Gradient-like Structure Research Articles

Alleviating the strength-ductility trade-off dilemma in high manganese steels after hydrogen charging by adjusting the gradient distribution of twins

The twinning and hydrogen embrittlement (HE) behavior of twinning-induced plasticity (TWIP) steels with various gradient twin structures were investigated by slow strain rate tensile testing. The results revealed that the gradient design can be very beneficial for alleviating the strength-ductility trade-off dilemma in TWIP steels, especially under hydrogen environment. Among the different structures after cathodic hydrogen charging, the most outstanding was the inverse gradient-like structure material, which comprehensively utilized fine recrystallized grains at sample’s edge region to effectively lower HE (EIloss: 4.01% and UTSloss: 1.61%) and gradient-design of the central coarse grain to improve the comprehensive mechanical properties.

Blends of gelatin and hyaluronic acid stratified by stereolithographic bioprinting approximate cartilaginous matrix gradients.

Stereolithographic bioprinting holds great promise in the quest for creating artificial, biomimetic cartilage-like tissue. To introduce a more biomimetic approach, we examined blending and stratifying methacrylated hyaluronic acid (HAMA) and methacrylated gelatin (GelMA) bioinks to mimic the zonal structure of articular cartilage. Bioinks were suspended with porcine chondrocytes before being printed in a digital light processing approach. Homogenous constructs made from hybrid bioinks of varying polymer ratios as well as stratified constructs combining different bioink blends were cultivated over 14 days and analyzed by histochemical staining for proteoglycans/collagen type II, cartilage marker expression analysis, and for cellular viability. The stiffness of blended bioinks increased gradually with HAMA content, from 2.41 ± 0.58 kPa (5% GelMA, 0% HAMA) to 8.84 ± 0.11 kPa (0% GelMA, 2% HAMA). Cell-laden constructs maintained vital chondrocytes and supported the formation of proteoglycans and collagen type II. Higher concentrations of GelMA resulted in increased formation of cartilaginous matrix proteins and a more premature phenotype. However, decreased matrix production in central areas of constructs was observed in higher GelMA content constructs. Biomimetically stratified constructs retained their gradient-like structure even after ECM formation, and exclusively exhibited a significant increase in COL2A1 gene expression (+178%). Concluding, we showed the feasibility of blending and stratifying photopolymerizable, natural biopolymers by SLA bioprinting to modulate chondrocyte attributes and to create zonally segmented ECM structures, contributing to improved modeling of cartilaginous tissue for regenerative therapies or in vitro models.

Continuity of pullback attractors for evolution processes associated with semilinear damped wave equations with time-dependent coefficients

In this paper we consider the semilinear damped wave problem of the form { ( α ( t ) u t ) t − β ( t ) Δ u + γ ( t ) u t + δ ( t ) u = β ( t ) f ( u ) , x ∈ Ω , t > τ , u ( x , t ) = 0 , x ∈ ∂ Ω , t ⩾ τ , u ( x , τ ) = u τ ( x ) , u t ( x , τ ) = v τ ( x ) , x ∈ Ω , where Ω is a bounded smooth domain in R N , N ⩾ 3 , τ ∈ R , f is a real valued function of a real variable with some suitable conditions of growth, regularity and dissipativity, and α , β , γ and δ are continuous real valued functions of a real variable with some suitable conditions of growth, regularity and signs. Using rescaling of time we prove existence, regularity, gradient-like structure, upper and lower semicontinuity of the pullback attractors for the evolution processes associated with this boundary initial value problem in a suitable phase space.

Lower semicontinuity of the pullback attractors of non-autonomous damped wave equations with terms concentrating on the boundary

In this paper we analyze the asymptotic behavior of the pullback attractors for non-autonomous dynamical systems generated by a family of non-autonomous damped wave equations when some reaction terms are concentrated in a neighbourhood of the boundary and this neighbourhood shrinks to boundary as a parameter $\varepsilon$ goes to zero. We show the gradient-like structure of the limit pullback attractor, the existence and continuity of global hyperbolic solutions and the lower semicontinuity of the pullback attractors at $\varepsilon=0$. Finally, we obtain the continuity of the pullback attractors at $\varepsilon=0$.

