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Fourth-order phase-field modeling for brittle fracture in piezoelectric materials
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作者 Yu TAN Fan PENG +2 位作者 Chang LIU Daiming PENG Xiangyu LI 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI CSCD 2024年第5期837-856,共20页
Failure analyses of piezoelectric structures and devices are of engineering and scientific significance.In this paper,a fourth-order phase-field fracture model for piezoelectric solids is developed based on the Hamilt... Failure analyses of piezoelectric structures and devices are of engineering and scientific significance.In this paper,a fourth-order phase-field fracture model for piezoelectric solids is developed based on the Hamilton principle.Three typical electric boundary conditions are involved in the present model to characterize the fracture behaviors in various physical situations.A staggered algorithm is used to simulate the crack propagation.The polynomial splines over hierarchical T-meshes(PHT-splines)are adopted as the basis function,which owns the C1continuity.Systematic numerical simulations are performed to study the influence of the electric boundary conditions and the applied electric field on the fracture behaviors of piezoelectric materials.The electric boundary conditions may influence crack paths and fracture loads significantly.The present research may be helpful for the reliability evaluation of the piezoelectric structure in the future applications. 展开更多
关键词 isogeometric analysis(IGA) brittle fracture fourth-order phase-field model piezoelectric solid
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Existence of Monotone Positive Solution for a Fourth-Order Three-Point BVP with Sign-Changing Green’s Function
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作者 Junrui Yue Yun Zhang Qingyue Bai 《Open Journal of Applied Sciences》 2024年第1期63-69,共7页
This paper is concerned with the following fourth-order three-point boundary value problem , where , we discuss the existence of positive solutions to the above problem by applying to the fixed point theory in cones a... This paper is concerned with the following fourth-order three-point boundary value problem , where , we discuss the existence of positive solutions to the above problem by applying to the fixed point theory in cones and iterative technique. 展开更多
关键词 fourth-order Three-Point Boundary Value Problem Sign-Changing Green’s Function Fixed Point Index Iterative Technique Monotone Positive Solution EXISTENCE
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ALMOST SURE GLOBAL WELL-POSEDNESS FOR THE FOURTH-ORDER NONLINEAR SCHR?DINGER EQUATION WITH LARGE INITIAL DATA
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作者 陈明娟 张帅 《Acta Mathematica Scientia》 SCIE CSCD 2023年第5期2215-2233,共19页
We consider the fourth-order nonlinear Schr?dinger equation(4NLS)(i?t+εΔ+Δ2)u=c1um+c2(?u)um-1+c3(?u)2um-2,and establish the conditional almost sure global well-posedness for random initial data in Hs(Rd)for s∈(sc-... We consider the fourth-order nonlinear Schr?dinger equation(4NLS)(i?t+εΔ+Δ2)u=c1um+c2(?u)um-1+c3(?u)2um-2,and establish the conditional almost sure global well-posedness for random initial data in Hs(Rd)for s∈(sc-1/2,sc],when d≥3 and m≥5,where sc:=d/2-2/(m-1)is the scaling critical regularity of 4NLS with the second order derivative nonlinearities.Our proof relies on the nonlinear estimates in a new M-norm and the stability theory in the probabilistic setting.Similar supercritical global well-posedness results also hold for d=2,m≥4 and d≥3,3≤m<5. 展开更多
关键词 fourth-order Schrodinger equation random initial data almost sure global well-posedness M-norm stability theory
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Fourth-Order Predictive Modelling: II. 4th-BERRU-PM Methodology for Combining Measurements with Computations to Obtain Best-Estimate Results with Reduced Uncertainties
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作者 Dan Gabriel Cacuci 《American Journal of Computational Mathematics》 2023年第4期439-475,共37页
This work presents a comprehensive fourth-order predictive modeling (PM) methodology that uses the MaxEnt principle to incorporate fourth-order moments (means, covariances, skewness, kurtosis) of model parameters, com... This work presents a comprehensive fourth-order predictive modeling (PM) methodology that uses the MaxEnt principle to incorporate fourth-order moments (means, covariances, skewness, kurtosis) of model parameters, computed and measured model responses, as well as fourth (and higher) order sensitivities of computed model responses to model parameters. This new methodology is designated by the acronym 4<sup>th</sup>-BERRU-PM, which stands for “fourth-order best-estimate results with reduced uncertainties.” The results predicted by the 4<sup>th</sup>-BERRU-PM incorporates, as particular cases, the results previously predicted by the second-order predictive modeling methodology 2<sup>nd</sup>-BERRU-PM, and vastly generalizes the results produced by extant data assimilation and data adjustment procedures. 展开更多
