Modeling of Cracks with Nonlinear Effects at the Tip Zones and the Generalized Energy Criterion

Starting with a plane anisotropic homogeneous elasticity problem in a domain with an interior crack, we develop a mathematical frame where nonlinear effects in the tip zones like crack kinking or plastic zones can be modeled in an enlarged state space with the help of additional conditions at the crack tips. Using generalized Green’s formulae, we show that the solutions to these problems turn out to minimize energy functionals which contain terms additional to the classical elastic energy and work of external forces. They can be interpreted as performed work and energy stored in the crack tips. Within the theory of matched asymptotic expansions, the general type of these energy functionals can be characterized in a form applicable to mechanical problems.     

Two‐objective optimization strategies using the adjoint method and game theory in inverse natural convection problems

This paper considers the problem of estimating the strengths of two time‐varying heat sources simultaneously, from measurements of the temperature inside the square domain in a porous medium, when prior knowledge of the source functions is not available. This problem is an inverse natural convection problem. In order to circumvent this problem, we define several optimization criteria (objective functionals) that measure discrepancies between model and measured data, where objective functionals depend on two heat sources and use multi‐criteria optimization to identify Nash equilibria, which are solutions to the non‐cooperative game according to game theory. Two non‐cooperative game strategies are considered: competitive (Nash) game and hierarchical (modified Stackelberg) game. The methodology that we employ relies on a combination of mixed finite element space approximations, finite difference time discretizations, adjoint equation and sensitivity equation techniques, and nonlinear conjugate gradient algorithms for the solutions of estimating two heat sources. Applying the Sobolev gradient for the noise removal is investigated. The performance of the present technique of inverse analysis is evaluated, by means of several numerical experiments, and is found to be very accurate as well as efficient. Copyright © 2012 John Wiley & Sons, Ltd.     

A boundary-field equation method for a nonlinear exterior elasticity problem in the plane

This paper presents a new method for solving a nonlinear exterior boundary value problem arising in two-dimensional elasto-plasticity. The procedure is based on the introduction of a sufficiently large circle that divides the exterior domain into a bounded region and an unbounded one. This allows us to consider the Dirichlet-Neumann mapping on the circle, which provides an explicit formula for the stress in terms of the displacement by using an appropriate infinite Fourier series. In this way we can reduce the original problem to an equivalent nonlinear boundary value problem on the bounded domain with a natural boundary condition on the circle. Hence, the resulting weak formulation includes boundary and field terms, which yields the so called boundary-field equation method. Next, we employ the finite Fourier series to obtain a sequence of approximating nonlinear problems from which the actual Galerkin schemes are derived. Finally, we apply some tools from monotone operators to prove existence, uniqueness and approximation results, including Cea type error estimates for the corresponding discrete solutions.     

Dynamic stability of weak equations of continuous systems

The stability analysis method is developed for distributed dynamic problems with relaxed assumptions imposed on solutions. The problem is motivated by structural vibrations with external time-dependent parametric excitations which are controlled using surface-mounted or -embedded actuators and sensors. The strong form of equations involves irregularities which lead to computational difficulties for estimation and control problems. In order to avoid irregular terms resulting from differentiation of the force and moment terms the dynamics equations are written in a weak form. The weak form of dynamics equations of linear mechanical structures is obtained using variational calculus. The study of stability of stochastic weak system is based on examining properties of Liapunov functional along a weak solution.     

Asymptotic Linear Stability of Solitary Water Waves

In this paper, we study the well-posedness of Cahn–Hilliard equations with degenerate phase-dependent diffusion mobility. We consider a popular form of the equations which has been used in phase field simulations of phase separation and microstructure evolution in binary systems. We define a notion of weak solutions for the nonlinear equation. The existence of such solutions is obtained by considering the limits of Cahn–Hilliard equations with non-degenerate mobilities.     

A control problem for Burgers' equation with bounded input/output

A stabilization problem for Burgers' equation is considered. Using linearization, various controllers are constructed which minimize certain weighted energy functionals. These controllers produce the desired degree of stability for the closed-loop nonlinear system. A numerical scheme for computing the feedback gain functional is developed and several numerical experiments are performed to illustrate the theoretical results.     

A Class of High Order Nonlocal Operators

We study a class of nonlocal operators that may be seen as high order generalizations of the well known nonlocal diffusion operators. We present properties of the associated nonlocal functionals and nonlocal function spaces including nonlocal versions of Sobolev inequalities such as the nonlocal Poincaré and nonlocal Gagliardo–Nirenberg inequalities. Nonlocal characterizations of high order Sobolev spaces in the spirit of Bourgain–Brezis–Mironescu are provided. Applications of nonlocal calculus of variations to the well-posedness of linear nonlocal models of elastic beams and plates are also considered.     

