Level Set–Based Topological Shape Optimization of Heat Conduction Problems Considering Design-Dependent Convection Boundary

A level set–based topological shape optimization method considering design-dependent convection boundaries is developed for steady-state heat conduction problems. We embed the level set function obtained from a Hamilton-Jacobi type of equation into a fixed initial domain to implicitly represent thermal boundaries. The effects of the implicit convection boundary obtained from topological shape variations are represented by numerical Dirac delta and Heaviside functions. The method minimizes the thermal compliance of systems by varying the implicit boundary, satisfying the constraint of allowable material volume. During design optimization, the boundary velocity to integrate the Hamilton-Jacobi equation is derived from an optimality condition.     

Level Set-based Topological Shape Optimization of Nonlinear Heat Conduction Problems

A level set-based topological shape optimization method is developed for nonlinear heat conduction problems. While minimizing the objective function of instantaneous thermal compliance and satisfying the constraint of allowable volume, solution of the Hamilton-Jacobi equation leads the initial boundary to an optimal one according to the normal velocity field determined from the descent direction of the Lagrangian. To overcome the convergence difficulty in nonlinear problems resulting from introduction of an approximate boundary, an actual boundary is identified by tracking the level set functions and remeshing using Delaunay triangulation. The velocity field outside the actual domain is determined through a velocity extension scheme.     

SENSITIVITY ANALYSIS AND OPTIMAL DESIGN FOR STEADY CONDUCTION PROBLEM WITH RADIATIVE HEAT TRANSFER

ABSTRACT

A steady heat conduction problem is considered, that is described by the heat conduction equation and the thermal boundary conditions (i.e., Dirichlet, Neumann, Henkel, and radiation conditions on the external boundary, and radiation condition on the hole boundary). An arbitrary behavioral functional is defined and its first-order sensitivity is derived using both the direct and the adjoint approaches. The shape optimization problem is next formulated and two optimization functionals are discussed. The simple numerical example is presented.     

A Numerical Method for Multiphase Incompressible Thermal Flows with Solid–Liquid and Liquid–Vapor Phase Transformations

ABSTRACT

A numerical method for multiphase incompressible thermal flows with solid–liquid and liquid–vapor phase transformations is presented. The flow is mainly driven by thermocapillary force and vaporization. Based on the level set method and mixture continuum model, a set of governing equations valid for solid, liquid, and vapor phases is derived, considering phase boundary conditions as source terms in the transport equations. The vaporization process is treated as a source term in the continuity equation. The model developed is applied to the laser welding process, where the flow is coupled with optical phenomena. Formation and collapse of a laser-created hole is simulated.     

A novel numerical method for steady-state thermal simulation based on loop-tree and HBRWG basis functions

Abstract

This article presents a novel numerical method for steady-state thermal simulation. This method firstly solves the heat flux efficiently by applying the loop-tree basis functions. Then, the temperature is obtained by finding solutions of the gradient equation. The half boundary Rao-Wilton-Glisson (HBRWG) basis functions are employed for handling arbitrary boundary conditions. In addition, the triangulation-based interpolation technique is utilized to interpolate temperature profile with obtained results in post-processing. Three examples with mixed boundary conditions are studied to validate the accuracy of proposed method for simulating steady-state thermal problems. Numerical results show that our method has a good accuracy and is well capable of handling arbitrary boundary conditions.     

Development of a mass-preserving level set redistancing algorithm for simulation of rising bubble

Abstract

This article is aimed to simulate the gas-liquid flow of rising bubbles with a mass-preserving level set method. To resolve the topological changes of gas-liquid interface where the classic finite difference scheme may yield oscillation solutions, the spatial terms in the level set advection equation will be approximated by an optimized compact reconstruction weighted essentially non-oscillatory (OCRWENO) scheme. This scheme achieves high-order accuracy in smooth regions, and meanwhile avoid numerical oscillation near discontinuities. Two benchmark problems including vortex flow and deforming field are chosen to compare the present simulation with previous numerical researches. Several rising bubble problems are validated by the proposed level set method.     