Vibration transfer characteristic of gradient-like structure based on magnetorheological fluid

In this paper, a gradient-like structure composed of magnetorheological (MR) fluids is proposed, and its vibration transfer characteristic is studied through the modeling, numerical calculation and experimental test. Under the action of an externally applied magnetic field, the MR fluid exhibits the liquid-solid transformation property: the process of transformation between solid and liquid in fact is the change of vibration-transfer impedance. Therefore, based on this property, the gradient-like structure is constructed by controlling the external magnetic field. Based on the wave equation of one-dimensional elastic wave propagation, the wave equation of elastic wave transfer in the gradient-like structure is established. In order to describe the relationship between the complex shear modulus and Lame constant of MR fluid and magnetic field intensity in the wave equation, the equivalent parameter model of MR fluid is established based on the theory of elasticity and viscoelastic materials. Then, the experimental set-up is built to modify this model through experiments. Afterward, the discretization method of continuous medium and transfer matrix method are adopted to solve the wave equation, and the expression of vibration level drop is obtained. Through the numerical calculation, the trend of vibration level drop varying with the frequency of incident elastic wave and the intensity of magnetic field for the gradient-like structure is obtained. Finally, the vibration transfer characteristic of the gradient-like structure is studied experimentally, and the influence of magnetic field intensity on the vibration transfer characteristic of the gradient-like structure is analyzed. The results show that the numerical results are in good accordance with the experimental results, thereby verifying that the numerical model is accurate. And the gradient-like structure has a better attenuation effect on the elastic wave than the MR fluid under the action of a uniform magnetic field, and has an excellent tunable property as well.

A Lyapunov function for fully nonlinear parabolic equations in one spatial variable

Lyapunov functions are used in order to prove stability of equilibria, or to indicate a gradient-like structure of a dynamical system. Zelenyak (1968) and Matano (1988) constructed a Lyapunov function for quasilinear parabolic equations. We modify Matano’s method to construct a Lyapunov function for fully nonlinear parabolic equations under Dirichlet and mixed nonlinear boundary conditions of Robin type.

Global existence and boundedness in a Keller–Segel–Stokes system involving a tensor-valued sensitivity with saturation

This paper deals with a boundary-value problem in two-dimensional smoothly bounded domains for the coupled Keller–Segel–Stokes system{nt+u⋅∇n=Δn−∇⋅(nS(x,n,c)⋅∇c), (x,t)∈Ω×(0,T),ct+u⋅∇c=Δc−c+n, (x,t)∈Ω×(0,T),ut+∇P=Δu+n∇ϕ, (x,t)∈Ω×(0,T),∇⋅u=0, (x,t)∈Ω×(0,T). Here, one of the novelties is that the chemotactic sensitivity S is not a scalar function but rather attains values in R2×2, and satisfies |S(x,n,c)|≤CS(1+n)−α with some CS>0 and α>0. We shall establish the existence of global bounded classical solutions for arbitrarily large initial data. In contrast to the corresponding case of scalar-valued sensitivities, this system does not possess any gradient-like structure due to the appearance of such matrix-valued S. To overcome this difficulty, we will derive a series of a priori estimates involving a new interpolation inequality.To the best of our knowledge, this is the first result on global existence and boundedness in a Keller–Segel–Stokes system with tensor-valued sensitivity, in which production of the chemical signal is involved.

Large-Data Global Generalized Solutions in a Chemotaxis System with Tensor-Valued Sensitivities

The chemotaxis system $u_t=\Delta u - \nabla \cdot (uS(x,u,v)\cdot\nabla v); v_t=\Delta v - uf(v)$ (referred to as ($\star$) in this abstract), for the density $u=u(x,t)$ of a cell population and the concentration $v=v(x,t)$ of an attractive chemical consumed by the former, is considered under no-flux boundary conditions in a bounded domain $\Omega\subset{\mathbb{R}}^n$, $n\ge 1$, with smooth boundary, where $f \in C^1([0,\infty);[0,\infty))$ and $S \in C^2(\bar\Omega\times [0,\infty)^2;{\mathbb{R}}^{n\times n})$ are given functions such that f(0)=0. In contrast to related Keller--Segel-type problems with scalar sensitivities, in the presence of such matrix-valued $S$ the system ($\star$) in general apparently does not possess any useful gradient-like structure. Accordingly, its analysis needs to be based on new types of a priori bounds. Using a spatio-temporal $L^2$ estimate for $\nabla \ln (u+1)$ as a starting point, we derive a series of compactness properties of solutions to suitably regularized vers...