Nonlinear Schrodinger equation with combined power-type nonlinearities and harmonic potential

This paper discusses a class of nonlinear Schrodinger equations with combined power-type nonlinearities and harmonic potential.By constructing a variational problem the potential well method is applied.The structure of the potential well and the properties of the depth function are given.The invariance of some sets for the problem is shown.It is proven that,if the initial data are in the potential well or out of it,the solutions will lie in the potential well or lie out of it,respectively.By the convexity method,the sharp condition of the global well-posedness is given.     

Nonlinear Schrdinger equation with combined power-type nonlinearities and harmonic potential

This paper discusses a class of nonlinear Schrdinger equations with combined power-type nonlinearities and harmonic potential. By constructing a variational problem the potential well method is applied. The structure of the potential well and the properties of the depth function are given. The invariance of some sets for the problem is shown. It is proven that, if the initial data are in the potential well or out of it, the solutions will lie in the potential well or lie out of it, respectively. By the convexity method, the sharp condition of the global well-posedness is given.     

非稳态Hamilton-Jacobi方程的7阶加权紧致非线性格式

Hamilton-Jacobi (HJ) 方程是一类重要的非线性偏微分方程, 在物理学、流体力学、图像处理、微分几何、金融数学、最优化控制理论等方面有着广泛的应用. 由于HJ方程的弱解存在但不唯一, 且解的导数可能出现间断, 导致其数值求解具有一定的难度. 本文提出了非稳态HJ方程的7阶精度加权紧致非线性格式 (WCNS). 该格式结合了Hamilton函数的Lax-Friedrichs型通量分裂方法和一阶空间导数左、右极限值的高阶精度混合节点和半节点型中心差分格式. 基于7点全局模板和4个4点子模板推导了半节点函数值的高阶线性逼近和4个低阶线性逼近, 以及全局模板和子模板的光滑度量指标. 为避免间断附近数值解产生非物理振荡以及提高格式稳定性, 采用WENO型非线性插值方法计算半节点函数值. 时间离散采用3阶TVD型Runge-Kutta方法. 通过理论分析验证了WCNS格式对于光滑解具有最佳的7阶精度. 为方便比较, 经典的7阶WENO格式也被推广用于求解HJ方程. 数值结果表明, 本文提出的WCNS格式能够很好地模拟HJ方程的精确解, 且在光滑区域能够达到7阶精度; 与经典的同阶WENO格式相比, WCNS格式在精度、收敛性和分辨率方面更优, 计算效率略高.      

Complex-potential method in the nonlinear theory of elasticity

For materials characterized by a linear relation between Almansi strains and Cauchy stresses, relations between stresses and complex potentials are obtained and the plane static problem of the theory of elasticity is thus reduced to a boundary-value problem for the potentials. The resulting relations are nonlinear in the potentials; they generalize well-known Kolosov's formulas of linear elasticity. A condition under which the results of the linear theory of elasticity follow from the nonlinear theory considered is established. An approximate solution of the nonlinear problem for the potentials is obtained by the small-parameter method, which reduces the problem to a sequence of linear problems of the same type, in which the zeroth approximation corresponds to the problem of linear elasticity. The method is used to obtain both exact and approximate solutions for the problem of the extension of a plate with an elliptic hole. In these solutions, the behavior of stresses on the hole contour is illustrated by graphs. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1, pp. 133–143, January–February, 2000.     

Variational principles and optimal solutions of the inverse problems of creep bending of plates

It is shown that inverse problems of steady-state creep bending of plates in both the geometrically linear and nonlinear formulations can be represented in a variational formulation. Steady-state values of the obtained functionals corresponding to the solutions of the problems of inelastic deformation and elastic unloading are determined by applying a finite element procedure to the functionals. Optimal laws of creep deformation are formulated using the criterion of minimizing damage in the functionals of the inverse problems. The formulated problems are reduced to the problems solved by the finite element method using MSC.Marc software.     

Maximal Dissipation and Well-posedness for the Compressible Euler System

We discuss the problem of well-posedness of the compressible (barotropic) Euler system in the framework of weak solutions. The principle of maximal dissipation introduced by C.M. Dafermos is adapted and combined with the concept of admissible weak solutions. We use the method of convex integration in the spirit of the recent work of C.DeLellis and L.Székelyhidi to show various counterexamples to well-posedness. On the other hand, we conjecture that the principle of maximal dissipation should be retained as a possible criterion of uniqueness as it is violated by the oscillatory solutions obtained in the process of convex integration.     

