Convex Gradient Research Articles

Optimizing Fatigue Performance in Gradient Structural Steels by Manipulating the Grain Size Gradient Rate.

As a new type of high-performance material, gradient structural steel is widely used in engineering fields due to its unique microstructure and excellent mechanical properties. For the prevalent fatigue failure problem, the rate of change in the local grain size gradients along the structure (referred to as the gradient rate) is a key parameter in the design of gradient structures, which significantly affects the fatigue performance of gradient structural steel. In this study, a new method of 'Voronoi primary + secondary modeling' is adopted to successfully establish three typical high-strength steel models corresponding to the convex-, linear-, and concave-type gradient rates for gradient structures, focusing on the stress-strain response and crack propagation in structural steel with different gradient rates under cyclic loading. It was found that the concave gradient rate structural model is dominated by finer grains with larger volume fraction, which is conducive to hindering fatigue crack propagation and has the longest fatigue life, which is 16.16% longer than that of the linear gradient rate structure and 23.66% longer than that of the convex gradient rate structure. The simulation results in this study are consistent with the relevant experimental phenomena. Therefore, when regulating the gradient rate, priority should be given to increasing the volume fraction of fine grains and designing a gradient rate structure dominated by fine grains to improve the fatigue life of the material. This study presents a new strategy for designing engineering materials with better service performance.

Welding Penetration Monitoring for Ship Robotic GMAW Using Arc Sound Sensing Based on Improved Wavelet Denoising

The arc sound signal is one of the most important aspects of information related to pattern identification regarding the penetration state of ship robotic GMAW; however, arc sound is inevitably affected by noise interference during the signal acquisition process. In this paper, an improved wavelet threshold denoising method is proposed to eliminate interference and purify the arc sound signal. The non-stationary random distribution characteristics of GMAW noise interference are also estimated by using the high-frequency detail coefficients in different domains after wavelet transformation, and a mode of measuring scale that is logarithmically negatively correlated with the wavelet decomposition scale is created to update the threshold. The gradient convergent threshold function is established using the natural logarithmic function structure and concave–convex gradient to enable the nonlinear adjustment of the asymptotic rate. Further, some property theorems related to the optimized threshold function are proposed and theoretically proven, and the effectiveness and adaptability of the improved method are verified via the denoising simulation of speech synthesis signals. The four traditional denoising methods and our improved version are applied in the pretreatment of the GMAW arc sound signal, respectively. Statistical analysis and short-time Fourier transform are used to extract eight-dimensional time and frequency domain feature parameters from the denoised signals with randomly time-varying characteristics, and the extracted joint feature parameters are used to establish a nonlinear mapping model of penetration state identification for ship robotic GMAW using the pattern classifiers of RBFNN, PNN and PSO-SVM. The simulation results yielded by visual penetration classification and the multi-dimensional evaluation index of the confusion matrix indicate that the improved denoising method proposed in this paper achieves a higher accuracy in the extraction of penetration state features and greater precision in the identification of pattern classification.

Convergence to the thermodynamic limit for random-field random surfaces

We study random surfaces with a uniformly convex gradient interaction in the presence of quenched disorder taking the form of a random independent external field. Previous work on the model has focused on proving existence and uniqueness of infinite-volume gradient Gibbs measures with a given tilt and on studying the fluctuations of the surface and its discrete gradient. In this work we focus on the convergence of the thermodynamic limit, establishing convergence of the finite-volume distributions with Dirichlet boundary conditions to translation-covariant (gradient) Gibbs measures. Specifically, it is shown that, when the law of the random field has finite second moment and is symmetric, the distribution of the gradient of the surface converges in dimensions $d\geq4$ while the distribution of the surface itself converges in dimensions $d\geq 5$. Moreover, a power-law upper bound on the rate of convergence in Wasserstein distance is obtained. The results partially answer a question discussed by Cotar and Kulske

Global optimal regularity for variational problems with nonsmooth non-strictly convex gradient constraints

We prove the optimal W2,∞ regularity for variational problems with convex gradient constraints. We do not assume any regularity of the constraints; so the constraints can be nonsmooth, and they need not be strictly convex. When the domain is smooth enough, we show that the optimal regularity holds up to the boundary. In this process, we also characterize the set of singular points of the viscosity solutions to some Hamilton-Jacobi equations. Furthermore, we obtain an explicit formula for the second derivative of these viscosity solutions; and we show that the second derivatives satisfy a monotonicity property.