关键词 fourth-order Predictive Modeling Data Assimilation Data Adjustment Uncertainty Quantification Reduced Predicted Uncertainties
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Fourth-Order Predictive Modelling: I. General-Purpose Closed-Form Fourth-Order Moments-Constrained MaxEnt Distribution
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作者 Dan Gabriel Cacuci 《American Journal of Computational Mathematics》 2023年第4期413-438,共26页
This work (in two parts) will present a novel predictive modeling methodology aimed at obtaining “best-estimate results with reduced uncertainties” for the first four moments (mean values, covariance, skewness and k... This work (in two parts) will present a novel predictive modeling methodology aimed at obtaining “best-estimate results with reduced uncertainties” for the first four moments (mean values, covariance, skewness and kurtosis) of the optimally predicted distribution of model results and calibrated model parameters, by combining fourth-order experimental and computational information, including fourth (and higher) order sensitivities of computed model responses to model parameters. Underlying the construction of this fourth-order predictive modeling methodology is the “maximum entropy principle” which is initially used to obtain a novel closed-form expression of the (moments-constrained) fourth-order Maximum Entropy (MaxEnt) probability distribution constructed from the first four moments (means, covariances, skewness, kurtosis), which are assumed to be known, of an otherwise unknown distribution of a high-dimensional multivariate uncertain quantity of interest. This fourth-order MaxEnt distribution provides optimal compatibility of the available information while simultaneously ensuring minimal spurious information content, yielding an estimate of a probability density with the highest uncertainty among all densities satisfying the known moment constraints. Since this novel generic fourth-order MaxEnt distribution is of interest in its own right for applications in addition to predictive modeling, its construction is presented separately, in this first part of a two-part work. The fourth-order predictive modeling methodology that will be constructed by particularizing this generic fourth-order MaxEnt distribution will be presented in the accompanying work (Part-2). 展开更多
关键词 Maximum Entropy Principle fourth-order Predictive Modeling Data Assimilation Data Adjustment Reduced Predicted Uncertainties Model Parameter Calibration
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Domain-based noise removal method using fourth-order partial differential equation
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作者 曾维理 谭湘花 路小波 《Journal of Southeast University(English Edition)》 EI CAS 2011年第2期154-158,共5页
Due to the fact that the fourth-order partial differential equation (PDE) for noise removal can provide a good trade-off between noise removal and edge preservation and avoid blocky effects often caused by the secon... Due to the fact that the fourth-order partial differential equation (PDE) for noise removal can provide a good trade-off between noise removal and edge preservation and avoid blocky effects often caused by the second-order PDE, a domain-based fourth-order PDE method for noise removal is proposed. First, the proposed method segments the image domain into two domains, a speckle domain and a non-speckle domain, based on the statistical properties of isolated speckles in the Laplacian domain. Then, depending on the domain type, different conductance coefficients in the proposed fourth-order PDE are adopted. Moreover, the frequency approach is used to determine the optimum iteration stopping time. Compared with the existing fourth-order PDEs, the proposed fourth-order PDE can remove isolated speckles and keeps the edges from being blurred. Experimental results show the effectiveness of the proposed method. 展开更多
关键词 fourth-order partial differential equation conductance coefficient speckle domain image denoising
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Existence of solutions and positive solutions to a fourth-order two-point BVP with second derivative 被引量:12
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作者 姚庆六 《Journal of Zhejiang University-Science A(Applied Physics & Engineering)》 SCIE EI CAS CSCD 2004年第3期104-108,共5页
