N.S. Mera

AN INVERSION METHOD WITH DECREASING REGULARIZATION FOR THE BACKWARD HEAT CONDUCTION PROBLEM

In this article an iterative algorithm is proposed for solving numerically the one-dimensional backward heat conduction problem (BHCP). The algorithm consists of using iteratively a least-squares fitting of the given data with a regularization term that decreases as the number of iterations increases. The algorithm is implemented using the boundary-element method (BEM). The convergence and the stability of the method are investigated for a severe test example, hence revealing the computational performance and limitations of the method proposed.

The boundary element solution of the Cauchy steady heat conduction problem in an anisotropic medium

In this paper the iterative algorithm proposed by Kozlov et al. for the Cauchy problem for the Laplace equation is extended in order to solve the Cauchy steady-state heat conduction problem in an anisotropic medium. The iterative algorithm is numerically implemented using the boundary element method (BEM). The convergence and the stability of the numerical method, as well as various types of accuracy, convergence and stopping criteria, are investigated. The numerical results obtained confirm that provided an appropriate stopping regularization criterion is imposed, then the iterative BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data. An efficient stopping regularization criterion to cease the iterative process is proposed and the rate of convergence of the algorithm is improved by using various relaxation procedures between iterations. A new concept of a variable relaxation factor is proposed. Analytical formulae for the coefficients of the matrices resulting from the direct application of the BEM in an anisotropic medium are also presented. Copyright

A comparison of boundary element method formulations for steady state anisotropic heat conduction problems

In this paper the boundary element method (BEM) is numerically implemented in order to solve steady state anisotropic heat conduction problems. Various types of elements, namely, constant elements, continuous and discontinuous linear elements and continuous and discontinuous quadratic elements are used. The performances of these various BEM formulations are compared for both the direct well-posed Dirichlet problem and the inverse ill-posed Cauchy problem, revealing several features of the BEM. Furthermore, previously undetermined analytical solutions for the integrals associated with linear and quadratic elements are presented.

On the use of Genetic Algorithms for Solving Ill-Posed Problems

In this article we investigate the use of genetic algorithms for regularizing ill-posed problems. Various approaches of genetic algorithms are considered in order to deal with the numerical problems caused by the instability of the solution of ill-posed problems. Two test examples, a simple one-dimensional ill-posed problem and the Cauchy problem for the Laplace equation are considered in order to illustrate the performance and limitations of genetic algorithms when applied to inverse and ill-posed problems. The combination of genetic algorithms with other regularization methods such as the function specification method is also investigated.

Reaction mechanism reduction and optimisation for modelling aviation fuel oxidation using standard and hybrid genetic algorithms

Abstract This study describes the development of a new binary encoded genetic algorithm for the combinatorial problem of determining a subset of species and their associated reactions that best represent the full starting point reaction mechanism in modelling aviation fuel oxidation. The genetic algorithm has a dual objective in finding a reduced mechanism that best represents aviation fuel oxidation in both a laminar premixed flame and perfectly stirred reactor systems. The number of species in the subset chosen is kept fixed and is specified at the start of the procedure. The genetic algorithm chooses ever improving mechanisms based on an objective function which measures how well the new reduced mechanisms predict a set of species’ profiles simulated by the full mechanism. In order to verify the validity of our approach, a full enumeration was performed on a reduced problem and it was found that the genetic algorithm was able to find the optimum solution to this reduced problem after a few generations. The reduction involved going from 338 reactions involving 67 species to 215 reactions involving 50 species. This corresponded to a 90% CPU time saving in each function evaluation. A second step was to take the reduced reaction mechanism and to use a second real encoded genetic algorithm for the parameter optimisation problem of determining the optimal reaction rate parameters that best model an experimental set of premixed flame and jet stirred reactor species’ profiles. A significant improvement could be seen in the species profiles obtained using the mechanism with the GA optimised rates over those obtained from the original reduced mechanism. Further, in order to increase the efficiency of the second reaction rate coefficient optimisation step, a new hybrid method was developed which incorporates a direct optimisation method (Rosenbrock method) into the GA. A significant improvement in both accuracy and efficiency was apparent in using this new hybrid approach.

