Foundations of the wald space for phylogenetic trees

\ua9 2024 The Authors. Journal of the London Mathematical Society is copyright \ua9 London Mathematical Society.Evolutionary relationships between species are represented by phylogenetic trees, but these relationships are subject to uncertainty due to the random nature of evolution. A geometry for the space of phylogenetic trees is necessary in order to properly quantify this uncertainty during the statistical analysis of collections of possible evolutionary trees inferred from biological data. Recently, the wald space has been introduced: a length space for trees which is a certain subset of the manifold of symmetric positive definite matrices. In this work, the wald space is introduced formally and its topology and structure is studied in detail. In particular, we show that wald space has the topology of a disjoint union of open cubes, it is contractible, and by careful characterisation of cube boundaries, we demonstrate that wald space is a Whitney stratified space of type (A). Imposing the metric induced by the affine invariant metric on symmetric positive definite matrices, we prove that wald space is a geodesic Riemann stratified space. A new numerical method is proposed and investigated for construction of geodesics, computation of Fr\ue9chet means and calculation of curvature in wald space. This work is intended to serve as a mathematical foundation for further geometric and statistical research on this space

Effects of Surface Coatings on the Joint Formation During Magnetic Pulse Welding in Tube-to-Cylinder Configuration

Magnetic Pulse Welding (MPW) is a joining technique favorable for the generation of strong atomic bonded areas between different metals, e.g. aluminum and steel. Brittle intermetallic phases can be avoided due to the high-speed collision and the absence of external heat. The demand for the use of this technique in industries like automotive and plant engineering rises. However, workpieces used in these fields are often coated, e.g. in order to improve the corrosion resistance. Since the weld quality depends on the material’s behavior at the collision zone, surface layers in that region have to be taken into account as well. This work investigates the influences of different coating types. Aluminum to steel welding is used as an example system. On the inner steel part (C45) coatings like zinc, nickel and chrome are applied, while the aluminum flyer tubes (EN AW-6060) are anodized, chromated and passivated. Welding tests are performed using two different welding systems with varying discharging frequencies and four geometrical part setups. For all combinations, the flyer velocity during the process is measured by Photon Doppler Velocimetry (PDV). By using the uncoated material combination as a reference, the removal of surface layers due to jetting is analyzed. Finally, the weld quality is characterized in peel tests, shear-push tests and by the help of metallographic analysis. It is found that certain coatings improve the joint formation, while others are obstructive for the performance of MPW. Some coatings have no influence on the joining process at all

Influence of the Wall Thicknesses on the Joint Quality During Magnetic Pulse Welding in Tube-to-Tube Configuration

The implementation of multi-material concepts, for example, in automotive engineering or aerospace technologies, requires adequate joining techniques. The Magnetic Pulse Welding (MPW) process allows for joining both similar and dissimilar materials without additional mechanical elements, chemical binders, or adverse influences of heat on the joining partners. In this process, an electro-conductive at (‘flyer’) part is accelerated by Lorentz forces and impacts the inner (‘parent’) part under high velocity and high pressure, leading to the formation of a metallurgical joint. Besides joining of sheets and tubes to solid cylinders, the connection of two tubes is of particular interest due to the increased lightweight potential. The present paper focuses on the MPW of aluminum (EN AW-6060) to steel (C45) tubes. An experimental study was performed, in which the wall thickness of the parent part was reduced successively. The deformation behavior of both the flyer and parent parts was recorded during the experiments by a two-probe Photon Doppler Velocimeter (PDV). The final shape of the joined specimens was analyzed by a 3D digitizer. An instrumented peel test was used for the determination of the weld quality. It was found that defect-free MPW of aluminum tubes on steel tubes without supporting mandrel is possible

Influence of the Free Compression Stage on Magnetic Pulse Welding of Tubes

In magnetic pulse welding (MPW) of tubular parts, the acceleration of the ‘flyer’ part typically corresponds to a free electromagnetic compression (EMC) process over the distance of the initial standoff between the outer and inner tube. During this process stage, already significant plastic strains occur. In addition, wrinkling is a phenomenon frequently observed during EMC. In this manuscript, influencing factors on the wrinkling effect are identified, taking the initial geometry of the flyer tube and its manufacturing process into account. Moreover, a link between the strains and wrinkles caused by the tube compression and the MPW process is made. An experimental study is performed aiming for the quantification of the plastic deformation during EMC. The effect of this deformation on the stability and adhesion of brittle surface layers is analyzed. Accompanying numerical simulations help to understand the wrinkle formation and its influencing factors. Based on the results, hints for an improved process design of MPW are given

Influence of Axial Workpiece Positioning during Magnetic Pulse Welding of Aluminum-Steel Joints

Magnetic Pulse Welding (MPW) offers a method to economically join similar and dissimilar metals without the need for external physical or chemical binders, while avoiding the adverse heating effects seen in many welding techniques. MPW allows for the fabrication of joints via the harnessing of Lorentz forces, which result from discharging a current pulse through a coil. In the process an outer piece (flyer) is accelerated onto an inner piece (parent), and welding is achieved using propagating impact fronts. There are several geometrical factors to be considered including the flyer-coil distance, the parentflyer distance, as well as the axial relationship between flyer and coil (working length). Various shapes of the front are possible and each configuration has its own advantages and drawbacks. The goal of this work is to show not only how the aforementioned parameters are related, but also ways to optimize front propagations, which are vital to the welding result. This is done primarily by determining the influence of the working length of tubular MPW specimens. It is shown that for steel-aluminum joints in the given arrangements, three different front regimes exist, which are related to geometrical factors. These results are especially useful to avoid seemingly favorable but nevertheless suboptimal conditions for flyer movement that would reduce weld quality and energy efficiency of the process

Influence of axial workpiece position in the coil for the electromagnetic pulse joining

Magnetic Pulse Welding (MPW) enables the fabrication of joints via the harnessing of Lorentz forces, which result from discharging a current pulse through a coil. In the process an outer piece (flyer) is accelerated onto an inner piece (parent), and welding is achieved using propagating impact fronts. The working length of the experimental setup allows for various shapes of the deformation front, and each configuration has its own advantages and drawbacks. The objective of this work is to show how the working length of tubular MPW specimens affects the front propagation as well as to indicate ways to optimize the front propagations, which are vital to the welding result. It is shown that for steel-aluminum joints, three different front regimes exist, which are related to geometrical factors. These results may be used to avoid seemingly favorable but nevertheless suboptimal conditions for flyer movement, which reduce the weld quality and the energy efficiency of the process. Additionally, the methodology presented here may allow for faster process optimization without the need for time-consuming metallographic analyses