The paper proposes a modification of the multigrid method for solving the main classes of inverse heat and mass transfer problems, which belong to the class of ill-posed problems of mathematical physics, which in turn are of great importance for applied heat engineering research. The modification of the multigrid method is applied to direct heat and mass transfer problems, due to the fact that solving one inverse problem requires repeated solving of the direct problem, regardless of which optimizer is used to find the global extremum of the quadratic functional, which underlies the inverse heat and mass transfer problems. This approach allows formalizing the problem of parameter identification and ensuring the possibility of its solution by optimization methods of global multidimensional optimization. This paper also proposes numerical schemes based on difference approximations, which guarantee the stability of the computational process and the convergence of the obtained results. The effectiveness of the developed approach was tested on model examples, which showed its ability to significantly reduce the number of calculations to find a solution to a specific problem, while leaving the same error that arises during approximate calculations. The results obtained demonstrate the possibility of using the proposed approach for modeling and optimizing heat engineering processes in industrial and muffle furnaces, the mathematical model for which is used in work where high-precision control of temperature distribution is required. The practical significance of the study is to create a basis for the development of new systems for monitoring and controlling thermal regimes. This will ensure increased efficiency and reliability of energy technology processes. Also, the practical significance of the obtained results lies in the application of the developed methods, models and tools in the tasks of monitoring and diagnosing complex processes, including in the modeling and localization of environmental pollutants.
In the conditions of rapid development of modern information technologies and computing facilities, the demand for the development of new and modification of existing methods and algorithms for solving inverse problems in the optimization mathematical formulation has grown rapidly [1]. Such problems make it possible to automate the process of designing complex objects, their systems and components of these systems in order to obtain reliable systems in various difficult conditions of use. It should be noted that such problems make it possible to perform an analysis of existing physical objects in order to determine their residual service life.
Inverse problems are an important class of problems in applied mathematics and mathematical physics, because they allow, based on known observations, experimental data or monitoring results, to restore unknown characteristics of the studied object, process or environment [1, 2]. Unlike direct problems, in which the consequences are determined based on given parameters, inverse problems are aimed at finding unknown input data from known output observations. This approach is of fundamental importance in science and technology, as it allows for the analysis and reproduction of those characteristics of objects that cannot be directly measured or observed. In modern conditions, inverse problems are widely used in many industries: from technical diagnostics and non-destructive testing to medicine, environmental monitoring, and energy [2–4]. For example, using inverse problems, it is possible to restore the temperature distribution or heat flows in solids, determine the physical and mechanical properties of materials, detect defects in structures, track the state of energy infrastructure objects, or determine environmental parameters based on sensor system data [1, 5]. On the other hand, solving inverse problems is traditionally associated with a number of difficulties. Such problems belong to the class of incorrectly posed problems in the Hadamard sense, since even minor errors in the initial or experimental data can lead to significant distortions in the results [1]. This necessitates the development of special methods of regularization, stabilization and optimization, which would ensure the correctness of solutions and the stability of algorithms to errors and noise in the data. Therefore, mathematical models and algorithmic approaches for solving inverse problems require constant improvement, taking into account the latest achievements in the field of computational mathematics, optimization and intelligent information technologies [6].
This paper considers a special example of the application of methods for solving inverse problems, which consists in the analysis and optimization of operating modes of muffle and industrial furnaces, examples of which are given in Figure 1. Muffle and industrial furnaces are widely used in metallurgy, mechanical engineering, ceramic production and other industries where prolonged heating of materials to high temperatures under controlled conditions is required [5–7]. For such units, an important task is to maintain a uniform temperature distribution in the working volume of the furnace, which ensures the uniformity of the physical and mechanical properties of the finished product. The formulation of the problem in the inverse formulation allows us to determine the optimal heat transfer parameters, design characteristics of the muffle or thermal insulation, as well as energy supply modes to achieve the specified technological indicators.
