We introduce a novel liquid simulation approach that combines a spatially adaptive pressure projection solver with the Particle-in-Cell (PIC) method. The solver relies on a generalized version of the Finite Difference (FD) method to approximate the pressure field and its gradients in tree-based grid discretizations, possibly non-graded. In our approach, FD stencils are computed by using meshfree interpolations provided by a variant of Radial Basis Function (RBF), known as RBF-Finite-Difference (RBF-FD). This meshfree version of the FD produces differentiation weights on scattered nodes with high-order accuracy. Our method adapts a quadtree/octree dynamically in a narrow-band around the liquid interface, providing an adaptive particle sampling for the PIC advection step. Furthermore, RBF affords an accurate scheme for velocity transfer between the grid and particles, keeping the system's stability and avoiding numerical dissipation. We also present a data structure that connects the spatial subdivision of a quadtree/octree with the topology of its corresponding dual-graph. Our data structure makes the setup of stencils straightforward, allowing its updating without the need to rebuild it from scratch at each time-step. We show the effectiveness and accuracy of our solver by simulating incompressible inviscid fluids and comparing results with regular PIC-based solvers available in the literature.

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Os agradecimentos principais são direcionados à minha família, meus pais e minha irmã, amigos e todos aqueles que contribuíram para o desenvolvimento desse trabalho. Em especial, Bruno, Kelly, Filipe, Camila e Matheus, pelo suporte técnico e emocional durante os longos anos de doutorado.Agradecimentos especiais são direcionados ao Instituto de Ciências Matemáticas e de Computação, ao grupo de pesquisa VICG e ao LMACC. Agradecimentos aos técnicos responsáveis pelos equipamentos do laboratório, Leonardo e Gabriel, e ao meu orientador, Afonso, pelo suporte e auxílio durante o desenvolvimento do projeto dessa tese.Agradeço à CAPES pelo auxílio financeiro e ao CNPq pelo financiamento do período sanduíche realizado no Japão sob orientação do professor Dr. Ryoichi Ando. Agradecimentos à equipe do ICMC, Aline, Carol e Marcos pela ajuda com os procedimentos do estágio sanduíche. Agradecimentos também à equipe do National Institute of Informatics pelo suporte e auxílio durante o período de estágio.

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