Several existence theorems were established for a nonlinear fourth-order two-point boundary value problem with second derivative by using Leray-Schauder fixed point theorem, equivalent norm and technique on system of ... Several existence theorems were established for a nonlinear fourth-order two-point boundary value problem with second derivative by using Leray-Schauder fixed point theorem, equivalent norm and technique on system of integral equations. The main conditions of our results are local. In other words, the existence of the solution can be determined by considering the height of the nonlinear term on a bounded set. This class of problems usually describes the equilibrium state of an elastic beam which is simply supported at both ends. 展开更多
关键词 Nonlinear fourth-order equation Two-point boundary value problem Solution and positive solution EXISTENCE Fixed point theorem
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An Efficient Technique for Finding the Eigenvalues of Fourth-Order Sturm-Liouville Problems 被引量:5
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作者 Mohamed El-Gamel Mona Sameeh 《Applied Mathematics》 2012年第8期920-925,共6页
In this work, we present a computational method for solving eigenvalue problems of fourth-order ordinary differential equations which based on the use of Chebychev method. The efficiency of the method is demonstrated ... In this work, we present a computational method for solving eigenvalue problems of fourth-order ordinary differential equations which based on the use of Chebychev method. The efficiency of the method is demonstrated by three numerical examples. Comparison results with others will be presented. 展开更多
关键词 Chebychev POLYNOMIAL fourth-order STURM-LIOUVILLE EIGENVALUE
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A Fourth-order Covergence Newton-type Method 被引量:3
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作者 WANG Xia ZHAO Ling-ling 《Chinese Quarterly Journal of Mathematics》 CSCD 北大核心 2008年第4期589-593,共5页
A fourth-order convergence method of solving roots for nonlinear equation, which is a variant of Newton's method given. Its convergence properties is proved. It is at least fourth-order convergence near simple roots ... A fourth-order convergence method of solving roots for nonlinear equation, which is a variant of Newton's method given. Its convergence properties is proved. It is at least fourth-order convergence near simple roots and one order convergence near multiple roots. In the end, numerical tests are given and compared with other known Newton and Newton-type methods. The results show that the proposed method has some more advantages than others. It enriches the methods to find the roots of non-linear equations and it is important in both theory and application. 展开更多
关键词 Newton iteration method root-finding method fourth-order convergence numerical test
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POSITIVE SOLUTIONS OF FOURTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS 被引量:2
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作者 马如云 张凤然 《Acta Mathematica Scientia》 SCIE CSCD 1998年第S1期124-128,共5页
Under suitable conditions on h(x) and f(u), the authors show that the following boundary value problem has at least one positive solution. Moreover, the authors also establish several existence theorems of multiple po... Under suitable conditions on h(x) and f(u), the authors show that the following boundary value problem has at least one positive solution. Moreover, the authors also establish several existence theorems of multiple positive solutions. 展开更多
关键词 fourth-order BVP positive solution CONE fixed point
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Symmetry Reductions of Cauchy Problems for Fourth-Order Quasi-Linear Parabolic Equations 被引量:2
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作者 李吉娜 张顺利 苏敬蕊 《Communications in Theoretical Physics》 SCIE CAS CSCD 2010年第1期28-36,共9页
This paper is devoted to studying symmetry reduction of Cauchy problems for the fourth-order quasi-linear parabolic equations that admit certain generalized conditional symmetries (GCSs). Complete group classificati... This paper is devoted to studying symmetry reduction of Cauchy problems for the fourth-order quasi-linear parabolic equations that admit certain generalized conditional symmetries (GCSs). Complete group classification results are presented, and some examples are given to show the main reduction procedure. 展开更多
关键词 fourth-order quasi-linear parabolic equation symmetry reduction Cauchy problem
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Symmetry Reduction and Cauchy Problems for a Class of Fourth-Order Evolution Equations 被引量:2
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作者 LI Ji-Na ZHANG Shun-Li 《Communications in Theoretical Physics》 SCIE CAS CSCD 2008年第7期31-38,共8页