Double obstacle problems and fully nonlinear PDE with non-strictly convex gradient constraints

We prove the optimal W2,∞ regularity for fully nonlinear elliptic equations with convex gradient constraints. We do not assume any regularity about the constraints; so the constraints need not be C1 or strictly convex. We also show that the optimal regularity holds up to the boundary. Our approach is to show that these elliptic equations with gradient constraints are related to some fully nonlinear double obstacle problems. Then we prove the optimal W2,∞ regularity for the double obstacle problems. In this process, we also employ the monotonicity property for the second derivative of obstacles, which we have obtained in a previous work.

ELECTROHYDRODYNAMIC MINI FLAT HEAT PIPE FOR ADVANCED ELECTRONICS COOLING

Mini-grooved flat heat pipe with small size, compact structure, high thermal conductivity, and close contact with chips is very promising to be applied for electronic devices cooling under small space and high heat flux condition. However, capillary limit always restricts the heat transfer capacity of heat pipe. Adopting electrohydrodynamic (EHD) technology with simple structure, sensitive control, and low-energy consumption to the heat pipe can lower the reliance of heat pipe on capillary force and improve the heat transfer performance of heat pipe. In this paper, a one-dimensional mathematical model is established for mini-grooved flat heat pipe under dielectrophoretic force to study the working fluid flow and heat transfer along axial direction under EHD effect. The performance of mini-grooved flat heat pipe under different electric field intensity and electrodes distribution is analyzed. It is found that adding electric field to the mini-grooved flat heat pipe can lessen the demand for capillary force on the heat pipe, reduce the required working fluid filling amount, improve the wall temperature uniformity, and enlarge the range of negative working inclined angle, which enhances the heat transfer performance of heat pipe significantly. Under the electric field intensity 5 kV·mm-1, the critical input heat of Case 1 heat pipe is 4.8 times higher than that without electric field. And, when electrodes are arranged at the whole heat pipe with Case 2 convex gradient V-type grooved wick, the heat transfer performance of heat pipe is optimal. Considering the practical condition, fabricating sloped grooved heat pipe under EHD is more convenient. However, the electric field intensity cannot be larger than 5.5 kV·mm-1 in case of reaching entrainment limit and electrical properties of working fluid being destroyed.

Convergence rates for an inertial algorithm of gradient type associated to a smooth non-convex minimization

We investigate an inertial algorithm of gradient type in connection with the minimization of a non-convex differentiable function. The algorithm is formulated in the spirit of Nesterov’s accelerated convex gradient method. We prove some abstract convergence results which applied to our numerical scheme allow us to show that the generated sequences converge to a critical point of the objective function, provided a regularization of the objective function satisfies the Kurdyka–Łojasiewicz property. Further, we obtain convergence rates for the generated sequences and the objective function values formulated in terms of the Łojasiewicz exponent of a regularization of the objective function. Finally, some numerical experiments are presented in order to compare our numerical scheme and some algorithms well known in the literature.

An energy stable Runge–Kutta method for convex gradient problems

We propose a class of Runge–Kutta methods which provide a simple unified framework to solve the gradient flow of a convex functional in an unconditionally energy stable manner. Stiffly accurate Runge–Kutta methods are high order accurate in terms of time and also assure the energy stability for any time step size when they satisfy the positive definite condition. We provide a detailed proof of the unconditional energy stability as well as unique solvability of the proposed scheme. We demonstrate the accuracy and stability of the proposed methods using numerical experiments for a specific example.

