scholarly journals One numerical approach to optimal control the linear heat conduction processes

2020 ◽  
Keyword(s):  
Optimal Control ◽  
Cauchy Problem ◽  
Heat Conduction ◽  
Mathematical Formulation ◽  
Grid Method ◽  
Numerical Approach ◽  
Temperature Fields ◽  
Control Vector ◽  
Transient Time ◽  
Unknown Control

It is proposed the generalized mathematical formulation of the problem about the optimal control for the heat conduction processes representing by the partial differential equation. The proposed formulation not includes the necessary clarifications about the conditions which must be satisfied by the current and required temperature fields. But, during the generalized solving of the formulated problem, it is established that the current and required temperature fields must be agreed with the mathematical model of the heat conduction so that to have possibilities to provide uniquely these temperature fields by means the control vector. To solve the problem about the optimal control for the heat conduction processes it is developed the numerical approaches based on reducing to the especially built ordinary differential equations and minimization problem. This reducing is based on discretisation the heat conduction by using the grid method and on defining the unknown control vector as the numerical solution of the especially built Cauchy problem. To satisfy the all limitations it is proposed to build the permissible velocity of the unknown control vector considering with the requirements of necessary switching in some moments of the time. The particular example of using the proposed generalized approaches is considered to illustrate their application technique. It is shown that the proposed generalized mathematical formulation is fully corresponded with the considered particular example. In this considered particular example, the resolving Cauchy problem can be built and the switching time can be found in the depending on the grid node choosing. It is shown that the transient time can be decrease almost twice due to optimizing the control in the particular example at least. All these results will allow giving the clear representation of the proposed approaches and the technique of their using to solve the engineering problems about the optimal control of the heat conduction processes in different industrial systems.

scholarly journals On the optimal control of coronavirus (2019-nCov) mathematical model; a numerical approach

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
N. H. Sweilam ◽  
S. M. Al-Mekhlafi ◽  
A. O. Albalawi ◽  
D. Baleanu
Keyword(s):  
Mathematical Model ◽  
Optimal Control ◽  
Weighted Average ◽  
Necessary Optimality Conditions ◽  
Numerical Approach ◽  
Population Number ◽  
Infected People ◽  
The Stability ◽  
Novel Coronavirus ◽  
Theoretical Results

Abstract In this paper, a novel coronavirus (2019-nCov) mathematical model with modified parameters is presented. This model consists of six nonlinear fractional order differential equations. Optimal control of the suggested model is the main objective of this work. Two control variables are presented in this model to minimize the population number of infected and asymptotically infected people. Necessary optimality conditions are derived. The Grünwald–Letnikov nonstandard weighted average finite difference method is constructed for simulating the proposed optimal control system. The stability of the proposed method is proved. In order to validate the theoretical results, numerical simulations and comparative studies are given.

scholarly journals Optimization of two opposing neutron beams parameters in dynamical (B/Gd) neutron cancer therapy

2019 ◽  
Vol 5 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Nassar H. S. Haidar
Keyword(s):  
Optimal Control ◽  
Cancer Therapy ◽  
Nonlinear Optimization ◽  
Optimization Problem ◽  
Pareto Optimal ◽  
Control Vector ◽  
Relative Time ◽  
Neutron Beams ◽  
Utility Index ◽  
First Time

We demonstrate how the therapeutic utility index and the ballistic index for dynamical neutron cancer therapy (NCT) with two opposing neutron beams form a nonlinear optimization problem. In this problem, the modulation frequencies ω and ϖ of the beams and the relative time advance ε are the control variables. A Pareto optimal control vector ω* = (ω*, ϖ*, ε*) for this problem is identified and reported for the first time. The utility index is shown to be remarkably periodically discontinuous in ε, even in the neighborhood of ε*.

