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- Using the finite element method to understand calculusPublication . Rodrigues, José Alberto; Loja, Amélia; Barbosa, JoaquimThis paper presents a complementary, alternative teaching and learning methodology based on the use of the finite element method to illustrate mathematical models and to explore their (numerical) solutions in the context of vector calculus properties understanding. This methodology is illustrated via a set of examples focused on specific engineering problems, but its scope can also be widened to other scientific areas. The examples presented on this paper allow concluding that this approach may be an interesting way to re-think and complement the perspective usually considered in the transmission of mathematical concepts. The use of a freeware finite element method computational package, FreeFEM++ may also be an important issue to stimulate the dissemination of this phenomena modelling and comprehension approach.
- Developments on finite element methods for medical image supported diagnosticsPublication . Almeida, Ana; Barbosa, Joaquim; Carvalho, André; Loja, Amélia; Portal, Ricardo; Rodrigues, José Alberto; Vieira, Lina OliveiraVariational image-processing models offer high-quality processing capabilities for imaging. They have been widely developed and used in the last two decades, enriching the fields of mathematics as well as information science. Mathematically, several tools are needed: energy optimization, regularization, partial differential equations, level set functions, and numerical algorithms. For this work we consider a second-order variational model for solving medical image problems. The aim is to obtain as far as possible fine features of the initial image and identify medical pathologies. The approach consists of constructing a regularized functional and to locally analyse the obtained solution. Some parameters selection is performed at the discrete level in the framework of the finite element method. We present several numerical simulations to test the efficiency of the proposed approach.
- Finite element techniques for medical image processingPublication . Almeida, A.; Barbosa, Joaquim; Carvalho, A.; Loja, Amélia; Portal, Ricardo; Rodrigues, José Alberto; Vieira, L.We consider some second-order variational model for solving medical image problems. The aim is to obtain as far as possible fine features of the initial image and identify medical pathologies. The approach consists of constructing a family of regularized functionals and to select locally and adaptively the regularization parameters. The parameters selection is performed at the discrete level in the framework of the finite element method. We present several numerical simulations to test the efficiency of the proposed approach.