We exploit higher-order conditional symmetry to reduce initial-value problems for evolution equations toCauchy problems for systems of ordinary differential equations (ODEs).We classify a class of fourth-order evoluti... We exploit higher-order conditional symmetry to reduce initial-value problems for evolution equations toCauchy problems for systems of ordinary differential equations (ODEs).We classify a class of fourth-order evolutionequations which admit certain higher-order generalized conditional symmetries (GCSs) and give some examples to showthe main reduction procedure.These reductions cannot be derived within the framework of the standard Lie approach,which hints that the technique presented here is something essential for the dimensional reduction of evolu tion equations. 展开更多
关键词 fourth-order evolution equation generalized conditional symmetry Cauchy problem
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Superlinear Fourth-order Elliptic Problem without Ambrosetti and Rabinowitz Growth Condition 被引量:2
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作者 Wei Yuan-hong Chang Xiao-jun +1 位作者 L Yue Li Yong 《Communications in Mathematical Research》 CSCD 2013年第1期23-31,共9页
This paper deals with superlinear fourth-order elliptic problem under Navier boundary condition. By using the mountain pass theorem and suitable truncation, a multiplicity result is established for all λ〉 0 and some... This paper deals with superlinear fourth-order elliptic problem under Navier boundary condition. By using the mountain pass theorem and suitable truncation, a multiplicity result is established for all λ〉 0 and some previous result is extended. 展开更多
关键词 fourth-order elliptic problem variational method mountain pass theorem Navier boundary condition
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Adiabatic Shear Localization for Steels Based on Johnson-Cook Model and Second-and Fourth-Order Gradient Plasticity Models 被引量:2
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作者 WANG Xue-bin 《Journal of Iron and Steel Research International》 SCIE EI CAS CSCD 2007年第5期56-61,共6页
To consider the effects of the interactions and interplay among microstructures, gradient-dependent models of second- and fourth-order are included in the widely used phenomenological Johnson-Cook model where the effe... To consider the effects of the interactions and interplay among microstructures, gradient-dependent models of second- and fourth-order are included in the widely used phenomenological Johnson-Cook model where the effects of strain-hardening, strain rate sensitivity, and thermal-softening are successfully described. The various parameters for 1006 steel, 4340 steel and S-7 tool steel are assigned. The distributions and evolutions of the local plastic shear strain and deformation in adiabatic shear band (ASB) are predicted. The calculated results of the second- and fourth- order gradient plasticity models are compared. S-7 tool steel possesses the steepest profile of local plastic shear strain in ASB, whereas 1006 steel has the least profile. The peak local plastic shear strain in ASB for S-7 tool steel is slightly higher than that for 4340 steel and is higher than that for 1006 steel. The extent of the nonlinear distribution of the local plastic shear deformation in ASB is more apparent for the S-7 tool steel, whereas it is the least apparent for 1006 steel. In fourth-order gradient plasticity model, the profile of the local plastic shear strain in the middle of ASB has a pronounced plateau whose width decreases with increasing average plastic shear strain, leading to a shrink of the portion of linear distribution of the profile of the local plastic shear deformation. When compared with the sec- ond-order gradient plasticity model, the fourth-order gradient plasticity model shows a lower peak local plastic shear strain in ASB and a higher magnitude of plastic shear deformation at the top or base of ASB, which is due to wider ASB. The present numerical results of the second- and fourth-order gradient plasticity models are consistent with the previous numerical and experimental results at least qualitatively. 展开更多
关键词 adiabatic shear band steel gradient-dependent plasticity Johnson-Cook model second-order gradient fourth-order gradient
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Application of the Adomian Decomposition Method (ADM) for Solving the Singular Fourth-Order Parabolic Partial Differential Equation 被引量:1
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作者 Béyi Boukary Justin Loufouilou-Mouyedo +1 位作者 Joseph Bonazebi-Yindoula Gabriel Bissanga 《Journal of Applied Mathematics and Physics》 2018年第7期1476-1480,共5页
In this paper, the ADM method is used to construct the solution of the singular fourth-order partial differential equation.