Industrial furnaces are characterized by high energy consumption and complex dynamics of thermal processes, so their modeling requires taking into account nonlinear equations of heat and mass transfer, convection and radiation [8, 9]. The use of inverse problems in combination with optimization methods allows you to identify thermal parameters, diagnose defects and increase the energy efficiency of equipment. In particular, thanks to modern optimization algorithms, it is possible not only to predict the condition of furnaces and the residual resource of their components, but also to provide adaptive control of heating processes in order to reduce energy consumption and improve the quality of the final product.
Modern engineering systems and technological processes are characterized by multiparametricity, nonlinearity and complex dynamics. Performing the analysis of such systems in direct formulation is resource-intensive and often impractical. Instead, formulating the problem in the form of an inverse problem allows you to determine the optimal parameters or structure of the object, focusing directly on the desired output characteristics. This opens up wide opportunities for building reliable and durable systems capable of operating in difficult operating conditions.
An equally important aspect is the use of inverse problem solving methods for analyzing the residual resource of physical objects and structures [8–10]. In modern conditions, when a significant part of the infrastructure and industrial equipment is in operation for a long time, the task of predicting the residual resource acquires strategic importance. The use of identification methods and mathematical modeling based on experimental data allows not only to assess the current state of the object, but also to predict possible failures, prevent emergency situations and increase operational safety [1, 11, 12].
An important direction of modern research is the combination of classical optimization methods with the latest approaches of artificial intelligence and swarm algorithms (Figure 2) [10].
It should be noted that in today's conditions, the use of genetic algorithms, particle swarm methods, ant colonies, differential evolution and other metaheuristic algorithms for global optimization makes it possible to significantly expand the fields of science and technology, in particular, in complex design problems, combinatorial optimization problems, etc. [10]. Such methods are flexible, avoid getting stuck in local minima, and also provide high-quality optimization results even in cases where analytical or classical numerical approaches become ineffective.
The technical task is to build a mathematical model for optimizing the operation of an industrial / muffle furnace, which contains an object in the area G, which is located in a fixed position of the furnace itself. In order to increase its efficiency, this furnace contains heat sources, which are specified in the form of point heaters on an electric basis. Electric furnace heaters are also located in the furnace in predetermined geometric coordinates. It is necessary to determine what temperatures need to be applied to these point heaters so that the temperature on the object furnace is as close as possible to the predetermined Tactual. The constructed mathematical model must be implemented programmatically, choosing an optimization method. Since the constructed mathematical model refers to inverse models, since the cause-and-effect relationships of the temperature distribution in the industrial furnace are lost, it is reduced to the repeated use of the procedure for solving direct problems, which are described in the form of second-order partial differential equations. It is necessary to construct an additional method for solving the direct problem based on the multi-grid difference method in order to reduce the number of calculations required to obtain a solution to the problem of determining the optimal temperatures of the furnace heaters. For simplicity, the number of furnace heaters can be taken as fixed. In the following description, their number will be equal to six.
The optimization problem is to find a global functional that has the classical form [8]:
The partial differential equation that describes the process of heat propagation has the form:
The calculation area in (
The first two equalities in (
To solve such a problem in this mathematical formulation, it is necessary to have sufficiently effective methods for finding numerical solutions to second-order elliptic-type mathematical physics equations, as well as high-speed optimization methods to accelerate the search for optimal modes in heat and mass transfer processes. The next part of this article contains a description of these methods.