THE MATHEMATICAL MODELLING OF NEAR COAST SHALLOW WATER CIRCULATION

This paper presents a finite element method to solve the shallow water circulation problem numerically. Considering the Coriolis effect, bottom friction and eddy viscosity, the continuity equation and momentum equation are integrated vertically. Using Galerkin weighted residual method, the weak variational formulation is derived for the finite element analysis. The split-time method is applied for the numerical integration instead of iteration for nonlinear terms. Moreover, an artificial smooth approach is proposed to suppress the short wavelength noise. In order to save Computer storage units, a densed storage scheme is set up, where all the zero elements in large scaled and sparse matrices are excluded.     

Linear <Emphasis Type="Italic">Γ</Emphasis>-limits of multiwell energies in nonlinear elasticity theory

We derive linearized theories from nonlinear elasticity theory for multiwell energies. Under natural assumptions on the nonlinear stored energy densities, the properly rescaled nonlinear energy functionals are shown to Γ-converge to the relaxation of a corresponding linearized model. Minimizing sequences of problems with displacement boundary conditions and body forces are investigated and found to correspond to minimizing sequences of the linearized problems. As applications of our results we discuss the validity and failure of a formula that is widely used to model multiwell energies in the regime of linear elasticity. Applying our convergence results to the special case of single well densities, we also obtain a new strong convergence result for the sequence of minimizers of the nonlinear problem.      

A class of higher order compact schemes for the unsteady two‐dimensional convection–diffusion equation with variable convection coefficients

A class of higher order compact (HOC) schemes has been developed with weighted time discretization for the two‐dimensional unsteady convection–diffusion equation with variable convection coefficients. The schemes are second or lower order accurate in time depending on the choice of the weighted average parameter μ and fourth order accurate in space. For 0.5?μ?1, the schemes are unconditionally stable. Unlike usual HOC schemes, these schemes are capable of using a grid aspect ratio other than unity. They efficiently capture both transient and steady solutions of linear and nonlinear convection–diffusion equations with Dirichlet as well as Neumann boundary condition. They are applied to one linear convection–diffusion problem and three flows of varying complexities governed by the two‐dimensional incompressible Navier–Stokes equations. Results obtained are in excellent agreement with analytical and established numerical results. Overall the schemes are found to be robust, efficient and accurate. Copyright © 2002 John Wiley & Sons, Ltd.     

Isotropic Hypoellipticity and Trend to Equilibrium for the Fokker-Planck Equation with a High-Degree Potential

We consider the Fokker-Planck equation with a confining or anti-confining potential which behaves at infinity like a possibly high-degree homogeneous function. Hypoellipticity techniques provide the well-posedness of the weak Cauchy problem in both cases as well as instantaneous smoothing and exponential trend to equilibrium. Lower and upper bounds for the rate of convergence to equilibrium are obtained in terms of the lowest positive eigenvalue of the corresponding Witten Laplacian, with detailed applications.     

基于泰勒展开的二阶层次分析法初探

经典AHP常用于武器装备效能或能力评估,然而经典AHP采用线性加权模型,即通过底层指标效能线性加权得到顶层指标效能,对非线性体系往往会引入较大评估误差,易产生决策性错误。为解决这一问题,本文以底层指标效能值等于1/2为展开点,给出基于泰勒展开的二阶加权模型,对线性体系,二阶加权模型可退化为经典线性加权模型;推导出二阶AHP的数学表达式,给出二阶AHP效能评估步骤,构建二阶AHP的基本框架。算例表明,二阶AHP评估结果相比经典AHP更接近于解析解,产生决策性错误的机率远小于经典AHP。     

Finite-time non-fragile boundary feedback control for a class of nonlinear parabolic systems

In this paper, the finite-time non-fragile boundary feedback control problem is investigated for a class of nonlinear parabolic systems, where both the multiplicative and additive controller gain variations are considered to describe the actuator parameter perturbation. Non-fragile boundary control strategies are designed with respect to two controller gain variations via collocated or non-collocated boundary measurement, respectively. In light of the finite-time stability and Lyapunov-based techniques, some sufficient conditions are presented in terms of linear matrix inequalities such that the resulting closed-loop system is well-posedness and practically finite-time stable. Finally, numerical examples are given to verify the effectiveness of the proposed design method.