Developing a ghost fluid lattice Boltzmann method for simulation of thermal Dirichlet and Neumann conditions at curved boundaries

ABSTRACT

In this paper, a ghost fluid thermal lattice Boltzmann method is developed to simulate Dirichlet and Neumann thermal boundary conditions at curved boundaries. As such, a new formulation for both thermal boundary conditions is developed using a bilinear interpolation method. The presented method is also formulated to address the special cases that arise when the values of the macroscopic variables are interpolated at the image points surrounded by many solid nodes as well as the fluid nodes. The results of the presented method are compared to those available in the literature from conventional numerical methods, and excellent agreement is observed.     

Topology optimization of heat and mass transfer problems in two fluids—one solid domains

Abstract

Among various topology optimization methods used in fluid flow problems, density approach has gained more interest compared to other techniques as level set approach, topological derivative technique, and phase field method. The key part of density approach is the penalized interpolation function, which forces progressively porous cells made of fluid and solid simultaneously to belong discretely to fluid or solid sub-domains. However this type of problem was only solved in mono-fluid domains, in which the method accounts for the distribution of a single fluid and a single solid. The actual work aims to extend topology optimization in fluid flow problems to bi-fluid domain. A new interpolation function was developed for this purpose. Furthermore a penalization function was integrated in the multiobjective function, which ensure that each fluid takes its own path in the device, while maintaining a minimal required solid thickness between the channels of different fluids. The results showed the capacity of the proposed method to deal with multiple fluid phases in minimizing the pressure drop while maximizing heat exchange between different flows. The main conclusion is the potential of density approach to be applied on optimization of heat exchangers.     

A variational approach for free vibrating micro-rods with classical and non-classical new boundary conditions accounting for nonlocal strengthening and temperature effects

Abstract

Transverse vibration of a circular cross sectional micro-rod subjected to a new kind of boundary constraints with elastic torsional springs is presented based on nonlocal elasticity. A nonlocal strengthening beam model is utilized and the effect of temperature changing is taken into consideration. The variational method and Hamilton’s principle are applied to derive the governing equation of motion and corresponding boundary conditions. A higher-order partial differential equation that is a typical characteristic of nonlocal strengthening model is developed, and the boundary conditions contain not only classical conditions but also non-classical higher-order conditions. Unlike previous studies which were only concerned with some conventional boundary constraints, we consider more general boundary conditions named elastic torsional spring supports. Such boundary conditions are between the simply supported and clamped ones, and they are closer to the actual constraints of existing engineering structures. Natural frequencies of micro-rods with new boundary constraints are determined via an eigenvalue method and compared with other results in the literature. It is shown that the nonlocal scale factor, thermal parameter, rigidity parameter and torsional spring coefficient play significant roles in free vibration of micro-rods. The research can provide a reference for a large class of boundary conditions ranging from simply supported to clamped micro-rods.     

Sensitivity analysis and shape optimization for transient heat conduction with radiation

Transient heat conduction problem is stated by the differential heat conduction equation, thermal boundary conditions on the external and internal boundary portions and the initial condition within the domain. Next an arbitrary behavioral functional is defined and its first-order sensitivities are determined using the material derivative concept as well as both the direct and adjoint approaches. The most used shape domain modifications are discussed in order to investigate the effect of design parameters on the integral radiation condition. The shape optimization problem is next formulated applying the obtained sensitivities. The illustration is the simple example of the shape optimization.     

Global,nonparametric, noniterative optimization of time-averaged quantities under small,time-varying forcing: An application to a thermal convection field

Abstract

This study demonstrates a global, nonparametric, noniterative optimization of time-mean value of a kind of index vibrated by time-varying forcing. It is based on the fact that the (steady) forced vibration of nonautonomous ordinary differential equation systems is well approximated by an analytical solution when the amplitude of forcing is sufficiently small and its base state without forcing is linearly stable and steady. It is applied to optimize a time-averaged heat-transfer rate on a two-dimensional thermal convection field in a square cavity with horizontal temperature difference, and the globally optimal way of vibrational forcing, i.e. the globally optimal, spatial distribution of vibrational heat and vorticity sources, is first obtained. The maximized vibrational thermal convection corresponds well to the state of internal gravity wave resonance. In contrast, the minimized thermal convection is weak, keeping the boundary layers on both sidewalls thick.     