Equation (
Taking into account (
Taking into account (7), we can write down the classical implicit difference scheme for equation
(
It should be noted that in (8) a 5-point difference scheme is used. This difference scheme has a significant drawback. This drawback is manifested in the fact that when finding four valueson a new time layer, data on one point from the previous time layer are used. The difference scheme is stable, but does not provide the necessary accuracy of calculations due to damping. To increase the accuracy of calculations, a high-precision scheme is used. It is based on the Crank-Nicholson difference scheme:
It can be seen from equation (9) that it contains factors of the form ai,±0.5,j and ai,j±0.5 . These factors, as an option, can be approximately replaced by the following formulas: (10)
After such a replacement in (10), we solve the posed direct nonlinear direct heat and mass transfer problem using a difference scheme containing 8 points: 4 on the new (searched layer) and 4 on the old (known layer). But the resulting system of equations is nonlinear. For its implementation, we can use Newton linearization with the subsequent application of classical iterative methods for solving linear equations (for example, the conjugate or biconjugate gradient method). Let us proceed to the process of linearization of the terms of the nonlinear heat and mass transfer equation. To perform the procedure for linearization of functions, we take into account: (9) (11) (12) Then we will have
Let us assume that we need to find the temperature distribution at time (k+1). The procedure for obtaining this layer will be performed iteratively. To obtain this temperature distribution at layer (k+1), we use the iterative process (9) based on (11), taking into account (10). As a result, we obtain the following iterative formula:
After condition (11) is satisfied, the solution (13) is obtained T (i m,j+1) accepted as wanted T (i k,j+1)
The beginning of the iterative process is provided by the initial condition for equation (
Let's return to equation (
In (15) it is necessary to perform linearization for the first and third terms of the right-hand side. Linearization by Newton's method for any function f (x) is carried out according to the following formula:
Let ρ (T )=Σ aiρ T i and C (T )=Σ aiC T i. Then we will have:
Taking into account (17), we obtain an expression for the first nonlinear term on the right-hand side of equation (15).
Similarly to (18), linearization is performed for any function with the aim of constructing an
efficient iterative process for solving equation (
In the study of inverse heat and mass transfer problems, an important place is occupied by optimization methods aimed at restoring unknown parameters or boundary conditions from available experimental or calculated data. The most common approach is to reduce the identification problem to the problem of minimizing the quadratic residual functional, which characterizes the difference between the calculated and experimentally measured temperature field. Such functionals have, as a rule, a strictly convex structure, which makes the use of deterministic optimization methods effective.
Classical deterministic methods include gradient methods (the method of steepest descent, the gradient design method), which are based on the use of the first derivative of the functional. Their effectiveness in heat and mass transfer problems is due to the relative simplicity of calculating derivatives, in particular using sensitivity methods or the adjoint approach. However, such methods are often characterized by a slow convergence rate in conditions of poorly conditioned problems. (16) (17) (18)
Second-order methods, in particular Newton's method and quasi-Newton approaches, are more efficient. They involve the use of information about the Hessian matrix of the functional or its approximation. For quadratic functionals, these methods guarantee quadratic convergence and have proven themselves particularly well in restoring the parameters of thermophysical processes.
The necessary condition for the existence of a minimum of the functional (
It is obvious that all derivatives given in (19) must be given numerically based on approximation methods, for example, using the left difference scheme: or using the left difference scheme:
If second-order methods are used, for example, Newton's method and its modifications, it is necessary to use the Hessian matrix to approximate the second derivatives. Here it is better to use the central cutting scheme to approximate the second derivatives, in particular, the mixed secondorder derivatives will have the following form:
Similarly, we obtain expressions for specifying the second-order derivatives of the functional (
To obtain modifications of the methods, it is necessary to use the multi-grid principle, which allows obtaining faster solutions without loss of accuracy.
In practical applications of inverse heat and mass transfer problems, iterative deterministic algorithms with combined strategies, which combine gradient information with regularization approaches, have become widespread. Such algorithms are reliable, have guaranteed convergence and relatively high speed of solving problems.
