System Of Heat Equations Research Articles

Simulation and multi-objective optimization to improve the final shape and process efficiency of a laser-based material accumulation process

Common goals of modern production processes are precision and efficiency. Typically, they are conflicting and cannot be optimized at the same time. Multi-objective optimization methods are able to compute a set of good parameters, from which a decision maker can make a choice for practical situations. For complex processes, the use of physical experiments and/or extensive process simulations can be too costly or even unfeasible, so the use of surrogate models based on few simulations is a good alternative.In this work, we present an integrated framework to find optimal process parameters for a laser-based material accumulation process (thermal upsetting) using a combination of meta-heuristic optimization models and finite element simulations. In order to effectively simulate the coupled system of heat equation with solid-liquid phase transitions and melt flow with capillary free surface in three space dimensions for a wide range of process parameters, we introduce a new coupled numerical 3d finite element method. We use a multi-objective optimization method based on surrogate models. Thus, with only few direct simulations necessary, we are able to select Pareto sets of process parameters which can be used to optimize three or six different performance measures.

The Discrete Model for the System of the Myocardium and Coronary Vessels

Background. The numerical heat transfer model for a system of myocardium coronary vessels is considered. Objective. The goal is to develop a discrete model for the physical system of myocardium and coronary vessels that would make it possible to explore the process of hypo- and hyperthermia with cardiopulmonary bypass. Methods. To solve the differential equation of heat conduction in the MSC Sinda thermal system the network method (TNM – Thermal Network Method) is used, in which system of heat equations is presented in the form of cellular-centered nodes and resistances between the nodes using the finite difference method. In constructing the model of myocardial in the MSC Sinda system the thermal contact between three-dimensional bodies is implemented – the myocardium, coronary arteries, a liquid cooling of heart. Results. Implementation of the model of heat exchange in the MSC Sinda system for infarction cooling process gives on the final process step in establishing the heat balance the temperature difference at the boundary between the myocardium and coronary vessels not more than 0,5 °C. However, in the areas of the myocardium that are removed from the coronary vessels the temperature difference exceeds 1,0 °C. The use of additional cooling for hearts allows for the cooling of myocardium with using of ice surface, that provides the unevenness reduction of the heart temperature during its cooling with cardiopulmonary bypass. This result allows exploring the dynamics of the process of hypo- and hyperthermia with cardiopulmonary bypass. Conclusions. The discrete 3D-model of heat transfer in the layer structure of the myocardium and coronary vessels allows us to investigate the process of hypo- and hyperthermia with cardiopulmonary bypass. The simulation results also make it possible to perform the analysis of the temperature distribution on the surface of the myocardium provided free convection of heat between the layers.

Existence and Uniqueness of Solutions to Heat Equations with Hysteresis Coupled with Navier–Stokes Equations in 2D and 3D

This paper is concerned with a system of heat equations with hysteresis and Navier–Stokes equations. In Tsuzuki (J Math Anal Appl 423:877–897, 2015) an existence result is obtained for the problem in a 2-dimensional domain with the Navier–Stokes equation in a weak sense. However the result does not include uniqueness for the problem due to the low regularity for solutions. This paper establishes existence and uniqueness in 2- and 3-dimensional domains with the Navier–Stokes equation in a stronger sense. Moreover this work decides required height of regularity for the initial data by introducing the fractional power of the Stokes operator.

Mathematical modeling of a water-in-oil emulsion droplet behavior under the microwave impact

The problem of microwave (MW) electromagnetic radiation impact on a single water-in-oil droplet is considered. The system of heat equations within the droplet and in the surrounding liquid, incompressible Navier-Stokes equations within the droplet and in the surrounding liquid, and equation of state are considered. The formulated problem is solved numerically using TDMA (Tri-diagonal-matrix algorithm), SIMPLE algorithm and VOF method (volume of fluid method for the dynamics of free boundaries) in Euler coordinates. The results in the form of the dependence of the temperature within the droplet and in the surrounding liquid on the time of microwave impact and streamlines thermal convection are represented; dependence of the velocity of droplet's moving on the power of the of the microwave impact is shown. The obtained results can help to establish criteria for the efficient applicable of the microwave method for the water-in-oil emulsions destruction.

Existence of solutions to heat equations with hysteresis coupled with Navier–Stokes equations in 2D domains

This paper is concerned with a system of heat equations with hysteresis and Navier–Stokes equations in two-dimensional domains. There are some studies on the system of heat equations and Navier–Stokes equations in which the temperature (the solution to heat equations) is controlled by obstacle functions. In this paper, the control of the temperature is precisely described by introducing hysteresis. This paper provides the existence of solutions along the abstract theory for evolution equations with subdifferential operators.

Optimal boundary control of a hyperbolic system of heat equations

Optimal boundary control of a hyperbolic system of heat equations

Numerical quenching of a system of equations coupled at the boundary

AbstractWe study numerical approximations of solutions of the following system of heat equations, coupled at the boundary through a nonlinear flux: where p and q are parameters. We prove that the solutions of a semidiscretization in space quench in finite time. Moreover, we describe in terms of p and q the simultaneous versus non‐simultaneous quenching phenomena. We also find the numerical extinction sets. Finally, in order to obtain the correct quenching rates in the non‐simultaneous case we present some adaptive methods. Copyright © 2009 John Wiley & Sons, Ltd.

