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"Structural stability Mathematics."
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Optimization and anti-optimization of structures under uncertainty
The volume presents a collaboration between internationally recognized experts on anti-optimization and structural optimization, and summarizes various novel ideas, methodologies and results studied over 20 years. The book vividly demonstrates how the concept of uncertainty should be incorporated in a rigorous manner during the process of designing real-world structures. The necessity of anti-optimization approach is first demonstrated, then the anti-optimization techniques are applied to static, dynamic and buckling problems, thus covering the broadest possible set of applications. Finally, anti-optimization is fully utilized by a combination of structural optimization to produce the optimal design considering the worst-case scenario. This is currently the only book that covers the combination of optimization and anti-optimization. It shows how various optimization techniques are used in the novel anti-optimization technique, and how the structural optimization can be exponentially enhanced by incorporating the concept of worst-case scenario, thereby increasing the safety of the structures designed in various fields of engineering.
eBook
Shadowing and structural stability for operators
A well-known result in the area of dynamical systems asserts that any invertible hyperbolic operator on any Banach space is structurally stable. This result was originally obtained by Hartman in 1960 for operators on finite-dimensional spaces. The general case was independently obtained by Palis and Pugh around 1968. We will exhibit a class of examples of structurally stable operators that are not hyperbolic, thereby showing that the converse of the above-mentioned result is false in general. We will also prove that an invertible operator on a Banach space is hyperbolic if and only if it is expansive and has the shadowing property. Moreover, we will show that if a structurally stable operator is expansive, then it must be uniformly expansive. Finally, we will characterize the weighted shifts on the spaces
$c_{0}(\\mathbb{Z})$
and
$\\ell _{p}(\\mathbb{Z})$
(
$1\\leq p<\\infty$
) that satisfy the shadowing property.
Journal Article
Instability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs
Consider a general linear Hamiltonian system
eBook
A new directional stability transformation method of chaos control for first order reliability analysis
The HL-RF iterative algorithm of the first order reliability method (FORM) is popularly applied to evaluate reliability index in structural reliability analysis and reliability-based design optimization. However, it sometimes suffers from non-convergence problems, such as bifurcation, periodic oscillation, and chaos for nonlinear limit state functions. This paper derives the formulation of the Lyapunov exponents for the HL-RF iterative algorithm in order to identify these complicated numerical instability phenomena of discrete chaotic dynamic systems. Moreover, the essential cause of low efficiency for the stability transform method (STM) of convergence control of FORM is revealed. Then, a novel method, directional stability transformation method (DSTM), is proposed to reduce the number of function evaluations of original STM as a chaos feedback control approach. The efficiency and convergence of different reliability evaluation methods, including the HL-RF algorithm, STM and DSTM, are analyzed and compared by several numerical examples. It is indicated that the proposed DSTM method is versatile, efficient and robust, and the bifurcation, periodic oscillation, and chaos of FORM is controlled effectively.
Journal Article
Dynamic stability analysis of porous functionally graded beams under hygro-thermal loading using nonlocal strain gradient integral model
We present a study on the dynamic stability of porous functionally graded (PFG) beams under hygro-thermal loading. The variations of the properties of the beams across the beam thicknesses are described by the power-law model. Unlike most studies on this topic, we consider both the bending deformation of the beams and the hygro-thermal load as size-dependent, simultaneously, by adopting the equivalent differential forms of the well-posed nonlocal strain gradient integral theory (NSGIT) which are strictly equipped with a set of constitutive boundary conditions (CBCs), and through which both the stiffness-hardening and stiffness-softening effects of the structures can be observed with the length-scale parameters changed. All the variables presented in the differential problem formulation are discretized. The numerical solution of the dynamic instability region (DIR) of various bounded beams is then developed via the generalized differential quadrature method (GDQM). After verifying the present formulation and results, we examine the effects of different parameters such as the nonlocal/gradient length-scale parameters, the static force factor, the functionally graded (FG) parameter, and the porosity parameter on the DIR. Furthermore, the influence of considering the size-dependent hygro-thermal load is also presented.
