Harmonic mappings in the plane are univalent complex-valued harmonic functions of a complex variable. Conformal mappings are a special case where the real and imaginary parts are conjugate harmonic functions, satisfying the Cauchy-Riemann equations. Harmonic mappings were studied classically by differential geometers because they provide isothermal (or conformal) parameters for minimal surfaces. More recently they have been actively investigated by complex analysts as generalizations of univalent analytic functions, or conformal mappings. Many classical results of geometric function theory extend to harmonic mappings, but basic questions remain unresolved. This book is the first comprehensive account of the theory of planar harmonic mappings, treating both the generalizations of univalent analytic functions and the connections with minimal surfaces.  Essentially self-contained, the book contains background material in complex analysis and a full development of the classical theory of minimal surfaces, including the Weierstrass-Enneper representation. It is designed to introduce non-specialists to a beautiful area of complex analysis and geometry.

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Harmonic Mappings in the Plane

Micro and Nano Machining by Electro-Physical and Chemical Processes

Multi-objective optimization using Taguchi based grey relational analysis for micro-milling of Al 7075 material with ball nose end mill

The capability of machining intricate features with high dimensional accuracy in hard and difficult-to-cut material has made electrodischarge machining (EDM) process as an inevitable and one of the most popular non-conventional machining processes. In recent years, both EDM and micro-EDM processes are being used extensively in the field of mould making, production of dies, cavities and complex 3D structures using difficult-to-cut tungsten carbide and its composites. The objective of this paper is to provide a state of the art in the field of EDM and micro-EDM of tungsten carbide and its composites. The review begins with a brief introduction on the EDM and micro-EDM processes. The research and developments in electrodischarge machining of tungsten carbide are grouped broadly into conventional EDM of tungsten carbide, micro-EDM of tungsten carbide and current research trends in EDM and micro-EDM of tungsten carbide. The problems and challenges in the area of conventional and micro-EDM of tungsten carbide and the importance of compound and hybrid machining processes are discussed. A summary of the future research directions based on the review is presented at the final section.

A review on the conventional and micro-electrodischarge machining of tungsten carbide

Micro milling cutters or micro mills are unique and important micro tools for fabricating miniaturised devices with sufficient geometrical and dimensional accuracy and machined surface integrity. Micro milling cutters, compared to conventional macro tools, have significantly different material removal mechanisms. They are also made of different raw materials and structures and exhibit distinctive machining characteristics and performances. Herein, we present the first comprehensive and up-to-date review of micro milling cutters in terms of their uniqueness, material removal mechanisms, materials and compositions, structures and design, fabrication techniques and machining performances, to provide adequate guidance for interested involvers. We also outline and discuss several possible future research directions to offer potential insights for the micro milling community and future researchers.

Advances in micro milling: From tool fabrication to process outcomes

Brittle materials are difficult to mechanically micro machine due to damage resulting from material removal by brittle fracture, cutting force-induced tool deflection or breakage and tool wear. This paper demonstrates the feasibility of micro machining glass materials with polycrystalline diamond (PCD) micro tools that are prepared in a variety of shapes using non-contact micro electro discharge machining. The PCD tools contain randomly distributed sharp protrusions of diamond with dimensions around 1 µm that serve as cutting edges for micro machining grooves in soda-lime glass and pockets in ultra-low expansion glass. Results indicate that smooth surfaces are obtained with process conditions allowing material removal by ductile regime cutting instead of brittle fractures, and the PCD tools show very little wear. With further improvements in material removal rate, micro machining with PCD tools is a promising approach for producing micro molds and micro fluidic devices in glass materials.

https://iopscience.iop.org/article/10.1088/0960-1317/14/12/013/pdf

Micro machining glass with polycrystalline diamond tools shaped by micro electro discharge machining

This paper provides an overview of several approaches to micro-machining by mechanical and electro-discharge means of material removal. Two steps are required in machining micro features. Firstly, a custom-shaped tool is created from suitable stock. In many cases, this is carried out using a small-scale version of wire Electro-Discharge Machining (EDM) in tool materials such as sintered PolyCrystalline Diamond (PCD) or tungsten carbide. Then the micro-tool can be used as a miniature end mill, drill or abrasive wheel. Each of these mechanical machining methods can be combined with EDM to achieve a customisable surface finish and feature accuracy. Trade-offs such as tool wear, Material Removal Rate (MRR) and machining time are discussed in this paper within the context of several examples.

Micro-machining and micro-grinding with tools fabricated by micro electro-discharge machining

In multi-station assembly systems, common for mass-customization manufacturing strategies, the product being assembled is held in a fixture attached to a pallet, and the pallet is conveyed between workstations. In high-precision assembly systems, variation in the position of the pallet is one of the largest sources of variation within the error budget, reducing quality and yields. Conventional approaches to locating pallets use pins and bushings, and a method for predicting their repeatability is presented. This paper also presents an exact constraint approach using a split-groove kinematic coupling, which reduces variation in pallet location by an order of magnitude.

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Precisely positioning pallets in multi-station assembly systems

Micro-electro-discharge machining produces tools for micro-machining processes with diameters down to 3 ?m and lengths up to fifty times their diameter. This paper describes a metrology technique for characterizing the geometry of these tools by the standardized components of roughness, waviness and form. In this technique two-dimensional (2D) profiles are extracted from three-dimensional (3D) surface measurements acquired by scanning white light interferometry with height resolution below 1 nm. The profiles are transformed into the frequency domain and separated into form (wavelengths longer than 100 ?m), waviness (wavelengths between 10 ?m and 100 ?m) and roughness (wavelengths shorter than 10 ?m). Standard height and shape parameters are computed from the roughness, waviness and form profiles for 81 tools, enabling relative comparison of tool geometry in a quantitative and reproducible manner. This approach provides a more comprehensive quantification of geometry than alternatives such as measuring the diameter at a few points along the tool or scanning probe microscopy over small areas. The utility of the technique is demonstrated by characterizing the precision of three micro-tools produced by wire electro-discharge grinding (WEDG).

http://iopscience.iop.org/article/10.1088/0960-1317/14/8/017/pdf

Micro-tool characterization using scanning white light interferometry

A family of sense–preserving complex valued harmonic polynomials that are close-to-convex on the open unit disk is introduced. By taking limits as the degree of the polynomials tends to infinity, a famiiy of close-to-convex harmonic mappings is also obtained. Similar results are considered for mappings of the punctured disk.

Chris Morgan

Papers

Micro machining glass with polycrystalline diamond tools shaped by micro electro discharge machining

Micro-machining and micro-grinding with tools fabricated by micro electro-discharge machining

Precisely positioning pallets in multi-station assembly systems

Micro-tool characterization using scanning white light interferometry

Construction of close-to-convex harmonic polynomials