A 2D domain boundary estimation

This paper presents a numerical method for solving a problem usually encountered in thermal imaging. The goal is to estimate an interior boundary of a material by applying a known heat flux and measuring the induced temperature response on its external boundary. The interior boundary is assumed to be under a homogeneous Neumann condition. This boundary is first parameterized by a finite-term of Fourrier series and the corresponding approximate inverse problem is numerically optimized using an iterative Newton method. The required gradient is established using the domain derivative techniques. The system of heat equations is treated using finite element method for space and implicit scheme for time. Some numerical tests are provided to illustrate the performances of the proposed method.

Incomplete quenching of heat equations with absorption

This article deals with a system of heat equation with absorption. This system exhibits simultaneous and non-simultaneous quenching. We give a natural continuation for problem beyond quenching time T for the case of non-simultaneous quenching.

Blow-up for a system ofheat equations with nonlocal sources and absorptions

In this paper we establish the local existence of the nonnegative solution and the finite time blow-up results for the following parabolic system: u t = Δ u + ∫ Ω v p d x − a u r , x ∈ Ω , t > 0 , v t = Δ v + ∫ Ω v q d x − b v s , x ∈ Ω , t > 0 , subject to homogeneous Dirichlet conditions and nonnegative initial data, where p, q, r, s ≥ 1 and a, b > 0. In the situation when the nonnegative solution ( u, v) of the above problem blows up in finite time, we show that the blow-up is global and this differs from the local source case. Moreover, it is proved that for the special case r = s = 1, if ( u, v) blows up then the limits lim ⁡ t → T * ( T * − t ) ( p + 1 ) / ( p q − 1 ) u ( x , t ) = ( p + 1 ) 1 / ( p q − 1 ) ( q + 1 ) p / ( p q − 1 ) × ( ( p q − 1 ) | Ω | ) − ( p + 1 ) / ( p q − 1 ) , lim ⁡ t → T * ( T * − t ) ( q + 1 ) / ( p q − 1 ) v ( x , t ) = ( p + 1 ) q / ( p q − 1 ) ( q + 1 ) 1 / ( p q − 1 ) × ( ( p q − 1 ) | Ω | ) − ( q + 1 ) / ( p q − 1 ) converge uniformly on compact subsets of Ω, where T * is the blow-up time.

On the Statistics of Partial Quotients of Finite Continued Fractions

A system of heat equations in a nonholonomic frame is considered. Solutions of the system are constructed in the form of general sigma functions of Abelian tori. As a corollary, we solve the problem (of general interest) to describe the generators of the ring of differential operators annihilating the sigma functions of families of plane algebraic curves.

Blowup behaviors for diffusion system coupled through nonlinear boundary conditions in a half space

This paper deals with the blowup estimates near the blowup time for the system of heat equations in a half space coupled through nonlinear boundary conditions. The upper and lower bounds of blowup rate are established. The uniqueness and nonuniqueness results for the system with vanishing initial value are given.

The blow-up estimate for heat equations with non-linear boundary conditions

This paper deals with the blow-up rate estimates of positive solutions for a system of heat equations with non-linear boundary conditions. The upper and lower bounds of blow-up rate are obtained.

The blow-up rates for systems of heat equations with nonlinear boundary conditions

This paper deals with the blow-up rate estimates of positive solutions for systems of heat equations with nonlinear boundary conditions. The upper and lower bounds of blow-up rate are obtained.

The blow-up rate for a system of heat equations with neumann boundary conditions

This paper deals with the blow-up properties of solutions to a system of heat equations ut=Δu, vt=Δv in BR×(0, T) with the Neumann boundary conditions eu/eη=ev, ev/eη=eu on SR×[0, T). The exact blow-up rates are established. It is also proved that the blow-up will occur only on the boundary.

Uniqueness and non-uniqueness for a system of heat equations with nonlinear coupling at the boundary

We study the uniqueness problem for non negative solutions of a system of two heat equations, ut = ∆u, vt = ∆v, in a bounded smooth domain Ω, with nonlinear boundary conditions, ∂u ∂η = v, ∂v ∂η = u. We prove that for identically zero initial data, (u(x, 0), v(x, 0)) ≡ (0, 0), the zero solution is unique if and only if if pq ≥ 1. Moreover, in the case of non-negative non-trivial initial data the solution is always unique.

The blow-up rate for a system of heat equations with nonlinear boundary conditions

This paper deals with the blow-up rate estimates of positive solutions for systems of heat equations with nonlinear boundary conditions. The upper and lower bounds of blow-up rate are obtained.

The blow-up rate for a system of heat equations completely coupled in the boundary conditions

The blow-up rate for a system of heat equations completely coupled in the boundary conditions

The blow-up property for a system of heat equations with nonlinear boundary conditions

This paper deals with the blow-up properties of solutions to the systems u t=Δu,vt=Δv in B RX(0,T) subject to nonlinear boundary conditions $$\frac{{\partial u}}{{\partial \eta }} = \upsilon ^p ,\frac{{\partial \upsilon }}{{\partial \eta }} = u^q $$ , in S RX(0,T). It is shown that under certain conditions the solution blows up at a finite time and the blow-up only occurs on the boundary. The self-similar solution for the one-dimensional case has been studied. Moreover, the exact blow-up rates are also derived.

Positive solutions to systems of heat equations with a semilinear boundary condition

Positive solutions to systems of heat equations with a semilinear boundary condition