Nonlinear Sturm global attractors: Unstable manifold decompositions as regular CW-complexes

We study global attractors $\mathcal{A}_f$ of scalar partial differential equations $u_t=u_{xx}+f(x,u,u_x)$ on the unit interval with, say, Neumann boundary. Due to nodal properties of differences of solutions, which amount to a nonlinear Sturm property, we call $\mathcal{A}_f$ a Sturm global attractor. We assume all equilibria $v$ to be hyperbolic. Due to a gradient-like structure we can then write \begin{equation} \mathcal{A}_f = \bigcup\limits_{v}\, W^u(v) (*) \end{equation} as a dynamic decomposition into finitely many disjoint invariant sets: the unstable manifolds $W^u(v)$ of the equilibria $v$. Based on our previous Schoenflies result [17], we prove that the dynamic decomposition $(*)$ is in fact a regular finite CW-complex with cells $W^u(v)$, in the Sturm case. We call this complex the regular dynamic complex or Sturm complex of the Sturm attractor $\mathcal{A}_f$. We characterize the planar Sturm complexes by bipolar orientations of their 1-skeletons. We also show that any regular finite CW-complex which is the closure of a single 3-cell arises as a Sturm complex. We include a preliminary discussion of the tetrahedron and the octahedron as Sturm complexes.

The recurrence and the gradient-like structure of a flow

In this paper, some new properties of the nonwandering set, the partial order set, the chain recurrent set and the generalized recurrent set of a flow are proved. A new characterization of the gradient-like part of a flow is given. Some results of Conley are generalized.

Global dynamics of mechanical systems with cubic potentials

We study the behavior of solutions of mechanical systems with polynomial potentials of degree 3 by using a blow up of McGehee type. We first state some general properties for positive degree homogeneous potentials. In particular, we prove a very general property of transversality of the invariant manifolds of the flow along the homothetic orbit. The paper focuses in the study of global flow in the case of homogeneous polynomial potentials of degree 3 for negative energy. The flow is fairly simple because of its gradient-like structure, although for some values of the polynomial coefficients we have diverse behaviour of the separatrices on the infinity manifold, which are essential to describe the global flow.

Large patterns of elliptic systems in infinite cylinders

We consider systems of elliptic equations ∂ t 2 u + Δ x u + γ∂ t u + f( u) = 0, u(t, x) ∈ R N in unbounded cylinders ( t, x) ∈ R × Ω with bounded cross-section Ω ⊂ R n and Dirichlet boundary conditions. We establish existence of bounded solutions u(t, x) with non-trivial dependence on t ∈ R , ∂ t u( t, x) n= 0. Our main assumptions are dissipativity of the nonlinearity f and the existence of at least two t-independent solutions w 1( x), w 2( x) which solve Δ x w j + f( w j ) = 0, j = 1, 2. The proof exploits the dynamical systems structure of the equations: solutions can be translated along the axis of the cylinder. We first prove existence and compactness of attractors for the dynamical system induced by this translation. We then compute Conley indices for cross-sectional Galerkin approximations to conclude that the attractor does not consist of only the two solutions w j ( x), j = 1,2. We also prove existence of solutions converging for t → +∞ or t → −∞. If the system possesses a gradient-like structure, in addition, solutions will converge on both sides of the cylinder.

Gradient-like structure and Morse decompositions for time-periodic one-dimensional parabolic equations

Scalar one-dimensional parabolic equations with periodically time-dependent nonlinearities are considered. For each such equation, the associated discretetime dynamical system is shown to admit Morse decompositions of the global attractor whose Morse sets are contained in a given, arbitrarily small neighborhood of the set of fixed points. Existence of such Morse decompositions implies that the chain recurrent set coincides with the set of fixed points. In particular, the dynamical system has a gradient-like structure. As an application of these results, a description of the asymptotic behavior of solutions of asymptotically periodic equations is given: Any bounded solution approaches a set of periodic solutions of the limiting equation. Other possible applications to nonlocal equations and thin domain problems are discussed.

Existence of dichotomies and invariant splittings for linear differential systems, II

This paper is primarily concerned with linear time-varying ordinary differential equations. Sufficient conditions are given for the existence of an exponential dichotomy or equivalently an invariant splitting. The conditions are more general than those given in Part I of this paper and include the case in which the coefficients lie in a base space which is chain-recurrent under the translation flow and also the case in which compatible splittings are known to exist over invariant subsets of the base space. When the compatibility fails, the flow in the base space is shown to exhibit a gradient-like structure with attractors and repellers. Sufficient conditions are given guaranteeing the existence of bounded solutions of a linear system. The problem is treated in the unified setting of a skew-product dynamical system and the results apply to discrete systems including those generated by diffeomorphisms of manifolds. Sufficient conditions are given for a diffeomorphism to be an Anosov diffeomorphism.