Journal Article
A low-frequency pure metal metamaterial absorber with continuously tunable stiffness
To address the incompatibility between high environmental adaptability and deep subwavelength characteristics in conventional local resonance metamaterials, and overcome the deficiencies in the stability of existing active control techniques for band gaps, this paper proposes a design method of pure metal vibration damping metamaterial with continuously tunable stiffness for wideband elastic wave absorption. We design a dual-helix narrow-slit pure metal metamaterial unit, which possesses the triple advantage of high spatial compactness, low stiffness characteristics, and high structural stability, enabling the opening of elastic flexural band gaps in the low-frequency range. Similar to the principle of a sliding rheostat, the introduction of continuously sliding plug-ins into the helical slits enables the continuous variation of the stiffness of the metamaterial unit, achieving a continuously tunable band gap effect. This successfully extends the effective band gap by more than ten times. The experimental results indicate that this metamaterial unit can be used as an additional vibration absorber to absorb the low-frequency vibration energy effectively. Furthermore, it advances the metamaterial absorbers from a purely passive narrowband design to a wideband tunable one. The pure metal double-helix metamaterials retain the subwavelength properties of metamaterials and are suitable for deployment in harsh environments. Simultaneously, by adjusting its stiffness, it substantially broadens the effective band gap range, presenting promising potential applications in various mechanical equipment operating under adverse conditions.
Journal Article
Existence and Stability of Fuzzy Slightly Altruistic Equilibrium for a Class of Generalized Multiobjective Fuzzy Games
We mainly study the existence, structural stability and robustness of fuzzy slightly altruistic equilibria for a class of generalized multiobjective fuzzy games which are expressed as ϖ. Firstly, we introduce the concept of fuzzy slightly altruistic equilibrium and prove the existence of equilibrium for the ϖ by Fan–Glicksberg fixed point theorem. Secondly, the connections between ϖ and bounded rationality are discussed by an abstract rationality functions. Moreover, we construct the problem space of ϖ which is represented by Λ and show that most ϖ∈Λ are structurally stable and robust to ε-equilibrium on the meaning of Baire category. These results are new and extend some existing results in recent literature.
Journal Article
Buckling phenomenon of vertical beam/column of variable density carrying a top mass
This study focuses on modeling ideal nonuniform standing beams and towers supporting a constant top mass. We also analyze their dynamical stability, as determining the design parameters influencing their shape and stability holds significant value for structural engineering. Initially, we employ a statical mechanics approach to balance the mechanical and gravitational forces. By solving an initial-value problem, we derive the cross-sectional areas of the columns. Our findings reveal that these areas, rather than the shapes, are the primary contributors to the engineering performance of the columns. Additionally, the top mass acts as a multiplying factor for the cross-sectional areas, and the density distribution along the column determines whether the top should be heavier or lighter. Furthermore, we demonstrate that exponential, parabolic, or linear cross-sections with significantly wider base profiles are crucial for accommodating heavier top loads. Moving on to the dynamical analysis, we consider two ideal tower configurations: FC and SC. Numerical and analytical results reveal that higher modes exhibit shorter amplitudes. FC modes necessitate higher design parameters to resist buckling phenomena, whereas SC modes show lower resistance to vibrational deflections. In terms of stability, a heavier top mass enhances the vertical beam’s stability, while towers with parabolic bases are more susceptible to instabilities.
Journal Article
Long-Time Behavior of a Nonlinearly-Damped Three-Layer Rao–Nakra Sandwich Beam
In this paper, a three-layer Rao–Nakra sandwich beam is considered where the core viscoelastic layer is constrained by the purely elastic or piezoelectric outer layers. In the model, uniform bending motions of the overall laminate are coupled to the longitudinal motions of the outer layers, and the shear of the middle layer contributes to the overall motion. Together with nonlinear damping injection and nonlinear source terms, the existence and uniqueness of local and global weak solutions are obtained by the nonlinear semigroup theory and the theory of monotone operators. The global existence of potential well solutions and the uniform energy decay rates of such a solution, given as a solution to a certain nonlinear ODE, are shown are proved under certain assumptions of the parameters and by the Nehari manifold. Finally, the existence of a smooth global attractor with finite fractal dimension, which is characterized as an unstable manifold of the set of stationary solutions, and exponential attractors for the associated dynamical system are proved. The present paper extends the linear analysis of the stability of the Rao–Nakra sandwich beam to nonlinear analysis in the existing literature.
Journal Article