An inverse analysis of fluid flow through granular media using differentiable lattice Boltzmann method

This study presents a method for the inverse analysis of fluid flow problems.The focus is put on accurately determining boundary conditions and characterizing the physical properties of granular media,such as permeability,and fluid components,like viscosity.The primary aim is to deduce either constant pressure head or pressure profiles,given the known velocity field at a steady-state flow through a conduit containing obstacles,including walls,spheres,and grains.The lattice Boltzmann method(LBM)combined with automatic differentiation(AD)(AD-LBM)is employed,with the help of the GPU-capable Taichi programming language.A lightweight tape is used to generate gradients for the entire LBM simulation,enabling end-to-end backpropagation.Our AD-LBM approach accurately estimates the boundary conditions for complex flow paths in porous media,leading to observed steady-state velocity fields and deriving macro-scale permeability and fluid viscosity.The method demonstrates significant advantages in terms of prediction accuracy and computational efficiency,making it a powerful tool for solving inverse fluid flow problems in various applications.

Analytical solution of a double moving boundary problem for nonlinear flows in one-dimensional semi-infinite long porous media with low permeability

Based on Huang＇s accurate tri-sectional nonlin- ear kinematic equation （1997）, a dimensionless simplified mathematical model for nonlinear flow in one-dimensional semi-infinite long porous media with low permeability is presented for the case of a constant flow rate on the inner boundary. This model contains double moving boundaries, including an internal moving boundary and an external mov- ing boundary, which are different from the classical Stefan problem in heat conduction： The velocity of the external moving boundary is proportional to the second derivative of the unknown pressure function with respect to the distance parameter on this boundary. Through a similarity transfor- mation, the nonlinear partial differential equation （PDE） sys- tem is transformed into a linear PDE system. Then an ana- lytical solution is obtained for the dimensionless simplified mathematical model. This solution can be used for strictly checking the validity of numerical methods in solving such nonlinear mathematical models for flows in low-permeable porous media for petroleum engineering applications. Finally, through plotted comparison curves from the exact an- alytical solution, the sensitive effects of three characteristic parameters are discussed. It is concluded that with a decrease in the dimensionless critical pressure gradient, the sensi- tive effects of the dimensionless variable on the dimension- less pressure distribution and dimensionless pressure gradi- ent distribution become more serious; with an increase in the dimensionless pseudo threshold pressure gradient, the sensi- tive effects of the dimensionless variable become more serious; the dimensionless threshold pressure gradient （TPG） has a great effect on the external moving boundary but has little effect on the internal moving boundary.

The Construction Method for Solving Radial Flow Problem through the Homogeneous Reservoir

On the basis of similar structure of solutions of ordinary differential equation (ODE) boundary value problem, the similar construction method was put forward by solving problems of fluid flow in porous media through the homogeneous reservoir. It is indicate that the pressure distribution of dimensionless reservoir and bottom hole in Laplace space, which take on the radial flow, also shows similar structure, and the internal relationship between the above solutions were illustrated in detail.

Analysis and management strategies for the low voltage problems of rural power grid based on the distribution network flow calculation method

For a long time, because of the lack of investment capital and enough attentions, the overall constructions of rural power grid were far behind than the urban power grid in Chongqing Jiangbei Power Company. The low voltage problems were highlighted in the rural power grid due to the characteristics of rural power grid. Using the distribution network flow calculation method, we evaluated the low voltage problems of the rural power grid which belongs to Chongqing Jiangbei Power Company. In addition, we collected the data of distribution transformers in electricity consumption peak period. Some practical management strategies were proposed by the analysis and evaluation of potential and appeared low voltage problems.

One-dimensional non-Darcy flow in a semi-infinite porous media:a multiphase implicit Stefan problem with phases divided by hydraulic gradients

One-dimensional non-Darcy flow in a semi-infinite porous media is investigated. We indicate that the non-Darcy relation which is usually determined from experimental results can always be described by a piecewise linear function, and the problem can be equivalently transformed to a multiphase implicit Stefan problem. The novel feature of this Stefan problem is that the phases of the porous media are divided by hydraulic gradients, not the excess pore water pressures. Using the similarity transformation technique, an exact solution for the situation that the external load increases in proportion to the square root of time is developed. The study on the existence and uniqueness of the solution leads to the requirement of a group of inequalities. A similar Stefan problem considering constant surface seepage velocity is also investigated, and the solution, which we indicate to be uniquely existent under all conditions, is established. Meanwhile, the relation between our Stefan problem and the traditional multiphase Stefan problem is demonstrated. In the end, computational examples of the solution are presented and discussed. The solution provides a useful benchmark for verifying the accuracy of general approximate algorithms of Stefan problems, and it is also attractive in the context of inverse problem analysis.