Direct integration of three-dimensional thermoelasticity equations for a transversely isotropic layer

Abstract

An application of the direct integration method is presented for analytical solution of a three-dimensional thermoelasticity problem for a transversely isotropic layer subjected to general thermal loadings on the limiting planes and internal heat sources. By making use of relations between the stress-tensor components, derived from the equilibrium equations, the original thermoelasticity problem was reduced to a set of governing equations for individual stresses with corresponding boundary conditions. In the mapping domain of the Fourier double-integral transform, the explicit analytical solutions for the stresses and temperature are obtained and then reproduced in physical domain.     

Economic evaluation and optimization of solar heating systems

A procedure is developed for assessing the economic viability of a solar heating system in terms of the life cycle savings of a solar heating system over a conventional heating system. The life cycle savings is expressed in a generalized formby introducing two economic parameters, P₁ and P₂, which relate all life cycle cost considerations to the first year fuel cost or the initial solar system investment cost. Using the generalized life cycle savings equation, a method is developed for calculating the solar heating system design which maximizes the life cycle savings. A similar method is developed for determining the set of economic conditions at which the optimal solar heating system design is just competitive with the conventional heating system. The results of these optimization methods can be presented in tabular or graphical form. The sensitivity of the economic evaluation and optimization calculations to uncertainties in constituent thermal and economic variables is also investigated.     

Identifying heat conductivity and source functions for a nonlinear convective-diffusive equation by energetic boundary functional methods

Abstract

In the article, we solve the inverse problems to recover unknown space-time dependent functions of heat conductivity and heat source for a nonlinear convective-diffusive equation, without needing of initial temperature, final time temperature, and internal temperature data. After adopting a homogenization technique, a set of spatial boundary functions are derived, which satisfy the homogeneous boundary conditions. The homogeneous boundary functions and zero element constitute a linear space, and then a new energetic functional is derived in the linear space, which preserves the time-dependent energy. The linear systems and iterative algorithms to recover the unknown parameters with energetic boundary functions as the bases are developed, which are convergent fast at each time marching step. The data required for the recovery of unknown functions are parsimonious, including the boundary data of temperatures and heat fluxes and the boundary data of unknown functions to be recovered. The accuracy and robustness of present methods are confirmed by comparing the exact solutions with the identified results, which are obtained under large noisy disturbance.     

Stress constrained thermo-elastic topology optimization based on stabilizing control schemes

Abstract

This article proposes two effective stabilizing control schemes for addressing the stress constrained thermo-elastic topology optimization in a non-uniform temperature field. Based on the density interpolation scheme, two linear elastic equations for coupling a thermo-elastic problem are considered. For comparison, different topology problem formulations for minimizing compliance or volume subject to stress constraints are solved. By virtue of a stabilization transform method, two stabilizing control schemes combined with the grouped aggregation method are developed to handle the challenging difficulties stemming from the local nature of highly nonlinear stress constraints. Moreover, the adjoint method is adopted to perform the sensitivity analysis. The design variables are updated by utilizing the method of moving asymptotes. The results of several typical numerical examples verify the validity of the proposed methodology, including the present stabilizing control schemes which can be employed to obtain clear topological design and fast convergence rate for thermo-elastic coupling problems. Meanwhile, compliance minimization design with stress constraints is appropriate to achieve balance between stress level and stiffness.     

