Some practical remarks in solving partial differential equations using reduced order schemes obtained through the POD method

SOLOMON, Alexandru; OLTEI, Valentin Claudiu; Bogoi, Alina

doi:10.13111/2066-8201.2022.14.1.15

Cited by 2 publications

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“…In this way, everything comes down to solving the problem of numerical modeling of a dynamic system, which consists of a large number of boundary and initial conditions. To do this, we can use a system of differential equations, but their implementation requires significant computer system resources and time [3].…”

Section: Introductionmentioning

confidence: 99%

The use of cellular automata in the simulation of wood drying processes in a wood drying chamber of periodic action

Sokolovskyy¹,

Sinkevych²

2022

CDS

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In this work, research the essence of the wood drying process in a periodic wood drying chamber. This paper provides a mathematical model of a wood drying chamber, which describes the general essence of physical drying processes using the equipment available in the wood drying chamber. This approach allows to take into account the physical parameters of the necessary equipment, such as heaters, fans, humidifying nozzles or other. This approach also allows to ignore some design characteristics that may differ depending on the type of wood drying chamber. Considering this, the main task in this work is to determine the temperature and humidity of the drying agent and lumber in the stack, as well as the temperature of the main components of the wood drying chamber. Taking into account such a large number of input parameters and describing a complex non-stationary process of heat transfer, there is a need to create complicated mathematical models. The presence of such mathematical models greatly complicates their application and requires significant computer resources for their calculation. In this way, the mathematical description is reduced to the description of non-linear partial differential equations. To simplify and speed up the calculations of this mathematical model, the use of cellular automata is suggested. To do this, the 3D model of the wood drying chamber is represented as a cell-automatic field, which consists of cells of the same size but different types. As a result, neighboring cells contain local relationships that describe their general behavior. This behavior depends on the type of tangent cells and is described by transition rules based on a mathematical model. Through the use of the developed cell-automatic model and transition rules, it is possible to obtain the values of the temperature and moisture content of the wood in the stack, the drying agent in the chamber, as well as the temperature of the main components of the chamber. The work also shows the corresponding graphs of changes in temperature and moisture content. To check the adequacy and reliability, the obtained results were compared with the results of other authors' experiments. As a result of the verification, the values of the average absolute error aren't high, which confirms the adequacy of the mathematical model and the prospects of using the developed cell-automatic model.

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Section: Introductionmentioning

confidence: 99%

The use of cellular automata in the simulation of wood drying processes in a wood drying chamber of periodic action

Sokolovskyy¹,

Sinkevych²

2022

CDS

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Applying Proper Orthogonal Decomposition to Parabolic Equations: A Reduced Order Numerical Approach

MORARU

2024

INCAS BULLETIN

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n this paper we present a low-order numerical scheme developed using the Proper Orthogonal (POD) method to address non-homogeneous parabolic equations in both one and two dimensions. The proposed schemes leverage the POD technique to reduce the computational complexity associated with solving these equations while maintaining accuracy. By employing POD, the high-dimensional problem is approximated by a reduced set of models, allowing for a more efficient representation of the system dynamics. The application of this method to non-homogenous parabolic equations offers a promising approach for enhancing the computational efficiency of simulations in diverse fields, such as fluid dynamics, heat conduction, and reaction-diffusion processes. The presented numerical scheme demonstrates its efficacy in achieving accurate results with significantly reduced computational costs, making it a valuable tool for applications demanding efficient solutions to non-homogeneous parabolic equations in one and two dimensions.

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Some practical remarks in solving partial differential equations using reduced order schemes obtained through the POD method

Cited by 2 publications

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The use of cellular automata in the simulation of wood drying processes in a wood drying chamber of periodic action

The use of cellular automata in the simulation of wood drying processes in a wood drying chamber of periodic action

Applying Proper Orthogonal Decomposition to Parabolic Equations: A Reduced Order Numerical Approach

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