Convergence and Non-convergence of Many-Particle Evolutions with Multiple Signs

We address the question of the convergence of evolving interacting particle systems as the number of particles tends to infinity. We consider two types of particles, called positive and negative. Same-sign particles repel each other, and opposite-sign particles attract each other. The interaction potential is the same for all particles, up to the sign, and has a logarithmic singularity at zero. The central example of such systems is that of dislocations in crystals. Because of the singularity in the interaction potential, the discrete evolution leads to blow-up in finite time. We remedy this situation by regularising the interaction potential at a length-scale delta _n>0, which converges to zero as the number of particles n tends to infinity. We establish two main results. The first one is an evolutionary convergence result showing that the empirical measures of the positive and of the negative particles converge to a solution of a set of coupled PDEs which describe the evolution of their continuum densities. In the setting of dislocations these PDEs are known as the Groma–Balogh equations. In the proof we rely on both the theory of lambda -convex gradient flows, to establish a quantitative bound on the distance between the empirical measures and the continuum solution to a delta _n-regularised version of the Groma–Balogh equations, and a priori estimates for the Groma–Balogh equations to pass to the small-regularisation limit in a functional setting based on Orlicz spaces. In order for the quantitative bound not to degenerate too fast in the limit n rightarrow infty we require delta _n to converge to zero sufficiently slowly. The second result is a counterexample, demonstrating that if delta _n converges to zero sufficiently fast, then the limits of the empirical measures of the positive and the negative dislocations do not satisfy the Groma–Balogh equations. These results show how the validity of the Groma–Balogh equations as the limit of many-particle systems depends in a subtle way on the scale at which the singularity of the potential is regularised.

A second-order dynamical approach with variable damping to nonconvex smooth minimization

ABSTRACTWe investigate a second-order dynamical system with variable damping in connection with the minimization of a nonconvex differentiable function. The dynamical system is formulated in the spirit of the differential equation which models Nesterov's accelerated convex gradient method. We show that the generated trajectory converges to a critical point, if a regularization of the objective function satisfies the Kurdyka- Lojasiewicz property. We also provide convergence rates for the trajectory formulated in terms of the Lojasiewicz exponent.

Nanocrystalline gradient engineering: Grain evolution and grain boundary networks

We study materials with spatial gradients in nanoscale grain size (5–120nm), and quantitatively examine the effect of spatial gradient on microstructure evolution and thermal stability using mesoscale Monte Carlo modeling and statistical analysis. The spatial grain size gradient weakens and the grain size distribution widens at elevated temperatures, accompanied by grain rounding and movement of grains along the gradient direction. Introducing heterogeneous grain boundary networks into gradient materials leads to better preservation of the spatial grain size gradient but less equiaxed grains. Coarsening in small grain regions is accompanied by an increase in the local fraction of low-energy grain boundaries, as these are competing mechanisms for reducing total energy, and spatial gradients in grain boundary character distribution and triple junction character distribution develop in the material. We further compare concave, linear, and convex gradient materials with increasing grain size gradient for small grains. Grains in convex gradient materials have the highest grain growth rate compared to grains of the same size in linear and concave gradient materials. The accelerated grain growth in the presence of a steeper grain size gradient is attributed to a change in local grain neighbor environment that promotes grain boundary curvature (and pressure) and enhances the driving force for grain growth.

On Hamilton–Jacobi–Bellman equations with convex gradient constraints

We study PDE of the form \max\{F(D^2u,x)-f(x), H(Du)\}=0 where F is uniformly elliptic and convex in its first argument, H is convex, f is a given function and u is the unknown. These equations are derived from dynamic programming in a wide class of stochastic singular control problems. In particular, examples of these equations arise in mathematical finance models involving transaction costs, in queuing theory, and spacecraft control problems. The main aspects of this work are to identify conditions under which solutions are uniquely defined and have Lipschitz continuous gradients.

