This paper discusses possible procedures to improve finite element meshes in order to enable an accurate and stable numerical analysis processes. In general, the process of mesh improvement has to address three main aspects: mesh untangling (removal or fix of inverted finite elements), improvement of element shape, and making the element sizes more uniform. This paper focusses on the last two aspects: improving shape and size uniformity. This problem is addressed from purely geometrical aspect and by engaging optimization methods; thus, stress/strain related finite element quality measures are not considered. Two various element quality measures are discussed with emphasis on their coupling with an adequate optimization procedure. These two measures are the inverse mean ratio measure and the size measure. The first one addresses the shape and size of finite elements, but concentrates more on the shape. The later one, on the other hand, addresses only finite element size. Since mesh improvement tasks are typical multi-objective optimization problems, the paper also addresses briefly the procedure how to transform the multi-objective problem into a usual single-objective one that can be solved by employing standard optimization techniques. In this work a gradient-based approximation method was employed to do the optimization. The discussed theory is numerically tested on simply deformed meshes.
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