A modified Kacanov iteration scheme with application to quasilinear diffusion models

The classical Kacanov scheme for the solution of nonlinear variational problems can be interpreted as a fixed point iteration method that updates a given approximation by solving a linear problem in each step. Based on this observation, we introduce a modified Kacanov method, which allows for (adaptive) damping, and, thereby, to derive a new convergence analysis under more general assumptions and for a wider range of applications. For instance, in the specific context of quasilinear diffusion models, our new approach does no longer require a standard monotonicity condition on the nonlinear diffusion coefficient to hold. Moreover, we propose two different adaptive strategies for the practical selection of the damping parameters involved.

A Robust Optimization Model to the Day-Ahead Operation of an Electric Vehicle Aggregator Providing Reliable Reserve

This paper presents a robust optimization model to find out the day-ahead energy and reserve to be scheduled by an electric vehicle (EV) aggregator. Energy can be purchased from, and injected to, the distribution network, while upward and downward reserves can be also provided by the EV aggregator. Although it is an economically driven model, the focus of this work relies on the actual availability of the scheduled reserves in a future real-time. To this end, two main features stand out: on one hand, the uncertainty regarding the EV driven pattern is modeled through a robust approach and, on the other hand, a set of non-anticipativity constraints are included to prevent from unavailable future states. The proposed model is posed as a mixed-integer robust linear problem in which binary variables are used to consider the charging, discharging or idle status of the EV aggregator. Results over a 24-h case study show the capability of the proposed model.

Post-buckling behavior estimation of rigidly supported cylindrical composite panels in case shear

We present the solution of the geometrically nonlinear problem of the shear-critical behavior of a thin composite cylindrical panel of small curvature of orthotropic structure. The obtained solution considers the conditions of all-round rigid support. The expression for determining the membrane stresses arising in the supercritical state is given. When considering a linear problem, expressions for determining the critical shear flow are given. A method for determining the nonlinear stress-strain state in the overcritical state for a given thickness and stacking of an orthotropic panel is presented. The obtained solutions can be used in the design of load-bearing cylindrical panels, as well as in the analysis of geometrically nonlinear behavior of defects such as delaminations.

A variational sheath model for gyrokinetic Vlasov-Poisson equations

We construct a stationary gyrokinetic variational model for sheaths close to the metallic wall of a magnetized plasma, following a physical extremalization principle for the natural energy. By considering a reduced set of parameters we show that our model has a unique minimal solution, and that the resulting electric potential has an infinite number of oscillations as it propagates towards the core of the plasma. We prove this result for the non linear problem and also provide a simpler analysis for a linearized problem, based on the construction of exact solutions. Some numerical illustrations show the well-posedness of the model after numerical discretization. They also exhibit the oscillating behavior.

On solution of one linear problem with initial and boundary conditions

We solve one boundary problem of fourth order with initial conditions, that appears, for example, when one solves the problem about lateral oscillations of elastic-viscous-relaxating rod of variable profile with variable momentum of inertia with freely supported ends.

WKB periods for higher order ODE and TBA equations

We study the WKB periods for the (r + 1)-th order ordinary differential equation (ODE) which is obtained by the conformal limit of the linear problem associated with the $$ {A}_r^{(1)} $$ A r 1 affine Toda field equation. We compute the quantum corrections by using the Picard-Fuchs operators. The ODE/IM correspondence provides a relation between the Wronskians of the solutions and the Y-functions which satisfy the thermodynamic Bethe ansatz (TBA) equation related to the Lie algebra Ar. For the quadratic potential, we propose a formula to show the equivalence between the logarithm of the Y-function and the WKB period, which is confirmed by solving the TBA equation numerically.

Tire noise optimization problem: a mixed integer linear programming approach

We present a Mixed Integer Linear Programming (MILP) approach in order to model the non-linear problem of minimizing the tire noise function. In a recent work, we proposed an exact solution for the Tire Noise Optimization Problem, dealing with an APproximation of the noise (TNOP-AP). Here we study the original non-linear problem modeling the EXact - or real - noise (TNOP-EX) and propose a new scheme to obtain a solution for the TNOP-EX. Relying on the solution for the TNOP-AP, we use a Branch&Cut framework and develop an exact algorithm to solve the TNOP-EX. We also take more industrial constraints into account. Finally, we compare our experimental results with those obtained by other methods.

An effective high-order element for analysis of two-dimensional linear problem using SBFEM

The scaled boundary finite element method (SBFEM) is a semi-analytical method, whose versatility, accuracy, and efficiency are not only equal to, but potentially better than the finite element method and the boundary element method for certain problems. This paper investigates the possibility of using an efficient high-order polynomial element in the SBFEM to form the approximation in the circumferential direction. The governing equations are formulated from the classical linear elasticity theory via the SBFEM technique. The scaled boundary finite element equations are formulated within a general framework integrating the influence of the distributed body source, mixed boundary conditions, contributions the side face with either prescribed surface load or prescribed displacement. The position of scaling center is considered for modeling problem. The proposed method is evaluated by solving two-dimensional linear problem. A selected set of results is reported to demonstrate the accuracy and convergence of the proposed method for solving problems in general boundary conditions.

Vessel Scheduling Optimization Model Based on Variable Speed in a Seaport with One-Way Navigation Channel

To improve the efficiency of in-wharf vessels and out-wharf vessels in seaports, taking into account the characteristics of vessel speeds that are not fixed, a vessel scheduling method with whole voyage constraints is proposed. Based on multi-time constraints, the concept of a minimum safety time interval (MSTI) is clarified to make the mathematical formula more compact and easier to understand. Combining the time window concept, a calculation method for the navigable time window constrained by tidal height and drafts for vessels is proposed. In addition, the nonlinear global constraint problem is converted into a linear problem discretely. With the minimum average waiting time as the goal, the genetic algorithm (GA) is designed to optimize the reformulated vessel scheduling problem (VSP). The scheduling methods under different priorities, such as the first-in-first-out principle, the largest-draft-vessel-first-service principle, and the random service principle are compared and analyzed experimentally with the simulation data. The results indicate that the reformulated and simplified VSP model has a smaller relative error compared with the general priority scheduling rules and is versatile, can effectively improve the efficiency of vessel optimization scheduling, and can ensure traffic safety.

On a Nonlinear Mixed Problem for a Parabolic Equation with a Nonlocal Condition

The aim of this work is to prove the well-posedness of some linear and nonlinear mixed problems with integral conditions defined only on two parts of the considered boundary. First, we establish for the associated linear problem a priori estimate and prove that the range of the operator generated by the considered problem is dense using a functional analysis method. Then by applying an iterative process based on the obtained results for the linear problem, we establish the existence, uniqueness and continuous dependence of the weak solution of the nonlinear problem.