A numerical method for differential equations of second order

A new algorithm of order five is presented for the solution of the initial value problem where the system of ordinary differential equations is of second order and does not contain the first derivative.     

Numerical modelling of the impact crater depth-diameter dependence in an acoustically fluidized target

2D numerical modelling of impact cratering has been utilized to quantify an important depth-diameter relationship for different crater morphologies, simple and complex. It is generally accepted that the final crater shape is the result of a gravity-driven collapse of the transient crater, which is formed immediately after the impact. Numerical models allow a quantification of the formation of simple craters, which are bowl-shaped depressions with a lens of rock debris inside, and complex craters, which are characterized by a structural uplift. The computation of the cratering process starts with the first contact of the impactor and the planetary surface and ends with the morphology of the final crater. Using different rheological models for the sub-crater rocks, we quantify the influence on crater mechanics. To explain the formation of complex craters in accordance to the threshold diameter between simple and complex craters, we utilize the Acoustic Fluidization model. We carried out a series of simulations over a broad parameter range with the goal to fit the observed depth/diameter relationships as well as the observed threshold diameters on the Moon, Earth and Venus.     

Restricted Three-Body Problem: An Approximation of its General Solution – Part One – the Manifold of Symmetric Periodic Solutions

This paper gives the results of a programme attempting to exploit ‘la seule bréche’ (Poincaré, 1892, p. 82) of non-integrable systems, namely to develop an approximate general solution for the three out of its four component-solutions of the planar restricted three-body problem. This is accomplished by computing a large number of families of ‘solutions précieuses’ (periodic solutions) covering densely the space of initial conditions of this problem. More specifically, we calculated numerically and only for μ = 0.4, all families of symmetric periodic solutions (1st component of the general solution) existing in the domain D:(x ₀ ∊ [−2,2],C ∊ [−2,5]) of the (x ₀, C) space and consisting of symmetric solutions re-entering after 1 up to 50 revolutions (see graph in Fig. 4). Then we tested the parts of the domain D that is void of such families and established that they belong to the category of escape motions (2nd component of the general solution). The approximation of the 3rd component (asymmetric solutions) we shall present in a future publication. The 4th component of the general solution of the problem, namely the one consisting of the bounded non-periodic solutions, is considered as approximated by those of the 1st or the 2nd component on account of the `Last Geometric Theorem of Poincaré' (Birkhoff, 1913). The results obtained provoked interest to repeat the same work inside the larger closed domain D:(x ₀ ∊ [−6,2], C ∊ [−5,5]) and the results are presented in Fig. 15. A test run of the programme developed led to reproduction of the results presented by Hénon (1965) with better accuracy and many additional families not included in the sited paper. Pointer directions construed from the main body of results led to the definition of useful concepts of the basic family of order n, n = 1, 2,… and the completeness criterion of the solution inside a compact sub-domain of the (x ₀, C) space. The same results inspired the ‘partition theorem’, which conjectures the possibility of partitioning an initial conditions domain D into a finite set of sub-domains D i that fulfill the completeness criterion and allow complete approximation of the general solution of this problem by computing a relatively small number of family curves. The numerical results of this project include a large number of families that were computed in detail covering their natural termination, the morphology, and stability of their member solutions. Zooming into sub-domains of D permitted clear presentation of the families of symmetric solutions contained in them. Such zooming was made for various values of the parameter N, which defines the re-entrance revolutions number, which was selected to be from 50 to 500. The areas generating escape solutions have being investigated. In Appendix A we present families of symmetric solutions terminating at asymptotic solutions, and in Appendix B the morphology of large period symmetric solutions though examples of orbits that re-enter after from 8 to 500 revolutions. The paper concludes that approximations of the general solution of the planar restricted problem is possible and presents such approximations, only for some sub-domains that fulfill the completeness criterion, on the basis of sufficiently large number of families.     

Numerical modeling for analyzing thermal surface anomalies induced by underground coal fires

Coal seams burning underneath the surface are recognized all over the world and have drawn increasing public attention in the past years. Frequently, such fires are analyzed by detecting anomalies like increased exhaust gas concentrations and soil temperatures at the surface. A proper analysis presumes the understanding of involved processes, which determine the spatial distribution and dynamic behavior of the anomalies.In this paper, we explain the relevance of mechanical and energy transport processes with respect to the occurrence of temperature anomalies at the surface. Two approaches are presented, aiming to obtain insight into the underground coal fire situation: In-situ temperature mapping and numerical simulation. In 2000 to 2005, annual temperature mapping in the Wuda (Inner Mongolia, PR China) coal fire area showed that most thermal anomalies on the surface are closely related to fractures, where hot exhaust gases from the coal fire are released. Those fractures develop due to rock mechanical failure after volume reduction in the seams. The measured signals at the surface are therefore strongly affected by mechanical processes.More insight into causes and effects of involved energy transport processes is obtained by numerical simulation of the dynamic behavior of coal fires. Simulations show the inter-relation between release and transport of thermal energy in and around underground coal fires. Our simulation results show a time delay between the coal fire propagation and the observed appearance of the surface temperature signal. Additionally, the overall energy flux away from the burning coal seam into the surrounding bedrock is about 30-times higher than the flux through the surface. This is of particular importance for an estimation of the energy released based on surface temperature measurements. Finally, the simulation results also prove that a fire propagation rate estimated from the interpretation of surface anomalies can differ from the actual rate in the seam.     