On functionally graded composite structures for crashworthiness

The foam-filled thin-walled composite structures have proven an ideal energy absorber in automotive engineering for its extraordinary energy absorption ability and lightweight features. Unlike existing uniform foam and thickness (UFT) structure, this paper introduces functionally graded foam (FGF) to fill into functionally graded thickness (FGT) thin-walled structure, named as double functionally graded (DFG) tube, where different configurations of foam and wall thickness gradients are taken into account. To systematically explore the crashworthiness of DFG structures, first, experimental results were performed to validate finite element (FE) models. Second, a comparison of crashworthiness was carried out for (1) four different DFG structures, (2) four single functionally-graded (SFG) structures and (3) one traditional UFT structure. The results showed that the DFG structures have better energy absorption capacity than the SFG and UFT structures, especially with a convex gradient configuration. In addition, the specific energy absorption (SEA) values of these four DFG structures are fairly close to each other, while their loading responses highly depend on the combination of gradients. Of these DFG structures, Ascending–Ascending configuration exhibits best overall crashworthiness characteristics. Finally, parametric studies were performed and the results indicated that widening the ranges of foam density and tube wall thickness can improve the energy absorption of the Ascending–Ascending DFG structures without increasing the initial peak load. Therefore, the DFG structure of Ascending–Ascending gradient is recommended for a potential absorber.

Sobolev regularity for a class of second order elliptic PDE’s in infinite dimension

We consider an elliptic Kolmogorov equationu Ku = f in a separable Hilbert space H. The Kolmogorov operator K is associated to an infinite dimensional convex gradient system: dX = (AX DU(X))dt + dW(t), where A is a self-adjoint operator in H and U is a convex lower semicontinuous function. Under mild assumptions we prove that for � > 0 and f 2 L 2 (H,�) the weak solution u belongs to the Sobolev space W 2,2 (H,�), whereis the log-concave probability measure of the system. Moreover maximal estimates on the gradient of u are proved. The maximal regularity results are used in the study of perturbed non gradient systems, for which we prove that there exists an invariant measure. The general results are applied to Kolmogorov equations associated to reaction-diffusion and Cahn-Hilliard stochastic PDEs.

Convex gradient optimization for increased spatiotemporal resolution and improved accuracy in phase contrast MRI

Purpose: To evaluate convex gradient optimization (CVX) for increased spatiotemporal resolution and improved accuracy for phase-contrast MRI (PC-MRI). Methods: A conventional flow-compensated and flowencoded (FCFE) PC-MRI sequence was compared with a CVX PC-MRI sequence using numerical simulations, flow phantom experiments, and in vivo experiments. Flow measurements within the ascending aorta, main pulmonary artery, and right/ left pulmonary arteries of normal volunteers (N ¼ 10) were acquired at 3T and analyzed using a conventional FCFE sequence and a CVX sequence with either higher spatial resolution or higher temporal resolution. All sequences mitigated chemical shift–induced phase errors and used equivalent breath-hold durations. Results: Chemical shift–optimized PC-MRI has increased sequence efficiency when using CVX, which can provide either higher spatial or higher temporal resolution compared with conventional FCFE PC-MRI. Numerical simulations, flow phantom experiments, and in vivo experiments indicate that CVX measurements of total flow and peak velocity are increased and more accurate when compared with FCFE. Conclusion: CVX PC-MRI increases sequence efficiency while reducing chemical shift–induced phase errors. This can be used to provide either higher spatial or higher temporal resolution than conventional chemical shift–mitigated PC-MRI methods to provide more accurate measurements of blood flow and peak velocity. Magn Reson Med 72:1552–1564, 2014. V C 2013 Wiley Periodicals, Inc.

Convex gradient optimization for increased spatiotemporal resolution and improved accuracy in phase contrast MRI

