Adaptive numerical simulations for a moving boundary problem related to pitting corrosion

Sarker, Abu Naser (2023) Adaptive numerical simulations for a moving boundary problem related to pitting corrosion. Doctoral (PhD) thesis, Memorial University of Newfoundland.

[English] PDF - Accepted Version Available under License - The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission. Download (7MB)

Abstract

Many real-world science and engineering problems are described mathematically by partial differential equations (PDEs). Most of these PDEs can not be solved analytically and instead their solutions must be approximated on a computer. One numerical method capable of high accuracy solutions is the finite element method where the physical system is discretized using set of elements called a mesh. These elements consist of nodes or points, and the PDE is solved at each node of the mesh. We need high-quality meshes in order to achieve accurate numerical solutions to PDEs and an adaptive mesh that moves as the system evolves has many desirable properties. Moving meshes are now widely used in the numerical solution of PDEs, especially when dealing with problems that involve significant changes in the solution, such as fast-moving fronts, or moving boundary problems. A nonuniform mesh can maintain accuracy and also boost the efficiency of existing methods by automatically adjusting to solution behaviour and concentrating mesh points in critical areas, while minimizing the number of mesh nodes. The main focus of this thesis is the design and implementation of an adaptive moving mesh method for a moving boundary problem related to pitting corrosion with homogeneous and heterogeneous materials. Pitting corrosion is one of the most devastating localized forms of corrosion generating a small pit, cavity or hole in the metal. Damage due to pitting corrosion of metals cost governments and industry billions of dollars per year and can put human lives at risk. The first part of this research develops an adaptive moving mesh method for simulating pitting corrosion. The adaptive mesh is generated automatically by solving a mesh PDE coupled to the pitting corrosion PDE model. The moving mesh approach is shown to enable initial mesh generation, provide mesh recovery, and is able to smoothly tackle changing pit geometry. Materials with varying crystallography are considered as are single and multiple pits. A procedure is presented which allows pits to merge without a change in mesh topology, allowing computation to proceed withi out restarting. We have presented a robust, fully automatic, moving mesh solution framework for pitting corrosion. The second part of this research is aimed at developing an adaptive moving mesh method for simulating pitting corrosion in materials containing heterogeneous inclusions. Inclusions are regions of a material that have a different composition or properties than the surrounding material. This makes for a challenging task due to the presence of the inclusion-type domains. In order to handle moving boundary domains with an inclusion, the metric is modified according to the location of the inclusions. The moving mesh approach using r-refinement is shown to handle changing pit geometry, including materials with varying crystallography, corrosion-resistant inclusions, and material voids. r-refinement alone was not able to provide high mesh density near the inclusion(s) for long simulation times due to the obstacle(s) and the moving front. To overcome this issue, we propose a combination of h- and r- refinement, which is the focus of the third part of the research. h-refinement adds mesh elements by dividing each existing element into two or more elements and maintaining the type of element used. We design and implement an adaptive hr-refinement procedure for the simulation of pitting corrosion with heterogeneous materials. The adaptive hr-refinement is demonstrated to handle changing pit geometry, including materials with varying crystallography and corrosion-resistant inclusions. The three main components of the research include theory, modeling, and application, which aim to provide effective and efficient meshes over complex moving domains in the solution of the pitting corrosion problem. The research is also focused on the development of software providing a new extension of MMPDElab with hr–refinement.

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