Numerical evaluation of hypersingular integrals

AbstractBoundary integral equations which employ integrals which exist only if defined in the Cauchy principal value sense or as the Hadamard finite part are currently used with success to solve many two- and three-dimensional problems of applied mechanics. We will recall definitions and main properties of these integrals, examine some numerical approaches for their evaluation and present several new results

Glacial dynamics in pre-Alpine narrow valleys during the Last Glacial Maximum inferred by lowland fluvial records (northeast Italy)

During the Last Glacial Maximum (LGM), most of the major glaciated basins of the European Southern Alps had piedmont lobes with large outwash plains; only a few glaciers remained within the valley. Piedmont glaciers have left well-preserved terminal moraines, which allow for investigations to be carried out and inferences to be made regarding their evolution and chronology. Valley glaciers\u2019 remnants, on the contrary, are often scantly preserved, and changes can only be detected through correlations with glaciofluvial deposits in downstream alluvial basins. The Brenta glacial system\u2019s dynamics in the glacier\u2019s terminal tract have been inferred through a wide range of sediment analysis techniques on an alluvial stratigraphic record of the Brenta megafan (northeast Italy), and via the mapping of in-valley glacial/glaciofluvial remnants. Glaciers flowing across narrow gorges could possibly be slowed/blocked by such morphology, and glacial/sediment fluxes may then be diverted to lateral valleys. Moreover, narrow valleys may induce glaciers to bulge and form icefalls at their front, preventing the formation of terminal moraines. The Brenta Glacier was probably slowed/blocked by the narrow Valsugana Gorge downstream of Primolano and was effectively diverted eastwards across a wind gap (Canal La Menor Valley), joining the Cismon/Piave glaciers near Rocca and ending 2 km downstream. The Cismon and Piave catchments started to contribute to the Brenta system just after 27 ka cal BP until at least 19:5 ka cal BP. After the glaciers collapsed, the Piave River once again flowed into its main valley, whilst the Cismon continued to merge with the Brenta. This investigation shows that glacial catchments may vary significantly over time during a single glaciation in rugged Alpine terrains. Sand petrography and the chemical/mineralogical composition of sediments are powerful proxies for tracing such variations, as they propagate through the glacial and glaciofluvial systems and can be recognized in the alluvial stratigraphic record far downstream from the glacier front

Properties and numerical solution of an integral equation system to minimize airplane drag for a multiwing system

We consider an open multiwing system, composed of N 2 disjoint openplane curves, not necessarily symmetric, and examine the corresponding (con-strained) induced drag minimization problem. To this end, we first derive theassociated Euler-Lagrange system of equations, which is then reduced to anequivalent system of Cauchy singular integral equations. By generalizing a pre-vious approach of ours for the case of a single open wing, we obtain existenceand uniqueness results for the problem solution in a product of weighted Sobolevtype spaces. This system is then solved by applying to it a collocation-quadraturemethod. For this, we prove stability and derive corresponding error estimates.Finally, to test the efficiency of the proposed numerical method, we apply it tosome multiwing system

Late Quaternary evolution of the sedimentary environment in Modrejce near Most na Soči (Soča Valley, Julian Alps)

Geomorphological and geological mapping have long been used to study the glacial history of the Slovenian Alps, but many uncertainties remain regarding the time and extent of Pleistocene glaciations there. Glacial landforms and undisturbed glacial deposits are rare in the areas of the former glacier terminus, especially in the Soča Valley, where large discrepancies in the interpretation of the extent of the former Soča Glacier have been reported. Early studies proved inconclusive as to whether one or two glaciations extended into the Soča Valley as far as Most na Soči. In order to answer this question, the Quaternary sedimentary succession and landforms in the Modrejce Valley near Most na Soči were investigated. New geological and geomorphological field data allow the interpretation of the sedimentary environment and the stratigraphic relationships between different units. In response to glacial dynamics, the sedimentation developed from glaciofluvial and glaciolacustrine to fully glacial environments, followed by slope deposition. At higher altitudes lateral moraines are preserved, while the staircase-like slope below has been carved into older glacial, glaciofluvial and glaciolacustrine deposits by glacial and post-glacial processes, including fluvial erosion and slope dynamics. We conclude that the succession studied here was deposited over the course of two different glacial advances – LGM and pre-LGM. Our study thus suggests that the Soča Glacier extended as far as the area of Most na Soči twice over the course of the late Quaternary

Two boundary integral equation methods for linear elastodynamics problems on unbounded domains

We consider (transient) 3D elastic wave propagation problems in unbounded isotropic homogeneous media, which can be reduced to corresponding 2D ones. For their solution, we propose and compare two boundary integral equation approaches, both based on the coupling of a discrete time convolution quadrature with a classical space collocation discretization. In the first approach, the PDE problem is preliminary replaced by the equivalent well known (vector) space-time boundary integral equation formulation, while in the second, the same PDE is replaced by a system of two (coupled) wave equations, each one of which is then represented by the associated boundary integral equation. The construction of these two approaches is described and discussed. Some numerical testing are also presented