Three-dimensional discontinuous deformation analysis (3-D DDA) using a new contact resolution algorithm

This paper presents a new contact calculating algorithm for contacts between two polyhedra with planar boundaries in the three-dimensional discontinuous deformation analysis (3-D DDA). In this algorithm, all six type contacts in 3-D (vertex-to-face, vertex-to-edge, vertex-to-vertex, face-to-face, edge-to-edge, and edge-to-face) are simply transformed into the form of point-to-face contacts. The presented algorithm is a simple and efficient method and it can be easily coded into a computer program. In this paper, formulations of normal contact, shear contact and frictional force submatrices based on the new method are derived and the algorithm has been programmed in VC⁺⁺. Examples are provided to demonstrate the new contact rule between two blocks.     

On CO<Subscript>2</Subscript> fluid flow and heat transfer behavior in the subsurface,following leakage from a geologic storage reservoir

Geologic storage of CO₂ is expected to produce plumes of large areal extent, and some leakage may occur along fractures, fault zones, or improperly plugged pre-existing wellbores. A review of physical and chemical processes accompanying leakage suggests a potential for self-enhancement. The numerical simulations presented here confirm this expectation, but reveal self-limiting features as well. It seems unlikely that CO₂ leakage could trigger a high-energy run-away discharge, a so-called “pneumatic eruption,” but present understanding is insufficient to rule out this possibility. The most promising avenue for increasing understanding of CO₂ leakage behavior is the study of natural analogues.     

Determination of historical emission of heavy metals into the atmosphere: Celje case study

The research shows that in the Celje area (Slovenia), the historical anthropogenical emissions are 1,712 tons of Zn and 9.1 tons of Cd. For Zn, this value represents approximately 0.3% of the total Zn production in that area. Close to the former zinc smelting plant, the “Zn precipitation” has been estimated to be up to 0.036 mm. The 100-year Zn production left behind a heavily contaminated area with maximum concentrations of Zn of up to 5.6% in attic dust and 0.85% in the soil, and 456 mg/kg of Cd in attic dust and 59.1 mg/kg in the soil. The calculation of historical emissions is based on the data of heavy metals concentration in the attic dust at 98 sampling points and on the data from 19 measurement sites of the weight of total monthly air deposit. The main idea behind determining past emissions is that when the weight of the deposited dust on a small area is multiplied by the concentration of the element in that area, the mass of the polluter which has been transported to the place of interest by air can be calculated. If we sum up all the weight over the whole geochemical anomaly, we get the quantity of historical emissions.     

An operational multiscale system for hazards prediction, mapping, and response

By definition, a crisis is a situation that requires assistance to be managed. Hence, response to a crisis involves the merging of local and non-local emergency response personnel. In this situation, it is critical that each participant: (1) know the roles and responsibilities of each of the other participants; (2) know the capabilities of each of the participants; and (3) have a common basis for action. For many types of natural disasters, this entails having a common operational picture of the unfolding events, including detailed information on the weather, both current and forecasted, that may impact on either the emergency itself or on response activities. The Consequences Assessment Tool Set (CATS) is a comprehensive package of hazard prediction models and casualty and damage assessment tools that provides a linkage between a modeled or observed effect and the attendant consequences for populations, infrastructure, and resources, and, hence, provides the common operational picture for emergency response. The Operational Multiscale Environment model with Grid Adaptivity (OMEGA) is an atmospheric simulation system that links the latest methods in computational fluid dynamics and high-resolution gridding technologies with numerical weather prediction to provide specific weather analysis and forecast capability that can be merged into the geographic information system framework of CATS. This paper documents the problem of emergency response as an end-to-end system and presents the integrated CATS–OMEGA system as a prototype of such a system that has been used successfully in a number of different situations.     

Numerical modeling of tide-surge interaction along Orissa coast of India

The Orissa coast of India is one of the most vulnerable regions of extreme sea levels associated with severe tropical cyclones. There was extensive loss of life and property due to the October 1999 super cyclone, which devastated large part of the Orissa coast. The shallow nature of the head bay, presence of a large number of deltas formed by major rivers of Orissa such as Mahanadi and Dhamra, and high tidal range are responsible for storm surge flooding in the region. Specifically, rising and falling tidal phases influence the height, duration, and arrival time of peak surge along the coast. The objective of the present study is to evaluate the tide-surge interaction during the 1999 Orissa cyclone by using nonlinear vertically integrated numerical models. The pure tidal solution for the head bay region of the Bay of Bengal provides the initial condition for the fine resolution nested grid Orissa model. However, the feedback from the Orissa model does not affect the head bay model as the study provides a one-way interaction. Numerical experiments are performed to study the tide-surge interaction by considering various relative phases of the tidal waves with the surge-wave produced by 1999 Orissa cyclone. The comparison, although utilizing only the limited estimates of tidal data, appears adequate to assert that the principal features are reproduced correctly.     

Detection of free vertical convection and double-diffusion in groundwater monitoring wells with geophysical borehole measurements

Two algorithms for in-situ detection and identification of vertical free convective and double-diffusive flows in groundwater monitoring wells or boreholes are proposed. With one algorithm the causes (driving forces) and with the other one the effects (convection or double-diffusion) of vertical transport processes can be detected based on geophysical borehole measurements in the water column. Five density-driven flow processes are identified: thermal, solutal, and thermosolutal convection leading to an equalization, as well as saltfingers and diffusive layering leading to an intensification of a vertical density gradient. The occurrence of density-driven transport processes could be proven in many groundwater monitoring wells and boreholes; especially shallow sections of boreholes or groundwater monitoring wells are affected dramatically by such vertical flows. Deep sections are also impaired as the critical threshold for the onset of a density-driven flow is considerably low. In monitoring wells or boreholes, several sections with different types of density-driven vertical flows may exist at the same time. Results from experimental investigations in a medium-scale testing facility with high aspect ratio (height/radius = 19) and from numerical modeling of a water column agree well with paramters of in-situ detected convection cells.