experimental data. The proposed method uses a chirp signal as input to produce the output in the eigensystem realization algorithm (ERA). This method estimates the system's Markov parameters that accurately reproduce the output. Balanced truncation is used to show that model reduction is still effective on ERA produced approximated systems. The method is applied to a prototype convective flow on obstacle geometry. AnH∞feedback flow controller is designed based on the reduced model to achieve tracking and then applied to the full-order model with excellent performance.

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Dumbser, Michael, and Eleuterio F. Toro. "On Universal Osher-Type Schemes for General Nonlinear Hyperbolic Conservation Laws." Communications in Computational Physics 10, no. 3 (September 2011): 635–71. http://dx.doi.org/10.4208/cicp.170610.021210a.

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This paper is concerned with a new version of the Osher-Solomon Riemann solver and is based on a numerical integration of the path-dependent dissipation matrix. The resulting scheme is much simpler than the original one and is applicable to general hyperbolic conservation laws, while retaining the attractive features of the original solver: the method is entropy-satisfying, differentiable and complete in the sense that it attributes a different numerical viscosity to each characteristic field, in particular to the intermediate ones, since the full eigenstructure of the underlying hyperbolic system is used. To illustrate the potential of the proposed scheme we show applications to the following hyperbolic conservation laws: Euler equations of compressible gas-dynamics with ideal gas and real gas equation of state, classical and relativistic MHD equations as well as the equations of nonlinear elasticity. To the knowledge of the authors, apart from the Euler equations with ideal gas, an Osher-type scheme has never been devised before for any of these complicated PDE systems. Since our new general Riemann solver can be directly used as a building block of high order finite volume and discontinuous Galerkin schemes we also show the extension to higher order of accuracy and multiple space dimensions in the new framework of PNPM schemes on unstructured meshes recently proposed in [9].

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Kraus, Johannes, Svetoslav Nakov, and Sergey I. Repin. "Reliable Numerical Solution of a Class of Nonlinear Elliptic Problems Generated by the Poisson–Boltzmann Equation." Computational Methods in Applied Mathematics 20, no. 2 (April 1, 2020): 293–319. http://dx.doi.org/10.1515/cmam-2018-0252.

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AbstractWe consider a class of nonlinear elliptic problems associated with models in biophysics, which are described by the Poisson–Boltzmann equation (PBE). We prove mathematical correctness of the problem, study a suitable class of approximations, and deduce guaranteed and fully computable bounds of approximation errors. The latter goal is achieved by means of the approach suggested in [19] for convex variational problems. Moreover, we establish the error identity, which defines the error measure natural for the considered class of problems and show that it yields computable majorants and minorants of the global error as well as indicators of local errors that provide efficient adaptation of meshes. Theoretical results are confirmed by a collection of numerical tests that includes problems on 2D and 3D Lipschitz domains.

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COMAN, Calin-Dumitru. "Influence of Geometry on Failure Modes of Hybrid Metal-Composite Protruding Bolted Joints." INCAS BULLETIN 13, no. 3 (September 4, 2021): 29–44. http://dx.doi.org/10.13111/2066-8201.2021.13.3.3.

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This article presents the influence of joint geometry on the damage mode in the CFRP (Carbon Fiber Reinforced Polymer) composite plate of the single-lap, protruding, hybrid metal-composite joints. A detailed 3D finite element model incorporating geometric, material and friction-based contact full nonlinearities is developed to numerically investigate the geometry effects on the progressive damage analysis (PDA) of the orthotropic material model. The PDA material model integrates the nonlinear shear response, Hashin-tape failure criteria and strain-based continuum degradation rules being developed using the UMAT user subroutine in Nastran commercial software. In order to validate the geometry effects on the failure modes of the joints with hexagonal head bolts, experiments were conducted using the SHM (Structural Health Monitoring) technique. The results showed that the plate geometry is an important parameter in the design process of an adequate bolted joint and its effects on damage initiation and failure modes were quite accurately predicted by the PDA material model, which proved to be computational efficient and can predict failure propagation and damage mechanism in hybrid metal-composite bolted joints.

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CHRISTOV, C. I., and M. G. VELARDE. "INELASTIC INTERACTION OF BOUSSINESQ SOLITONS." International Journal of Bifurcation and Chaos 04, no. 05 (October 1994): 1095–112. http://dx.doi.org/10.1142/s0218127494000800.

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Two improved versions of Boussinesq equation (Boussinesq paradigm) have been considered which are well-posed (correct in the sense of Hadamard) as an initial value problem: the Proper Boussinesq Equation (PBE) and the Regularized Long Wave Equation (RLWE). Fully implicit difference schemes have been developed strictly representing, on difference level, the conservation or balance laws for the mass, pseudoenergy or pseudomomentum of the wave system. Thresholds of possible nonlinear blow-up are identified for both PBE and RLWE. The head-on collisions of solitary waves of the sech type (Boussinesq solitons) have been investigated. They are subsonic and negative (surface depressions) for PBE and supersonic and positive (surface elevations) for RLWE. The numerically recovered sign and sizes of the phase shifts are in very good quantitative agreement with analytical results for the two-soliton solution of PBE. The subsonic surface elevations are found to be not permanent but to gradually transform into oscillatory pulses whose support increases and amplitude decreases with time although the total pseudoenergy is conserved within 10−10. The latter allows us to claim that the pulses are solitons despite their “aging” (which is felt on times several times the time-scale of collision). For supersonic phase speeds, the collision of Boussinesq solitons has inelastic character exhibiting not only a significant phase shift but also a residual signal of sizable amplitude but negligible pseudoenergy. The evolution of the residual signal is investigated numerically for very long times.

