Resolução da equação de Laplace aplicada a problemas diretos e inversos de transferência de calor por condução

Abstract

The study of heat transfer mechanisms configures an area of great interest due to various applications that can be developed. Mathematically, these phenomena are usually represented by partial differential equations associated with initial and boundary conditions related to domain of interest. In general, the resolution of these problems requires the application of numerical techniques through the discretization of contour and internal points of domain considered, resulting in a great computational cost to solve the system obtained. As alternative to reduce the computational cost, in last years, various studies based on Meshless (Meshfree) Methods have been developed. Basically, in these methods there is no need to generate meshes at points inside the domain, simplifying the treatment of problems with complex geometries, as well the reduction of computational cost related to need to reconstruct the computational mesh in each iteration. However, the systems resulting from the application of this type of methodology are inherently illconditioned, being necessary the application of regularization techniques to obtain a reliable solution. In this contribution, the aim is to formulate and to solve direct and inverse problems applied to Laplace Equation in steady state and bi-dimensional system considering different geometries. For this purpose, the Method of Fundamental Solutions (MFS) is considered as methodology for solving the direct problem and the Differential Evolution (DE) algorithm as optimization tool for solving the inverse problem. In addition, the influence of parameters required by using MSF on quality of solution obtained and the methodology used for the treatment of ill-conditioned problems is also evaluated. From the obtained results it was possible to observe that the MFS was able to obtain equivalent results when compared to the Finite Differences Method. In addition, the size of radius required by MSF is one of factors that most influence the precision of numerical approach and the use of a regularization technique is very important for obtaining a reliable solution. In relation to inverse problem, it was possible to conclude that the results obtained by proposed methodology (MFS+DE+Tikhonov Regularization Technique) were considered satisfactory, as even with different levels of noise, good estimates for design variables in proposed inverse problem were obtained