An Extrema Approach to Probabilistic Creep Modeling in Finite Element Analysis

This paper introduces a computationally efficient extremum condition-based Reduced Order Modeling (ROM) approach for the probabilistic predictions of creep in finite element (FE). Component-level probabilistic simulations are needed to assess the reliability and safety of high-temperature components. Full-scale probabilistic creep models in FE are computationally expensive, requiring many hundreds of simulations to replicate the uncertainty of component failure. In this study, an extremum condition-based ROM approach is proposed. In the extremum approach, full-scale probabilistic simulations are completed in 1D across a wide range of stresses, the data is processed and extremum conditions extracted, and those conditions alone are applied in 2D/3D FE to predict the mean and range of creep-failure. The probabilistic Sinh model, calibrated for alloy 304 stainless steel, is selected . The uncertainty sources (i.e. test condition, pre-existing damage, and model constants) are evaluated and pdfs sampling are performed via Monte carlo method. The extremum conditions are chosen from numerous 1D model simulations. These conditions include extremum cases of creep ductility, rupture, and area under creep curves. Only the extremum cases are simulated for 2D model saving significant computational time and memory. The goodness-of-fit of the predicted creep response for 1D and 2D model shows satisfactory agreement with the experimental data. The accuracy of the extremum condition-based ROM will reduce significant computational burden of simulating complex engineering systems. Introduction of multi-stage Sinh, stochasticity, and spatial uncertainty will further improve the prediction.

“Stationary” Schrödinger equation in quantum cosmology

The conditional principle of extremum in quantum cosmology is formulated for a positive functional of the energy density of space, in which gravitational constraints serve as additional conditions. The extremum conditions determine the discrete spectrum of the “stationary” state of the universe with the corresponding values of the energy density of space. A dynamic interpretation of solutions is proposed, in which the quantum number of the energy density plays the role of cosmic time. In the self-consistent harmonic approximation, the quantum dynamics of the anisotropic model of the Bianchi IX universe is considered.

Optimal regular reflection of shock and blast waves

The regular reflection of an oblique steady shock in supersonic gas flow is considered. The static pressure extremum conditions after the point of reflection of the shock with fixed strength depending on oncoming flow Mach number are determined analytically. The obtained results are applied to solution of the mechanically equivalent problem of the reflection of a propagating shock from an inclined surface. Non-monotonic variation of the mechanical loads on the obstacle with respect to its inclination angle is shown; the obstacle slope angles that correspond to pressure minima downwards of the unsteady shock reflection point are determined analytically.

Boundary Element Approach in Impedance Cloaking Problem

The cloaking problem is considered for a 2-D wave scattering model in an unbounded homogenous medium containing an impenetrable covered (cloaked) boundary. The control is a surface impedance which enters the boundary condition as a coefficient. The problem is reduced to the inverse extremal problem of choosing the surface impedance. The solvability of the original scattering problem for 2-D Helmholtz equation and of the extremal problem is proved. Optimality system describing the necessary extremum conditions is derived. The algorithm for numerical solving of the control problem based on the optimality system and boundary element method is designed.

Analysis of 2-D Impedance Cloaking Problem Based on Boundary Element Method

Control problems are considered for a two-dimensional model describing wave scattering in an unbounded homogenous medium containing an impenetrable covered (cloaked) boundary. The control is a surface impedance which enters the boundary condition as a coefficient. The solvability of the original scattering problem for 2-D Helmholtz equation and of the control problem is proved. Optimality system dгescribing the necessary extremum conditions are derived. The algorithm for numerical solving of the control problem based on the optimality system and boundary element method is designed.

Global Dynamics of the Hastings-Powell System

This paper studies the problem of bounding a domain that contains all compact invariant sets of the Hastings-Powell system. The results were obtained using the first-order extremum conditions and the iterative theorem to a biologically meaningful model. As a result, we calculate the bounds given by a tetrahedron with excisions, described by several inequalities of the state variables and system parameters. Therefore, a region is identified where all the system dynamics are located, that is, its compact invariant sets: equilibrium points, periodic-homoclinic-heteroclinic orbits, and chaotic attractors. It was also possible to formulate a nonexistence condition of the compact invariant sets. Additionally, numerical simulations provide examples of the calculated boundaries for the chaotic attractors or periodic orbits. The results provide insights regarding the global dynamics of the system.

THERMODYNAMIC POTENTIAL WITH CORRECT ASYMPTOTICS FOR PNJL MODEL

An attempt is made to resolve certain incongruities within the Nambu–Jona-Lasinio (NJL) and Polyakov loop extended NJL models (PNJL) which currently are used to extract the thermodynamic characteristics of the quark–gluon system. It is argued that the most attractive resolution of these incongruities is the possibility to obtain the thermodynamic potential directly from the corresponding extremum conditions (gap equations) by integrating them, an integration constant being fixed in accordance with the Stefan–Boltzmann law. The advantage of the approach is that the regulator is kept finite both in divergent and finite valued integrals at finite temperature and chemical potential. The Pauli–Villars regularization is used, although a standard 3D sharp cutoff can be applied as well.

The polylogarithm and the Lambert W functions in thermoelectrics

In this work, we determine the conditions for the extremum of the figure of merit, θ2, in a degenerate semiconductor for thermoelectric (TE) applications. We study the variation of the function θ2 with respect to the reduced chemical potential μ* using relations involving polylogarithms of both integral and nonintegral orders. We present the relevant equations for the thermopower, thermal, and electrical conductivities that result in optimizing θ2 and obtaining the extremum equations. We discuss the different cases that arise for various values of r, which depends on the type of carrier scattering mechanism present in the semiconductor. We also present the important extremum conditions for θ2 obtained by extremizing the TE power factor and the thermal conductivity separately. In this case, simple functional equations, which lead to solutions in terms of the Lambert W function, result. We also present some solutions for the zeros of the polylogarithms. Our analysis allows for the possibility of considering the reduced chemical potential and the index r of the polylogarithm as complex variables.