关键词 SBA Method SINGULAR fourth-order PARTIAL DIFFERENTIAL Equation
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Initial value problem for a class of fourth-order nonlinear wave equations 被引量:1
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作者 陈国旺 侯长顺 Shi-qiang DAI 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2009年第3期391-401,共11页
In this paper, existence and uniqueness of the generalized global solution and the classical global solution to the initial value problem for a class of fourth-order nonlinear wave equations are studied in the fractio... In this paper, existence and uniqueness of the generalized global solution and the classical global solution to the initial value problem for a class of fourth-order nonlinear wave equations are studied in the fractional order Sobolev space using the contraction mapping principle and the extension theorem. The sufficient conditions for the blow up of the solution to the initial value problem are given. 展开更多
关键词 fourth-order nonlinear wave equation initial value problem global solution blow up of solution
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An FEM approximation for a fourth-order variational inequality of second kind 被引量:1
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作者 QIAN Fu-bin DING Rui 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2008年第1期19-24,共6页
A fourth-order variational inequality of the second kind arising in a plate frictional bending problem is considered. By using regularization method, the original problem can be formulated as a differentiable variatio... A fourth-order variational inequality of the second kind arising in a plate frictional bending problem is considered. By using regularization method, the original problem can be formulated as a differentiable variational equation, and the corresponding discrete FEM variational equation is presented afterwards. Abstract error estimates and error estimates of the approximation are derived in terms of energy norm and L^2-norm. 展开更多
关键词 plate theory variational inequality of the second kind fourth-order problem FEM regularization error estimate.
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ASYMPTOTIC SOLUTION OF SINGULAR PERTURBATION PROBLEMS FOR THE FOURTH-ORDER ELLIPTIC DIFFERENTIAL EQUATIONS 被引量:1
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作者 苏煜城 刘国庆 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 1990年第7期637-650,共14页
In this paper we consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the energy estimates of the solutionand its derivatives and construct the formal... In this paper we consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the energy estimates of the solutionand its derivatives and construct the formal asymptotic solution by Lyuternik- Vishik 's method. Finally, by means of the energy estimates we obtain the bound of the remainder of the asymptotic solution. 展开更多
关键词 ASYMPTOTIC SOLUTION OF SINGULAR PERTURBATION PROBLEMS FOR THE fourth-order ELLIPTIC DIFFERENTIAL EQUATIONS
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High Accurate Fourth-Order Finite Difference Solutions of the Three Dimensional Poisson’s Equation in Cylindrical Coordinate 被引量:1
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作者 Alemayehu Shiferaw Ramesh Chand Mittal 《American Journal of Computational Mathematics》 2014年第2期73-86,共14页
In this work, by extending the method of Hockney into three dimensions, the Poisson’s equation in cylindrical coordinates system with the Dirichlet’s boundary conditions in a portion of a cylinder for is solved dire... In this work, by extending the method of Hockney into three dimensions, the Poisson’s equation in cylindrical coordinates system with the Dirichlet’s boundary conditions in a portion of a cylinder for is solved directly. The Poisson equation is approximated by fourth-order finite differences and the resulting large algebraic system of linear equations is treated systematically in order to get a block tri-diagonal system. The accuracy of this method is tested for some Poisson’s equations with known analytical solutions and the numerical results obtained show that the method produces accurate results. 展开更多
关键词 Poisson’s EQUATION Tri-Diagonal Matrix fourth-order FINITE DIFFERENCE APPROXIMATION Hockney’s Method Thomas Algorithm
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CONDITIONS FOR FOURTH-ORDER STATIONARITY AND ERGODICITY OF A HARMONIC RANDOM PROCESS
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作者 Mao Yongcai(institute of Electronic Engineering, Xidian University, Xi’an, 710071) 《Journal of Electronics(China)》 1996年第3期235-241,共7页
The finite data estimates of the complex fourth-order moments of a signal consisting of random harmonics are analyzed. Conditions for the fourth-order stationarity and ergodicity are obtained. Explicit formulas for th... The finite data estimates of the complex fourth-order moments of a signal consisting of random harmonics are analyzed. Conditions for the fourth-order stationarity and ergodicity are obtained. Explicit formulas for the estimation error and its variance, as well as their limiting large sample values are derived. Finally, a special case relevant to cubic phase coupling is considered, and these results are stated for this case, the variance is shown to comprise an ergodic and a nonergodic part. 展开更多
关键词 HARMONIC RANDOM PROCESS fourth-order stationarity fourth-order ERGODICITY
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