To evaluate convex gradient optimization (CVX) for increased spatiotemporal resolution and improved accuracy for phase-contrast MRI (PC-MRI). A conventional flow-compensated and flow-encoded (FCFE) PC-MRI sequence was compared with a CVX PC-MRI sequence using numerical simulations, flow phantom experiments, and in vivo experiments. Flow measurements within the ascending aorta, main pulmonary artery, and right/left pulmonary arteries of normal volunteers (N = 10) were acquired at 3T and analyzed using a conventional FCFE sequence and a CVX sequence with either higher spatial resolution or higher temporal resolution. All sequences mitigated chemical shift-induced phase errors and used equivalent breath-hold durations. Chemical shift-optimized PC-MRI has increased sequence efficiency when using CVX, which can provide either higher spatial or higher temporal resolution compared with conventional FCFE PC-MRI. Numerical simulations, flow phantom experiments, and in vivo experiments indicate that CVX measurements of total flow and peak velocity are increased and more accurate when compared with FCFE. CVX PC-MRI increases sequence efficiency while reducing chemical shift-induced phase errors. This can be used to provide either higher spatial or higher temporal resolution than conventional chemical shift-mitigated PC-MRI methods to provide more accurate measurements of blood flow and peak velocity.

Chromatographic models to predict the elution of ionizable analytes by organic modifier gradient in reversed phase liquid chromatography

The retention of an ionizable analyte under RP-HPLC organic modifier gradient elution is strongly affected by its ionization degree which, in turn, depends on its pKa and on the pH of the mobile phase. The values of both parameters change depending on the mobile phase composition and thus retention becomes a parameter quite difficult to predict, particularly when working in gradient mode. In this work, an equation describing the retention of ionizable analytes has been combined with three different models of different complexity, developed for gradient elution of neutral compounds (1, 2, or 3 fitting parameters), in order to predict retention of compounds with acid–base properties with particular buffers. All models have been tested under 16 different gradient patterns (4 linear gradients, 4 concave gradients, 4 convex gradients and 4 combinations between them) for the prediction of the retention time of 12 acid–base compounds (pKa values from 4 to 9) in 3 different buffered mobile phases (pH 5, pH 7 and pH 9) with acetonitrile as organic modifier. The agreement between the experimental and calculated retention times is good for all models. The best results are obtained through the model that depends on three parameters and the accuracy of the two-parameter model is slightly lower but very acceptable too. On the other hand, the predictions performed with the one-parameter model are the less accurate, but good enough to become a valid model taking into account that it requires very little experimental work.

Convex Hamilton-Jacobi Equations Under Superlinear Growth Conditions on Data

Unbounded stochastic control problems may lead to Hamilton-Jacobi-Bellman equations whose Hamiltonians are not always defined, especially when the diffusion term is unbounded with respect to the control. We obtain existence and uniqueness of viscosity solutions growing at most like o(1+|x|p) at infinity for such HJB equations and more generally for degenerate parabolic equations with a superlinear convex gradient nonlinearity. If the corresponding control problem has a bounded diffusion with respect to the control, then our results apply to a larger class of solutions, namely those growing like O(1+|x|p) at infinity. This latter case encompasses some equations related to backward stochastic differential equations.

Designing the energy absorption capacity of functionally graded foam materials

In this paper, a functionally graded foam model is proposed in order to improve the energy absorption characteristics offered by uniform foams. In this novel model, the characteristics of the foam (e.g., density) are varied through the thickness according to various gradient functions. The energy absorption ability of the novel foam is explored by performing finite element simulations of physical impact tests on flat specimens of the functionally graded foam materials. Energy absorbing capacity with respect to parameters including gradient functions, density difference, average density, and impact energy, is explored in detail. It is illustrated that the functionally graded foam is superior in energy absorption to the uniform foam and that convex gradients perform better than concave gradients. The performance of such foams can be improved more if the density difference is enlarged. These findings provide valuable suggestions in the design of high performance energy absorption polymeric foams.

Towards a Nonperturbative Renormalization Group Analysis

We prove that a certain class of convex gradient models in high dimensional spaces without the presence of a small parameter renormalizes to a free field. As a consequence we establish a certain asymptotic formula for the partition function. In some ways, this is a realization of Gawedzki and Kupiainen’s idea to use correlation inequalities to augment the rigorous renormalization group methods. We use the more particular suggestion of Spencer to use certain inequalities of Brascamp and Lieb and also the formulation of the correlation functions in terms of the solutions to some partial differential equations given by Helffer and Sjostrand.