A geoscientifically relevant wavelet approach is established for the classical (inner) displacement problem corresponding to a regular surface (such as sphere, ellipsoid, actual earth's surface). Basic tools are the limit and jump relations of (linear) elastostatics. Scaling functions and wavelets are formulated within the framework of the vectorial Cauchy-Navier equation. Based on appropriate numerical integration rules a pyramid scheme is developed providing fast wavelet transform (FWT). Finally multiscale deformation analysis is investigated numerically for the case of a spherical boundary.
This survey paper deals with multiresolution analysis from geodetically relevant data and its numerical realization for functions harmonic outside a (Bjerhammar) sphere inside the Earth. Harmonic wavelets are introduced within a suit- able framework of a Sobolev-like Hilbert space. Scaling functions and wavelets are defined by means of convolutions. A pyramid scheme provides efficient implementation und economical computation. Essential tools are the multiplicative Schwarz alternating algorithm (providing domain decomposition procedures) and fast multipole techniques (accelerating iterative solvers of linear systems).
By means of the limit and jump relations of classical potential theory the framework of a wavelet approach on a regular surface is established. The properties of a multiresolution analysis are verified, and a tree algorithm for fast computation is developed based on numerical integration. As applications of the wavelet approach some numerical examples are presented, including the zoom-in property as well as the detection of high frequency perturbations. At the end we discuss a fast multiscale representation of the solution of (exterior) Dirichlet's or Neumann's boundary-value problem corresponding to regular surfaces.
In micro milling, size effects such as the ratio of uncut chip thickness to cutting edge radius result to high mechanical stresses. The tools need to be able to withstand these, with as little tool wear as possible. Cemented carbides are currently the tool substrates of choice. Technical ceramics are highly wear resis- tant as well, but they are not yet used in micro milling. To utilize their potential in micro cutting pro- cesses, we previously identified Y-TZP as the best ceramic for this purpose. Compared to cemented carbide, they exhibit only marginal tool wear when micro milling PMMA. To investigate whether the 3Y-TZP characteristics influence the performance of all-ceramic micro end mills, three different substrate materials were used to manufacture tools that were tested by micro milling of PMMA. Further varied factors were the feed per tooth and the spindle speed. The initial cutting edge sharpness of the tools and the tool wear were used to quantify the results. One substrate was found to result in lower cutting edge radii and a more stable manufacturing process than the others. Also, a feed per tooth dependent wear behavior was observed.
Micro milling is a very flexible micro cutting process widely deployed to manufacture miniaturized parts. However, size effects occur when downscaling the cutting processes. They lead to higher mechanical loads on the tools and therefore increased tool wear. Micro milling tools are usually made of cemented carbides due to their mechanical strength and fine grain structure. Technical ceramics as alternative tool materials offer very good mechanical properties as well, with grain sizes well below 1 μ m. In conventional machining, they have proven to be able to reduce tool wear. To transfer these wear improvements to the micro scale, we manufactured all-ceramic micro end mills in previous studies ( ∅ 50 and ∅ 100 μm). Tools made from zirconia (Y-TZP) showed the sharpest cutting edges, and were the best performing in micro milling trials amongst the substrates tested. However, the advantages of the ceramic substrate could not be utilized for the brass and titanium materials tested in those studies. Therefore, in this study the capabilities of all-ceramic micro end mills ( ∅ 50 μ m) in different workpiece materials (1.4404, 1.7225, 3.1325 and PMMA GS) were researched. For the two steels and the aluminum alloy, the ceramic tools did not offer an improvement over the cemented carbide tools used as reference. For the thermoplastic PMMA however, significant improvements could be achieved by utilizing the Y-TZP ceramic tools: Less tool wear, less and more stable cutting forces, and higher surface qualities.