Discrete adjoint methodology for general multiphysics problems

This article presents a methodology whereby adjoint solutions for partitioned multiphysics problems can be computed efficiently, in a way that is completely independent of the underlying physical sub-problems, the associated numerical solution methods, and the number and type of couplings between them. By applying the reverse mode of algorithmic differentiation to each discipline, and by using a specialized recording strategy, diagonal and cross terms can be evaluated individually, thereby allowing different solution methods for the generic coupled problem (for example block-Jacobi or block-Gauss-Seidel). Based on an implementation in the open-source multiphysics simulation and design software SU2, we demonstrate how the same algorithm can be applied for shape sensitivity analysis on a heat exchanger (conjugate heat transfer), a deforming wing (fluid–structure interaction), and a cooled turbine blade where both effects are simultaneously taken into account.

Inverse problems of the dynamics of observation interpretation systems

Based on the analysis and systematization of the inverse problems of dynamics, the study of the properties and features of the types of dynamic models under consideration, an approach is proposed for the development of appropriate methods of mathematical modeling based on the use and implementation of integral models in the form of Volterra equations of the I and II kind, their functional capabilities are determined in the study of various classes of problems, and also formulated the features that affect the choice of methods for their numerical solution. Methods for obtaining integral models are proposed, which are the basis for constructing algorithms for solving inverse problems of dynamics for a fairly wide class of dynamic objects. Integral methods for the identification of dynamic objects have been developed, which make it possible to obtain stable non-optimization algorithms for calculating the parameters of mathematical models. Recurrent methods of parametric identification of transfer functions of dynamic objects with an arbitrary input action are proposed (the obtained parameters of the transfer functions are also coefficients of the corresponding differential equations, which makes it possible to obtain equivalent mathematical models in the form of integral equations). The study of algorithms that implement the proposed identification methods allows us to conclude about their efficiency in terms of the amount of computation and ease of implementation, as well as the high accuracy of calculating the model parameters.

Diffusive Mass Transfer and Gaussian Pressure Transient Solutions for Porous Media

This study revisits the mathematical equations for diffusive mass transport in 1D, 2D and 3D space and highlights a widespread misconception about the meaning of the regular and cumulative probability of random-walk solutions for diffusive mass transport. Next, the regular probability solution for molecular diffusion is applied to pressure diffusion in porous media. The pressure drop (by fluid extraction) or increase (by fluid injection) due to the production system may start with a simple pressure step function. The pressure perturbation imposed by the step function (representing the engineering intervention) will instantaneously diffuse into the reservoir at a rate that is controlled by the hydraulic diffusivity. Traditionally, the advance of the pressure transient in porous media such as geological reservoirs is modeled by two distinct approaches: (1) scalar equations for well performance testing that do not attempt to solve for the spatial change or the position of the pressure transient without reference to a well rate; (2) advanced reservoir models based on numerical solution methods. The Gaussian pressure transient solution method presented in this study can compute the spatial pressure depletion in the reservoir at arbitrary times and is based on analytical expressions that give spatial resolution without gridding-meaning solutions that have infinite resolution. The Gaussian solution is efficient for quantifying the advance of the pressure transient and associated pressure depletion around single wells, multiple wells and hydraulic fractures. This work lays the basis for the development of advanced reservoir simulations based on the superposition of analytical pressure transient solutions.

Multi-leader-follower potential games

In this paper, we discuss a particular class of Nash games, where the participants of the game (the players) are divided into two groups (leaders and followers) according to their position or influence on the other players. Moreover, we consider the case, when the leaders’ and/or the followers’ game can be described as a potential game. This is a subclass of Nash games that has been introduced by Monderer and Shapley in 1996 and has beneficial properties to reformulate the bilevel Nash game. We develope necessary and sufficient conditions for Nash equilibria and present existence and uniqueness results. Furthermore, we discuss some Examples to illustrate our results. In this paper, we discussed analytical properties for multi-leader follower potential games, that form a subclass of hierarchical Nash games. The application of these theoretical results to various fields of applications are a future research topic. Moreover, they are meant to serve as a starting point for the developement of efficient numerical solution methods for multi-leader-follower games.

Numerical Solution Methods for a Nonlinear Operator Equation Arising in an Inverse Coefficient Problem

We consider the inverse problem of determining two unknown coefficients in a linear system of partial differential equations using additional information about one of the solution components. The problem is reduced to a nonlinear operator equation for one of the unknown coefficients. The successive approximation method and the Newton method are used to solve this operator equation numerically. Results of calculations illustrating the convergence of numerical methods for solving the inverse problem are presented.

Application of Numerical Methods for the Analysis of Damped Parallel RLC Circuit

A sudden application of sources results in time-varying currents and voltages in the circuit known as transients. This phenomenon occurs frequently during switching. A simple circuit constituting a resistor, an inductor, and a capacitor is termed an RLC circuit. It may be in parallel or series configuration or both. Different values of damping factors determine the different nature of the transient response. We applied different numerical solution methods such as explicit (forward) Euler method, third-order Runge-Kutta (RK3) method, and Butcher's fifth-order Runge-Kutta (BRK5) method to approximate the solution of second-order differential equation with initial value problem (IVP). We thoroughly compared the numerical solutions obtained by these methods with the necessary visualization and analysis of error. We also examined the superiority of these methods over one another and the appropriateness of numerical methods for different damping conditions is explored. With high accuracy of the approximation and thorough analysis of the observation, we found Butcher's fifth-order Runge-Kutta (BRK5) method to be the best numerical technique. Regarding the different values of damping factors, we considered the further possibility of discussion and analysis of this iterative method.

Hybrid of differential quadrature and sub-gradients methods for solving the system of Eikonal equations

Many important natural phenomena of wave propagations are modeled by Eikonal equations and a variety of new methods are needed to solve them. The differential quadrature method (DQM) is an effective numerical method for solving the system of differential equations that can achieve accurate numerical results using fewer grid points and therefore requires relatively little computational effort. In this paper, we focus on the implementation of the non-smooth Eikonal optimization by using a hybrid of polynomial differential quadrature (PDQ) or Fourier differential quadrature (FDQ) method and sub-gradients idea. Our goal is to develop a new Eikonal equation system design for wave propagation equations, as well as the efficiency and accuracy of new grid points to reduce errors and compare errors in various physical equation problems, especially wave propagation equations, and achieve their convergence. We explore the accuracy and stability of the Eikonal equation system by two-dimensional and three-dimensional numerical examples and the use of three types of grid points in a comprehensive manner. This article aims to create a new and innovative solution to the non-smooth Eikonal equation system. This new method is much more efficient and effective than traditional numerical solution methods same as DQ.