Solution Method for Singular Initial Value Problems of One-dimensional Steady Transonic Flow in a Dual-mode Scramjet

Singular initial value problems arise in solving one-dimensional steady transonic flow of dualmode scramjet. The existing solution method has the problems of large initial value errors in principles. This paper puts forward an improved algorithm based on variable transformation, and constructs a nonsingular one-dimensional steady transonic flow equation by defining a new variable. The improved algorithm can eliminate the singularity of the differential equation, and can solve the singular initial value problems of one-dimensional steady transonic flow of dual-mode scramjet.

Numerical method of the Riemann problem for two-dimensional multi-fluid flows with general equation of state

Based on the classical Roe method, we develop an interface capture method according to the general equation of state, and extend the single-fluid Roe method to the two-dimensional （2D） multi-fluid flows, as well as construct the continuous Roe matrix for the whole flow field. The interface capture equations and fluid dynamic conservative equations are coupled together and solved by using any high-resolution schemes that usually suit for the single-fluid flows. Some numerical examples are given to illustrate the solution of 1D and 2D multi-fluid Riemann problems.

An effective discrete artificial bee colony algorithm for flow shop scheduling problem with intermediate buffers

An effective discrete artificial bee colony(DABC) algorithm is proposed for the flow shop scheduling problem with intermediate buffers(IBFSP) in order to minimize the maximum completion time(i.e makespan). The effective combination of the insertion and swap operator is applied to producing neighborhood individual at the employed bee phase. The tournament selection is adopted to avoid falling into local optima, while, the optimized insert operator embeds in onlooker bee phase for further searching the neighborhood solution to enhance the local search ability of algorithm. The tournament selection with size 2 is again applied and a better selected solution will be performed destruction and construction of iterated greedy(IG) algorithm, and then the result replaces the worse one. Simulation results show that our algorithm has a better performance compared with the HDDE and CHS which were proposed recently. It provides the better known solutions for the makespan criterion to flow shop scheduling problem with limited buffers for the Car benchmark by Carlier and Rec benchmark by Reeves. The convergence curves show that the algorithm not only has faster convergence speed but also has better convergence value.

Linear Plus Linear Fractional Capacitated Transportation Problem with Restricted Flow

In this paper, a transportation problem with an objective function as the sum of a linear and fractional function is considered. The linear function represents the total transportation cost incurred when the goods are shipped from various sources to the destinations and the fractional function gives the ratio of sales tax to the total public expenditure. Our objective is to determine the transportation schedule which minimizes the sum of total transportation cost and ratio of total sales tax paid to the total public expenditure. Sometimes, situations arise where either reserve stocks have to be kept at the supply points, for emergencies or there may be extra demand in the markets. In such situations, the total flow needs to be controlled or enhanced. In this paper, a special class of transportation problems is studied where in the total transportation flow is restricted to a known specified level. A related transportation problem is formulated and it is shown that to each basic feasible solution which is called corner feasible solution to related transportation problem, there is a corresponding feasible solution to this restricted flow problem. The optimal solution to restricted flow problem may be obtained from the optimal solution to related transportation problem. An algorithm is presented to solve a capacitated linear plus linear fractional transportation problem with restricted flow. The algorithm is supported by a real life example of a manufacturing company.

AN EFFICIENT METHOD FOR SOLVING THE MIXED DIRECT-INVERSE PROBLEM OF THE TRANSONIC ROTATIONAL FLOW IN PLANE CASCADES

The method in [1] has been extended to the case of rotational flow in this paper. A new method for dealing with the shock wave is presented. This method has the advantages of both the shock-fitting and the shock capturing methods. The direct problem and the mixed direct-inverse prob- lem of the rotational flow in a transonic plane cascade at both design and off design conditions are solved, and the results show that the present method has rapid convergence rate and high accuracy even for the flow with moderately strong shocks. The calculations have been carried out on the DPS-8 computer, and for the direct problem, only 50-80 iterations are needed, and 50-80 seconds of CPU time are required.

Migrating Birds Optimization for Flow Shop Sequencing Problem

FSSP is a typical NP-Hard problem which is desired to be minimum makespan. This study consid- ers Migrating Birds Optimization (MBO) which is metaheuristic approach for the solution of Flow Shop Sequencing Problem (FSSP). As the basic MBO algorithm is designed for discrete problems. The performance of basic MBO algorithm is tested via some FSSP data sets exist in literature. Obtained results are compared with optimal results of related data sets.

Unsteady Inverse Problem of Type B for 2-D Transonic Flow: A Variational Formulation

A family of variational principles (VP) has been developed for the unsteady inverse problem of the second type I B. It opens new ways for the inverse shape design of unsteady airfoils and can serve as key basis of multipoint inverse shape design of steady airfoils and cascades.