Generalized Coupled Thermoelasticity of a Layer

The generalized thermoelasticity based on the Lord-Shulman (LS), Green-Lindsay (GL), and Green-Naghdi (GN) theories admit the second sound effect. By introducing some parameters all these theories are combined and a unified set of equations is rendered. These equations are then solved for a layer of isotropic and homogeneous material to study the thermal and mechanical wave propagations. The disturbances are generated by a sudden application of temperature to the boundary. The non-dimensionalized form of the governing equations are solved utilizing the Laplace transform method in time domain. Closed form solutions are obtained for the layer in Laplace transform domain, and a numerical inverse Laplace transform method is used to obtain the temperature, displacement, and stress fields in the physical time domain. The thermo-mechanical wave propagations and reflections from the layer boundaries are investigated.     

AN INVERSE SOURCE PROBLEM IN RADIATIVE TRANSFER FOR SPHERICAL MEDIA

Abstract

A method is presented to identify the source term in a one-dimensional, absorbing, emitting, scattering spherical medium from the knowledge of the exit radiation intensities. The inverse radiation problem is formulated as an optimization problem. The sensitivity problem and the gradient equation are derived. The conjugate gradient method is used for its solution. Although the source term is a function of the space variable, only radiation intensities exiting the outer boundary are required. Both data with and without measurement errors are used as input to identify the source term. The study shows that the estimation of the source term is more sensitive to increases in measurement errors as the optical thickness increases.     

Transient Thermal Analysis of Engine Exhaust Valve

ABSTRACT

Internal combustion engines produce exhaust gases at extremely high temperatures and pressures. As these hot gases pass through the exhaust valve, temperatures of the valve, valve seat, and stem increase. To avoid any damage to the exhaust valve assembly, heat is transferred from the exhaust valve through different parts, especially the valve seat insert during the opening and closing cycle as they come into contact with each other. In this article, a finite-element method is used for modeling the transient thermal analysis of an exhaust valve. The temperature distribution and resultant thermal stresses at each opening and closing time are obtained. Detailed analyses are performed to estimate the boundary conditions of an internal combustion engine. The model includes exhaust valve, seat, guide, and spring. The analysis continues until a steady-state condition is obtained. In this study, ANSYS is employed for modeling and analysis of the exhaust valve. A methodology is developed for transient thermal analysis of the exhaust valve.     

Novel approach of thermal modeling of partially dry–wet chilled water cooling coil under unit and nonunit Lewis number conditions

ABSTRACT

The main objective of this work is to present a new modeling approach of thermal performance and design of partially dry–wet cooling coils working under unit or nonunit Lewis number conditions. The innovative model is presented as a new simplified and practical correlation that interrelates the cooling coil effectiveness (ε) with its number of transfer unit, and vice versa. The simplified model was constructed on a basis of solving the heat and mass transfer equation “enthalpy potential method” simultaneously coupled with the thermodynamics equations. The validity of the new correlations was tested through predictions of its thermal performance. The output results of those correlations show satisfactory agreement with those obtained from the referenced data with deviation of less than 10%. The main feature of this novel correlation is its simplicity and easiness in calculation by knowing input Lewis number and some other key parameters. Also, the main benefit of this new model is to provide helpful guidelines for optimization of fully wet or partially dry–wet cooling coils’ performance and developing suitable control strategies to achieve higher thermal behavior of the cooling coil during its operation.     

EFFECT OF REYNOLDS AND PRANDTL NUMBERS ON TURBULENT CONVECTIVE HEAT TRANSFER IN A THREE-DIMENSIONAL SQUARE DUCT

Abstract

This paper presents the numerical prediction of thermal developing processes in a square cross-section duct for turbulent flow with isothermal walls. The algebraic stress model has been employed to predict the fully developed turbulent flow. This fully developed turbulent flow was numerically solved using the Teach Program. The three-dimensional energy equation was discretized and solved by the method of lines. According to this method, the energy equation is reformulated by a system of first-order differential equations control-ting the temperature along each line. A nonuniform grid of 8 × 8 nodes was used for calculating the temperature profile at each cross section. The effects of Prandtl and Reynolds numbers on the thermal behavior in the entrance region are investigated. The computed results for the fully developed region are shown to be in good agreement with the measured data publisked in the open literature