Sensitivity-Coefficient Based Inverse Heat Conduction Method for Identifying Hot Spots in Electronics Packages: A Comparison of Grid-Refinement Methods

2021 ◽  
Author(s):  
Patrick Krane ◽  
David Gonzalez Cuadrado ◽  
Francisco Lozano ◽  
Guillermo Paniagua ◽  
Amy Marconnet
Keyword(s):  
Heat Conduction ◽  
Hot Spots ◽  
Computation Time ◽  
Grid Method ◽  
Sensitivity Coefficient ◽  
Coarse Grid ◽  
Inverse Heat Conduction ◽  
Fine Grid ◽  
Temperature Maps ◽  
Electronics Packages

Abstract Estimating the distribution and magnitude of heat generation within electronics packages is pivotal for thermal packaging design and active thermal management systems. Inverse heat conduction methods can provide estimates using measured temperature profiles acquired using infrared imaging or discrete temperature sensors. However, if the heater locations are unknown, applying a fine grid of potential heater locations across the surface where heat generation is expected can result in prohibitively-large computation times. In contrast, using a more computationally-efficient coarse grid can reduce the accuracy of heat flux estimations. This paper evaluates two methods for reducing computation time using a sensitivity-coefficient method for solving the inverse heat conduction problem. One strategy uses a coarse grid that is refined near the hot spots, while the other uses a fine grid of potential heaters only near the hot spots. These grid-refinement methods are compared using both input temperature maps acquired from a "numerical experiment" (where the outputs of a 3D steady-state thermal model in FloTHERM are used for input temperatures) and temperature maps procured using infrared microscopy on a real electronics package. Compared to the coarse-grid method, the fine-grid method reduces computation time without significantly reducing accuracy, making it more convenient for designing and testing electronics packages. It also avoids the problem of "false hot spots" that occurs with the coarse-grid method. Overall, this approach provides a mechanism to predict hot spot locations during design and testing and a tool for active thermal management.

scholarly journals On polynomials of Sheffer type arising from a Cauchy problem

2003 ◽  
Vol 2003 (33) ◽  
pp. 2119-2137
Author(s):  
D. G. Meredith
Keyword(s):  
Cauchy Problem ◽  
Heat Conduction ◽  
Parabolic Equations ◽  
Analytic Solution ◽  
Heat Conduction Problem ◽  
Laguerre Polynomials ◽  
Inverse Heat Conduction Problem ◽  
Inverse Heat Conduction ◽  
Operator Calculus ◽  
Classical Polynomials

A new sequence of eigenfunctions is developed and studied in depth. These theta polynomials are derived from a recent analytic solution of the canonical Cauchy problem for parabolic equations, namely, the inverse heat conduction problem. By appealing to the methods of the operator calculus, it is possible to categorize the new functions as polynomials of binomial and Sheffer types. The connection of the new set with the classical polynomials of Laguerre is carefully examined. Some integral relations involving the Laguerre polynomials and the theta polynomials are presented along with a number of binomial identities. The inverse heat conduction problem is revisited and an analytic solution depending on the generalized theta polynomials is presented.

A Mathematical Representation of the ion Nitriding Process

1987 ◽  
Vol 98 ◽  
Author(s):  
J -L. Marchand ◽  
D. Ablitzer ◽  
J. Szekely ◽  
H. Michel ◽  
M. Gantois
Keyword(s):  
Electron Number Density ◽  
Mathematical Formulation ◽  
Temperature Fields ◽  
Nitriding Process ◽  
Mathematical Representation ◽  
Number Density ◽  
Ion Nitriding ◽  
Electron Number ◽  
Theoretical Predictions ◽  
Experimental Findings

ABSTRACTA mathematical formulation and computed results are presented to describe the velocity fields, temperature fields and concentration of the activated species in an ion nitriding process, operated at 1–5 torr pres-sure. The theoretical predictions, which are based on the two-dimensional trasnport equations and on a model for computing the electron number density, gave results in broad agreement with experimental findings reported by others for similar systems.

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