Incorporation of physical bounds on rate parameters for reaction mechanism optimization using genetic algorithms

In this study a genetic algorithm (GA) approach for determining new reaction rate parameters ( A , g , and E a in the non-Arrhenius expressions) for the combustion of a hydrogen/air mixture in a perfectly stirred reactor (PSR) is assessed. A new floating-point coded GA and fitness function have been developed that dramatically increase both the rate of convergence and the predictive accuracy of the algorithm, thus promising the extension of the method to more detailed reaction schemes. Output profiles of species for 20 sets of PSR conditions, obtained from an original set of rate constants, are reproduced following a GA optimization inversion process. The new sets of rate constants following each iteration are constrained to lie between predefined boundaries that represent the uncertainty associated with the experimental findings listed in the National Institute of Standards and Technology (NIST) database. Comparisons with previous optimization work have demonstrated that those mechanisms generated using the NIST constraints can be applied to combustion scenarios outside those used in the mechanisms construction. In addition, the flexibility of the GA has been demonstrated by its success in generating reaction rate coefficients that reproduce a set of randomly perturbed species profiles.

Numerical solution of a boundary detection problem using genetic algorithms

Abstract In this paper we consider the identification of the geometric structure of the boundary of the solution domain for the two-dimensional Laplace equation. We investigate the determination of the location, size and shape of an unknown portion γ⊂ ∂ Ω of the boundary ∂ Ω of a solution domain Ω⊂R 2 from Cauchy data on the remaining portion of the boundary ∂ Ω\γ or on a subset Γ⊂ ∂ Ω\γ of this part of the boundary. This problem arises in the study of quantitative non-destructive evaluation of corrosion in materials in which boundary measurements of currents and voltages are used to determine the material loss caused by corrosion. The domain identification problem is considered as a variational problem to minimise a defect functional, which utilises some additional data on certain known parts of the boundary. A real coded genetic algorithm, combined with a function specification method, is used in order to minimise the objective functional. The Laplace equation is discretised using the boundary element method. Numerical results are presented and discussed for several test examples.

A Dual Reciprocity Boundary Element Method for the Regularized Numerical Solution of the Inverse Source Problem Associated to the Poisson Equation

This article revisits the application of the Dual Reciprocity Method to a class of inverse problems governed by the Poisson equation in a thorough and careful manner. Here the term inverse refers to the fact that the non-homogenous part of the Poisson equation is unknown, i.e. the governing equation of the problem is unknown and has to be determined from Cauchy data at the boundary. We show that, although the inverse problem does not have a unique solution, by employing the Tikhonov regularization method we can recover the minimal norm solution. This is usually the solution of most practical interest from the many solutions of the ill-posed problem of source identification. Other different, more complex solutions might be recovered if estimates of these solutions are available at some particular points inside the solution domain.

A Novel Approach to Mechanism Reduction Optimization for an Aviation Fuel/Air Reaction Mechanism Using a Genetic Algorithm

This study presents a novel multiobjective genetic-algorithm approach to produce a new reduced chemical kinetic reaction mechanism to simulate aviation fuel combustion under various operating conditions. The mechanism is used to predict the flame structure of an aviation fuel/O 2 /N 2 flame in both spatially homogeneous and one-dimensional premixed combustion. Complex hydrocarbon fuels, such as aviation fuel, involve large numbers of reaction steps with many species. As all the reaction rate data are not well known, there is a high degree of uncertainty in the results obtained using these large detailed reaction mechanisms. In this study a genetic algorithm approach is employed for determining new reaction rate parameters for a reduced reaction mechanism for the combustion of aviation fuel-air mixtures. The genetic algorithm employed incorporates both perfectly stirred reactor and laminar premixed flame data in the inversion process, thus producing an efficient reaction mechanism. This study provides an optimized reduced aviation fuel-air reaction scheme whose performance in predicting experimental major species profiles and ignition delay times is not only an improvement on the starting reduced mechanism but also on the full mechanism.

Use of the boundary element method to determine the thermal conductivity tensor of an anisotropic medium

Abstract In this paper, we propose a numerical algorithm to simultaneously predict the unknown conductivity coefficients and the unknown boundary data for a steady-state heat conduction problem in an anisotropic medium. The algorithm is based on a classical boundary element method (BEM) which is combined with a least squares technique. The numerical convergence and stability of the method proposed is investigated with respect to increasing the number of additional measurements provided and decreasing the amount of noise added into the input data.