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Jovanović, Radiša, Vladimir Zarić, Zoran Bučevac, and Uglješa Bugarić. "Discrete-Time System Conditional Optimization Based on Takagi–Sugeno Fuzzy Model Using the Full Transfer Function." Applied Sciences 12, no. 15 (July 31, 2022): 7705. http://dx.doi.org/10.3390/app12157705.

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The study proposes a novel method for synthesis of a discrete-time parallel distributed compensation (PDC) controller for the nonlinear discrete-time Takagi–Sugeno (TS) fuzzy plant model. For each of the fuzzy plant model linear subsystems, a local linear first order proportional-sum (PS) controller is designed. The algebraic technique is used in two-dimensional parameter space, utilizing the characteristic polynomial of the row nondegenerate full transfer function matrix. Each system’s relative stability is accomplished in relation to the selected damping coefficient. The supplementary criterion is the minimal value of the performance index in the form of the sum of squared errors (SSE). However, unlike the traditional technique, output error is impacted by all simultaneous actions on the system: nonzero inputs and nonzero initial conditions. The full transfer function matrix of the system allows for the treatment of simultaneous actions of the input vector and unknown unpredictable initial conditions. In order to show the improvement caused by the application of a new optimization method that considers nonzero initial conditions, a comparison of PDC controllers designed under zero and nonzero initial conditions is given, where the system in both cases starts from the same nonzero initial conditions, which is often the case in practice. The simulation and experimental results on a DC servo motor are shown to demonstrate the suggested method efficiency.

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GUTIERREZ, GAIL, and WHADY FLOREZ. "COMPARISON BETWEEN GLOBAL, CLASSICAL DOMAIN DECOMPOSITION AND LOCAL, SINGLE AND DOUBLE COLLOCATION METHODS BASED ON RBF INTERPOLATION FOR SOLVING CONVECTION-DIFFUSION EQUATION." International Journal of Modern Physics C 19, no. 11 (November 2008): 1737–51. http://dx.doi.org/10.1142/s0129183108013102.

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This work presents a performance comparison of several meshless RBF formulations for convection-diffusion equation with moderate-to-high Peclet number regimes. For the solution of convection-diffusion problems, several comparisons between global (full-domain) meshless RBF methods and mesh-based methods have been presented in the literature. However, in depth studies between new local RBF collocation methods and full-domain symmetric RBF collocation methods are not reported yet. The RBF formulations included: global symmetric method, symmetric double boundary collocation method, additive Schwarz domain decomposition method (DDM) when it is incorporated into two anterior approaches, and local single and double collocation methods. It can be found that the accuracy of solutions deteriorates as Pe increases, if no special treatment is used. From the numerical tests, it seems that the local methods, especially the derived double collocation technique incorporating PDE operator, are more effective than full domain approaches even with iterative DDM in solving moderate-to-high Pe convection-diffusion problems subject to mixed boundary conditions.

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COMAN, Calin-Dumitru. "Influence of Preload on Failure Modes of Hybrid Metal-Composite Protruding Bolted Joints." INCAS BULLETIN 13, no. 1 (March 5, 2021): 29–41. http://dx.doi.org/10.13111/2066-8201.2021.13.1.4.

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This paper presents the effects of torque preload on the damage initiation and growth in the CFRP (Carbon Fiber Reinforced Polymer) composite laminated adherent of the single-lap, single-bolt, hybrid metal-composite joints. A detailed 3D finite element model incorporating geometric, material and friction-based contact full nonlinearities is developed to numerically investigate the preload effects on the progressive damage analysis (PDA) of the orthotropic material model. The PDA material model integrates the nonlinear shear response, Hashin-tape failure criteria and strain-based continuum elastic properties degradation laws being developed using the UMAT user subroutine in Nastran commercial software. In order to validate the preload effects on the failure modes of the joints with hexagonal head bolts, experiments were conducted using the SHM (Structural Health Monitoring) technique. The results showed that the adherent torque level is an important parameter in the design process of an adequate bolted joint and its effects on damage initiation and failure modes were quite accurately predicted by the PDA material model, which proved to be computational efficient and can predict failure propagation and damage mechanism in hybrid metal-composite bolted joints.

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Coman, Calin Dumitru, and Dan Mihai Constantinescu. "Preload Effects on Failure Mechanisms of Hybrid Metal-Composite Bolted Joints." Materials Science Forum 957 (June 2019): 293–302. http://dx.doi.org/10.4028/www.scientific.net/msf.957.293.

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This paper presents the effects of torque preload on the damage initiation and growth in the CFRP (Carbon Fiber Reinforced Polymer) composite laminated adherent of the single-lap, single-bolt, hybrid metal-composite joints. A detailed 3D finite element model incorporating geometric, material and friction-based contact full nonlinearities is developed to numerically investigate the preload effects on the progressive damage analysis (PDA) of the orthotropic material model. The PDA material model integrates the nonlinear shear response, Hashin-tape failure criteria and strain-based continuum elastic properties degradation laws being developed using the UMAT user subroutine in Nastran commercial software. In order to validate the preload effects on the failure modes of the joints with hexagonal head bolts, experiments were conducted using the SHM (Structural Health Monitoring) technique. The results showed that the adherent torque level is an important parameter in the design process of an adequate bolted joint and its effects on damage initiation and failure modes were quite accurately predicted by the PDA material model, which proved to be computational efficient and can predict failure propagation and damage mechanism in hybrid metal-composite bolted joints.

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Journal articles on the topic 'Fully nonlinear PDE'

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