Variable-Domain Variational Finite Element Solution of I_A Inverse Problem of 2-D Unsteady Compressible Flow around Oscillating Airfoils

In this paper, on the basis of the variational principles developed the finite element method (FEM) is employed for numerical solution of the inverse pro blem of 2 D unsteady compressible flow around oscillating airfoils by incorporating the non reflecting far field boundary conditions and a new unsteady Kutta condition. All unknown boundary (airfoil contour) and discontinuities(shocks and free trailing vortex sheets) are determined via the functional variation with variable domain and artificial density concept. For the numerical realization of the variable domain variation, a special finite element with self adjusting nodes is also suggested herein. The numerical results show that the present method is effective for the design of unsteady airfoil.

A TRUST REGION ALGORITHM VIA BILEVEL LINEAR PROGRAMMING FOR SOLVING THE GENERAL MULTICOMMODITY MINIMAL COST FLOW PROBLEMS

This paper proposes a nonmonotonic backtracking trust region algorithm via bilevel linear programming for solving the general multicommodity minimal cost flow problems.Using the duality theory of the linear programming and convex theory,the generalized directional derivative of the general multicommodity minimal cost flow problems is derived.The global convergence and superlinear convergence rate of the proposed algorithm are established under some mild conditions.

Design of Self-Assembling Molecules and Boundary Value Problem for Flows on a Space of <i>n</i>-Simplices

Self-assembling molecules are ubiquitous in nature, among which are proteins, nucleic acids (DNA and RNA), peptides and lipids. Recognizing the ability of biomolecules to self-assemble into various 3D shapes at the nanoscale, researchers are mimicking the self-assembly strategy for engineering of complex nanostructures. However, the general principles underlying the design of self-assembled molecules have not yet been identified. The question is “How to obtain a well-defined shape with desired properties by folding a chain of subunits (such as amino acids and nucleic acids)”, where properties are determined by the precise spatial arrangement of the subunits on the surface. In this paper, we consider the question from the viewpoint of the discrete differential geometry of n-simplices. Self-assembling molecules are then represented as a union of trajectories of 3-simplices (i.e., tetrahedrons), and the question is rephrased as a “boundary value problem” for flows on a space of tetrahedrons. Also considered is a characterization of two types of surface flows of n-simplices. It is a rough classification of surface flows, but may be essential in characterizing important properties of biomolecules such as allosteric regulation. The author believes this paper not only provides a new perspective for the engineering of self-assembling molecules, but also promotes further collaboration between mathematics and other disciplines in life science.

Generalized Riemann Problem for Isentropic Flow with Dissipative Term

The paper concerns with generalized Riemann problem for isentropic flow with dissipation, and show that if the similarity solution to Riemann problem is composed of a backward centered rarefaction wave and a forward centered rarefaction wave, then generalized Riemann problem admits a unique global solution on t≥0. This solution is composed of backward centered wave and a forward centered wave with the origin as their center and then continuous for t 〉0.

Some Remarks on Application of Sandwich Methods in the Minimum Cost Flow Problem

In this paper, two new sandwich algorithms for the convex curve approximation are introduced. The proofs of the linear convergence property of the first method and the quadratic convergence property of the second method are given. The methods are applied to approximate the efficient frontier of the stochastic minimum cost flow problem with the moment bicriterion. Two numerical examples including the comparison of the proposed algorithms with two other literature derivative free methods are given.

Application of He’s Variational Iterative Method for Solving Thin Film Flow Problem Arising in Non-Newtonian Fluid Mechanics

In this paper, He’s variational iteration method is successfully employed to solve a nonlinear boundary value problem arising in the study of thin film flow of a third grade fluid down an inclined plane. For comparison, the same problem is solved by the Adomian decomposition method. The results show that the difference between the two solutions is negligible. The conclusion is that this technique may be considered an alternative and efficient method for finding approximate solutions of both linear and nonlinear boundary value problems. Furthermore, the variational iteration method has an advantage over the decomposition method in that it solves the nonlinear problems without using the Adomian polynomials.

Modified EDMONDS-KARP Algorithm to Solve Maximum Flow Problems

Maximum Flow Problem (MFP) discusses the maximum amount of flow that can be sent from the source to sink. Edmonds-Karp algorithm is the modified version of Ford-Fulkerson algorithm to solve the MFP. This paper presents some modifications of Edmonds-Karp algorithm for solving MFP. Solution of MFP has also been illustrated by using the proposed algorithm to justify the usefulness of proposed method.

Solving Load Flow Problems of Power System by Explicit Pseudo-Transient Continuation (E-ψtc) Method

This paper is a further study of two papers [1] and [2], which were related to Ill-Conditioned Load Flow Problems and were published by IEEE Trans. PAS. The authors of this paper have some different opinions, for example, the 11-bus system is not an ill-conditioned system. In addition, a new approach to solve Load Flow Problems, E-&psi;tc, is introduced. It is an explicit method;solving linear equations is not needed. It can handle very tough and very large systems. The advantage of this method has been fully proved by two examples. The authors give this new method a detailed description of how to use it to solve Load Flow Problems and successfully apply it to the 43-bus and the 11-bus systems. The authors also propose a strategy to test the reliability, and by solving gradient equations, this new method can answer if the solution exists or not.