Marcelo J. Colaço

Inverse and Optimization Problems in Heat Transfer

This paper presents basic concepts of inverse and optimization problems. Deterministic and stochastic minimization techniques in finite and infinite dimensional spaces are revised; advantages and disadvantages of each of them are discussed and a hybrid technique is introduced. Applications of the techniques discussed for inverse and optimization problems in heat transfer are presented. Keywords : Inverse problems, optimization, heat transfer

COMPARISON OF DIFFERENT VERSIONS OF THE CONJUGATE GRADIENT METHOD OF FUNCTION ESTIMATION

The inverse problem of estimating the spatial and transient variations of the heat transfer coefficient at the surface of a plate, with no information regarding its functional form, is solved by applying the conjugate gradient method with adjoint problem. Three different versions of this method, corresponding to different procedures of computing the search direction, are applied to the solution of the present inverse problem. They include the Fletcher-Reeves, Polak-Ribiere, and Powell-Beale versions. Such versions are compared for test cases involving different numbers of sensors, levels of measurement errors, and initial guesses used for the iterative procedure.

Recovering the source term in a linear diffusion problem by the method of fundamental solutions

This work considers the detection of the spatial source term distribution in a multidimensional linear diffusion problem with constant (and known) thermal conductivity. This work can be physically associated with the detection of non-homogeneities in a material that are inclusion sources in a heat conduction problem. The uniqueness of the inverse problem is discussed in terms of classes of identifiable sources. Numerically, we propose to solve these inverse source problems using fundamental solution-based methods, namely an extension of the method of fundamental solutions to domain problems. Several examples are presented and the numerical reconstructions are discussed.

Approximation of the likelihood function in the Bayesian technique for the solution of inverse problems

This work deals with the use of radial basis functions for the interpolation of the likelihood function in parameter estimation problems. The focus is on the use of Bayesian techniques based on Markov Chain Monte Carlo (MCMC) methods. The proposed interpolation of the likelihood function is applied to test cases of inverse problems in heat and mass transfer, solved with the Metropolis–Hastings algorithm. The use of the interpolated likelihood function reduces significantly the computational cost associated with the implementation of such Markov Chain Monte Carlo method without loss of accuracy in the estimated parameters.

A response surface method-based hybrid optimizer

In this article, we describe a hybrid optimizer based on a highly accurate response surface method, which uses several radial basis functions and polynomials as interpolants. The response surface is capable to interpolate linear as well as highly non-linear functions in multi-dimensional spaces having up to 500 dimensions. The accuracy, robustness, efficiency, transparency and conceptual simplicity are discussed. Based on the extensive testing performed on 296 test functions, the radial basis functions (RBFs) approach seems computationally easy to implement and results are superior, requiring small computing time. The performance of the RBF approximation is compared with wavelets neural networks for several selected test cases and the optimizer is compared with other hybrid optimizers, as well as with the IOSO commercial code.

CONTROL OF UNSTEADY SOLIDIFICATION VIA OPTIMIZED MAGNETIC FIELDS

ABSTRACT This article presents a numerical procedure for automatically controlling desired features of a melt undergoing solidification by applying an external magnetic field whose time-varying intensity and spatial distribution are obtained by the use of a hybrid optimization algorithm. The intensities of the magnets along the boundaries of the container were discretized by using B-splines. The inverse problem is then formulated to find the magnetic boundary conditions (the coefficients of the B-splines) in such a way that the gradients of temperature along the gravity direction are minimized at each instant as the solidification front advances through a moving melt. For this task, a hybrid optimization code was used that automatically switches among the following six optimization modules; the Davidon-Fletcher-Powell (DFP) gradient method, a genetic algorithm (GA), the Nelder-Mead (NM) simplex method, quasi-Newton algorithm of Pshenichny-Danilin (LM), differential evolution (DE), and sequential quadratic programming (SQP). Transient Navier-Stokes and Maxwells equations were discretized by using a finite volume method in a generalized curvilinear nonorthogonal coordinate system. For the phase change problems, an enthalpy formulation was used. The computer code was validated against analytical and numerical benchmark results with very good agreements in both cases.