When solving inverse heat and mass transfer problems, it is necessary to solve a significant number of direct heat and mass transfer problems. Therefore, the proposed method will be applied specifically to solving direct heat and mass transfer problems, since they are the basis for solving inverse heat and mass transfer problems. For the sake of simplicity, let us consider a certain twodimensional domain in which the inverse heat and mass transfer problems are solved. The solution of a two-dimensional problem is represented as a table of values of the desired function at each point of a certain grid into which the computational domain is divided. Let the differential equation be solved in partial derivatives by the difference method L( y )=0 . A system of linear algebraic equations is solved by an iterative method. The main idea of the multigrid method is as follows. Let the initial condition for solving systems of equations be in the form
Here T is the exact solution. Then the iterative procedure for finding a numerical solution is considered as a process of error reduction.
It is obvious that the components with a half-period equal to the grid step are destroyed most quickly, the slowest of all are the long-wave components with a half-period equal to the length of the entire calculation area. From solving the algebraic system of equations with respect to the values of the desired quantity, it is necessary to proceed to the system of equations with respect to the error.
Let Ah T h=bh is a system of equations built on a grid with a step h, Th is its exact solution. It is clear that the exact solution of this system of equations can be represented in the form where uhk is the result of the kth step of the iterative method and errhk is the error of this step, rh=bh− Ah uhk its residual, then, substituting into the ratio Ah T h=bh expression T h=errhk+uhk, we k get Ah(errhk+uhk)=bh or , Ah errk=bh− Ah uh=rhk that is, the equation for the error. Solving this k equation, we finderrhk, and therefore . ThT h=errhk+uhke next step is to choose the initial conditions for solving the system of equations. To calculate the error, we will use the system Ah(errh0)=rh0, where rh=bh− Ah u0h, а uh0 is the initial condition. The right-hand side of the system must be 0 related to the original problem. Therefore, the algorithm looks as follows. On a dense grid, we build a system of difference equations and perform several iterations. Based on the obtained result, we build a system of equations for the error. To suppress the long-wave components of the error, we project the system onto a sparse grid, solve it, and obtain a solution without the long-wave component of the error. We project the result onto a dense grid and solve the problem finally. After finding the error, we add it to the initial conditions and obtain a solution to the original problem. This method was used when testing the following stationary direct heat and mass transfer problems.
Problem 1. Find a numerical solution to the following boundary value problem: (20) where G is the boundary of the computational domain. The numerical solution of problem (6) is given in Figure 3: Problem 2. Find a numerical solution to the following boundary value problem:
Task number (20) (21) Below (Table 1) are the results of the multi-grid method based on the V-cycle according to the criterion of the number of iterations.
Based on the results obtained, it can be stated that the application of multigrid methods to find a numerical solution of nonlinear inverse problems is a rational approach. Using multigrid methods, a numerical solution to the problem can be obtained tens of times faster compared to the time of solving the same problem by classical methods on the same grid. As already mentioned, the
multigrid method is used when solving direct heat conduction problems, which are not used once
to solve one inverse heat conduction problem. Below in Table 2, the main results of modeling with
and without the multigrid method described in the work are presented.
Table 2 shows that the total number of iterations when using the modified multigrid method is
reduced by a factor of 2–4. This is a significant result, considering that such problems as (20) and
(21) are used to solve a problem of type (
The article builds a mathematical model for solving inverse heat and mass transfer problems based on the minimization of a quadratic functional that takes into account the differences between the actual and model temperature fields. To find optimal modes, the basic equation that mathematically describes the process of heat propagation in an industrial / muffle furnace is discretized. A modification of the method for solving the main classes of inverse heat and mass transfer problems using multi-grid methods based on the already known V-cycle is proposed. The method is tested in comparison with the classical multi-grid V-cycle. The results obtained confirm the effectiveness of the proposed approach for modeling and analyzing heat engineering processes, which creates the prerequisites for further use of the developed methods in applied problems of energy and industrial heat and mass transfer. The obtained modification of the multi-grid method speeds up calculations for solving one direct problem by 2–4 times. This means that in practice, the number of calculations when solving a single inverse problem can be up to 10 times.
The author(s) have not employed any Generative AI tools.