Optimizing chemistry of bulk metallic glasses for improved thermal stability

Thermo-mechanical-physical properties of bulk metallic glasses (BMGs) depend strongly on the concentrations of each of the chemical elements in a given alloy. The proposed methodology for simultaneously optimizing these multiple properties by accurately determining proper concentrations of each of the alloying elements is based on the use of computational algorithms rather than on traditional experimentation, expert experience and intuition. Specifically, the proposed BMG design method combines an advanced stochastic multi-objective evolutionary optimization algorithm based on self-adapting response surface methodology and an existing database of experimentally evaluated BMG properties. During the iterative computational design procedure, a relatively small number of new BMGs need to be manufactured and experimentally evaluated for their properties in order to continuously verify the accuracy of the entire design methodology. Concentrations of the most important alloying elements can be predicted so that new BMGs have multiple properties optimized in a Pareto sense. This design concept was verified for superalloys using strictly experimental data. Thus, the key innovation here lies in arriving at the BMG compositions which will have the highest glass forming ability by utilizing an advanced multi-objective optimization algorithm while requiring a minimum number of BMGs to be manufactured and tested in order to verify the predicted performance of the predicted BMG compositions.

Optimization of Intensities and Orientations of Magnets Controlling Melt Flow During Solidification

Abstract When growing large single crystals from a melt, it is desirable to minimize thermally induced convection effects so that solidification is achieved predominantly by thermal conduction. It is expected that under such conditions any impurities that originate from the walls of the crucible will be less likely to migrate into the mushy region and consequently deposit in the crystal. It is also desirable to achieve a distribution of the dopant in the crystal that is as uniform as possible. A finite volume method and a least-squares spectral finite element method were used to develop accurate computer codes for prediction of solidification from a melt under the influence of externally applied magnetic fields. A hybrid constrained optimization algorithm and a semi-stochastic self-adapting response surface optimizer were then used with these solidification analysis codes to determine the distributions of the magnets that will minimize the convective flow throughout the melt or in desired regions of the melt only.

Accelerated Bayesian Inference for the Estimation of Spatially Varying Heat Flux in a Heat Conduction Problem

This article aims at the acceleration of an inverse heat transfer problem solution within the Bayesian framework. The physical problem involves a spatially varying heat flux, which can reach very large magnitudes in small regions, such as in the heating imposed by high-power lasers. The inverse problem of estimating the imposed heat flux is solved by using the Markov chain Monte Carlo method, with simulated transient temperature measurements. The solution of the inverse problem is based on a reduced model, which consists of an improved lumped formulation of a linearized version of the original nonlinear problem. Two different priors are considered for the sought heat flux, including a total variation density and a Gaussian density. The Gaussian prior is based on the physics of the heat conduction problem. Parameters appearing in both priors are also estimated as part of the inference problem in hyperprior models. The Delayed Acceptance Metropolis-Hastings (DAMH) Algorithm and the Enhanced Approximation Error Model (AEM) are applied with the objective to improve the accuracy of the inverse problem solution.

Solidification of Double-Diffusive Flows Using Thermo-Magneto-Hydrodynamics and Optimization

A multilevel approach, based on our previously developed hybrid optimizer, is presented for solving a problem that consists of a solidifying thermosolutal flow in a square cavity subjected to variable thermal and magnetic boundary conditions. The objective is to reduce the standard deviations of the vorticity within the liquid region as well as reduce the liquid area over the entire domain. Thus, the optimization problem is formulated to simultaneously find thermal and magnetic boundary conditions that must induce such prescribed solute concentration and velocity profiles. The optimizer is based on several deterministic and evolutionary techniques with automatic switching among them, combining the best feature of each algorithm. A radial basis function based response surface scheme is implemented to reduce the overall computing time.